On the multi-mode behavior of vibrating rods attached to nonlinear springs

Abstract

This paper investigates the harmonic response of vibrating rods with an array of nonlinear springs. The proposed analysis is multi-mode in the sense that the response functions are plotted over wide frequency bands where several resonances can be observed. Particularly, this study aims at investigating the way the vibration modes interact with each other, given the occurrence of local nonlinearities. Also, it aims at investigating the effect of periodic local nonlinearities on the dynamic behavior of the rods, which is closely related to the topic of nonlinear metamaterials. Two approaches are proposed, namely the polynomial method and the perturbation method. The polynomial method uses closed-form solutions of the equation of motion of a rod attached to a small number of springs. This yields a scalar polynomial equation which is well suited for accurately computing the receptance functions at some point of the rod. On the other hand, the proposed perturbation method invokes a subspace projection, which consists in expanding the displacement of the rod on a reduced (finite) basis of vibration modes. This yields a cubic matrix equation which can be easily solved using appropriate solvers. Numerical experiments are carried out which highlight the relevance of both approaches. It is found that the resonance peaks of the rod, once coupled to the nonlinear springs, shift to the high frequencies. This appears to be an interesting feature for the passive control of these systems in the low-frequency range where the vibration levels can be strongly reduced, i.e., compared to the case where purely linear springs are only considered.

A Expressions of the coefficients of the polynomial equation (20)

$$\begin{aligned} \alpha _{0}=&\frac{{D_2} F}{{D_1}}, \end{aligned}$$

(49)

$$\begin{aligned} \alpha _{1} =&-\frac{\left( {s_1}+2 {D_1}\right) \left( {D_1} {s_1}-{{{D_2}}^{2}}+{{{D_1}}^{2}}\right) }{{D_1} {D_2}}, \end{aligned}$$

(50)

$$\begin{aligned} \alpha _{3} =&-3\frac{{D_1} {{{s_1}}^{3}}+3 {{{D_1}}^{2}} {{{s_1}}^{2}}+3 {{{D_1}}^{3}} {s_1}-{{{D_2}}^{4}}}{4D_{1}D_{2}^{3}}\nonumber \\&-\,3\frac{2 {{{D_1}}^{2}} {{{D_2}}^{2}}+{D_1} {{{D_2}}^{2}}s_1+{{{D_1}}^{4}}}{4D_{1}D_{2}^{3}}, \end{aligned}$$

(51)

$$\begin{aligned} \alpha _{5} =&-\frac{27 {{\left( {s_1}+{D_1}\right) }^{2}} {{{s_3}}^{2}}}{16 {{{D_2}}^{3}}}, \end{aligned}$$

(52)

$$\begin{aligned} \alpha _{7} =&-\frac{81 \left( {s_1}-{D_1}\right) {{{s_3}}^{3}}}{64 {{{D_2}}^{3}}}, \end{aligned}$$

(53)

$$\begin{aligned} \alpha _{9} =&-\frac{81 {{{s_3}}^{4}}}{256 {{{D_2}}^{3}}}. \end{aligned}$$

(54)