1 Introduction

The vibration response study of microelectromechanical system (MEMS) devices is very important to the design and optimization of a small component of the equipment. Today, manufacturing of small size devices in the field of the MEMS is possible by the development of new knowledge and technologies. Microbeams are one of the most common elements which are used in the field of the MEMS sensors [1,2,3,4,5,6,7,8,9].

Several of the investigators studied the vibration response of the microcomponents in the field of MEMS based on the classical continuum mechanics theories or using the finite element method (FEM) [6, 10, 11]. In recent years, it has been observed that the classical continuum mechanics theories are unable to predict and explain the static and dynamic behaviors of the materials in small sizes such as microbeams [12,13,14,15,16]. In other words, capturing the size effect is a significant challenge in the study of the vibration response of the small size structures. Therefore, several of the non-classical continuum theories such as nonlocal elasticity theory, strain gradient theory, and couple stress theory have been offered to capture the size effect in dynamic response for microcomponents [12, 17,18,19].

The couple stress theory was presented by Mindlin and Tiersten [18] as a non-classical continuum theory to consider the size dependency effect using two material length-scale parameters. Then, Yang et al. [19] proposed the modified couple stress theory (MCST) based on using one material length-scale parameter to capture the size effect in microcomponents.

The MCST has been used by some researchers to study the size dependency of the static and dynamic behaviors of materials in micro components. Liang et al. [20]; Park and Gao [21]; Dai et al. [22]; Ma et al. [23]; Ghiasi [24]; and Asghari et al. [25] investigated static or dynamic behaviors of micro-/nanobeams based on the MCST to capture the size dependency. Also, some researchers such as Simsek et al. [26]; He et al. [27]; Guo et al. [28]; Akbas [29]; Alinaghizadeh et al. [30]; Askari and Tahani [31]; and Ghayesh and Farokhi [32] studied mechanical behaviors of microplates based on the MCST.

Shoaib et al. [33,34,35] investigated the dynamic response and frequency analysis and of electrostatic cantilever-based MEMS sensors without any fault. Also, Shoaib et al. [35] studied the effect of crack faults on the dynamic behavior of a piezoelectric cantilever-based MEMS sensor using the FEM approach of COMSOL tool.

In the present work, the dynamic response of a multiple-layer microbeam is investigated using an analytical approach based on the MCST. The multiple-layer microbeam is modeled by a cantilever beam with several layers that are connected together. The governing equations of the lateral vibration and the associated boundary conditions are derived based on the MCST and using Hamilton’s principle. Then, the obtained governing equations are solved using an analytical approach to determine the natural frequencies of the system. Finally, the first, second, and third natural frequencies of the three-layers microbeam (one silicon layer and two piezoelectric layers) are investigated with respect to the different values of the system parameters such as the thickness of the layers, dimensionless material length-scale parameters and dimensionless parameters of the length and width of the system. The obtained results show that the material length-scale parameter values and also the length, width, and thickness of each layer have significant and interesting effects on the natural frequencies of the system.

2 Modeling and governing equations of motion

2.1 Modeling of the system

The modeling and geometry of the multiple-layer microbeam are shown in Fig. 1. The layer No. ith has length L, width b, thickness \(h_{i}\), density \(\rho _{i}\), Young’s modulus \(E_{i}\), and Poisson’s ratio \(\vartheta _{i}\). Also, the coordinate system X–Y–Z, middle surface and neutral surface have been shown in Fig. 1.

Fig. 1
figure 1

Modeling of the microbeam with K layers

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It should be noted that for multi-layer microplates with different materials and different elasticity modulus, the middle surface (mid-plane) and the neutral surface of the cantilever beam do not coincide with each other. The \(z_{m}\) and \(z_{n}\) are the transverse coordinates defined with respect to the middle surface and neutral surface of the cantilever microplate, respectively (please see Fig. 1). The position of the neutral surface with respect to the middle surface (e) can be calculated as:

$$\begin{aligned} e=\frac{\int _\frac{{-h}}{2}^\frac{{+h}}{2} {(E)(z_{m})}\, \mathrm{d}z_{m}}{\int _\frac{{-h}}{2}^\frac{{+h}}{2} E\, \mathrm{d}z_{m}}\, \end{aligned}$$
(1)

For a microbeam with k layer, Eq. (1) can be written as follows:

$$\begin{aligned} e=\frac{1}{2}\, \frac{N_{1}}{\sum \nolimits _{i=1}^k \left( E_{i}h_{i} \right) } \end{aligned}$$
(2)

where

$$\begin{aligned} N_{1}=\sum \limits _{j=1}^k {E_{j}\left[ \left( \frac{-h}{2}+\sum \limits _{i=j}^k h_{i} \right) ^{2}-\left( \frac{-h}{2}+\sum \limits _{i=j+1}^k h_{i} \right) ^{2} \right] } \, \end{aligned}$$
(3)

2.2 The modified couple stress theory

The strain energy of a linear elastic isotropic material based on the modified couple stress theory can be written as follows [21, 27]:

$$\begin{aligned} \pi _{s}=\frac{1}{2}\,\int _V\left( {\, \sigma }_{ij}\varepsilon _{ij}+m_{ij}\,\chi _{ij}^{s} \right) \mathrm{d}V\,;\, i=x,\, y,\, z\,and\, j=x,\, y,\, z\, \end{aligned}$$
(4)

where V denotes the volume of the system. In addition, the components of the stress tensor \(\sigma _{ij}\), the strain tensor \(\varepsilon _{ij}\), the deviatoric part of the symmetric couple stress tensor \(m_{ij}\) and the symmetric curvature tensor \(\chi _{ij}^{s}\) can be expressed as follows:

$$\begin{aligned} \sigma _{ij}= & {} \, 2\mu \varepsilon _{ij}+\lambda \varepsilon _{kk}\delta _{ij}\,\, ;\,\, m_{ij}=\, 2\mu l^{2}\chi _{ij}^{s} \end{aligned}$$
(5)
$$\begin{aligned} \varepsilon _{ij}= & {} \frac{1}{2}\, \left( u_{i,j}+u_{j,i} \right) \,\, ;\,\, \chi _{ij}^{s}=\frac{1}{2}\, \left( \theta _{i,j}+\theta _{j,i} \right) \end{aligned}$$
(6)
$$\begin{aligned} \theta _{i}= & {} \frac{1}{2}\epsilon _{ijk}\, u_{k,j} \end{aligned}$$
(7)

In the above equations, \(\theta _{i,j}\) is the gradient of rotation, \(\theta _{i}\) is infinitesimal rotation vector, \(\epsilon _{ijk}\) is alternating tensor (or permutation symbol), and \(u_{i}\) is components of the displacement vector. The parameters \(\left( \lambda ,\, \mu \right) \) and \(l\, \)are called Lamé constants and the material length-scale parameter, respectively. The Lamé constants can also be written regarding Young’s modulus E and Poisson’s ratio \(\vartheta \) as follows:

$$\begin{aligned} \lambda =\frac{\vartheta \, E}{\left( 1+\vartheta \right) \left( 1-2\vartheta \right) }\,\,\, ;\,\,\, \mu =\frac{E}{2\left( 1+\vartheta \right) }\, \end{aligned}$$
(8)

2.3 Governing equations of the system

Consider that the\(\, w(x,t)\) denotes the transverse deflection of the neutral line of the multiple-layer microbeam with k layers at any point x along the length of the sensor in the Z direction (please see Fig. 1). By using Euler–Bernoulli beam theory, the displacement field at any material point in the microbeam can be written as follows:

$$\begin{aligned} u_{1}=z_{n}\frac{\partial w\left( x,t \right) }{\partial x}\, ;\, u_{2}=0\, ;\, u_{3}=w\left( x,t \right) \end{aligned}$$
(9)

Assuming small transverse deflection in the multiple-layer microbeams, the nonzero components of the strain and the stress tensors can be expressed as follows:

$$\begin{aligned} \varepsilon _{xx}= & {} z_{n}\frac{\partial ^{2}w}{{\partial x}^{2}}\nonumber \\ \sigma _{xx}= & {} \, E\varepsilon _{xx}=E_{i}z_{n}\frac{\partial ^{2}w}{{\partial x}^{2}}\, ;\, \mathrm {(for\, layer\, No.\, }i\mathrm {)}\, \end{aligned}$$
(10)

The non-zero components of the symmetric curvature tensor and the deviatoric part of the symmetric couple stress tensor can be obtained as follows:

$$\begin{aligned} \chi _{xy}^{s}= & {} \chi _{yx}^{s}=\frac{-1}{2}\, \frac{\partial ^{2}w}{{\partial x}^{2}}\,\nonumber \\ m_{xy}= & {} m_{yx}=-\frac{E_{i}\, l_{i}^{2}}{2\left( 1+\vartheta _{i} \right) }\, \frac{\partial ^{2}w}{{\partial x}^{2}}\, ;\, \left( \mathrm {for\, layer\, No.\, }i \right) \, \end{aligned}$$
(11)

where \(l_{i}\) is the material length-scale parameter of the layers No. i.

Therefore, from Eq. (4), the total strain energy of the system can be written as follows:

$$\begin{aligned} \pi _{s}= & {} \frac{1}{2}\, \int _L \left[ \int _A \left( {\, \sigma }_{ij}\varepsilon _{ij}+m_{ij}\, \chi _{ij}^{s} \right) \mathrm{d}A \right] \mathrm{d}x\,\nonumber \\= & {} \frac{1}{2}\, \int _L \left[ \int _A \left( {\, \sigma }_{xx}\varepsilon _{xx}+{2\, m}_{xy}\, \chi _{xy}^{s} \right) \mathrm{d}A \right] \mathrm{d}x\nonumber \\= & {} \frac{1}{2}\int _0^L \left[ \left( \sum \limits _{i=1}^K {E_{i}\, \tilde{I}_{i}} \right) \left( \frac{\partial ^{2}w}{{\partial x}^{2}} \right) ^{2} \right] \, \mathrm{d}x\nonumber \\&+\,\frac{1}{2}\int _0^L {\left[ \left( \sum \limits _{i=1}^K {\frac{E_{i}\, l_{i}^{2}\, }{2(1+\vartheta _{i})}A_{i}} \right) \left( \frac{\partial ^{2}w}{{\partial x}^{2}} \right) ^{2} \right] \, \mathrm{d}x} \end{aligned}$$
(12)

where

$$\begin{aligned} \tilde{I}_{i}=\bar{I}_{i}+A_{i}\,\bar{z}_{ni}^{2}\,\bar{I}_{i}=\frac{b\,h_{i}^{3}}{12}\,\, ;\,\, A_{i}=bh_{i}\, \end{aligned}$$
(13)

where

$$\begin{aligned} \bar{z}_{n1}= & {} \bar{z}_{1}=\frac{1}{2}\left( h-2e-h_{1} \right) \,\nonumber \\ \bar{z}_{ni}= & {} \bar{z}_{i}=\frac{1}{2}\left( h-2e-h_{i} \right) -\sum \limits _{j=1}^{i-1} h_{j} \,\, ;\,\, for\, i=2,\, ... ,\, k\, \end{aligned}$$
(14)

In the above equations, \(A_{i}\) is the cross section area of the layers No. i at position x in length of the sensor.

It is noted that for a uniform and isotropic elastic with rectangular section \((b\times h_{i})\) , the cross-sectional area–moment of inertia about yy axis can be calculated from \(\bar{I}_{i}=\, \frac{bh_{i}^{3}}{12}=\frac{A_{i}\, h_{i}^{2}}{12}\, \mathrm {for}\, i=1,\, 2,\, ... ,\, K\). Therefore, according to Eq. (10), the total strain energy of the system can be written as follows:

$$\begin{aligned} \pi _{s}=\frac{1}{2}\, \left\{ \left( \sum \limits _{i=1}^K {E_{i}\, \tilde{I}_{i}} \right) +\left[ \sum \limits _{i=1}^K {\frac{{6\, E}_{i}\, \bar{I}_{i}\, }{(1+\vartheta _{i})}\left( \frac{l_{i}}{h_{i}} \right) ^{2}_{\, }} \right] \right\} \int _0^L \left[ \left( \frac{\partial ^{2}w}{{\partial x}^{2}} \right) ^{2} \right] \, \mathrm{d}x \end{aligned}$$
(15)

The kinetic energy of the system T can be written as follows:

$$\begin{aligned} T=\frac{1}{2}\int _0^{L\, } {\left[ \left( \sum \limits _{i=1}^K {\rho _{i}A_{i}} \right) \, \dot{w}^{2}+\left( \sum \limits _{i=1}^K {\rho _{i}\, \tilde{I}_{i}} \right) \left( \frac{\partial \dot{w}}{\partial x} \right) ^{\mathrm {2}} \right] \, } \mathrm{d}x \end{aligned}$$
(16)

where the dot over variables is the derivative of variable relative to time.

Hamilton’s principle is considered as follows:

$$\begin{aligned} \int ^{t2}_{t1} {\delta \left( T-\pi _{s}+W \right) \, \mathrm{d}t} =0 \end{aligned}$$
(17)

where \(\delta W\) is the virtual work done by external non-conservative forces on the system. For free vibration analysis of the considered system, the \(\delta W\) is zero.

By substituting Eqs. (15) and (16) into (17), and then using variational calculus, governing equations of motion of the system can be derived as follows:

$$\begin{aligned} S\frac{\partial ^{4}w}{{\partial x}^{4}}+B\, \ddot{w}\, -D\frac{\partial ^{2}\ddot{w}}{{\partial x}^{2}}=0 \end{aligned}$$
(18)

where

$$\begin{aligned} S= & {} \sum \limits _{i=1}^K {E_{i}\, \tilde{I}_{i}} +\sum \limits _{i=1}^K {\frac{{6\, E}_{i}\, \bar{I}_{i}\, }{(1+\vartheta _{i})}\left( \frac{l_{i}}{h_{i}} \right) ^{2}_{\, }}\nonumber \\ B= & {} \sum \limits _{i=1}^K {\rho _{i}A_{i}} =\, \sum \limits _{i=1}^K {\rho _{i}\, {b\, h}_{i}} \,\nonumber \\ D= & {} \, \sum \limits _{i=1}^K {\rho _{i}\, \tilde{I}_{i}} \end{aligned}$$
(19)

By neglecting \(\frac{\partial ^{2}\ddot{w}}{{\partial x}^{2}}\) , the Eq. (18) can be simplified as follows:

$$\begin{aligned} S\frac{\partial ^{4}w}{{\partial x}^{4}}+B\, \ddot{w}\, =0 \end{aligned}$$
(20)

The solution of Eq. (17) can be expressed as follows:

$$\begin{aligned} w\left( x,t \right) =W\left( x \right) \times sin\left( \omega t \right) \end{aligned}$$
(21)

By substituting Eq. (21) into Eq. (20) and to have some algebraic simplification, we have

$$\begin{aligned}&\frac{\mathrm{d}^{4}W}{{\mathrm{d}x}^{4}}-\beta ^{4}W\left( x \right) =0\,\nonumber \\&\beta ^{4}=\frac{{B\, \omega }^{2}}{S} \end{aligned}$$
(22)

where \(\omega \) is the natural frequency. Also, the general solution of Eq. (22) can be obtained as follows:

$$\begin{aligned} W\left( x \right) =\tilde{B}_{1}\sin \left( \beta x \right) +\tilde{B}_{2}\cos \left( \beta x \right) +\tilde{B}_{3}\sinh \left( \beta x \right) +\tilde{B}_{4}\cosh \left( \beta x \right) \end{aligned}$$
(23)

where \(\tilde{B}_{1}\, ,\, \tilde{B}_{2}\, ,\, \tilde{B}_{3}\) , and \(\tilde{B}_{4}\) are constants.

Also, the boundary conditions of the system can be expressed as follows:

$$\begin{aligned} W\left( 0 \right)= & {} 0\,\, ;\, \frac{\mathrm{d}W}{{\mathrm{d}x}}\left( 0 \right) =0\,\nonumber \\ \frac{\mathrm{d}^{2}W}{{\mathrm{d}x}^{2}}\left( L \right)= & {} 0\,\, ;\, \frac{\mathrm{d}^{3}W}{{\mathrm{d}x}^{3}}\left( L \right) =0 \end{aligned}$$
(24)

By substituting Eq. (24) into Eq. (23), a set of four algebraic equations resulting in matrix form can be obtained as follows:

$$\begin{aligned} \left[ Q_{ij} \right] \left\{ \tilde{B}_{j} \right\} =0\,\,\, ;\,\,\, i,\, j=1,\, 2,\, 3,\, 4 \end{aligned}$$
(25)

where

$$\begin{aligned} Q_{11}= & {} Q_{13}=0\, ;\, Q_{12}=Q_{14}=1\,\nonumber \\ Q_{22}= & {} Q_{24}=0\, ;\, Q_{21}=Q_{23}=-\beta \,\nonumber \\ Q_{31}= & {} -\beta ^{2}\, \sin \left( \beta L \right) \, ;\, Q_{32}=-\beta ^{2}\, \cos \left( \beta L \right) \nonumber \\ Q_{33}= & {} \beta ^{2}\, \sinh \left( \beta L \right) \, ;\, Q_{34}=\beta ^{2}\, \cosh (\beta L)\nonumber \\ Q_{41}= & {} -\beta ^{3}\, \cos \left( \beta L \right) \, ;\, Q_{42}=\beta ^{3}\, \sin \left( \beta L \right) \nonumber \\ Q_{43}= & {} \beta ^{3}\, \cosh \left( \beta L \right) \, ;\, Q_{44}=\beta ^{3}\, \sinh (\beta L) \end{aligned}$$
(26)

For nontrivial solution of Eq. (25), the determinant of the matrix \(\left[ Q_{ij} \right] \) must be zero. Also, if the determinant of the matrix \(\left[ Q_{ij} \right] \) be zero, the result and the first, second, third, and fourth roots of it can be calculated as follows:

$$\begin{aligned}&\textit{det} \, \left[ Q_{ij} \right] =0\, \Rightarrow \, {\mathrm {cos}}\left( \beta _{n}L \right) \cosh \left( \beta _{n}L \right) =-1\,\, ;\,\, n=1,\, ... ,\, \infty \nonumber \\&\beta _{1}L=1.87510\,\, ;\,\, \beta _{2}L=4.69409\,\, ;\,\, \beta _{3}L=7.85476\,\, ;\,\, \beta _{4}L=10.99554\, \end{aligned}$$
(27)

Therefore, according to Eqs. (22) and (27), the natural frequencies of the multiple-layer microbeams with k layers can be determined as follows:

$$\begin{aligned} \omega _{n}=\left( \beta _{n}L \right) ^{2}\left[ \, \frac{S}{B\, L^{4}}\, \right] ^{\frac{1}{2}}=\left( \beta _{n}L \right) ^{2}\left[ \frac{S}{\left( \sum \nolimits _{i=1}^K {\rho _{i}h_{i}} \right) L^{4}} \right] ^{\frac{1}{2}}\, \end{aligned}$$
(28)

where

$$\begin{aligned} S=\sum \limits _{i=1}^K {E_{i}h_{i}\left( \frac{h_{i}^{2}}{12}+\bar{z}_{i}^{2} \right) } +\sum \limits _{i=1}^K {\frac{{\, E}_{i}h_{i}^{3}\, }{2(1+\vartheta _{i})}\left( \frac{l_{i}}{h_{i}} \right) ^{2}_{\, }} \end{aligned}$$
(29)

3 Verification

In the previous section, the governing equation of motion for lateral vibration of the cantilever multiple-layer microbeams with k layers which are connected together was derived analytically based on the MCST as follows:

$$\begin{aligned} \left[ \sum \limits _{i=1}^K {E_{i}\, \tilde{I}_{i}} +\sum \limits _{i=1}^K {\frac{{6\, E}_{i}\, \bar{I}_{i}\, }{(1+\vartheta _{i})}\left( \frac{l_{i}}{h_{i}} \right) ^{2}_{\, }} \right] \, \frac{\partial ^{4}w}{{\partial x}^{4}}+\left( \sum \limits _{i=1}^K {\rho _{i}A_{i}} \right) \ddot{w}\, =0 \end{aligned}$$
(30)

where

$$\begin{aligned} \tilde{I}_{i}= & {} \bar{I}_{i}+A_{i}\,\bar{z}_{ni}^{2}\,\, ;\,\, \mathrm {for}\, i=1,\, 2,\, ... ,\, K\nonumber \\ \bar{I}_{i}= & {} \frac{b\, h_{i}^{3}}{12}\,\, ;\,\, A_{i}=bh_{i}\, ;\, \mathrm {for}\, i=1,\, 2,\, ... ,\, K\nonumber \\ \bar{z}_{n1}= & {} \frac{1}{2}\left( h-2e-h_{1} \right) \,\nonumber \\ \bar{z}_{ni}= & {} \frac{1}{2}\left( h-2e-h_{i} \right) -\sum \limits _{j=1}^{i-1} h_{j} \,\, ;\,\, \mathrm {for}\, i=2,\, ... ,\, k\, \end{aligned}$$
(31)

In special case \(\frac{l_{i}}{h_{i}}\approx \mathrm {0}\), for the macrosystem, Eq. (30) is simplified as

$$\begin{aligned} \left( \sum \limits _{i=1}^K {E_{i}\, \tilde{I}_{i}} \right) \, \frac{\partial ^{4}w}{{\partial x}^{4}}+\left( \sum \limits _{i=1}^K {\rho _{i}A_{i}} \right) \, \ddot{w}\, =0 \end{aligned}$$
(32)

Also, It can be easily seen that for the special case \(h_{i}=0\, (for\, i\ne 1)\) (when we have only one layer), Eq. (32) is simplified as

$$\begin{aligned} \left( E_{1}\, \bar{I}_{1} \right) \, \frac{\partial ^{4}w}{{\partial x}^{4}}+\left( \rho _{1}A_{1} \right) \, \ddot{w}\, =0\, \end{aligned}$$
(33)

Equation (33) is compatible with the classical form of the governing equations of free vibration of a cantilever beam in macro size.

4 Results and discussion

In this section, the effect of size and also dimensions of the multiple-layer microbeam with three layers (two layers of piezoelectric and one layer of silicon) on the natural frequencies of the system are studied. In numerical analyses, according to Fig.  1, the nominal dimensions, materials, and geometry of the multiple-layer microbeam are mentioned in Table 1 [35].

Table 1 Dimensions and material parameters of the microbeam with 3 layers
Full size table
Fig. 2
figure 2

Variation of the first natural frequency of the multiple-layer microbeam versus the ratio thickness of layers \(\frac{h_{2}}{h_{1}}\) for different values of the layer thickness \(h_{1}\) and \(h_{3}\)

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Fig. 3
figure 3

Variation of the second natural frequency of the multiple-layer microbeam versus the ratio thickness of layers \(\frac{h_{2}}{h_{1}}\) for different values of the layer thickness \(h_{1}\) and \(h_{3}\)

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Fig. 4
figure 4

Variation of the third natural frequency of the multiple-layer microbeam versus the ratio thickness of layers \(\frac{h_{2}}{h_{1}}\) for different values of the layer thickness \(h_{1}\) and \(h_{3}\)

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Fig. 5
figure 5

Variation of the first natural frequency of the multiple-layer microbeam versus thickness of the piezoelectric layer \(h_{\mathrm {3}}\) for different values of dimensionless material length-scale parameter \(l_{2}\mathrm {/}h_{2}\)

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Fig. 6
figure 6

Variation of the first natural frequency of the multiple-layer microbeam versus the parameter \(h_{2}\) for different values of the thickness \({of\, the\, layers\, h}_{1}\) and \(h_{3}\)

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In the present paper, the material length-scale parameter \(l_{i}\) for silicon and piezoelectric layers has been considered 1.0\(\upmu \mathrm {m}\) and 2.4\(\upmu \mathrm {m}\), respectively [35].

The first natural frequency of the multiple-layer microbeam versus the ratio thickness of the silicon layer to thickness of the piezoelectric layer \(\left( \frac{h_{\mathrm {2}}}{h_{\mathrm {1}}}\right) \) has been presented in Fig. 2, for different values of the layer thickness \(h_{\mathrm {1}}\) and \(h_{\mathrm {3}}\). The obtained results in Fig. 2 show that there is interesting behavior on the natural frequencies with the increase in dimensionless thickness parameter \(h_{\mathrm {2}}\mathrm {/}h_{\mathrm {1}}\) . It should be noted that this behavior can be considered in the optimization design of the multiple-layer microbeam such as microsensors system. The above behavior for second and third natural frequencies of the multiple-layer microbeam can also be observed in Figs. 3 and 4, respectively.

Also, the first natural frequency of the multiple-layer microbeam versus piezoelectric thickness \(h_{\mathrm {3}}\) has been depicted in Fig. 5 for various values of dimensionless material length-scale parameter \(l_{\mathrm {2}}\mathrm {/}h_{\mathrm {2}}\), at silicon thickness \(h_{1}=10\, \upmu \)m and \(h_{2}=20\, \upmu \)m.

In addition, the variation of the first natural frequency of the system versus the parameter \(h_{\mathrm {2}}\) has been studied in Fig. 6 for different values of the thickness \({of\,the\,layers\,h}_{\mathrm {1}}\) and \(h_{\mathrm {3}}\) , at microbeam length \(L=800\, \upmu \)m. The obtained results in Fig. 6 show that the first natural frequency of the multiple-layer microbeam will increase when dimensionless parameter \(h_{\mathrm {2}}\) increases.

In Figures 2, 3, 4, 5 and 6, the results of the higher-order theory have been presented with solid lines, and dashed lines have been used for the curves with the vanishing the material length-scale parameters \(l_{i}\).

5 Summary and conclusion

In this article, the dynamic response of a multiple-layer microbeam was studied by an analytical approach. To capture size effects and also to predict the vibration response of the multiple-layer microbeam, governing equations were derived based on Hamilton’s principle and modified couple stress theory (MCST). The analytical approach was used in the solution of the partial differential equations. The first, second, and third natural frequencies of the system for various values of system parameters such as dimensionless material length-scale parameter, the thickness of the silicon, and piezoelectric layers, were investigated.

The obtained results of the natural frequencies were presented for a multiple-layer microbeam with one silicon layer and two piezoelectric layers. The results show that material length-scale parameter values and also the length, width, and thickness of each layer have significant effects on the vibration behavior of the multiple-layer microbeam system.