Application of smart electro-rheological dampers in semi-active control of electro-rheological sandwich plates with nanocomposite facesheets rested on orthotropic visco-Pasternak foundation

Abstract

In this research, semi-active control of sandwich plates with electro-rheological (ER) core and carbon nanotubes-reinforced composite facesheets using smart ER dampers is studied. Sandwich plate is subjected to the external electric field and rested on orthotropic visco-Pasternak foundation. The material properties of ER core and nanocomposite facesheets are obtained by Yalcintas model and Eshelby–Mori–Tanaka approach, respectively. The governing equations of motion are solved by a combination between finite element and Newmark methods for clamped and simply supported boundary condition. The effects of various parameters such as applied voltage, controlled electric field and initial gap of the electrodes on the vibration suppression time are discussed. The results show that the settling time of system introduced in this work is much less than previous researches in this field which is a very important advantage.

Appendices

Appendix 1

$$\left[ {N_{c} } \right] = \left[ {N_{g} } \right]\left\{ {q\left( t \right)} \right\},\begin{array}{*{20}c} {} & {} \\ \end{array} \left[ {N_{g} } \right] = \frac{d}{{h_{c} }}\left[ {\begin{array}{*{20}c} {\frac{{N_{ut} - N_{ub} }}{d} + N_{w,x} } \\ {\frac{{N_{vt} - N_{vb} }}{d} + N_{w,y} } \\ \end{array} } \right]$$

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$$\begin{array}{*{20}l} {\varepsilon_{x}^{i} = \left( {\left[ {N_{u\alpha ,x} } \right] - z\left[ {N_{w,xx} } \right]} \right)\left\{ {q\left( t \right)} \right\},\quad \varepsilon_{y}^{i} = \left( {\left[ {N_{v\alpha ,y} } \right] - z\left[ {N_{w,yy} } \right]} \right)\left\{ {q\left( t \right)} \right\},} \hfill \\ {\gamma_{xy}^{i} = \left( {\left[ {N_{u\alpha ,y} } \right] + \left[ {N_{v\alpha ,x} } \right] - 2z\left[ {N_{w,xy} } \right]} \right)\left\{ {q\left( t \right)} \right\}} \hfill \\ \end{array}$$

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$$E_{x}^{t} = \left( { - \sin \left( {\frac{{\pi {\mkern 1mu} \left( {z - \frac{{h_{c} }}{2}} \right)}}{{h_{t} }}} \right)\left[ {N_{\varphi ,x} } \right]} \right)\left\{ {q\left( t \right)} \right\},\quad E_{y}^{t} = \left( { - \sin \left( {\frac{{\pi {\mkern 1mu} \left( {z - \frac{{h_{c} }}{2}} \right)}}{{h_{t} }}} \right)\left[ {N_{\varphi ,y} } \right]} \right)\left\{ {q\left( t \right)} \right\},\quad E_{z}^{t} = \left( { - \frac{\pi }{{h_{t} }}\cos \left( {\frac{{\pi {\mkern 1mu} \left( {z - \frac{{h_{c} }}{2}} \right)}}{{h_{t} }}} \right)\left[ {N_{\varphi } } \right]} \right)\left\{ {q\left( t \right)} \right\},$$

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$$E_{x}^{b} = \left( { - \sin \left( {\frac{{\pi {\mkern 1mu} \left( { - z - \frac{{h_{c} }}{2}} \right)}}{{h_{b} }}} \right)\left[ {N_{\varphi ,x} } \right]} \right)\left\{ {q\left( t \right)} \right\},\quad E_{y}^{b} = \left( { - \sin \left( {\frac{{\pi {\mkern 1mu} \left( { - z - \frac{{h_{c} }}{2}} \right)}}{{h_{b} }}} \right)\left[ {N_{\varphi ,y} } \right]} \right)\left\{ {q\left( t \right)} \right\},\quad E_{z}^{b} = \left( {\frac{\pi }{{h_{b} }}\cos \left( {\frac{{\pi {\mkern 1mu} \left( { - z - \frac{{h_{c} }}{2}} \right)}}{{h_{b} }}} \right)\left[ {N_{\varphi } } \right]} \right)\left\{ {q\left( t \right)} \right\},$$

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$$\left[ {\varepsilon^{i} } \right] = \left[ {\begin{array}{*{20}c} {\varepsilon_{x}^{i} } & {\varepsilon_{y}^{i} } & {\gamma_{xy}^{i} } & {E_{x}^{i} } & {E_{y}^{i} } & {E_{z}^{i} } \\ \end{array} } \right]^{T} ,$$

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$$\left[ {M_{1} } \right] = \rho_{c} h_{c} \int_{A} {\left[ {N_{w} } \right]^{T} \left[ {N_{w} } \right]} {\text{d}}A,\quad \left[ {M_{2} } \right] = I_{c} \int_{A} {\left[ {N_{g} } \right]^{T} \left[ {N_{g} } \right]} {\text{d}}A,\quad I_{c} = \rho_{c} \int\limits_{{ - h_{c} /2}}^{{h_{c} /2}} {z^{2} {\text{d}}z} ,$$

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$$\left[ {M_{3} } \right] = \rho_{i} h_{i} \int_{A} {\left[ {N_{w} } \right]^{T} \left[ {N_{w} } \right]} {\text{d}}A,\left[ {M_{4} } \right] = \rho_{i} h_{i} \int_{A} {\left[ {N_{u}^{i} } \right]^{T} \left[ {N_{u}^{i} } \right]} {\text{d}}A,\left[ {M_{5} } \right] = \rho_{i} h_{i} \int_{A} {\left[ {N_{v}^{i} } \right]^{T} \left[ {N_{v}^{i} } \right]} {\text{d}}A.$$

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Appendix 2

$$A_{1} = \left[ {\begin{array}{*{20}c} {\left[ {N_{u}^{t} } \right]^{T} } & {\left[ {N_{v}^{t} } \right]^{T} } & 0 & 0 & {\left[ {N_{w} } \right]^{T} } & 0 \\ \end{array} } \right]\left( {\frac{3}{2}\frac{{\pi \eta R_{el}^{4} }}{{\left[ {h_{0} + \delta_{\text{Damper}} (t)} \right]^{3} }}} \right)\left[ {\begin{array}{*{20}c} {\cos \psi^{\prime}\cos \phi^{\prime}\left[ {N_{ut} } \right]} \\ {\cos \psi^{\prime}\sin \phi^{\prime}\left[ {N_{vt} } \right]} \\ 0 \\ 0 \\ {sin\psi^{\prime}\left[ {N_{w} } \right]} \\ 0 \\ \end{array} } \right]$$

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$$A_{2} = \left[ {\begin{array}{*{20}c} {\left[ {N_{u}^{t} } \right]^{T} } & {\left[ {N_{v}^{t} } \right]^{T} } & 0 & 0 & {\left[ {N_{w} } \right]^{T} } & 0 \\ \end{array} } \right]\left( {\frac{4}{3}\frac{{\pi R_{el}^{3} }}{{h_{0} + \delta_{\text{Damper}} (t)}}\tau_{y} (E)} \right)\left[ {\begin{array}{*{20}c} {\cos \psi^{\prime}\cos \phi^{\prime}\left[ {N_{ut} } \right]} \\ {\cos \psi^{\prime}\sin \phi^{\prime}\left[ {N_{vt} } \right]} \\ 0 \\ 0 \\ {sin\psi^{\prime}\left[ {N_{w} } \right]} \\ 0 \\ \end{array} } \right]\text{sgn} \left\{ {\dot{q}(t)} \right\}$$

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$$\begin{aligned} K_{f} = K_{{w{\mkern 1mu} }} - K_{gx} {\mkern 1mu} \left( {\cos^{2} \theta \frac{{\partial^{2} }}{{\partial x^{2} }} + \sin 2\theta \frac{{\partial^{2} }}{\partial y\partial x} + \sin^{2} \theta \frac{{\partial^{2} }}{{\partial y^{2} }}} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - K_{gy} {\mkern 1mu} \left( {\sin^{2} \theta \frac{{\partial^{2} }}{{\partial x^{2} }} - \sin 2\theta \frac{{\partial^{2} }}{\partial y\partial x} + \cos^{2} \theta \frac{{\partial^{2} }}{{\partial y^{2} }}} \right), \hfill \\ \end{aligned}$$

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$$K_{e} = - 2\,{\mkern 1mu} e_{31} \,V_{{0{\mkern 1mu} }} \frac{{\partial^{2} }}{{\partial x^{2} }} - 2\,{\mkern 1mu} e_{32} \,V_{0} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial y^{2} }}.$$

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