# Mechanical Behaviors of Electrostatic Microbeams with Nonideal Supports

Authors: Yunxia Chang; Qichang Zhang; Wei Wang; Jianxin Han
Journal: Shock and Vibration
 (2020)

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## Abstract

Deviation of the actual system from the ideal supporting conditions caused by micromachining errors and manufacturing defects or the requirement of innovative design and optimization of microelectromechanical systems (MEMS) make the nonideal boundary in the micro-/nanoresonator system receive wide attention. In this paper, we consider the neutral plane tension, fringing field, and nonideal boundary factors to establish a continuum model of electrostatically driven microbeam resonators. The convergent static solution with nine-order Galerkin decomposition is calculated. Then, based on the static solution, a 1-DOF dynamic equation of up to the fifth-order of the dynamic displacement using a Taylor expansion is derived. The method of multiple scales is used to study the effect of spring stiffness coefficients on the primary frequency response characteristics and hardening-softening conversion phenomena in four cases. The various law of the system’s static and dynamic performances with the spring stiffness coefficients is obtained. The conditions for judging the hardening-softening transition are derived. So, adjusting the support stiffness values can be a measure of optimizing the resonator performance.


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## Body

## 1. Introduction

Electrostatic microbeam resonators have the advantages of small size and light weight and are widely used in many fields, for instance, MEMS resonant sensor [1] and actuator [2]. Their small size allows sensitive systems to consume minimal energy and have low fabrication costs. However, due to the nonlinearity of the electrostatic force and the neutral plane tension, MEMS resonators can exhibit typical nonlinear dynamic characteristics. So far, the effects of nonlinearities on the static and dynamic performances of MEMS devices have been discussed in many works of literature [3–19].But the above-mentioned works of literature only considered the ideal boundary conditions. In fact, both macro- and microstructures have errors and manufacturing defects, such as structure with elastically restrained, structure with nonuniform [20, 21], overcutting near anchor points [22], and initial deformation [23–26] of microstructures caused by residual stress. These cause the boundary conditions of the actual system to deviate from the ideal support conditions such that the displacement and rotation angle of the two fixed ends are not equal to zero. Bambill et al. [27] drew the conclusion that the characterization of real boundary conditions was much more important for microscale beams than for macroscale beams. In addition, Muthukumaran et al. [28] proposed that the boundary condition became one of the techniques responsible for the structural tuning of particular note in order to obtain a desired harmonic relation among its natural frequencies. Hence, the nonideal boundary has become a way of innovative design and optimization of electrostatic resonators [29]. Therefore, the nonideal boundary in the micro-/nanoresonator system has received wide attention from relevant scholars.Rinaldi et al. [30] studied the boundary features of microcantilever beams through experiments. Results showed that nonclassical support boundaries could reduce the pull-in voltage of the microstructure. Rezaei et al. [31] analyzed the principal resonance behavior of piezoelectric actuated microcantilever and clamped beam under nonideal boundary conditions by using a two-dimensional multiscale method, and obtained the influence of nonideal boundary on the amplitude and frequency. Zhong et al. [32, 33] calculated the equivalent stiffness of nonideal support with fixed microbeam and microcantilever beam, respectively, and obtained the influence of nonideal boundary on vibration mode and frequency. Alkharabsheh and Younis [24] studied the fixed microarches with nonideal boundaries and compared the numerical results with experiments. Results demonstrated that nonideal boundary conditions had a significant effect on the qualitative dynamical behavior of the MEMS arch such as lowering its natural frequencies and causing unpredictable snap through or dynamic pull-in. Ekici et al. [34] used a multiscale method to study the superharmonic and subharmonic vibration of nonideal boundary microbeams. The nonideal boundary was simulated by helical spring. The conclusion was drawn that nonideal boundary conditions could cause shifting to the left or right side or no shifting in the graphs, depending on the mode numbers, axial force, deflections, and moments on the boundaries. Zeng et al. [35] studied the simulation of the nonideal boundary of the inclined supported beam. The finite element method and Galerkin method were used to study the static characteristics of the microbeam, which was in good agreement with the experimental results.In recent years, the static and dynamic characteristics of microbeam were studied by considering nonideal support along with edge field, or other factors. In Pallay and Towfighian’s literature [29], a MEMS-parametric resonator was introduced, which used nonideal supports and electrostatic edge fields to achieve an innovative design of energy-efficient resonators. Bashma et al. [36] used the finite element method based on wavelet transform to obtain the influence of nonideal support and edge effect on static attraction voltage and first-order natural frequency of the microcantilever beam. Chuang et al. [37] obtained an approximate analytical solution to the pull-in voltage of a microcurled cantilever beam considering the nonideal boundary and edge effect. Yayli [38] investigated the lateral free vibration of microbeams under various boundary conditions on the basis of the gradient theory of elasticity. Lishchynska et al. [39]established a model for predicting the static behavior of an electrostatic microcantilever beam considering the residual stress gradient and nonideal anchors. The analytical results were in good agreement with the experimental results. Tadi Beni et al. [40] introduced the modified couple stress theory, in conjunction with the MAD solving method, to investigate the effect of the Casimir attraction, Elastic boundary conditions, and size dependency on nonlinear pull-in behavior of the supported beam. Shojaeian et al. [41] studied the electromechanical instabilities of micro-/nanobeams with an initial curved shape and subjected to the electrostatic field and Casimir intermolecular force using a modified couple stress theory. However, they seldom analyze their effects on dynamic response.The purpose of this article is to explore the influence law of nonideal boundaries on the dynamics of resonators and to provide theoretical support for the optimal design of MEMS. In this paper, a continuum model of electrostatic microbeam resonators is established. The convergent static solution with nine-order Galerkin decomposition [42] is calculated. It is used to investigate the influences of nonideal boundaries on the system’s static characteristics. Based on the static solution, a 1-DOF dynamic equation of up to five orders of the dynamic displacement using Taylor expansion is derived. This quintic equation is a powerful complement to the cubic equation. Then, the method of multiple scales [43] is used to study the effects of nonideal boundaries on the primary frequency response characteristics and hardening-softening conversion phenomena in four cases. The various law of the system’s static and dynamic performances with the spring stiffness coefficients is obtained. The details are going to be discussed in the following sections.The paper is organized as follows: an introduction including the literature review and motivation of the research, equation of motion and methods, results and discussion, and lastly, conclusions.


## 2. Equation of Motion and Methods

Due to micromachining errors and manufacturing defects or the requirement of innovative design, the displacements and rotation angles of the two fixed ends of the microbeam are not equal to zero. To simulate and quantify these small rotation angles and deflections, artificial rotational and translational springs are introduced [24, 30–33, 44]. In this paper, only symmetrical springs are considered. That is, the stiffness coefficients of left and right springs, KR and KT, are the same. The schematic diagram is shown in Figure 1. With the application of DC and AC voltage across the beam and electrode backplate, a distributed electrostatic force between the beam and electrode is generated to deform and vibrate the microbeam.Figure 1
Schematic diagram of one-electrode microbeam with nonideal boundary condition.The first-order fringing field correction of electrostatic force per unit length, namely, Palmer model [45, 46], is expressed as follows: (1)Fef=0.65⋅g0−w^b0⋅ε0b0VD+VAsinΩ^⋅t^22g0−w^2.Hence, the governing equation per unit length along the microbeam [24, 32, 42, 47] can be given by the following equation:(2)E^I∂4w^∂x^4+μ^∂w^∂t^+ρb0h0∂2w^∂t^2=N^+E^b0h02l0∫0l0∂w^∂x^2dx^∂2w^∂x^2+ε0b0VD+VAsinΩ^⋅t^22g0−w^21+0.65⋅g0−w^b0,where x^ is the position along the microbeam length, wx^,t^ represents transverse deflection, I is the moment of inertia of the cross section (I=b0h03/12, where b0 and h0 represent, respectively, the microbeam’s width and thickness), l0 and g0 are, respectively, the microbeam’s length and the initial gap, μ^ and ρrepresent, respectively, the viscous damping per unit length and the material density, ε0 is the dielectric constant of the gap medium, t^ is time, t^ represents the effective Young’s modulus with E^=E/1−υ2 for a wide microbeam b0≥5h0 and E^=E for a narrow microbeam b0<5h0, in which E^ represents Young’s modulus while υ represents the Poisson’s ratio.The system’s nonideal boundary conditions are given by the following equation:(3)atx^=0:E^I∂2w^∂x^2−KR∂w^∂x^=0,E^I∂3w^∂x^3−N^∂w^∂x^+KTw^=0,atx^=l0:E^I∂2w^∂x^2+KR∂w^∂x^=0,E^I∂3w^∂x^3−N^∂w^∂x^−KTw^=0,where KR and KT represent the rotational and translational stiffness.For convenience, introduce the following nondimensional variables:(4)w=w^g0,x=x^l0,t=t^T,where T=ρb0h0l04/E^I. Thus, the governing equation of motion and the system’s boundary conditions become as follows:(5)∂4w∂x4+μ∂w∂t+∂2w∂t2=N+α1∫01∂w∂x2dx∂2w∂x2+α2VD+VAsinΩ⋅t21−w21+η1−w,(6)atx=0:∂2w∂x2−κR∂w∂x=0,∂3w∂x3−N∂w∂x+κTw=0,(7)atx=1:∂2w∂x2+κR∂w∂x=0,∂3w∂x3−N∂w∂x−κTw=0,where μ^=μ^l04/E^IT, N=N^l02/E^I, Ω=Ω^/T, α1=6g0/h02, α2=6ε0l04/E^h03g03, η=0.65⋅g0/b0, κR=KRl0/E^I, κT=KTl03/E^I.Galerkin method is a powerful method, which is capable of handling static and dynamic problems of nonlinear systems. It is limited by solving the mode shapes. See Appendix A for the expression of the mode shapes.In a microbeam resonator, the DC voltage causes the microbeam to form a new static equilibrium position, and the AC voltage excites the microbeam to vibrate around the equilibrium position. Thus, microbeam deflection can be given by the following equation:(8)wx,t=wsx+wdx,t,where the static deflection due to the DC voltage is denoted by wsx, and the dynamic deflection due to the AC voltage is denoted by wdx,t.To calculate the static deflection, set the time-varying terms in (5) equal to zero and obtain the following: (9)∂4ws∂x4=N+α1∫01∂ws∂x2dx∂2ws∂x2+α2VD21−ws21+η1−ws.Here the static deflection is expressed asws=∑As,iϕix,i=1,2,3,L…, where As,i is the ith generalized coordinate (here is a constant), ϕix is the ith mode shape, and i is the order of Galerkin discreteness. Substituting the expression into (9) and applying the Galerkin method yield the system of nonlinear algebraic equations.Substituting the above (8) and (9) into (5), setting VD≫VA, omitting the parameter exciting terms, and keeping the fifth power terms about wd, yield the following:(10)∂4wd∂x4+∂2wd∂t2+μ∂wd∂t=α1∫01∂ws∂x2dx+N∂2wd∂x2+2α1∂2ws∂x2∫01∂ws∂x∂wd∂xdx+α2VD22+η1−ws1−ws3wd+α1∂2ws∂x2∫01∂wd∂x2dx+2α1∂2wd∂x2∫01∂ws∂x∂wd∂xdx+α2VD23+η1−ws1−ws4wd2+α1∂2wd∂x2∫01∂wd∂x2dx+α2VD24+η1−ws1−ws5wd3+α2VD25+η1−ws1−ws6wd4+α2VD26+η1−ws1−ws7wd5+2α2VDVA1+η1−ws1−ws2sinΩt.Omitting the dynamic forcing term, neglecting damping term, and keeping only linear terms aboutwd in (10) yield the following:(11)∂4wd∂x4+∂2wd∂t2=N∂2wd∂x2+α1∂2ws∂x2∫012∂ws∂x∂wd∂xdx+α1∂2wd∂x2∫01∂ws∂x2dx+α2VD22+η1−ws1−ws3wd.Letwdx,t=ϕxeiϖt, where ϕx is mode shape and ϖ is the frequency of the system. Then, one can obtain the frequency of the system for various conditions.


## 3. Results and Discussion

Next, this paper analyzes the static and dynamic characteristics of the systems.


### 3.1. Static Analysis

In this section, the influences of the support spring coefficients on the static deflection and the natural frequency of the system are discussed by the numerical solution.


#### 3.1.1. Convergence Analysis

Because the microbeam maintains symmetric shapes during motion, only symmetric modes (namely the modes wheni equals even number) are considered to verify the convergence using the data in literature [4].SettingκR=+∞,κT=+∞regardless of fringing field effect, the outcome obtained by the first five symmetric modes (namely, i = 9) is shown in Figure 2. Compared with Nayfeh et al. [4], the convergence effect is good, and the research in this paper is feasible.Figure 2
Maximum static deflection of a microbeam-based resonator.


#### 3.1.2. Static Deflection

Next, calculate the static deflection of a microbeam with the parameters listed in Table1 and the various spring stiffness coefficients κR and κT.Table 1
Physical parameters of microbeam-based resonator [4].
ParametersValuesYoung’s modulusE (GPa)166Densityρ (kg/m3)2332Poisson’s ratioυ0.06Lengthl0 (μm)510Widthb0 (μm)100Thicknessh0 (μm)1.5The dielectric constantε0 (F/m)8.854 × 10−12Initial gap widthg0 (μm)1.18The effects of rotational stiffness on the system’s static features are discussed by changing the value of rotational stiffness, as shown in Figure3. The red dot dash line is the corresponding stable value when the rotational stiffness κR is infinite. The deflection-DC voltage curve is divided into the upper branch and lower one. The solid line of the lower branch represents the steady state, and the black dotted line of the upper branch represents the unstable state. Only the stable values make sense. With the increase of κR before pull-in phenomenon occurs, the static deflection corresponding to the same voltage decreases, the static pull-in voltage increases and eventually reaches the stable value.Figure 3
The effect of rotational stiffness on static features by applying the first five-order symmetry modes. (a) Relationship between maximum static deflection and DC voltage under different rotational stiffness. (b) Variation trend of the pull-in voltage with rotational stiffness.
(a)(b)Figure4 shows the effect of the translational stiffness on the static characteristics of the system. The red dot dash line is the corresponding stable value when the translational stiffness κT is infinite. As the κT goes up before pull-in phenomenon occurs, the static deflection corresponding to the same voltage goes down, the static pull-in voltage goes up, and eventually reaches the stable value.Figure 4
The effect of translational stiffness on static features by applying the first five-order symmetry modes. (a) Relationship between maximum displacement and DC voltage under different translational stiffness. (b) Variation trend of the pull-in voltage with translational stiffness.
(a)(b)In a word, with the growth of support stiffness (refers to rotational stiffness and translational stiffness) before pull-in phenomenon occurs, the static deflection corresponding to the same voltage goes down, the static pull-in voltage goes up, and eventually reaches a stable value.


#### 3.1.3. Natural Frequency

The influence of the rotational coefficientκR on the first-order frequency is shown in Figure 5. The red dot dash line is the corresponding stable value when the rotational stiffness κR is infinite. In the case of the same DC voltage, the greater the rotational stiffness is, the higher the first-order frequency is until it finally tends to the stable value.Figure 5
The effect of rotational stiffness on first-order frequency natural frequency. (a) Relationship between natural frequency and DC voltage under different rotational stiffness. (b) Variation trend of the natural frequency with rotational stiffness.
(a)(b)Figure6 shows the effect of translational stiffness κT on first-order frequency by applying the numerical method. The red dot dash line is the corresponding value when the translational stiffness κT is infinite. In the case of the same DC voltage, the greater the translational stiffness is, the higher the first-order frequency is until it eventually tends to the stable value.Figure 6
The effect of translational stiffness on first-order frequency fundamental frequency. (a) Relationship between natural frequency and DC voltage under different translational stiffness. (b) Variation trend of the natural frequency with translational stiffness.
(a)(b)In short, as the support stiffness (refers to rotational stiffness and translational stiffness) goes up, the first-order natural frequency increases, until it eventually tends to the stable value.


### 3.2. Dynamic Analysis

Introducingwdx,t≈ut⋅ϕ1x and applying the Galerkin method, obtain the following dynamic equation via equation (10):(12)u¨+μu˙+ωn2u+k2u2+k3u3+k4u4+k5u5=f1sinΩt,where ωn2=k1,(13)k1=∫01ϕ1ϕ14−N⋅ϕ12−2α1ws2∫01ws1ϕ11dx−α1ϕ12∫01ws12dx−α2VD22+η−η⋅wsϕ1/1−ws3dx∫01ϕ12dx,k2=∫01ϕ1−α2VD23+η−η⋅wsϕ1/1−ws4−α1ws2∫01ϕ112dx−2α1ϕ12∫01ws1ϕ11dxdx∫01ϕ12dx,k3=∫01ϕ1−α2VD24+η−η⋅wsϕ1/1−ws5−α1ϕ12∫01ϕ112dxdx∫01ϕ12dx,k4=∫01ϕ1−α2VD25+η−η⋅wsϕ1/1−ws6dx∫01ϕ12dx,k5=∫01ϕ1−α2VD26+η−η⋅wsϕ1/1−ws7dx∫01ϕ12dx,f1=∫012α2VDVA1+η−η⋅wsϕ1/1−ws2dx∫01ϕ12dx,wsj represents the corresponding Jth derivative of the function wsx, ϕ1j is the corresponding Jth derivative of the mode shape ϕ1x.See Appendix B for the detailed solving process by the method of multiple scales.Hence, frequency response equation is obtained as follows:(14)a2p1−p3a2−p5a42+μ22=f12ωn2,where(15)p1=Ω−ωn,p3=−10k22−9k3ωn224ωn3,p5=−1940k24+6228k22k3ωn2−405k32ωn4−6048k2k4ωn4+2160k5ωn66912ωn7.The peak amplitude is as follows:(16)amax=f1μωn.It is found through analysis thatamax is the function of DC voltage and AC voltage as independent variables for a given microbeam resonator.The backbone curve is as follows:(17)ωc=ωn+p3a2+p5a4.Whenp3>0, the nonlinearity has a hardening effect that tends to bend the frequency response curves to higher frequencies. When p3<0, the nonlinearity has a softening effect, which tends to bend the frequency response curves to lower frequencies.Whendωc/da=0, a real threshold can be obtained as follows:(18)as=−p32p5.If the real thresholdasexists and satisfies the following condition, the system has a hardening-softening effect transition phenomenon.(19)as≤amax.


#### 3.2.1. Convergence Verification

Assuming that the maximum microbeam deflection is given bywmax=ws+maxϕ1⋅u, where ws is the approximate solution of the static deflection obtained by the nine-order Galerkin discretization. Given the voltage value, calculate the system’s response, as shown in Figure 7.Figure 7
Convergence verification of frequency response curve regardless of the fringing field effect:κR=+∞, κT=+∞, VD=2V, VA=0.01V.It can be seen from Figure7 that the regularity trend of the frequency response curve is consistent and the convergence is good. Therefore, the above-mentioned approximate solution of maximum microbeam deflection can be used to determine the vibration characteristics of the system.


#### 3.2.2. Softening and Hardening Effects

Nonlinear hardening and softening characters have been reported in many works of literature. The system’s nonlinear softening and hardening characteristics are determined by parameterp3, which is determined by the DC voltage VD, static deflection ws , and natural frequency ωn. Furtherly, since the static displacement and natural frequency depend on VD and the boundary support stiffness values, the parameter p3 is ultimately affected by the DC voltage and the boundary support stiffness values. So, the boundary support stiffness values do affect the softening and hardening characteristics of the system.To investigate how the nonideal support stiffness to affect the system's frequency response curve in this section, there are four cases to discuss as seen in Table2.Table 2
Analysis cases.
Torsional spring stiffness valueκRTranslational spring stiffness valueκTCase 11050+∞1050+∞Case 2102104Case 31050+∞103,104Case 41,51050+∞(1) Comparison of Case 1 and Case 2. The stiffness values in Case 1 are used to simulate the ideal fixed boundary system, and the data in Case 2 simulate a nonideal boundary system. Other physical parameters refer to Table 1.For a given electrostatic microbeam resonator in Cases 1 and 2, the parameterp3 is the function of the DC voltage. Hence, Figure 8(a) shows the relationship between the parameter p3 and DC voltage VD under different boundary conditions. As the DC voltage increases, the parameter p3 decreases. The dotted line p3=0 is the boundary of the hardening and softening features. As shown in Figure 8(a), when VD=2V, hardening characteristics that ideal boundary system demonstrates are predicted. According to Figure 7, this prediction is true. Meanwhile, the nonideal boundary system at the same DC voltage also shows hardening characteristics.Figure 8
Key parameters of local soft and hard characteristics. (a) Relationship between parameterp3 and DC voltage under different boundary conditions. (b) Partial enlarged detail.
(a)(b)It can be seen clearly from Figure8(b) that the nonideal boundary has an influence on the softening and hardening features. The abscissa value corresponding to the intersection point S1 is smaller than the value of S0. Approximate maximum deflection at the time of the transition can be obtained according to the formula wmax,s≈maxwmax+ϕ1as, as shown in Figure 9. Obviously, the nonideal boundary reduces the critical deflection of the transition. Therefore, under the same DC voltage, the transition occurs first.Figure 9
Maximum deflection at the time of the transition-DC voltage curve.For the sake of illustration, select the three groupsVD corresponding to the dotted line in the above Figure 8(b) to analyze the amplitude-frequency response. Assume that the dimensionless damping coefficient is μ=0.01.WhenVD<S1, the real threshold as of the ideal boundary system exists, and the system is in hardening effect region. Nevertheless, whether there is a hardening-softening effect, conversion needs to be furtherly analyzed and judged by the second condition of equation (19). When VD is given, the system’s excitation amplitude f1 depends on the AC voltage VA, so the parameter amax in equation (19) is actually determined by the parameter VA. Therefore, given a suitable VA to meet with the condition of equation (19), there is a hardening-softening effect conversion in the ideal boundary system, as shown in Figure 10(b). In this case, the nonideal boundary system has a hardening-softening conversion phenomenon, but the maximum deflection of the transition is less than that of the ideal boundary system, as shown in Figure 10(a).Figure 10
Amplitude-frequency curve:VD = 2.9 V. (a) nonideal; (b) ideal.
(a)(b)WhenS1<VD<S0, the two boundary systems behave differently. The real threshold asof the ideal boundary system exists, and there is a hardening-softening conversion if a suitable VA to meet with the condition of equation (19) is selected, as shown in Figure 11(b). While the real threshold as of the nonideal boundary system does not exist, the system shows a softening effect and there is not a hardening-softening conversion, as shown in Figure 11(a).Figure 11
Amplitude-frequency curve:VD = 3.15 V. (a) nonideal; (b) ideal.
(a)(b)WhenVD>S0, the real threshold as of both boundary systems do not exist. So there are no hardening-softening conversion phenomena in both boundary systems. Both boundary systems show a softening effect, as shown in Figure 12.Figure 12
Amplitude-frequency curve:VD = 3.4 V. (a) nonideal; (b) ideal.
(a)(b)In summary, the laws of the dynamic performance of the system with the spring stiffness coefficients are obtained.(i)
Under the same DC voltage, as the spring stiffness goes down, the vibration frequency of the system becomes smaller and the amplitude becomes larger.(ii)
If the real thresholdas does not exist, there will be no hardening-softening transition. If the real threshold as exists and as≤amax is satisfied, the system will have a hardening-softening transition phenomenon.(iii)
If there is hardening-softening transition, the nonideal boundary reduces the critical deflection of the transition. So, under the same DC and AC voltage, the transition occurs first.(2) Case 3. In order to study the influence of translational spring on the system’s softening and hardening characteristics, the spring stiffness coefficient values are taken as follows:κR=1050, κT=103,104. Similarly, select the three groups VD to analyze the amplitude-frequency response. They are 1.5 V, 2.8 V, and 3.4 V, respectively.WhenκT=103, VD=1.5V, the real threshold as exists, but does not meet with (19), and the system is in the hardening effect region. So, the system shows hardening characteristic, as shown in Figure 13(a). Likewise, the system with κT=104also shows hardening characteristics, as shown in Figure 13(b).Figure 13
Amplitude-frequency curve:VD = 1.5 V. (a) κT=103; (b) κT=104.
(a)(b)WhenκT=103, VD=2.8V, the real threshold as does not exist and the system is in softening effect region. So the system shows softening characteristic, as shown in Figure 14(a). However, the real threshold as of the system with κT=104 exists and the system is in hardening effect region. So there is hardening-softening conversion if a suitable VA to meet with the condition of (19) is selected, as shown in Figure 14(b).Figure 14
Amplitude-frequency curve:VD = 2.8 V. (a) κT=103; (b) κT=104.
(a)(b)WhenκT=103, VD=3.4V, the real threshold asdoes not exist and the system is in the softening effect region. So the system shows softening characteristic, as shown in Figure 15(a). Likewise, the system with κT=104also shows softening characteristics, as shown in Figure 15(b).Figure 15
Amplitude-frequency curve:VD = 3.4 V. (a) κT=103; (b) κT=104.
(a)(b)In a word, the laws of the dynamic performance of the system with the translational spring stiffness coefficient are the same as that of the comparison of Case 1 and Case 2.(3) Case 4. In order to study the influence of rotational spring on the system’s softening and hardening characteristics, the spring stiffness coefficient values are taken as follows: κT=1050,κR=1,5. Similarly, select the three groups VD to analyze the amplitude-frequency response. They are 2 V, 2.6 V, and 2.75 V, respectively.WhenκR=1,VD=1.5V the real threshold asexists and the system is in the hardening effect region. So there is hardening-softening conversion if a suitable VAto meet with the condition of (19) is selected, as shown in Figure 16(a). Likewise, the system with κR=5 shows hardening-softening conversion if a suitable VAto meet with the condition of (19) is selected, for example VA=0.006V. However, because VAselected in Figure 16(b) is too small, the system with κR=5 only shows the hardening characteristic.Figure 16
Amplitude-frequency curve:VD = 2V. (a) κR=1; (b) κR=5.
(a)(b)WhenκR=1, VD=2.6V, the real threshold as does not exist and the system is in the softening effect region. So the system shows softening characteristics, as shown in Figure 17(a). However, the real threshold as of the system with κR=5 exists and the system is in the hardening effect region. So there is hardening-softening conversion if a suitable VA to meet with the condition of (19) is selected, as shown in Figure 17(b).Figure 17
Amplitude-frequency curve:VD = 2.6 V. (a) κR=1; (b) κR=5.
(a)(b)WhenκR=1, VD=2.75V, the real threshold as does not exist and the system is in the softening effect region. So the system shows softening characteristic, as shown in Figure 18(a). Likewise, the system with κR=5 also shows softening characteristics, as shown in Figure 18(b).Figure 18
Amplitude-frequency curve:VD = 2.75 V. (a) κR=1; (b) κR=5.
(a)(b)In a word, the laws of the dynamic performance of the system with the rotational spring stiffness coefficient are the same as that of the comparison of Case 1 and Case 2, too.


## 3.1. Static Analysis

In this section, the influences of the support spring coefficients on the static deflection and the natural frequency of the system are discussed by the numerical solution.


### 3.1.1. Convergence Analysis

Because the microbeam maintains symmetric shapes during motion, only symmetric modes (namely the modes wheni equals even number) are considered to verify the convergence using the data in literature [4].SettingκR=+∞,κT=+∞regardless of fringing field effect, the outcome obtained by the first five symmetric modes (namely, i = 9) is shown in Figure 2. Compared with Nayfeh et al. [4], the convergence effect is good, and the research in this paper is feasible.Figure 2
Maximum static deflection of a microbeam-based resonator.


### 3.1.2. Static Deflection

Next, calculate the static deflection of a microbeam with the parameters listed in Table1 and the various spring stiffness coefficients κR and κT.Table 1
Physical parameters of microbeam-based resonator [4].
ParametersValuesYoung’s modulusE (GPa)166Densityρ (kg/m3)2332Poisson’s ratioυ0.06Lengthl0 (μm)510Widthb0 (μm)100Thicknessh0 (μm)1.5The dielectric constantε0 (F/m)8.854 × 10−12Initial gap widthg0 (μm)1.18The effects of rotational stiffness on the system’s static features are discussed by changing the value of rotational stiffness, as shown in Figure3. The red dot dash line is the corresponding stable value when the rotational stiffness κR is infinite. The deflection-DC voltage curve is divided into the upper branch and lower one. The solid line of the lower branch represents the steady state, and the black dotted line of the upper branch represents the unstable state. Only the stable values make sense. With the increase of κR before pull-in phenomenon occurs, the static deflection corresponding to the same voltage decreases, the static pull-in voltage increases and eventually reaches the stable value.Figure 3
The effect of rotational stiffness on static features by applying the first five-order symmetry modes. (a) Relationship between maximum static deflection and DC voltage under different rotational stiffness. (b) Variation trend of the pull-in voltage with rotational stiffness.
(a)(b)Figure4 shows the effect of the translational stiffness on the static characteristics of the system. The red dot dash line is the corresponding stable value when the translational stiffness κT is infinite. As the κT goes up before pull-in phenomenon occurs, the static deflection corresponding to the same voltage goes down, the static pull-in voltage goes up, and eventually reaches the stable value.Figure 4
The effect of translational stiffness on static features by applying the first five-order symmetry modes. (a) Relationship between maximum displacement and DC voltage under different translational stiffness. (b) Variation trend of the pull-in voltage with translational stiffness.
(a)(b)In a word, with the growth of support stiffness (refers to rotational stiffness and translational stiffness) before pull-in phenomenon occurs, the static deflection corresponding to the same voltage goes down, the static pull-in voltage goes up, and eventually reaches a stable value.


### 3.1.3. Natural Frequency

The influence of the rotational coefficientκR on the first-order frequency is shown in Figure 5. The red dot dash line is the corresponding stable value when the rotational stiffness κR is infinite. In the case of the same DC voltage, the greater the rotational stiffness is, the higher the first-order frequency is until it finally tends to the stable value.Figure 5
The effect of rotational stiffness on first-order frequency natural frequency. (a) Relationship between natural frequency and DC voltage under different rotational stiffness. (b) Variation trend of the natural frequency with rotational stiffness.
(a)(b)Figure6 shows the effect of translational stiffness κT on first-order frequency by applying the numerical method. The red dot dash line is the corresponding value when the translational stiffness κT is infinite. In the case of the same DC voltage, the greater the translational stiffness is, the higher the first-order frequency is until it eventually tends to the stable value.Figure 6
The effect of translational stiffness on first-order frequency fundamental frequency. (a) Relationship between natural frequency and DC voltage under different translational stiffness. (b) Variation trend of the natural frequency with translational stiffness.
(a)(b)In short, as the support stiffness (refers to rotational stiffness and translational stiffness) goes up, the first-order natural frequency increases, until it eventually tends to the stable value.


## 3.1.1. Convergence Analysis

Because the microbeam maintains symmetric shapes during motion, only symmetric modes (namely the modes wheni equals even number) are considered to verify the convergence using the data in literature [4].SettingκR=+∞,κT=+∞regardless of fringing field effect, the outcome obtained by the first five symmetric modes (namely, i = 9) is shown in Figure 2. Compared with Nayfeh et al. [4], the convergence effect is good, and the research in this paper is feasible.Figure 2
Maximum static deflection of a microbeam-based resonator.


## 3.1.2. Static Deflection

Next, calculate the static deflection of a microbeam with the parameters listed in Table1 and the various spring stiffness coefficients κR and κT.Table 1
Physical parameters of microbeam-based resonator [4].
ParametersValuesYoung’s modulusE (GPa)166Densityρ (kg/m3)2332Poisson’s ratioυ0.06Lengthl0 (μm)510Widthb0 (μm)100Thicknessh0 (μm)1.5The dielectric constantε0 (F/m)8.854 × 10−12Initial gap widthg0 (μm)1.18The effects of rotational stiffness on the system’s static features are discussed by changing the value of rotational stiffness, as shown in Figure3. The red dot dash line is the corresponding stable value when the rotational stiffness κR is infinite. The deflection-DC voltage curve is divided into the upper branch and lower one. The solid line of the lower branch represents the steady state, and the black dotted line of the upper branch represents the unstable state. Only the stable values make sense. With the increase of κR before pull-in phenomenon occurs, the static deflection corresponding to the same voltage decreases, the static pull-in voltage increases and eventually reaches the stable value.Figure 3
The effect of rotational stiffness on static features by applying the first five-order symmetry modes. (a) Relationship between maximum static deflection and DC voltage under different rotational stiffness. (b) Variation trend of the pull-in voltage with rotational stiffness.
(a)(b)Figure4 shows the effect of the translational stiffness on the static characteristics of the system. The red dot dash line is the corresponding stable value when the translational stiffness κT is infinite. As the κT goes up before pull-in phenomenon occurs, the static deflection corresponding to the same voltage goes down, the static pull-in voltage goes up, and eventually reaches the stable value.Figure 4
The effect of translational stiffness on static features by applying the first five-order symmetry modes. (a) Relationship between maximum displacement and DC voltage under different translational stiffness. (b) Variation trend of the pull-in voltage with translational stiffness.
(a)(b)In a word, with the growth of support stiffness (refers to rotational stiffness and translational stiffness) before pull-in phenomenon occurs, the static deflection corresponding to the same voltage goes down, the static pull-in voltage goes up, and eventually reaches a stable value.


## 3.1.3. Natural Frequency

The influence of the rotational coefficientκR on the first-order frequency is shown in Figure 5. The red dot dash line is the corresponding stable value when the rotational stiffness κR is infinite. In the case of the same DC voltage, the greater the rotational stiffness is, the higher the first-order frequency is until it finally tends to the stable value.Figure 5
The effect of rotational stiffness on first-order frequency natural frequency. (a) Relationship between natural frequency and DC voltage under different rotational stiffness. (b) Variation trend of the natural frequency with rotational stiffness.
(a)(b)Figure6 shows the effect of translational stiffness κT on first-order frequency by applying the numerical method. The red dot dash line is the corresponding value when the translational stiffness κT is infinite. In the case of the same DC voltage, the greater the translational stiffness is, the higher the first-order frequency is until it eventually tends to the stable value.Figure 6
The effect of translational stiffness on first-order frequency fundamental frequency. (a) Relationship between natural frequency and DC voltage under different translational stiffness. (b) Variation trend of the natural frequency with translational stiffness.
(a)(b)In short, as the support stiffness (refers to rotational stiffness and translational stiffness) goes up, the first-order natural frequency increases, until it eventually tends to the stable value.


## 3.2. Dynamic Analysis

Introducingwdx,t≈ut⋅ϕ1x and applying the Galerkin method, obtain the following dynamic equation via equation (10):(12)u¨+μu˙+ωn2u+k2u2+k3u3+k4u4+k5u5=f1sinΩt,where ωn2=k1,(13)k1=∫01ϕ1ϕ14−N⋅ϕ12−2α1ws2∫01ws1ϕ11dx−α1ϕ12∫01ws12dx−α2VD22+η−η⋅wsϕ1/1−ws3dx∫01ϕ12dx,k2=∫01ϕ1−α2VD23+η−η⋅wsϕ1/1−ws4−α1ws2∫01ϕ112dx−2α1ϕ12∫01ws1ϕ11dxdx∫01ϕ12dx,k3=∫01ϕ1−α2VD24+η−η⋅wsϕ1/1−ws5−α1ϕ12∫01ϕ112dxdx∫01ϕ12dx,k4=∫01ϕ1−α2VD25+η−η⋅wsϕ1/1−ws6dx∫01ϕ12dx,k5=∫01ϕ1−α2VD26+η−η⋅wsϕ1/1−ws7dx∫01ϕ12dx,f1=∫012α2VDVA1+η−η⋅wsϕ1/1−ws2dx∫01ϕ12dx,wsj represents the corresponding Jth derivative of the function wsx, ϕ1j is the corresponding Jth derivative of the mode shape ϕ1x.See Appendix B for the detailed solving process by the method of multiple scales.Hence, frequency response equation is obtained as follows:(14)a2p1−p3a2−p5a42+μ22=f12ωn2,where(15)p1=Ω−ωn,p3=−10k22−9k3ωn224ωn3,p5=−1940k24+6228k22k3ωn2−405k32ωn4−6048k2k4ωn4+2160k5ωn66912ωn7.The peak amplitude is as follows:(16)amax=f1μωn.It is found through analysis thatamax is the function of DC voltage and AC voltage as independent variables for a given microbeam resonator.The backbone curve is as follows:(17)ωc=ωn+p3a2+p5a4.Whenp3>0, the nonlinearity has a hardening effect that tends to bend the frequency response curves to higher frequencies. When p3<0, the nonlinearity has a softening effect, which tends to bend the frequency response curves to lower frequencies.Whendωc/da=0, a real threshold can be obtained as follows:(18)as=−p32p5.If the real thresholdasexists and satisfies the following condition, the system has a hardening-softening effect transition phenomenon.(19)as≤amax.


### 3.2.1. Convergence Verification

Assuming that the maximum microbeam deflection is given bywmax=ws+maxϕ1⋅u, where ws is the approximate solution of the static deflection obtained by the nine-order Galerkin discretization. Given the voltage value, calculate the system’s response, as shown in Figure 7.Figure 7
Convergence verification of frequency response curve regardless of the fringing field effect:κR=+∞, κT=+∞, VD=2V, VA=0.01V.It can be seen from Figure7 that the regularity trend of the frequency response curve is consistent and the convergence is good. Therefore, the above-mentioned approximate solution of maximum microbeam deflection can be used to determine the vibration characteristics of the system.


### 3.2.2. Softening and Hardening Effects

Nonlinear hardening and softening characters have been reported in many works of literature. The system’s nonlinear softening and hardening characteristics are determined by parameterp3, which is determined by the DC voltage VD, static deflection ws , and natural frequency ωn. Furtherly, since the static displacement and natural frequency depend on VD and the boundary support stiffness values, the parameter p3 is ultimately affected by the DC voltage and the boundary support stiffness values. So, the boundary support stiffness values do affect the softening and hardening characteristics of the system.To investigate how the nonideal support stiffness to affect the system's frequency response curve in this section, there are four cases to discuss as seen in Table2.Table 2
Analysis cases.
Torsional spring stiffness valueκRTranslational spring stiffness valueκTCase 11050+∞1050+∞Case 2102104Case 31050+∞103,104Case 41,51050+∞(1) Comparison of Case 1 and Case 2. The stiffness values in Case 1 are used to simulate the ideal fixed boundary system, and the data in Case 2 simulate a nonideal boundary system. Other physical parameters refer to Table 1.For a given electrostatic microbeam resonator in Cases 1 and 2, the parameterp3 is the function of the DC voltage. Hence, Figure 8(a) shows the relationship between the parameter p3 and DC voltage VD under different boundary conditions. As the DC voltage increases, the parameter p3 decreases. The dotted line p3=0 is the boundary of the hardening and softening features. As shown in Figure 8(a), when VD=2V, hardening characteristics that ideal boundary system demonstrates are predicted. According to Figure 7, this prediction is true. Meanwhile, the nonideal boundary system at the same DC voltage also shows hardening characteristics.Figure 8
Key parameters of local soft and hard characteristics. (a) Relationship between parameterp3 and DC voltage under different boundary conditions. (b) Partial enlarged detail.
(a)(b)It can be seen clearly from Figure8(b) that the nonideal boundary has an influence on the softening and hardening features. The abscissa value corresponding to the intersection point S1 is smaller than the value of S0. Approximate maximum deflection at the time of the transition can be obtained according to the formula wmax,s≈maxwmax+ϕ1as, as shown in Figure 9. Obviously, the nonideal boundary reduces the critical deflection of the transition. Therefore, under the same DC voltage, the transition occurs first.Figure 9
Maximum deflection at the time of the transition-DC voltage curve.For the sake of illustration, select the three groupsVD corresponding to the dotted line in the above Figure 8(b) to analyze the amplitude-frequency response. Assume that the dimensionless damping coefficient is μ=0.01.WhenVD<S1, the real threshold as of the ideal boundary system exists, and the system is in hardening effect region. Nevertheless, whether there is a hardening-softening effect, conversion needs to be furtherly analyzed and judged by the second condition of equation (19). When VD is given, the system’s excitation amplitude f1 depends on the AC voltage VA, so the parameter amax in equation (19) is actually determined by the parameter VA. Therefore, given a suitable VA to meet with the condition of equation (19), there is a hardening-softening effect conversion in the ideal boundary system, as shown in Figure 10(b). In this case, the nonideal boundary system has a hardening-softening conversion phenomenon, but the maximum deflection of the transition is less than that of the ideal boundary system, as shown in Figure 10(a).Figure 10
Amplitude-frequency curve:VD = 2.9 V. (a) nonideal; (b) ideal.
(a)(b)WhenS1<VD<S0, the two boundary systems behave differently. The real threshold asof the ideal boundary system exists, and there is a hardening-softening conversion if a suitable VA to meet with the condition of equation (19) is selected, as shown in Figure 11(b). While the real threshold as of the nonideal boundary system does not exist, the system shows a softening effect and there is not a hardening-softening conversion, as shown in Figure 11(a).Figure 11
Amplitude-frequency curve:VD = 3.15 V. (a) nonideal; (b) ideal.
(a)(b)WhenVD>S0, the real threshold as of both boundary systems do not exist. So there are no hardening-softening conversion phenomena in both boundary systems. Both boundary systems show a softening effect, as shown in Figure 12.Figure 12
Amplitude-frequency curve:VD = 3.4 V. (a) nonideal; (b) ideal.
(a)(b)In summary, the laws of the dynamic performance of the system with the spring stiffness coefficients are obtained.(i)
Under the same DC voltage, as the spring stiffness goes down, the vibration frequency of the system becomes smaller and the amplitude becomes larger.(ii)
If the real thresholdas does not exist, there will be no hardening-softening transition. If the real threshold as exists and as≤amax is satisfied, the system will have a hardening-softening transition phenomenon.(iii)
If there is hardening-softening transition, the nonideal boundary reduces the critical deflection of the transition. So, under the same DC and AC voltage, the transition occurs first.(2) Case 3. In order to study the influence of translational spring on the system’s softening and hardening characteristics, the spring stiffness coefficient values are taken as follows:κR=1050, κT=103,104. Similarly, select the three groups VD to analyze the amplitude-frequency response. They are 1.5 V, 2.8 V, and 3.4 V, respectively.WhenκT=103, VD=1.5V, the real threshold as exists, but does not meet with (19), and the system is in the hardening effect region. So, the system shows hardening characteristic, as shown in Figure 13(a). Likewise, the system with κT=104also shows hardening characteristics, as shown in Figure 13(b).Figure 13
Amplitude-frequency curve:VD = 1.5 V. (a) κT=103; (b) κT=104.
(a)(b)WhenκT=103, VD=2.8V, the real threshold as does not exist and the system is in softening effect region. So the system shows softening characteristic, as shown in Figure 14(a). However, the real threshold as of the system with κT=104 exists and the system is in hardening effect region. So there is hardening-softening conversion if a suitable VA to meet with the condition of (19) is selected, as shown in Figure 14(b).Figure 14
Amplitude-frequency curve:VD = 2.8 V. (a) κT=103; (b) κT=104.
(a)(b)WhenκT=103, VD=3.4V, the real threshold asdoes not exist and the system is in the softening effect region. So the system shows softening characteristic, as shown in Figure 15(a). Likewise, the system with κT=104also shows softening characteristics, as shown in Figure 15(b).Figure 15
Amplitude-frequency curve:VD = 3.4 V. (a) κT=103; (b) κT=104.
(a)(b)In a word, the laws of the dynamic performance of the system with the translational spring stiffness coefficient are the same as that of the comparison of Case 1 and Case 2.(3) Case 4. In order to study the influence of rotational spring on the system’s softening and hardening characteristics, the spring stiffness coefficient values are taken as follows: κT=1050,κR=1,5. Similarly, select the three groups VD to analyze the amplitude-frequency response. They are 2 V, 2.6 V, and 2.75 V, respectively.WhenκR=1,VD=1.5V the real threshold asexists and the system is in the hardening effect region. So there is hardening-softening conversion if a suitable VAto meet with the condition of (19) is selected, as shown in Figure 16(a). Likewise, the system with κR=5 shows hardening-softening conversion if a suitable VAto meet with the condition of (19) is selected, for example VA=0.006V. However, because VAselected in Figure 16(b) is too small, the system with κR=5 only shows the hardening characteristic.Figure 16
Amplitude-frequency curve:VD = 2V. (a) κR=1; (b) κR=5.
(a)(b)WhenκR=1, VD=2.6V, the real threshold as does not exist and the system is in the softening effect region. So the system shows softening characteristics, as shown in Figure 17(a). However, the real threshold as of the system with κR=5 exists and the system is in the hardening effect region. So there is hardening-softening conversion if a suitable VA to meet with the condition of (19) is selected, as shown in Figure 17(b).Figure 17
Amplitude-frequency curve:VD = 2.6 V. (a) κR=1; (b) κR=5.
(a)(b)WhenκR=1, VD=2.75V, the real threshold as does not exist and the system is in the softening effect region. So the system shows softening characteristic, as shown in Figure 18(a). Likewise, the system with κR=5 also shows softening characteristics, as shown in Figure 18(b).Figure 18
Amplitude-frequency curve:VD = 2.75 V. (a) κR=1; (b) κR=5.
(a)(b)In a word, the laws of the dynamic performance of the system with the rotational spring stiffness coefficient are the same as that of the comparison of Case 1 and Case 2, too.


## 3.2.1. Convergence Verification

Assuming that the maximum microbeam deflection is given bywmax=ws+maxϕ1⋅u, where ws is the approximate solution of the static deflection obtained by the nine-order Galerkin discretization. Given the voltage value, calculate the system’s response, as shown in Figure 7.Figure 7
Convergence verification of frequency response curve regardless of the fringing field effect:κR=+∞, κT=+∞, VD=2V, VA=0.01V.It can be seen from Figure7 that the regularity trend of the frequency response curve is consistent and the convergence is good. Therefore, the above-mentioned approximate solution of maximum microbeam deflection can be used to determine the vibration characteristics of the system.


## 3.2.2. Softening and Hardening Effects

Nonlinear hardening and softening characters have been reported in many works of literature. The system’s nonlinear softening and hardening characteristics are determined by parameterp3, which is determined by the DC voltage VD, static deflection ws , and natural frequency ωn. Furtherly, since the static displacement and natural frequency depend on VD and the boundary support stiffness values, the parameter p3 is ultimately affected by the DC voltage and the boundary support stiffness values. So, the boundary support stiffness values do affect the softening and hardening characteristics of the system.To investigate how the nonideal support stiffness to affect the system's frequency response curve in this section, there are four cases to discuss as seen in Table2.Table 2
Analysis cases.
Torsional spring stiffness valueκRTranslational spring stiffness valueκTCase 11050+∞1050+∞Case 2102104Case 31050+∞103,104Case 41,51050+∞(1) Comparison of Case 1 and Case 2. The stiffness values in Case 1 are used to simulate the ideal fixed boundary system, and the data in Case 2 simulate a nonideal boundary system. Other physical parameters refer to Table 1.For a given electrostatic microbeam resonator in Cases 1 and 2, the parameterp3 is the function of the DC voltage. Hence, Figure 8(a) shows the relationship between the parameter p3 and DC voltage VD under different boundary conditions. As the DC voltage increases, the parameter p3 decreases. The dotted line p3=0 is the boundary of the hardening and softening features. As shown in Figure 8(a), when VD=2V, hardening characteristics that ideal boundary system demonstrates are predicted. According to Figure 7, this prediction is true. Meanwhile, the nonideal boundary system at the same DC voltage also shows hardening characteristics.Figure 8
Key parameters of local soft and hard characteristics. (a) Relationship between parameterp3 and DC voltage under different boundary conditions. (b) Partial enlarged detail.
(a)(b)It can be seen clearly from Figure8(b) that the nonideal boundary has an influence on the softening and hardening features. The abscissa value corresponding to the intersection point S1 is smaller than the value of S0. Approximate maximum deflection at the time of the transition can be obtained according to the formula wmax,s≈maxwmax+ϕ1as, as shown in Figure 9. Obviously, the nonideal boundary reduces the critical deflection of the transition. Therefore, under the same DC voltage, the transition occurs first.Figure 9
Maximum deflection at the time of the transition-DC voltage curve.For the sake of illustration, select the three groupsVD corresponding to the dotted line in the above Figure 8(b) to analyze the amplitude-frequency response. Assume that the dimensionless damping coefficient is μ=0.01.WhenVD<S1, the real threshold as of the ideal boundary system exists, and the system is in hardening effect region. Nevertheless, whether there is a hardening-softening effect, conversion needs to be furtherly analyzed and judged by the second condition of equation (19). When VD is given, the system’s excitation amplitude f1 depends on the AC voltage VA, so the parameter amax in equation (19) is actually determined by the parameter VA. Therefore, given a suitable VA to meet with the condition of equation (19), there is a hardening-softening effect conversion in the ideal boundary system, as shown in Figure 10(b). In this case, the nonideal boundary system has a hardening-softening conversion phenomenon, but the maximum deflection of the transition is less than that of the ideal boundary system, as shown in Figure 10(a).Figure 10
Amplitude-frequency curve:VD = 2.9 V. (a) nonideal; (b) ideal.
(a)(b)WhenS1<VD<S0, the two boundary systems behave differently. The real threshold asof the ideal boundary system exists, and there is a hardening-softening conversion if a suitable VA to meet with the condition of equation (19) is selected, as shown in Figure 11(b). While the real threshold as of the nonideal boundary system does not exist, the system shows a softening effect and there is not a hardening-softening conversion, as shown in Figure 11(a).Figure 11
Amplitude-frequency curve:VD = 3.15 V. (a) nonideal; (b) ideal.
(a)(b)WhenVD>S0, the real threshold as of both boundary systems do not exist. So there are no hardening-softening conversion phenomena in both boundary systems. Both boundary systems show a softening effect, as shown in Figure 12.Figure 12
Amplitude-frequency curve:VD = 3.4 V. (a) nonideal; (b) ideal.
(a)(b)In summary, the laws of the dynamic performance of the system with the spring stiffness coefficients are obtained.(i)
Under the same DC voltage, as the spring stiffness goes down, the vibration frequency of the system becomes smaller and the amplitude becomes larger.(ii)
If the real thresholdas does not exist, there will be no hardening-softening transition. If the real threshold as exists and as≤amax is satisfied, the system will have a hardening-softening transition phenomenon.(iii)
If there is hardening-softening transition, the nonideal boundary reduces the critical deflection of the transition. So, under the same DC and AC voltage, the transition occurs first.(2) Case 3. In order to study the influence of translational spring on the system’s softening and hardening characteristics, the spring stiffness coefficient values are taken as follows:κR=1050, κT=103,104. Similarly, select the three groups VD to analyze the amplitude-frequency response. They are 1.5 V, 2.8 V, and 3.4 V, respectively.WhenκT=103, VD=1.5V, the real threshold as exists, but does not meet with (19), and the system is in the hardening effect region. So, the system shows hardening characteristic, as shown in Figure 13(a). Likewise, the system with κT=104also shows hardening characteristics, as shown in Figure 13(b).Figure 13
Amplitude-frequency curve:VD = 1.5 V. (a) κT=103; (b) κT=104.
(a)(b)WhenκT=103, VD=2.8V, the real threshold as does not exist and the system is in softening effect region. So the system shows softening characteristic, as shown in Figure 14(a). However, the real threshold as of the system with κT=104 exists and the system is in hardening effect region. So there is hardening-softening conversion if a suitable VA to meet with the condition of (19) is selected, as shown in Figure 14(b).Figure 14
Amplitude-frequency curve:VD = 2.8 V. (a) κT=103; (b) κT=104.
(a)(b)WhenκT=103, VD=3.4V, the real threshold asdoes not exist and the system is in the softening effect region. So the system shows softening characteristic, as shown in Figure 15(a). Likewise, the system with κT=104also shows softening characteristics, as shown in Figure 15(b).Figure 15
Amplitude-frequency curve:VD = 3.4 V. (a) κT=103; (b) κT=104.
(a)(b)In a word, the laws of the dynamic performance of the system with the translational spring stiffness coefficient are the same as that of the comparison of Case 1 and Case 2.(3) Case 4. In order to study the influence of rotational spring on the system’s softening and hardening characteristics, the spring stiffness coefficient values are taken as follows: κT=1050,κR=1,5. Similarly, select the three groups VD to analyze the amplitude-frequency response. They are 2 V, 2.6 V, and 2.75 V, respectively.WhenκR=1,VD=1.5V the real threshold asexists and the system is in the hardening effect region. So there is hardening-softening conversion if a suitable VAto meet with the condition of (19) is selected, as shown in Figure 16(a). Likewise, the system with κR=5 shows hardening-softening conversion if a suitable VAto meet with the condition of (19) is selected, for example VA=0.006V. However, because VAselected in Figure 16(b) is too small, the system with κR=5 only shows the hardening characteristic.Figure 16
Amplitude-frequency curve:VD = 2V. (a) κR=1; (b) κR=5.
(a)(b)WhenκR=1, VD=2.6V, the real threshold as does not exist and the system is in the softening effect region. So the system shows softening characteristics, as shown in Figure 17(a). However, the real threshold as of the system with κR=5 exists and the system is in the hardening effect region. So there is hardening-softening conversion if a suitable VA to meet with the condition of (19) is selected, as shown in Figure 17(b).Figure 17
Amplitude-frequency curve:VD = 2.6 V. (a) κR=1; (b) κR=5.
(a)(b)WhenκR=1, VD=2.75V, the real threshold as does not exist and the system is in the softening effect region. So the system shows softening characteristic, as shown in Figure 18(a). Likewise, the system with κR=5 also shows softening characteristics, as shown in Figure 18(b).Figure 18
Amplitude-frequency curve:VD = 2.75 V. (a) κR=1; (b) κR=5.
(a)(b)In a word, the laws of the dynamic performance of the system with the rotational spring stiffness coefficient are the same as that of the comparison of Case 1 and Case 2, too.


## 4. Conclusions

In this paper, the Galerkin method and the method of multiple scales are used to theoretically investigate the mechanical behavior of electrostatic microbeam resonator with nonideal supports under forced excitation and study the feasibility of increasing vibration amplitude and reducing excitation voltage. The following main conclusions can be drawn.(i)
The laws of the system’s static performances with the spring stiffness coefficients are obtained; with the growth of support stiffness before pull-in phenomenon occurs, the static deflection corresponding to the same voltage goes down, the static pull-in voltage increases, and eventually reaches a stable value.(ii)
The laws of the system’s dynamic performances with the spring stiffness coefficients are obtained. Under the same DC voltage, as the support stiffness decreases, the vibration frequency of the system becomes smaller and the amplitude becomes larger.(iii)
The conditions for judging the hardening-softening transition are derived. If the real thresholdas does not exist, there will be no hardening-softening transition. If the real threshold as exists and as≤amax is satisfied, the system will have a hardening-softening transition phenomenon.(iv)
If there is hardening-softening transition, the nonideal boundary reduces the critical deflection of the transition. So, under the same DC and AC voltage, the transition occurs first.The analysis provides a theoretical basis for implementing energy-saving resonators by adjusting the support stiffness values. The dynamic design of specific parameters can be further carried out to achieve frequency, amplitude regulation, and vibration form regulation.So far, many studies have focused on the effect of Casimir gravity and size effect on the mechanical behavior of the system. The results show that both of them have a significant influence on the static and dynamic performance of the system. Then, the size effect on the system with nonideal boundary and Casimir force can be our next concern.



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