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Scale Effect on the Nonlinear Vibration of Piezoelectric Sandwich Nanobeams on Winkler Foundation

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Abstract

Purpose

Goal for the present research is investigating the effect of scale effect on free vibration of piezoelectric sandwich nanobeams on Winkler foundation. For this purpose, the effects of nonlocal parameters and strain gradient parameters on the free vibration of the model are studied.

Methods

Based on the nonlocal strain gradient theory and Timoshenko beam theory, the nonlinear vibration of piezoelectric sandwich nanobeams on Winkler foundation is investigated. The nonlinear governing equations and boundary conditions are derived using the Hamilton's principle. The partial differential equation is transformed into ordinary differential equation by Galerkin's method, and then the nonlinear vibration of piezoelectric nanobeam is numerically analyzed using the Runge-Kutta method.

Results and Conclusions

The results show that the nonlinear frequency ratio decreases with the increase of length-to-thickness ratio. When the nonlocal parameter is not less than the strain gradient length scale parameter, the piezoelectric nanobeam exhibits stiffness softening effect. When the nonlocal parameter is not greater than the strain gradient length scale parameter, the piezoelectric nanobeam exhibits stiffness hardening effect. It is also observed that both large length-to-thickness ratios and shear deformation can attenuate the nonlocal strain gradient effect. In addition, changes in the external applied voltage have a significant effect on the natural frequency of the piezoelectric nanobeams and increasing the thickness of the piezoelectric layer can enhance the structural stiffness.

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (Grant Nos. 51965041, 51975266, 11972137, 11602072), the Foundation of Jiangxi Educational Committee (Grant No. GJJ190522) and the Jiangxi Provincial Natural Science Foundation (Grant No. 20202BAB211005) and the Shenzhen (China) Science and technology innovation committee (Project No: JSGG20180504170449754).

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Appendices

Appendix A

$$ Z_{1} = A_{11} \left[ {\frac{\partial u}{{\partial x}}\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{1}{2}\frac{{\partial^{2} w}}{{\partial x^{2} }}\left( {\frac{\partial w}{{\partial x}}} \right)^{2} - l^{2} \left( {\frac{{\partial^{3} u}}{{\partial x^{3} }}\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{\partial w}{{\partial x}}\frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{3} w}}{{\partial x^{3} }} + \left( {\frac{{\partial^{2} w}}{{\partial x^{2} }}} \right)^{3} } \right)} \right] $$
$$ Z_{2} = A_{11} \left[ {\frac{{\partial u^{2} }}{{\partial x^{2} }}\frac{\partial w}{{\partial x}} + \left( {\frac{\partial w}{{\partial x}}} \right)^{2} \frac{{\partial^{2} w}}{{\partial x^{2} }} - l^{2} \left( \begin{gathered} \frac{{\partial^{4} u}}{{\partial x^{4} }}\frac{\partial w}{{\partial x}} + 3\frac{\partial w}{{\partial x}}\frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{3} w}}{{\partial x^{3} }} + \hfill \\ \left( {\frac{\partial w}{{\partial x}}} \right)^{2} \frac{{\partial^{4} w}}{{\partial x^{4} }} \hfill \\ \end{gathered} \right)} \right] $$
$$ \begin{gathered} \alpha_{22} = - \frac{{n^{2} \pi^{2} A_{44} K_{s} \left( {l^{2} n^{2} \pi^{2} + L^{2} } \right) + k\left( {L^{4} + \mu^{2} n^{2} \pi^{2} L^{2} } \right) + n^{2} \pi^{2} N_{e} \left( {\mu^{2} n^{2} \pi^{2} + L^{2} } \right)}}{{2L^{3} }} \hfill \\ \alpha_{23} = - \frac{{K_{s} n\pi A_{44} \left( {l^{2} n^{2} \pi^{2} + L^{2} } \right)}}{{2L^{2} }} \hfill \\ \alpha_{24} = \frac{{K_{s} n^{2} \pi^{2} E_{15} }}{2L}\rm{, }\alpha_{uw} = 0 \hfill \\ \end{gathered} $$
$$ \alpha_{11} = - \frac{{n^{2} \pi^{2} A_{11} }}{{2L^{3} }}\left( {L^{2} + l^{2} n^{2} \pi^{2} } \right),\;\alpha_{12} = 0,\rm{ }\lambda_{1} = - \frac{{I_{0} (L - \mu \pi n)(L + \mu \pi n)}}{2L} $$
$$ \alpha_{223} = - \frac{{n^{4} \pi^{4} A_{11} \left( {10l^{2} n^{2} \pi^{2} + 3L^{2} } \right)}}{{16L^{5} }} $$
$$ \lambda_{2} = - \frac{{L^{2} + \mu^{2} n^{2} \pi^{2} }}{2L} $$
$$ \begin{gathered} \alpha_{32} = - \frac{{n\pi K_{s} A_{44} \left( {l^{2} n^{2} \pi^{2} + L^{2} } \right)}}{{2L^{2} }} \hfill \\ \alpha_{33} = - \frac{{\left( {\pi^{2} l^{2} n^{2} + L^{2} } \right)\left( {A_{44} K_{s} L^{2} + \pi^{2} D_{11} n^{2} } \right)}}{{2L^{3} }} \hfill \\ \alpha_{34} = \frac{{n\pi \left( {E_{15} K_{s} + F_{31} } \right)}}{2} \hfill \\ \end{gathered} $$
$$ \begin{gathered} \lambda_{3} = - \frac{{\left( {L^{2} + \pi^{2} \mu^{2} n^{2} } \right)}}{2L},\rm{ }\alpha_{42} = - \frac{{n^{2} \pi^{2} E_{15} }}{2L} \hfill \\ \alpha_{43} = - \frac{{n\pi (E_{15} + F_{31} )}}{2},\rm{ }\alpha_{44} = - \frac{{LS_{33} + n^{2} \pi^{2} S_{11} }}{2} \hfill \\ a_{1} = \frac{{\left( {\alpha_{22} - \frac{{\alpha_{24} \alpha_{42} }}{{\alpha_{44} }}} \right)}}{{\beta_{2} I_{0} }},\rm{ }a_{2} = \frac{{\left( {\alpha_{23} - \frac{{\alpha_{24} \alpha_{43} }}{{\alpha_{44} }}} \right)}}{{\beta_{2} I_{0} }} \hfill \\ a_{3} = \frac{{\alpha_{223} }}{{\beta_{2} I_{0} }},\rm{ }a_{4} = \frac{{\left( {\alpha_{32} - \frac{{\alpha_{34} \alpha_{42} }}{{\alpha_{44} }}} \right)}}{{\beta_{3} I_{2} }},\rm{ }a_{5} = \frac{{\left( {\alpha_{33} - \frac{{\alpha_{34} \alpha_{43} }}{{\alpha_{44} }}} \right)}}{{\beta_{3} I_{2} }} \hfill \\ \end{gathered} $$

Appendix B

The displacement field is expressed by the following form

$$ \begin{gathered} u_{x} \left( {x,z,t} \right) = u\left( {x,t} \right) - z\frac{\partial w}{{\partial x}} \hfill \\ u_{z} \left( {x,z,t} \right) = w\left( {x,t} \right) \hfill \\ \end{gathered} $$
(B1)

According to the Von Karman's nonlinear strain theory, the nonzero strain components are expressed as

$$ \varepsilon_{xx} = \frac{\partial u}{{\partial x}} - z\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{1}{2}\left( {\frac{\partial w}{{\partial x}}} \right)^{2} $$
(B2)

Using the Hamilton’s principle, we can obtain the following differential equations of motion

$$ \frac{{\partial N_{xx} }}{\partial x} = I_{0} \frac{{\partial^{2} u}}{{\partial t^{2} }} $$
(B3)
$$ \frac{{\partial M_{xx} }}{\partial x} + \frac{\partial }{\partial x}\left( {N_{xx} \frac{\partial w}{{\partial x}}} \right) - kw = I_{0} \frac{{\partial^{2} w}}{{\partial t^{2} }} - I_{2} \frac{{\partial^{4} w}}{{\partial x^{2} \partial t^{2} }} $$
(B4)
$$ \int_{ - h/2}^{h/2} {\left[ {\cos \left( {\beta z} \right)\frac{{\partial D_{x} }}{\partial x} + \beta \sin \left( {\beta z} \right)D_{z} } \right]} dz = 0 $$
(B5)

The corresponding boundary conditions are

$$ U = 0\rm{ }or\rm{ }N_{xx} = 0,\rm{ }at\rm{ }x = 0,L $$
(B6)
$$ W = 0\rm{ }or\rm{ }\frac{{\partial M_{xx} }}{\partial x} + N_{xx} \frac{\partial w}{{\partial x}} = 0,\rm{ }at\rm{ }x = 0,L $$
(B7)
$$ \frac{\partial w}{{\partial x}} = 0\rm{ }or\rm{ }M_{xx} = 0,\rm{ }at\rm{ }x = 0,L $$
(B8)
$$ \phi = 0\rm{ }{\text{or}}\rm{ }\int\limits_{{\frac{ - h}{2}}}^{\frac{h}{2}} {D_{x} } \cos \left( {\beta z} \right)dz = 0,\rm{ }{\text{at}}\rm{ }x = 0,L $$
(B9)

Then, based on nonlocal strain gradient theory and Euler beam theory, the nonlinear governing equation Eqs. (B3)–(B5) of piezoelectric nanobeam is transformed into displacement function as follows

$$ \begin{gathered} A_{11} \left[ {\frac{{\partial u^{2} }}{{\partial x^{2} }} + \frac{\partial w}{{\partial x}}\frac{{\partial^{2} w}}{{\partial x^{2} }} - l^{2} \left( {\frac{{\partial^{4} u}}{{\partial x^{4} }} + 3\frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{3} w}}{{\partial x^{3} }} + \frac{\partial w}{{\partial x}}\frac{{\partial^{4} w}}{{\partial x^{4} }}} \right)} \right] \hfill \\ \quad = I_{0} \frac{{\partial^{2} u}}{{\partial t^{2} }} - \mu^{2} I_{0} \frac{{\partial^{4} u}}{{\partial x^{2} \partial t^{2} }} \hfill \\ \end{gathered} $$
(B10)
$$ \begin{gathered} - D_{11} \left( {\frac{{\partial^{4} w}}{{\partial x^{4} }} - l^{2} \frac{{\partial^{6} w}}{{\partial x^{6} }}} \right) + F_{31} \frac{{\partial^{2} \phi }}{{\partial x^{2} }} - k\left( {w - \mu^{2} \frac{{\partial^{2} w}}{{\partial x^{2} }}} \right) + Z_{1} + Z_{2} \hfill \\ N_{e} \frac{{\partial^{2} w}}{{\partial x^{2} }} - \mu^{2} N_{e} \frac{{\partial^{4} w}}{{\partial x^{4} }} = I_{0} \frac{{\partial^{2} w}}{{\partial t^{2} }} - \mu^{2} I_{0} \frac{{\partial^{4} w}}{{\partial x^{2} \partial t^{2} }} - I_{2} \frac{{\partial^{4} w}}{{\partial x^{2} \partial t^{2} }} + \mu^{2} I_{2} \frac{{\partial^{6} w}}{{\partial x^{4} \partial t^{2} }} \hfill \\ \end{gathered} $$
(B11)
$$ S_{11} \frac{{\partial^{2} \phi }}{{\partial x^{2} }} - F_{31} \frac{{\partial^{2} w}}{{\partial x^{2} }} - S_{33} \phi = 0 $$
(B12)

Then, the nonlinear equations are obtained by Galerkin's method and solved by adaptive fourth-order Runge–Kutta method.

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Luo, T., Mao, Q., Zeng, S. et al. Scale Effect on the Nonlinear Vibration of Piezoelectric Sandwich Nanobeams on Winkler Foundation. J. Vib. Eng. Technol. 9, 1289–1303 (2021). https://doi.org/10.1007/s42417-021-00297-8

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