Analytical Study of a Nonlinear Beam Including a Piezoelectric Patch

Abstract

The chapter presents modal responses and nonlinear interactions of a multi-physics nonlinear beam. It is composed of a piezoelectric material which is patched on a two-dimensional nonlinear Euler–Bernoulli beam. The spatio-temporal variables of governing equations of the composite beam are separated. Traced frequencies and mode shapes of the overall beam show modifications of its modal response due to the piezoelectric patch and its position on the beam. Studying the system in time domain via a multiple scale technique, leads to detection of its responses as function of frequency of excitation which present strongly nonlinear response due to nonlinearity of the beam and also presence of the piezoelectric patch.

Appendices

Appendix 1

$$\begin{aligned} M&=\int _{0}^{L}m(s)\phi _n(s)^2 \ ds \end{aligned}$$

(57)

$$\begin{aligned} \mathcal {C}_v&=\int _{0}^{L}c_v\phi _n(s)^2 \ ds \end{aligned}$$

(58)

$$\begin{aligned} T_v&=\int _{0}^{L}-EI(s)\phi _n^{iv}(s)\phi _n(s) \ ds \end{aligned}$$

(59)

$$\begin{aligned} T_{vvv}&=\int _{0}^{L}-EI(s)[\phi _n'(s)\phi _{n}''^2(s)-\phi _n'^2(s)\phi _n'''(s)]'\phi _n(s) \ ds \end{aligned}$$

(60)

$$\begin{aligned} T_{vv1}&=\int _{0}^{L}[-\phi _n'(s)\int _{0}^{L}\frac{c_u}{2} \int _{0}^{s}\phi _n'^2(s) \ ds \ ds]' \phi _n(s)\ ds \end{aligned}$$

(61)

$$\begin{aligned} T_{vv2}&=\int _{0}^{L}[-\phi _n'(s)\int _{0}^{L}\frac{-m(s)}{2}\int _{0}^{s}\phi _n'^2(s)\ ds \ ds]'\phi _n(s) \ ds \end{aligned}$$

(62)

$$\begin{aligned} T_{vvp}&=\int _{0}^{L}6V_c[\phi _n''(s)\phi _n'''(s)]'\phi _n(s)\ ds \end{aligned}$$

(63)

Appendix 2

$$\begin{aligned} \alpha _1&=\bigg (\frac{w_{vn}}{2}T_{vv2}-\frac{3}{8}T_{vvv}-\frac{T_{vvp}}{4}\bigg (\frac{2}{T_v}+\frac{T_{vvp}}{T_v-4Mw_{vn}^2}\bigg )\bigg )^2\end{aligned}$$

(64)

$$\begin{aligned} \alpha _2&=2Mw_{vn}\sigma \sqrt{\alpha _1}\end{aligned}$$

(65)

$$\begin{aligned} \alpha _3&=(Mw_{vn}\sigma )^2+\frac{w_{vn}^2\mathcal {C}_v^2}{4}\end{aligned}$$

(66)

$$\begin{aligned} \alpha _4&=\bigg (\frac{m_b\omega _{vn}f}{2}\bigg )^2 \end{aligned}$$

(67)