Effects of Nonlinear Damping on Vibrations of Microbeam

Abstract

The present paper develops a new Bernoulli–Euler theory of microbeams for the consideration of small-scale effects and nonlinear terms, which are induced by the axial elongation of the beam and Kelvin–Voigt damping. The non-resonance and primary resonance of microbeams are researched through the application of Galerkin and multiple scale methods to the new model. The results suggest the following: (1) Nonlinear damping slightly affects the vibration amplitudes under the non-resonance condition; (2) nonlinear damping can significantly change the bifurcation points that induce a jump in the vibration amplitudes under the primary resonance condition. The current researches indicate that nonlinear damping is necessary for an accurate description of microbeam vibrations.

1. Introduction

A study of the dynamics and energy dissipation of microbeams is necessary, since slender beams are frequently used in micro-scale devices and systems, such as microelectromechanical systems (MEMS) [1,2,3].

A crucial issue of the mechanical properties of microbeams is the size-independent behavior observed in micro-scale experiments [4,5,6]. McFarland and Colton observed that the microbeam’s flexural rigidity is at least four times larger than the one predicted by the classical beam theory [4]. Fleck found that the dimensionless torque increases the classical torque theory by three times when the wire diameter decreases from 170 to 12 μm [5]. Stölken and Evans found that the dimensionless flexural rigidity increases significantly as the beam thickness decreases from 50 to 12.5 μm [6]. However, the classical slender beam models based on the classical elasticity cannot describe this size effect due to the lack of material length scale parameters. This disadvantage motivated the development of beam theories using nonlocal elasticity theories to model nanobeams [7,8] and higher-order strain gradient theories to model microbeams [3,9,10,11]. Moreover, almost all of the existing microbeam models are based on linear constitutive relations. However, the material nonlinearity also influences the mechanical properties of nanobeams [12].

The modified couple stress theory is widely accepted to model the mechanical properties of microbeams [13]. The couple stress tensor is symmetric and only involves one internal material length scale parameter in the modified couple stress theory. Based on this theory, the linear and nonlinear Bernoulli–Euler and Timoshenko microbeam models have been proposed [14,15,16,17,18,19,20]. Nevertheless, most of the studies on microbeam vibrations have overlooked the nonlinear terms, which are induced by damping. For example, the authors of [3,11,19] only considered the linear term of damping, while the authors of [11,16] completely overlooked it. The energy dissipation is related to the vibration frequencies of the structures [21]. In macrostructures, nonlinear damping is almost smaller than the geometrical nonlinear, which is induced by the finite deformations [22]. However, since the vibration frequencies of microbeams are significantly higher than the macrobeams, nonlinear damping significantly affects the microstructure vibrations. Based on the modified couple stress theory, the present paper will propose a new Bernoulli–Euler model that includes the small-size effect and nonlinear terms, which are induced by the axial elongation of the beam and Kelvin–Voigt damping. Then, the effects of nonlinear damping are studied for the non-resonance and primary resonance.

2. Methods

In the modified couple stress theory, the symmetric couple stress characterizes the small-scale effect, and the strain energy density can be written as [13,23]:

$$U=\frac{1}{2}\left({\sigma }_{ij}{\epsilon }_{ij}+m_{ij}{\chi }_{ij}\right),\hspace{1em}\left(i,j=x,y,z\right)$$

(1)

where ${\epsilon }_{ij}$ and ${\sigma }_{ij}$ are the stress and strain, $m_{ij}=2G{\xi }^2{\chi }_{ij}$ is the couple stress tensor, ${\chi }_{ij}=\frac{1}{2}\left[\nabla \mathsf{\omega }+{\left(\nabla \mathsf{\omega }\right)}^T\right]$ is the symmetrical curvature tensor, $\mathsf{\omega }=\left(\nabla \times u\right)/2$ with $u$ being the displacement vector [13], and $G=E/\left[2\left(1+\nu \right)\right]$. Here, $E$ and $\nu$ are the Young’s modulus and Poisson’s ratio, $\xi$ is the material length scale parameter obtained from the experimentations. Based on the modified couple stress theory, Xia established an integral differential equation to model the mechanical properties of microbeams, as shown in Figure 1. This theory takes into account the small-scale effect and nonlinear damping, which are induced by the elongation of the microbeam [16]:

$$m\frac{{\partial }^{2}w}{\partial t^2}+\left(EI+GS{\xi }^2\right)\frac{{\partial }^{4}w}{\partial x^4}+\left[P_0-\frac{ES}{2L}{\displaystyle \underset{0}{\overset{L}{\int }}{\left(\frac{\partial w}{\partial x}\right)}^{2}dx}\right]\frac{{\partial }^{2}w}{\partial x^2}=F\left(x,t\right)$$

(2)

where $w$ is the vertical deflection, $P_0$ and $m$ are the initial axial load and mass per unit length, respectively, $S$ and $I$ are the cross-sectional area and moment of inertia. From Equation (2), it is found that the small-scale effect significantly affects the static bending and vibration frequencies of microbeams [16]. However, Equation (2) is an integral differential equation, and it is not easy to find solutions to the equation. As shown below, we will develop a new differential equation to replace Equation (2).

Suppose that the average axial force, which is induced by the axial elongation of the beam, is ${\overline{P}}_1$, thus we obtain:

$${\overline{P}}_1=\frac{ES\left(\tilde{L}-L\right)}{L}\approx \frac{ES}{2L}{\displaystyle \underset{0}{\overset{L}{\int }}{\left(\partial w/\partial x\right)}^{2}dx}$$

(3)

where $\tilde{L}={\displaystyle \underset{0}{\overset{L}{\int }}\left[1+{\left(\partial w/\partial x\right)}^2/2\right]dx}$ is the deformed length of the beam axle. On the beam element, the $z$ component of the axial force increment is $\left(d{P}_1\right)\cdot \sin \theta$, and $\theta$ is the rotation angle of the cross-section, as shown in Figure 1c. Considering ${\overline{P}}_1\approx P_1$, $\sin \theta \approx \theta$, $\theta \approx \partial w/\partial x$, and using Equation (3), we obtain:

$$\frac{\partial }{\partial x}\left(P_1\sin \theta \right)dx\approx \frac{\partial }{\partial x}\left({\overline{P}}_1\frac{\partial w}{\partial x}\right)dx\approx \frac{ES}{2L}\left\{\left[\frac{{\partial }^{2}w}{\partial x^2}{\displaystyle \underset{0}{\overset{L}{\int }}{\left(\frac{\partial w}{\partial x}\right)}^{2}dx}\right]\right\}dx$$

(4)

The right part of Equation (4) is also the nonlinear term of Equation (2). Equation (3) indicates that the accurate axial force $P_1$ is replaced by the average axial force ${\overline{P}}_1$ in Equation (2). The accurate force, which is induced by the axial elongation, can be written as $P_1=ES\left(\Delta L/dx\right)\approx \left(ES/2\right){\left(\partial w/\partial x\right)}^2$, where $\Delta L$ is the elongation of the element. Considering $d\left(P_1\sin \theta \right)\approx \left(3ES/2\right){\left(\partial w/\partial x\right)}^2\left({\partial }^{2}w/\partial x^2\right)dx$, and Equation (4), we obtain:

$$\left[\frac{ES}{2L}{\displaystyle \underset{0}{\overset{L}{\int }}{\left(\frac{\partial w}{\partial x}\right)}^{2}dx}\right]\frac{{\partial }^{2}w}{\partial x^2}\approx \frac{3ES}{2}{\left(\frac{\partial w}{\partial x}\right)}^2\frac{{\partial }^{2}w}{\partial x^2}$$

(5)

Equation (5) indicates that the nonlinear differential term can replace the nonlinear term of Equation (2). Furthermore, Equation (2) does not consider the structural damping. Here, we suppose that Kelvin–Voigt damping is in the beams, thus $E$ in Equation (2) will be replaced by $E+\tilde{E}\partial /\partial t$ [21,24], where $\tilde{E}$ is the viscous damping coefficient. Substituting Equation (5) into Equation (2), the nonlinear equation, which includes the small-scale effect and nonlinear damping term, can be obtained as:

$$\begin{array}{l}m\frac{{\partial }^{2}w}{\partial t^2}+\tilde{E}\frac{\partial }{\partial t}\left[I\frac{{\partial }^{4}w}{\partial x^4}-\frac{3S}{2}{\left(\frac{\partial w}{\partial x}\right)}^2\frac{{\partial }^{2}w}{\partial x^2}\right]+P_0\frac{{\partial }^{2}w}{\partial x^2}\\ +\left(EI+GS{\xi }^2\right)\frac{{\partial }^{4}w}{\partial x^4}-\frac{3ES}{2}{\left(\frac{\partial w}{\partial x}\right)}^2\frac{{\partial }^{2}w}{\partial x^2}=F\end{array}$$

(6)

Kahrobaiyan et al. studied the primary resonance, super-resonance, and sub-resonance of microbeams using Equation (2) with linear damping $\tilde{E}\partial w/\partial t$ [11]. In the present paper, we will focus on the effects of nonlinear damping. Let $\tilde{x}=x/L$, $\tilde{w}=w/L$, $\tilde{t}={\mathsf{\omega }}_{0}t$ normalize the variables in Equation (6), thus we obtain:

$$\begin{array}{l}\frac{{\partial }^2\overline{w}}{\partial {\overline{t}}^2}+{\pi }^{- 4}\frac{{\partial }^4\overline{w}}{\partial {\overline{x}}^4}+P\frac{{\partial }^2\overline{w}}{\partial {\overline{x}}^2}+C_1\frac{{\partial }^5\overline{w}}{\partial \overline{t}\partial {\overline{x}}^4}\\ -D{\left(\frac{\partial \overline{w}}{\partial \overline{x}}\right)}^2\frac{{\partial }^2\overline{w}}{\partial {\overline{x}}^2}-C_2\frac{\partial }{\partial \overline{t}}\left[{\left(\frac{\partial \overline{w}}{\partial \overline{x}}\right)}^2\frac{{\partial }^2\overline{w}}{\partial {\overline{x}}^2}\right]=\overline{f}\end{array}$$

(7)

The parameters in Equation (7) can be written as:

$$P=\frac{P_0}{m{L}^2{\mathsf{\omega }}_0^2},\hspace{1em}{C}_1=\frac{\tilde{E}I}{m{L}^4{\mathsf{\omega }}_0},\hspace{1em}{C}_2=\frac{3\tilde{E}S}{2m{L}^2{\mathsf{\omega }}_0},\hspace{1em}D=\frac{3ES}{2m{L}^2{\mathsf{\omega }}_0^2},\hspace{1em}\overline{f}=\frac{F}{Lm{\mathsf{\omega }}_0^2}$$

(8)

where ${\mathsf{\omega }}_0=\sqrt{{\pi }^4\left(EI+GS{\xi }^2\right){\left(m{L}^4\right)}^{-1}}$ is the natural frequency of the microbeam. For the same $I/L^{ 4}$, the microbeam’s natural frequency ${\mathsf{\omega }}_0$ is more than the macrobeam, since the macrobeam’s mass per unit length is less than the microbeam. Therefore, the nonlinear damping term of the microbeam is larger than the correspondent term of the macrobeam due to $D/C_2=E/\left(\tilde{E}{\mathsf{\omega }}_0\right)$. The boundary conditions for hinged–hinged microbeams can be written as:

$$\overline{w}\left(0,\overline{t}\right)=\overline{w}\left(1,\overline{t}\right)=0,\hspace{1em}\frac{{\partial }^2\overline{w}}{\partial {\overline{x}}^2}\left(0,\overline{t}\right)=\frac{{\partial }^2\overline{w}}{\partial {\overline{x}}^2}\left(1,\overline{t}\right)=0$$

(9)

Moreover, it is not easy to obtain an accurate analytical solution, since Equation (7) is the nonlinear differential equation. Therefore, we will use the Galerkin method to obtain an approximate analytical solution [24,25,26]. Suppose that the solution of Equation (7) can be written as:

$$\overline{w}=\sum _{j=1}^{\infty }\sin \left(j\pi \overline{x}\right) {\eta }_j\left(\overline{t}\right)$$

(10)

Substituting Equation (10) into Equation (7), then multiplying $\sin \pi \overline{x}$ by both sides of the equations, and integrating it in $\left[0,1\right]$, the first-mode truncation of Galerkin method [24], which is an ordinary differential equation in time, can be obtained as:

$$\ddot{\eta }+2{\tilde{c}}_1\dot{\eta }+{\mathsf{\omega }}^2\eta +d{\eta }^3+c_2\dot{\eta }{\eta }^2=f\left(\overline{t}\right).$$

(11)

Here, we let $\eta ={\eta }_1$ and ${\mathsf{\omega }}^2=1-N{\pi }^2$, ${\tilde{c}}_1={\pi }^4{C}_1/2$, $d={\pi }^{4}D/4$, $c_2=3{\pi }^4{C}_2/4$.

3. Results

In this section, the multiple scale method [24,25,26], which is widely used to solve weak nonlinear differential equations [27,28,29,30], will be applied to Equation (11). Let ${\tilde{c}}_1={\epsilon }^2{c}_1$, $f\left(t\right)=\epsilon f_1\cos \left({\Omega }_1\overline{t}\right)+{\epsilon }^3{f}_1\cos \left({\Omega }_2\overline{t}\right)$, then we obtain:

$$\ddot{\eta }+{\mathsf{\omega }}^2\eta +{\epsilon }^2{c}_1\dot{\eta }+d{\eta }^3+c_2\dot{\eta }{\eta }^2=\epsilon f_1\cos \left({\Omega }_{1}t\right)+{\epsilon }^3{f}_2\cos \left({\Omega }_{2}t\right)$$

(12)

Suppose

$$\eta \left(t;\epsilon \right)=\epsilon {\eta }_1\left(T_0,T_2\right)+{\epsilon }^3{\eta }_3\left(T_0,T_2\right)$$

(13)

where $T_0=\overline{t}$ and $T_2={\epsilon }^2\overline{t}$. Substituting Equation (13) into Equation (12), then equating the coefficients of $\epsilon$ and ${\epsilon }^3$ on both sides, we obtain:

$$\epsilon :\hspace{1em}\frac{{\partial }^2{\eta }_1}{\partial T_0^2}+{\mathsf{\omega }}^2{\eta }_1=\frac{1}{2}{f}_1\exp \left(i{\Omega }_1{T}_0\right)$$

(14)

$$\begin{array}{l}{\epsilon }^3:\frac{{\partial }^2{\eta }_3}{\partial T_0^2}+{\mathsf{\omega }}^2{\eta }_3=-2\frac{\partial }{\partial T_0}\left(\frac{\partial {\eta }_1}{\partial T_2}\right)\\ \hspace{1em}\hspace{1em}-2c_1\frac{\partial {\eta }_1}{\partial T_0}-d{\eta }_1^3-c_2{\eta }_1^2\frac{\partial {\eta }_1}{\partial T_0}+\frac{1}{2}{f}_2\exp \left(i {\Omega }_2{T}_0\right)\end{array}$$

(15)

The solution of Equation (14) can be written as:

$${\eta }_1=A\left(T_2\right)\exp \left(i\mathsf{\omega }{T}_0\right)+\Lambda \exp \left(i{\Omega }_1{T}_0\right)+cc$$

(16)

where $\Lambda =\frac{1}{2}{f}_1{\left({\mathsf{\omega }}^2-{\Omega }_1^2\right)}^{-1}$. Substituting Equation (16) into Equation (15), we obtain:

$$\begin{array}{l}\frac{{\partial }^2{\eta }_3}{\partial T_0^2}+{\mathsf{\omega }}^2{\eta }_3=\frac{1}{2}{f}_2\exp \left(i {\Omega }_2{T}_0\right)-[i2\mathsf{\omega }\left(A^{\prime }+c_{1}A\right)\\ +\left(i2c_2\mathsf{\omega }+6d\right){\Lambda }^{2}A+\left(i{c}_2\mathsf{\omega }+3d\right)A^2\overline{A}]\exp \left(i \mathsf{\omega }{T}_0\right)+NST\end{array}$$

(17)

where NST denotes non-secular terms [26], and the prime indicates differentiation with respect to $T_2$ for simplicity.

3.1. Non-Resonant Hard Excitation

When the frequency of the load ${\Omega }_1$ is further than the modal frequency $\mathsf{\omega }$, the condition for eliminating secular terms of Equation (17) can be written as:

$$i2\mathsf{\omega }\left(A^{\prime }+c_{1}A\right)+\left(i2c_2\mathsf{\omega }+6d\right){\Lambda }^{2}A+\left(i{c}_2\mathsf{\omega }+3d\right)A^2\overline{A}=0$$

(18)

Suppose

$$A=\frac{1}{2}\alpha \exp \left(i\beta \right)$$

(19)

where $\alpha$ and $\beta$ are real. Substituting Equation (19) into Equation (18), and separating the real and imaginary parts, we obtain:

$${\alpha }^{\prime }=\left(-c_1-c_2{\Lambda }^2\right)\alpha -\frac{1}{8}{c}_2{\alpha }^3,\hspace{1em}\alpha {\beta }^{\prime }=\frac{3d}{\mathsf{\omega }}\left({\Lambda }^2+\frac{1}{8}{\alpha }^2\right)\alpha$$

(20)

For steady-state solutions of Equation (20), we obtain ${\alpha }^{\prime }={\beta }^{\prime }=0$. Therefore, $\alpha$ and $\beta$ are the solutions of

$$\left(-c_1-c_2{\Lambda }^2\right)\alpha -\frac{1}{8}{c}_2{\alpha }^3=0,\hspace{1em}\frac{3d}{\mathsf{\omega }}\left({\Lambda }^2+\frac{1}{8}{\alpha }^2\right)\alpha =0$$

(21)

The first-order approximate solution of the non-resonant hard excitation can be written as:

$$\eta =\epsilon \left[ \alpha \cos \left(\mathsf{\omega }{T}_0+\beta \right)+\Lambda \cos \left({\Omega }_1{T}_0\right)\right]$$

(22)

From Equation (21), we obtain $\left(-c_1-c_2{\Lambda }^2\right)\alpha -c_2{\alpha }^3/8=0$. This indicates $\alpha =0$ due to $c_1>0$ and $c_2>0$. Therefore, the free-oscillation solution will decay with time and the first-order approximate solution will only include the composition excited by the external loads. This outcome can be confirmed by numerical simulations of Equation (11), as shown in Figure 2. While the free-oscillation term is decaying, its frequency is a function of the amplitude $\alpha$.

3.2. Primary Resonance

When the frequency of the load ${\Omega }_2$ approaches the modal frequency of the microbeam $\mathsf{\omega }$, the beam will appear as a relatively large amplitude response. Under this condition, let $f_1=0$ and ${\Omega }_2=\mathsf{\omega }+{\epsilon }^2\sigma$, thus the solvable condition of Equation (17) can be written as:

$$-i2\mathsf{\omega }\left(A^{\prime }+cA\right)-\left(i{c}_2\mathsf{\omega }+3d\right)A^2\overline{A}+\frac{1}{2}{f}_2\exp \left(i\sigma T_2\right)=0$$

(23)

Substituting Equation (19) into Equation (23), then separating the real and imaginary parts, we obtain:

$${\alpha }^{\prime }=-c_1\alpha -\frac{1}{8}{c}_2{\alpha }^3+\frac{f_2}{2\mathsf{\omega }}\sin \gamma ,\hspace{1em}\alpha {\gamma }^{\prime }=\sigma \alpha -\frac{3}{8\mathsf{\omega }}d{\alpha }^3+\frac{f_2}{2\mathsf{\omega }}\cos \gamma$$

(24)

where $\gamma =\sigma T_2-\beta$. The steady-state motions occur when ${\alpha }^{\prime }={\gamma }^{\prime }=0$, which correspond to the singular points of Equation (24). The steady-state solutions can be obtained by the following algebraic equation [26]:

$$[c_1^2+{\sigma }^2+\frac{1}{4}\left(c_1{c}_2-\frac{3d\sigma }{\mathsf{\omega }}\right)a^2+\frac{1}{64}\left(c_2^2+\frac{9d^2}{{\mathsf{\omega }}^2}\right){\alpha }^4]a^2={\left(\frac{f_2}{2\mathsf{\omega }}\right)}^2$$

(25)

The stability of the steady-state solutions can be judged by an investigation of the nature of the singular points of Equation (24). Letting $a=a_0+a_1$ and $\gamma ={\gamma }_0+{\gamma }_1$, substituting them into Equation (24), then expanding the small $a_0$ and ${\gamma }_0$, and maintaining the linear terms in $a_1$ and ${\gamma }_1$, we obtain:

$$\begin{array}{l}{{\alpha }^{\prime }}_1=\left(-c_1-\frac{3}{8}{c}_2{\alpha }_0^2\right){\alpha }_1+\frac{{\gamma }_1{f}_2\cos {\gamma }_0}{2\mathsf{\omega }}\\ {{\gamma }^{\prime }}_1=-\left(\frac{f\cos {\gamma }_0{}_1}{2\mathsf{\omega }{\alpha }_0^2}+\frac{3d{\alpha }_0}{4}\right){\alpha }_1-\frac{{\gamma }_1{f}_2\sin {\gamma }_0}{2\mathsf{\omega }{\alpha }_0}\end{array}$$

(26)

where $a_0$ and ${\gamma }_0$ are the singular points of Equation (24), which are used in Equation (26). The stability of the steady-state motions depends on the eigenvalues of the coefficient matrix on the right-hand side of Equation (26). If the real parts of the eigenvalues are greater than zero, the solutions are unstable [25]. Therefore, the steady-state motions are unstable when

$$\left(\frac{3c_2{\alpha }_0^2}{8}+c_1\right)\left(\frac{c_2{\alpha }_0^2}{8}+c_1\right)+\left(\frac{3d{\alpha }_0^2}{8\mathsf{\omega }}-\sigma \right)\left(\frac{9d{\alpha }_0^2}{8\mathsf{\omega }}-\sigma \right)<0$$

(27)

Equation (27) indicates that nonlinear damping affects the stability of the steady-state solutions.

4. Discussion

In this section, we use the following parameters as an example to study the effects of nonlinear damping on microbeam vibrations [16]: $E=1.44 \mathrm{G}\mathrm{P}\mathrm{a}$, $G=0.5127 \mathrm{G}\mathrm{P}\mathrm{a}$, and mass density $\rho =1220 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$. The beam’s geometrical parameters are $L=100 \mathsf{\mu }\mathrm{m}$, $b=10 \mathsf{\mu }\mathrm{m}$, and $h=20 \mathsf{\mu }\mathrm{m}$, thus we have $S=2\times {10}^2 {\left(\mathsf{\mu }\mathrm{m}\right)}^2$, $m=\rho S=2.44\times {10}^{-7} \mathrm{k}\mathrm{g}/\mathrm{m}$, $I=2\times {10}^4/3 {\left(\mathsf{\mu }\mathrm{m}\right)}^4$. Let $\xi =20 \mathsf{\mu }\mathrm{m}$ and $P_0=0.188 \mathrm{N}$, thus we obtain ${\mathsf{\omega }}_0=1.42\times {10}^7$, $\mathsf{\omega }=0.901$, $d=21.3$, and $C_2=450C_1$. Suppose that ${\overline{c}}_1=0.01$, thus we obtain $c_1=6.75$.

In Section 3.1, it is theoretically demonstrated that nonlinear damping does not affect the vibration amplitudes under the non-resonant condition. Here, the numerical calculations of Equation (11) with $\left(f_1,{\Omega }_1\right)=\left(0.5,1.5\right)$ confirm this prediction, as shown in Figure 2. However, when the exciting amplitude increases, for example $\left(f_1,{\Omega }_1\right)=\left(10,1.5\right)$, multi-frequency vibrations will appear in the system without nonlinear damping, as shown in Figure 3. In this case, the perturbation methods are no longer applicable, since the strong excitations allow for Equation (11) to appear nonlinearly strong. When the large-amplitude vibrations of non-resonant excitations appear, nonlinear damping significantly reduces the vibration amplitudes, and suppresses the vibration components of non-excited frequencies, as shown in Figure 3.

The influence of nonlinear damping on primary resonance is mainly shown in the following two aspects: First, when the exciting frequency is greater than the modal frequency of the microbeam, nonlinear damping can significantly change the bifurcation points of the load and strongly affects the dynamic behavior of the microbeam, as shown in Figure 4 and Figure 5. Second, when the exciting frequency is less than the modal frequency of the microbeam, the impact of nonlinear damping is slight, as shown in Figure 5 and Figure 6. The numerical calculations of Equation (11) confirm the aforementioned results, as shown in Figure 7 and Figure 8. For example, when $f_2=3$ and $\sigma =10$, the vibration amplitude with $c_2=0$ is about six times larger than the amplitude with $c_2=6.75$, as shown in Figure 7. This difference is essential in practice.

Nonlinear elastic effects play an essential role in the dynamics of MEMS [1,2]. Nonlinear dissipation effects in micromechanical oscillators are often overlooked. However, our work shows that arbitrarily overlooking nonlinear damping may lead to qualitative mistakes. In this paper, we especially focus on the system’s behavior near the bifurcation points of primary resonance. The results show that nonlinear damping has a significant impact on the dynamics of microbeams, as shown in Figure 4 and Figure 6. Moreover, it is necessary to research particular vibrations, such as the super-resonance and sub-resonance. Furthermore, the experimental results suggest that linear Kelvin–Voigt damping cannot fully account for all of the experimentally observed nonlinear dissipation [22]. We believe that the effects of nonlinear damping in microbeams necessitate further studies, both experimentally and theoretically.

5. Conclusions

In the present paper, we propose a new partial differential equation to represent nonlinear oscillations of the microbeam. The new model considers the small-scale effect and nonlinear terms, which are induced by the axial elongation of the beam and Kelvin–Voigt damping. Here, we solved the model using the Galerkin and multiple scale methods. The results show that nonlinear damping affects the microbeam vibrations in the following three aspects: First, under the non-resonant condition, nonlinear damping scarcely affects the small-amplitude vibrations. However, in the large-amplitude vibrations, nonlinear damping remarkably reduces the amplitudes and suppresses the multi-period vibrations. Second, nonlinear damping slightly decreases the vibration amplitudes when the frequency of the load is less than the modal frequency of the microbeam under the primary resonance condition. Third, when the frequency of the load is greater than the modal frequency under the primary resonance condition, nonlinear damping can significantly change the bifurcation points of the load and strongly affects the vibrations. The present research suggests that the accurate description of microbeam vibrations must consider nonlinear damping.