Vibration suppression in MEMS devices using electrostatic forces

Transcription

1 Vibration suppression in MEMS devices using electrostatic forces Hamed Haddad Khodaparast a, Hadi Madinei a, Micheal I. Friswell a, and Sondipon Adhikari a a College of Engineering, Swansea University, Bay Campus, Fabian Way, Swansea, United Kingdom, SA1 8EN ABSTRACT This paper investigates the use of electrostatic forces for vibration control of MEMS devices. A micro beam subject to electrostatic loading is considered. The electrostatic forces cause softening nonlinearity and their amplitudes are proportional to the square of applied DC voltages. An optimization problem is set up to minimize the vibration level of the micro-beam at given excitation frequencies. A new method based on incrementing nonlinear control parameters of the system and Harmonic Balance is used to obtain the required DC voltages that suppress unwanted vibration of the micro-beam. The results are illustrated using numerical simulations. Keywords: Vibration suppression, electrostatic forces, MicroElectroMechanical Systems MEMS, Harmonic Balance 1. INTRODUCTION MicroElectroMechanical Systems MEMS are widely used in various applications such as automotive components, mobile phones, gaming devices and toys. They could undergo large vibrations which can lead to failure due to resonant excitation or fatigue. Moreover, these effects degrade the performance of MEMS devices. Therefore, there has been great interest and effort in developing technologies that protect the MEMS devices from unwanted vibration. There are different ways of vibration suppression in MEMS devices. Perhaps the easiest way is to re-size the structural elements to increase the natural frequency and avoid resonance. However, this is not often possible due to design restrictions. There are several methods reported in the literature for vibration suppression in MEMS devices. Yoon et al. 1 proposed the use of nonlinear springs and soft coating shock stops for the reduction of impact forces. This was carried out through decoupling the shock-protection device from MEMS device. Pan and Cho exploited a shape memory alloy SMA based micro-damper. In their proposed SMA damper, the pseudoelasticity of NiTi wires is used to dissipate the vibration energy. Diveyev et al. 3 improved MEMS dynamic vibration absorber performance by adding damping layers in the joint and optimizing the device. Kaur et al. 4 used shape memory alloy NiTiCu thin films coupled with a piezoelectric AlN layer to produce an intelligent material for vibration damping in MEMS. They showed that the NiTiCu/AlN/NiTiCu heterostructure exhibit enhanced mechanical and damping properties as compared to NiTiCu/AlN and pure NiTiCu. This enhancement in hardness and damping of the heterostructure can be used in dissipation of mechanical vibration. Iyer et al. 5 proposed a model based on using fluidic damping to absorb unwanted vibration in micro-scale systems. In this model they used shear thickening fluids STFs to create a passively damped system that adapts the damping force to the magnitude of the disturbing force. Meyer and Cumunel 6 investigated piezoelectric and geometrical nonlinear effects on an active vibration isolation MEMS device. They studied the impact of these nonlinearities on the vibration control efficiency and showed that it is necessary to consider the effect of these nonlinearities because they affect strongly the dynamical behaviour of the structure. Zhang et al. 7 proposed a dynamical model of the micro-cantilever beam with piezoelectric actuator. In this model, they used a rational linearizing feedback controller with a high gain observer to eliminate the unwanted deflection of the micro-cantilever beam system. Delahunty and Pike 8 proposed a concept for the shock protection of MEMS suspensions. They used Send correspondence to H. Haddad Khodaprast: H.HaddadKhodaparast@swansea.ac.uk, Telephone:

2 solder incorporated within the sidewalls of the suspension to produce protective metal armouring. Reid et al. 9 developed a micromachined system for reducing the vibration sensitivity of surface transverse wave STW resonators. Their isolation system consisted of a support platform for mounting the STW resonator, four support arms, and a support rim. The aforementioned studies are very elegant; however, their performance will be subject to the initial design of vibration absorber or require expensive materials. In this paper a new vibration absorber based on using electrostatic device is introduced. Electrodes which are subjected to DC voltages are used to control the resonance frequency of the system. The main motivation is to minimize the vibration amplitude of a MEMS device by controlling the resonance frequency of the system. It will be shown that by applying the DC voltages to the electrodes, the dynamical behavior of the system is changed. This change in dynamical behavior can be exploited in vibration suppression of MEMS devices, particularly when they are excited by frequencies close to their resonance frequencies.. MODEL DESCRIPTION AND PROBLEM DEFINITION Figure 1 shows the model of MEMS cantilever beam with electrodes that will be used for vibration control. The isotropic micro-beam has the length L, thickness t, mass density ρ and Young s modulus E. The equation of motion of the beam can be expressed as: EI 4 w x, t w x, t x 4 + c a + ρa w x, t t t = ϵ ah x d 1 V1 g 1 w x, t ϵ ah x d 1 V1 g 1 + w x, t + ϵ a H x d H x d 3 V g w x, t ϵ a H x d H x d 3 V g + w x, t ρa z t t 1 subject to: w, t w, t = = x w L, t x = 3 w L, t x 3 = In the above equation, w x, t is the transverse deflection of the beam at the position x and time t, c a is the damping constant, ϵ is the permittivity of free space, H x is the Heaveside function, V i i = 1, is the applied DC voltage to the ith electrode and g i is the air gap between electrode i and the beam. The system is assumed to be perfectly symmetric. The base excitation function is z t = z cos Ωt. The electrostatic forces cause attraction of the surfaces and therefore will change the resonance frequency of the system. The electrostatic force functions in Eq.1 may be expressed in terms of its Taylor series. Therefore the nondimensionalised form of Eq.1 with the truncated cubic terms of electrostatic force becomes 4 ŵ ˆx, ˆt ˆx 4 + c ŵ ˆx, ˆt + ŵ ˆx, ˆt + α ˆt ˆt 1 ŵ + α 3 ŵ 3 + O ŵ 5 = γexp iˆωˆt + cc.

3 z 3 1 zt=zcos Ωt x d 1 d 3 V V 1 Micro-beam g 1 g 4 V V1 d Figure 1. Schematic of the MEMS device. where c = c al 4 ρal EIT, T = 4 g ŵ = w/g 1, ˆx = x/l, ˆt = t/t, ˆΩ = ΩT, λ =, g 1 i = 1 EI, η = ϵ al 4 EIg1 3, γ = z ρal 4 Ω, cc. : Complex Conjugate EIg 1 α 1 = ηv1 H ˆx d 1 /L + ηv H ˆx d /L H ˆx d 3 /L λ 3 α 3 = 4 ηv1 H ˆx d 1 /L + ηv H ˆx d /L H ˆx d 3 /L λ 5 Only the terms up to cubic order are included in the series. This is valid for a certain level of vibration and higher order terms need to be included or numerical integration 1 should be used if higher accuracy is required. Figure demonstrates a comparison between the time responses of the non-dimensionalized beam tip displacement due to the true electrostatic forces and the truncated cubic terms of electrostatic forces. It is assumed that the beam is excited at its first non-dimensionalized resonance frequency and V 1 = V = 7 V and z =.1 µm. The error is about.7% which shows that the cubic non-linearity in Eq. is the most dominant non-linear term for the given amplitude of excitation and the voltages. The objective of this paper is to reduce the vibration level of MEMS devices to an acceptable level by applying the DC voltages, i.e. V 1 and V. Assume that the acceptable vibration level is ŵ lim.. In this paper, the first and second bending modes of the micro-beam are included in the analysis and the maximum tip amplitude of the micro-beam is considered. Consequently, the objective is to determine the voltages V 1 and V that satisfy the following equation at the given excitation frequency, To this end, the following objective function can be defined, max ŵ ˆx, ˆt = ŵ lim. at ˆΩ = ˆΩ i 3 minimize max ŵ ˆx, ˆt ŵ lim. V 1,V 4 subject to V 1 and V < V p, ˆΩ = ˆΩi

4 Nondimensionalized tip displacement V DC = 7 V Numerical integration True electrostatic forces Harmonic Balance Electrostatic forces up to order Nondimensionalized time ˆt a Time histories Nondimensionalized tip displacement V DC = 7 V Numerical integration True electrostatic forces Harmonic Balance Electrostatic forces up to order 3 Error=.7 % Nondimensionalized time ˆt b Zoomed area Figure. Non-dimensionalized tip displacement time histories for the system with only cubic non-linearity solid line and the system with true electrostatic force dotted line. In the above equation, the pull-in voltage V p is the voltage at which the MEMS device becomes unstable. This is due to the softening effects of nonlinear electrostatic forces. This optimization problem requires the solution to the nonlinear partial differential equation of the micro beam given in Eq.. A method for the solution of this nonlinear problem is developed in this parer and will be described in the following subsection..1 Solution to the optimisation problem In this paper, it is assumed that the nonlinearity is controlled by a set of parameters called nonlinear control parameters. When these parameters are zero, the system is linear. In our example, these parameters are the applied DC voltages and Eq. is linear when V 1 = V =. In this case, the solution of Eq. can be described as ŵ = Y j x Q j exp iˆωˆt + cc. where Y j x is the jth linear mode shape of linear system and N is the number of linear mode shapes retained for the analysis. Substituting Eq.5 into Eq., applying the standard Galerkin approach and solving for q = {Q j } R N note that exp iωˆt is cancelled out from both sides yields q = [ ˆΩ M + iˆωc + K] 1 F 6 where M = [M ij ] R N N, C = [C ij ] R N N, K = [K ij ] R N N are the mass, damping and structural stiffness matrices of underlying linear system and F = {F j } R N is the vector of force amplitudes and, 5 M ij = C ij = Y i ˆx dˆx if i = j if i j cy i ˆx dˆx if i = j if i j K ij = Y i ˆx d4 Y jˆx dˆx if i = j 4 if i j 7 8 9

5 F j = γy j ˆx dˆx 1 It is assumed that the nonlinear parameters are normalised in a way that they vary from zero to one. If all the normalised nonlinear parameters are perturbed by δθ, the steady state solution of Eq. may be expressed by where ŵ 1 = ŵ θ 1 + ŵ θ ŵ ŵ 1 = ŵ + + ŵ δθ + O δθ ŵ + θ 1 θ ŵ 1 δθ 11, θ i = Vi V pi is the nondimensionalized control parameter in which V pi is pull-in voltage of ith electrode and δθ = δθ 1 = δθ. In the above equation, we assume that δθ is small enough so that the higher order terms are neglected. substituting Eq.11 into Eq. and neglecting the higher order terms of δθ yield 4 ŵ 1 x 4 + c ŵ 1 + ŵ 1 ˆt ˆt + α 1 θ ŵ 1 δθ + α 1 θ ŵ + α 3 θ ŵ 3 + 3ŵ ŵ1 δθ = 1 where θ = [θ 1, θ ]. Eq.1 is a linear function in terms of ŵ1. According to Eq.1, the steady state solution of ŵ1 includes primary and higher harmonics of the excitation frequency. One may ignore the higher harmonics and assume ŵ 1 = m Y j x Q 1j exp iˆωˆt + cc. 13 This is a reasonable assumption if no internal resonances are present. Substituting Eq.13 and Eq.5 into Eq.1 and balancing the harmonic terms and applying standard Galerkin projection gives where q 1 = { Q 1j } R N and [ and K L = k L ij b 1 i A 1 = ˆΩ M + iˆωc + K + K L + K NL δθ ] [ ] [ ] R N N, K NL = R N N, b 1 = R N where = k NL ij = k NL ij k L ij = A 1 q 1 = b 1 14 b 1 i α 1 θ 1 Yi ˆx dˆx if i = j if i j 6α 3 θ 1 Y i ˆx Y j ˆx Y j ˆx Q j Y j ˆx Q j dˆx α 1 θ 1 Y i ˆx dˆx 3α 3 θ 1 Y i ˆx Y j ˆx Q j Y j ˆx Q j dˆx where subscript 1 in θ 1 denotes the values of θ at the first iteration. Once the vector q 1 is determined, the solution of the weakly nonlinear system can be obtained from Eq.11. In the following iterations, the steadystate solution of the non-linear system is calculated through a recursive set of linear equations. Similar to the first step, it is supposed that an estimate ŵ k+1 may be obtained by the previous solution ŵ k as

6 ŵ k+1 = ŵ k + ŵ k δ θ 15 The following recursive system of linear equations are obtained by using the same method explained for the first iteration where q k = { Q kj } R N and [ and K Lk = k Lk ij A k q k = b k 16 A k = ˆΩ M + iˆωc + K + K Lk + K NLk δθ ] [ ] [ ] R N N, K NLk = R N N, b k = R N where k NLk ij = k Lk ij = k NLk ij b k i α 1 θ k+1 Yi ˆx dˆx if i = j if i j 6α 3 θ k+1 Y j ˆx Y j ˆx Q kj Y j ˆx Q kj dˆx b 1 i = α 1 θ k α 1 θ k+1 Y i ˆx dˆx + 3 α 3 θ k α 3 θ k+1 Y j ˆx Q kj Y j ˆx Q kj dˆx where q k and Q kj are defined in a similar way as q 1 and Q j. The above procedure can now be used to solve the optimization problem defined in Eq.4. Depending on how the DC voltages applied to the device, different optimal parameters can be obtained. In the following section, two cases are considered. In the first case, it is assumed that V 1 = V and in the second case V = 4V 1. In each case, the system has different pull-in voltages and therefore the solution described in this section will use different parameter paths to obtain the optimal voltages. 3. RESULTS AND DISCUSSION The characteristics of the micro-beam shown in Figure 1 are introduced in Table 1. The first two bending modes of the beam are considered in this analysis. For V 1 = V =, the first two nondimensionalized resonance frequencies of the beam are 3.5 and.3 respectively. It is assumed that the acceptable vibration level of the device is one percent of g 1. This assumption gives ŵ lim. =.1. Two cases are considered in this section. In the first case, it is assumed that V 1 = V while in the second case V = 4V 1. The pull-in voltages are found to be V p1 = V p = 1 V for the first case and V p = 4V p1 = 16 V. Note that these voltages are actually 1.4 V and 16. V but have been slightly reduced for safety of the system and also allowing us to increase the voltages up to pull-in voltage. The problem is considered for the cases that the MEMS device is excited by a harmonic base excitation with the amplitude z =.1µm at frequencies of ˆΩ = 3.3, 3.5, 3.7, 1.73,.3,.43. Figures 3 and 4 illustrate how the level of vibration at prescribed frequencies can be reduced by the application of DC voltages to the electrodes. The results are also tabulated in Table. Figures 3 and 4 and Table show that the voltage required to suppress the vibration level depends on the excitation frequency and the choice of relation between V 1 and V. The first observation is the fact that the vibration level at frequency points around the second mode did not reach the acceptable vibration level in neither of the two cases. Because of this, the voltages are increased to their corresponding pull-in voltages for vibration control of the frequency points adjacent to the second resonance frequency. However, what is interesting in these results is the fact that the choice of relation between V 1 and V has a significant effect on the percentage reduction in vibration level. The results shown in Table indicate that %, 8% and 64% reductions at frequencies of 1.73,. and.43

7 Table 1. Geometrical and material properties of the micro-beam. Model parameter Value Length of the beam L 3 µm Width of the beam a 1 µm Thickness of the beam h 4 µm Beam material Young s modulus E GPa Beam material density ρ 33 kg/m Viscous damping coefficient c.3 N.s/m 3 Permittivity component ϵ nf/m g 1 3 µm g 15 µm are respectively achieved in case 1 V 1 = V while in the second case V = 4V 1, these are 63%, 88% and 75% respectively. Figures 3 and 4 show that in both cases, at frequency points about the first resonance, the proposed method is capable of reducing the vibration level of the MEMS device to acceptable level, i.e..1. For the frequency points adjacent to the first resonance frequency, 71% reduction at ˆΩ = 3.3, 84% reduction at ˆΩ = 3.5 and 74% reduction at ˆΩ = 3.7 are achieved. However, the system requires less total voltages in case to obtain this objective and this highlights the importance of choosing the relation between V 1 and V. The vibration control of the system in case requires about 17% less total voltage V 1 + V than case 1. The final observation is that the frequencies which are below the resonance frequencies require higher voltages than those which are greater. This is mainly due to the softening effects of electrostatic forces V1 = V = V1 = V = 8.4 V. V1 = V = V1 = V = 1. V Tip displacement ŵ1, ˆΩ V1 = V = 8.88 V V1 = V = 9.36 V Tip displacement ŵ1, ˆΩ Frequency ˆΩ a Frequency ˆΩ Figure 3. Vibration suppression using the applied voltages in case 1 V 1 = V a mode 1 and b mode. b In this paper, only two cases are considered but it is obvious that there are infinite number of choices for relation between the two voltage sources. Therefore, an optimization problem can be set up to obtain the optimal relation between V 1 and V that minimizes the total voltage required for vibration control. Of course the parameters such as distances between electrodes and the micro-beam and the locations, the lengths and the number of electrodes can be included in this optimization problem. This will be considered in the future work. 4. CONCLUSION This paper proposed the use of electrostatic forces in vibration suppression of MEMS devices. A micro-beam with four electrodes and with two voltage sources was considered. The aim was to reduce the vibration levels of a MEMS device when it is excited by frequencies near to its first and second resonances. To this end, an extended Harmonic Balance method for calculation of steady state responses of nonlinear equation motion of the micro-beam was developed and exploited. It was observed that the total voltage that is required to achieve

8 V1 = V = V = 4V1 = 11.4 V. V1 = V = V = 4V1 = 16. V Tip displacement ŵ1, ˆΩ V = 4V1 = V V = 4V1 = 1.16 V Tip displacement ŵ1, ˆΩ Frequency ˆΩ Frequency ˆΩ a Figure 4. Vibration suppression using the applied voltages in case V = 4V 1 a mode 1 and b mode. Table. Micro-beam tip Displacement Disp. at different excitation frequencies and the applied DC voltages. Frequency Tip Disp. at V 1 = V = Tip Disp. at V 1 = V = V Tip Disp. at V = 4V 1 = V at V = 8.4 V.1 at V = 11.4 V at V = 8.88 V.1 at V = V at V = 9.36 V.1 at V = 1.16 V at V = 1 V.36 at V = 16 V...44 at V = 1 V.6 at V = 16 V at V = 1 V.18 at V = 16 V b this objective, depends on the initial assumption on the relation between the voltage sources and also frequency of excitation. Two different cases are considered in this paper. In the first case, it is assumed that the two voltages are equal and in the second case, the voltage which is applied to the electrodes at the mid-point of the beam are four times greater than the voltage provided to the electrodes located at the tip. The results showed that the proposed method is capable of reducing the vibration level of the MEMS device to an acceptable level at frequencies around the first resonance in both cases, but case needed about 17% less total voltage than case 1. It was also observed that the vibration level at frequency points around the second mode did not reach the acceptable vibration level in neither of the two cases. Therefore, the voltages are increased to their maximum possible values, i.e. pull-in voltages. However, the percentage reduction in vibration levels in cases were significantly higher than case 1. This highlights the importance of the choice of the relation between the voltages. Future work includes the solution to an optimization problem that find the optimal relation between the voltages in which the total voltage is minimized. ACKNOWLEDGMENTS Financial support from the Royal Academy of Engineering and the Ser Cymru National Research Network through an industrial secondment award is gratefully acknowledged. Hadi Madinei acknowledges the financial support from the Swansea University through the award of the Zienkiewicz scholarship. REFERENCES [1] Yoon, S., Lee, S., Perkins, N., and Najafi, K., Shock-protection improvement using integrated novel shockprotection technologies, J. Microelectromech. Syst. 4, [] Pan, Q. and Cho, C., The investigation of a shape memory alloy micro-damper for mems applications, Sensors Basel, Switzerland 79,

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