Dynamic Analysis of a Multiple-Span Euler–Bernoulli Beam Supported by Pneumatic Quasi-zero-stiffness System

Abstract

Purpose

To overcome the influence of supporting stiffness and realize the consistency of modal results between ground tests and its space missions, a new supporting method for modal test of a geometrical pneumatic quasi-zero-stiffness (PQZS) system is proposed based on the dynamic characteristics of rolling lobe air-springs and dipteran mechanism.

Methods

First, the dynamic model is constructed for a multiple-span Euler–Bernoulli beam with concentrated mass supported by the PQZS system according to the Second Newton’s Law. Moreover, the modal analysis is carried out theoretically via transformation matrix, the Galerkin and the averaging method for the first four orders including mode shapes, frequencies and its bifurcation with different supporting parameters. Meanwhile, based on the general single-freedom system for modal 1, the dynamic behaviors are also investigated including bifurcation, chaos and co-existing periodic solutions theoretically and numerically by Melnikov and the forth order Runge–Kutta method. Furthermore, a modal analysis system is set up to verify the performance of PQZS supports experimentally.

Results

The theoretical analysis shows the modalities obtained from PQZS supports is more consistent with free-constraint condition than linear supports with an easy requirement about assembling position. The system will vibrate in chaotic state through subharmonic and double-periodic bifurcation for the general single-freedom system of modal 1. The experimental results also indicate the more accurate results can be obtained by PQZS supports which is agree with the modalities under free-constraint condition than linear boundary.

Conclusion

The findings in this work illustrate a better performance of PQZS supporting system for modal test than traditional ways especially in low-order modalities, which can realize consistency of modal results between ground tests and its space missions, and provide reference for engineering application.

Appendix A: Expression for Some Matrix in the Text

$$\begin{aligned} {{T}_{k}}=\left[ \begin{matrix} t_{11} &{} t_{12} &{} t_{13} &{} t_{14} \\ t_{21} &{} t_{22} &{} 0 &{} 0 \\ t_{31} &{} t_{32} &{} t_{33} &{} t_{34} \\ 0 &{} 0 &{} t_{43} &{} t_{44} \\ \end{matrix} \right] \end{aligned}$$

(58)

where: $t_{11}=\cos \left( {{p}_{n}}{{l}_{k}} \right) +\frac{{{k}_{1}}}{2{{k}^{3}}}\sin \left( {{p}_{n}}{{l}_{k}} \right),$ ${t_{12}}=-\sin \left( {{p}_{n}}{{l}_{k}} \right) +\frac{{{k}_{1}}}{2{{p}^{3}}}\cos \left( {{p}_{n}}{{l}_{k}} \right),$ $t_{13}=\frac{{{k}_{1}}}{2{{p}^{3}}}\sinh \left( {{p}_{n}}{{l}_{k}} \right),$ $t_{14}=\frac{{{k}_{1}}}{2{{p}^{3}}}\cosh \left( {{p}_{n}}{{l}_{k}} \right),$ $t_{21}=\sin \left( {{p}_{n}}{{l}_{k}} \right),$ $t_{22}=\cos \left( {{p}_{n}}{{l}_{k}} \right),$ $t_{31}=-\frac{{{k}_{1}}}{2{{p}^{3}}}\sin \left( {{p}_{n}}{{l}_{k}} \right),$ $t_{32}=-\frac{{{k}_{1}}}{2{{p}^{3}}}\cos \left( {{p}_{n}}{{l}_{k}} \right),$ $t_{33}=\cosh \left( {{p}_{n}}{{l}_{k}} \right) -\frac{{{k}_{1}}}{2{{p}^{3}}}\sinh \left( {{p}_{n}}{{l}_{k}} \right),$ $t_{34}=\sinh \left( {{p}_{n}}{{l}_{k}} \right) -\frac{{{k}_{1}}}{2{{p}^{3}}}\cosh \left( {{p}_{n}}{{l}_{k}} \right),$ $t_{43}=\sinh \left( {{p}_{n}}{{l}_{k}} \right),$ $t_{44}=\cosh \left( {{p}_{n}}{{l}_{k}} \right).$

$$\begin{aligned} {{U}_{2\times 4}}=\left( \begin{matrix} U_{11}&{} U_{12} &{} U_{13} &{} U_{14} \\ U_{21}&{} U_{22} &{} U_{23} &{} U_{24} \\ \end{matrix} \right) \end{aligned}$$

(59)

where: $U_{11}=-\sin \left( {{p}_{n}}{{l}_{k+1}} \right),$ $U_{12}=-\cos \left( {{p}_{n}}{{l}_{k+1}} \right),$ $U_{13}=\sinh \left( {{p}_{n}}{{l}_{k+1}} \right),$ $U_{14}=\cosh \left( {{p}_{n}}{{l}_{k+1}} \right),$ $U_{21}=-{{p}_{n}}^{3}\cos \left( {{p}_{n}}{{l}_{k+1}} \right) +m{{\omega }^{2}}\sin \left( {{p}_{n}}{{l}_{k+1}} \right),$ $U_{22}={{p}_{n}}^{3}\sin \left( {{p}_{n}}{{l}_{k+1}} \right) + m{{\omega }^{2}}\cos \left( {{p}_{n}}{{l}_{k+1}} \right),$ $U_{23}={{p}_{n}}^{3}\cosh \left( {{p}_{n}}{{l}_{k+1}} \right) +m{{\omega }^{2}}\sinh \left( {{p}_{n}}{{l}_{k+1}} \right),$ $U_{24}={{p}_{n}}^{3}\sinh \left( {{p}_{n}}{{l}_{k+1}} \right) +m{{\omega }^{2}}\cosh \left( {{p}_{n}}{{l}_{k+1}} \right).$