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{{short description|Mechanical oscillations about an equilibrium point}}
{{other uses|Vibration (disambiguation)|Vibrate (disambiguation)}}
[[File:Drum vibration mode21.gif|thumb|One of the possible modes of [[Vibrations of a circular drum|vibration of a circular drum]] (see [[:commons:Category:Drum vibration animations|other modes]]).]]▼
[[
{{Classical mechanics|core}}
Vibration can be desirable: for example, the motion of a [[tuning fork]], the [[Reed (music)|reed]] in a [[woodwind instrument]] or [[harmonica]], a [[mobile phone]], or the cone of a [[loudspeaker]].
In many cases, however, vibration is undesirable,
The studies of sound and vibration are closely related (both fall under [[acoustics]]). Sound, or pressure [[wave]]s, are generated by vibrating structures (e.g. [[vocal cords]]); these pressure waves can also induce the vibration of structures (e.g. [[ear drum]]). Hence, attempts to reduce noise are often related to issues of vibration.<ref name="Tustin06" />
▲[[File:Drum vibration mode21.gif|thumb|One of the possible modes of [[Vibrations of a circular drum|vibration of a circular drum]] (see [[:commons:Category:Drum vibration animations|other modes]]).]]
▲[[Image:suspension.jpg|thumb|upright|Car suspension: Designing vibration control is undertaken as part of [[Acoustical engineering|acoustic]], [[Automotive engineering|automotive]] or [[Mechanical engineering|mechanical]] [[engineering]].]]
[[Machining vibrations]] are common in the process of [[subtractive manufacturing]].
==Types of vibration==▼
'''Free vibration''' or '''natural vibration''' occurs when a mechanical system is set in motion with an initial input and allowed to vibrate freely. Examples of this type of vibration are pulling a child back on a swing and letting it go, or hitting a tuning fork and letting it ring. The mechanical system vibrates at one or more of its [[Natural Frequency|natural frequencies]] and [[Damping ratio|damps]] down to motionlessness.
'''Forced vibration''' is when a time-varying disturbance (load, displacement, velocity, or '''Damped vibration:''' When the energy of a vibrating system is gradually dissipated by friction and other resistances, the vibrations are said to be damped. The vibrations gradually reduce or change in frequency or intensity or cease and the system rests in its equilibrium position. An example of this type of vibration is the [[Vehicle suspension|vehicular suspension]] dampened by the [[shock absorber]].
==
{{excerpt|Vibration isolation}}
== Testing ==
Vibration testing is accomplished by introducing a forcing function into a structure, usually with some type of shaker. Alternately, a DUT (device under test) is attached to the "table" of a shaker. Vibration testing is performed to examine the response of a device under test (DUT) to a defined vibration environment. The measured response may be ability to function in the vibration environment, fatigue life, resonant frequencies or squeak and rattle sound output ([[Noise, vibration, and harshness|NVH]]). Squeak and rattle testing is performed with a special type of ''quiet shaker'' that produces very low sound levels while under operation.
For relatively low frequency forcing (typically less than 100
The most common types of vibration testing services conducted by vibration test labs are sinusoidal and random. Sine (one-frequency-at-a-time) tests are performed to survey the structural response of the device under test (DUT). During the early history of vibration testing, vibration machine controllers were limited only to controlling sine motion so only sine testing was performed. Later, more sophisticated analog and then digital controllers were able to provide random control (all frequencies at once). A random (all frequencies at once) test is generally considered to more closely replicate a real world environment, such as road inputs to a moving automobile.
Most vibration testing is conducted in a 'single DUT axis' at a time, even though most real-world vibration occurs in various axes simultaneously. MIL-STD-810G, released in late 2008, Test Method 527, calls for multiple exciter testing. The ''vibration test fixture<ref name="TonyAraujo">Tony Araujo. ''[https://www.evaluationengineering.com/applications/automotive-test/article/21093894/october-automotive-article The evolution of automotive vibration fixturing]'', EE-Evaluation Engineering, 2019</ref>'' used to attach the DUT to the shaker table must be designed for the frequency range of the vibration test spectrum. It is difficult to design a vibration test fixture which duplicates the dynamic response (mechanical impedance)<ref name="SVIC Notes">Blanks, H.S., "Equivalence Techniques for Vibration Testing," SVIC Notes, pp 17.</ref> of the actual in-use mounting. For this reason, to ensure repeatability between vibration tests, vibration fixtures are designed to be resonance free<ref name="SVIC Notes"
Generally for smaller fixtures and lower frequency ranges, the designer can target a fixture design that is free of resonances in the test frequency range. This becomes more difficult as the DUT gets larger and as the test frequency increases. In these cases multi-point control strategies<ref name="Araujo, T. and Yao, B.,">Araujo, T. and Yao, B., ''"Vibration Fixture Performance Qualification
Some vibration test methods limit the amount of crosstalk (movement of a response point in a mutually perpendicular direction to the axis under test) permitted to be exhibited by the vibration test fixture.
Devices specifically designed to trace or record vibrations are called [[vibroscope]]s.
==
Vibration analysis (VA), applied in an industrial or maintenance environment aims to reduce maintenance costs and equipment downtime by detecting equipment faults.<ref>Crawford, Art; Simplified Handbook of Vibration Analysis</ref><ref>Eshleman, R 1999, Basic machinery vibrations: An introduction to machine testing, analysis, and monitoring</ref> VA is a key component of a [[condition monitoring]] (CM) program, and is often referred to as [[predictive maintenance]] (PdM).<ref>Mobius Institute; Vibration Analyst Category 2
▲Vibration analysis (VA), applied in an industrial or maintenance environment aims to reduce maintenance costs and equipment downtime by detecting equipment faults.<ref>Crawford, Art; Simplified Handbook of Vibration Analysis</ref><ref>Eshleman, R 1999, Basic machinery vibrations: An introduction to machine testing, analysis, and monitoring</ref> VA is a key component of a [[condition monitoring]] (CM) program, and is often referred to as [[predictive maintenance]] (PdM).<ref>Mobius Institute; Vibration Analyst Category 2 - Course Notes 2013</ref> Most commonly VA is used to detect faults in rotating equipment (Fans, Motors, Pumps, and Gearboxes etc.) such as imbalance, misalignment, rolling element bearing faults and resonance conditions.<ref>{{Cite web|last=|first=|date=2021-01-05|title=Importance of Vibration Analysis in Maintenance|url=https://rms-reliability.com/vibration/vibration-analysis-in-maintenance/|url-status=live|archive-url=|archive-date=|access-date=2021-01-08|website=|language=en-US}}</ref>
VA can use the units of Displacement, Velocity and Acceleration displayed as a [[Waveform|time waveform]] (TWF), but most commonly the spectrum is used, derived from a [[fast Fourier transform]] of the TWF. The vibration spectrum provides important frequency information that can pinpoint the faulty component.
The fundamentals of vibration analysis can be understood by studying the simple [[Mass-spring-damper]] model. Indeed, even a complex structure such as an automobile body can be modeled as a "summation" of simple
Note: This article does not include the step-by-step mathematical derivations, but focuses on major vibration analysis equations and concepts. Please refer to the references at the end of the article for detailed derivations.
=== Free vibration without damping ===
[[File:Mass spring.svg|thumb|upright=0.9|Simple mass spring model]] To start the investigation of the mass–spring–damper assume the damping is negligible and that there is no external force applied to the mass (i.e. free vibration). The force applied to the mass by the spring is proportional to the amount the spring is stretched "x" (assuming the spring is already compressed due to the weight of the mass). The proportionality constant, k, is the stiffness of the spring and has units of force/distance (e.g. lbf/in or N/m). The negative sign indicates that the force is always opposing the motion of the mass attached to it:
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</math>
The force generated by the mass is proportional to the acceleration of the mass as given by [[Newton's laws of motion|Newton's second law of motion]]:
:<math>
\Sigma\ F = ma = m \ddot{x} = m \frac{d^2x}{dt^2}.
</math>
The sum of the forces on the mass then generates this [[ordinary differential equation]]: <math> \ m \ddot{x} + k x = 0.</math>
[[File:Simple harmonic oscillator.gif|thumb|upright=0.5|Simple harmonic motion of the mass–spring system]] Assuming that the initiation of vibration begins by stretching the spring by the distance of ''A'' and releasing, the solution to the above equation that describes the motion of mass is:
:<math>
x(t) = A \cos (2 \pi f_n t). \!
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</math>
Note: [[angular frequency]] ω (ω=2 π ''f'') with the units of radians per second is often used in equations because it simplifies the equations, but is normally converted to [[ordinary frequency]] (units of [[hertz|Hz]] or equivalently cycles per second) when stating the frequency of a system. If the mass and stiffness of the system is known, the formula above can determine the frequency at which the system vibrates once set in motion by an initial disturbance. Every vibrating system has one or more natural frequencies that it vibrates at once disturbed. This simple relation can be used to understand in general what happens to a more complex system once we add mass or stiffness. For example, the above formula explains why, when a car or truck is fully loaded, the suspension feels
==== What causes the system to vibrate: from conservation of energy point of view ====
Vibrational motion could be understood in terms of [[conservation of energy]]. In the above example the spring has been extended by a value of x and therefore some [[potential energy]] (<math>\tfrac {1}{2} k x^2</math>) is stored in the spring. Once released, the spring tends to return to its un-stretched state (which is the minimum potential energy state) and in the process accelerates the mass. At the point where the spring has reached its un-stretched state all the potential energy that we supplied by stretching it has been transformed into [[kinetic energy]] (<math>\tfrac {1}{2} m v^2</math>). The mass then begins to decelerate because it is now compressing the spring and in the process transferring the kinetic energy back to its potential. Thus oscillation of the spring amounts to the transferring back and forth of the kinetic energy into potential energy. In this simple model the mass continues to oscillate forever at the same magnitude—but in a real system, ''damping'' always dissipates the energy, eventually bringing the spring to rest.
=== Free vibration with damping ===
[[File:Mass spring damper.svg|thumb|upright=0.9|Mass–spring–damper model]]
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:<math>\zeta = { c \over 2 \sqrt{\text{km}} }.</math>
For example, metal structures (e.g., airplane fuselages, engine crankshafts) have damping factors less than 0.05, while automotive suspensions are in the range of 0.
:<math>x(t)=X e^{-\zeta \omega_n t} \cos\left( \sqrt{1-\zeta^2} \omega_n t - \phi \right) , \qquad \omega_n = 2\pi f_n. </math>
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The value of ''X'', the initial magnitude, and <math> \phi, </math> the [[Phase (waves)#Phase shift|phase shift]], are determined by the amount the spring is stretched. The formulas for these values can be found in the references.
==== Damped and undamped natural frequencies ====
<!--A link from Damping links here-->
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The plots to the side present how 0.1 and 0.3 damping ratios effect how the system “rings” down over time. What is often done in practice is to experimentally measure the free vibration after an impact (for example by a hammer) and then determine the natural frequency of the system by measuring the rate of oscillation, as well as the damping ratio by measuring the rate of decay. The natural frequency and damping ratio are not only important in free vibration, but also characterize how a system behaves under forced vibration.
{{multiple image
|align = left
|width1 = 400
|image1 = Spring-mass undamped.gif
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}}
{{multiple image
|align = left
|width3 = 400
|image3 = Spring-mass critically-damped.gif
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|image4 = Spring-mass over-damped.gif
|caption4 = Spring mass overdamped
}}<ref name="Simionescu 2014">{{cite book|last=Simionescu|first=P.A.|title=Computer Aided Graphing and Simulation Tools for AutoCAD Users|year=2014|publisher=CRC Press|location=Boca Raton, FL|isbn=978-1-4822-5290-3|edition=1st}}</ref>▼
▲{{cite book|last=Simionescu|first=P.A.|title=Computer Aided Graphing and Simulation Tools for AutoCAD Users|year=2014|publisher=CRC Press|location=Boca Raton, FL|isbn=978-1-4822-5290-3|edition=1st}}</ref>
=== Forced vibration with damping ===
The behavior of the spring mass damper model varies with the addition of a harmonic force. A force of this type could, for example, be generated by a rotating imbalance.
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:<math>X= {F_0 \over k} {1 \over \sqrt{(1-r^2)^2 + (2 \zeta r)^2}}.</math>
Where “r” is defined as the ratio of the harmonic force frequency over the undamped natural frequency of the
:<math>r=\frac{f}{f_n}.</math>
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*With little or no damping, the vibration is in phase with the forcing frequency when the frequency ratio ''r'' < 1 and 180 degrees out of phase when the frequency ratio ''r'' > 1
*When ''r'' ≪ 1 the amplitude is just the deflection of the spring under the static force <math>F_0. </math> This deflection is called the static deflection <math>\delta_{st}.</math> Hence, when ''r'' ≪ 1 the effects of the damper and the mass are minimal.
*When ''r'' ≫ 1 the amplitude of the vibration is actually less than the static deflection <math>\delta_{st}.</math> In this region the force generated by the mass (''F'' = ''ma'') is dominating because the acceleration seen by the mass increases with the frequency. Since the deflection seen in the spring, ''X'', is reduced in this region, the force transmitted by the spring (''F'' = ''kx'') to the base is reduced. Therefore, the
* Whatever the damping is, the vibration is 90 degrees out of phase with the forcing frequency when the frequency ratio ''r'' = 1, which is very helpful when it comes to determining the natural frequency of the system.
* Whatever the damping is, when ''r'' ≫ 1, the vibration is 180 degrees out of phase with the forcing frequency
* Whatever the damping is, when ''r'' ≪ 1, the vibration is in phase with the forcing frequency
==== Resonance causes ====
Resonance is simple to understand if the spring and mass are viewed as energy storage elements – with the mass storing kinetic energy and the spring storing potential energy. As discussed earlier, when the mass and spring have no external force acting on them they transfer energy back and forth at a rate equal to the natural frequency. In other words, to efficiently pump energy into both mass and spring requires that the energy source feed the energy in at a rate equal to the natural frequency. Applying a force to the mass and spring is similar to pushing a child on swing, a push is needed at the correct moment to make the swing get higher and higher. As in the case of the swing, the force applied need not be high to get large motions, but must just add energy to the system.
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The damper, instead of storing energy, dissipates energy. Since the damping force is proportional to the velocity, the more the motion, the more the damper dissipates the energy. Therefore, there is a point when the energy dissipated by the damper equals the energy added by the force. At this point, the system has reached its maximum amplitude and will continue to vibrate at this level as long as the force applied stays the same. If no damping exists, there is nothing to dissipate the energy and, theoretically, the motion will continue to grow into infinity.
==== Applying "complex" forces to the
In a previous section only a simple harmonic force was applied to the model, but this can be extended considerably using two powerful mathematical tools. The first is the [[Fourier transform]] that takes a signal as a function of time ([[time domain]]) and breaks it down into its harmonic components as a function of frequency ([[frequency domain]]). For example, by applying a force to the
[[File:Square wave frequency spectrum animation.gif|thumb|upright=1.75|How a 1
The Fourier transform of the square wave generates a [[frequency spectrum]] that presents the magnitude of the harmonics that make up the square wave (the phase is also generated, but is typically of less concern and therefore is often not plotted). The Fourier transform can also be used to analyze non-[[periodic function|periodic]] functions such as transients (e.g. impulses) and random functions. The Fourier transform is almost always computed using the [[fast Fourier transform]] (FFT) computer algorithm in combination with a [[window function]].
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In the previous section, the vibration solution was given for a single harmonic force, but the Fourier transform in general gives multiple harmonic forces. The second mathematical tool, the [[superposition principle]], allows the summation of the solutions from multiple forces if the system is [[linear system|linear]]. In the case of the spring–mass–damper model, the system is linear if the spring force is proportional to the displacement and the damping is proportional to the velocity over the range of motion of interest. Hence, the solution to the problem with a square wave is summing the predicted vibration from each one of the harmonic forces found in the frequency spectrum of the square wave.
==== Frequency response model ====
The solution of a vibration problem can be viewed as an input/output relation – where the force is the input and the output is the vibration. Representing the force and vibration in the frequency domain (magnitude and phase) allows the following relation:
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:<math>X(i\omega)=H(i\omega)\cdot F(i\omega) \text{ or } H(i\omega)= {X(i\omega) \over F(i\omega)}.</math>
<math>H(i\omega)</math> is called the [[frequency response]] function (also referred to as the [[transfer function]], but not technically as accurate) and has both a magnitude and phase component (if represented as a [[complex number]], a real and imaginary component). The magnitude of the frequency response function (FRF) was presented earlier for the
:<math>|H(i\omega)|=\left |{X(i\omega) \over F(i\omega)} \right|= {1 \over k} {1 \over \sqrt{(1-r^2)^2 + (2 \zeta r)^2}}, \text{ where } r=\frac{f}{f_n}=\frac{\omega}{\omega_n}.</math>
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[[File:Frequency response example.png|thumb|upright=1.95|Frequency response model]]
For example, calculating the FRF for a
The figure also shows the time domain representation of the resulting vibration. This is done by performing an inverse Fourier Transform that converts frequency domain data to time domain. In practice, this is rarely done because the frequency spectrum provides all the necessary information.
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The frequency response function (FRF) does not necessarily have to be calculated from the knowledge of the mass, damping, and stiffness of the system—but can be measured experimentally. For example, if a known force over a range of frequencies is applied, and if the associated vibrations are measured, the frequency response function can be calculated, thereby characterizing the system. This technique is used in the field of experimental [[modal analysis]] to determine the vibration characteristics of a structure.
=== Multiple degrees of freedom systems and mode shapes ===
[[File:2dof model.gif|thumb|upright=1.15|Two degrees of freedom model]]
The simple
The equations of motion of the 2DOF system are found to be:
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:<math>\begin{bmatrix}\begin{bmatrix}K\end{bmatrix}-\omega^2 \begin{bmatrix} M \end{bmatrix} \end{bmatrix} \begin{Bmatrix} X \end{Bmatrix}=0.</math>
=== Eigenvalue problem ===
This is referred to an [[eigenvalue]] problem in mathematics and can be put in the standard format by pre-multiplying the equation by <math>\begin{bmatrix}M\end{bmatrix}^{-1}</math>
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Since the system is a 2 DOF system, there are two modes with their respective natural frequencies and shapes. The mode shape vectors are not the absolute motion, but just describe relative motion of the degrees of freedom. In our case the first mode shape vector is saying that the masses are moving together in phase since they have the same value and sign. In the case of the second mode shape vector, each mass is moving in opposite direction at the same rate.
=== Illustration of a multiple DOF problem ===
When there are many degrees of freedom, one method of visualizing the mode shapes is by animating them using structural analysis software such as [[Femap]], [[ANSYS]] or VA One by [[ESI Group]]. An example of animating mode shapes is shown in the figure below for a [[cantilever]]ed [[I-beam|{{ibeam}}-beam]] as demonstrated using [[modal analysis]] on ANSYS. In this case, the [[finite element method]] was used to generate an approximation of the mass and stiffness matrices by meshing the object of interest in order to solve a [[Eigenvalues and eigenvectors#Vibration analysis|discrete eigenvalue problem]]. Note that, in this case, the finite element method provides an approximation of the meshed surface (for which there exists an infinite number of vibration modes and frequencies). Therefore, this relatively simple model that has over 100 degrees of freedom and hence as many natural frequencies and mode shapes, provides a good approximation for the first natural frequencies and modes{{ref|1|†}}. Generally, only the first few modes are important for practical applications.
{| width="1000" class="wikitable" style="margin:1em auto;"▼
| colspan=3|In this table the first and second (top and bottom respectively) [[Horizontal plane|horizontal]] [[bending]] (left), [[Torsion (mechanics)|
▲{| width="1000" class="wikitable"
▲| colspan=3|In this table the first and second (top and bottom respectively) [[Horizontal plane|horizontal]] [[bending]] (left), [[Torsion (mechanics)|torsion]]al (middle), and [[Vertical direction|vertical]] bending (right) vibrational modes of an [[I-beam|{{ibeam}}-beam]] are visualized. There also exist other kinds of vibrational modes in which the beam gets [[Compression (physical)|compressed]]/[[Stress (mechanics)|stretched]] out in the height, width and length directions respectively.
|-
! colspan=3|The mode shapes of a cantilevered I-beam
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|align="center"|[[File:beam mode 5.gif|200px|center]]
|align="center"|[[File:beam mode 6.gif|200px|center]]
|}
{{note|1}} Note that when performing a numerical approximation of any mathematical model, convergence of the parameters of interest must be ascertained.
=== Multiple DOF problem converted to a single DOF problem ===
The eigenvectors have very important properties called orthogonality properties. These properties can be used to greatly simplify the solution of multi-degree of freedom models. It can be shown that the eigenvectors have the following properties:
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:<math>\begin{bmatrix} ^\diagdown m_{r\diagdown} \end{bmatrix} \begin{Bmatrix}\ddot{q}\end{Bmatrix}+\begin{bmatrix}^\diagdown k_{r\diagdown}\end{bmatrix} \begin{Bmatrix} q\end{Bmatrix}=0.</math>
This equation is the foundation of vibration analysis for multiple degree of freedom systems. A similar type of result can be derived for damped systems.<ref name="MaiaSilva97" /> The key is that the modal mass and stiffness matrices are diagonal matrices and therefore the equations have been "decoupled". In other words, the problem has been transformed from a large unwieldy multiple degree of freedom problem into many single degree of freedom problems that can be solved using the same methods outlined above.
Solving for ''x'' is replaced by solving for ''q'', referred to as the modal coordinates or modal participation factors.
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=== Rigid-body mode ===
An unrestrained multi-degree of freedom system experiences both rigid-body translation and/or rotation and vibration. The existence of a rigid-body mode results in a zero natural frequency. The corresponding mode shape is called the rigid-body mode.
== See also ==
{{Div col|colwidth=22em}}
*[[Acoustical engineering|Acoustic engineering]]
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{{div col end}}
== References ==
{{reflist}}
== Further reading ==
{{refbegin}}
*Tongue, Benson, ''Principles of Vibration'', Oxford University Press, 2001, {{ISBN|0-19-514246-2}}
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*Thompson, W.T., ''Theory of Vibrations'', Nelson Thornes Ltd, 1996, {{ISBN|0-412-78390-8}}
*Hartog, Den, ''Mechanical Vibrations'', Dover Publications, 1985, {{ISBN|0-486-64785-4}}
*{{cite book|last1=Reynolds|first1=Douglas D.|title=Engineering Principles of Mechanical Vibration.|date=2016|publisher=Trafford On Demand Publishing|location=Bloomington, Indiana, USA|isbn=978-
*[https://www.cbmconnect.com/when-vibration-from-trains-damages-buildings]
* [[Institute for Occupational Safety and Health of the German Social Accident Insurance]]: [http://www.dguv.de/ifa/fachinfos/vibrationen/index-2.jsp Whole-body and hand-arm vibration]
* Manarikkal, I., Elsaha, F., Mba, D. and Laila, D. Dynamic Modelling of Planetary Gearboxes with Cracked Tooth Using Vibrational Analysis, (2019) Advances in Condition Monitoring of Machinery in Non-Stationary Operations, p
{{refend}}
== External links ==
{{
{{wikidata property|P2806}}
*[http://www.noisestructure.com/products/MPE_SDOF.php Free Excel sheets to estimate modal parameters]
*[https://www.mobiusinstitute.com/site2/detail.asp?LinkID=55 Vibration Analysis Reference
*[https://www.siemens.com/diagnostic-monitoring-and-protection Condition Monitoring and Machinery Protection
[[Category:Mechanical vibrations| ]]
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