Tuning inter-modal energy exchanges of a nonlinear electromechanical beam by a nonlinear circuit

Abstract

A model of a nonlinear beam with a piezoelectric patch linked to a nonlinear circuit is considered. The physical and mechanical parameters of the system are such that it presents a 1:3 internal resonance. The aim is to study, for different time scales, the effect of the nonlinearity of the electrical circuit on energy exchanges between the resonant modes. In fact, we would like to master energy channelling between two internally resonant modes of a composite beam via a nonlinear circuit. The investigations are carried out on the projected system equations on its internally resonant modes. The system behaviours at different scales of time are studied by coupled methods of complexification and multiple scale. Analytical developments permit to reveal different system dynamics characterized by periodic and/or non-periodic regimes. Finally, the paper is accompanied by comparisons between systems equipped with a resonant circuit (linear) and a nonlinear one.

Appendices

Definition of the parameters of Eq. 4

The parameters of Eq. 4 are as follows:

$$\begin{aligned} \mu _n= & {} \displaystyle {\int _{0}^{L_b}c_v\phi _n^2 \ \mathrm{d}s } \\ \omega _n^2= & {} \displaystyle {\int _{0}^{L_b}EI\phi _n^{(iv)}(s)\phi _n\ \mathrm{d}s}\\ F_n= & {} \displaystyle {\int _{0}^{L_b}\mu F\phi _n\ \mathrm{d}s }\\ De_{nnn}= & {} \displaystyle {\int _{0}^{L_b}(-EI)(\phi _n'(\phi _n'\phi _n'')')' \ \phi _n\ \mathrm{d}s}\\ De_{nnm}= & {} \displaystyle {\int _{0}^{L_b}(-EI)(\phi _n'(\phi _m'\phi _n'')'+\phi _n'(\phi _m'\phi _n'')'+\phi _n'(\phi _n'\phi _m'')')' \ \phi _n\ \mathrm{d}s} \end{aligned}$$

$$\begin{aligned} De_{nmm}= & {} \displaystyle {\int _{0}^{L_b}(-EI)(\phi _m'(\phi _m'\phi _n'')'+\phi _n'(\phi _m'\phi _m'')'+\phi _m'(\phi _n'\phi _m'')')' \ \phi _n\ \mathrm{d}s}\nonumber \\ De_{mmm}= & {} \displaystyle {\int _{0}^{L_b}(-EI)(\phi _m'(\phi _m'\phi _m'')')' \ \phi _n\ \mathrm{d}s}\nonumber \\ G_{nnn}= & {} \displaystyle {\int _{0}^{L_b} \left( \phi _n'\int _{s}^{L_b}\frac{-\mu }{2}\int _{s}^{0}\phi _n'^2\ \mathrm{d}s \ \mathrm{d}s\right) '\ \phi _n \ \mathrm{d}s}\nonumber \\ G_{nnm}= & {} \displaystyle {\int _{0}^{L_b} \left( \phi _n'\int _{s}^{L_b}\frac{-\mu }{2}\int _{s}^{0}\phi _n'\phi _m'\ \mathrm{d}s \ \mathrm{d}s\right) '\ \phi _n \ \mathrm{d}s}\nonumber \\ G_{nmm}= & {} \displaystyle {\int _{0}^{L_b} \left( \phi _n'\int _{s}^{L_b}\frac{-\mu }{2}\int _{s}^{0}\phi _m'^2\ \mathrm{d}s \ \mathrm{d}s\right) '\ \phi _n \ \mathrm{d}s}\nonumber \\ G_{mnn}= & {} \displaystyle {\int _{0}^{L_b} \left( \phi _m'\int _{s}^{L_b}\frac{-\mu }{2}\int _{s}^{0}\phi _n'^2\ \mathrm{d}s \ \mathrm{d}s\right) '\ \phi _n \ \mathrm{d}s}\nonumber \\ G_{mnm}= & {} \displaystyle {\int _{0}^{L_b} \left( \phi _m'\int _{s}^{L_b}\frac{-\mu }{2}\int _{s}^{0}\phi _n'\phi _m'\ \mathrm{d}s \ \mathrm{d}s\right) '\ \phi _n \ \mathrm{d}s}\nonumber \\ G_{mmm}= & {} \displaystyle {\int _{0}^{L_b} \left( \phi _m'\int _{s}^{L_b}\frac{-\mu }{2}\int _{s}^{0}\phi _m'^2\ \mathrm{d}s \ \mathrm{d}s\right) '\ \phi _n \ \mathrm{d}s}\nonumber \\ De_V= & {} \displaystyle {-\frac{b_pd_{31}(y_2^2-y_1^2)}{2h_p}\int _{x_1}^{x_2}\phi _n''\ \mathrm{d}s} \end{aligned}$$

(46)

$$\begin{aligned} \mu _m= & {} \displaystyle {\int _{0}^{L_b}c_v\phi _m^2 \ \mathrm{d}s } \nonumber \\ \omega _m^2= & {} \displaystyle {\int _{0}^{L_b}EI\phi _m^{(iv)}(s)\phi _m\ \mathrm{d}s}\nonumber \\ F_m= & {} \displaystyle {\int _{0}^{L_b}\mu F\phi _m\ \mathrm{d}s }\nonumber \\ Ae_{nnn}= & {} \displaystyle {\int _{0}^{L_b}(-EI)(\phi _n'(\phi _n'\phi _n'')')' \ \phi _m\ \mathrm{d}s}\nonumber \\ Ae_{nnm}= & {} \displaystyle {\int _{0}^{L_b}(-EI)(\phi _n'(\phi _m'\phi _n'')'+\phi _n'(\phi _m'\phi _n'')'+\phi _n'(\phi _n'\phi _m'')')' \ \phi _m\ \mathrm{d}s}\nonumber \\ Ae_{nnm}= & {} \displaystyle {\int _{0}^{L_b}(-EI)(\phi _m'(\phi _m'\phi _n'')'+\phi _n'(\phi _m'\phi _m'')'+\phi _m'(\phi _n'\phi _m'')')' \ \phi _m\ \mathrm{d}s}\nonumber \\ Ae_{mmm}= & {} \displaystyle {\int _{0}^{L_b}(-EI)(\phi _m'(\phi _m'\phi _m'')')' \ \phi _m\ \mathrm{d}s}\nonumber \\ Ga_{nnn}= & {} \displaystyle {\int _{0}^{L_b} \left( \phi _n'\int _{s}^{L_b}\frac{-\mu }{2}\int _{s}^{0}\phi _n'^2\ \mathrm{d}s \ \mathrm{d}s\right) '\ \phi _m \ \mathrm{d}s}\nonumber \\ Ga_{nnm}= & {} \displaystyle {\int _{0}^{L_b} \left( \phi _n'\int _{s}^{L_b}\frac{-\mu }{2}\int _{s}^{0}\phi _n'\phi _m'\ \mathrm{d}s \ \mathrm{d}s\right) '\ \phi _m \ \mathrm{d}s}\nonumber \\ Ga_{nmm}= & {} \displaystyle {\int _{0}^{L_b} \left( \phi _n'\int _{s}^{L_b}\frac{-\mu }{2}\int _{s}^{0}\phi _m'^2\ \mathrm{d}s \ \mathrm{d}s\right) '\ \phi _m \ \mathrm{d}s}\nonumber \\ Ga_{mnn}= & {} \displaystyle {\int _{0}^{L_b} \left( \phi _m'\int _{s}^{L_b}\frac{-\mu }{2}\int _{s}^{0}\phi _n'^2\ \mathrm{d}s \ \mathrm{d}s\right) '\ \phi _m \ \mathrm{d}s}\nonumber \\ Ga_{mnm}= & {} \displaystyle {\int _{0}^{L_b} \left( \phi _m'\int _{s}^{L_b}\frac{-\mu }{2}\int _{s}^{0}\phi _n'\phi _m'\ \mathrm{d}s \ \mathrm{d}s\right) '\ \phi _m \ \mathrm{d}s}\nonumber \\ Ga_{mmm}= & {} \displaystyle {\int _{0}^{L_b} \left( \phi _m'\int _{s}^{L_b}\frac{-\mu }{2}\int _{s}^{0}\phi _m'^2\ \mathrm{d}s \ \mathrm{d}s\right) '\ \phi _m \ \mathrm{d}s}\nonumber \\ Ae_V= & {} \displaystyle {-\frac{b_pd_{31}(y_2^2-y_1^2)}{2h_p}\int _{x_1}^{x_2}\phi _m''\mathrm{d}s} \end{aligned}$$

(47)

The parameters of Eq. 9 are defined as:

$$\begin{aligned} \begin{array}{rlrll} a_1&{}=\displaystyle {\frac{\mu _n}{\omega _n}}, &{} \nu &{}=\displaystyle {\frac{\varOmega }{\omega _n}}, \\ \\ \varLambda _{nnn}&{}=\displaystyle {\frac{De_{nnn}}{\omega _n^2}},&{} \varLambda _{nnm}&{}=\displaystyle {\frac{De_{nnm}}{\omega _n^2}}, \\ \varLambda _{nmm}&{}=\displaystyle {\frac{De_{nmm}}{\omega _n^2}},&{} \varLambda _{mmm}&{}=\displaystyle {\frac{De_{mmm}}{\omega _n^2}}, \\ L_{nnn}&{}=\displaystyle {G_{nnn}}, &{} L_{nnm}&{}=\displaystyle {G_{nnm}}, \\ L_{nmm}&{}=\displaystyle {G_{nmm}}, &{} L_{mnn}&{}=\displaystyle {G_{mnn}}, \\ L_{mnm}&{}=\displaystyle {G_{mnm}}, &{} L_{mmm}&{}=\displaystyle {G_{mmm}}, \\ \gamma _V&{}=\displaystyle {\frac{De_V}{L_V\omega _n^2}}, \\ \gamma _n&{}=-\displaystyle {\frac{De_VL_n}{L_V\omega _n^2}}, &{} \gamma _m&{}=-\displaystyle {\frac{De_VL_m}{L_V\omega _n^2}} \end{array} \end{aligned}$$

(48)

$$\begin{aligned} \begin{array}{rlrll} a_2&{}{}=\displaystyle {\frac{\mu _m}{\omega _m}}, &{}{} \varGamma _{nnn}&{}{}=\displaystyle {\frac{Ae_{nnn}}{\omega _n^2}}, \\ \\ \varGamma _{nnm}&{}{}=\displaystyle {\frac{Ae_{nnm}}{\omega _n^2}},&{}{} \varGamma _{nmm}&{}{}=\displaystyle {\frac{Ae_{nmm}}{\omega _n^2}}, \\ \\ T_{nnn}&{}{}=\displaystyle {Ga_{nnn}}, &{}{} T_{nnm}&{}{}=\displaystyle {Ga_{nnm}}, \\ \\ T_{nmm}&{}{}=\displaystyle {Ga_{nmm}}, &{}{} T_{mnn}&{}{}=\displaystyle {Ga_{mnn}}, \\ \\ T_{mnm}&{}{}=\displaystyle {Ga_{mnm}}, &{}{} T_{mmm}&{}{}=\displaystyle {Ga_{mmm}}, \\ \\ \varGamma _{mmm}&{}{}=\displaystyle {\frac{Ae_{mmm}}{\omega _n^2}}, \\ \\ \beta _V&{}{}=\displaystyle {\frac{De_V}{L_V\omega _n^2}},&{}{} \beta _n&{}{}=-\displaystyle {\frac{De_VL_n}{L_V\omega _n^2}}, \\ \\ \beta _m&{}{}=-\displaystyle {\frac{De_VL_m}{L_V\omega _n^2}}, &{}{} a_3&{}{}=\displaystyle {\frac{R}{L_0\omega _n}},\\ \\ \gamma &{}{}=\displaystyle {\frac{1}{C_{NL}L_0\omega _n^2}}, &{}{} \varTheta _V&{}{}=\displaystyle {\frac{1}{L_VL_0\omega _n^2}}-\displaystyle {\frac{1}{C_{neg}L_0\omega _n^2}}, \\ \\ \varTheta _n&{}{}=-\displaystyle {\frac{L_n}{L_VL_0\omega _n^2}}, &{}{} \varTheta _m&{}{}=-\displaystyle {\frac{L_m}{L_VL_0\omega _n^2}} \end{array} \end{aligned}$$

(49)

It is noted that $\displaystyle {\zeta =1-\frac{L_V}{C_{neg}}}$ as $\varTheta _V=\displaystyle {\frac{\zeta }{L_VL_0\omega _n^2}}$.

Definition of the matrix $\varvec{A}$ from Eq. 39

$$\begin{aligned} \begin{array}{ll} \varvec{A}= \left( \begin{array}{ll} \displaystyle {\frac{\partial \mathcal {F}(\phi ,\overline{\phi },\psi ,\overline{\psi })}{\partial \phi }} &{} \displaystyle {\frac{\partial \mathcal {F}(\phi ,\overline{\phi },\psi ,\overline{\psi })}{\partial \overline{\phi }}} \\ \\ \displaystyle {\frac{\partial \mathcal {\overline{F}}(\phi ,\overline{\phi },\psi ,\overline{\psi })}{\partial \phi }} &{} \displaystyle {\frac{\partial \mathcal {\overline{F}}(\phi ,\overline{\phi },\psi ,\overline{\psi })}{\partial \overline{\phi }}} \end{array} \right) \end{array} \end{aligned}$$

(50)

Definition of the matrices $M_j$ with $j=1,\ldots ,5$ from Eq. 41

$$\begin{aligned} \begin{array}{ll} \varvec{M_1}= \left( \begin{array}{ll} \displaystyle {\frac{\partial \mathcal {G}}{\partial \phi }} &{} \displaystyle {\frac{\partial \mathcal {G}}{\partial \overline{\phi }}} \\ \\ \displaystyle {\frac{\partial \mathcal {\overline{G}}}{\partial \phi }} &{} \displaystyle {\frac{\partial \mathcal {\overline{G}}}{\partial \overline{\phi }}} \end{array} \right) \end{array} \end{aligned}$$

(51)

$$\begin{aligned} \begin{array}{ll} \varvec{M_2}= \left( \begin{array}{ll} \displaystyle {\frac{\partial \mathcal {G}}{\partial \phi _3}} &{} \displaystyle {\frac{\partial \mathcal {G}}{\partial \overline{\phi }_3}} \\ \\ \displaystyle {\frac{\partial \mathcal {\overline{G}}}{\partial \phi _3}} &{} \displaystyle {\frac{\partial \mathcal {\overline{G}}}{\partial \overline{\phi }_3}} \end{array} \right) \end{array} \end{aligned}$$

(52)

$$\begin{aligned} \begin{array}{ll} \varvec{M_3}= \left( \begin{array}{ll} \displaystyle {\frac{\partial \mathcal {G}}{\partial \psi }} &{} \displaystyle {\frac{\partial \mathcal {G}}{\partial \overline{\psi }}} \\ \\ \displaystyle {\frac{\partial \mathcal {\overline{G}}}{\partial \psi }} &{} \displaystyle {\frac{\partial \mathcal {\overline{G}}}{\partial \overline{\psi }}} \end{array} \right) \end{array} \end{aligned}$$

(53)

$$\begin{aligned} \begin{array}{ll} \varvec{M_4}= \left( \begin{array}{ll} \displaystyle {\frac{\partial \mathcal {H}}{\partial \phi }} &{} \displaystyle {\frac{\partial \mathcal {H}}{\partial \overline{\phi }}} \\ \\ \displaystyle {\frac{\partial \mathcal {\overline{H}}}{\partial \phi }} &{} \displaystyle {\frac{\partial \mathcal {\overline{H}}}{\partial \overline{\phi }}} \end{array} \right) \end{array} \end{aligned}$$

(54)

$$\begin{aligned} \begin{array}{ll} \varvec{M_5}= \left( \begin{array}{ll} \displaystyle {\frac{\partial \mathcal {H}}{\partial \phi _3}} &{} \displaystyle {\frac{\partial \mathcal {H}}{\partial \overline{\phi }_3}} \\ \\ \displaystyle {\frac{\partial \mathcal {\overline{H}}}{\partial \phi _3}} &{} \displaystyle {\frac{\partial \mathcal {\overline{H}}}{\partial \overline{\phi }_3}} \end{array} \right) \end{array} \end{aligned}$$

(55)