A reduced-order model for electrically actuated ... - IEEE Xplore

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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 12, NO. 5, OCTOBER 2003

A Reduced-Order Model for Electrically Actuated Microbeam-Based MEMS Mohammad I. Younis, Student Member, ASME, Eihab M. Abdel-Rahman, Member, ASME, and Ali Nayfeh, Fellow, ASME

Abstract—We present an analytical approach and a reduced-order model (macromodel) to investigate the behavior of electrically actuated microbeam-based MEMS. The macromodel provides an effective and accurate design tool for this class of MEMS devices. The macromodel is obtained by discretizing the distributed-parameter system using a Galerkin procedure into a finite-degree-of-freedom system consisting of ordinary-differential equations in time. The macromodel accounts for moderately large deflections, dynamic loads, and the coupling between the mechanical and electrical forces. It accounts for linear and nonlinear elastic restoring forces and the nonlinear electric forces generated by the capacitors. A new technique is developed to represent the electric force in the equations of motion. The new approach allows the use of few linear-undamped mode shapes of a microbeam in its straight position as basis functions in a Galerkin procedure. The macromodel is validated by comparing its results with experimental results and finite-element solutions available in the literature. Our approach shows attractive features compared to finite-element softwares used in the literature. It is robust over the whole device operation range up to the instability limit of the device (i.e., pull-in). Moreover, it has low computational cost and allows for an easier understanding of the influence of the various design parameters. As a result, it can be of significant benefit to the development of MEMS design software. [883] Index Terms—Capacitive microswitches, microelectromechanical systems (MEMS), pressure sensors, pull-in time, reduced-order models.

I. INTRODUCTION

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ICROELECTROMECHANICAL devices and systems (MEMS) drew attention in the 1980s as sensors and actuators. The fact that they could be manufactured using existing manufacturing techniques and infrastructure of the semiconductor industry meant that they could be produced at low cost and in large volumes, making their commercialization quite attractive. Their light weight, small size, low-energy consumption, and durability made them even more attractive. By the mid-nineties, MEMS pressure sensors and accelerometers were widely used in the automotive industry. A wider array of MEMS sensors (e.g., blood pressure sensors) and actuators (e.g., micropumps) were also adopted for various biomedical applications. Currently MEMS devices and systems are being actively developed for various applications in a wide spectrum of indusManuscript received June 4, 2002; revised April 18, 2003. Subject Editor G. K. Fedder. The authors are with the Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/JMEMS.2003.818069

tries. However, design tools have not kept up pace with this growth. MEMS devices are still being designed by trial and error. A prototype of a proposed device is fabricated and tested in the laboratory and then modified until the desired performance is achieved. While it would be more effective to design, tune, and verify the performance of proposed devices using software tools before building and testing them, the state-of-the-art of these tools does not lend itself to this task. This paper presents an analytical approach and a reduced-order model to investigate the mechanical behavior of microbeam-based MEMS devices under electric actuation. We utilize the model to study the behavior of electrostatic resonators and capacitive microswitches and pressure sensors. The behavior of electrically actuated microbeams has been studied using different models and approaches. We can identify two groups of studies on this topic. The first group focuses on manufacturing techniques, introducing new designs, and testing the performance of proposed devices. It uses simple lumped-mass models or generic finite-element softwares, which are not designed to address MEMS devices, to predict the mechanical behavior of microbeams. The second group, on the other hand, gives more attention to modeling the devices and predicting their behaviors rather than to building and testing them. We begin by summarizing contributions of the first group. Zavracky et al. [1] studied the static behavior of a cantilever microswitch. They calculated the microbeam deflection at various DC voltages by numerically solving the fourth-order differential equation of a beam and by using a spring-mass model. They indicated that the spring-mass model is inaccurate since it predicts the pull-in voltage at a maximum deflection near one-third the gap, while the numerical solution predicts a higher value. Chan et al. [2] utilized the finite-element package ABAQUS to create a two-dimensional (2-D) model to study the static behavior of a fixed-fixed microbeam. Residual stresses were included in the model. They compared the computed maximum deflections using their model with those obtained by using the finite-element programs three-dimensional (3-D) IntelliCAD and Quasi 2-D. The Quasi 2-D simulator model included the effects of midplane stretching. The results of all three programs were in good agreement. They indicated that the 3-D simulations took much more time compared with the other methods and that the observed out-of-plane bending was very small. They calculated the pull-in voltages using ABAQUS for microbeams of different lengths and compared the results with experimental results, the agreement was good.

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YOUNIS et al.: REDUCED-ORDER MODEL FOR ELECTRICALLY ACTUATED MICROBEAM-BASED MEMS

Yao et al. [3] fabricated a capacitive microswitch utilizing a fixed-fixed microbeam. They stated that the use of a 3-D finite element model was necessary to accurately predict the pull-in voltage since lumped mass models and 2-D finite-element models were inadequate. For that purpose, they used the 3-D finite-element software IntelliCAD. They compared the theoretical results with experimental results and found good agreement. In the second group of contributions, Senturia et al. [4] developed a hybrid finite-element-boundary-element code MEMCAD to simulate electrostatic MEMS. They proposed a scheme to determine the deflection of a microbeam actuated by an electric force. It starts by computing the electric force at the undeflected position, which is then used to deform the microbeam. Then, the electric force on the deformed shape is recomputed and used to redeform the structure, and so forth. Gilbert et al. [5] used a similar 3-D electromechanical solver, CoSolve-EM, to predict the pull-in voltage of electrostatic MEMS. It is based on calculating the microbeam deflections, starting from a zero voltage and then increasing it in steps until the deflection reaches a large value, where the solution starts to diverge. At that point, the corresponding voltage is considered to be the pull-in voltage. Grtétillat et al. [6] studied the electromechanical behavior of asymmetric fixed-fixed microbeams. They simulated the pull-in voltages using the MEMCAD system [4], the resonance frequencies of undeflected microbeams using ABAQUS, and the dynamic motion using a macromodel. The calculated pull-in voltages and resonance frequencies were compared to experimental results and were found to be in good agreement. They also calculated the time that an undeflected microbeam takes to reach pull-in (the pull-in time) for various actuation voltages and found good agreement with experimental results. The parameters involved in the macromodel were adjusted to fit the experimental pull-in voltage by either adjusting the stiffness of the microbeam or by adding external axial loads. Hung and Senturia [7] used a macromodel to simulate the dynamics of a fixed-fixed microbeam proposed as a pressure sensor. They used the Galerkin method to discretize two coupled partial-differential equations (PDE): the linear beam equation under the electric forcing and the nonlinear Reynolds equation. They assumed an axial load applied to the microbeam to match the experimental pull-in voltage. The global basis functions were extracted using data produced from a few runs of a fully meshed and slow finite-difference method. They computed the pull-in time using two models: the macromodel and a model that assumes a linear damping, in which they estimated the damping coefficient by matching the experimental pull-in time to the theoretical one. The results of the two models were then compared to experimental results and results obtained from a full slow finite-difference simulation. The model based on the linear damping gave poor results, whereas the macromodel gave results in good agreement with the finite-difference and experimental results. Gabbay et al. [8] developed an automated procedure for generating a macromodel from a 3-D finite-element simulation. The procedure is limited to conservative systems and to nonstress-stiffened problems. Their method is based on using

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the linear mode shapes of a microbeam as basis functions. The electric forces are represented by curve-fitting data produced from a 3-D finite-element package. They reported that the linear mode shapes fail to correctly predict the dynamics of a clamped-clamped microbeam at a voltage near 30% of the pull-in value. Mehner et al. [9] modified the procedure of Gabbay et al. [8] to address problems involving midplane stretching. They modified the data, which are used to extract the basis functions, to include the effect of midplane stretching. The extracted basis functions were close to, but different from, the linear mode shapes of the undeflected microbeams. The procedure was not automated and does not apply to damped systems. Thus far, the literature lacks comprehensive models and efficient design tools to simulate the behavior of electrically actuated microbeams in MEMS devices over general operating conditions. The use of FEM codes in this field remains limited for two reasons. First, the use of FEM codes to simulate MEMS devices is prohibitively cumbersome, expensive, and time consuming. Consequently, it is very expensive to close the loop on an FEM model of a device to allow for the design of feedback control laws or to use the model in system-level simulations. As a result, FEM models are mostly used to analyze the performance of finished devices rather than to design them. Second, FEM models use numerous variables to represent the device state. This approach makes the process of mapping the design space complex. Also, the relationship between each of these variables and the overall device performance is not clear to designers. It would be easier and more intuitive for the designer to explore the design space if the model had only a few variables with a clear relationship between them and the overall device performance. Reduced-order models, also called macromodels, lend themselves very well to these purposes. Models constructed using this approach seek to capture the most significant characteristics of a device behavior in a few variables governed by a few ordinary-differential equations of motion. The variables can be selected to represent physically meaningful quantities. The resulting system is typically easy to simulate as a standalone model or integrate into system-level simulations. The reduced-order models in the literature are either lumped-mass models or based on modal analysis methods, such as the method of weighted residuals, especially, the Rayleigh-Ritz method [10]. Some of these models do not account for the coupling between the electric forces and the structural element displacement. Other models use approximations valid only for small displacements. Most models do not capture enough of the characteristics of the devices to be effective design tools. The predicted device performance progressively diverges from that experimentally observed as the applied load (voltages) increases. In this work, we introduce a novel technique to generate reduced-order models in MEMS devices geared for efficient, accurate, and fast simulation. Design parameters are included in the model by lumping them into nondimensional parameters, thereby allowing for an easier understanding of their effects and the interaction between the mechanical and electrical forces. The model treats the devices as distributed-parameter systems

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Fig. 1.


Schematic of an electrically actuated microbeam.

and accounts for moderately large deflections, dynamic loads, and coupling between the mechanical and electrical forces. It accounts for linear and nonlinear elastic restoring forces and nonlinear electric forces generated by capacitors. The developed model is validated by comparing its results with experimental results and finite-element solutions available in the literature.

where the functional

is given by (6)

The parameters appearing in (4) are

II. PROBLEM FORMULATION We consider a clamped-clamped microbeam, Fig. 1, actuated and subject to a viscous damping per by an electric force unit length. The equation of motion that governs the transverse is written as [11], [12] deflection

(1) where is the position along the plate length, and are the area and moment of inertia of the cross section, is Young’s modulus, is time, is the material density, is the microbeam thickness, is the gap width, and is the dielectric constant corresponds to a tensile of the gap medium. The parameter or compressive axial load, depending on whether it is positive or negative. The last term in (1) represents the parallel-plate electric forces assuming complete overlap area between the microbeam and the stationary electrode. The boundary conditions are (2) For convenience, we introduce the nondimensional variables (denoted by hats) (3) where is a time scale, defined below. Substituting (3) into (1) and (2) and dropping the hats, we obtain

(4)

(5)

(7) . and is chosen as We derive the equation governing the microbeam static deflection by setting the time derivatives in (4) equal to zero, assuming a constant electric load , dropping the time dependence of in (5), and obtaining (8) (9)

III. REDUCED-ORDER MODELS To generate a reduced-order model, we discretize (4) and (5) into a finite-degree-of-freedom system consisting of ordinarydifferential equations in time. Then, we integrate this system to simulate the dynamic behavior. This technique and its application to nonlinear systems is discussed by Nayfeh and Mook [13] and Nayfeh [14]. We use the undamped linear mode shapes of the undeflected microbeam as basis functions in the Galerkin procedure. However, the mode shapes of a microbeam vary with its deflection [11], [12]. Hence, we need to investigate the accuracy of the motions predicted using the mode shapes of the undeflected microbeam close to pull-in. To this end, we compare the obtained results with experimental results and results based on simulating the distributed-parameter system. We investigate two methods to treat the electric-force term in the discretization procedure. In the first method, we expand it in up to fifth order and rewrite (4) as a Taylor series around

(10)


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We express the solution of (10) and (5) as (11) is the th linear undamped mode shape of where , normalized such that the straight microbeam and governed by (12) (13) is the th natural frequency of the microbeam. Equawhere tion (12) and (13) represent a boundary-value problem that can be solved using a combination of shooting method and a bisection procedure for each pair of mode shape and natural frequency [11], [12]. Substituting (11) into (10), using (12), multiplying the result , and integrating the outcome from to 1, we by obtain (15) orEquations (14) and (15) represent discretized systems of dinary-differential equations, which contain all nonlinearities up to fifth order. We note that the coupling of the system of (15) is stronger than that of (14) because of the nonlinear inertia terms. The mass matrix of this system is no longer diagonal, and hence the numerical integration of the model is more complicated. On the other hand, the nonlinearity in this system always has the same form, however the nonlinear terms in (14) depend on where the electric force expansion is truncated. IV. STATIC BEHAVIOR AND PULL-IN INSTABILITY A. First Macromodel

(14) so that In the second method, we multiply (4) by the electric-force term is represented exactly. Substituting (11) , and and (12) into the resulting equation, multiplying by to 1, we obtain integrating the outcome from

We utilize the derived reduced-order models to simulate the static behavior of microbeam-based MEMS devices under a constant DC loading . Because the microbeam maintains a symmetric shape during motion, we use symmetric modes only. We obtain these modes by solving the linear undamped eigenvalue problem, (12) and (13), for the symmetric mode and their corresponding natural frequencies . We shapes generate the discretized static equations of the first macromodel and independent of time and setting by making the all of the time derivatives equal to zero in (14), which result in a nonlinear algebraic equations. We plug the calculated set of mode shapes into this algebraic system and solve it numerically using a pseudo-arclength continuation for the fixed points scheme [15]. Finally, we use (11) to calculate the microbeam static deflection. We determined the needed number of terms in the Taylor-series expansion in (10) by calculating the microbeam deflection near the pull-in voltage retaining four, five, and six terms in the expansion and comparing the results with those obtained by solving directly the boundary-value problem, (8) and (9), using a shooting method [11], [12]. The maximum deflection obtained using four and five terms are less than that obtained using the shooting method, indicating that the electric force is

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Fig. 2. Comparison of the calculated maximum nondimensional deflection (dashed line) using a Taylor-series expansion and a three-symmetric-mode discretization with that obtained in [11], [12] (solid line) for various values of .

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incompletely represented. On the other hand, the results of the six-term expansion are almost the same as those obtained with the shooting method. Thus, we adopt a six-term expansion of the forcing term. Very close to pull-in however, we cannot find a good initial guess of the static deflection, which is essential for the shooting method [11], [12] to converge. Therfore, we cannot compare results in that range. In Fig. 2, we compare the calculated maximum nondimenof an electrostatic microdevice with sional deflection , , , the specifications , and subject to a compression axial stress of 3.7 MPa using the first three symmetric modes with results obtained using the shooting method [11], [12]. Results are presented for values of ranging from 8 V to pull-in voltage. We note from Fig. 2 that the results obtained using this macromodel are in poor agreement with those obtained by solving the boundary-value problem. The predicted lower branches are very close away from pull-in, but they seem to deviate increasingly as pull-in is approached. The predicted upper branches are far away from each other. This is because approaches unity there, and thus the Taylor-series expansion breaks down and the approximation of the electric force becomes inadequate. We conclude that expanding the forcing term in a Taylor seis an improper way to represent the elecries around tric force. Discretizing the equation of motion using this method may lead to incorrect results. B. Second Macromodel We obtain the discretized static equations of the second and be independent of macromodel by letting the time and setting all of the time derivatives equal to zero in (15), which result in a system of nonlinear algebraic equations.

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v

Fig. 3. The influence of the number of symmetric modes retained in the with . discretization on the variation of

We follow the same procedure used in the first macromodel to calculate the static deflection . obtained by solving the algeFig. 3 shows a plot of braic system for various values of using one-, two-, three-, four-, and five-mode discretizations. All the modes used in the discretization are symmetric. We used the same device specifications used in Fig. 2. We note that increasing the number of modes leads to results that converge on the lower branch, which is stable. On the other hand, using an even number of modes predicts an erroneous form of the unstable upper branch. Fig. 3 also shows that the predicted upper branch converges with increasing number of modes as long as this number is odd. As of now, we do not have a clear explanation of this strange behavior. calculated using the second In Fig. 4, we compare macromodel and employing the first three (dashed line) and the first five (solid line) symmetric modes with results obtained by solving the static boundary-value problem using a shooting method [11], [12] (diamonds). The five-mode solution is in excellent agreement with the results of the shooting method for both the upper and lower branches. We conclude that the second macromodel predicts accurately the static behavior of microbeams by using no more than five modes. This significant result demonstrates a way of using the linear undamped mode shapes of a straight microbeam to generate a reduced-order model for MEMS devices. Mehner et al. [9] concluded that linear normal modes do not provide an adequate basis set to compute the elastic stored energy in stressstiffened microbeams. Gabbay et al. [8] indicated, based on their nonstress-stiffened model, that using the linear normal modes does not yield correct results beyond 30% of the pull-in voltage. Therefore, they recommended a procedure to extract a new set of basis functions from dynamic simulations using a finite-element code. However, our results show that the linear normal modes yield correct results up to the pull-in voltage, provided


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Fig. 4. Variation of calculated using the second macromodel for two cases: three symmetric modes (dashed line) and five symmetric modes (solid line). The discrete points are results obtained by solving the static boundary-value problem using a shooting method [11], [12].

that the model accounts for the midplane stretching and that the electric force is properly treated in the discretization procedure. The use of the linear mode shapes of an undeflected microbeam as basis functions is much more convenient than extracting these functions from running finite-element codes. Solving an algebraic system of equations to simulate the static behavior has the advantage of being stable and unaffected by the pull-in instability, unlike the shooting method presented in [11], [12] or other finite-element-based software used in the literature. Further, using the second discretization method, we can simulate the device behavior very close to pull-in, thereby enabling a precise prediction of the pull-in voltage, corresponding to the coalescence of the two branches of static solutions. In previous works [11], [12], we mentioned that solutions of the static boundary-value problem using the shooting method and are very sensitive to the initial guesses at high values of close to pull-in because the problem becomes very stiff. The developed macromodel however does not have such a problem. Using the second macromodel, we simulated the static behavior , , and . We found of a device of that the stable and unstable solutions coalesce approximately at , corresponding to the pull-in voltage. Using the shooting method [11], [12] , however, we were unable to obtain . Hence, assuming the pull-in voltage results beyond to be the value of voltage beyond which the numerical solver does not converge, which is the basis of many available finiteelement-based packages, may give erroneous results. To verify the results obtained by the second macromodel near pull-in, we substituted them into (8) and found that they satisfy it within the predefined tolerance of the solution algorithm [11], [12]. Using the discretized system of equations one can study the stability of the obtained deflections and, in general, investigate

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Fig. 5. Comparison of the normalized fundamental natural frequency calculated using the macromodel and employing five symmetric modes in the discretization (solid line) with results obtained in [11], [12] (triangles) and the experimental results (circles) obtained by Tilmans and Legtenberg [10].

possible nonlinear phenomena. As an example, we studied the local stability of the two branches of solutions by calculating the Jacobian matrix of the system of (15) and evaluating the eigenvalues corresponding to each static solution. The results show that one of the eigenvalues corresponding to the upper branch is always positive, and hence it is unstable (saddles). All of the eigenvalues of the lower branch are always real and negative, and hence they correspond to stable solutions (sinks). At pull-in, both branches collide and destroy each other with one eigenvalue tending to zero. Thus, the pull-in voltage corresponds to a saddle-node bifurcation. V. NATURAL FREQUENCIES Next, we use the macromodel to calculate the natural frequencies of a resonant microsensor. For a given voltage , we substitute the static solution corresponding to the lower branch into the Jacobian matrix of (15) and find the corresponding eigenvalues. Then by taking the square root of the magnitudes of the individual eigenvalues, we obtain the natural frequencies of the device. In Fig. 5, we compare the normalized fundamental natural frequency calculated using the macromodel and employing five symmetric modes in the discretization (solid line) with results obtained by solving the eigenvalue problem of the distributed-parameter system (triangles) using a shooting method [11], [12] and the experimental results (circles) obtained by Tilmans and Legtenberg [10] for a resonator with the specifi, , , , cations , and . There is an excellent agreement among the results. The macromodel shows robustness in predicting the natural frequency over the whole range even as the microbeam approaches its stability limit where the frequency approaches zero.

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Fig. 6. The first four natural frequencies, corresponding to the symmetric modes for the microbeam of Fig. 5. The solid lines are the results calculated retaining five symmetric modes in the discretization and the diamonds are obtained using four symmetric modes.

Fig. 6 shows four natural frequencies, corresponding to the first four symmetric modes, calculated using the macromodel for the same device of Fig. 5 for various values of . The solid lines are the results calculated using five symmetric modes in the discretization whereas the diamonds are obtained using four symmetric modes. Both sets of results are almost the same, indicating convergence of the macromodel. We note that the macromodel does not suffer from increasing stiffness as the actuation level approaches the pull-in voltage. VI. DYNAMIC BEHAVIOR AND PULL-IN TIME We use the macromodel to predict the dynamic behavior of electrically actuated capacitive microswitches and pressure sensors. As an example, we calculate the pull-in time of a pressure sensor and validate the results with available experimental data. and corresponding to the first symmetric We plug . To obmodes into (15) and integrate them in time for the tain the deflection variation with time, we use (11) with the caland . We find the pull-in time by monitoring culated the beam response over time for a sudden rise in the displacement, at that point we report the time as the pull-in time. We use the device specifications of Hung and Senturia [7] for a pressure , , , , sensor with . The reported experimental pull-in voltage and . Hung and Senturia [7] assumed the pressure is sensor to be subject to a compression axial stress of 3.7 MPa to match their theoretical pull-in voltage to the experimental value. Because is given as a nominal value, we modify it to match . the pull-in voltage. Accordingly, we let Fig. 7 shows the evolution of , the dominant coefficient, with the nondimensional time obtained by integrating (15)

Fig. 7. Evolution of u with the nondimensional time demonstrating the onset of pull-in.

using the first five symmetric modes. We use a nondimensional and a voltage of damping coefficient . The value of the a nondimensional rise time equal to damping coefficient is estimated using our model and the data of Fig. 8. The nondimensional pull-in time is approximately , where a sudden rise in occurs. In Fig. 8, we compare the calculated pull-in time obtained using our model with the theoretical and experimental results of a nondiof Hung and Senturia [7] for various values of . The discrete points, triangles mensional rise time equal to and squares, are the experimental and theoretical results of Hung and Senturia [7], respectively. We determined the damping coefficient by matching the pull-in time obtained using (15) at to the experimental value of Hung and Senturia and the results obtained [7]. The extracted coefficient is using this value are shown in Fig. 8 as a solid line. We note that the results are in excellent agreement with the experimental results. We conclude that using the first five symmetric modes in the proposed macromodel accurately simulates the mechanical behavior of electrically actuated MEMS devices. VII. SUMMARY AND CONCLUSION We proposed a novel approach to generate reduced-order models (macromodels) for electrically actuated microbeam-based MEMS and used them to study the static and dynamic behavior of these devices. The macromodel uses few linear-undamped mode shapes of a straight microbeam as basis functions in a Galerkin procedure, and hence it reduces the complexity of the simulation and reduces significantly the computational time. The macromodel accounts for the nonlinear elastic restoring forces and the nonlinear electric forcing.


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pull-in voltage. Another important advantage of the developed macromodel is the ability to study the local and global dynamics and their stability. Thus, one is able to investigate many possible nonlinear phenomena, which may adversely affect the performance of microbeam-based MEMS. REFERENCES

Fig. 8. Comparison of the pull-in time t obtained using our model (solid line) with those obtained theoretically (squares) and experimentally (triangles) by Hung and Senturia [7].

We found two static solutions at each voltage value and studied their local stability. The results show that the larger solution is unstable with one eigenvalue being positive. At pull-in, both solutions coalesce with an eigenvalue approaching zero. Beyond the pull-in voltage, both solutions disappear in a saddle-node bifurcation. We showed that a Taylor-series expansion of the electricforce term fails to represent correctly the electric force because the expansion diverges as grows because the neglected higherorder terms become significant. We introduced a new method in and which the equation of motion is multiplied by hence the electric force in the discretized equations is represented exactly. We validated this macromodel by simulating the static behavior of the microbeam and comparing the results with solutions of the static boundary-value problem. We simulated the dynamic behavior and compared the results with numerical solutions and experimental results available in the literature. In both cases, we found excellent agreement among all of the results. This macromodel possesses several attractive features. The use of linear mode shapes of a straight microbeam in the discretization is very attractive compared with running finite-element packages to extract the basis functions. Further, the model needs a few modes to converge. Hence it reduces considerably the computational time. Also, the macromodel predicts the static behavior and the pull-in voltage accurately as solutions of an algebraic system of equations, which is computationally easier to implement compared to solving a boundary-value problem. Numerical methods for solving the boundary-value problem usually suffer from the stiffness of the equations and may become unstable, especially as the pull-in voltage is approached. We demonstrated that the reduced-order model can handle very stiff problems and accurately predict the

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Mohammad I. Younis received the B.S. degree in mechanical engineering from Jordan University of Science and Technology in 1999 and the M.S. degree in engineering mechanics from Virginia Polytechnic Institute and State University, Blacksburg, in 2001. His M.S. thesis included work on the analysis and design of electrostatic MEMS. Currently, he is a doctoral candidate in the Department of Engineering Science and Mechanics at Virginia Polytechnic Institute and State University. His research is focused on the modeling and simulation of MEMS under the effect of squeeze-film damping. Mr. Younis is the recipient of the Paul E. Torgersen Graduate Research Excellence Award in 2002. He is a Student Member of the American Society of Mechanical Engineers (ASME).

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Eihab M. Abdel-Rahman received the B.S. degree in mechanical engineering from Kuwait University in 1988 and the M.S. degree in mechanical engineering and the Ph.D. degree in engineering mechanics from the University of Toledo, OH, in 1991 and 1997, respectively. He has been a Research Associate at the Department of Engineering Science and Mechanics at Virginia Polytechnic Institute and State University, Blacksburg. His research interests are in MEMS, biomechanics, structural dynamics, nonlinear dynamics, vibrations, and control. He is coauthor of several papers on MEMS, mechanics of the human knee, and dynamics of cranes. Dr. Abdel-Rahman is a Member of the American Society of Mechanical Engineers (ASME).

Ali Nayfeh received the B.S. degree in engineering science in 1962 and the M.S. and Ph.D. degrees in aeronautics and astronautics in 1963 and 1964, respectively, all from Stanford University, Stanford, CA. He is the author of the Wiley-Interscience books Perturbation Methods, Introduction to Perturbation Techniques, Problems in Perturbation, Method of Normal Forms, and Nonlinear Interactions. He is coauthor of the Wiley-Interscience books Nonlinear Oscillations, Applied Nonlinear Dynamics, Perturbation Methods with Maple, and Perturbation Methods with Mathematics. He is the Editor of the Wiley Book Series on Nonlinear Science and the Editor-in-Chief of Nonlinear Dynamics and Journal of Vibration and Control. He established and served as Dean of the College of Engineering, Yarmouk University, Jordan from 1980 to 1984. He is currently University Distinguished Professor of Engineering in the Department of Engineering Science and Mechanics at Virginia Polytechnic Institute and State University, Blacksburg. His research interests include nonlinear dynamics and chaos, linear and nonlinear control, aerodynamics, aeroelasticity, structural dynamics, dynamics of ships and submarines, and MEMS. Dr. Nayfeh is a Fellow of the American Physical Society, the American Institute of Aeronautics and Astronautics, the American Society of Mechanical Engineers (ASME), and the American Academy of Mechanics. He is the recipient of the Kuwait Prize in Basic Sciences (Physics), 1981; American Institute of Aeronautics and Astronautics Pendray Aerospace Literature Award, 1995; American Society of Mechanical Engineers J. P. Den Hartog Award, 1997; Honorary Doctorate, St. Petersburg University, Russia, 1996; Frank J. Maher Award for Excellence in Engineering Education, 1997; College of Engineering Dean’s Award for Excellence in Research, 1998; and Honorary Doctorate, Technical University of Munchen, Germany, 1999.