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Free vibration analysis of rotating piezoelectric/flexoelectric microbeams

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Abstract

In recent years, flexoelectric and piezoelectric effects have been investigated in many studies, but these effects have not been studied simultaneously in a rotating microbeam, taking into account the size effects. The simultaneous investigation of these effects is necessary to better understand the vibrational behavior of flexoelectric microstructures. It will be possible to identify the flexoelectric and size-dependent effects and their influence on the vibration frequency as a result of this analysis. Furthermore, it can lead to improved design of microstructures. Under the influence of the flexoelectric effect, the current research analyzes the free vibrations of a rotating piezoelectric microbeam. Also, all rotating microbeams are affected by the flexoelectric effect, and therefore the study of this phenomenon is of great importance. Due to the small strain and moderate rotation of the beam, Euler–Bernoulli beam theory and von Karman strain–displacement relations are employed in this article. The Gibbs energy function presented in this research includes the effects of couple stress, piezoelectric and flexoelectric. Hamilton's principle is used to obtain the equations of motion and boundary conditions of the rotating microbeam. In the final step, it has been examined in this study how parameters such as slenderness ratios, changes in rotation speed, and coefficients related to flexoelectric and piezoelectric properties affect the vibration of a rotating microbeam. The results show that the natural frequency increases as the rotation speed increases. Further, the use of piezoelectric and flexoelectric effects has increased the structure's stiffness and, as a result, increased its natural frequency.

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Authors and Affiliations

  1. Mechanical Engineering Department, Shahrekord University, Shahrekord, Iran

    S. M. H. Hosseini

  2. Faculty of Engineering, Shahrekord University, Shahrekord, Iran

    Yaghoub Tadi Beni

  3. Nanotechnology Research Center, Shahrekord University, Shahrekord, 8818634141, Iran

    Yaghoub Tadi Beni

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Appendix

Appendix

$$\left[{C}_{11}A\frac{\partial {u}_{0}}{\partial x}+\frac{{C}_{11}A}{2}{\left(\frac{\partial W}{\partial x}\right)}^{2}+\frac{{e}_{31}{\mu }_{31}A}{2{k}_{33}}\frac{{\partial }^{2}W}{\partial {x}^{2}}\right]\delta {u}_{0}\left|\begin{array}{c}x=L\\ x=0\end{array}\right.=0,$$
(31)
$$\left[{-C}_{11}I\frac{{\partial }^{3}W}{\partial {x}^{3}}-\frac{{e}_{31}^{2}I}{{k}_{33}}\frac{{\partial }^{3}W}{\partial {x}^{3}}-\mu A{l}^{2}\frac{{\partial }^{3}W}{\partial {x}^{3}}-\frac{{\mu }_{31}^{2}A}{2{k}_{33}}\frac{{\partial }^{3}W}{\partial {x}^{3}}+{C}_{11}A\frac{\partial {u}_{0}}{\partial x}\frac{\partial W}{\partial x}+\frac{A}{2}{\left(\frac{\partial W}{\partial x}\right)}^{3}+{e}_{31}Vb\frac{\partial W}{\partial x}-J{\Omega }^{2}\frac{\partial W}{\partial x}+J\frac{{\partial }^{3}W}{\partial x\partial {t}^{2}}\right]\delta W\left|\begin{array}{c}x=L\\ x=0\end{array}\right.=0,$$
(32)
$$\left[{C}_{11}I\frac{{\partial }^{2}W}{\partial {x}^{2}}+\frac{{e}_{31}^{2}I}{{k}_{33}}\frac{{\partial }^{2}W}{\partial {x}^{2}}+\mu A{l}^{2}\frac{{\partial }^{2}W}{\partial {x}^{2}}+\frac{{\mu }_{31}^{2}A}{2{k}_{33}}\frac{{\partial }^{2}W}{\partial {x}^{2}}\right]\frac{\partial W}{\partial x}\left|\begin{array}{c}x=L\\ x=0\end{array}\right.=0.$$
(33)

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Hosseini, S.M.H., Beni, Y.T. Free vibration analysis of rotating piezoelectric/flexoelectric microbeams. Appl. Phys. A 129, 330 (2023). https://doi.org/10.1007/s00339-023-06615-z

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