Generalized parametric resonance in electrostatically actuated microelectromechanical oscillators

Transcription

1 Journal of Sound and Vibration 296 (26) JOURNAL OF SOUND AND VIBRATION Generalized parametric resonance in electrostaticall actuated microelectromechanical oscillators Jeffre F. Rhoads a,, Steven W. Shaw a, Kimberl L. Turner b, Jeff Moehlis b, Barr E. DeMartini b, Wenhua Zhang b a Department of Mechanical Engineering, Michigan State Universit, East Lansing, MI 48824, USA b Department of Mechanical Engineering, Universit of California - Santa Barbara, Santa Barbara, CA 9316, USA Received 8 June 25; received in revised form 28 Februar 26; accepted 8 March 26 Available online 23 Ma 26 Abstract This paper investigates the dnamic response of a class of electrostaticall driven microelectromechanical (MEM) oscillators. The particular sstems of interest are those which feature parametric ecitation that arises from forces produced b fluctuating voltages applied across comb drives. These sstems are known to ehibit a wide range of behaviors, some of which have escaped eplanation or prediction. In this paper we eamine a general governing equation of motion for these sstems and use it to provide a complete description of the dnamic response and its dependence on the sstem parameters. The defining feature of this equation is that both the linear and cubic terms feature parametric ecitation which, in comparison to the case of purel linear parametric ecitation (e.g. the Mathieu equation), significantl complicates the sstem s dnamics. One consequence is that an effective nonlinearit for the overall sstem cannot be defined. Instead, the sstem features separate effective nonlinearities for each branch of its nontrivial response. As such, it can ehibit not onl hardening and softening nonlinearities, but also mied nonlinearities, wherein the response branches in the sstem s frequenc response bend toward or awa from one another near resonance. This paper includes some brief background information on the equation of motion under consideration, an outline of the analtical techniques used to reach the aforementioned results, stabilit results for the responses in question, a numerical eample, eplored using simulation, of a MEM oscillator which features this nonlinear behavior, and preliminar eperimental results, taken from an actual MEM device, which show evidence of the analticall predicted behavior. Practical issues pertaining to the design of parametricall ecited MEM devices are also considered. r 26 Elsevier Ltd. All rights reserved. 1. Introduction The recent analsis of parametricall ecited microelectromechanical (MEM) oscillators, b the authors and others, has led to the consideration of a certain class of nonlinear equations of motion [1 3]. The defining feature of these equations is that parametric ecitation acts on both the linear and cubic terms. As such, the dnamics of these oscillators are quite comple in comparison to a tpical Mathieu sstem, which features Corresponding author. Tel.: ; fa: address: rhoadsje@msu.edu (J.F. Rhoads) X/$ - see front matter r 26 Elsevier Ltd. All rights reserved. doi:1.116/j.jsv

2 798 ARTICLE IN PRESS J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) onl linear parametric ecitation. One interesting result of this complication is that a single effective nonlinearit for the overall sstem cannot be defined. Rather, the sstem features separate effective nonlinearities for each branch of its nontrivial response. Accordingl, the sstem can ehibit not onl the tpical hardening and softening nonlinearities, but also mied nonlinearities wherein the response branches in the sstem s frequenc response bend toward or awa from one another near resonance. In fact, such behavior has been observed in eperiments, as described subsequentl. The paper begins in Section 2 with a brief description of the tpes of MEM devices of interest, and a derivation of their governing equation of motion. General analtical results for the undamped sstem, including information on the eistence, stabilit, and bifurcations of the stead-state solutions, are given in Sections 3 and 4. These results include detailed descriptions of the resulting frequenc response curves and the associated phase planes. To facilitate understanding of the aforementioned results, a representative numerical eample is eplored in Section 5. In Section 6, the effects of light damping on the sstem are considered. Section 7 eplores the given results in the contet of a specific MEM oscillator, provides the pertinent results of preliminar eperimental investigations, and discusses practical design issues associated with parametricall ecited MEM devices. The paper concludes in Section 8 with a summar of the present work and some comments on possible directions of future work in this area. It should be noted that the bulk of this paper describes a detailed analsis of the governing equation of motion; this ma be of interest to the nonlinear vibrations communit, but of less direct interest to the MEMS communit. Those readers not interested in the details of the analsis can focus on Section 2, in which the MEMS model is emploed, and Section 7, which presents the general response features of parametricall ecited MEMS, including an approach for evaluating and predicting the response features of these devices. 2. Parametricall ecited MEM oscillators The impetus of the present stud was the analsis of single-degree-of-freedom (sdof) parametricall ecited MEM oscillators, such as the one shown in Fig. 1. Such oscillators consist of a shuttle mass, namel the oscillator s backbone, B, connected to a substrate via four folded beam springs, S, and ecited b a pair of non-interdigitated electrostatic comb drives, N, which are powered b a voltage source. These devices are Fig. 1. A representative parametricall ecited MEM oscillator. The oscillator s backbone is labeled B, S designates the mechanical beam springs, and N designates the non-interdigitated comb drives, which are used to provide the parametric ecitation.

3 J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) being proposed for use in a number of sensing and filtering applications [1,3,4]. Previous works, including those b the authors, have verified that these parametricall ecited oscillators ehibit motion consistent with a rather simple-looking equation of motion, presented in Eq. (1) [1,3,5,6]. This is based on the following development. To begin, the oscillator s motion is assumed to be described b the movement of the shuttle mass in one direction. The forces acting on the sstem include those from the springs, the electrostatic forces from the comb drives, and a force that represents dissipative effects, which arise primaril from aerodnamic effects. These dissipative effects are generall small, especiall when these devices are operated in a vacuum, and are tpicall modeled b a simple viscous damping force, although more complicated models have been proposed (see, for eample, Refs. [7 1]). The resulting equation of motion is given b m þ c _ þ F r ðþþf es ð; tþ ¼, (1) where F r ðþ represents the elastic restoring force provided b the springs and F es ð; tþ represents the electrostatic restoring and driving forces provided b the non-interdigitated electrostatic comb drives. As previous works have shown, the aforementioned forces can be accuratel represented b cubic functions of displacement [1,3]. The restoring force from the mechanical springs is modeled b F r ðþ ¼k 1 þ k 3 3, (2) where k 1 and k 3 are the effective linear and nonlinear spring coefficients from the beams. The electrostatic forces arising from the comb drives can accuratel be modeled for small displacements as F es ð; tþ ¼ðr 1A þ r 3A 3 ÞV 2 ðtþ, (3) where r 1A and r 3A are electrostatic coefficients that depend on the phsical dimensions of the electrostatic comb drives, and VðtÞ is the voltage applied across the drives. In order to provide harmonic ecitation to the device, an AC voltage of the form pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VðtÞ ¼V A 1 þ cos ot (4) is applied across the comb drives [1,3,6]. Substituting these forces back into Eq. (1) results in an equation of motion for the oscillator s backbone of the form [1,6]: m þ c _ þ k 1 þ k 3 3 þðr 1A þ r 3A 3 ÞV 2 Að1 þ cos otþ ¼. (5) Time and displacement in Eq. (5) are rescaled according to t ¼ o t, (6) where o is the purel elastic natural frequenc defined as rffiffiffiffiffi k 1 o ¼ (7) m and z ¼, (8) where is a characteristic length of the sstem (e.g. the length of the backbone). This scaling, assuming that the nondimensional damping, mechanical nonlinearities, and electrostatic forces are small (which is entirel consistent with the operation of such oscillators, especiall near resonance), ields a nondimensional equation of motion of the form z þ 2ezz þ zð1 þ el 1 þ el 1 cos OtÞþez 3 ðw þ l 3 þ l 3 cos OtÞ ¼, (9) where the new derivative operator and nondimensional parameters are defined as stated in Table 1 [1,6] and e represents a small parameter introduced for analtical purposes. The equation of motion considered in this work is a generalization of Eq. (9), wherein the effective stiffness and ecitation coefficients (the l s) are allowed to differ, in both the linear and nonlinear terms. This arises

4 8 ARTICLE IN PRESS J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) Table 1 Nondimensional parameter definitions [1,6] Definition ðþ ¼ dðþ dt ez ¼ c 2mo el 1 ¼ r 1AV 2 A k 1 O ¼ o o ew ¼ k 3 2 k 1 el 3 ¼ 2 r 3AV 2 A k 1 Nondimensional parameter Scaled time derivative Scaled damping ratio Linear electrostatic ecitation amplitude Nondimensional ecitation frequenc Nonlinear mechanical stiffness coefficient Nonlinear electrostatic ecitation coefficient naturall in the analsis of parametricall ecited MEM oscillators with two sets of non-interdigitated comb drives, one of which provides an AC voltage, as described above, while the other provides a DC bias to the sstem. This bias does not offset the sstem equilibrium in the usual sense, but allows one to independentl var the effective electrostatic stiffness and ecitation coefficients, resulting in Eq. (1). This arrangement of comb drives allows for more fleibilit in tuning the response of such devices, so that designers can achieve certain desirable response features [1,5,6]. More generall, this governing equation of motion is representative of a number of parametricall ecited dnamical sstems, as highlighted in Refs. [11 13]. While works such as [1,3,5,6] eamine the nonlinear dnamics of these parametricall ecited MEM oscillators, their nonlinear response has not been full eplored, and a number of apparent anomalies have been observed in eperimental investigations of these devices [2,3,14]. The analsis in the following sections provides a description for the entire range of possible responses for these sstems (under the small damping and small response amplitude assumptions). This is followed b a specific eample using the parameters from a device similar to that shown in Fig. 1. One of the more interesting features of the equation of motion presented in Eq. (9) is that several of the coefficients of the equation, including the effective linear and nonlinear stiffnesses, depend on the amplitude of the AC voltage, V A. This results in the fact that the qualitative nature of the sstem s response generall depends on the amplitude of the ecitation, and ma differ for different input amplitudes. The analsis presented here describes a complete picture of this behavior, and provides designers of these devices with useful predictive tools. In Section 7, an eample of this process and preliminar eperimental results that verif the utilit of the approach are given. 3. The equation of motion and perturbation analsis The equation of motion of interest here is of the form z þ 2ezz þ zð1 þ en 1 þ el 1 cos OtÞþez 3 ðg 3 þ l 3 cos OtÞ ¼, (1) where e is a scaling parameter introduced for analsis and prime designates the derivative with respect to t [1,5,6,15,16]. Note that this is a special case of generalized parametric ecitation in that there is not a phase difference in the ecitation seen b the linear and nonlinear terms. However, in man sstems the ecitation arises from a single source, resulting in zero phase difference (see, for eample, Refs. [11 13,17,18]). This is the case for the present stud, and this equation proves sufficient for capturing the dnamic response of the MEM devices under consideration.

5 J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) The method of averaging is emploed to analze Eq. (1). To aid this, a standard coordinate transformation is introduced to convert the equation to amplitude and phase coordinates, namel, zðtþ ¼aðtÞ cos Ot 2 þ cðtþ, (11) z ðtþ ¼ aðtþ O Ot sin 2 2 þ cðtþ. (12) Note that this conversion requires that a certain constraint equation involving a and c be satisfied, just as in the method of variation of parameters; see, for eample, Ref. [19]. In order to capture the dnamics near the principal parametric resonance, an ecitation frequenc detuning parameter s is introduced, defined b, O ¼ 2 þ es, (13) where s measures the closeness of the ecitation frequenc to the principal parametric resonance condition. Averaging the resulting equations, now in terms of amplitude and phase coordinates, over the period 4p=O in the t-domain produces the sstem s averaged equations [1,5,6]. These are given b a ¼ 1 8 ae½ 8z þð2l 1 þ a 2 l 3 Þ sin 2cŠþOðe 2 Þ, (14) c ¼ 1 8 e½3a2 g 3 þ 4n 1 4s þ 2ðl 1 þ a 2 l 3 Þ cos 2cŠþOðe 2 Þ. (15) Note that the presence of parametric ecitation in the cubic term, captured b the parameter l 3, introduces additional coupling in these equations that is not present in the case of purel linear parametric ecitation. Consequentl, the nontrivial stead-state solutions of the sstem take on a more complicated form and the sstem has much more interesting response characteristics. 4. Stead-state responses, their stabilit, and their effective nonlinearities Since the characteristic form of the sstem nonlinearit (i.e., hardening or softening) is a critical feature of the response for current purposes, and this is largel unaffected b damping, zero damping ðz ¼ Þ is assumed to simplif the analsis. (Note that while closed-form solutions are obtainable for nonzero damping, the are inherentl complicated, and thus not well suited for detailed analsis or eas interpretation.) We will consider the effects of small damping subsequentl. Stead-state responses are captured b setting ða ; c Þ¼ð; Þ. Eamining the resulting equations reveals that the sstem has a trivial solution and three distinct sets of nontrivial solution branches. The first two nontrivial sets appear in pairs with amplitudes and associated phases given b sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4s þ 2l 1 4n 1 a 1 ¼, (16) 3g 3 2l 3 c 1 ¼ p 2, (17) and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 ¼ 4s 2l 1 4n 1, 3g 3 þ 2l 3 (18) c 2 ¼. (19) Note that each pair represents the same phsical response; the appear this wa due to the phase relation of this subharmonic response relative to the ecitation. Likewise, all responses with magnitude p phase shifts represent the same phsical response, and thus are not included. For each of these solutions, the signs of the terms under the square root determine the frequenc range over which these branches are real valued, and thus phsicall meaningful. The numerators of these terms are dictated b linear frequenc terms, and the

6 82 ARTICLE IN PRESS J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) denominators are set b nonlinear terms. Of direct interest here is the role plaed b the nonlinearities, which differ for each of these frequenc-dependent branches. To describe these effects, effective nonlinearities Z 1 and Z 2 are defined that dictate the behavior of these branches. These nonlinearities are taken from the denominator of the epressions above and are given b [1,5] Z 1 ¼ 3g 3 2l 3 (2) and Z 2 ¼ 3g 3 þ 2l 3. (21) The case Z 1 o and Z 2 o results in the usual softening nonlinearit. Likewise, the usual hardening nonlinearit occurs for Z 1 4 and Z 2 4. However, due to the nature of this equation of motion, two mied cases also eist, namel, Z 1 4 and Z 2 o, and Z 1 o and Z 2 4, which correspond to the two branches delineated in Eqs. (16) and (18) bending toward or awa from one another, as determined b l 1 and n 1. This result is summarized in Fig. 2, which delineates the various response regions within the g 3 l 3 parameter space. Note that a transition between response tpes occurs when one or both of the effective nonlinearities equals zero, in which case the bifurcation associated with crossing the boundar of the parametric instabilit (given b an Arnold tongue) becomes degenerate. Accordingl, phenomena near these degeneracies cannot be captured without the inclusion of high-order nonlinear terms, which are not considered here. The solutions for the third set of nontrivial responses have constant amplitude (for zero damping) and are given, with their phase, b sffiffiffiffiffiffiffiffiffiffiffi 2l 1 a 3 ¼, (22) l 3 c 3 ¼ 1 2 arccos 3g 3l 1 þ 2l 3 n 1 2l 3 s. (23) l 1 l 3 Note that there will be four such solutions, onl two of which will be phsicall distinct (as evident in the net section), corresponding to each branch of the respective inverse cosine function. Accordingl, these solutions will appear in either the upper or lower half-plane of the parameter space delineated in Fig. 2, as dictated b the sign of l 1. As such, the sstem undergoes a bifurcation across the l 3 ¼ ais in Fig. 2, which corresponds to the creation or annihilation of the constant amplitude solution. It should be noted however, that this creation (or annihilation) is a smooth process, as the solution appears from positive or negative infinit due to the presence of l 3 in the solution s denominator. As will be shown shortl, the presence of these additional solution branches significantl complicates the sstem s dnamics within certain parameter regimes, especiall λ η2 < η2> III IV IIb Va Vb γ 3 IIa I VI η2 < η2> η 1 < η 1 < η 1 > η 1 > Fig. 2. The g 3 l 3 parameter space [1]. Those regions designated I and VI ehibit hardening (or quasi-hardening) nonlinear characteristics, those designated III and IV ehibit softening (or quasi-softening) nonlinear characteristics, and those designated IIa, IIb, Va, and Vb ehibit mied nonlinear characteristics.

7 J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) when viewed from a frequenc response perspective. Before turning to the generation of frequenc response curves, the stabilit of the various response branches is first determined. Though the local stabilit of stead-state responses is generall addressed b considering the local linear behavior of the averaged equations near the stead-state responses in the ða; cþ polar coordinate space, singularities at the origin require that a coordinate change be introduced. As such, the following coordinate change is evoked, which converts the polar coordinates ða; cþ into Cartesian coordinates ð; Þ: ðtþ ¼aðtÞ cos½cðtþš, (24) ðtþ ¼aðtÞ sin½cðtþš. (25) This coordinate change results in averaged equations of the form ¼ 1 8 e½ð2l 1 4n 1 þ 4sÞ 3g 3 2 þð 3g 3 þ 2l 3 Þ 3 8zŠþOðe 2 Þ, (26) ¼ 1 8 e½ð2l 1 þ 4n 1 4sÞ þ 3g 3 2 þð3g 3 þ 2l 3 Þ 3 8zŠþOðe 2 Þ. (27) Using these equations, with z ¼, local stabilit results are inferred b considering the response of the sstem when linearized about the stead states. Letting the state vector be " XðtÞ ¼ ðtþ # (28) ðtþ and the stead-state values be given b X ¼ " #, (29) the local linearized equation of motion for the sstem can be written as Y ðtþ ¼Jj X YðtÞ, (3) where YðtÞ ¼XðtÞ X (31) and J is the Jacobian matri of the averaged equations, Eqs. (26) and (27), evaluated at stead state, X. The stabilit of the stead-state response is governed b the eigenvalues of the corresponding Jacobian. To simplif the analsis, we will consider the stabilit in terms of the trace and determinant of the 2 2 Jacobian [2]. In particular, the eigenvalues can be epressed in terms of the trace, T, and the determinant, D, as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L 1;2 ¼ 1 2 ðt T 2 4DÞ. (32) For the undamped case under consideration here, the trace of the Jacobian for each of the stead-state responses is zero, thus onl two generic equilibrium tpes are possible for nontrivial eigenvalues: saddles (unstable) and centers (marginall stable). Which of these equilibrium tpes eists depends solel on the sign of the determinant. In particular, when D4 the equilibrium is a center, and when Do the equilibrium is a saddle. The remaining case, D ¼, corresponds to the degenerate case of two identicall zero eigenvalues, and as such is used here onl to identif where stabilit changes occur. The trivial solution features a determinant of the form D ¼ 1 16 e2 ½l 2 1 4ðn 1 sþ 2 Š. (33) B finding the roots of D ¼, the critical frequenc values (detuning values) at which stabilit changes occur can be shown to be s 1 ¼ n 1 l 1 2 (34)

8 84 ARTICLE IN PRESS J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) and s 2 ¼ n 1 þ l 1 2, (35) respectivel. The stabilit in each frequenc and detuning region is then determined b evaluating the determinant at a single point within the domain or b calculating the derivative of the determinant at the critical frequenc value. This shows that the trivial solution appears as a center for so minðs 1 ; s 2 Þ and s4 maðs 1 ; s 2 Þ, and as a saddle for minðs 1 ; s 2 Þoso maðs 1 ; s 2 Þ. These results are, of course, unaffected b the nonlinear ecitation term and are closel related to the results obtained for a standard Mathieu equation. The distinct difference here, however, is that wedge of instabilit which appears in the l 1 O parameter space is generall distorted awa from the tpical configuration seen in Refs. [21,22], for eample, due to the presence of the frequenc-shifting n 1 term (this is considered etensivel in Ref. [1]). Using similar techniques, the constant amplitude solutions of Eq. (22) can be shown to be saddles for all frequenc values where the solutions eist, as determined b considering parameter values where their phases are real-valued. It is found that these solutions eist between the detuning values and s ¼ 3l 1g 3 l 1 l 3 þ 2l 3 n 1 2l 3 (36) s ¼ 3l 1g 3 þ l 1 l 3 þ 2l 3 n 1. (37) 2l 3 Note that each of these values ma serve as an upper or lower bound, depending on the sign of l 1. The remaining two branches are slightl more complicated in that their stabilit is heavil dependent on which region of the g 3 l 3 parameter space (see Fig. 2) the sstem falls within. Regardless, the stabilit for each of the branches can be found b evoking the fact that the response branch presented in Eq. (16) has a determinant given b D ¼ e2 ðl 1 2n 1 þ 2sÞð3l 1 g 3 l 1 l 3 2l 3 n 1 þ 2l 3 sþ, (38) 12g 3 8l 3 which features critical frequenc values of s ¼ n 1 l 1 2, (39) corresponding to the bifurcation at which this branch emanates from the origin, and s ¼ 3l 1g 3 þ l 1 l 3 þ 2l 3 n 1, (4) 2l 3 corresponding to the bifurcation which connects this branch to the constant amplitude branches. The response branch presented in Eq. (18) has a determinant given b D ¼ e2 ðl 1 þ 2n 1 2sÞð3l 1 g 3 þ l 1 l 3 2l 3 n 1 þ 2l 3 sþ, (41) 12g 3 þ 8l 3 which features critical frequenc values of s ¼ n 1 þ l 1 2, (42) corresponding to the bifurcation at which this branch emanates from the origin, and s ¼ 3l 1g 3 l 1 l 3 þ 2l 3 n 1, (43) 2l 3 corresponding to the bifurcation which connects this branch to the constant amplitude branches. Combining these results with the stead-state solutions derived above ields a complete picture of the oscillator s possible frequenc responses, as described in detail in Section 5.

9 J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) A B C 5 4 a Fig. 3. Representative frequenc response plot showing response amplitude versus frequenc detuning for Region I (g 3 ¼ :5, l 3 ¼ :5) in Fig. 2. Note the response s hardening nonlinear characteristics. Also note that here, as well as in Figs. 4 14, solid lines indicate stable response branches and dashed lines unstable response branches. 5. Sstem frequenc responses To facilitate understanding of the analsis presented in Section 4, consider a representative sdof dnamical sstem modeled b Eq. (1). In particular, consider an undamped version (z ¼ ) with n 1 ¼ l 1 ¼ 1. (The results can easil be etended for other values of n 1 and l 1, both positive and negative.) B appling the results of the aforementioned section to this tpical case, frequenc response plots, such as those shown in Figs. 3 14, are easil generated for each of the regions depicted in Fig. 2. A brief eplanation of each of these plots follows. To begin, consider the amplitude versus frequenc response curves depicted in Figs. 3 and 6, corresponding to Regions I and III in Fig. 2, respectivel. As a quick eamination reveals, these regions appear consistent with normal softening and hardening nonlinear behavior. That is, the nontrivial solutions branch off in two distinct pitchfork bifurcations, one subcritical and the other supercritical, and all solutions remain globall bounded. (Note that the term globall bounded is used here to designate the fact that all initial conditions result in bounded responses.) Though not shown, the phase response will be equall mundane in that it remains constant across the branches, as dictated b Eqs. (17) and (19). The frequenc responses shown in Figs. 4 and 5, corresponding to the topologicall equivalent regions labeled IIa and IIb in Fig. 2, are slightl more interesting. 1 Here the nontrivial solutions also branch off, but in each instance a subcritical bifurcation occurs. The net result is that the two response curves actuall bend awa from one another and some solutions are globall unbounded (specificall, all nonzero initial conditions lead to unbounded motions over the frequenc range where the trivial solution is unstable, and where the trivial solution is stable, initial conditions with sufficientl large energies lead to unbounded responses). Of course, here, and elsewhere, these globall unbounded solutions will be bounded b higher-order nonlinear effects not included here. The phase responses of the solutions within these regions, much like those considered above, remain invariant in frequenc and thus are omitted. 1 Note that these regions are given different designations onl due to the fact that the nontrivial response branches to the left and right of the resonance have different amplitudes, which are reversed in these two regions. This is not phsicall important, since both branches are unstable. However, in other regions, this distinction will have consequences in terms of the observed frequenc response behavior.

10 86 ARTICLE IN PRESS J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) D E F a Fig. 4. Representative frequenc response plot showing response amplitude versus frequenc detuning for Region IIa (g 3 ¼ :5, l 3 ¼ :1) in Fig. 2. Note the response s mied nonlinear characteristics G H I a Fig. 5. Representative frequenc response plot showing response amplitude versus frequenc detuning for Region IIb (g 3 ¼ :5, l 3 ¼ :1) in Fig. 2. Note the response s mied nonlinear characteristics. In the lower half-plane of the parameter space depicted in Fig. 2, the eistence of the additional nontrivial, constant amplitude, solutions complicates matters. First, consider Figs. 7 and 13, corresponding to Regions IV and VI in Fig. 2, respectivel. Here the near zero amplitude responses resemble those of Figs. 3 and 6, respectivel. However, as the frequenc is swept further awa from zero detuning (s ¼ ), the sstem undergoes two additional local bifurcations, corresponding to the creation and annihilation of the constant amplitude response, which connects the other two nontrivial branches. Though this still results in a globall

11 J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) J K L 5 4 a Fig. 6. Representative frequenc response plot showing response amplitude versus frequenc detuning for Region III (g 3 ¼ :5, l 3 ¼ :5) in Fig. 2. Note the response s softening nonlinear characteristics C M J K L 3 25 a Fig. 7. Representative frequenc response plot showing response amplitude versus frequenc detuning for Region IV (g 3 ¼ :5, l 3 ¼ :5) in Fig. 2. Note the response s quasi-softening nonlinear characteristics. bounded sstem with quasi-softening or hardening characteristics, it is distinct from the usual hardening and softening response curves of the upper half-plane, due to the fact that the stabilit in the two nonconstant branches switches via the constant amplitude branch. This switching can be further reconciled b considering the phase response plots shown in Figs. 8 and 14, which show the transition from one phase to another as the constant amplitude branch is traversed. The responses shown in Figs. 9 and 11, corresponding to the regions labeled Va and Vb in Fig. 2, are also quite interesting. Again, the response features four distinct local bifurcations, two of the pitchfork variet

12 88 ARTICLE IN PRESS J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) C M J K L ψ (rad) Fig. 8. Representative frequenc response plot showing phase angle versus frequenc detuning for Region IV (g 3 ¼ :5, l 3 ¼ :5) in Fig F N Q D 5 O P 4 a Fig. 9. Representative frequenc response plot showing response amplitude versus frequenc detuning for Region Va (g 3 ¼ :5, l 3 ¼ :1) in Fig. 2. Note the response s mied nonlinear characteristics. where the nontrivial solutions branch off from zero, and two corresponding to the creation and annihilation of the constant amplitude solution, again connecting the other two nontrivial branches. The net result is a globall unbounded sstem (that is, sufficientl large initial energies will lead to unbounded responses) with nontrivial response branches which bend towards one another and are both initiall stable. Both of these branches lose stabilit via bifurcations where the connect with the constant amplitude branch. It should also be noted that a global bifurcation, in particular, a saddle connection, occurs within these two regimes as well.

13 J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) F N Q D O P ψ (rad) Fig. 1. Representative frequenc response plot showing phase angle versus frequenc detuning for Region Va (g 3 ¼ :5, l 3 ¼ :1) in Fig I R U G 5 S T 4 a Fig. 11. Representative frequenc response plot showing response amplitude versus frequenc detuning for Region Vb (g 3 ¼ :5, l 3 ¼ :1) in Fig. 2. Note the response s mied nonlinear characteristics. This can be verified geometricall b considering the phase portraits presented in Fig. 15 below, which clearl show the connection of the manifolds originating from the saddle at the origin with the manifolds originating from the saddles corresponding to the constant amplitude solutions. This can also be confirmed analticall b performing a simple invariant manifold calculation. In particular, it is observed that when the saddle connections eist, the are linear manifolds, and the attendant and dnamics of the sstem at the specified

14 81 ARTICLE IN PRESS J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) I R U G S T ψ (rad) Fig. 12. Representative frequenc response plot showing phase angle versus frequenc detuning for Region Vb (g 3 ¼ :5, l 3 ¼ :1) in Fig A B C V J a Fig. 13. Representative frequenc response plot showing response amplitude versus frequenc detuning for Region VI (g 3 ¼ :5, l 3 ¼ :5) in Fig. 2. Note the response s quasi-hardening nonlinear characteristics. bifurcation point can be related according to ¼ a (44) and ¼ a, (45)

15 J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) A B C V J ψ (rad) Fig. 14. Representative frequenc response plot showing phase angle versus frequenc detuning for Region VI (g 3 ¼ :5, l 3 ¼ :5) in Fig (a) (b) Fig. 15. Representative phase planes corresponding to the frequenc (detuning) boundaries (a) OP and (b) ST where the saddle connection global bifurcations occur. where a is a to be determined constant that represents the slopes of the saddle-connection invariant manifolds. Substituting these relationships into the undamped versions of Eqs. (26) and (27) and equating like powers of (or ) ields both a constraint on the magnitude of a and a frequenc, or more accuratel, a detuning condition, at which the bifurcation occurs. In particular, it can be shown that the global bifurcation occurs for s ¼ 3g 3l 1 þ 4l 3 n 1, (46) 4l 3 which interestingl enough also corresponds to the frequenc value at which the two stable nontrivial branches intersect in the specified amplitude versus frequenc response plots (for Regions Va and Vb). It should be noted that the phase dnamics within these regions follow the general trends outlined above for Regions IV and VI.

16 812 ARTICLE IN PRESS J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) (a) (b) (c) (d) (e) (f) Fig. 16. Representative phase planes corresponding to (a) Region A, (b) Region B, (c) Region C, (d) Region D, (e) Region E, and (f) Region F in Figs To provide additional insight into the bifurcations described above, representative phase planes corresponding to each of the frequenc response regimes delineated in Figs are presented. These plots, corresponding to the conservative sstem eplored in this section, are shown in

17 J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) (a) (b) (c) (d) (e) (f) Fig. 17. Representative phase planes corresponding to (a) Region G, (b) Region H, (c) Region I, (d) Region J, (e) Region K, and (f) Region L in Figs Figs Note that the capital letters designate topologicall distinct phase portraits, and that these are associated with the similarl labeled frequenc ranges in the frequenc response plots shown in Figs

18 814 ARTICLE IN PRESS J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) (a) (b) (c) (d) (e) (f) Fig. 18. Representative phase planes corresponding to (a) Region M, (b) Region N, (c) Region O, (d) Region P, (e) Region Q, and (f) Region R in Figs While the results presented above are for a special case of the equation of motion in question, it should be noted, as previousl mentioned, that the results are easil etendible for arbitrar n 1 and l 1. In particular, other values of these parameters tpicall result in two notable differences. First, selecting different values for

19 J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) (a) (b) (c) (d) Fig. 19. Representative phase planes corresponding to (a) Region S, (b) Region T, (c) Region U, and (d) Region V in Figs n 1 and l 1 results in a distortion of the Arnold tongue associated with the stabilit of the solution s trivial solution. This is easil reconciled through consideration of Eqs. (34) and (35). In addition, specifing the value of l 1 to be negative results in a smmetric reflection of the g 3 l 3 parameter space about the l 3 ¼ ais. This can be reconciled b noting that the eistence of a 3 depends on the sign of the ratio l 1 =l 3, and thus for l 1 o the constant amplitude solutions will appear in the upper half-plane of the parameter space instead of the lower half-plane. It should also be noted that, in addition to the phenomena described above, the sstem ehibits a number of additional interesting features. First, note that in the absence of damping the phase portraits show a distinct smmetr. The breaking of this smmetr is considered in the following section, where the effects of small sstem damping on the sstem response are considered. Also, and related to the smmetr, over certain frequenc (detuning) ranges, an invariant circle eists, given b 2 þ 2 ¼ 2l 1. (47) l 3 Note that the radius of this circle corresponds to the magnitude of the constant amplitude branches and, therefore, these solutions lie on this invariant circle. Furthermore, these solutions are connected to one another via invariant manifolds consisting of circular arcs. These analtical results, and the graphical representations epressed in terms of frequenc response curves and phase portraits, give a complete picture of the response of the sstem for zero damping. Small damping will destro the smmetr that permits such a thorough analsis, but the resulting responses can be inferred b

20 816 J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) straightforward etrapolations of the undamped results. Steps in this direction, including results from numerical analsis of the damped equations of motion, are considered in the net section. 6. The effects of damping Thus far, our analsis of Eq. (1) has considered the case of zero damping (z ¼ ). To gain a better understanding of the full oscillator dnamics, we now present numerical bifurcation results for the sstem with nonzero damping and with the detuning (s) treated as a bifurcation parameter. These results are obtained using AUTO [23], which requires that we rewrite the non-autonomous Eq. (1) as a sstem of autonomous first-order equations. Rewriting this as first-order equations is accomplished b defining v ¼ z and considering equations for z and v. We make this sstem autonomous b augmenting Eq. (1) with equations for a nonlinear oscillator with a stable periodic orbit with one variable equal to cos Ot, and replacing the appropriate terms of Eq. (1) with this variable. Most of our attention will be on Region IV from Fig. 2, as this displas the most interesting bifurcation behavior. We will also present tpical results for Regions IIb, III, and Va; Regions I, IIa, Vb, and VI are related to the presented regions b smmetr. As a tpical point in Region IV, we take l 1 ¼ n 1 ¼ 1 and g 3 ¼ l 3 ¼ :5, as in Fig. 7. Throughout this section, we take ¼ :1. The no-motion state (z ¼ z ¼ ) eists for all values of z and s, and undergoes bifurcations giving rise to branches of periodic orbits which ma be followed with AUTO. These periodic orbits displa response at half the driving frequenc O, and have the smmetr propert ðz; z Þðt þ T=2Þ ¼ð z; z ÞðtÞ, (48) where T ¼ 2p=O is the period of the response. Alread for small damping we find the interesting result that the constant amplitude branch shown in Fig. 7 splits such that the branches with the distinct phases now have slightl different amplitudes; see Fig. 2 for results with z ¼ :1. Here the stable periodic orbit branch which bifurcates from the trivial solution at s 1:5 undergoes a saddle-node bifurcation at s 1 before returning to the trivial state in the bifurcation at s :5. The branch containing the stable periodic orbit for s ¼ 1 undergoes a saddle-node bifurcation at s, and is disconnected from the no-motion state. Periodic orbits on this disconnected branch also displa the smmetr given in Eq. (48) z ma Fig. 2. Bifurcation diagram showing the maimum value of the displacement (z ma ) for z ¼ :1 and other parameters as described in the tet. In this and other bifurcation diagrams, diamonds, squares, and triangles indicate crossing the boundar of an Arnold tongue, period-doubling, and smmetr-breaking pitchfork bifurcations, respectivel. Furthermore, solid and dashed lines represent stable and unstable solutions, respectivel.

21 J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) z ma Fig. 21. Bifurcation diagram for z ¼ :1 and other parameters corresponding to Region IV, as described in the tet. For larger damping, we find additional bifurcations on the disconnected branch; see Fig. 21 for z ¼ :1 (a ver large value for a MEM resonator). A periodic orbit with the smmetr of Eq. (48) is stable from :57oso 3:739; a tpical orbit is shown in Fig. 22(a). At the upper limit it undergoes a saddle-node bifurcation, while at the lower limit it undergoes a smmetr-breaking pitchfork bifurcation. The smmetrbreaking bifurcation leads to two smmetr-related asmmetric periodic orbits, which lack the smmetr given in Eq. (48); see Fig. 22(b) for an eample of one of these (the other is obtained b letting ðz; z Þ!ð z; z Þ). Each of these smmetr-related asmmetric periodic orbits undergoes a period-doubling bifurcation at s ¼ 3:884; one of the period-doubled orbits is shown in Fig. 22(c). This is the beginning of a period-doubling cascade to chaos as s decreases, with chaotic attractor as shown in Fig. 22(d). This is one of two smmetr-related chaotic attractors (the other is obtained b letting ðz; z Þ!ð z; z Þ). We note that the basin of attraction for this chaotic attractor is quite small, making it somewhat difficult to locate (and likel unobservable in phsical eperiments). We also note that from [24], it is necessar for a smmetric periodic orbit such as that shown in Fig. 22(a) to first undergo a smmetr-breaking bifurcation before a perioddoubling cascade can occur. Fig. 23 shows results from a two-parameter (s and z) bifurcation stud for the Region IV parameters used above. The no-motion state is unstable inside the dashed curve. The width of this instabilit region decreases as the damping z increases, consistent with the effect of damping for the related Mathieu equation [21]. The dot-dashed lines show bifurcation sets for the saddle-node bifurcations of the smmetric periodic orbits. The crossing of these at ðs; zþ ð :3; :5Þ does not correspond to a codimension-two bifurcation; the two saddlenode bifurcations are just (independentl) occurring at the same parameter values. One dash-dotted curve (corresponding to a saddle-node bifurcation of the periodic orbit branch which bifurcates from the no-motion branch) terminates on the dashed curve at ðs; zþ ð:796; :222Þ. Here the periodic orbit branch emerging from the no-motion branch bifurcates verticall. For smaller z values, there is an unstable periodic orbit branch bifurcating to smaller s values (which then turns around in a saddle-node bifurcation), while for larger z values there is a stable branch which bifurcates to larger s values. We now consider the point in Region Va with l 1 ¼ n 1 ¼ 1 and g 3 ¼ :5; l 3 ¼ :1, as in Fig. 9. For small z we similarl find that the constant amplitude solution branch shown in Fig. 9 splits, giving a periodic orbit branch which is disconnected from the no-motion state; see Fig. 24 for z ¼ :1. This disconnected branch persists for larger z; see Fig. 25 for z ¼ :1, for which we do not find an further bifurcations. Note that we obtained unstable periodic orbits on the disconnected branch using a shooting method, and these were then followed with AUTO.

22 818 J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) z z -3-4 z (a) z τ z z z (b) z τ z z z -3-4 (c) z τ z z (d) z z Fig. 22. Stable solutions for parameters as in Fig. 21 and with forcing cos Ot: (a) smmetric periodic orbit for s ¼ 1:5; (b) asmmetric periodic orbit for s ¼ 3:85; (c) period doubled orbit for s ¼ 3:895; (d) chaotic solution for s ¼ 3:9235, with right panel zoomed in. The timeseries in (a), (b), and (c) show one period.

23 J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) ζ Fig. 23. Bifurcation sets for the parameters corresponding to Region IV. The no-motion state is unstable inside the dashed curve, while the dot-dashed lines show bifurcation sets for the saddle-node bifurcations of the smmetric periodic orbits. z ma Fig. 24. Bifurcation diagram for z ¼ :1 and other parameters corresponding to Region Va, as described in the tet. A similar period-doubling cascade to chaos, as found above for Region IV, is present for z ¼ :1 for the point in Region III with l 1 ¼ n 1 ¼ 1 and g 3 ¼ :5; l 3 ¼ :5, as in Fig. 6; see Fig. 26. In this region, there are no constant amplitude solutions for z ¼, as there were for Region IV. We close this section b noting that no interesting bifurcation behavior is found for z ¼ :1 for the point in Region IIb with l 1 ¼ n 1 ¼ 1andg 3 ¼ :5; l 3 ¼ :1, as in Fig. 5.

24 82 ARTICLE IN PRESS J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) z ma Fig. 25. Bifurcation diagram for z ¼ :1 and other parameters corresponding to Region Va, as described in the tet z ma Fig. 26. Bifurcation diagram for z ¼ :1 and other parameters corresponding to Region III, as described in the tet. 7. Application to MEM resonators The results developed in the previous sections are directl applicable to the equations of motion for parametricall ecited MEM oscillators of the tpe developed in Section 2. However, in this case, the coefficients in the equation of motion that are related to the electrostatic forces depend on the amplitude of the AC voltage, V A. Accordingl, the qualitative nature of both the linear and nonlinear behavior of the oscillator depend on V A. The effect of varing V A on the linear behavior of the sstem is easil noted b considering the undamped MEM oscillator s linear stabilit chart, as presented in Fig. 27. As clearl shown, as V A is increased, the width (in terms of detuning) of the linear instabilit zone increases. As noted in Section 4, this is largel consistent

25 J.F. Rhoads et al. / Journal of Sound and Vibration 296 (26) Stable Unstable Stable 25 V A (V) f (Hz) Fig. 27. Linear stabilit chart in the AC ecitation voltage amplitude (V A ) versus ecitation frequenc (f) parameter space for an undamped MEM oscillator with sstem parameters as specified in Table 2. Eperimentall derived instabilit points are also shown. Note that in each case the noted stabilit corresponds to the trivial solution. with the behavior of a tpical Mathieu sstem, but with two notable eceptions. First, due to the nonlinear relationship between l 1 and V A the boundaries of the instabilit feature a distinct curvature. In addition, due the presence of the frequenc shift caused b the electrostatic stiffness term (l 1 ), the instabilit region is distorted from its nominall smmetric configuration. Each of these differences is eamined etensivel in Ref. [1]. From a practical point of view, it is also important to note that in the presence of damping, the bottom of the instabilit wedge in Fig. 27 pulls up from zero voltage and becomes rounded off, as shown eperimentall in both Fig. 27 and Ref. [15]. The effect of V A on the nonlinear behavior of the sstem can be realized b noting that as the ecitation amplitude is varied, the sstem can transition from one response regime to another (in Fig. 2). This can be confirmed b eamining the form of the nonlinearit in the g 3 l 3 parameter space. Taking the definitions of l 3 and g 3 from Table 1 it is seen that at zero voltage the sstem starts at ðg 3 ; l 3 Þ¼ðw; Þ (the point corresponding to the purel mechanical hardening nonlinearit), which lies on the boundar between Regions I and VI, and moves along a straight line in the g 3 l 3 parameter space given b l 3 ¼ g 3 w (49) as V A is increased [1,5]. To validate this, consider the oscillator design described b the parameters in Table 2, which were obtained, as subsequentl described, from a device similar to that shown in Fig. 1. AsFig. 28 shows, the nonlinearit for this device will move through regions VI, Vb, Va, and IV as the amplitude of the AC voltage is varied. Thus, the sstem transitions from a quasi-hardening nonlinearit, to a mied nonlinearit, and finall to a quasi-softening nonlinearit as the oscillator s input voltage increases. Figs show response curves for the eperimental parameter values for three different values of V A.Note that the response indeed transitions from hardening, to mied, to softening, as predicted. With the aforementioned behavior in mind, a designer can account for these transitions and estimate whether or not the will occur in a given device s operating voltage range. The following relationships, derived b setting Z 1 and Z 2 equal to zero, provide epressions for the transition voltages, that is, the critical voltage values corresponding to qualitative changes in the sstem s nonlinearit. For a given oscillator, sffiffiffiffiffiffiffiffiffiffiffi 3k 3 V A;C1 ¼ (5) 5r 3A