1911 Encyclopædia Britannica/Mechanics/Applied

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1911 Encyclopædia Britannica, Volume 17

- Mechanics Applied by William J. M. Rankine & William E. Dalby

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37383291911 Encyclopædia Britannica, Volume 17 — - Mechanics AppliedWilliam J. M. Rankine & William E. Dalby

II.—Applied Mechanics^[1]

§ 1. The practical application of mechanics may be divided into two classes, according as the assemblages of material objects to which they relate are intended to remain fixed or to move relatively to each other—the former class being comprehended under the term “Theory of Structures” and the latter under the term “Theory of Machines.”

PART I.—OUTLINE OF THE THEORY OF STRUCTURES

§ 2. Support of Structures.—Every structure, as a whole, is maintained in equilibrium by the joint action of its own weight, of the external load or pressure applied to it from without and tending to displace it, and of the resistance of the material which supports it. A structure is supported either by resting on the solid crust of the earth, as buildings do, or by floating in a fluid, as ships do in water and balloons in air. The principles of the support of a floating structure form an important part of Hydromechanics (q.v.). The principles of the support, as a whole, of a structure resting on the land, are so far identical with those which regulate the equilibrium and stability of the several parts of that structure that the only principle which seems to require special mention here is one which comprehends in one statement the power both of liquids and of loose earth to support structures. This was first demonstrated in a paper “On the Stability of Loose Earth,” read to the Royal Society on the 19th of June 1856 (Phil. Trans. 1856), as follows:—

Let E represent the weight of the portion of a horizontal stratum of earth which is displaced by the foundation of a structure, S the utmost weight of that structure consistently with the power of the earth to resist displacement, φ the angle of repose of the earth; then

S	= ${\Big (}$	1 + sin φ	${\Big )}$ 2.
E		1 − sin φ

To apply this to liquids φ must be made zero, and then S/E = 1, as is well known. For a proof of this expression see Rankine’s Applied Mechanics, 17th ed., p. 219.

§ 3. Composition of a Structure, and Connexion of its Pieces.—A structure is composed of pieces,—such as the stones of a building in masonry, the beams of a timber frame-work, the bars, plates and bolts of an iron bridge. Those pieces are connected at their joints or surfaces of mutual contact, either by simple pressure and friction (as in masonry with moist mortar or without mortar), by pressure and adhesion (as in masonry with cement or with hardened mortar, and timber with glue), or by the resistance of fastenings of different kinds, whether made by means of the form of the joint (as dovetails, notches, mortices and tenons) or by separate fastening pieces (as trenails, pins, spikes, nails, holdfasts, screws, bolts, rivets, hoops, straps and sockets.)

§ 4. Stability, Stiffness and Strength.—A structure may be damaged or destroyed in three ways:—first, by displacement of its pieces from their proper positions relatively to each other or to the earth; secondly by disfigurement of one or more of those pieces, owing to their being unable to preserve their proper shapes under the pressures to which they are subjected; thirdly, by breaking of one or more of those pieces. The power of resisting displacement constitutes stability, the power of each piece to resist disfigurement is its stiffness; and its power to resist breaking, its strength.

§ 5. Conditions of Stability.—The principles of the stability of a structure can be to a certain extent investigated independently of the stiffness and strength, by assuming, in the first instance, that each piece has strength sufficient to be safe against being broken, and stiffness sufficient to prevent its being disfigured to an extent inconsistent with the purposes of the structure, by the greatest forces which are to be applied to it. The condition that each piece of the structure is to be maintained in equilibrium by having its gross load, consisting of its own weight and of the external pressure applied to it, balanced by the resistances or pressures exerted between it and the contiguous pieces, furnishes the means of determining the magnitude, position and direction of the resistances required at each joint in order to produce equilibrium; and the conditions of stability are, first, that the position, and, secondly, that the direction, of the resistance required at each joint shall, under all the variations to which the load is subject, be such as the joint is capable of exerting—conditions which are fulfilled by suitably adjusting the figures and positions of the joints, and the ratios of the gross loads of the pieces. As for the magnitude of the resistance, it is limited by conditions, not of stability, but of strength and stiffness.

§ 6. Principle of Least Resistance.—Where more than one system of resistances are alike capable of balancing the same system of loads applied to a given structure, the smallest of those alternative systems, as was demonstrated by the Rev. Henry Moseley in his Mechanics of Engineering and Architecture, is that which will actually be exerted—because the resistances to displacement are the effect of a strained state of the pieces, which strained state is the effect of the load, and when the load is applied the strained state and the resistances produced by it increase until the resistances acquire just those magnitudes which are sufficient to balance the load, after which they increase no further.

This principle of least resistance renders determinate many problems in the statics of structures which were formerly considered indeterminate.

§ 7. Relations between Polygons of Loads and of Resistances.—In a structure in which each piece is supported at two joints only, the well-known laws of statics show that the directions of the gross load on each piece and of the two resistances by which it is supported must lie in one plane, must either be parallel or meet in one point, and must bear to each other, if not parallel, the proportions of the sides of a triangle respectively parallel to their directions, and, if parallel, such proportions that each of the three forces shall be proportional to the distance between the other two,—all the three distances being measured along one direction.

Fig. 86.

Considering, in the first place, the case in which the load and the two resistances by which each piece is balanced meet in one point, which may be called the centre of load, there will be as many such points of intersection, or centres of load, as there are pieces in the structure; and the directions and positions of the resistances or mutual pressures exerted between the pieces will be represented by the sides of a polygon joining those points, as in fig. 86 where P₁, P₂, P₃, P₄ represent the centres of load in a structure of four pieces, and the sides of the polygon of resistances P₁ P₂ P₃ P₄ represent respectively the directions and positions of the resistances exerted at the joints. Further, at any one of the centres of load let PL represent the magnitude and direction of the gross load, and Pa, Pb the two resistances by which the piece to which that load is applied is supported; then will those three lines be respectively the diagonal and sides of a parallelogram; or, what is the same thing, they will be equal to the three sides of a triangle; and they must be in the same plane, although the sides of the polygon of resistances may be in different planes.

Fig. 87.

According to a well-known principle of statics, because the loads or external pressures P₁L₁, &c., balance each other, they must be proportional to the sides of a closed polygon drawn respectively parallel to their directions. In fig. 87 construct such a polygon of loads by drawing the lines L₁, &c., parallel and proportional to, and joined end to end in the order of, the gross loads on the pieces of the structure. Then from the proportionality and parallelism of the load and the two resistances applied to each piece of the structure to the three sides of a triangle, there results the following theorem (originally due to Rankine):—

If from the angles of the polygon of loads there be drawn lines (R₁, R₂, &c.), each of which is parallel to the resistance (as P₁P₂, &c.) exerted at the joint between the pieces to which the two loads represented by the contiguous sides of the polygon of loads (such as L₁, L₂, &c.) are applied; then will all those lines meet in one point (O), and their lengths, measured from that point to the angles of the polygon, will represent the magnitudes of the resistances to which they are respectively parallel.

When the load on one of the pieces is parallel to the resistances which balance it, the polygon of resistances ceases to be closed, two of the sides becoming parallel to each other and to the load in question, and extending indefinitely. In the polygon of loads the direction of a load sustained by parallel resistances traverses the point O.^[2]

§ 8. How the Earth’s Resistance is to be treated . . . When the pressure exerted by a structure on the earth (to which the earth’s resistance is equal and opposite) consists either of one pressure, which is necessarily the resultant of the weight of the structure and of all the other forces applied to it, or of two or more parallel vertical forces, whose amount can be determined at the outset of the investigation, the resistance of the earth can be treated as one or more upward loads applied to the structure. But in other cases the earth is to be treated as one of the pieces of the structure, loaded with a force equal and opposite in direction and position to the resultant of the weight of the structure and of the other pressures applied to it.

§ 9. Partial Polygons of Resistance.—In a structure in which there are pieces supported at more than two joints, let a polygon be constructed of lines connecting the centres of load of any continuous series of pieces. This may be called a partial polygon of resistances. In considering its properties, the load at each centre of load is to be held to include the resistances of those joints which are not comprehended in the partial polygon of resistances, to which the theorem of § 7 will then apply in every respect. By constructing several partial polygons, and computing the relations between the loads and resistances which are determined by the application of that theorem to each of them, with the aid, if necessary, of Moseley’s principle of the least resistance, the whole of the relations amongst the loads and resistances may be found.

§ 10. Line of Pressures—Centres and Line of Resistance.—The line of pressures is a line to which the directions of all the resistances in one polygon are tangents. The centre of resistance at any joint is the point where the line representing the total resistance exerted at that joint intersects the joint. The line of resistance is a line traversing all the centres of resistance of a series of joints,—its form, in the positions intermediate between the actual joints of the structure, being determined by supposing the pieces and their loads to be subdivided by the introduction of intermediate joints ad infinitum, and finding the continuous line, curved or straight, in which the intermediate centres of resistance are all situated, however great their number. The difference between the line of resistance and the line of pressures was first pointed out by Moseley.

Fig. 88.

§ 11.* The principles of the two preceding sections may be illustrated by the consideration of a particular case of a buttress of blocks forming a continuous series of pieces (fig. 88), where aa, bb, cc, dd represent plane joints. Let the centre of pressure C at the first joint aa be known, and also the pressure P acting at C in direction and magnitude. Find R₁ the resultant of this pressure, the weight of the block aabb acting through its centre of gravity, and any other external force which may be acting on the block, and produce its line of action to cut the joint bb in C₁. C₁ is then the centre of pressure for the joint bb, and R₁ is the total force acting there. Repeating this process for each block in succession there will be found the centres of pressure C₂, C₃, &c., and also the resultant pressures R₂, R₃, &c., acting at these respective centres. The centres of pressure at the joints are also called centres of resistance, and the curve passing through these points is called a line of resistance. Let all the resultants acting at the several centres of resistance be produced until they cut one another in a series of points so as to form an unclosed polygon. This polygon is the partial polygon of resistance. A curve tangential to all the sides of the polygon is the line of pressures.

§ 12. Stability of Position, and Stability of Friction.—The resistances at the several joints having been determined by the principles set forth in §§ 6, 7, 8, 9 and 10, not only under the ordinary load of the structure, but under all the variations to which the load is subject as to amount and distribution, the joints are now to be placed and shaped so that the pieces shall not suffer relative displacement under any of those loads. The relative displacement of the two pieces which abut against each other at a joint may take place either by turning or by sliding. Safety against displacement by turning is called stability of position; safety against displacement by sliding, stability of friction.

§ 13. Condition of Stability of Position.—If the materials of a structure were infinitely stiff and strong, stability of position at any joint would be insured simply by making the centre of resistance fall within the joint under all possible variations of load. In order to allow for the finite stiffness and strength of materials, the least distance of the centre of resistance inward from the nearest edge of the joint is made to bear a definite proportion to the depth of the joint measured in the same direction, which proportion is fixed, sometimes empirically, sometimes by theoretical deduction from the laws of the strength of materials. That least distance is called by Moseley the modulus of stability. The following are some of the ratios of the modulus of stability to the depth of the joint which occur in practice:—

Retaining walls, as designed by British engineers	1 : 8
Retaining walls, as designed by French engineers	1 : 5
Rectangular piers of bridges and other buildings, and arch-stones	1 : 3
Rectangular foundations, firm ground	1 : 3
Rectangular foundations, very soft ground	1 : 2
Rectangular foundations, intermediate kinds of ground	1 : 3 to 1 : 2
Thin, hollow towers (such as furnace chimneys exposed to high winds), square	1 : 6
Thin, hollow towers, circular	1 : 4
Frames of timber or metal, under their ordinary or average distribution of load	1 : 3
Frames of timber or metal, under the greatest irregularities of load	1 : 3

In the case of the towers, the depth of the joint is to be understood to mean the diameter of the tower.

Fig. 89.

§ 14. Condition of Stability of Friction.—If the resistance to be exerted at a joint is always perpendicular to the surfaces which abut at and form that joint, there is no tendency of the pieces to be displaced by sliding. If the resistance be oblique, let JK (fig. 89) be the joint, C its centre of resistance, CR a line representing the resistance, CN a perpendicular to the joint at the centre of resistance. The angle NCR is the obliquity of the resistance. From R draw RP parallel and RQ perpendicular to the joint; then, by the principles of statics, the component of the resistance normal to the joint is—

CP = CR · cos PCR;

and the component tangential to the joint is—

CQ = CR · sin PCR = CP · tan PCR.

If the joint be provided either with projections and recesses, such as mortises and tenons, or with fastenings, such as pins or bolts, so as to resist displacement by sliding, the question of the utmost amount of the tangential resistance CQ which it is capable of exerting depends on the strength of such projections, recesses, or fastenings; and belongs to the subject of strength, and not to that of stability. In other cases the safety of the joint against displacement by sliding depends on its power of exerting friction, and that power depends on the law, known by experiment, that the friction between two surfaces bears a constant ratio, depending on the nature of the surfaces, to the force by which they are pressed together. In order that the surfaces which abut at the joint JK may be pressed together, the resistance required by the conditions of equilibrium CR, must be a thrust and not a pull; and in that case the force by which the surfaces are pressed together is equal and opposite to the normal component CP of the resistance. The condition of stability of friction is that the tangential component CQ of the resistance required shall not exceed the friction due to the normal component; that is, that

CQ ≯ ƒ · CP,

where ƒ denotes the coefficient of friction for the surfaces in question. The angle whose tangent is the coefficient of friction is called the angle of repose, and is expressed symbolically by—

φ = tan ⁻¹ ƒ.

Now CQ = CP · tan PCR;

consequently the condition of stability of friction is fulfilled if the angle PCR is not greater than φ; that is to say, if the obliquity of the resistance required at the joint does not exceed the angle of repose; and this condition ought to be fulfilled under all possible variations of the load.

It is chiefly in masonry and earthwork that stability of friction is relied on.

§ 15. Stability of Friction in Earth.—The grains of a mass of loose earth are to be regarded as so many separate pieces abutting against each other at joints in all possible positions, and depending for their stability on friction. To determine whether a mass of earth is stable at a given point, conceive that point to be traversed by planes in all possible positions, and determine which position gives the greatest obliquity to the total pressure exerted between the portions of the mass which abut against each other at the plane. The condition of stability is that this obliquity shall not exceed the angle of repose of the earth. The consequences of this principle are developed in a paper, “On the Stability of Loose Earth,” already cited in § 2.

§ 16. Parallel Projections of Figures.—If any figure be referred to a system of co-ordinates, rectangular or oblique, and if a second figure be constructed by means of a second system of co-ordinates, rectangular or oblique, and either agreeing with or differing from the first system in rectangularity or obliquity, but so related to the co-ordinates of the first figure that for each point in the first figure there shall be a corresponding point in the second figure, the lengths of whose co-ordinates shall bear respectively to the three corresponding co-ordinates of the corresponding point in the first figure three ratios which are the same for every pair of corresponding points in the two figures, these corresponding figures are called parallel projections of each other. The properties of parallel projections of most importance to the subject of the present article are the following:—

(1) A parallel projection of a straight line is a straight line.

(2) A parallel projection of a plane is a plane.

(3) A parallel projection of a straight line or a plane surface divided in a given ratio is a straight line or a plane surface divided in the same ratio.

(4) A parallel projection of a pair of equal and parallel straight lines, or plain surfaces, is a pair of equal and parallel straight lines, or plane surfaces; whence it follows

(5) That a parallel projection of a parallelogram is a parallelogram, and

(6) That a parallel projection of a parallelepiped is a parallelepiped.

(7) A parallel projection of a pair of solids having a given ratio is a pair of solids having the same ratio.

Though not essential for the purposes of the present article, the following consequence will serve to illustrate the principle of parallel projections:—

(8) A parallel projection of a curve, or of a surface of a given algebraical order, is a curve or a surface of the same order.

For example, all ellipsoids referred to co-ordinates parallel to any three conjugate diameters are parallel projections of each other and of a sphere referred to rectangular co-ordinates.

§ 17. Parallel Projections of Systems of Forces.—If a balanced system of forces be represented by a system of lines, then will every parallel projection of that system of lines represent a balanced system of forces.

For the condition of equilibrium of forces not parallel is that they shall be represented in direction and magnitude by the sides and diagonals of certain parallelograms, and of parallel forces that they shall divide certain straight lines in certain ratios; and the parallel projection of a parallelogram is a parallelogram, and that of a straight line divided in a given ratio is a straight line divided in the same ratio.

The resultant of a parallel projection of any system of forces is the projection of their resultant; and the centre of gravity of a parallel projection of a solid is the projection of the centre of gravity of the first solid.

§ 18. Principle of the Transformation of Structures.—Here we have the following theorem: If a structure of a given figure have stability of position under a system of forces represented by a given system of lines, then will any structure whose figure is a parallel projection of that of the first structure have stability of position under a system of forces represented by the corresponding projection of the first system of lines.

For in the second structure the weights, external pressures, and resistances will balance each other as in the first structure; the weights of the pieces and all other parallel systems of forces will have the same ratios as in the first structure; and the several centres of resistance will divide the depths of the joints in the same proportions as in the first structure.

If the first structure have stability of friction, the second structure will have stability of friction also, so long as the effect of the projection is not to increase the obliquity of the resistance at any joint beyond the angle of repose.

The lines representing the forces in the second figure show their relative directions and magnitudes. To find their absolute directions and magnitudes, a vertical line is to be drawn in the first figure, of such a length as to represent the weight of a particular portion of the structure. Then will the projection of that line in the projected figure indicate the vertical direction, and represent the weight of the part of the second structure corresponding to the before-mentioned portion of the first structure.

The foregoing “principle of the transformation of structures” was first announced, though in a somewhat less comprehensive form, to the Royal Society on the 6th of March 1856. It is useful in practice, by enabling the engineer easily to deduce the conditions of equilibrium and stability of structures of complex and unsymmetrical figures from those of structures of simple and symmetrical figures. By its aid, for example, the whole of the properties of elliptical arches, whether square or skew, whether level or sloping in their span, are at once deduced by projection from those of symmetrical circular arches, and the properties of ellipsoidal and elliptic-conoidal domes from those of hemispherical and circular-conoidal domes; and the figures of arches fitted to resist the thrust of earth, which is less horizontally than vertically in a certain given ratio, can be deduced by a projection from those of arches fitted to resist the thrust of a liquid, which is of equal intensity, horizontally and vertically.

§ 19. Conditions of Stiffness and Strength.—After the arrangement of the pieces of a structure and the size and figure of their joints or surfaces of contact have been determined so as to fulfil the conditions of stability,—conditions which depend mainly on the position and direction of the resultant or total load on each piece, and the relative magnitude of the loads on the different pieces—the dimensions of each piece singly have to be adjusted so as to fulfil the conditions of stiffness and strength—conditions which depend not only on the absolute magnitude of the load on each piece, and of the resistances by which it is balanced, but also on the mode of distribution of the load over the piece, and of the resistances over the joints.

The effect of the pressures applied to a piece, consisting of the load and the supporting resistances, is to force the piece into a state of strain or disfigurement, which increases until the elasticity, or resistance to strain, of the material causes it to exert a stress, or effort to recover its figure, equal and opposite to the system of applied pressures. The condition of stiffness is that the strain or disfigurement shall not be greater than is consistent with the purposes of the structure; and the condition of strength is that the stress shall be within the limits of that which the material can bear with safety against breaking. The ratio in which the utmost stress before breaking exceeds the safe working stress is called the factor of safety, and is determined empirically. It varies from three to twelve for various materials and structures. (See Strength of Materials.)

PART II. THEORY OF MACHINES

§ 20. Parts of a Machine: Frame and Mechanism.—The parts of a machine may be distinguished into two principal divisions,—the frame, or fixed parts, and the mechanism, or moving parts. The frame is a structure which supports the pieces of the mechanism, and to a certain extent determines the nature of their motions.

The form and arrangement of the pieces of the frame depend upon the arrangement and the motions of the mechanism; the dimensions of the pieces of the frame required in order to give it stability and strength are determined from the pressures applied to it by means of the mechanism. It appears therefore that in general the mechanism is to be designed first and the frame afterwards, and that the designing of the frame is regulated by the principles of the stability of structures and of the strength and stiffness of materials,—care being taken to adapt the frame to the most severe load which can be thrown upon it at any period of the action of the mechanism.

Each independent piece of the mechanism also is a structure, and its dimensions are to be adapted, according to the principles of the strength and stiffness of materials, to the most severe load to which it can be subjected during the action of the machine.

§ 21. Definition and Division of the Theory of Machines.—From what has been said in the last section it appears that the department of the art of designing machines which has reference to the stability of the frame and to the stiffness and strength of the frame and mechanism is a branch of the art of construction. It is therefore to be separated from the theory of machines, properly speaking, which has reference to the action of machines considered as moving. In the action of a machine the following three things take place:—

Firstly, Some natural source of energy communicates motion and force to a piece or pieces of the mechanism, called the receiver of power or prime mover.

Secondly, The motion and force are transmitted from the prime mover through the train of mechanism to the working piece or pieces, and during that transmission the motion and force are modified in amount and direction, so as to be rendered suitable for the purpose to which they are to be applied.

Thirdly, The working piece or pieces by their motion, or by their motion and force combined, produce some useful effect.

Such are the phenomena of the action of a machine, arranged in the order of causation. But in studying or treating of the theory of machines, the order of simplicity is the best; and in this order the first branch of the subject is the modification of motion and force by the train of mechanism; the next is the effect or purpose of the machine; and the last, or most complex, is the action of the prime mover.

The modification of motion and the modification of force take place together, and are connected by certain laws; but in the study of the theory of machines, as well as in that of pure mechanics, much advantage has been gained in point of clearness and simplicity by first considering alone the principles of the modification of motion, which are founded upon what is now known as Kinematics, and afterwards considering the principles of the combined modification of motion and force, which are founded both on geometry and on the laws of dynamics. The separation of kinematics from dynamics is due mainly to G. Monge, Ampère and R. Willis.

The theory of machines in the present article will be considered under the following heads:—

I. Pure Mechanism, or Applied Kinematics; being the theory of machines considered simply as modifying motion.

II. Applied Dynamics; being the theory of machines considered as modifying both motion and force.

Chap. I. On Pure Mechanism

§ 22. Division of the Subject.—Proceeding in the order of simplicity, the subject of Pure Mechanism, or Applied Kinematics, may be thus divided:—

Division 1.—Motion of a point.
Division 2.—Motion of the surface of a fluid.
Division 3.—Motion of a rigid solid.
Division 4.—Motions of a pair of connected pieces, or of an “elementary combination” in mechanism.
Division 5.—Motions of trains of pieces of mechanism.
Division 6.—Motions of sets of more than two connected pieces, or of “aggregate combinations.”

A point is the boundary of a line, which is the boundary of a surface, which is the boundary of a volume. Points, lines and surfaces have no independent existence, and consequently those divisions of this chapter which relate to their motions are only preliminary to the subsequent divisions, which relate to the motions of bodies.

Division 1. Motion of a Point.

§ 23. Comparative Motion.—The comparative motion of two points is the relation which exists between their motions, without having regard to their absolute amounts. It consists of two elements,—the velocity ratio, which is the ratio of any two magnitudes bearing to each other the proportions of the respective velocities of the two points at a given instant, and the directional relation, which is the relation borne to each other by the respective directions of the motions of the two points at the same given instant.

It is obvious that the motions of a pair of points may be varied in any manner, whether by direct or by lateral deviation, and yet that their comparative motion may remain constant, in consequence of the deviations taking place in the same proportions, in the same directions and at the same instants for both points.

Robert Willis (1800–1875) has the merit of having been the first to simplify considerably the theory of pure mechanism, by pointing out that that branch of mechanics relates wholly to comparative motions.

The comparative motion of two points at a given instant is capable of being completely expressed by one of Sir William Hamilton’s Quaternions,—the “tensor” expressing the velocity ratio, and the “versor” the directional relation.

Graphical methods of analysis founded on this way of representing velocity and acceleration were developed by R. H. Smith in a paper communicated to the Royal Society of Edinburgh in 1885, and illustrations of the method will be found below.

Division 2. Motion of the Surface of a Fluid Mass.

§ 24. General Principle.—A mass of fluid is used in mechanism to transmit motion and force between two or more movable portions (called pistons or plungers) of the solid envelope or vessel in which the fluid is contained; and, when such transmission is the sole action, or the only appreciable action of the fluid mass, its volume is either absolutely constant, by reason of its temperature and pressure being maintained constant, or not sensibly varied.

Let a represent the area of the section of a piston made by a plane perpendicular to its direction of motion, and v its velocity, which is to be considered as positive when outward, and negative when inward. Then the variation of the cubic contents of the vessel in a unit of time by reason of the motion of one piston is va. The condition that the volume of the fluid mass shall remain unchanged requires that there shall be more than one piston, and that the velocities and areas of the pistons shall be connected by the equation—

Σ · va = 0.

(1)

§ 25. Comparative Motion of Two Pistons.—If there be but two pistons, whose areas are a₁ and a₂, and their velocities v₁ and v₂, their comparative motion is expressed by the equation—

v₂/v₁ = −a₁/a₂;

(2)

that is to say, their velocities are opposite as to inwardness and outwardness and inversely proportional to their areas.

§ 26. Applications: Hydraulic Press: Pneumatic Power-Transmitter.—In the hydraulic press the vessel consists of two cylinders, viz. the pump-barrel and the press-barrel, each having its piston, and of a passage connecting them having a valve opening towards the press-barrel. The action of the enclosed water in transmitting motion takes place during the inward stroke of the pump-plunger, when the above-mentioned valve is open; and at that time the press-plunger moves outwards with a velocity which is less than the inward velocity of the pump-plunger, in the same ratio that the area of the pump-plunger is less than the area of the press-plunger. (See Hydraulics.)

In the pneumatic power-transmitter the motion of one piston is transmitted to another at a distance by means of a mass of air contained in two cylinders and an intervening tube. When the pressure and temperature of the air can be maintained constant, this machine fulfils equation (2), like the hydraulic press. The amount and effect of the variations of pressure and temperature undergone by the air depend on the principles of the mechanical action of heat, or Thermodynamics (q.v.), and are foreign to the subject of pure mechanism.

Division 3. Motion of a Rigid Solid.

§ 27. Motions Classed.—In problems of mechanism, each solid piece of the machine is supposed to be so stiff and strong as not to undergo any sensible change of figure or dimensions by the forces applied to it—a supposition which is realized in practice if the machine is skilfully designed.

This being the case, the various possible motions of a rigid solid body may all be classed under the following heads: (1) Shifting or Translation; (2) Turning or Rotation; (3) Motions compounded of Shifting and Turning.

The most common forms for the paths of the points of a piece of mechanism, whose motion is simple shifting, are the straight line and the circle.

Shifting in a straight line is regulated either by straight fixed guides, in contact with which the moving piece slides, or by combinations of link-work, called parallel motions, which will be described in the sequel. Shifting in a straight line is usually reciprocating; that is to say, the piece, after shifting through a certain distance, returns to its original position by reversing its motion.

Circular shifting is regulated by attaching two or more points of the shifting piece to ends of equal and parallel rotating cranks, or by combinations of wheel-work to be afterwards described. As an example of circular shifting may be cited the motion of the coupling rod, by which the parallel and equal cranks upon two or more axles of a locomotive engine are connected and made to rotate simultaneously. The coupling rod remains always parallel to itself, and all its points describe equal and similar circles relatively to the frame of the engine, and move in parallel directions with equal velocities at the same instant.

§ 28. Rotation about a Fixed Axis: Lever, Wheel and Axle.—The fixed axis of a turning body is a line fixed relatively to the body and relatively to the fixed space in which the body turns. In mechanism it is usually the central line either of a rotating shaft or axle having journals, gudgeons, or pivots turning in fixed bearings, or of a fixed spindle or dead centre round which a rotating bush turns; but it may sometimes be entirely beyond the limits of the turning body. For example, if a sliding piece moves in circular fixed guides, that piece rotates about an ideal fixed axis traversing the centre of those guides.

Let the angular velocity of the rotation be denoted by α = dθ/dt, then the linear velocity of any point A at the distance r from the axis is αr; and the path of that point is a circle of the radius r described about the axis.

This is the principle of the modification of motion by the lever, which consists of a rigid body turning about a fixed axis called a fulcrum, and having two points at the same or different distances from that axis, and in the same or different directions, one of which receives motion and the other transmits motion, modified in direction and velocity according to the above law.

In the wheel and axle, motion is received and transmitted by two cylindrical surfaces of different radii described about their common fixed axis of turning, their velocity-ratio being that of their radii.

Fig. 90.

§ 29. Velocity Ratio of Components of Motion.—As the distance between any two points in a rigid body is invariable, the projections of their velocities upon the line joining them must be equal. Hence it follows that, if A in fig. 90 be a point in a rigid body CD, rotating round the fixed axis F, the component of the velocity of A in any direction AP parallel to the plane of rotation is equal to the total velocity of the point m, found by letting fall Fm perpendicular to AP; that is to say, is equal to

α · Fm.

Hence also the ratio of the components of the velocities of two points A and B in the directions AP and BW respectively, both in the plane of rotation, is equal to the ratio of the perpendiculars Fm and Fn.

§ 30. Instantaneous Axis of a Cylinder rolling on a Cylinder.—Let a cylinder bbb, whose axis of figure is B and angular velocity γ, roll on a fixed cylinder aaa, whose axis of figure is A, either outside (as in fig. 91), when the rolling will be towards the same hand as the rotation, or inside (as in fig. 92), when the rolling will be towards the opposite hand; and at a given instant let T be the line of contact of the two cylindrical surfaces, which is at their common intersection with the plane AB traversing the two axes of figure.

The line T on the surface bbb has for the instant no velocity in a direction perpendicular to AB; because for the instant it touches, without sliding, the line T on the fixed surface aaa.

The line T on the surface bbb has also for the instant no velocity in the plane AB; for it has just ceased to move towards the fixed surface aaa, and is just about to begin to move away from that surface.

The line of contact T, therefore, on the surface of the cylinder bbb, is for the instant at rest, and is the “instantaneous axis” about which the cylinder bbb turns, together with any body rigidly attached to that cylinder.


Fig. 91.	Fig. 92.

To find, then, the direction and velocity at the given instant of any point P, either in or rigidly attached to the rolling cylinder T, draw the plane PT; the direction of motion of P will be perpendicular to that plane, and towards the right or left hand according to the direction of the rotation of bbb; and the velocity of P will be

v_P = γ·PT,

(3)

PT denoting the perpendicular distance of P from T. The path of P is a curve of the kind called epitrochoids. If P is in the circumference of bbb, that path becomes an epicycloid.

The velocity of any point in the axis of figure B is

v_B = γ·TB;

(4)

and the path of such a point is a circle described about A with the radius AB, being for outside rolling the sum, and for inside rolling the difference, of the radii of the cylinders.

Let α denote the angular velocity with which the plane of axes AB rotates about the fixed axis A. Then it is evident that

v_B = α·AB,

(5)

and consequently that

α = γ·TB/AB.

(6)

For internal rolling, as in fig. 92, AB is to be treated as negative, which will give a negative value to α, indicating that in this case the rotation of AB round A is contrary to that of the cylinder bbb.

The angular velocity of the rolling cylinder, relatively to the plane of axes AB, is obviously given by the equation—

β = γ − α	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$ ,
whence β = γ · TA/AB

(7)

care being taken to attend to the sign of α, so that when that is negative the arithmetical values of γ and α are to be added in order to give that of β.

The whole of the foregoing reasonings are applicable, not merely when aaa and bbb are actual cylinders, but also when they are the osculating cylinders of a pair of cylindroidal surfaces of varying curvature, A and B being the axes of curvature of the parts of those surfaces which are in contact for the instant under consideration.

Fig. 93.

§ 31. Instantaneous Axis of a Cone rolling on a Cone.—Let Oaa (fig. 93) be a fixed cone, OA its axis, Obb a cone rolling on it, OB the axis of the rolling cone, OT the line of contact of the two cones at the instant under consideration. By reasoning similar to that of § 30, it appears that OT is the instantaneous axis of rotation of the rolling cone.

Let γ denote the total angular velocity of the rotation of the cone B about the instantaneous axis, β its angular velocity about the axis OB relatively to the plane AOB, and α the angular velocity with which the plane AOB turns round the axis OA. It is required to find the ratios of those angular velocities.

Solution.—In OT take any point E, from which draw EC parallel to OA, and ED parallel to OB, so as to construct the parallelogram OCED. Then

OD : OC : OE : : α : β : γ.

(8)

Or because of the proportionality of the sides of triangles to the sines of the opposite angles,

sin TOB : sin TOA : sin AOB : : α : β : γ,

(8 a)

that is to say, the angular velocity about each axis is proportional to the sine of the angle between the other two.

Demonstration.—From C draw CF perpendicular to OA, and CG perpendicular to OE

Then CF = 2 ×	area EC	,
	CE

and CG = 2 ×	area ECO	;
	OE

∴∴ CG : CF :: CE = OD : OE.

Let v_c denote the linear velocity of the point C. Then

v_c = α · CF = γ · CG
∴ γ : α :: CF : CG :: OE : OD,

which is one part of the solution above stated. From E draw EH perpendicular to OB, and EK to OA. Then it can be shown as before that

EK : EH :: OC : OD.

Let v_E be the linear velocity of the point E fixed in the plane of axes AOB. Then

v_K = α · EK.

Now, as the line of contact OT is for the instant at rest on the rolling cone as well as on the fixed cone, the linear velocity of the point E fixed to the plane AOB relatively to the rolling cone is the same with its velocity relatively to the fixed cone. That is to say,

β · EH = v_E = α · EK;

therefore

α : β :: EH : EK :: OD : OC,

which is the remainder of the solution.

The path of a point P in or attached to the rolling cone is a spherical epitrochoid traced on the surface of a sphere of the radius OP. From P draw PQ perpendicular to the instantaneous axis. Then the motion of P is perpendicular to the plane OPQ, and its velocity is

v_P = γ · PQ.

(9)

The whole of the foregoing reasonings are applicable, not merely when A and B are actual regular cones, but also when they are the osculating regular cones of a pair of irregular conical surfaces, having a common apex at O.

§ 32. Screw-like or Helical Motion.—Since any displacement in a plane can be represented in general by a rotation, it follows that the only combination of translation and rotation, in which a complex movement which is not a mere rotation is produced, occurs when there is a translation perpendicular to the plane and parallel to the axis of rotation.

Fig. 94.

Such a complex motion is called screw-like or helical motion; for each point in the body describes a helix or screw round the axis of rotation, fixed or instantaneous as the case may be. To cause a body to move in this manner it is usually made of a helical or screw-like figure, and moves in a guide of a corresponding figure. Helical motion and screws adapted to it are said to be right- or left-handed according to the appearance presented by the rotation to an observer looking towards the direction of the translation. Thus the screw G in fig. 94 is right-handed.

The translation of a body in helical motion is called its advance. Let v_x denote the velocity of advance at a given instant, which of course is common to all the particles of the body; α the angular velocity of the rotation at the same instant; 2π = 6.2832 nearly, the circumference of a circle of the radius unity. Then

T = 2π/α

(10)

is the time of one turn at the rate α; and

p = v_xT = 2πv_x/α

(11)

is the pitch or advance per turn—a length which expresses the comparative motion of the translation and the rotation.

The pitch of a screw is the distance, measured parallel to its axis, between two successive turns of the same thread or helical projection.

Let r denote the perpendicular distance of a point in a body moving helically from the axis. Then

v_r = αr

(12)

is the component of the velocity of that point in a plane perpendicular to the axis, and its total velocity is

v = √ {v_x² + v_r²}.

(13)

The ratio of the two components of that velocity is

v_x/v_r = p/2πr = tan θ.

(14)

where θ denotes the angle made by the helical path of the point with a plane perpendicular to the axis.

Division 4. Elementary Combinations in Mechanism

§ 33. Definitions.—An elementary combination in mechanism consists of two pieces whose kinds of motion are determined by their connexion with the frame, and their comparative motion by their connexion with each other—that connexion being effected either by direct contact of the pieces, or by a connecting piece, which is not connected with the frame, and whose motion depends entirely on the motions of the pieces which it connects.

The piece whose motion is the cause is called the driver; the piece whose motion is the effect, the follower.

The connexion of each of those two pieces with the frame is in general such as to determine the path of every point in it. In the investigation, therefore, of the comparative motion of the driver and follower, in an elementary combination, it is unnecessary to consider relations of angular direction, which are already fixed by the connexion of each piece with the frame; so that the inquiry is confined to the determination of the velocity ratio, and of the directional relation, so far only as it expresses the connexion between forward and backward movements of the driver and follower. When a continuous motion of the driver produces a continuous motion of the follower, forward or backward, and a reciprocating motion a motion reciprocating at the same instant, the directional relation is said to be constant. When a continuous motion produces a reciprocating motion, or vice versa, or when a reciprocating motion produces a motion not reciprocating at the same instant, the directional relation is said to be variable.

The line of action or of connexion of the driver and follower is a line traversing a pair of points in the driver and follower respectively, which are so connected that the component of their velocity relatively to each other, resolved along the line of connexion, is null. There may be several or an indefinite number of lines of connexion, or there may be but one; and a line of connexion may connect either the same pair of points or a succession of different pairs.

§ 34. General Principle.—From the definition of a line of connexion it follows that the components of the velocities of a pair of connected points along their line of connexion are equal. And from this, and from the property of a rigid body, already stated in § 29, it follows, that the components along a line of connexion of all the points traversed by that line, whether in the driver or in the follower, are equal; and consequently, that the velocities of any pair of points traversed by a line of connexion are to each other inversely as the cosines, or directly as the secants, of the angles made by the paths of those points with the line of connexion.

The general principle stated above in different forms serves to solve every problem in which—the mode of connexion of a pair of pieces being given—it is required to find their comparative motion at a given instant, or vice versa.

Fig. 95.

§ 35. Application to a Pair of Shifting Pieces.—In fig. 95, let P₁P₂ be the line of connexion of a pair of pieces, each of which has a motion of translation or shifting. Through any point T in that line draw TV₁, TV₂, respectively parallel to the simultaneous direction of motion of the pieces; through any other point A in the line of connexion draw a plane perpendicular to that line, cutting TV₁, TV₂ in V₁, V₂; then, velocity of piece 1 : velocity of piece 2 :: TV₁ : TV₂. Also TA represents the equal components of the velocities of the pieces parallel to their line of connexion, and the line V₁V₂ represents their velocity relatively to each other.

§ 36. Application to a Pair of Turning Pieces.—Let α₁, α₂ be the angular velocities of a pair of turning pieces; θ₁, θ₂ the angles which their line of connexion makes with their respective planes of rotation; r₁, r₂ the common perpendiculars let fall from the line of connexion upon the respective axes of rotation of the pieces. Then the equal components, along the line of connexion, of the velocities of the points where those perpendiculars meet that line are—

α₁r₁ cos θ₁ = α₂r₂ cos θ₂;

consequently, the comparative motion of the pieces is given by the equation

α₂	=	r₁ cos θ₁	.
α₁		r₂ cos θ₂

(15)

§ 37. Application to a Shifting Piece and a Turning Piece.—Let a shifting piece be connected with a turning piece, and at a given instant let α₁ be the angular velocity of the turning piece, r₁ the common perpendicular of its axis of rotation and the line of connexion, θ₁ the angle made by the line of connexion with the plane of rotation, θ₂ the angle made by the line of connexion with the direction of motion of the shifting piece, v₂ the linear velocity of that piece. Then

α₁r₁ cos θ₁ = v₂ cos θ₂;

(16)

which equation expresses the comparative motion of the two pieces.

§ 38. Classification of Elementary Combinations in Mechanism.—The first systematic classification of elementary combinations in mechanism was that founded by Monge, and fully developed by Lanz and Bétancourt, which has been generally received, and has been adopted in most treatises on applied mechanics. But that classification is founded on the absolute instead of the comparative motions of the pieces, and is, for that reason, defective, as Willis pointed out in his admirable treatise On the Principles of Mechanism.

Willis’s classification is founded, in the first place, on comparative motion, as expressed by velocity ratio and directional relation, and in the second place, on the mode of connexion of the driver and follower. He divides the elementary combinations in mechanism into three classes, of which the characters are as follows:—

Class A: Directional relation constant; velocity ratio constant.

Class B: Directional relation constant; velocity ratio varying.

Class C: Directional relation changing periodically; velocity ratio constant or varying.

Each of those classes is subdivided by Willis into five divisions, of which the characters are as follows:—

Division	A:	Connexion	by	rolling contact.
”	B:	”	”	sliding contact.
”	C:	”	”	wrapping connectors.
”	D:	”	”	link-work.
”	E:	”	”	reduplication.

In the Reuleaux system of analysis of mechanisms the principle of comparative motion is generalized, and mechanisms apparently very diverse in character are shown to be founded on the same sequence of elementary combinations forming a kinematic chain. A short description of this system is given in § 80, but in the present article the principle of Willis’s classification is followed mainly. The arrangement is, however, modified by taking the mode of connexion as the basis of the primary classification, and by removing the subject of connexion by reduplication to the section of aggregate combinations. This modified arrangement is adopted as being better suited than the original arrangement to the limits of an article in an encyclopaedia; but it is not disputed that the original arrangement may be the best for a separate treatise.

§ 39. Rolling Contact: Smooth Wheels and Racks.—In order that two pieces may move in rolling contact, it is necessary that each pair of points in the two pieces which touch each other should at the instant of contact be moving in the same direction with the same velocity. In the case of two shifting pieces this would involve equal and parallel velocities for all the points of each piece, so that there could be no rolling, and, in fact, the two pieces would move like one; hence, in the case of rolling contact, either one or both of the pieces must rotate.

The direction of motion of a point in a turning piece being perpendicular to a plane passing through its axis, the condition that each pair of points in contact with each other must move in the same direction leads to the following consequences:—

I. That, when both pieces rotate, their axes, and all their points of contact, lie in the same plane.

II. That, when one piece rotates, and the other shifts, the axis of the rotating piece, and all the points of contact, lie in a plane perpendicular to the direction of motion of the shifting piece.

The condition that the velocity of each pair of points of contact must be equal leads to the following consequences:—

III. That the angular velocities of a pair of turning pieces in rolling contact must be inversely as the perpendicular distances of any pair of points of contact from the respective axes.

IV. That the linear velocity of a shifting piece in rolling contact with a turning piece is equal to the product of the angular velocity of the turning piece by the perpendicular distance from its axis to a pair of points of contact.

The line of contact is that line in which the points of contact are all situated. Respecting this line, the above Principles III. and IV. lead to the following conclusions:—

V. That for a pair of turning pieces with parallel axes, and for a turning piece and a shifting piece, the line of contact is straight, and parallel to the axes or axis; and hence that the rolling surfaces are either plane or cylindrical (the term “cylindrical” including all surfaces generated by the motion of a straight line parallel to itself).

VI. That for a pair of turning pieces with intersecting axes the line of contact is also straight, and traverses the point of intersection of the axes; and hence that the rolling surfaces are conical, with a common apex (the term “conical” including all surfaces generated by the motion of a straight line which traverses a fixed point).

Turning pieces in rolling contact are called smooth or toothless wheels. Shifting pieces in rolling contact with turning pieces may be called smooth or toothless racks.

VII. In a pair of pieces in rolling contact every straight line traversing the line of contact is a line of connexion.

§ 40. Cylindrical Wheels and Smooth Racks.—In designing cylindrical wheels and smooth racks, and determining their comparative motion, it is sufficient to consider a section of the pair of pieces made by a plane perpendicular to the axis or axes.

The points where axes intersect the plane of section are called centres; the point where the line of contact intersects it, the point of contact, or pitch-point; and the wheels are described as circular, elliptical, &c., according to the forms of their sections made by that plane.

When the point of contact of two wheels lies between their centres, they are said to be in outside gearing; when beyond their centres, in inside gearing, because the rolling surface of the larger wheel must in this case be turned inward or towards its centre.

From Principle III. of § 39 it appears that the angular velocity-ratio of a pair of wheels is the inverse ratio of the distances of the point of contact from the centres respectively.

Fig. 96.

For outside gearing that ratio is negative, because the wheels turn contrary ways; for inside gearing it is positive, because they turn the same way.

If the velocity ratio is to be constant, as in Willis’s Class A, the wheels must be circular; and this is the most common form for wheels.

If the velocity ratio is to be variable, as in Willis’s Class B, the figures of the wheels are a pair of rolling curves, subject to the condition that the distance between their poles (which are the centres of rotation) shall be constant.

The following is the geometrical relation which must exist between such a pair of curves:—

Let C₁, C₂ (fig. 96) be the poles of a pair of rolling curves; T₁, T₂ any pair of points of contact; U₁, U₂ any other pair of points of contact. Then, for every possible pair of points of contact, the two following equations must be simultaneously fulfilled:—

Sum of radii, C₁U₁ + C₂U₂ = C₁T₁ + C₂T₂ = constant;
arc, T₂U₂ = T₁U₁.

(17)

A condition equivalent to the above, and necessarily connected with it, is, that at each pair of points of contact the inclinations of the curves to their radii-vectores shall be equal and contrary; or, denoting by r₁, r₂ the radii-vectores at any given pair of points of contact, and s the length of the equal arcs measured from a certain fixed pair of points of contact—

dr₂/ds = −dr₁/ds;

(18)

which is the differential equation of a pair of rolling curves whose poles are at a constant distance apart.

For full details as to rolling curves, see Willis’s work, already mentioned, and Clerk Maxwell’s paper on Rolling Curves, Trans. Roy. Soc. Edin., 1849.

A rack, to work with a circular wheel, must be straight. To work with a wheel of any other figure, its section must be a rolling curve, subject to the condition that the perpendicular distance from the pole or centre of the wheel to a straight line parallel to the direction of the motion of the rack shall be constant. Let r₁ be the radius-vector of a point of contact on the wheel, x₂ the ordinate from the straight line before mentioned to the corresponding point of contact on the rack. Then

dx₂/ds = −dr₁/ds

(19)

is the differential equation of the pair of rolling curves.

To illustrate this subject, it may be mentioned that an ellipse rotating about one focus rolls completely round in outside gearing with an equal and similar ellipse also rotating about one focus, the distance between the axes of rotation being equal to the major axis of the ellipses, and the velocity ratio varying from (1 + eccentricity)/(1 − eccentricity) to (1 − eccentricity)/(1 + eccentricity); an hyperbola rotating about its further focus rolls in inside gearing, through a limited arc, with an equal and similar hyperbola rotating about its nearer focus, the distance between the axes of rotation being equal to the axis of the hyperbolas, and the velocity ratio varying between (eccentricity + 1)/(eccentricity − 1) and unity; and a parabola rotating about its focus rolls with an equal and similar parabola, shifting parallel to its directrix.

Fig. 97.

§ 41. Conical or Bevel and Disk Wheels.—From Principles III. and VI. of § 39 it appears that the angular velocities of a pair of wheels whose axes meet in a point are to each other inversely as the sines of the angles which the axes of the wheels make with the line of contact. Hence we have the following construction (figs. 97 and 98).—Let O be the apex or point of intersection of the two axes OC₁, OC₂. The angular velocity ratio being given, it is required to find the line of contact. On OC₁, OC₂ take lengths OA₁, OA₂, respectively proportional to the angular velocities of the pieces on whose axes they are taken. Complete the parallelogram OA₁EA₂; the diagonal OET will be the line of contact required.

When the velocity ratio is variable, the line of contact will shift its position in the plane C₁OC₂, and the wheels will be cones, with eccentric or irregular bases. In every case which occurs in practice, however, the velocity ratio is constant; the line of contact is constant in position, and the rolling surfaces of the wheels are regular circular cones (when they are called bevel wheels); or one of a pair of wheels may have a flat disk for its rolling surface, as W₂ in fig. 98, in which case it is a disk wheel. The rolling surfaces of actual wheels consist of frusta or zones of the complete cones or disks, as shown by W₁, W₂ in figs. 97 and 98.

Fig. 98.

Fig. 99.

Fig. 100.

§ 42. Sliding Contact (lateral): Skew-Bevel Wheels.—An hyperboloid of revolution is a surface resembling a sheaf or a dice box, generated by the rotation of a straight line round an axis from which it is at a constant distance, and to which it is inclined at a constant angle. If two such hyperboloids E, F, equal or unequal, be placed in the closest possible contact, as in fig. 99, they will touch each other along one of the generating straight lines of each, which will form their line of contact, and will be inclined to the axes AG, BH in opposite directions. The axes will not be parallel, nor will they intersect each other.

The motion of two such hyperboloids, turning in contact with each other, has hitherto been classed amongst cases of rolling contact; but that classification is not strictly correct, for, although the component velocities of a pair of points of contact in a direction at right angles to the line of contact are equal, still, as the axes are parallel neither to each other nor to the line of contact, the velocities of a pair of points of contact have components along the line of contact which are unequal, and their difference constitutes a lateral sliding.

The directions and positions of the axes being given, and the required angular velocity ratio, the following construction serves to determine the line of contact, by whose rotation round the two axes respectively the hyperboloids are generated:—

In fig. 100, let B₁C₁, B₂C₂ be the two axes; B₁B₂ their common perpendicular. Through any point O in this common perpendicular draw OA₁ parallel to B₁C₁ and OA₂ parallel to B₂C₂; make those lines proportional to the angular velocities about the axes to which they are respectively parallel; complete the parallelogram OA₁EA₂, and draw the diagonal OE; divide B₁B₂ in D into two parts, inversely proportional to the angular velocities about the axes which they respectively adjoin; through D parallel to OE draw DT. This will be the line of contact.

A pair of thin frusta of a pair of hyperboloids are used in practice to communicate motion between a pair of axes neither parallel nor intersecting, and are called skew-bevel wheels.

In skew-bevel wheels the properties of a line of connexion are not possessed by every line traversing the line of contact, but only by every line traversing the line of contact at right angles.

If the velocity ratio to be communicated were variable, the point D would alter its position, and the line DT its direction, at different periods of the motion, and the wheels would be hyperboloids of an eccentric or irregular cross-section; but forms of this kind are not used in practice.

§ 43. Sliding Contact (circular): Grooved Wheels.—As the adhesion or friction between a pair of smooth wheels is seldom sufficient to prevent their slipping on each other, contrivances are used to increase their mutual hold. One of those consists in forming the rim of each wheel into a series of alternate ridges and grooves parallel to the plane of rotation; it is applicable to cylindrical and bevel wheels, but not to skew-bevel wheels. The comparative motion of a pair of wheels so ridged and grooved is the same as that of a pair of smooth wheels in rolling contact, whose cylindrical or conical surfaces lie midway between the tops of the ridges and bottoms of the grooves, and those ideal smooth surfaces are called the pitch surfaces of the wheels.

The relative motion of the faces of contact of the ridges and grooves is a rotatory sliding or grinding motion, about the line of contact of the pitch-surfaces as an instantaneous axis.

Grooved wheels have hitherto been but little used.

§ 44. Sliding Contact (direct): Teeth of Wheels, their Number and Pitch.—The ordinary method of connecting a pair of wheels, or a wheel and a rack, and the only method which ensures the exact maintenance of a given numerical velocity ratio, is by means of a series of alternate ridges and hollows parallel or nearly parallel to the successive lines of contact of the ideal smooth wheels whose velocity ratio would be the same with that of the toothed wheels. The ridges are called teeth; the hollows, spaces. The teeth of the driver push those of the follower before them, and in so doing sliding takes place between them in a direction across their lines of contact.

The pitch-surfaces of a pair of toothed wheels are the ideal smooth surfaces which would have the same comparative motion by rolling contact that the actual wheels have by the sliding contact of their teeth. The pitch-circles of a pair of circular toothed wheels are sections of their pitch-surfaces, made for spur-wheels (that is, for wheels whose axes are parallel) by a plane at right angles to the axes, and for bevel wheels by a sphere described about the common apex. For a pair of skew-bevel wheels the pitch-circles are a pair of contiguous rectangular sections of the pitch-surfaces. The pitch-point is the point of contact of the pitch-circles.

The pitch-surface of a wheel lies intermediate between the points of the teeth and the bottoms of the hollows between them. That part of the acting surface of a tooth which projects beyond the pitch-surface is called the face; that part which lies within the pitch-surface, the flank.

Teeth, when not otherwise specified, are understood to be made in one piece with the wheel, the material being generally cast-iron, brass or bronze. Separate teeth, fixed into mortises in the rim of the wheel, are called cogs. A pinion is a small toothed wheel; a trundle is a pinion with cylindrical staves for teeth.

The radius of the pitch-circle of a wheel is called the geometrical radius; a circle touching the ends of the teeth is called the addendum circle, and its radius the real radius; the difference between these radii, being the projection of the teeth beyond the pitch-surface, is called the addendum.

The distance, measured along the pitch-circle, from the face of one tooth to the face of the next, is called the pitch. The pitch and the number of teeth in wheels are regulated by the following principles:—

I. In wheels which rotate continuously for one revolution or more, it is obviously necessary that the pitch should be an aliquot part of the circumference.

In wheels which reciprocate without performing a complete revolution this condition is not necessary. Such wheels are called sectors.

II. In order that a pair of wheels, or a wheel and a rack, may work correctly together, it is in all cases essential that the pitch should be the same in each.

III. Hence, in any pair of circular wheels which work together, the numbers of teeth in a complete circumference are directly as the radii and inversely as the angular velocities.

IV. Hence also, in any pair of circular wheels which rotate continuously for one revolution or more, the ratio of the numbers of teeth and its reciprocal the angular velocity ratio must be expressible in whole numbers.

From this principle arise problems of a kind which will be referred to in treating of Trains of Mechanism.

V. Let n, N be the respective numbers of teeth in a pair of wheels, N being the greater. Let t, T be a pair of teeth in the smaller and larger wheel respectively, which at a particular instant work together. It is required to find, first, how many pairs of teeth must pass the line of contact of the pitch-surfaces before t and T work together again (let this number be called a); and, secondly, with how many different teeth of the larger wheel the tooth t will work at different times (let this number be called b); thirdly, with how many different teeth of the smaller wheel the tooth T will work at different times (let this be called c).

Case 1. If n is a divisor of N,

a = N; b = N/n; c = 1.

(20)

Case 2. If the greatest common divisor of N and n be d, a number less than n, so that n = md, N = Md; then

a = mN = Mn = Mmd; b = M; c = m.

(21)

Case 3. If N and n be prime to each other,

a = nN; b = N; c = n.

(22)

It is considered desirable by millwrights, with a view to the preservation of the uniformity of shape of the teeth of a pair of wheels, that each given tooth in one wheel should work with as many different teeth in the other wheel as possible. They therefore study that the numbers of teeth in each pair of wheels which work together shall either be prime to each other, or shall have their greatest common divisor as small as is consistent with a velocity ratio suited for the purposes of the machine.

§ 45. Sliding Contact: Forms of the Teeth of Spur-wheels and Racks.—A line of connexion of two pieces in sliding contact is a line perpendicular to their surfaces at a point where they touch. Bearing this in mind, the principle of the comparative motion of a pair of teeth belonging to a pair of spur-wheels, or to a spur-wheel and a rack, is found by applying the principles stated generally in §§ 36 and 37 to the case of parallel axes for a pair of spur-wheels, and to the case of an axis perpendicular to the direction of shifting for a wheel and a rack.

In fig. 101, let C₁, C₂ be the centres of a pair of spur-wheels; B₁IB₁′, B₂IB₂′ portions of their pitch-circles, touching at I, the pitch-point. Let the wheel 1 be the driver, and the wheel 2 the follower.

Let D₁TB₁A₁, D₂TB₂A₂ be the positions, at a given instant, of the acting surfaces of a pair of teeth in the driver and follower respectively, touching each other at T; the line of connexion of those teeth is P₁P₂, perpendicular to their surfaces at T. Let C₁P₁, C₂P₂ be perpendiculars let
Fig. 101.fall from the centres of the wheels on the line of contact. Then, by § 36, the angular velocity-ratio is

a₂/a₁ = C₁P₁/C₂P₂.

(23)

The following principles regulate the forms of the teeth and their relative motions:—

I. The angular velocity ratio due to the sliding contact of the teeth will be the same with that due to the rolling contact of the pitch-circles, if the line of connexion of the teeth cuts the line of centres at the pitch-point.

For, let P₁P₂ cut the line of centres at I; then, by similar triangles,

α₁ : α₂ : : C₂P₂ : C₁P₁ : : IC₂ : : IC₁;

(24)

which is also the angular velocity ratio due to the rolling contact of the circles B₁IB₁′, B₂IB₂′.

This principle determines the forms of all teeth of spur-wheels. It also determines the forms of the teeth of straight racks, if one of the centres be removed, and a straight line EIE′, parallel to the direction of motion of the rack, and perpendicular to C₁IC₂, be substituted for a pitch-circle.

II. The component of the velocity of the point of contact of the teeth T along the line of connexion is

α₁ · C₁P₁ = α₂ · C₂P₂.

(25)

III. The relative velocity perpendicular to P₁P₂ of the teeth at their point of contact—that is, their velocity of sliding on each other—is found by supposing one of the wheels, such as 1, to be fixed, the line of centres C₁C₂ to rotate backwards round C₁ with the angular velocity α₁, and the wheel 2 to rotate round C₂ as before, with the angular velocity α₂ relatively to the line of centres C₁C₂, so as to have the same motion as if its pitch-circle rolled on the pitch-circle of the first wheel. Thus the relative motion of the wheels is unchanged; but 1 is considered as fixed, and 2 has the total motion, that is, a rotation about the instantaneous axis I, with the angular velocity α₁ + α₂. Hence the velocity of sliding is that due to this rotation about I, with the radius IT; that is to say, its value is

(α₁ + α₂) · IT;

(26)

so that it is greater the farther the point of contact is from the line of centres; and at the instant when that point passes the line of centres, and coincides with the pitch-point, the velocity of sliding is null, and the action of the teeth is, for the instant, that of rolling contact.

IV. The path of contact is the line traversing the various positions of the point T. If the line of connexion preserves always the same position, the path of contact coincides with it, and is straight; in other cases the path of contact is curved.

It is divided by the pitch-point I into two parts—the arc or line of approach described by T in approaching the line of centres, and the arc or line of recess described by T after having passed the line of centres.

During the approach, the flank D₁B₁ of the driving tooth drives the face D₂B₂ of the following tooth, and the teeth are sliding towards each other. During the recess (in which the position of the teeth is exemplified in the figure by curves marked with accented letters), the face B₁′A₁′ of the driving tooth drives the flank B₂′A₂′ of the following tooth, and the teeth are sliding from each other.

The path of contact is bounded where the approach commences by the addendum-circle of the follower, and where the recess terminates by the addendum-circle of the driver. The length of the path of contact should be such that there shall always be at least one pair of teeth in contact; and it is better still to make it so long that there shall always be at least two pairs of teeth in contact.

V. The obliquity of the action of the teeth is the angle EIT = IC₁, P₁ = IC₂P₂.

In practice it is found desirable that the mean value of the obliquity of action during the contact of teeth should not exceed 15°, nor the maximum value 30°.

It is unnecessary to give separate figures and demonstrations for inside gearing. The only modification required in the formulae is, that in equation (26) the difference of the angular velocities should be substituted for their sum.

§ 46. Involute Teeth.—The simplest form of tooth which fulfils the conditions of § 45 is obtained in the following manner (see fig. 102). Let C₁, C₂ be the centres of two wheels, B₁IB₁′, B₂IB₂′ their pitch-circles, I the pitch-point; let the obliquity of action of the teeth be constant, so that the same straight line P₁IP₂ shall represent at once the constant line of connexion of teeth and the path of contact. Draw C₁P₁, C₂P₂ perpendicular to P₁IP₂, and with those lines as radii describe about the centres of the wheels the circles D₁D₁′, D₂D₂′, called base-circles. It is evident that the radii of the base-circles bear to each other the same proportions as the radii of the pitch-circles, and also that

C₁P₁ = IC₁ · cos obliquity
C₂P₂ = IC₂ · cos obliquity.

\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}

(27)

Fig. 102.

(The obliquity which is found to answer best in practice is about 141/2°; its cosine is about 31/32, and its sine about 1/4. These values though not absolutely exact, are near enough to the truth for practical purposes.)

Suppose the base-circles to be a pair of circular pulleys connected by means of a cord whose course from pulley to pulley is P₁IP₂. As the line of connexion of those pulleys is the same as that of the proposed teeth, they will rotate with the required velocity ratio. Now, suppose a tracing point T to be fixed to the cord, so as to be carried along the path of contact P₁IP₂, that point will trace on a plane rotating along with the wheel 1 part of the involute of the base-circle D₁D₁′, and on a plane rotating along with the wheel 2 part of the involute of the base-circle D₂D₂′; and the two curves so traced will always touch each other in the required point of contact T, and will therefore fulfil the condition required by Principle I. of § 45.

Consequently, one of the forms suitable for the teeth of wheels is the involute of a circle; and the obliquity of the action of such teeth is the angle whose cosine is the ratio of the radius of their base-circle to that of the pitch-circle of the wheel.

All involute teeth of the same pitch work smoothly together.

To find the length of the path of contact on either side of the pitch-point I, it is to be observed that the distance between the fronts of two successive teeth, as measured along P₁IP₂, is less than the pitch in the ratio of cos obliquity : I; and consequently that, if distances equal to the pitch be marked off either way from I towards P₁ and P₂ respectively, as the extremities of the path of contact, and if, according to Principle IV. of § 45, the addendum-circles be described through the points so found, there will always be at least two pairs of teeth in action at once. In practice it is usual to make the path of contact somewhat longer, viz. about 2.4 times the pitch; and with this length of path, and the obliquity already mentioned of 141/2°, the addendum is about 3.1 of the pitch.

The teeth of a rack, to work correctly with wheels having involute teeth, should have plane surfaces perpendicular to the line of connexion, and consequently making with the direction of motion of the rack angles equal to the complement of the obliquity of action.

§ 47. Teeth for a given Path of Contact: Sang’s Method.—In the preceding section the form of the teeth is found by assuming a figure for the path of contact, viz. the straight line. Any other convenient figure may be assumed for the path of contact, and the corresponding forms of the teeth found by determining what curves a point T, moving along the assumed path of contact, will trace on two disks rotating round the centres of the wheels with angular velocities bearing that relation to the component velocity of T along TI, which is given by Principle II. of § 45, and by equation (25). This method of finding the forms of the teeth of wheels forms the subject of an elaborate and most interesting treatise by Edward Sang.

All wheels having teeth of the same pitch, traced from the same path of contact, work correctly together, and are said to belong to the same set.

Fig. 103.

§ 48. Teeth traced by Rolling Curves.—If any curve R (fig. 103) be rolled on the inside of the pitch-circle BB of a wheel, it appears, from § 30, that the instantaneous axis of the rolling curve at any instant will be at the point I, where it touches the pitch-circle for the moment, and that consequently the line AT, traced by a tracing-point T, fixed to the rolling curve upon the plane of the wheel, will be everywhere perpendicular to the straight line TI; so that the traced curve AT will be suitable for the flank of a tooth, in which T is the point of contact corresponding to the position I of the pitch-point. If the same rolling curve R, with the same tracing-point T, be rolled on the outside of any other pitch-circle, it will have the face of a tooth suitable to work with the flank AT.

In like manner, if either the same or any other rolling curve R′ be rolled the opposite way, on the outside of the pitch-circle BB, so that the tracing point T′ shall start from A, it will trace the face AT′ of a tooth suitable to work with a flank traced by rolling the same curve R′ with the same tracing-point T′ inside any other pitch-circle.

The figure of the path of contact is that traced on a fixed plane by the tracing-point, when the rolling curve is rotated in such a manner as always to touch a fixed straight line EIE (or E′I′E′, as the case may be) at a fixed point I (or I′).

If the same rolling curve and tracing-point be used to trace both the faces and the flanks of the teeth of a number of wheels of different sizes but of the same pitch, all those wheels will work correctly together, and will form a set. The teeth of a rack, of the same set, are traced by rolling the rolling curve on both sides of a straight line.

The teeth of wheels of any figure, as well as of circular wheels, may be traced by rolling curves on their pitch-surfaces; and all teeth of the same pitch, traced by the same rolling curve with the same tracing-point, will work together correctly if their pitch-surfaces are in rolling contact.

Fig. 104.

§ 49. Epicycloidal Teeth.—The most convenient rolling curve is the circle. The path of contact which it traces is identical with itself; and the flanks of the teeth are internal and their faces external epicycloids for wheels, and both flanks and faces are cycloids for a rack.

For a pitch-circle of twice the radius of the rolling or describing circle (as it is called) the internal epicycloid is a straight line, being, in fact, a diameter of the pitch-circle, so that the flanks of the teeth for such a pitch-circle are planes radiating from the axis. For a smaller pitch-circle the flanks would be convex and in-curved or under-cut, which would be inconvenient; therefore the smallest wheel of a set should have its pitch-circle of twice the radius of the describing circle, so that the flanks may be either straight or concave.

In fig. 104 let BB′ be part of the pitch-circle of a wheel with epicycloidal teeth; CIC′ the line of centres; I the pitch-point; EIE′ a straight tangent to the pitch-circle at that point; R the internal and R′ the equal external describing circles, so placed as to touch the pitch-circle and each other at I. Let DID′ be the path of contact, consisting of the arc of approach DI and the arc of recess ID′. In order that there may always be at least two pairs of teeth in action, each of those arcs should be equal to the pitch.

The obliquity of the action in passing the line of centres is nothing; the maximum obliquity is the angle EID = E′ID; and the mean obliquity is one-half of that angle.

It appears from experience that the mean obliquity should not exceed 15°; therefore the maximum obliquity should be about 30°; therefore the equal arcs DI and ID′ should each be one-sixth of a circumference; therefore the circumference of the describing circle should be six times the pitch.

It follows that the smallest pinion of a set in which pinion the flanks are straight should have twelve teeth.

§ 50. Nearly Epicycloidal Teeth: Willis’s Method.—To facilitate the drawing of epicycloidal teeth in practice, Willis showed how to approximate to their figure by means of two circular arcs—one concave, for the flank, and the other convex, for the face—and each having for its radius the mean radius of curvature of the epicycloidal arc. Willis’s formulae are founded on the following properties of epicycloids:—

Let R be the radius of the pitch-circle; r that of the describing circle; θ the angle made by the normal TI to the epicycloid at a given point T, with a tangent to the circle at I—that is, the obliquity of the action at T.

Then the radius of curvature of the epicycloid at T is—

For an internal epicycloid, ρ = 4r sin θ R − r/R − 2r	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$
For an external epicycloid, ρ′ = 4r sin θ R + r/R + 2r

(28)

Also, to find the position of the centres of curvature relatively to the pitch-circle, we have, denoting the chord of the describing circle TI by c, c = 2r sin θ; and therefore

For the flank, ρ − c = 2r sin θ R/R − 2r	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$
For the face, ρ′ − c = 2r sin θ R/R + 2r

(29)

For the proportions approved of by Willis, sin θ = 1/4 nearly; r = p (the pitch) nearly; c = 1/2p nearly; and, if N be the number of teeth in the wheel, r/R = 6/N nearly; therefore, approximately,

ρ − c = p/2 · N/N − 12	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$
ρ′ − c = p/2 · N/N + 12

(30)

Fig. 105.

Hence the following construction (fig. 105). Let BB be part of the pitch-circle, and a the point where a tooth is to cross it. Set off ab = ac − 1/2p. Draw radii bd, ce; draw fb, cg, making angles of 751/2° with those radii. Make bf = p′ − c, cg = p − c. From f, with the radius fa, draw the circular arc ah; from g, with the radius ga, draw the circular arc ak. Then ah is the face and ak the flank of the tooth required.

To facilitate the application of this rule, Willis published tables of ρ − c and ρ′ − c, and invented an instrument called the “odontograph.”

§ 51. Trundles and Pin-Wheels.—If a wheel or trundle have cylindrical pins or staves for teeth, the faces of the teeth of a wheel suitable for driving it are described by first tracing external epicycloids, by rolling the pitch-circle of the pin-wheel or trundle on the pitch-circle of the driving-wheel, with the centre of a stave for a tracing-point, and then drawing curves parallel to, and within the epicycloids, at a distance from them equal to the radius of a stave. Trundles having only six staves will work with large wheels.

§ 52. Backs of Teeth and Spaces.—Toothed wheels being in general intended to rotate either way, the backs of the teeth are made similar to the fronts. The space between two teeth, measured on the pitch-circle, is made about 1/6th part wider than the thickness of the tooth on the pitch-circle—that is to say,

Thickness of tooth	= 5/11 pitch;
Width of space	= 6/11 pitch.

The difference of 1/11 of the pitch is called the back-lash. The clearance allowed between the points of teeth and the bottoms of the spaces between the teeth of the other wheel is about one-tenth of the pitch.

§ 53. Stepped and Helical Teeth.—R. J. Hooke invented the making of the fronts of teeth in a series of steps with a view to increase the smoothness of action. A wheel thus formed resembles in shape a series of equal and similar toothed disks placed side by side, with the teeth of each a little behind those of the preceding disk. He also invented, with the same object, teeth whose fronts, instead of being parallel to the line of contact of the pitch-circles, cross it obliquely, so as to be of a screw-like or helical form. In wheel-work of this kind the contact of each pair of teeth commences at the foremost end of the helical front, and terminates at the aftermost end; and the helix is of such a pitch that the contact of one pair of teeth shall not terminate until that of the next pair has commenced.

Stepped and helical teeth have the desired effect of increasing the smoothness of motion, but they require more difficult and expensive workmanship than common teeth; and helical teeth are, besides, open to the objection that they exert a laterally oblique pressure, which tends to increase resistance, and unduly strain the machinery.

§ 54. Teeth of Bevel-Wheels.—The acting surfaces of the teeth of bevel-wheels are of the conical kind, generated by the motion of a line passing through the common apex of the pitch-cones, while its extremity is carried round the outlines of the cross section of the teeth made by a sphere described about that apex.

Fig. 106.

The operations of describing the exact figures of the teeth of bevel-wheels, whether by involutes or by rolling curves, are in every respect analogous to those for describing the figures of the teeth of spur-wheels, except that in the case of bevel-wheels all those operations are to be performed on the surface of a sphere described about the apex instead of on a plane, substituting poles for centres, and great circles for straight lines.

In consideration of the practical difficulty, especially in the case of large wheels, of obtaining an accurate spherical surface, and of drawing upon it when obtained, the following approximate method, proposed originally by Tredgold, is generally used:—

Let O (fig. 106) be the common apex of a pair of bevel-wheels; OB₁I, OB₂I their pitch cones; OC₁, OC₂ their axes; OI their line of contact. Perpendicular to OI draw A₁IA₂, cutting the axes in A₁, A₂; make the outer rims of the patterns and of the wheels portions of the cones A₁B₁I, A₂B₂I, of which the narrow zones occupied by the teeth will be sufficiently near to a spherical surface described about O for practical purposes. To find the figures of the teeth, draw on a flat surface circular arcs ID₁, ID₂, with the radii A₁I, A₂I; those arcs will be the developments of arcs of the pitch-circles B₁I, B₂I, when the conical surfaces A₁B₁I, A₂B₂I are spread out flat. Describe the figures of teeth for the developed arcs as for a pair of spur-wheels; then wrap the developed arcs on the cones, so as to make them coincide with the pitch-circles, and trace the teeth on the conical surfaces.

§ 55. Teeth of Skew-Bevel Wheels.—The crests of the teeth of a skew-bevel wheel are parallel to the generating straight line of the hyperboloidal pitch-surface; and the transverse sections of the teeth at a given pitch-circle are similar to those of the teeth of a bevel-wheel whose pitch surface is a cone touching the hyperboloidal surface at the given circle.

§ 56. Cams.—A cam is a single tooth, either rotating continuously or oscillating, and driving a sliding or turning piece either constantly or at intervals. All the principles which have been stated in § 45 as being applicable to teeth are applicable to cams; but in designing cams it is not usual to determine or take into consideration the form of the ideal pitch-surface, which would give the same comparative motion by rolling contact that the cam gives by sliding contact.

§ 57. Screws.—The figure of a screw is that of a convex or concave cylinder, with one or more helical projections, called threads, winding round it. Convex and concave screws are distinguished technically by the respective names of male and female; a short concave screw is called a nut; and when a screw is spoken of without qualification a convex screw is usually understood.

The relation between the advance and the rotation, which compose the motion of a screw working in contact with a fixed screw or helical guide, has already been demonstrated in § 32; and the same relation exists between the magnitudes of the rotation of a screw about a fixed axis and the advance of a shifting nut in which it rotates. The advance of the nut takes place in the opposite direction to that of the advance of the screw in the case in which the nut is fixed. The pitch or axial pitch of a screw has the meaning assigned to it in that section, viz. the distance, measured parallel to the axis, between the corresponding points in two successive turns of the same thread. If, therefore, the screw has several equidistant threads, the true pitch is equal to the divided axial pitch, as measured between two adjacent threads, multiplied by the number of threads.

If a helix be described round the screw, crossing each turn of the thread at right angles, the distance between two corresponding points on two successive turns of the same thread, measured along this normal helix, may be called the normal pitch; and when the screw has more than one thread the normal pitch from thread to thread may be called the normal divided pitch.

The distance from thread to thread, measured on a circle described about the axis of the screw, called the pitch-circle, may be called the circumferential pitch; for a screw of one thread it is one circumference; for a screw of n threads, (one circumference)/n.

Let r denote the radius of the pitch circle;
n the number of threads;
θ the obliquity of the threads to the pitch circle, and of the normal helix to the axis;

P_a	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ the axial $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$	pitch,
P_a/n = p_a		divided pitch;

P_n	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ the normal $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$	pitch,
P_n/n = p_n		divided pitch;

P_c the circumferential pitch;

then

p_c = p_a cot θ = p_n cos θ =	2πr/n,	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
p_a = p_n sec θ = p_c tan θ =	2πr tan θ/n,
p_n = p_c sin θ = p_a cos θ =	2πr sin θ/n.

(31)

If a screw rotates, the number of threads which pass a fixed point in one revolution is the number of threads in the screw.

A pair of convex screws, each rotating about its axis, are used as an elementary combination to transmit motion by the sliding contact of their threads. Such screws are commonly called endless screws. At the point of contact of the screws their threads must be parallel; and their line of connexion is the common perpendicular to the acting surfaces of the threads at their point of contact. Hence the following principles:—

I. If the screws are both right-handed or both left-handed, the angle between the directions of their axes is the sum of their obliquities; if one is right-handed and the other left-handed, that angle is the difference of their obliquities.

II. The normal pitch for a screw of one thread, and the normal divided pitch for a screw of more than one thread, must be the same in each screw.

III. The angular velocities of the screws are inversely as their numbers of threads.

Hooke’s wheels with oblique or helical teeth are in fact screws of many threads, and of large diameters as compared with their lengths.

The ordinary position of a pair of endless screws is with their axes at right angles to each other. When one is of considerably greater diameter than the other, the larger is commonly called in practice a wheel, the name screw being applied to the smaller only; but they are nevertheless both screws in fact.

To make the teeth of a pair of endless screws fit correctly and work smoothly, a hardened steel screw is made of the figure of the smaller screw, with its thread or threads notched so as to form a cutting tool; the larger screw, or “wheel,” is cast approximately of the required figure; the larger screw and the steel screw are fitted up in their proper relative position, and made to rotate in contact with each other by turning the steel screw, which cuts the threads of the larger screw to their true figure.

Fig. 107.

§ 58. Coupling of Parallel Axes—Oldham’s Coupling.—A coupling is a mode of connecting a pair of shafts so that they shall rotate in the same direction with the same mean angular velocity. If the axes of the shafts are in the same straight line, the coupling consists in so connecting their contiguous ends that they shall rotate as one piece; but if the axes are not in the same straight line combinations of mechanism are required. A coupling for parallel shafts which acts by sliding contact was invented by Oldham, and is represented in fig. 107. C₁, C₂ are the axes of the two parallel shafts; D₁, D₂ two disks facing each other, fixed on the ends of the two shafts respectively; E₁E₁ a bar sliding in a diametral groove in the face of D₁; E₂E₂ a bar sliding in a diametral groove in the face of D₂: those bars are fixed together at A, so as to form a rigid cross. The angular velocities of the two disks and of the cross are all equal at every instant; the middle point of the cross, at A, revolves in the dotted circle described upon the line of centres C₁C₂ as a diameter twice for each turn of the disks and cross; the instantaneous axis of rotation of the cross at any instant is at I, the point in the circle C₁C₂ diametrically opposite to A.

Oldham’s coupling may be used with advantage where the axes of the shafts are intended to be as nearly in the same straight line as is possible, but where there is some doubt as to the practibility or permanency of their exact continuity.

§ 59. Wrapping Connectors—Belts, Cords and Chains.—Flat belts of leather or of gutta percha, round cords of catgut, hemp or other material, and metal chains are used as wrapping connectors to transmit rotatory motion between pairs of pulleys and drums.

Belts (the most frequently used of all wrapping connectors) require nearly cylindrical pulleys. A belt tends to move towards that part of a pulley whose radius is greatest; pulleys for belts, therefore, are slightly swelled in the middle, in order that the belt may remain on the pulley, unless forcibly shifted. A belt when in motion is shifted off a pulley, or from one pulley on to another of equal size alongside of it, by pressing against that part of the belt which is moving towards the pulley.

Cords require either cylindrical drums with ledges or grooved pulleys.

Chains require pulleys or drums, grooved, notched and toothed, so as to fit the links of the chain.

Wrapping connectors for communicating continuous motion are endless.

Wrapping connectors for communicating reciprocating motion have usually their ends made fast to the pulleys or drums which they connect, and which in this case may be sectors.

Fig. 108.

The line of connexion of two pieces connected by a wrapping connector is the centre line of the belt, cord or chain; and the comparative motions of the pieces are determined by the principles of § 36 if both pieces turn, and of § 37 if one turns and the other shifts, in which latter case the motion must be reciprocating.

The pitch-line of a pulley or drum is a curve to which the line of connexion is always a tangent—that is to say, it is a curve parallel to the acting surface of the pulley or drum, and distant from it by half the thickness of the wrapping connector.

Pulleys and drums for communicating a constant velocity ratio are circular. The effective radius, or radius of the pitch-circle of a circular pulley or drum, is equal to the real radius added to half the thickness of the connector. The angular velocities of a pair of connected circular pulleys or drums are inversely as the effective radii.

A crossed belt, as in fig. 108, A, reverses the direction of the rotation communicated; an uncrossed belt, as in fig. 108, B, preserves that direction.

The length L of an endless belt connecting a pair of pulleys whose effective radii are r₁, r₂, with parallel axes whose distance apart is c, is given by the following formulae, in each of which the first term, containing the radical, expresses the length of the straight parts of the belt, and the remainder of the formula the length of the curved parts.

For a crossed belt:—

L = 2 √{c² − (r₁ + r₂)²} + (r₁ + r₂) ${\Big (}$ π − 2 sin⁻¹	r₁ + r₂	${\Big )}$ ;
	c

(32 A)

and for an uncrossed belt:—

L = 2 √{c² − (r₁ − r₂)² } + π (r₁ + r₂ + 2 (r₁ − r₂) sin⁻¹	r₁ − r₂	;
	c

(32 B)

in which r₁ is the greater radius, and r₂ the less.

When the axes of a pair of pulleys are not parallel, the pulleys should be so placed that the part of the belt which is approaching each pulley shall be in the plane of the pulley.

§ 60. Speed-Cones.—A pair of speed-cones (fig. 109) is a contrivance for varying and adjusting the velocity ratio communicated between a pair of parallel shafts by means of a belt. The speed-cones are either continuous cones or conoids, as A, B, whose velocity ratio can be varied gradually while they are in motion by shifting the belt, or sets of pulleys whose radii vary by steps, as C, D, in which case the velocity ratio can be changed by shifting the belt from one pair of pulleys to another.

Fig. 109.

In order that the belt may fit accurately in every possible position on a pair of speed-cones, the quantity L must be constant, in equations (32 A) or (32 B), according as the belt is crossed or uncrossed.

For a crossed belt, as in A and C, fig. 109, L depends solely on c and on r₁ + r₂. Now c is constant because the axes are parallel; therefore the sum of the radii of the pitch-circles connected in every position of the belt is to be constant. That condition is fulfilled by a pair of continuous cones generated by the revolution of two straight lines inclined opposite ways to their respective axes at equal angles.

For an uncrossed belt, the quantity L in equation (32 B) is to be made constant. The exact fulfilment of this condition requires the solution of a transcendental equation; but it may be fulfilled with accuracy sufficient for practical purposes by using, instead of (32 B) the following approximate equation:—

L nearly = 2c + π (r₁ + r₂) + (r₁ − r₂)² / c.

(33)

The following is the most convenient practical rule for the application of this equation:—

Let the speed-cones be equal and similar conoids, as in B, fig. 109, but with their large and small ends turned opposite ways. Let r₁ be the radius of the large end of each, r₂ that of the small end, r₀ that of the middle; and let v be the sagitta, measured perpendicular to the axes, of the arc by whose revolution each of the conoids is generated, or, in other words, the bulging of the conoids in the middle of their length. Then

v = r₀ − (r₁ + r₂) / 2 = (r₁ − r₂)² / 2πc.

(34)

2π = 6.2832; but 6 may be used in most practical cases without sensible error.

The radii at the middle and end being thus determined, make the generating curve an arc either of a circle or of a parabola.

§ 61. Linkwork in General.—The pieces which are connected by linkwork, if they rotate or oscillate, are usually called cranks, beams and levers. The link by which they are connected is a rigid rod or bar, which may be straight or of any other figure; the straight figure being the most favourable to strength, is always used when there is no special reason to the contrary. The link is known by various names in various circumstances, such as coupling-rod, connecting-rod, crank-rod, eccentric-rod, &c. It is attached to the pieces which it connects by two pins, about which it is free to turn. The effect of the link is to maintain the distance between the axes of those pins invariable; hence the common perpendicular of the axes of the pins is the line of connexion, and its extremities may be called the connected points. In a turning piece, the perpendicular let fall from its connected point upon its axis of rotation is the arm or crank-arm.

The axes of rotation of a pair of turning pieces connected by a link are almost always parallel, and perpendicular to the line of connexion in which case the angular velocity ratio at any instant is the reciprocal of the ratio of the common perpendiculars let fall from the line of connexion upon the respective axes of rotation.

If at any instant the direction of one of the crank-arms coincides with the line of connexion, the common perpendicular of the line of connexion and the axis of that crank-arm vanishes, and the directional relation of the motions becomes indeterminate. The position of the connected point of the crank-arm in question at such an instant is called a dead-point. The velocity of the other connected point at such an instant is null, unless it also reaches a dead-point at the same instant, so that the line of connexion is in the plane of the two axes of rotation, in which case the velocity ratio is indeterminate. Examples of dead-points, and of the means of preventing the inconvenience which they tend to occasion, will appear in the sequel.

§ 62. Coupling of Parallel Axes.—Two or more parallel shafts (such as those of a locomotive engine, with two or more pairs of driving wheels) are made to rotate with constantly equal angular velocities by having equal cranks, which are maintained parallel by a coupling-rod of such a length that the line of connexion is equal to the distance between the axes. The cranks pass their dead-points simultaneously. To obviate the unsteadiness of motion which this tends to cause, the shafts are provided with a second set of cranks at right angles to the first, connected by means of a similar coupling-rod, so that one set of cranks pass their dead points at the instant when the other set are farthest from theirs.

§ 63. Comparative Motion of Connected Points.—As the link is a rigid body, it is obvious that its action in communicating motion may be determined by finding the comparative motion of the connected points, and this is often the most convenient method of proceeding.

If a connected point belongs to a turning piece, the direction of its motion at a given instant is perpendicular to the plane containing the axis and crank-arm of the piece. If a connected point belongs to a shifting piece, the direction of its motion at any instant is given, and a plane can be drawn perpendicular to that direction.

The line of intersection of the planes perpendicular to the paths of the two connected points at a given instant is the instantaneous axis of the link at that instant; and the velocities of the connected points are directly as their distances from that axis.

Fig. 110.

In drawing on a plane surface, the two planes perpendicular to the paths of the connected points are represented by two lines (being their sections by a plane normal to them), and the instantaneous axis by a point (fig. 110); and, should the length of the two lines render it impracticable to produce them until they actually intersect, the velocity ratio of the connected points may be found by the principle that it is equal to the ratio of the segments which a line parallel to the line of connexion cuts off from any two lines drawn from a given point, perpendicular respectively to the paths of the connected points.

To illustrate this by one example. Let C₁ be the axis, and T₁ the connected point of the beam of a steam-engine; T₁T₂ the connecting or crank-rod; T₂ the other connected point, and the centre of the crank-pin; C₂ the axis of the crank and its shaft. Let v₁ denote the velocity of T₁ at any given instant; v₂ that of T₂. To find the ratio of these velocities, produce C₁T₁, C₂T₂ till they intersect in K; K is the instantaneous axis of the connecting rod, and the velocity ratio is

v₁ : v₂ : : KT₁ : KT₂.

(35)

Should K be inconveniently far off, draw any triangle with its sides respectively parallel to C₁T₁, C₂T₂ and T₁T₂; the ratio of the two sides first mentioned will be the velocity ratio required. For example, draw C₂A parallel to C₁T₁, cutting T₁T₂ in A; then

v₁ : v₂ : : C₂A : C₂T₂.

(36)

§ 64. Eccentric.—An eccentric circular disk fixed on a shaft, and used to give a reciprocating motion to a rod, is in effect a crank-pin of sufficiently large diameter to surround the shaft, and so to avoid the weakening of the shaft which would arise from bending it so as to form an ordinary crank. The centre of the eccentric is its connected point; and its eccentricity, or the distance from that centre to the axis of the shaft, is its crank-arm.

An eccentric may be made capable of having its eccentricity altered by means of an adjusting screw, so as to vary the extent of the reciprocating motion which it communicates.

§ 65. Reciprocating Pieces—Stroke—Dead-Points.—The distance between the extremities of the path of the connected point in a reciprocating piece (such as the piston of a steam-engine) is called the stroke or length of stroke of that piece. When it is connected with a continuously turning piece (such as the crank of a steam-engine) the ends of the stroke of the reciprocating piece correspond to the dead-points of the path of the connected point of the turning piece, where the line of connexion is continuous with or coincides with the crank-arm.

Let S be the length of stroke of the reciprocating piece, L the length of the line of connexion, and R the crank-arm of the continuously turning piece. Then, if the two ends of the stroke be in one straight line with the axis of the crank,

S = 2R;

(37)

and if these ends be not in one straight line with that axis, then S, L − R, and L + R, are the three sides of a triangle, having the angle opposite S at that axis; so that, if θ be the supplement of the arc between the dead-points,

S² = 2 (L² + R²) − 2 (L² − R²) cos θ, cos θ =	2L² + 2R² − S²	.	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$
	2 (L² − R²)

(38)

Fig. 111.

§ 66. Coupling of Intersecting Axes—Hooke’s Universal Joint.—Intersecting axes are coupled by a contrivance of Hooke’s, known as the “universal joint,” which belongs to the class of linkwork (see fig. 111). Let O be the point of intersection of the axes OC₁, OC₂, and θ their angle of inclination to each other. The pair of shafts C₁, C₂ terminate in a pair of forks F₁, F₂ in bearings at the extremities of which turn the gudgeons at the ends of the arms of a rectangular cross, having its centre at O. This cross is the link; the connected points are the centres of the bearings F₁, F₂. At each instant each of those points moves at right angles to the central plane of its shaft and fork, therefore the line of intersection of the central planes of the two forks at any instant is the instantaneous axis of the cross, and the velocity ratio of the points F₁, F₂ (which, as the forks are equal, is also the angular velocity ratio of the shafts) is equal to the ratio of the distances of those points from that instantaneous axis. The mean value of that velocity ratio is that of equality, for each successive quarter-turn is made by both shafts in the same time; but its actual value fluctuates between the limits:—

	α₂/α₁ = 1/cos θ when F₁ is the plane of OC₁C₂	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$
and	α₂/α₁ = cos θ when F₂ is in that plane.

(39)

Its value at intermediate instants is given by the following equations: let φ₁, φ₂ be the angles respectively made by the central planes of the forks and shafts with the plane OC₁C₂ at a given instant; then

cos θ	= tan φ₁ tan φ₂,	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$
α₂/α₁ = −dφ₂/dφ₁	= tan φ₁ + cot φ₁/tan φ₂ + cot φ₂.

(40)

§ 67. Intermittent Linkwork—Click and Ratchet.—A click acting upon a ratchet-wheel or rack, which it pushes or pulls through a certain arc at each forward stroke and leaves at rest at each backward stroke, is an example of intermittent linkwork. During the forward stroke the action of the click is governed by the principles of linkwork; during the backward stroke that action ceases. A catch or pall, turning on a fixed axis, prevents the ratchet-wheel or rack from reversing its motion.

Division 5.—Trains of Mechanism.

§ 68. General Principles..—A train of mechanism consists of a series of pieces each of which is follower to that which drives it and driver to that which follows it.

The comparative motion of the first driver and last follower is obtained by combining the proportions expressing by their terms the velocity ratios and by their signs the directional relations of the several elementary combinations of which the train consists.

§ 69. Trains of Wheelwork.—Let A₁, A₂, A₃, &c., A_m−1, A_m denote a series of axes, and α₁, α₂, α₃, &c., α_m−1, α_m their angular velocities. Let the axis A₁ carry a wheel of N₁ teeth, driving a wheel of n₂ teeth on the axis A₂, which carries also a wheel of N₂ teeth, driving a wheel of n₃ teeth on the axis A₃, and so on; the numbers of teeth in drivers being denoted by N′s, and in followers by n’s, and the axes to which the wheels are fixed being denoted by numbers. Then the resulting velocity ratio is denoted by

α_m	=	α₂	·	α₃	· &c. . . .	α_m	=	N₁ · N₂ . . . &c. . . . N_m−1	;
α₁		α₁		α₂		α_m−1		n₂ · n₃ . . . &c. . . . n_m

(41)

that is to say, the velocity ratio of the last and first axes is the ratio of the product of the numbers of teeth in the drivers to the product of the numbers of teeth in the followers.

Supposing all the wheels to be in outside gearing, then, as each elementary combination reverses the direction of rotation, and as the number of elementary combinations m − 1 is one less than the number of axes m, it is evident that if m is odd the direction of rotation is preserved, and if even reversed.

It is often a question of importance to determine the number of teeth in a train of wheels best suited for giving a determinate velocity ratio to two axes. It was shown by Young that, to do this with the least total number of teeth, the velocity ratio of each elementary combination should approximate as nearly as possible to 3.59. This would in many cases give too many axes; and, as a useful practical rule, it may be laid down that from 3 to 6 ought to be the limit of the velocity ratio of an elementary combination in wheel-work. The smallest number of teeth in a pinion for epicycloidal teeth ought to be twelve (see § 49)—but it is better, for smoothness of motion, not to go below fifteen; and for involute teeth the smallest number is about twenty-four.

Let B/C be the velocity ratio required, reduced to its least terms, and let B be greater than C. If B/C is not greater than 6, and C lies between the prescribed minimum number of teeth (which may be called t) and its double 2t, then one pair of wheels will answer the purpose, and B and C will themselves be the numbers required. Should B and C be inconveniently large, they are, if possible, to be resolved into factors, and those factors (or if they are too small, multiples of them) used for the number of teeth. Should B or C, or both, be at once inconveniently large and prime, then, instead of the exact ratio B/C some ratio approximating to that ratio, and capable of resolution into convenient factors, is to be found by the method of continued fractions.

Should B/C be greater than 6, the best number of elementary combinations m − 1 will lie between

1/log B − log Clog 6 and 1/log B − log Clog 6.

Then, if possible, B and C themselves are to be resolved each into m − 1 factors (counting 1 as a factor), which factors, or multiples of them, shall be not less than t nor greater than 6t; or if B and C contain inconveniently large prime factors, an approximate velocity ratio, found by the method of continued fractions, is to be substituted for B/C as before.

So far as the resultant velocity ratio is concerned, the order of the drivers N and of the followers n is immaterial: but to secure equable wear of the teeth, as explained in § 44, the wheels ought to be so arranged that, for each elementary combination, the greatest common divisor of N and n shall be either 1, or as small as possible.

§ 70. Double Hooke’s Coupling.—It has been shown in § 66 that the velocity ratio of a pair of shafts coupled by a universal joint fluctuates between the limits cos θ and 1/cos θ. Hence one or both of the shafts must have a vibratory and unsteady motion, injurious to the mechanism and framework. To obviate this evil a short intermediate shaft is introduced, making equal angles with the first and last shaft, coupled with each of them by a Hooke’s joint, and having its own two forks in the same plane. Let α₁, α₂, α₃ be the angular velocities of the first, intermediate, and last shaft in this train of two Hooke’s couplings. Then, from the principles of § 60 it is evident that at each instant α₂/α₁ = α₂/α₃, and consequently that α₃ = α₁; so that the fluctuations of angular velocity ratio caused by the first coupling are exactly neutralized by the second, and the first and last shafts have equal angular velocities at each instant.

§ 71. Converging and Diverging Trains of Mechanism.—Two or more trains of mechanism may converge into one—as when the two pistons of a pair of steam-engines, each through its own connecting-rod, act upon one crank-shaft. One train of mechanism may diverge into two or more—as when a single shaft, driven by a prime mover, carries several pulleys, each of which drives a different machine. The principles of comparative motion in such converging and diverging trains are the same as in simple trains.

Division 6.—Aggregate Combinations.

§ 72. General Principles.—Willis designated as “aggregate combinations” those assemblages of pieces of mechanism in which the motion of one follower is the resultant of component motions impressed on it by more than one driver. Two classes of aggregate combinations may be distinguished which, though not different in their actual nature, differ in the data which they present to the designer, and in the method of solution to be followed in questions respecting them.

Class I. comprises those cases in which a piece A is not carried directly by the frame C, but by another piece B, relatively to which the motion of A is given—the motion of the piece B relatively to the frame C being also given. Then the motion of A relatively to the frame C is the resultant of the motion of A relatively to B and of B relatively to C; and that resultant is to be found by the principles already explained in Division 3 of this Chapter §§ 27-32.

Class II. comprises those cases in which the motions of three points in one follower are determined by their connexions with two or with three different drivers.

This classification is founded on the kinds of problems arising from the combinations. Willis adopts another classification founded on the objects of the combinations, which objects he divides into two classes, viz. (1) to produce aggregate velocity, or a velocity which is the resultant of two or more components in the same path, and (2) to produce an aggregate path—that is, to make a given point in a rigid body move in an assigned path by communicating certain motions to other points in that body.

It is seldom that one of these effects is produced without at the same time producing the other; but the classification of Willis depends upon which of those two effects, even supposing them to occur together, is the practical object of the mechanism.

Fig. 112.

§ 73. Differential Windlass.—The axis C (fig. 112) carries a larger barrel AE and a smaller barrel DB, rotating as one piece with the angular velocity α₁ in the direction AE. The pulley or sheave FG has a weight W hung to its centre. A cord has one end made fast to and wrapped round the barrel AE; it passes from A under the sheave FG, and has the other end wrapped round and made fast to the barrel BD. Required the relation between the velocity of translation v₂ of W and the angular velocity α₁ of the differential barrel.

In this case v₂ is an aggregate velocity, produced by the joint action of the two drivers AE and BD, transmitted by wrapping connectors to FG, and combined by that sheave so as to act on the follower W, whose motion is the same with that of the centre of FG.

The velocity of the point F is α₁·AC, upward motion being considered positive. The velocity of the point G is −α₁·CB, downward motion being negative. Hence the instantaneous axis of the sheave FG is in the diameter FG, at the distance

FG2 · AC − BCAC + B

from the centre towards G; the angular velocity of the sheave is

α₂ = α₁ · AC + BCFG;

and, consequently, the velocity of its centre is

v₂ = α₂ ·	FG	·	AC − BC	=	α₁ (AC − BC)	,
	2		AC + BC		2

(42)

or the mean between the velocities of the two vertical parts of the cord.

If the cord be fixed to the framework at the point B, instead of being wound on a barrel, the velocity of W is half that of AF.

A case containing several sheaves is called a block. A fall-block is attached to a fixed point; a running-block is movable to and from a fall-block, with which it is connected by two or more plies of a rope. The whole combination constitutes a tackle or purchase. (See Pulleys for practical applications of these principles.)

§ 74. Differential Screw.—On the same axis let there be two screws of the respective pitches p₁ and p₂, made in one piece, and rotating with the angular velocity α. Let this piece be called B. Let the first screw turn in a fixed nut C, and the second in a sliding nut A. The velocity of advance of B relatively to C is (according to § 32) αp₁, and of A relatively to B (according to § 57) −αp₂; hence the velocity of A relatively to C is

α (p₁ − p₂),

(46)

being the same with the velocity of advance of a screw of the pitch p₁ − p₂. This combination, called Hunter’s or the differential screw, combines the strength of a large thread with the slowness of motion due to a small one.

Fig. 113.

§ 75. Epicyclic Trains.—The term epicyclic train is used by Willis to denote a train of wheels carried by an arm, and having certain rotations relatively to that arm, which itself rotates. The arm may either be driven by the wheels or assist in driving them. The comparative motions of the wheels and of the arm, and the aggregate paths traced by points in the wheels, are determined by the principles of the composition of rotations, and of the description of rolling curves, explained in §§ 30, 31.

§ 76. Link Motion.—A slide valve operated by a link motion receives an aggregate motion from the mechanism driving it. (See Steam-engine for a description of this and other types of mechanism of this class.)

§ 77. Parallel Motions.—A parallel motion is a combination of turning pieces in mechanism designed to guide the motion of a reciprocating piece either exactly or approximately in a straight line, so as to avoid the friction which arises from the use of straight guides for that purpose.

Fig. 113 represents an exact parallel motion, first proposed, it is believed, by Scott Russell. The arm CD turns on the axis C, and is jointed at D to the middle of the bar ADB, whose length is double of that of CD, and one of whose ends B is jointed to a slider, sliding in straight guides along the line CB. Draw BE perpendicular to CB, cutting CD produced in E, then E is the instantaneous axis of the bar ADB; and the direction of motion of A is at every instant perpendicular to EA—that is, along the straight line ACa. While the stroke of A is ACa, extending to equal distances on either side of C, and equal to twice the chord of the arc Dd, the stroke of B is only equal to twice the sagitta; and thus A is guided through a comparatively long stroke by the sliding of B through a comparatively short stroke, and by rotatory motions at the joints C, D, B.

§ 78.* An example of an approximate straight-line motion composed of three bars fixed to a frame is shown in fig. 114. It is due to P. L. Tchebichev of St Petersburg. The links AB and CD are equal in length and are centred respectively at A and C. The ends D and B are joined by a link DB.


Fig. 114.		Fig. 115.

If the respective lengths are made in the proportions AC : CD : DB = 1 : 1.3 : 0.4 the middle point P of DB will describe an approximately straight line parallel to AC within limits of length about equal to AC. C. N. Peaucellier, a French engineer officer, was the first, in 1864, to invent a linkwork with which an exact straight line could be drawn. The linkwork is shown in fig. 115, from which it will be seen that it consists of a rhombus of four equal bars ABCD, jointed at opposite corners with two equal bars BE and DE. The seventh link AF is equal in length to halt the distance EA when the mechanism is in its central position. The points E and F are fixed. It can be proved that the point C always moves in a straight line at right angles to the line EF. The more general property of the mechanism corresponding to proportions between the lengths FA and EF other than that of equality is that the curve described by the point C is the inverse of the curve described by A. There are other arrangements of bars giving straight-line motions, and these arrangements together with the general properties of mechanisms of this kind are discussed in How to Draw a Straight Line by A. B. Kempe (London, 1877).

Fig. 116.

Fig. 117.

§ 79.* The Pantograph.—If a parallelogram of links (fig. 116), be fixed at any one point a in any one of the links produced in either direction, and if any straight line be drawn from this point to cut the links in the points b and c, then the points a, b, c will be in a straight line for all positions of the mechanism, and if the point b be guided in any curve whatever, the point c will trace a similar curve to a scale enlarged in the ratio ab : ac. This property of the parallelogram is utilized in the construction of the pantograph, an instrument used for obtaining a copy of a map or drawing on a different scale. Professor J. J. Sylvester discovered that this property of the parallelogram is not confined to points lying in one line with the fixed point. Thus if b (fig. 117) be any point on the link CD, and if a point c be taken on the link DE such that the triangles CbD and DcE are similar and similarly situated with regard to their respective links, then the ratio of the distances ab and ac is constant, and the angle bac is constant for all positions of the mechanism; so that, if b is guided in any curve, the point c will describe a similar curve turned through an angle bac, the scales of the curves being in the ratio ab to ac. Sylvester called an instrument based on this property a plagiograph or a skew pantograph.

The combination of the parallelogram with a straight-line motion, for guiding one of the points in a straight line, is illustrated in Watt’s parallel motion for steam-engines. (See Steam-engine.)

§ 80.* The Reuleaux System of Analysis.—If two pieces, A and B, (fig. 118) are jointed together by a pin, the pin being fixed, say, to A, the only relative motion possible between the pieces is one of turning about the axis of the pin. Whatever motion the pair of pieces may have as a whole each separate piece shares in common, and this common motion in no way affects the relative motion of A and B. The motion of one piece is said to be completely constrained relatively to the other piece. Again, the pieces A and B (fig. 119) are paired together as a slide, and the only relative motion possible between them now is that of sliding, and therefore the motion of one relatively to the other is completely constrained. The pieces may be paired together as a screw and nut, in which case the relative motion is compounded of turning with sliding.


Fig. 118.		Fig. 119.

These combinations of pieces are known individually as kinematic pairs of elements, or briefly kinematic pairs. The three pairs mentioned above have each the peculiarity that contact between the two pieces forming the pair is distributed over a surface. Kinematic pairs which have surface contact are classified as lower pairs. Kinematic pairs in which contact takes place along a line only are classified as higher pairs. A pair of spur wheels in gear is an example of a higher pair, because the wheels have contact between their teeth along lines only.

A kinematic link of the simplest form is made by joining up the halves of two kinematic pairs by means of a rigid link. Thus if A₁B₁ represent a turning pair, and A₂B₂ a second turning pair, the rigid link formed by joining B₁ to B₂ is a kinematic link. Four links of this kind are shown in fig. 120 joined up to form a closed kinematic chain.

Fig. 120.

In order that a kinematic chain may be made the basis of a mechanism, every point in any link of it must be completely constrained with regard to every other link. Thus in fig. 120 the motion of a point a in the link A₁A₂ is completely constrained with regard to the link B₁B₄ by the turning pair A₁B₁, and it can be proved that the motion of a relatively to the non-adjacent link A₃A₄ is completely constrained, and therefore the four-bar chain, as it is called, can be and is used as the basis of many mechanisms. Another way of considering the question of constraint is to imagine any one link of the chain fixed; then, however the chain be moved, the path of a point, as a, will always remain the same. In a five-bar chain, if a is a point in a link non-adjacent to a fixed link, its path is indeterminate. Still another way of stating the matter is to say that, if any one link in the chain be fixed, any point in the chain must have only one degree of freedom. In a five-bar chain a point, as a, in a link non-adjacent to the fixed link has two degrees of freedom and the chain cannot therefore be used for a mechanism. These principles may be applied to examine any possible combination of links forming a kinematic chain in order to test its suitability for use as a mechanism. Compound chains are formed by the super-position of two or more simple chains, and in these more complex chains links will be found carrying three, or even more, halves of kinematic pairs. The Joy valve gear mechanism is a good example of a compound kinematic chain.

A chain built up of three turning pairs and one sliding pair, and known as the slider crank chain, is shown in fig. 121. It will be seen that the piece A₁ can only slide relatively to the piece B₁, and these two pieces therefore form the sliding pair. The piece A₁ carries the
Fig. 121. pin B₄, which is one half of the turning pair A₄ B₄. The piece A₁ together with the pin B₄ therefore form a kinematic link A₁B₄. The other links of the chain are, B₁A₂, B₂B₃, A₃A₄. In order to convert a chain into a mechanism it is necessary to fix one link in it. Any one of the links may be fixed. It follows therefore that there are as many possible mechanisms as there are links in the chain. For example, there is a well-known mechanism corresponding to the fixing of three of the four links of the slider crank chain (fig. 121). If the link d is fixed the chain at once becomes the mechanism of the ordinary steam engine; if the link e is fixed the mechanism obtained is that of the oscillating cylinder steam engine; if the link c is fixed the mechanism becomes either the Whitworth quick-return motion or the slot-bar motion, depending upon the proportion between the lengths of the links c and e. These different mechanisms are called inversions of the slider crank chain. What was the fixed framework of the mechanism in one case becomes a moving link in an inversion.

The Reuleaux system, therefore, consists essentially of the analysis of every mechanism into a kinematic chain, and since each link of the chain may be the fixed frame of a mechanism quite diverse mechanisms are found to be merely inversions of the same kinematic chain. Franz Reuleaux’s Kinematics of Machinery, translated by Sir A. B. W. Kennedy (London, 1876), is the book in which the system is set forth in all its completeness. In Mechanics of Machinery, by Sir A. B. W. Kennedy (London, 1886), the system was used for the first time in an English textbook, and now it has found its way into most modern textbooks relating to the subject of mechanism.

§ 81.* Centrodes, Instantaneous Centres, Velocity Image, Velocity Diagram.—Problems concerning the relative motion of the several parts of a kinematic chain may be considered in two ways, in addition to the way hitherto used in this article and based on the principle of § 34. The first is by the method of instantaneous centres, already exemplified in § 63, and rolling centroids, developed by Reuleaux in connexion with his method of analysis. The second is by means of Professor R. H. Smith’s method already referred to in § 23.

Method 1.—By reference to § 30 it will be seen that the motion of a cylinder rolling on a fixed cylinder is one of rotation about an instantaneous axis T, and that the velocity both as regards direction and magnitude is the same as if the rolling piece B were for the instant turning about a fixed axis coincident with the instantaneous axis. If the rolling cylinder B and its path A now be assumed to receive a common plane motion, what was before the velocity of the point P becomes the velocity of P relatively to the cylinder A, since the motion of B relatively to A still takes place about the instantaneous axis T. If B stops rolling, then the two cylinders continue to move as though they were parts of a rigid body. Notice that the shape of either rolling curve (fig. 91 or 92) may be found by considering each fixed in turn and then tracing out the locus of the instantaneous axis. These rolling cylinders are sometimes called axodes, and a section of an axode in a plane parallel to the plane of motion is called a centrode. The axode is hence the locus of the instantaneous axis, whilst the centrode is the locus of the instantaneous centre in any plane parallel to the plane of motion. There is no restriction on the shape of these rolling axodes; they may have any shape consistent with rolling (that is, no slipping is permitted), and the relative velocity of a point P is still found by considering it with regard to the instantaneous centre.

Reuleaux has shown that the relative motion of any pair of non-adjacent links of a kinematic chain is determined by the rolling together of two ideal cylindrical surfaces (cylindrical being used here in the general sense), each of which may be assumed to be formed by the extension of the material of the link to which it corresponds. These surfaces have contact at the instantaneous axis, which is now called the instantaneous axis of the two links concerned. To find the form of these surfaces corresponding to a particular pair of non-adjacent links, consider each link of the pair fixed in turn, then the locus of the instantaneous axis is the axode corresponding to the fixed link, or, considering a plane of motion only, the locus of the instantaneous centre is the centrode corresponding to the fixed link.

To find the instantaneous centre for a particular link corresponding to any given configuration of the kinematic chain, it is only necessary to know the direction of motion of any two points in the link, since lines through these points respectively at right angles to their directions of motion intersect in the instantaneous centre.

Fig. 122.

To illustrate this principle, consider the four-bar chain shown in fig. 122 made up of the four links, a, b, c, d. Let a be the fixed link, and consider the link c. Its extremities are moving respectively in directions at right angles to the links b and d; hence produce the links b and d to meet in the point O_ac. This point is the instantaneous centre of the motion of the link c relatively to the fixed link a, a fact indicated by the suffix ac placed after the letter O. The process being repeated for different values of the angle θ the curve through the several points Oac is the centroid which may be imagined as formed by an extension of the material of the link a. To find the corresponding centroid for the link c, fix c and repeat the process. Again, imagine d fixed, then the instantaneous centre O_bd of b with regard to d is found by producing the links c and a to intersect in O_bd, and the shapes of the centroids belonging respectively to the links b and d can be found as before. The axis about which a pair of adjacent links turn is a permanent axis, and is of course the axis of the pin which forms the point. Adding the centres corresponding to these several axes to the figure, it will be seen that there are six centres in connexion with the four-bar chain of which four are permanent and two are instantaneous or virtual centres; and, further, that whatever be the configuration of the chain these centres group themselves into three sets of three, each set lying on a straight line. This peculiarity is not an accident or a special property of the four-bar chain, but is an illustration of a general law regarding the subject discovered by Aronhold and Sir A. B. W. Kennedy independently, which may be thus stated: If any three bodies, a, b, c, have plane motion their three virtual centres, O_ab, O_bc, O_ac, are three points on one straight line. A proof of this will be found in The Mechanics of Machinery quoted above. Having obtained the set of instantaneous centres for a chain, suppose a is the fixed link of the chain and c any other link; then O_ac is the instantaneous centre of the two links and may be considered for the instant as the trace of an axis fixed to an extension of the link a about which c is turning, and thus problems of instantaneous velocity concerning the link c are solved as though the link c were merely rotating for the instant about a fixed axis coincident with the instantaneous axis.


Fig. 123.	Fig. 124.

Method 2.—The second method is based upon the vector representation of velocity, and may be illustrated by applying it to the four-bar chain. Let AD (fig. 123) be the fixed link. Consider the link BC, and let it be required to find the velocity of the point B having given the velocity of the point C. The principle upon which the solution is based is that the only motion which B can have relatively to an axis through C fixed to the link CD is one of turning about C. Choose any pole O (fig. 124). From this pole set out Oc to represent the velocity of the point C. The direction of this must be at right angles to the line CD, because this is the only direction possible to the point C. If the link BC moves without turning, Oc will also represent the velocity of the point B; but, if the link is turning, B can only move about the axis C, and its direction of motion is therefore at right angles to the line CB. Hence set out the possible direction of B′s motion in the velocity diagram, namely cb₁, at right angles to CB. But the point B must also move at right angles to AB in the case under consideration. Hence draw a line through O in the velocity diagram at right angles to AB to cut cb₁ in b. Then Ob is the velocity of the point b in magnitude and direction, and cb is the tangential velocity of B relatively to C. Moreover, whatever be the actual magnitudes of the velocities, the instantaneous velocity ratio of the points C and B is given by the ratio Oc/Ob.

A most important property of the diagram (figs. 123 and 124) is the following: If points X and x are taken dividing the link BC and the tangential velocity cb, so that cx:xb = CX:XB, then Ox represents the velocity of the point X in magnitude and direction. The line cb has been called the velocity image of the rod, since it may be looked upon as a scale drawing of the rod turned through 90° from the actual rod. Or, put in another way, if the link CB is drawn to scale on the new length cb in the velocity diagram (fig. 124), then a vector drawn from O to any point on the new drawing of the rod will represent the velocity of that point of the actual rod in magnitude and direction. It will be understood that there is a new velocity diagram for every new configuration of the mechanism, and that in each new diagram the image of the rod will be different in scale. Following the method indicated above for a kinematic chain in general, there will be obtained a velocity diagram similar to that of fig. 124 for each configuration of the mechanism, a diagram in which the velocity of the several points in the chain utilized for drawing the diagram will appear to the same scale, all radiating from the pole O. The lines joining the ends of these several velocities are the several tangential velocities, each being the velocity image of a link in the chain. These several images are not to the same scale, so that although the images may be considered to form collectively an image of the chain itself, the several members of this chain-image are to different scales in any one velocity diagram, and thus the chain-image is distorted from the actual proportions of the mechanism which it represents.

Fig. 125.

§ 82.* Acceleration Diagram. Acceleration Image.—Although it is possible to obtain the acceleration of points in a kinematic chain with one link fixed by methods which utilize the instantaneous centres of the chain, the vector method more readily lends itself to this purpose. It should be understood that the instantaneous centre considered in the preceding paragraphs is available only for estimating relative velocities; it cannot be used in a similar manner for questions regarding acceleration. That is to say, although the instantaneous centre is a centre of no velocity for the instant, it is not a centre of no acceleration, and in fact the centre of no acceleration is in general a quite different point. The general principle on which the method of drawing an acceleration diagram depends is that if a link CB (fig. 125) have plane motion and the acceleration of any point C be given in magnitude and direction, the acceleration of any other point B is the vector sum of the acceleration of C, the radial acceleration of B about C and the tangential acceleration of B about C. Let A be any origin, and let Ac represent the acceleration of the point C, ct the radial acceleration of B about C which must be in a direction parallel to BC, and tb the tangential acceleration of B about C, which must of course be at right angles to ct; then the vector sum of these three magnitudes is Ab, and this vector represents the acceleration of the point B. The directions of the radial and tangential accelerations of the point B are always known when the position of the link is assigned, since these are to be drawn respectively parallel to and at right angles to the link itself. The magnitude of the radial acceleration is given by the expression v ²/BC, v being the velocity of the point B about the point C. This velocity can always be found from the velocity diagram of the chain of which the link forms a part. If dω/dt is the angular acceleration of the link, dω/dt × CB is the tangential acceleration of the point B about the point C. Generally this tangential acceleration is unknown in magnitude, and it becomes part of the problem to find it. An important property of the diagram is that if points X and x are taken dividing the link CB and the whole acceleration of B about C, namely, cb in the same ratio, then Ax represents the acceleration of the point X in magnitude and direction; cb is called the acceleration image of the rod. In applying this principle to the drawing of an acceleration diagram for a mechanism, the velocity diagram of the mechanism must be first drawn in order to afford the means of calculating the several radial accelerations of the links. Then assuming that the acceleration of one point of a particular link of the mechanism is known together with the corresponding configuration of the mechanism, the two vectors Ac and ct can be drawn. The direction of tb, the third vector in the diagram, is also known, so that the problem is reduced to the condition that b is somewhere on the line tb. Then other conditions consequent upon the fact that the link forms part of a kinematic chain operate to enable b to be fixed. These methods are set forth and exemplified in Graphics, by R. H. Smith (London, 1889). Examples, completely worked out, of velocity and acceleration diagrams for the slider crank chain, the four-bar chain, and the mechanism of the Joy valve gear will be found in ch. ix. of Valves and Valve Gear Mechanism, by W. E. Dalby (London, 1906).

Chapter II. On Applied Dynamics.

§ 83. Laws of Motion.—The action of a machine in transmitting force and motion simultaneously, or performing work, is governed, in common with the phenomena of moving bodies in general, by two “laws of motion.”

Division 1. Balanced Forces in Machines of Uniform Velocity.

§ 84. Application of Force to Mechanism.—Forces are applied in units of weight; and the unit most commonly employed in Britain is the pound avoirdupois. The action of a force applied to a body is always in reality distributed over some definite space, either a volume of three dimensions or a surface of two. An example of a force distributed throughout a volume is the weight of the body itself, which acts on every particle, however small. The pressure exerted between two bodies at their surface of contact, or between the two parts of one body on either side of an ideal surface of separation, is an example of a force distributed over a surface. The mode of distribution of a force applied to a solid body requires to be considered when its stiffness and strength are treated of; but, in questions respecting the action of a force upon a rigid body considered as a whole, the resultant of the distributed force, determined according to the principles of statics, and considered as acting in a single line and applied at a single point, may, for the occasion, be substituted for the force as really distributed. Thus, the weight of each separate piece in a machine is treated as acting wholly at its centre of gravity, and each pressure applied to it as acting at a point called the centre of pressure of the surface to which the pressure is really applied.

§ 85. Forces applied to Mechanism Classed.—If θ be the obliquity of a force F applied to a piece of a machine—that is, the angle made by the direction of the force with the direction of motion of its point of application—then by the principles of statics, F may be resolved into two rectangular components, viz.:—

Along the direction of motion, P = F cos θ
Across the direction of motion, Q = F sin θ

\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}

(49)

If the component along the direction of motion acts with the motion, it is called an effort; if against the motion, a resistance. The component across the direction of motion is a lateral pressure; the unbalanced lateral pressure on any piece, or part of a piece, is deflecting force. A lateral pressure may increase resistance by causing friction; the friction so caused acts against the motion, and is a resistance, but the lateral pressure causing it is not a resistance. Resistances are distinguished into useful and prejudicial, according as they arise from the useful effect produced by the machine or from other causes.

§ 86. Work.—Work consists in moving against resistance. The work is said to be performed, and the resistance overcome. Work is measured by the product of the resistance into the distance through which its point of application is moved. The unit of work commonly used in Britain is a resistance of one pound overcome through a distance of one foot, and is called a foot-pound.

Work is distinguished into useful work and prejudicial or lost work, according as it is performed in producing the useful effect of the machine, or in overcoming prejudicial resistance.

§ 87. Energy: Potential Energy.—Energy means capacity for performing work. The energy of an effort, or potential energy, is measured by the product of the effort into the distance through which its point of application is capable of being moved. The unit of energy is the same with the unit of work.

When the point of application of an effort has been moved through a given distance, energy is said to have been exerted to an amount expressed by the product of the effort into the distance through which its point of application has been moved.

§ 88. Variable Effort and Resistance.—If an effort has different magnitudes during different portions of the motion of its point of application through a given distance, let each different magnitude of the effort P be multiplied by the length Δs of the corresponding portion of the path of the point of application; the sum

Σ · PΔs

(50)

is the whole energy exerted. If the effort varies by insensible gradations, the energy exerted is the integral or limit towards which that sum approaches continually as the divisions of the path are made smaller and more numerous, and is expressed by

\int

Pds.

(51)

Similar processes are applicable to the finding of the work performed in overcoming a varying resistance.

The work done by a machine can be actually measured by means of a dynamometer (q.v.).

§ 89. Principle of the Equality of Energy and Work.—From the first law of motion it follows that in a machine whose pieces move with uniform velocities the efforts and resistances must balance each other. Now from the laws of statics it is known that, in order that a system of forces applied to a system of connected points may be in equilibrium, it is necessary that the sum formed by putting together the products of the forces by the respective distances through which their points of application are capable of moving simultaneously, each along the direction of the force applied to it, shall be zero,—products being considered positive or negative according as the direction of the forces and the possible motions of their points of application are the same or opposite.

In other words, the sum of the negative products is equal to the sum of the positive products. This principle, applied to a machine whose parts move with uniform velocities, is equivalent to saying that in any given interval of time the energy exerted is equal to the work performed.

The symbolical expression of this law is as follows: let efforts be applied to one or any number of points of a machine; let any one of these efforts be represented by P, and the distance traversed by its point of application in a given interval of time by ds; let resistances be overcome at one or any number of points of the same machine; let any one of these resistances be denoted by R, and the distance traversed by its point of application in the given interval of time by ds′; then

Σ · Pds = Σ · Rds′.

(52)

The lengths ds, ds′ are proportional to the velocities of the points to whose paths they belong, and the proportions of those velocities to each other are deducible from the construction of the machine by the principles of pure mechanism explained in Chapter I.

§ 90. Static Equilibrium of Mechanisms.—The principle stated in the preceding section, namely, that the energy exerted is equal to the work performed, enables the ratio of the components of the forces acting in the respective directions of motion at two points of a mechanism, one being the point of application of the effort, and the other the point of application of the resistance, to be readily found. Removing the summation signs in equation (52) in order to restrict its application to two points and dividing by the common time interval during which the respective small displacements ds and ds′ were made, it becomes Pds/dt = Rds′/dt, that is, Pv = Rv′, which shows that the force ratio is the inverse of the velocity ratio. It follows at once that any method which may be available for the determination of the velocity ratio is equally available for the determination of the force ratio, it being clearly understood that the forces involved are the components of the actual forces resolved in the direction of motion of the points. The relation between the effort and the resistance may be found by means of this principle for all kinds of mechanisms, when the friction produced by the components of the forces across the direction of motion of the two points is neglected. Consider the following example:—

Fig. 126.

A four-bar chain having the configuration shown in fig. 126 supports a load P at the point x. What load is required at the point y to maintain the configuration shown, both loads being supposed to act vertically? Find the instantaneous centre O_bd, and resolve each load in the respective directions of motion of the points x and y; thus there are obtained the components P cos θ and R cos φ. Let the mechanism have a small motion; then, for the instant, the link b is turning about its instantaneous centre O_bd, and, if ω is its instantaneous angular velocity, the velocity of the point x is ωr, and the velocity of the point y is ωs. Hence, by the principle just stated, P cos θ × ωr = R cos φ × ωs. But, p and q being respectively the perpendiculars to the lines of action of the forces, this equation reduces to P_p = R_q, which shows that the ratio of the two forces may be found by taking moments about the instantaneous centre of the link on which they act.

The forces P and R may, however, act on different links. The general problem may then be thus stated: Given a mechanism of which r is the fixed link, and s and t any other two links, given also a force ƒ_s, acting on the link s, to find the force ƒ_t acting in a given direction on the link t, which will keep the mechanism in static equilibrium. The graphic solution of this problem may be effected thus:—

(1) Find the three virtual centres O_rs, O_rt, O_st, which must be three points in a line.

(2) Resolve ƒ_s into two components, one of which, namely, ƒ_q, passes through O_rs and may be neglected, and the other ƒ_p passes through O_st.

(3) Find the point M, where ƒ_p joins the given direction of ƒ_t, and resolve ƒ_p into two components, of which one is in the direction MO_rt, and may be neglected because it passes through O_rt, and the other is in the given direction of ƒ_t and is therefore the force required.

Fig. 127.

This statement of the problem and the solution is due to Sir A. B. W. Kennedy, and is given in ch. 8 of his Mechanics of Machinery. Another general solution of the problem is given in the Proc. Lond. Math. Soc. (1878–1879), by the same author. An example of the method of solution stated above, and taken from the Mechanics of Machinery, is illustrated by the mechanism fig. 127, which is an epicyclic train of three wheels with the first wheel r fixed. Let it be required to find the vertical force which must act at the pitch radius of the last wheel t to balance exactly a force ƒ_s acting vertically downwards on the arm at the point indicated in the figure. The two links concerned are the last wheel t and the arm s, the wheel r being the fixed link of the mechanism. The virtual centres O_rs, O_st are at the respective axes of the wheels r and t, and the centre O_rt divides the line through these two points externally in the ratio of the train of wheels. The figure sufficiently indicates the various steps of the solution.

The relation between the effort and the resistance in a machine to include the effect of friction at the joints has been investigated in a paper by Professor Fleeming Jenkin, “On the application of graphic methods to the determination of the efficiency of machinery” (Trans. Roy. Soc. Ed., vol. 28). It is shown that a machine may at any instant be represented by a frame of links the stresses in which are identical with the pressures at the joints of the mechanism. This self-strained frame is called the dynamic frame of the machine. The driving and resisting efforts are represented by elastic links in the dynamic frame, and when the frame with its elastic links is drawn the stresses in the several members of it may be determined by means of reciprocal figures. Incidentally the method gives the pressures at every joint of the mechanism.

§ 91. Efficiency.—The efficiency of a machine is the ratio of the useful work to the total work—that is, to the energy exerted—and is represented by

Σ · R_uds′	=	Σ · R_u ds′	=	Σ · R_u ds′	=	U	.
Σ · Rds′		Σ · R_u ds′ + Σ · R_p ds′		Σ · P ds		E

(53)

R_u being taken to represent useful and R_p prejudicial resistances. The more nearly the efficiency of a machine approaches to unity the better is the machine.

§ 92. Power and Effect.—The power of a machine is the energy exerted, and the effect the useful work performed, in some interval of time of definite length, such as a second, an hour, or a day.

The unit of power, called conventionally a horse-power, is 550 foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour.

§ 93. Modulus of a Machine.—In the investigation of the properties of a machine, the useful resistances to be overcome and the useful work to be performed are usually given. The prejudicial resistances arc generally functions of the useful resistances of the weights of the pieces of the mechanism, and of their form and arrangement; and, having been determined, they serve for the computation of the lost work, which, being added to the useful work, gives the expenditure of energy required. The result of this investigation, expressed in the form of an equation between this energy and the useful work, is called by Moseley the modulus of the machine. The general form of the modulus may be expressed thus—

E = U + φ (U, A) + ψ (A),

(54)

where A denotes some quantity or set of quantities depending on the form, arrangement, weight and other properties of the mechanism. Moseley, however, has pointed out that in most cases this equation takes the much more simple form of

E = (1 + A) U + B,

(55)

where A and B are constants, depending on the form, arrangement and weight of the mechanism. The efficiency corresponding to the last equation is

U	=	1	.
E		1 + A + B/U

(56)

§ 94. Trains of Mechanism.—In applying the preceding principles to a train of mechanism, it may either be treated as a whole, or it may be considered in sections consisting of single pieces, or of any convenient portion of the train—each section being treated as a machine, driven by the effort applied to it and energy exerted upon it through its line of connexion with the preceding section, performing useful work by driving the following section, and losing work by overcoming its own prejudicial resistances. It is evident that the efficiency of the whole train is the product of the efficiencies of its sections.

§ 95. Rotating Pieces: Couples of Forces.—It is often convenient to express the energy exerted upon and the work performed by a turning piece in a machine in terms of the moment of the couples of forces acting on it, and of the angular velocity. The ordinary British unit of moment is a foot-pound; but it is to be remembered that this is a foot-pound of a different sort from the unit of energy and work.

If a force be applied to a turning piece in a line not passing through its axis, the axis will press against its bearings with an equal and parallel force, and the equal and opposite reaction of the bearings will constitute, together with the first-mentioned force, a couple whose arm is the perpendicular distance from the axis to the line of action of the first force.

A couple is said to be right or left handed with reference to the observer, according to the direction in which it tends to turn the body, and is a driving couple or a resisting couple according as its tendency is with or against that of the actual rotation.

Let dt be an interval of time, α the angular velocity of the piece; then αdt is the angle through which it turns in the interval dt, and ds = vdt = rαdt is the distance through which the point of application of the force moves. Let P represent an effort, so that Pr is a driving couple, then

Pds = Pvdt = Prα dt = Mαdt

(57)

is the energy exerted by the couple M in the interval dt; and a similar equation gives the work performed in overcoming a resisting couple. When several couples act on one piece, the resultant of their moments is to be multiplied by the common angular velocity of the whole piece.

§ 96. Reduction of Forces to a given Point, and of Couples to the Axis of a given Piece.—In computations respecting machines it is often convenient to substitute for a force applied to a given point, or a couple applied to a given piece, the equivalent force or couple applied to some other point or piece; that is to say, the force or couple, which, if applied to the other point or piece, would exert equal energy or employ equal work. The principles of this reduction are that the ratio of the given to the equivalent force is the reciprocal of the ratio of the velocities of their points of application, and the ratio of the given to the equivalent couple is the reciprocal of the ratio of the angular velocities of the pieces to which they are applied.

These velocity ratios are known by the construction of the mechanism, and are independent of the absolute speed.

§ 97. Balanced Lateral Pressure of Guides and Bearings.—The most important part of the lateral pressure on a piece of mechanism is the reaction of its guides, if it is a sliding piece, or of the bearings of its axis, if it is a turning piece; and the balanced portion of this reaction is equal and opposite to the resultant of all the other forces applied to the piece, its own weight included. There may be or may not be an unbalanced component in this pressure, due to the deviated motion. Its laws will be considered in the sequel.

§ 98. Friction. Unguents.—The most important kind of resistance in machines is the friction or rubbing resistance of surfaces which slide over each other. The direction of the resistance of friction is opposite to that in which the sliding takes place. Its magnitude is the product of the normal pressure or force which presses the rubbing surfaces together in a direction perpendicular to themselves into a specific constant already mentioned in § 14, as the coefficient of friction, which depends on the nature and condition of the surfaces of the unguent, if any, with which they are covered. The total pressure exerted between the rubbing surfaces is the resultant of the normal pressure and of the friction, and its obliquity, or inclination to the common perpendicular of the surfaces, is the angle of repose formerly mentioned in § 14, whose tangent is the coefficient of friction. Thus, let N be the normal pressure, R the friction, T the total pressure, ƒ the coefficient of friction, and φ the angle of repose; then

ƒ = tan φ
R = ƒN = N tan φ = T sin φ

\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}

(58)

Experiments on friction have been made by Coulomb, Samuel Vince, John Rennie, James Wood, D. Rankine and others. The most complete and elaborate experiments are those of Morin, published in his Notions fondamentales de mécanique, and republished in Britain in the works of Moseley and Gordon.

The experiments of Beauchamp Tower (“Report of Friction Experiments,” Proc. Inst. Mech. Eng., 1883) showed that when oil is supplied to a journal by means of an oil bath the coefficient of friction varies nearly inversely as the load on the bearing, thus making the product of the load on the bearing and the coefficient of friction a constant. Mr Tower’s experiments were carried out at nearly constant temperature. The more recent experiments of Lasche (Zeitsch, Verein Deutsche Ingen., 1902, 46, 1881) show that the product of the coefficient of friction, the load on the bearing, and the temperature is approximately constant. For further information on this point and on Osborne Reynolds’s theory of lubrication see Bearings and Lubrication.

§ 99. Work of Friction. Moment of Friction.—The work performed in a unit of time in overcoming the friction of a pair of surfaces is the product of the friction by the velocity of sliding of the surfaces over each other, if that is the same throughout the whole extent of the rubbing surfaces. If that velocity is different for different portions of the rubbing surfaces, the velocity of each portion is to be multiplied by the friction of that portion, and the results summed or integrated.

When the relative motion of the rubbing surfaces is one of rotation, the work of friction in a unit of time, for a portion of the rubbing surfaces at a given distance from the axis of rotation, may be found by multiplying together the friction of that portion, its distance from the axis, and the angular velocity. The product of the force of friction by the distance at which it acts from the axis of rotation is called the moment of friction. The total moment of friction of a pair of rotating rubbing surfaces is the sum or integral of the moments of friction of their several portions.

To express this symbolically, let du represent the area of a portion of a pair of rubbing surfaces at a distance r from the axis of their relative rotation; p the intensity of the normal pressure at du per unit of area; and ƒ the coefficient of friction. Then the moment of friction of du is ƒprdu;

the total moment of friction is ƒ ∫ pr·du; and the work performed in a unit of time in overcoming friction, when the angular velocity is α, is αƒ ∫ pr·du.	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$

(59)

It is evident that the moment of friction, and the work lost by being performed in overcoming friction, are less in a rotating piece as the bearings are of smaller radius. But a limit is put to the diminution of the radii of journals and pivots by the conditions of durability and of proper lubrication, and also by conditions of strength and stiffness.

§ 100. Total Pressure between Journal and Bearing.—A single piece rotating with a uniform velocity has four mutually balanced forces applied to it: (l) the effort exerted on it by the piece which drives it; (2) the resistance of the piece which follows it—which may be considered for the purposes of the present question as useful resistance; (3) its weight; and (4) the reaction of its own cylindrical bearings. There are given the following data:—

The direction of the effort.

The direction of the useful resistance.

The weight of the piece and the direction in which it acts.

The magnitude of the useful resistance.

The radius of the bearing r.

The angle of repose φ, corresponding to the friction of the journal on the bearing.

And there are required the following:—

The direction of the reaction of the bearing.

The magnitude of that reaction.

The magnitude of the effort.

Let the useful resistance and the weight of the piece be compounded by the principles of statics into one force, and let this be called the given force.

Fig. 128.

The directions of the effort and of the given force are either parallel or meet in a point. If they are parallel, the direction of the reaction of the bearing is also parallel to them; if they meet in a point, the direction of the reaction traverses the same point.

Also, let AAA, fig. 128, be a section of the bearing, and C its axis; then the direction of the reaction, at the point where it intersects the circle AAA, must make the angle φ with the radius of that circle; that is to say, it must be a line such as PT touching the smaller circle BB, whose radius is r · sin φ. The side on which it touches that circle is determined by the fact that the obliquity of the reaction is such as to oppose the rotation.

Thus is determined the direction of the reaction of the bearing; and the magnitude of that reaction and of the effort are then found by the principles of the equilibrium of three forces already stated in § 7.

The work lost in overcoming the friction of the bearing is the same as that which would be performed in overcoming at the circumference of the small circle BB a resistance equal to the whole pressure between the journal and bearing.

In order to diminish that pressure to the smallest possible amount, the effort, and the resultant of the useful resistance, and the weight of the piece (called above the “given force”) ought to be opposed to each other as directly as is practicable consistently with the purposes of the machine.

An investigation of the forces acting on a bearing and journal lubricated by an oil bath will be found in a paper by Osborne Reynolds in the Phil. Trans. pt. i. (1886). (See also Bearings.)

§ 101. Friction of Pivots and Collars.—When a shaft is acted upon by a force tending to shift it lengthways, that force must be balanced by the reaction of a bearing against a pivot at the end of the shaft; or, if that be impossible, against one or more collars, or rings projecting from the body of the shaft. The bearing of the pivot is called a step or footstep. Pivots require great hardness, and are usually made of steel. The flat pivot is a cylinder of steel having a plane circular end as a rubbing surface. Let N be the total pressure sustained by a flat pivot of the radius r; if that pressure be uniformly distributed, which is the case when the rubbing surfaces of the pivot and its step are both true planes, the intensity of the pressure is

p = N / πr ²;

(60)

and, introducing this value into equation 59, the moment of friction of the flat pivot is found to be

23ƒNr

(61)

or two-thirds of that of a cylindrical journal of the same radius under the same normal pressure.

The friction of a conical pivot exceeds that of a flat pivot of the same radius, and under the same pressure, in the proportion of the side of the cone to the radius of its base.

The moment of friction of a collar is given by the formula—

2/3 ƒN	r ³ − r′³	,
	r ² − r′²

(62)

where r is the external and r′ the internal radius.

Fig. 129.

In the cup and ball pivot the end of the shaft and the step present two recesses facing each other, into which art fitted two shallow cups of steel or hard bronze. Between the concave spherical surfaces of those cups is placed a steel ball, being either a complete sphere or a lens having convex surfaces of a somewhat less radius than the concave surfaces of the cups. The moment of friction of this pivot is at first almost inappreciable from the extreme smallness of the radius of the circles of contact of the ball and cups, but, as they wear, that radius and the moment of friction increase.

It appears that the rapidity with which a rubbing surface wears away is proportional to the friction and to the velocity jointly, or nearly so. Hence the pivots already mentioned wear unequally at different points, and tend to alter their figures. Schiele has invented a pivot which preserves its original figure by wearing equally at all points in a direction parallel to its axis. The following are the principles on which this equality of wear depends:—

The rapidity of wear of a surface measured in an oblique direction is to the rapidity of wear measured normally as the secant of the obliquity is to unity. Let OX (fig. 129) be the axis of a pivot, and let RPC be a portion of a curve such that at any point P the secant of the obliquity to the normal of the curve of a line parallel to the axis is inversely proportional to the ordinate PY, to which the velocity of P is proportional. The rotation of that curve round OX will generate the form of pivot required. Now let PT be a tangent to the curve at P, cutting OX in T; PT = PY × secant obliquity, and this is to be a constant quantity; hence the curve is that known as the tractory of the straight line OX, in which PT = OR = constant. This curve is described by having a fixed straight edge parallel to OX, along which slides a slider carrying a pin whose centre is T. On that pin turns an arm, carrying at a point P a tracing-point, pencil or pen. Should the pen have a nib of two jaws, like those of an ordinary drawing-pen, the plane of the jaws must pass through PT. Then, while T is slid along the axis from O towards X, P will be drawn after it from R towards C along the tractory. This curve, being an asymptote to its axis, is capable of being indefinitely prolonged towards X; but in designing pivots it should stop before the angle PTY becomes less than the angle of repose of the rubbing surfaces, otherwise the pivot will be liable to stick in its bearing. The moment of friction of “Schiele’s anti-friction pivot,” as it is called, is equal to that of a cylindrical journal of the radius OR = PT the constant tangent, under the same pressure.

Records of experiments on the friction of a pivot bearing will be found in the Proc. Inst. Mech. Eng. (1891), and on the friction of a collar bearing ib. May 1888.

§ 102. Friction of Teeth.—Let N be the normal pressure exerted between a pair of teeth of a pair of wheels; s the total distance through which they slide upon each other; n the number of pairs of teeth which pass the plane of axis in a unit of time; then

nƒNs

(63)

is the work lost in unity of time by the friction of the teeth. The sliding s is composed of two parts, which take place during the approach and recess respectively. Let those be denoted by s₁ and s₂, so that s = s₁ + s₂. In § 45 the velocity of sliding at any instant has been given, viz. u = c (α₁ + α₂), where u is that velocity, c the distance T1 at any instant from the point of contact of the teeth to the pitch-point, and α₁, α₂ the respective angular velocities of the wheels.

Let v be the common velocity of the two pitch-circles, r₁, r₂, their radii; then the above equation becomes

u = cv ${\Big (}$	1	+	1	${\Big )}$ .
	r₁		r₂

To apply this to involute teeth, let c₁ be the length of the approach, c₂ that of the recess, u₁, the mean volocity of sliding during the approach, u₂ that during the recess; then

u₁ =

c₁v

{\Big (}

1

+

1

{\Big )}

; u₂ =

c₂v

{\Big (}

1

+

1

{\Big )}

2

r₁

r₂

2

r₁

r₂

also, let θ be the obliquity of the action; then the times occupied by the approach and recess are respectively

c₁	,	c₂	;
v cos θ		v cos θ

giving, finally, for the length of sliding between each pair of teeth,

s = s₁ + s₂ =	c₁² + c₂²	${\Big (}$	1	+	1	${\Big )}$
	2 cos θ		r₁		r₂

(64)

which, substituted in equation (63), gives the work lost in a unit of time by the friction of involute teeth. This result, which is exact for involute teeth, is approximately true for teeth of any figure.

For inside gearing, if r₁ be the less radius and r₂ the greater, 1/r₁ − 1/r₂ is to be substituted for 1/r₁ + 1/r₂.

§ 103. Friction of Cords and Belts.—A flexible band, such as a cord, rope, belt or strap, may be used either to exert an effort or a resistance upon a pulley round which it wraps. In either case the tangential force, whether effort or resistance, exerted between the band and the pulley is their mutual friction, caused by and proportional to the normal pressure between them.

Let T₁ be the tension of the free part of the band at that side towards which it tends to draw the pulley, or from which the pulley tends to draw it; T₂ the tension of the free part at the other side; T the tension of the band at any intermediate point of its arc of contact with the pulley; θ the ratio of the length of that arc to the radius of the pulley; dθ the ratio of an indefinitely small element of that arc to the radius; F = T₁ − T₂ the total friction between the band and the pulley; dF the elementary portion of that friction due to the elementary arc dθ; ƒ the coefficient of friction between the materials of the band and pulley.

Then, according to a well-known principle in statics, the normal pressure at the elementary arc dθ is T dθ, T being the mean tension of the band at that elementary arc; consequently the friction on that arc is dF = ƒT dθ. Now that friction is also the difference between the tensions of the band at the two ends of the elementary arc, or dT = dF = ƒT dθ; which equation, being integrated throughout the entire arc of contact, gives the following formulae:—

hyp log. T₁/T₂ = ƒθ	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
T₁/T₂ = eƒ^θ
F = T₁ − T₂ = T₁ (1 − e − ƒ^θ) = T₂ (eƒ^θ − 1)

(65)

When a belt connecting a pair of pulleys has the tensions of its two sides originally equal, the pulleys being at rest, and when the pulleys are next set in motion, so that one of them drives the other by means of the belt, it is found that the advancing side of the belt is exactly as much tightened as the returning side is slackened, so that the mean tension remains unchanged. Its value is given by this formula—

T₁ + T₂	=	eƒ^θ + 1
2	=	2 (eƒ^θ − 1)

(66)

which is useful in determining the original tension required to enable a belt to transmit a given force between two pulleys.

The equations 65 and 66 are applicable to a kind of brake called a friction-strap, used to stop or moderate the velocity of machines by being tightened round a pulley. The strap is usually of iron, and the pulley of hard wood.

Let α denote the arc of contact expressed in turns and fractions of a turn; then

θ = 6.2832a
eƒ^θ = number whose common logarithm is 2.7288ƒa $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$

(67)

See also Dynamometer for illustrations of the use of what are essentially friction-straps of different forms for the measurement of the brake horse-power of an engine or motor.

§ 104. Stiffness of Ropes.—Ropes offer a resistance to being bent, and, when bent, to being straightened again, which arises from the mutual friction of their fibres. It increases with the sectional area of the rope, and is inversely proportional to the radius of the curve into which it is bent.

The work lost in pulling a given length of rope over a pulley is found by multiplying the length of the rope in feet by its stiffness in pounds, that stiffness being the excess of the tension at the leading side of the rope above that at the following side, which is necessary to bend it into a curve fitting the pulley, and then to straighten it again.

The following empirical formulae for the stiffness of hempen ropes have been deduced by Morin from the experiments of Coulomb:—

Let F be the stiffness in pounds avoirdupois; d the diameter of the rope in inches, n = 48d ² for white ropes and 35d ² for tarred ropes; r the effective radius of the pulley in inches; T the tension in pounds. Then

For white ropes, F = n/r (0.0012 + 0.001026n + 0.0012T).	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$
For tarred ropes, F =n/r (0.006 + 0.001392n + 0.00168T).

(68)

§ 105. Friction-Couplings.—Friction is useful as a means of communicating motion where sudden changes either of force or velocity take place, because, being limited in amount, it may be so adjusted as to limit the forces which strain the pieces of the mechanism within the bounds of safety. Amongst contrivances for effecting this object are friction-cones. A rotating shaft carries upon a cylindrical portion of its figure a wheel or pulley turning loosely on it, and consequently capable of remaining at rest when the shaft is in motion. This pulley has fixed to one side, and concentric with it, a short frustum of a hollow cone. At a small distance from the pulley the shaft carries a short frustum of a solid cone accurately turned to fit the hollow cone. This frustum is made always to turn along with the shaft by being fitted on a square portion of it, or by means of a rib and groove, or otherwise, but is capable of a slight longitudinal motion, so as to be pressed into, or withdrawn from, the hollow cone by means of a lever. When the cones are pressed together or engaged, their friction causes the pulley to rotate along with the shaft; when they are disengaged, the pulley is free to stand still. The angle made by the sides of the cones with the axis should not be less than the angle of repose. In the friction-clutch, a pulley loose on a shaft has a hoop or gland made to embrace it more or less tightly by means of a screw; this hoop has short projecting arms or ears. A fork or clutch rotates along with the shaft, and is capable of being moved longitudinally by a handle. When the clutch is moved towards the hoop, its arms catch those of the hoop, and cause the hoop to rotate and to communicate its rotation to the pulley by friction. There are many other contrivances of the same class, but the two just mentioned may serve for examples.

§ 106. Heat of Friction: Unguents.—The work lost in friction is employed in producing heat. This fact is very obvious, and has been known from a remote period; but the exact determination of the proportion of the work lost to the heat produced, and the experimental proof that that proportion is the same under all circumstances and with all materials, solid, liquid and gaseous, are comparatively recent achievements of J. P. Joule. The quantity of work which produces a British unit of heat (or so much heat as elevates the temperature of one pound of pure water, at or near ordinary atmospheric temperatures, by 1° F.) is 772 foot-pounds. This constant, now designated as “Joule’s equivalent,” is the principal experimental datum of the science of thermodynamics.

A more recent determination (Phil. Trans., 1897), by Osborne Reynolds and W. M. Moorby, gives 778 as the mean value of Joule’s equivalent through the range of 32° to 212° F. See also the papers of Rowland in the Proc. Amer. Acad. (1879), and Griffiths, Phil. Trans. (1893).

The heat produced by friction, when moderate in amount, is useful in softening and liquefying thick unguents; but when excessive it is prejudicial, by decomposing the unguents, and sometimes even by softening the metal of the bearings, and raising their temperature so high as to set fire to neighbouring combustible matters.

Excessive heating is prevented by a constant and copious supply of a good unguent. The elevation of temperature produced by the friction of a journal is sometimes used as an experimental test of the quality of unguents. For modern methods of forced lubrication see Bearings.

§ 107. Rolling Resistance.—By the rolling of two surfaces over each other without sliding a resistance is caused which is called sometimes “rolling friction,” but more correctly rolling resistance. It is of the nature of a couple, resisting rotation. Its moment is found by multiplying the normal pressure between the rolling surfaces by an arm, whose length depends on the nature of the rolling surfaces, and the work lost in a unit of time in overcoming it is the product of its moment by the angular velocity of the rolling surfaces relatively to each other. The following are approximate values of the arm in decimals of a foot:—

Oak upon oak	0.006 (Coulomb).
Lignum vitae on oak	0.004 ”
Cast iron on cast iron	0.002 (Tredgold).

§ 108. Reciprocating Forces: Stored and Restored Energy.—When a force acts on a machine alternately as an effort and as a resistance, it may be called a reciprocating force. Of this kind is the weight of any piece in the mechanism whose centre of gravity alternately rises and falls; for during the rise of the centre of gravity that weight acts as a resistance, and energy is employed in lifting it to an amount expressed by the product of the weight into the vertical height of its rise; and during the fall of the centre of gravity the weight acts as an effort, and exerts in assisting to perform the work of the machine an amount of energy exactly equal to that which had previously been employed in lifting it. Thus that amount of energy is not lost, but has its operation deferred; and it is said to be stored when the weight is lifted, and restored when it falls.

In a machine of which each piece is to move with a uniform velocity, if the effort and the resistance be constant, the weight of each piece must be balanced on its axis, so that it may produce lateral pressure only, and not act as a reciprocating force. But if the effort and the resistance be alternately in excess, the uniformity of speed may still be preserved by so adjusting some moving weight in the mechanism that when the effort is in excess it may be lifted, and so balance and employ the excess of effort, and that when the resistance is in excess it may fall, and so balance and overcome the excess of resistance—thus storing the periodical excess of energy and restoring that energy to perform the periodical excess of work.

Other forces besides gravity may be used as reciprocating forces for storing and restoring energy—for example, the elasticity of a spring or of a mass of air.

In most of the delusive machines commonly called “perpetual motions,” of which so many are patented in each year, and which are expected by their inventors to perform work without receiving energy, the fundamental fallacy consists in an expectation that some reciprocating force shall restore more energy than it has been the means of storing.

Division 2. Deflecting Forces.

§ 109. Deflecting Force for Translation in a Curved Path.—In machinery, deflecting force is supplied by the tenacity of some piece, such as a crank, which guides the deflected body in its curved path, and is unbalanced, being employed in producing deflexion, and not in balancing another force.

§ 110. Centrifugal Force of a Rotating Body.—The centrifugal force exerted by a rotating body on its axis of rotation is the same in magnitude as if the mass of the body were concentrated at its centre of gravity, and acts in a plane passing through the axis of rotation and the centre of gravity of the body.

The particles of a rotating body exert centrifugal forces on each other, which strain the body, and tend to tear it asunder, but these forces balance each other, and do not affect the resultant centrifugal force exerted on the axis of rotation.^[3]

If the axis of rotation traverses the centre of gravity of the body, the centrifugal force exerted on that axis is nothing.

Hence, unless there be some reason to the contrary, each piece of a machine should be balanced on its axis of rotation; otherwise the centrifugal force will cause strains, vibration and increased friction, and a tendency of the shafts to jump out of their bearings.

§ 111. Centrifugal Couples of a Rotating Body.—Besides the tendency (if any) of the combined centrifugal forces of the particles of a rotating body to shift the axis of rotation, they may also tend to turn it out of its original direction. The latter tendency is called a centrifugal couple, and vanishes for rotation about a principal axis.

It is essential to the steady motion of every rapidly rotating piece in a machine that its axis of rotation should not merely traverse its centre of gravity, but should be a permanent axis; for otherwise the centrifugal couples will increase friction, produce oscillation of the shaft and tend to make it leave its bearings.

The principles of this and the preceding section are those which regulate the adjustment of the weight and position of the counterpoises which are placed between the spokes of the driving-wheels of locomotive engines.

(From Balancing of Engines, by
permission of Edward Arnold.)

Fig. 130.

§ 112.* Method of computing the position and magnitudes of balance weights which must be added to a given system of arbitrarily chosen rotating masses in order to make the common axis of rotation a permanent axis.—The method here briefly explained is taken from a paper by W. E. Dalby, “The Balancing of Engines with special reference to Marine Work,” Trans. Inst. Nav. Arch. (1899). Let the weight (fig. 130), attached to a truly turned disk, be rotated by the shaft OX, and conceive that the shaft is held in a bearing at one point, O. The force required to constrain the weight to move in a circle, that is the deviating force, produces an equal and opposite reaction on the shaft, whose amount F is equal to the centrifugal force Wa²r/g ℔, where r is the radius of the mass centre of the weight, and a is its angular velocity in radians per second. Transferring this force to the point O, it is equivalent to, (1) a force at O equal and parallel to F, and, (2) a centrifugal couple of Fa foot-pounds. In order that OX may be a permanent axis it is necessary that there should be a sufficient number of weights attached to the shaft and so distributed that when each is referred to the point O

(1) ΣF = 0
(2) ΣFa = 0 $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$

(a)

The plane through O to which the shaft is perpendicular is called the reference plane, because all the transferred forces act in that plane at the point O. The plane through the radius of the weight containing the axis OX is called the axial plane because it contains the forces forming the couple due to the transference of F to the reference plane. Substituting the values of F in (a) the two conditions become

(1) (W₁r₁ + W₂r₂ + W₃r₃ + . . .) α²g = 0

(2) (W₁a₁r₁ + W₂a₂r₂ + . . .) α²g = 0

(b)

In order that these conditions may obtain, the quantities in the brackets must be zero, since the factor α²/g is not zero. Hence finally the conditions which must be satisfied by the system of weights in order that the axis of rotation may be a permanent axis is

(1) (W₁r₁ + W₂r₂ + W₃r₃) = 0

(2) (W₁a₁r₁ + W₂a₂r₂ + W₃a₃r₃) = 0

(c)

It must be remembered that these are all directed quantities, and that their respective sums are to be taken by drawing vector polygons. In drawing these polygons the magnitude of the vector of the type Wr is the product Wr, and the direction of the vector is from the shaft outwards towards the weight W, parallel to the radius r. For the vector representing a couple of the type War, if the masses are all on the same side of the reference plane, the direction of drawing is from the axis outwards; if the masses are some on one side of the reference plane and some on the other side, the direction of drawing is from the axis outwards towards the weight for all masses on the one side, and from the mass inwards towards the axis for all weights on the other side, drawing always parallel to the direction defined by the radius r. The magnitude of the vector is the product War. The conditions (c) may thus be expressed: first, that the sum of the vectors Wr must form a closed polygon, and, second, that the sum of the vectors War must form a closed polygon. The general problem in practice is, given a system of weights attached to a shaft, to find the respective weights and positions of two balance weights or counterpoises which must be added to the system in order to make the shaft a permanent axis, the planes in which the balance weights are to revolve also being given. To solve this the reference plane must be chosen so that it coincides with the plane of revolution of one of the as yet unknown balance weights. The balance weight in this plane has therefore no couple corresponding to it. Hence by drawing a couple polygon for the given weights the vector which is required to close the polygon is at once found and from it the magnitude and position of the balance weight which must be added to the system to balance the couples follow at once. Then, transferring the product Wr corresponding with this balance weight to the reference plane, proceed to draw the force polygon. The vector required to close it will determine the second balance weight, the work may be checked by taking the reference plane to coincide with the plane of revolution of the second balance weight and then re-determining them, or by taking a reference plane anywhere and including the two balance weights trying if condition (c) is satisfied.

When a weight is reciprocated, the equal and opposite force required for its acceleration at any instant appears as an unbalanced force on the frame of the machine to which the weight belongs. In the particular case, where the motion is of the kind known as “simple harmonic” the disturbing force on the frame due to the reciprocation of the weight is equal to the component of the centrifugal force in the line of stroke due to a weight equal to the reciprocated weight supposed concentrated at the crank pin. Using this principle the method of finding the balance weights to be added to a given system of reciprocating weights in order to produce a system of forces on the frame continuously in equilibrium is exactly the same as that just explained for a system of revolving weights, because for the purpose of finding the balance weights each reciprocating weight may be supposed attached to the crank pin which operates it, thus forming an equivalent revolving system. The balance weights found as part of the equivalent revolving system when reciprocated by their respective crank pins form the balance weights for the given reciprocating system. These conditions may be exactly realized by a system of weights reciprocated by slotted bars, the crank shaft driving the slotted bars rotating uniformly. In practice reciprocation is usually effected through a connecting rod, as in the case of steam engines. In balancing the mechanism of a steam engine it is often sufficiently accurate to consider the motion of the pistons as simple harmonic, and the effect on the framework of the acceleration of the connecting rod may be approximately allowed for by distributing the weight of the rod between the crank pin and the piston inversely as the centre of gravity of the rod divides the distance between the centre of the cross head pin and the centre of the crank pin. The moving parts of the engine are then divided into two complete and independent systems, namely, one system of revolving weights consisting of crank pins, crank arms, &c., attached to and revolving with the crank shaft, and a second system of reciprocating weights consisting of the pistons, cross-heads, &c., supposed to be moving each in its line of stroke with simple harmonic motion. The balance weights are to be separately calculated for each system, the one set being added to the crank shaft as revolving weights, and the second set being included with the reciprocating weights and operated by a properly placed crank on the crank shaft. Balance weights added in this way to a set of reciprocating weights are sometimes called bob-weights. In the case of locomotives the balance weights required to balance the pistons are added as revolving weights to the crank shaft system, and in fact are generally combined with the weights required to balance the revolving system so as to form one weight, the counterpoise referred to in the preceding section, which is seen between the spokes of the wheels of a locomotive. Although this method balances the pistons in the horizontal plane, and thus allows the pull of the engine on the train to be exerted without the variation due to the reciprocation of the pistons, yet the force balanced horizontally is introduced vertically and appears as a variation of pressure on the rail. In practice about two-thirds of the reciprocating weight is balanced in order to keep this variation of rail pressure within safe limits. The assumption that the pistons of an engine move with simple harmonic motion is increasingly erroneous as the ratio of the length of the crank r, to the length of the connecting rod l increases. A more accurate though still approximate expression for the force on the frame due to the acceleration of the piston whose weight is W is given by

W	ω²r { cos θ +	r	cos 2θ }
g		l

The conditions regulating the balancing of a system of weights reciprocating under the action of accelerating forces given by the above expression are investigated in a paper by Otto Schlick, “On Balancing of Steam Engines,” Trans, Inst. Nav. Arch. (1900), and in a paper by W. E. Dalby, “On the Balancing of the Reciprocating Parts of Engines, including the Effect of the Connecting Rod” (ibid., 1901). A still more accurate expression than the above is obtained by expansion in a Fourier series, regarding which and its bearing on balancing engines see a paper by J. H. Macalpine, “A Solution of the Vibration Problem” (ibid., 1901). The whole subject is dealt with in a treatise, The Balancing of Engines, by W. E. Dalby (London, 1906). Most of the original papers on this subject of engine balancing are to be found in the Transactions of the Institution of Naval Architects.

§ 113.* Centrifugal Whirling of Shafts.—When a system of revolving masses is balanced so that the conditions of the preceding section are fulfilled, the centre of gravity of the system lies on the axis of revolution. If there is the slightest displacement of the centre of gravity of the system from the axis of revolution a force acts on the shaft tending to deflect it, and varies as the deflexion and as the square of the speed. If the shaft is therefore to revolve stably, this force must be balanced at any instant by the elastic resistance of the shaft to deflexion. To take a simple case, suppose a shaft, supported on two bearings to carry a disk of weight W at its centre, and let the centre of gravity of the disk be at a distance e from the axis of rotation, this small distance being due to imperfections of material or faulty construction. Neglecting the mass of the shaft itself, when the shaft rotates with an angular velocity a, the centrifugal force Wa²e/g will act upon the shaft and cause its axis to deflect from the axis of rotation a distance, y say. The elastic resistance evoked by this deflexion is proportional to the deflexion, so that if c is a constant depending upon the form, material and method of support of the shaft, the following equality must hold if the shaft is to rotate stably at the stated speed—

W	(y + e) a² = cy,
g

from which y = Wa²e / (gc − Wa²).

This expression shows that as a increases y increases until when Wa² = gc, y becomes infinitely large. The corresponding value of a, namely ${\sqrt {gc{\mbox{/W}}}}$ , is called the critical velocity of the shaft, and is the speed at which the shaft ceases to rotate stably and at which centrifugal whirling begins. The general problem is to find the value of a corresponding to all kinds of loadings on shafts supported in any manner. The question was investigated by Rankine in an article in the Engineer (April 9, 1869). Professor A. G. Greenhill treated the problem of the centrifugal whirling of an unloaded shaft with different supporting conditions in a paper “On the Strength of Shafting exposed both to torsion and to end thrust,” Proc. Inst. Mech. Eng. (1883). Professor S. Dunkerley (“On the Whirling and Vibration of Shafts,” Phil. Trans., 1894) investigated the question for the cases of loaded and unloaded shafts, and, owing to the complication arising from the application of the general theory to the cases of loaded shafts, devised empirical formulae for the critical speeds of shafts loaded with heavy pulleys, based generally upon the following assumption, which is stated for the case of a shaft carrying one pulley: If N₁, N₂ be the separate speeds of whirl of the shaft and pulley on the assumption that the effect of one is neglected when that of the other is under consideration, then the resulting speed of whirl due to both causes combined may be taken to be of the form N₁N₂ √(N²₁ + N₁²) where N means revolutions per minute. This form is extended to include the cases of several pulleys on the same shaft. The interesting and important part of the investigation is that a number of experiments were made on small shafts arranged in different ways and loaded in different ways, and the speed at which whirling actually occurred was compared with the speed calculated from formulae of the general type indicated above. The agreement between the observed and calculated values of the critical speeds was in most cases quite remarkable. In a paper by Dr C. Chree, “The Whirling and Transverse Vibrations of Rotating Shafts,” Proc. Phys. Soc. Lon., vol. 19 (1904); also Phil. Mag., vol. 7 (1904), the question is investigated from a new mathematical point of view, and expressions for the whirling of loaded shafts are obtained without the necessity of any assumption of the kind stated above. An elementary presentation of the problem from a practical point of view will be found in Steam Turbines, by Dr A. Stodola (London, 1905).

Fig. 131.

§ 114. Revolving Pendulum. Governors.—In fig. 131 AO represents an upright axis or spindle; B a weight called a bob, suspended by rod OB from a horizontal axis at O, carried by the vertical axis. When the spindle is at rest the bob hangs close to it; when the spindle rotates, the bob, being made to revolve round it, diverges until the resultant of the centrifugal force and the weight of the bob is a force acting at O in the direction OB, and then it revolves steadily in a circle. This combination is called a revolving, centrifugal, or conical pendulum. Revolving pendulums are usually constructed with pairs of rods and bobs, as OB, Ob, hung at opposite sides of the spindle, that the centrifugal forces exerted at the point O may balance each other.

In finding the position in which the bob will revolve with a given angular velocity, a, for most practical cases connected with machinery the mass of the rod may be considered as insensible compared with that of the bob. Let the bob be a sphere, and from the centre of that sphere draw BH = y perpendicular to OA. Let OH = z; let W be the weight of the bob, F its centrifugal force. Then the condition of its steady revolution is W : F :: z : y; that is to say, y/z = F/W = yα²/g; consequently

z = g/α²

(69)

Or, if n = α 2π = α/6.2832 be the number of turns or fractions of a turn in a second,

z =	g	=	0.8165 ft.	=	9.79771 in.	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$
	4π²n²		n²		n²

(70)

z is called the altitude of the pendulum.

Fig. 132.

If the rod of a revolving pendulum be jointed, as in fig. 132, not to a point in the vertical axis, but to the end of a projecting arm C, the position in which the bob will revolve will be the same as if the rod were jointed to the point O, where its prolongation cuts the vertical axis.

A revolving pendulum is an essential part of most of the contrivances called governors, for regulating the speed of prime movers, for further particulars of which see Steam Engine.

Division 3. Working of Machines of Varying Velocity.

§ 115. General Principles.—In order that the velocity of every piece of a machine may be uniform, it is necessary that the forces acting on each piece should be always exactly balanced. Also, in order that the forces acting on each piece of a machine may be always exactly balanced, it is necessary that the velocity of that piece should be uniform.

An excess of the effort exerted on any piece, above that which is necessary to balance the resistance, is accompanied with acceleration; a deficiency of the effort, with retardation.

When a machine is being started from a state of rest, and brought by degrees up to its proper speed, the effort must be in excess; when it is being retarded for the purpose of stopping it, the resistance must be in excess.

An excess of effort above resistance involves an excess of energy exerted above work performed; that excess of energy is employed in producing acceleration.

An excess of resistance above effort involves an excess of work performed above energy expended; that excess of work is performed by means of the retardation of the machinery.

When a machine undergoes alternate acceleration and retardation, so that at certain instants of time, occurring at the end of intervals called periods or cycles, it returns to its original speed, then in each of those periods or cycles the alternate excesses of energy and of work neutralize each other; and at the end of each cycle the principle of the equality of energy and work stated in § 87, with all its consequences, is verified exactly as in the case of machines of uniform speed.

At intermediate instants, however, other principles have also to be taken into account, which are deduced from the second law of motion, as applied to direct deviation, or acceleration and retardation.

§ 116. Energy of Acceleration and Work of Retardation for a Shifting Body.—Let w be the weight of a body which has a motion of translation in any path, and in the course of the interval of time Δt let its velocity be increased at a uniform rate of acceleration from v₁ to v₂. The rate of acceleration will be

dv/dt = const. = (v₂ − v₁) Δt;

and to produce this acceleration a uniform effort will be required, expressed by

P = w (v₂ − v₁) gΔt

(71)

(The product wv/g of the mass of a body by its velocity is called its momentum; so that the effort required is found by dividing the increase of momentum by the time in which it is produced.)

To find the energy which has to be exerted to produce the acceleration from v₁ to v₂, it is to be observed that the distance through which the effort P acts during the acceleration is

Δs = (v₂ + v₁) Δt/2;

consequently, the energy of acceleration is

PΔs = w (v₂ − v₁) (v₂ + v₁) / 2g = w (v₂² − v₁²) 2g,

(72)

being proportional to the increase in the square of the velocity, and independent of the time.

In order to produce a retardation from the greater velocity v₂ to the less velocity v₁, it is necessary to apply to the body a resistance connected with the retardation and the time by an equation identical in every respect with equation (71), except by the substitution of a in every respect with equation (71), except by the substitution of a resistance for an effort; and in overcoming that resistance the body performs work to an amount determined by equation (72), putting Rds for Pas.

§ 117. Energy Stored and Restored by Deviations of Velocity.—Thus a body alternately accelerated and retarded, so as to be brought back to its original speed, performs work during its retardation exactly equal in amount to the energy exerted upon it during its acceleration; so that that energy may be considered as stored during the acceleration, and restored during the retardation, in a manner analogous to the operation of a reciprocating force (§ 108).

Let there be given the mean velocity V = 1/2 (v₂ + v₁) of a body whose weight is w, and let it be required to determine the fluctuation of velocity v₂ − v₁, and the extreme velocities v₁, v₂, which that body must have, in order alternately to store and restore an amount of energy E. By equation (72) we have

E = w (v₂² − v₁²) / 2g

which, being divided by V = 1/2(v₂ + v₁), gives

E/V = w (v₂ − v₁) / g;

and consequently

v₂ − v₁ = gE / Vw

(73)

The ratio of this fluctuation to the mean velocity, sometimes called the unsteadiness of the motion of the body, is

(v₂ − v₁) V = gE / V²w.

(74)

§ 118. Actual Energy of a Shifting Body.—The energy which must be exerted on a body of the weight w, to accelerate it from a state of rest up to a given velocity of translation v, and the equal amount of work which that body is capable of performing by overcoming resistance while being retarded from the same velocity of translation v to a state of rest, is

wv ² / 2g.

(75)

This is called the actual energy of the motion of the body, and is half the quantity which in some treatises is called vis viva.

The energy stored or restored, as the case may be, by the deviations of velocity of a body or a system of bodies, is the amount by which the actual energy is increased or diminished.

§ 119. Principle of the Conservation of Energy in Machines.—The following principle, expressing the general law of the action of machines with a velocity uniform or varying, includes the law of the equality of energy and work stated in § 89 for machines of uniform speed.

In any given interval during the working of a machine, the energy exerted added to the energy restored is equal to the energy stored added to the work performed.

§ 120. Actual Energy of Circular Translation—Moment of Inertia.—Let a small body of the weight w undergo translation in a circular path of the radius ρ, with the angular velocity of deflexion α, so that the common linear velocity of all its particles is v = αρ. Then the actual energy of that body is

wv ² / 2g = wa²ρ² / 2g.

(76)

By comparing this with the expression for the centrifugal force (wa²ρ/g), it appears that the actual energy of a revolving body is equal to the potential energy Fρ/2 due to the action of the deflecting force along one-half of the radius of curvature of the path of the body.

The product wρ²/g, by which the half-square of the angular velocity is multiplied, is called the moment of inertia of the revolving body.

§ 121. Flywheels.—A flywheel is a rotating piece in a machine, generally shaped like a wheel (that is to say, consisting of a rim with spokes), and suited to store and restore energy by the periodical variations in its angular velocity.

The principles according to which variations of angular velocity store and restore energy are the same as those of § 117, only substituting moment of inertia for mass, and angular for linear velocity.

Let W be the weight of a flywheel, R its radius of gyration, a₂ its maximum, a₁ its minimum, and A = 1/2 (α₂ + α₁) its mean angular velocity. Let

I/S = (α₂ − α₂) / A

denote the unsteadiness of the motion of the flywheel; the denominator S of this fraction is called the steadiness. Let e denote the quantity by which the energy exerted in each cycle of the working of the machine alternately exceeds and falls short of the work performed, and which has consequently to be alternately stored by acceleration and restored by retardation of the flywheel. The value of this periodical excess is—

e = R²W (α₂² − α₁²), 2g,

(77)

from which, dividing both sides by A², we obtain the following equations:—

e / A² = R²W/gS
R²WA² / 2g = Se/2.

\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}

.

(78)

The latter of these equations may be thus expressed in words: The actual energy due to the rotation of the fly, with its mean angular velocity, is equal to one-half of the periodical excess of energy multiplied by the steadiness.

In ordinary machinery S = about 32; in machinery for fine purposes S = from 50 to 60; and when great steadiness is required S = from 100 to 150.

The periodical excess e may arise either from variations in the effort exerted by the prime mover, or from variations in the resistance of the work, or from both these causes combined. When but one flywheel is used, it should be placed in as direct connexion as possible with that part of the mechanism where the greatest amount of the periodical excess originates; but when it originates at two or more points, it is best to have a flywheel in connexion with each of these points. For example, in a machine-work, the steam-engine, which is the prime mover of the various tools, has a flywheel on the crank-shaft to store and restore the periodical excess of energy arising from the variations in the effort exerted by the connecting-rod upon the crank; and each of the slotting machines, punching machines, riveting machines, and other tools has a flywheel of its own to store and restore energy, so as to enable the very different resistances opposed to those tools at different times to be overcome without too great unsteadiness of motion. For tools performing useful work at intervals, and having only their own friction to overcome during the intermediate intervals, e should be assumed equal to the whole work performed at each separate operation.

§ 122. Brakes.—A brake is an apparatus for stopping and diminishing the velocity of a machine by friction, such as the friction-strap already referred to in § 103. To find the distance s through which a brake, exerting the friction F, must rub in order to stop a machine having the total actual energy E at the moment when the brake begins to act, reduce, by the principles of § 96, the various efforts and other resistances of the machine which act at the same time with the friction of the brake to the rubbing surface of the brake, and let R be their resultant—positive if resistance, negative if effort preponderates. Then

s = E / (F + R).

(79)

§ 123. Energy distributed between two Bodies: Projection and Propulsion.—Hitherto the effort by which a machine is moved has been treated as a force exerted between a movable body and a fixed body, so that the whole energy exerted by it is employed upon the movable body, and none upon the fixed body. This conception is sensibly realized in practice when one of the two bodies between which the effort acts is either so heavy as compared with the other, or has so great a resistance opposed to its motion, that it may, without sensible error, be treated as fixed. But there are cases in which the motions of both bodies are appreciable, and must be taken into account—such as the projection of projectiles, where the velocity of the recoil or backward motion of the gun bears an appreciable proportion to the forward motion of the projectile; and such as the propulsion of vessels, where the velocity of the water thrown backward by the paddle, screw or other propeller bears a very considerable proportion to the velocity of the water moved forwards and sideways by the ship. In cases of this kind the energy exerted by the effort is distributed between the two bodies between which the effort is exerted in shares proportional to the velocities of the two bodies during the action of the effort; and those velocities are to each other directly as the portions of the effort unbalanced by resistance on the respective bodies, and inversely as the weights of the bodies.

To express this symbolically, let W₁, W₂ be the weights of the bodies; P the effort exerted between them; S the distance through which it acts; R₁, R₂ the resistances opposed to the effort overcome by W₁, W₂ respectively; E₁, E₂ the shares of the whole energy E exerted upon W₁, W₂ respectively. Then

	E	:	E₁	:	E₂	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$ .
::	W₂ (P − R₁) + W₁ (P − R₂)	:	P − R₁	:	P − R₂
::	W₁W₂	:	W₁	:	W₂

(80)

If R₁ = R₂, which is the case when the resistance, as well as the effort, arises from the mutual actions of the two bodies, the above becomes,

E : E₁ : E₂
:: W₁ + W₂ : W₂ : W₁, $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$

(81)

that is to say, the energy is exerted on the bodies in shares inversely proportional to their weights; and they receive accelerations inversely proportional to their weights, according to the principle of dynamics, already quoted in a note to § 110, that the mutual actions of a system of bodies do not affect the motion of their common centre of gravity.

For example, if the weight of a gun be 160 times that of its ball 160/161 of the energy exerted by the powder in exploding will be employed in propelling the ball, and 1/161 in producing the recoil of the gun, provided the gun up to the instant of the ball’s quitting the muzzle meets with no resistance to its recoil except the friction of the ball.

§ 124. Centre of Percussion.—It is obviously desirable that the deviations or changes of motion of oscillating pieces in machinery should, as far as possible, be effected by forces applied at their centres of percussion.

If the deviation be a translation—that is, an equal change of motion of all the particles of the body—the centre of percussion is obviously the centre of gravity itself; and, according to the second law of motion, if dv be the deviation of velocity to be produced in the interval dt, and W the weight of the body, then

P =	W	·	dv
	g		dt

(82)

is the unbalanced effort required.

If the deviation be a rotation about an axis traversing the centre of gravity, there is no centre of percussion; for such a deviation can only be produced by a couple of forces, and not by any single force. Let dα be the deviation of angular velocity to be produced in the interval dt, and I the moment of the inertia of the body about an axis through its centre of gravity; then 1/2Id(α²) = Iα dα is the variation of the body’s actual energy. Let M be the moment of the unbalanced couple required to produce the deviation; then by equation 57, § 104, the energy exerted by this couple in the interval dt is Mα dt, which, being equated to the variation of energy, gives

M = I	dα	=	R²W	·	dα	.
	dt		g		dt

(83)

R is called the radius of gyration of the body with regard to an axis through its centre of gravity.

Fig. 133.

Now (fig. 133) let the required deviation be a rotation of the body BB about an axis O, not traversing the centre of gravity G, dα being, as before, the deviation of angular velocity to be produced in the interval dt. A rotation with the angular velocity α about an axis O may be considered as compounded of a rotation with the same angular velocity about an axis drawn through G parallel to O and a translation with the velocity α. OG, OG being the perpendicular distance between the two axes. Hence the required deviation may be regarded as compounded of a deviation of translation dv = OG · dα, to produce which there would be required, according to equation (82), a force applied at G perpendicular to the plane OG—

P = Wg · OG · dαdt

(84)

and a deviation dα of rotation about an axis drawn through G parallel to O, to produce which there would be required a couple of the moment M given by equation (83). According to the principles of statics, the resultant of the force P, applied at G perpendicular to the plane OG, and the couple M is a force equal and parallel to P, but applied at a distance GC from G, in the prolongation of the perpendicular OG, whose value is

GC = M / P = R² / OG.

(85)

Thus is determined the position of the centre of percussion C, corresponding to the axis of rotation O. It is obvious from this equation that, for an axis of rotation parallel to O traversing C, the centre of percussion is at the point where the perpendicular OG meets O.

§ 125.* To find the moment of inertia of a body about an axis through its centre of gravity experimentally.—Suspend the body from any conveniently selected axis O (fig. 48) and hang near it a small plumb bob. Adjust the length of the plumb-line until it and the body oscillate together in unison. The length of the plumb-line, measured from its point of suspension to the centre of the bob, is for all practical purposes equal to the length OC, C being therefore the centre of percussion corresponding to the selected axis O. From equation (85)

R² = CG × OG = (OC − OG) OG.

The position of G can be found experimentally; hence OG is known, and the quantity R² can be calculated, from which and the ascertained weight W of the body the moment of inertia about an axis through G, namely, W/g × R², can be computed.

Fig. 134.

§ 126.* To find the force competent to produce the instantaneous acceleration of any link of a mechanism.—In many practical problems it is necessary to know the magnitude and position of the forces acting to produce the accelerations of the several links of a mechanism. For a given link, this force is the resultant of all the accelerating forces distributed through the substance of the material of the link required to produce the requisite acceleration of each particle, and the determination of this force depends upon the principles of the two preceding sections. The investigation of the distribution of the forces through the material and the stress consequently produced belongs to the subject of the Strength of Materials (q.v.). Let BK (fig. 134) be any link moving in any manner in a plane, and let G be its centre of gravity. Then its motion may be analysed into (1) a translation of its centre of gravity; and (2) a rotation about an axis through its centre of gravity perpendicular to its plane of motion. Let α be the acceleration of the centre of gravity and let A be the angular acceleration about the axis through the centre of gravity; then the force required to produce the translation of the centre of gravity is F = Wα/g, and the couple required to produce the angular acceleration about the centre of gravity is M = IA/g, W and I being respectively the weight and the moment of inertia of the link about the axis through the centre of gravity. The couple M may be produced by shifting the force F parallel to itself through a distance x. such that Fx = M. When the link forms part of a mechanism the respective accelerations of two points in the link can be determined by means of the velocity and acceleration diagrams described in § 82, it being understood that the motion of one link in the mechanism is prescribed, for instance, in the steam-engine’s mechanism that the crank shall revolve uniformly. Let the acceleration of the two points B and K therefore be supposed known. The problem is now to find the acceleration α and A. Take any pole O (fig. 49), and set out Ob equal to the acceleration of B and Ok equal to the acceleration of K. Join bk and take the point g so that KG: GB = kg : gb. Og is then the acceleration of the centre of gravity and the force F can therefore be immediately calculated. To find the angular acceleration A, draw kt, bt respectively parallel to and at right angles to the link KB. Then tb represents the angular acceleration of the point B relatively to the point K and hence tb/KB is the value of A, the angular acceleration of the link. Its moment of inertia about G can be found experimentally by the method explained in § 125, and then the value of the couple M can be computed. The value of x is found immediately from the quotient M/F. Hence the magnitude F and the position of F relatively to the centre of gravity of the link, necessary to give rise to the couple M, are known, and this force is therefore the resultant force required.

Fig. 135.

§ 127.* Alternative construction for finding the position of F relatively to the centre of gravity of the link.—Let B and K be any two points in the link which for greater generality are taken in fig. 135, so that the centre of gravity G is not in the line joining them. First find the value of R experimentally. Then produce the given directions of acceleration of B and K to meet in O; draw a circle through the three points B, K and O; produce the line joining O and G to cut the circle in Y; and take a point Z on the line OY so that YG × GZ = R². Then Z is a point in the line of action of the force F. This useful theorem is due to G. T. Bennett, of Emmanuel College, Cambridge. A proof of it and three corollaries are given in appendix 4 of the second edition of Dalby’s Balancing of Engines (London, 1906). It is to be noticed that only the directions of the accelerations of two points are required to find the point Z.

For an example of the application of the principles of the two preceding sections to a practical problem see Valve and Valve Gear Mechanisms, by W. E. Dalby (London, 1906), where the inertia stresses brought upon the several links of a Joy valve gear, belonging to an express passenger engine of the Lancashire & Yorkshire railway, are investigated for an engine-speed of 68 m. an hour.

Fig. 136.

§ 128.* The Connecting Rod Problem.—A particular problem of practical importance is the determination of the force producing the motion of the connecting rod of a steam-engine mechanism of the usual type. The methods of the two preceding sections may be used when the acceleration of two points in the rod are known. In this problem it is usually assumed that the crank pin K (fig. 136) moves with uniform velocity, so that if α is its angular velocity and r its radius, the acceleration is α²r in a direction along the crank arm from the crank pin to the centre of the shaft. Thus the acceleration of one point K is known completely. The acceleration of a second point, usually taken at the centre of the crosshead pin, can be found by the principles of § 82, but several special geometrical constructions have been devised for this purpose, notably the construction of Klein,^[4] discovered also independently by Kirsch.^[5] But probably the most convenient is the construction due to G. T. Bennett^[6] which is as follows: Let OK be the crank and KB the connecting rod. On the connecting rod take a point L such that KL × KB = KO². Then, the crank standing at any angle with the line of stroke, draw LP at right angles to the connecting rod, PN at right angles to the line of stroke OB and NA at right angles to the connecting rod; then AO is the acceleration of the point B to the scale on which KO represents the acceleration of the point K. The proof of this construction is given in The Balancing of Engines.

The finding of F may be continued thus: join AK, then AK is the acceleration image of the rod, OKA being the acceleration diagram. Through G, the centre of gravity of the rod, draw Gg parallel to the line of stroke, thus dividing the image at g in the proportion that the connecting rod is divided by G. Hence Og represents the acceleration of the centre of gravity and, the weight of the connecting rod being ascertained, F can be immediately calculated. To find a point in its line of action, take a point Q on the rod such that KG × GQ = R², R having been determined experimentally by the method of § 125; join G with O and through Q draw a line parallel to BO to cut GO in Z. Z is a point in the line of action of the resultant force F; hence through Z draw a line parallel to Og. The force F acts in this line, and thus the problem is completely solved. The above construction for Z is a corollary of the general theorem given in § 127.

§ 129. Impact. Impact or collision is a pressure of short duration exerted between two bodies.

The effects of impact are sometimes an alteration of the distribution of actual energy between the two bodies, and always a loss of a portion of that energy, depending on the imperfection of the elasticity of the bodies, in permanently altering their figures, and producing heat. The determination of the distribution of the actual energy after collision and of the loss of energy is effected by means of the following principles:—

I. The motion of the common centre of gravity of the two bodies is unchanged by the collision.

II. The loss of energy consists of a certain proportion of that part of the actual energy of the bodies which is due to their motion relatively to their common centre of gravity.

Unless there is some special reason for using impact in machines, it ought to be avoided, on account not only of the waste of energy which it causes, but from the damage which it occasions to the frame and mechanism. (W. J. M. R.; W. E. D.)

↑ In view of the great authority of the author, the late Professor Macquorn Rankine, it has been thought desirable to retain the greater part of this article as it appeared in the 9th edition of the Encyclopaedia Britannica. Considerable additions, however, have been introduced in order to indicate subsequent developments of the subject; the new sections are numbered continuously with the old, but are distinguished by an asterisk. Also, two short chapters which concluded the original article have been omitted—ch. iii., “On Purposes and Effects of Machines,” which was really a classification of machines, because the classification of Franz Reuleaux is now usually followed, and ch. iv., “Applied Energetics, or Theory of Prime Movers,” because its subject matter is now treated in various special articles, e.g. Hydraulics, Steam Engine, Gas Engine, Oil Engine, and fully developed in Rankine’s The Steam Engine and Other Prime Movers (London, 1902). (Ed. E.B.)
↑ Since the relation discussed in § 7 was enunciated by Rankine, an enormous development has taken place in the subject of Graphic Statics, the first comprehensive textbook on the subject being Die Graphische Statik by K. Culmann, published at Zürich in 1866. Many of the graphical methods therein given have now passed into the textbooks usually studied by engineers. One of the most beautiful graphical constructions regularly used by engineers and known as “the method of reciprocal figures” is that for finding the loads supported by the several members of a braced structure, having given a system of external loads. The method was discovered by Clerk Maxwell, and the complete theory is discussed and exemplified in a paper “On Reciprocal Figures, Frames and Diagrams of Forces,” Trans. Roy. Soc. Ed., vol. xxvi. (1870). Professor M. W. Crofton read a paper on “Stress-Diagrams in Warren and Lattice Girders” at the meeting of the Mathematical Society (April 13, 1871), and Professor O. Henrici illustrated the subject by a simple and ingenious notation. The application of the method of reciprocal figures was facilitated by a system of notation published in Economics of Construction in relation to framed Structures, by Robert H. Bow (London, 1873). A notable work on the general subject is that of Luigi Cremona, translated from the Italian by Professor T. H. Beare (Oxford, 1890), and a discussion of the subject of reciprocal figures from the special point of view of the engineering student is given in Vectors and Rotors by Henrici and Turner (London, 1903). See also above under “Theoretical Mechanics,” Part 1. § 5.
↑ This is a particular case of a more general principle, that the motion of the centre of gravity of a body is not affected by the mutual actions of its parts.
↑ J. F. Klein, “New Constructions of the Force of Inertia of Connecting Rods and Couplers and Constructions of the Pressures on their Pins,” Journ. Franklin Inst., vol. 132 (Sept. and Oct., 1891).
↑ Prof. Kirsch, “Über die graphische Bestimmung der Kolbenbeschleunigung,” Zeitsch. Verein deutsche Ingen. (1890), p. 1320.
↑ Dalby, The Balancing of Engines (London, 1906), app. 1.

[f1-1] In view of the great authority of the author, the late Professor Macquorn Rankine, it has been thought desirable to retain the greater part of this article as it appeared in the 9th edition of the Encyclopaedia Britannica. Considerable additions, however, have been introduced in order to indicate subsequent developments of the subject; the new sections are numbered continuously with the old, but are distinguished by an asterisk. Also, two short chapters which concluded the original article have been omitted—ch. iii., “On Purposes and Effects of Machines,” which was really a classification of machines, because the classification of Franz Reuleaux is now usually followed, and ch. iv., “Applied Energetics, or Theory of Prime Movers,” because its subject matter is now treated in various special articles, e.g. Hydraulics, Steam Engine, Gas Engine, Oil Engine, and fully developed in Rankine’s The Steam Engine and Other Prime Movers (London, 1902). (Ed. E.B.)

[f2-2] Since the relation discussed in § 7 was enunciated by Rankine, an enormous development has taken place in the subject of Graphic Statics, the first comprehensive textbook on the subject being Die Graphische Statik by K. Culmann, published at Zürich in 1866. Many of the graphical methods therein given have now passed into the textbooks usually studied by engineers. One of the most beautiful graphical constructions regularly used by engineers and known as “the method of reciprocal figures” is that for finding the loads supported by the several members of a braced structure, having given a system of external loads. The method was discovered by Clerk Maxwell, and the complete theory is discussed and exemplified in a paper “On Reciprocal Figures, Frames and Diagrams of Forces,” Trans. Roy. Soc. Ed., vol. xxvi. (1870). Professor M. W. Crofton read a paper on “Stress-Diagrams in Warren and Lattice Girders” at the meeting of the Mathematical Society (April 13, 1871), and Professor O. Henrici illustrated the subject by a simple and ingenious notation. The application of the method of reciprocal figures was facilitated by a system of notation published in Economics of Construction in relation to framed Structures, by Robert H. Bow (London, 1873). A notable work on the general subject is that of Luigi Cremona, translated from the Italian by Professor T. H. Beare (Oxford, 1890), and a discussion of the subject of reciprocal figures from the special point of view of the engineering student is given in Vectors and Rotors by Henrici and Turner (London, 1903). See also above under “Theoretical Mechanics,” Part 1. § 5.

[f3-3] This is a particular case of a more general principle, that the motion of the centre of gravity of a body is not affected by the mutual actions of its parts.

[f4-4] J. F. Klein, “New Constructions of the Force of Inertia of Connecting Rods and Couplers and Constructions of the Pressures on their Pins,” Journ. Franklin Inst., vol. 132 (Sept. and Oct., 1891).

[f5-5] Prof. Kirsch, “Über die graphische Bestimmung der Kolbenbeschleunigung,” Zeitsch. Verein deutsche Ingen. (1890), p. 1320.

[f6-6] Dalby, The Balancing of Engines (London, 1906), app. 1.

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