Derivation of nonlinear damping from viscoelasticity in case of nonlinear vibrations

Abstract

Experiments show a strong increase in damping with the vibration amplitude during nonlinear vibrations of beams, plates and shells. This is observed for large size structures but also for micro- and nanodevices. The present study derives nonlinear damping from viscoelasticity by using a single-degree-of-freedom model obtained from standard linear solid material where geometric nonlinearity is inserted in. The solution of the problem is initially reached by a third-order harmonic balance method. Then, the equation of motion is obtained in differential form, which is extremely useful in applications. The damping model developed is nonlinear and the parameters are identified from experiments. Experimental and numerical results are compared for forced vibration responses measured for two different continuous structural elements: a free-edge plate and a shallow shell. The free-edge plate is interesting since it represents a case with no energy escape through the boundary.

Appendix A: Equation of motion for fractional damping

The standard linear solid viscoelastic material model in Eq. (1) can be modified by replacing the ordinary derivative with the fractional derivative of order $\alpha$, with $0<\alpha \le 1$, as done in [29]. This, mechanically speaking, corresponds to replacing the viscous dashpot with a spring-pot. In reference [29], the differential equation of motion in a form analogous to equation (31) was not derived. Therefore, it is reported here in order to complete the work presented in [29]

$$\begin{aligned}&\ddot{x}(t)+\frac{2\zeta \omega _n^{2-\alpha } /\sin (\alpha \pi /2)}{1+\left( {2\zeta /\sin (\alpha \pi /2)} \right) ^{2}\left( {\omega /\omega _n } \right) ^{2\alpha }+4\zeta \cot (\alpha \pi /2)\left( {\omega /\omega _n } \right) ^{\alpha }} \nonumber \\&\quad \left[ {D_t^\alpha x(t)+2\frac{\tilde{\beta }_2 }{h}x(t)D_t^\alpha x(t)+3\frac{\tilde{\beta }_3 }{h^{2}}x^{2}(t)D_t^\alpha x(t)} \right] +\,\omega _n^2 \left[ {x(t)+\frac{\beta _2 }{h}x^{2}(t)+\frac{\beta _3 }{h^{2}}x^{3}(t)} \right] \nonumber \\&\quad +\,\frac{4\zeta ^{2}\omega ^{2\alpha }\omega _n^{2-2\alpha } /\sin (\alpha \pi /2)}{1+\left( {2\zeta /\sin (\alpha \pi /2)} \right) ^{2}\left( {\omega /\omega _n } \right) ^{2\alpha }+4\zeta \cot (\alpha \pi /2)\left( {\omega /\omega _n } \right) ^{\alpha }} \nonumber \\&\quad \times \, \left[ {\left( {x(t)-\hbox {avg}(x)} \right) +\frac{\tilde{\beta }_2 }{h}\left( {x^{2}(t)-\hbox {avg}(x^{2})} \right) +\frac{\tilde{\beta }_3 }{h^{2}}\left( {x^{3}(t)-\hbox {avg}(x^{3})} \right) } \right] =\frac{\tilde{f} }{m}\sin (\omega t),\qquad \qquad \qquad \qquad \qquad \qquad \end{aligned}$$

(A1)

where the symbol avg indicates the average of the function in a period and

$$\begin{aligned} \tau _r = \frac{1}{\omega _n }\left[ {\frac{2 \zeta }{\sin (\alpha \pi /2)}} \right] ^{1/\alpha }. \end{aligned}$$

(A2)