Limb-inspired bionic quasi-zero stiffness vibration isolator

Abstract

Vibration reduction has always been one of hot and important topics in mechanical engineering, especially for the special measurement instrument. In this paper, a novel limb-inspired bionic structure is proposed to generate negative stiffness and design a new quasi-zero stiffness isolator via torsion springs, distinguishing from the existing tension spring structures in the literature. The nonlinear mathematical model of the proposed structure is developed and the corresponding dynamic properties are further investigated by using the Harmonic Balance method and ADAMS verification. To evaluate the vibration isolation performance, typical three-springs quasi-zero stiffness (TS QZS) system is selected to compare with the proposed bionic structure. And the graphical processing unit (GPU) parallel technology is applied to perform necessary two-parameter analyses, providing more insights into the effects of parameters on the transmissibility. It is shown that the proposed structure can show advantages over the typical TS QZS system in a wider vibration isolation range for harmonic excitation case and shorter decay time for the impact excitation case.

Graphic abstract

A novel limb-inspired bionic structure is proposed to generate negative stiffness and design a new quasi-zero stiffness isolator via torsion springs. To evaluate the vibration isolation performance, typical three-springs quasi-zero stiffness (TS QZS) system is selected to compare with the proposed bionic structure. It is shown that the proposed structure can show advantages over the typical TS QZS system in a wider vibration isolation range for harmonic excitation case and shorter decay time for the impact excitation case.

Appendices

Appendix 1

The coefficients in Eqs. (2)-(4) can be expressed as

$$\alpha_{1} { = }\arccos (\frac{{a^{2} + b^{2} - h^{2} }}{2ab}),\;\;\;\;\beta_{1} = \arccos (\frac{{h^{2} + b^{2} - a^{2} }}{2bh}),$$

$$\alpha_{2} (\Delta x) = \arccos \left[ {\frac{{a^{2} + b^{2} - (h - \Delta x)^{2} }}{2ab}} \right],\;\sin \left[ {\varphi (x + \Delta x)} \right] = \frac{{\left[ {h - (x + \Delta x)} \right]^{2} - b^{2} + a^{2} }}{{2a\left[ {h - (x + \Delta x)} \right]}},$$

$$\alpha_{2} (x + \Delta x) = \arccos \left\{ {\frac{{a^{2} + b^{2} - \left[ {h - (x + \Delta x)} \right]^{2} }}{2ab}} \right\}.$$

Appendix 2

The coefficients in Eq. (5) can be expressed as

$$\lambda_{1} = \overline{h} - \Delta \overline{x},\;\;\;\;\eta_{1} = \frac{{(2\sqrt 2 \lambda_{1} )^{2} }}{{\lambda_{2} \overline{\alpha }_{1} }},$$

$$\lambda_{2} = 4\overline{b}^{2} - (1 + \overline{b}^{2} - \lambda_{1}^{2} )^{2} ,\;\;\;\;\eta_{2} = \frac{{8\lambda_{1}^{2} \lambda_{3} \left[ {\overline{\alpha }_{2} (\Delta \overline{x}) - \overline{\alpha }_{1} } \right]}}{{\overline{\alpha }_{1} \lambda_{2}^{3/2} }},$$

$$\lambda_{3} { = }\overline{b}^{2} { + }1 - \lambda_{1}^{2} ,\;\;\;\;\eta_{3} = \frac{{4\left[ {\overline{\alpha }_{2} (\Delta \overline{x}) - \overline{\alpha }_{1} } \right]}}{{\overline{\alpha }_{1} \sqrt {\lambda_{2} } }},$$

$$\lambda_{4} { = }\overline{b}^{2} - 1 + \lambda_{1}^{2} ,\;\;\;\;\eta_{4} = \frac{{4\lambda_{7} }}{{\sqrt {\lambda_{2} } }},$$

$$\lambda_{5} { = }1{ + }\lambda_{1}^{2} - \overline{b}^{2} ,\;\;\;\;\eta_{5} = \frac{{4\lambda_{1} \lambda_{5} (4\lambda_{1}^{2} - 2\lambda_{4} )}}{{ - 4\overline{\alpha }_{1} \lambda_{1}^{3} \sqrt {\lambda_{2} (4\lambda_{1}^{2} \overline{b}^{2} - \lambda_{4}^{2} )} }},$$

$$\lambda_{6} { = }1 - \lambda_{1}^{2} - \overline{b}^{2} ,\;\;\;\;\eta_{6} = \frac{{4\lambda_{7} \lambda_{6} }}{{\overline{\alpha }_{1} \lambda_{2}^{3/2} }},$$

$$\lambda_{7} = \overline{\beta }_{1} + \overline{\beta }_{2} (\Delta \overline{x}) - \uppi ,\;\eta_{7} = \frac{{2\lambda_{7} \lambda_{5} }}{{ - \overline{\alpha }_{1} \lambda_{1} \sqrt {\lambda_{2} } }},$$

$$\overline{h} = \frac{h}{a},\;\;\;\;\Delta \overline{x} = \overline{h} - \sqrt {\overline{b}^{2} - 1} + \frac{{\overline{b}}}{12},$$

$$\overline{b} = \frac{b}{a}.$$

Appendix 3

The coefficients in Eq. (9) can be expressed as

$$\dot{\alpha} (x + \Delta x) = \frac{{(x + \Delta x - h)\dot{x} }}{{ab\sqrt {1 - \left\{ {\left[ {a^{2} + b^{2} - (x + \Delta x - h)^{2} } \right]/2ab} \right\}^{2} } }},$$

$$\dot{\beta} (x + \Delta x) = \frac{{h - (x + \Delta x) - (b^{2} - a^{2} )/\left[ {h - (x + \Delta x)} \right]}}{{\sqrt {4b^{2} \left[ {h - (x + \Delta x)} \right]^{2} - \left\{ {b^{2} + \left[ {h - (x + \Delta x)} \right]^{2} - a^{2} } \right\}^{2} } }}.$$