Size-dependent coupled bending–torsional vibration of Timoshenko microbeams

Abstract

In this paper, the coupled bending and torsional vibration analysis of microbeams under axial force based on Timoshenko’s beam theory is investigated. Modified non-classic coupled stress theory and the Hamilton principle used to establish the motion equations of the system. The generalized differential quadratures method is used to solve the obtained set of differential equations. After establishment of eigenvalue problem, two comparison studies are conducted to assure the validity and accuracy of the present solution and excellent agreement observed with the present results and those reported by other researchers in some specific cases by analytical solutions and classical beam theory. Afterwards, parametric studies are developed to examine the influences of boundary conditions, size effect, and various geometric characteristics of the beam on natural frequencies and the associated mode shapes are discussed. The results show that the non-compliance of the mass axis with the elastic axis reduces the natural frequency. Also, Poisson’s ratio have an opposite effect on the natural frequency.

Appendices

Appendix A

$${m}_{11}=2{l}^{2}G{\chi }_{11}=-{l}^{2}G {\psi }^{^{\prime}},$$

(21)

$${m}_{12}=2{l}^{2}G{\chi }_{12}=\frac{1}{2}{l}^{2}G \left({h}^{^{\prime\prime}}-x{\psi }^{{{\prime}}{^{\prime}}}+{\theta }^{^{\prime}}\right),$$

(22)

$${m}_{22}=2{l}^{2}G{\chi }_{22}=2{l}^{2}G {\psi }^{^{\prime}},$$

(23)

$${m}_{33}=2{l}^{2}G{\chi }_{33}=-{l}^{2}G {\psi }^{^{\prime}},$$

(24)

$${m}_{23}=2{l}^{2}G{\chi }_{23}=-\frac{1}{2}{l}^{2}Gz {\psi }^{^{\prime}},$$

(25)

$${\sigma }_{11}={\sigma }_{33}={\sigma }_{13}=0,$$

(26)

$${\sigma }_{22=-EZ{\theta }^{^{\prime}}},$$

(27)

$${\sigma }_{12}=-GZ{\psi }^{^{\prime}},$$

(28)

$${\sigma }_{23}=G\left({w}^{^{\prime}}-x{\psi }^{^{\prime}}- \theta \right).$$

(29)

Appendix B

The bending–torsion coupled governing motion equations of axially loaded Timoshenko microbeam after applying the GDQ method may be written in the following form

$$EI\mathop \sum \limits_{j = 1}^{N} C_{ij}^{\left( 2 \right)} \theta_{j} + kAG\mathop \sum \limits_{j = 1}^{N} C_{ij}^{\left( 1 \right)} w_{j} - kAG\theta_{i} + \frac{1}{4}GAl^{2} \mathop \sum \limits_{j = 1}^{N} C_{ij}^{\left( 3 \right)} w_{j} + \frac{1}{4}GAl^{2} \mathop \sum \limits_{j = 1}^{N} C_{ij}^{\left( 2 \right)} \theta_{j} - \frac{1}{4}GAl^{2} x_{\alpha } \mathop \sum \limits_{j = 1}^{N} C_{ij}^{\left( 3 \right)} \psi_{j} = \rho I\ddot{\theta },$$

(30)

$$kAG\sum_{j=1}^{N}{C}_{ij}^{\left(2\right)}{w}_{j}-kAG\sum_{j=1}^{N}{C}_{ij}^{\left(1\right)}{\theta }_{j}-P\sum_{j=1}^{N}{C}_{ij}^{\left(2\right)}{w}_{j}+P{x}_{\alpha }\sum_{j=1}^{N}{C}_{ij}^{\left(2\right)}{\psi }_{j}-\frac{1}{4}GA{l}^{2}\sum_{j=1}^{N}{C}_{ij}^{\left(4\right)}{w}_{j}-\frac{1}{4}GA{l}^{2}\sum_{j=1}^{N}{C}_{ij}^{\left(3\right)}{\theta }_{j}+\frac{1}{4}GA{l}^{2}{x}_{\alpha }\sum_{j=1}^{N}{C}_{ij}^{\left(4\right)}{\psi }_{j}=m\ddot{w}-m{x}_{\alpha }\ddot{\psi },$$

(31)

$$GJ\sum_{j=1}^{N}{C}_{ij}^{\left(2\right)}{\psi }_{j}-P\frac{{I}_{\alpha }}{m}\sum_{j=1}^{N}{C}_{ij}^{\left(2\right)}{\psi }_{j}+P{x}_{\alpha }\sum_{j=1}^{N}{C}_{ij}^{\left(2\right)}{w}_{j}+3GA{l}^{2}\sum_{j=1}^{N}{C}_{ij}^{\left(2\right)}{\psi }_{j}-\frac{1}{4}GJ{l}^{2}\sum_{j=1}^{N}{C}_{ij}^{\left(4\right)}{\psi }_{j}+\frac{1}{4}GA{l}^{2}{x}_{\alpha }\sum_{j=1}^{N}{C}_{ij}^{\left(3\right)}{\theta }_{j}+ \frac{1}{4}GA{l}^{2}{x}_{\alpha }\sum_{j=1}^{N}{C}_{ij}^{\left(4\right)}{w}_{j}={I}_{\alpha }\ddot{\psi }-m{x}_{\alpha }\ddot{w.}$$

(32)