Generalized Parametric Resonance in

Electrostatically-Actuated

Microelectromechanical Oscillators

Jeffrey F. Rhoads, Steven W. Shaw

Department of Mechanical Engineering, Michigan State University, East Lansing,MI 48824

Kimberly L. Turner, Jeff Moehlis,Barry E. DeMartini, and Wenhua Zhang

Department of Mechanical and Environmental Engineering, University ofCalifornia - Santa Barbara, Santa Barbara, CA 93106

Abstract

This paper investigates the dynamic response of a class of electrostatically-drivenmicroelectromechanical (MEM) oscillators. The particular systems of interest arethose which feature parametric excitation that arises from forces produced by fluc-tuating voltages applied across comb drives. These systems are known to exhibita wide range of behaviors, some of which have escaped explanation or prediction.In this paper we examine a general governing equation of motion for these systemsand use it to provide a complete description of the dynamic response and its de-pendence on the system parameters. The defining feature of this equation is thatboth the linear and cubic terms feature parametric excitation which, in compari-son to the case of purely linear parametric excitation (e.g. the Mathieu equation),significantly complicates the system’s dynamics. One consequence is that an ef-fective nonlinearity for the overall system cannot be defined. Instead, the systemfeatures separate effective nonlinearities for each branch of its nontrivial response.As such, it can exhibit not only hardening and softening nonlinearities, but alsomixed nonlinearities, wherein the response branches in the system’s frequency re-sponse bend toward or away from one another near resonance. This paper includessome brief background information on the equation of motion under consideration,an outline of the analytical techniques used to reach the aforementioned results,stability results for the responses in question, a numerical example, explored usingsimulation, of a MEM oscillator which features this nonlinear behavior, and prelim-inary experimental results, taken from an actual MEM device, which show evidenceof the analytically predicted behavior. Practical issues pertaining to the design ofparametrically-excited MEM devices are also considered.

Preprint submitted to Elsevier Science 15 April 2005

Key words: Effective Nonlinearities, Parametric Excitation, MEMS

1 Introduction

The recent analysis of parametrically-excited microelectromechanical (MEM)oscillators, by the authors and others, has led to the consideration of a certainclass of nonlinear equations of motion [1–3]. The defining feature of these equa-tions is that parametric excitation acts on both the linear and cubic terms. Assuch, the dynamics of these oscillators are quite complex in comparison to atypical Mathieu system, which features only linear parametric excitation. Oneinteresting result of this complication is that a single effective nonlinearity forthe overall system cannot be defined. Rather, the system features separateeffective nonlinearities for each branch of its nontrivial response. Accordingly,the system can exhibit not only the typical hardening and softening nonlin-earities, but also mixed nonlinearities wherein the response branches in thesystem’s frequency response bend toward or away from one another near res-onance. In fact, such behavior has been observed in experiments, as describedsubsequently.

The paper begins in Section 2 with a brief description of the types of MEMdevices of interest, and a derivation of their governing equation of motion.General analytical results for the undamped system, including informationon the existence, stability, and bifurcations of the steady-state solutions, aregiven in Sections 3 and 4. These results include detailed descriptions of theresulting frequency response curves and the associated phase planes. To facil-itate understanding of the aforementioned results, a representative numericalexample is explored in Section 5. In Section 6, the effects of light damping onthe system are considered. Section 7 explores the given results in the contextof a specific MEM oscillator, provides the pertinent results of preliminary ex-perimental investigations, and discusses practical design issues associated withparametrically-excited MEM devices. The paper concludes in Section 8 witha summary of the present work and some comments on possible directions offuture work in this area.

It should be noted that the bulk of this paper describes a detailed analysisof the governing equation of motion; this may be of interest to the nonlinearvibrations community, but of less direct interest to the MEMS community.Those readers not interested in the details of the analysis can focus on Sec-tion 2, which describes the MEMS model employed, and Section 7, whichpresents the general response features of parametrically-excited MEMS, in-cluding an approach for evaluating and predicting the response features ofthese devices.

2

the oscillator is no longer restricted to the frequency of the

externally applied force. In particular, there are excitations of

the natural frequency, when excited at certain integer fractions

or multiples of the resonant frequency. There is plenty of

literature on such systems modeled by Hill’s equation and

variations including the introductory book by Cartmell [13].

In the case of harmonic oscillators with time-modulated

stiffness, a sharp transition between zero response and a

large auto-parametric response (sub-harmonic resonance)

exists [14]. Since this transition is dependent on system

parameters, including the mass of the vibrating oscillators,

change in mass can be detected with such a system. In this

mass sensor implementation, the minimum detectable mass

change can be expressed as [5]:

dm ¼ k

4p2

1

f 30

df0 (1a)

where k is the system stiffness and f0 the natural frequency.

The sensitivity of a simple harmonic resonator based mass

sensor, such as a cantilever sensor, can be represented as

[15–17]:

dm ¼ k

4p2

1

f 020 1

f 20

¼ k

2p2

1

f 30

df0 (1b)

The sensitivity of these two cases is of the same order if

the smallest resolvable frequency shift (df0) due to mass

change is the same. However, the sensitivity of a normal

cantilever mass sensor is strongly dependent on pressure.

The minimum detectable frequency change is inversely

proportional to the quality factor [18]:

Df ¼ 1

A

f0kBTB

kQ(2)

where A is the amplitude of the oscillation, B the bandwidth,

Q the quality factor, T the temperature and k the stiffness of

the oscillator. It should be noted that damping cannot be

avoided in micro or nano scale [19]. In the case of a

parametrically driven oscillator, the sensitivity depends on

the transition between zero and large response and the

transition can be very sharp. At 7 mTorr, for the case of a

parametric torsional mode MEMS oscillator, the transition

was observed with an input frequency shift of 0.001 Hz [4],

which is the limit of the hardware used (Function generator

HP3245A). Because of the sharp transition, with the same

configuration, parametrically driven mass sensor can be two

orders of magnitude more sensitive than harmonic resonator

based mass sensor. In a cantilever mass senor with dimen-

sions as 22:37 mm 2 mm 0:5 mm, the theoretical sensi-

tivity can be 9:65e 17g [20]. If working in the parametric

mode, it can resolve as small a mass change as 3:62e 19 g.

The sharp transition is a reflection of a sub-harmonic

response pitchfork bifurcation in the driving voltage fre-

quency–amplitude parameter space. The occurrence of this

bifurcation will be shown to be independent of the ambient

pressure (modeled as viscous damping in the dynamics). We

have observed this sharp transition at 450 mTorr and higher.

Theoretically, good sensitivity can be achieved even at

atmospheric pressure.

In this paper, we describe the motivation for the need to

study nonlinear effects and present an introduction to the

bulk-micro-machined parametrically actuated mass sensor.

We then present the development of its model in the

electromechanical domain and present an analytical treat-

ment of the resulting nonlinear Mathieu equations. This

analysis is supported with experimental results. The impli-

cations of these results on the behavior of the mass sensor are

discussed.

2. Device

In this section, we describe the electromechanical system

fabricated using the bulk-micro-machining technique

SCREAM [21]. The device we have studied is an oscillator,

which was designed by Adams et al. [22] for the independent

tuning of linear and cubic stiffness terms. A scanning

electron micrograph of the oscillator is shown in Fig. 1.

The device size is about 500 mm 400 mm. It has two sets of

parallel interdigitated comb finger banks on either end of the

backbone and two sets of non-interdigitated comb fingers on

each side. The four folded beams provide elastic recovery

force for the oscillator. The beams, backbone and the fingers

are 2 mm wide and 12 mm deep. The backbone is 515 mm

long and 20 mm wide. Each of the four recovery folded

beams are 200 mm on the long side and 20 mm on the short

side. Either the interdigitated or the non-interdigitated comb

fingers may be used to drive the oscillator. Fig. 2 is a

schematic of these comb fingers. This oscillator is used to

Fig. 1. A scanning electron micrograph of the oscillator. Note the folded

beam springs (S), the two sets of interdigitated comb finger banks (C) on

both ends of backbone (B) and non-interdigitated comb fingers (N) on each

side of backbone (B).

140 W. Zhang et al. / Sensors and Actuators A 102 (2002) 139–150

Fig. 1. A representative parametrically-excited MEM oscillator. The oscillator’sbackbone is labeled B, S designates the mechanical beam springs, and N desig-nates the non-interdigitated comb drives, which are used to provide the parametricexcitation. Note that the interdigitated comb drives, labeled C, are used here onlyfor sensing. (From [2, 3])

2 Parametrically-Excited Microelectromechanical Oscillators

The impetus of the present study was the analysis of single degree of free-dom parametrically-excited microelectromechanical oscillators, such as the oneshown in Fig. 1. Such oscillators consist of a shuttle mass, namely the oscil-lator’s backbone, B, connected to a substrate via four folded beam springs,S, and excited by a pair of non-interdigitated electrostatic comb drives, N,which are powered by a voltage source. These devices are being proposed foruse in a number of sensing and filtering applications [1, 3, 4]. Previous works,including those by the authors, have verified that these parametrically-excitedoscillators exhibit motion consistent with a rather simple-looking equation ofmotion, presented in Eq. (10) below [1, 3, 5, 6]. This is based on the followingdevelopment.

To begin, the oscillator’s motion is assumed to be described by the movementof the shuttle mass in one direction. The forces acting on the system includethose from the springs, the electrostatic forces from the comb drive, and a forcethat represents dissipative effects, which arise primarily from aerodynamiceffects. These dissipative effects are generally small, especially when thesedevices are operated in a vacuum, and are typically modeled by a simpleviscous damping force, although more complicated models have been proposed

3

(see, for example, [7–10]). The resulting equation of motion is given by

mx+ cx+ Fr(x) + Fes(x, t) = 0, (1)

where Fr(x) represents the elastic restoring force provided by the springsand Fes(x, t) represents the electrostatic restoring and driving forces providedby the non-interdigitated electrostatic comb drives. As previous works haveshown, the aforementioned forces can be accurately represented by cubic func-tions of displacement [1, 3]. The restoring force from the mechanical springs ismodeled by,

Fr(x) = k1x+ k3x3 (2)

where k1 and k3 are the effective linear and nonlinear spring coefficients fromthe beams. The electrostatic forces arising from the comb drives can accuratelybe modeled for small displacements as,

Fes(x, t) =(r1Ax+ r3Ax

3)V 2(t), (3)

where r1A and r3A are electrostatic coefficient that depend on the physicaldimensions of the electrostatic comb drives, and V (t) is the voltage appliedacross the drives. In order to provide harmonic excitation to the device, anAC voltage of the form

V (t) = VA

√1 + cosωt (4)

is applied across the comb drives [1, 3, 6].

Substituting these forces back into Eq. (1) results in an equation of motionfor the oscillator’s backbone of the form [1, 6]:

mx+ cx+ k1x+ k3x3 +

(r1Ax+ r3Ax

3)VA

2 (1 + cosωt) = 0. (5)

Time and displacement in Eq. (5) are rescaled according to

τ = ω0t, (6)

where ω0 is the purely elastic natural frequency defined as,

4

ω0 =

√k1

m, (7)

and

ε1/2z =x

x0

, (8)

where x0 is a characteristic length of the system (e.g. the length of the back-bone) and ε is a scaling parameter introduced for the analysis. This yields anondimensional equation of motion of the form

z′′ + 2εζz′ + z (1 + ελ1 + ελ1 cos Ωτ) + εz3 (χ+ λ3 + λ3 cos Ωτ) = 0, (9)

with the new derivative operator and nondimensional parameters defined asin Table 1 [1, 6]. Note that the inclusion of the “small” scaling parameter εphysically equates to the assumption of small damping and small parametricexcitation; such assumptions are entirely consistent with the operation of suchoscillators, especially near resonance [1].

The equation of motion of considered in this work is a generalization of Eq. (9),wherein the effective stiffness and excitation coefficients (the λ’s) are allowedto differ, in both the linear and nonlinear terms. This arises naturally inthe analysis of parametrically-excited MEM oscillators with two sets of non-interdigitated comb drives, one of which provides an AC voltage, as describedabove, while the other provides a DC “bias” to the system. This “bias” doesnot offset the system equilibrium in the usual sense, but allows one to inde-pendently vary the effective electrostatic stiffness and excitation coefficients,resulting in Eq. (10) given below. This arrangement of comb drives allows formore flexibility in tuning the response of such devices, so that designers canachieve certain desirable response features [1, 5, 6]. More generally, this govern-ing equation of motion is representative of a number of parametrically-exciteddynamical systems, as highlighted in [11–13].

While works such as [1, 3, 5, 6], examine the nonlinear dynamics of these parametrically-excited MEM oscillators, their nonlinear response has not been fully explored,and a number of apparent anomalies have been observed in experimental in-vestigations of these devices [2, 3, 14]. The analysis in the following sectionsprovides a description for the entire range of possible responses for these sys-tems (under the small damping and small response amplitude assumptions).This is followed by a specific example using the parameters from a devicesimilar to that shown in Fig. 1.

5

Table 1Nondimensional Parameter Definitions [1, 6]

Definition Nondimensional Parameter

(•)′ =d(•)dτ

Scaled Time Derivative

εζ =c

2mω0Scaled Damping Ratio

ελ1 =r1AVA

2

k1Linear Electrostatic Excitation Amplitude

Ω =ω

ω0Nondimensional Excitation Frequency

χ =k3x0

2

k1Nonlinear Mechanical Stiffness Coefficient

λ3 =x0

2r3AVA2

k1Nonlinear Electrostatic Excitation Coefficient

One of the more interesting features of the equation of motion presented inEq. (9) is that several of the coefficients of the equation, including the ef-fective linear and nonlinear stiffnesses, depend on the amplitude of the ACvoltage, VA. This results in the fact that the qualitative nature of the sys-tem’s response generally depends on the amplitude of the excitation, and maydiffer for different input amplitudes. The analysis presented here describes acomplete picture of this behavior, and provides designers of these devices withuseful predictive tools. The example in Section 7 describes an example of thisprocess and includes preliminary experimental results that verify the utilityof the approach.

3 The Equation of Motion and Perturbation Analysis

The equation of motion of interest here is of the form

z′′ + 2εζz′ + z (1 + εν1 + ελ1 cos Ωτ) + εz3 (γ3 + λ3 cos Ωτ) = 0, (10)

where ε is a scaling parameter introduced for analysis and prime designates thederivative with respect to τ [1, 5, 6, 15, 16]. Note that this is a special case ofgeneralized parametric excitation in that there is not a phase difference in the

6

excitation seen by the linear and nonlinear terms. However, in many systemsthe excitation arises from a single source, resulting in zero phase difference(see, for example, [11–13, 17, 18]). This is the case for the present study, andthis equation proves sufficient for capturing the dynamic response of the MEMdevices under consideration.

The method of averaging is employed to analyze Eq. (10). To aid this, astandard coordinate transformation is introduced to convert the equation toamplitude and phase coordinates, namely,

z(τ) = a(τ) cos(

Ωτ

2+ ψ(τ)

), (11)

z′(τ) = −a(τ)Ω2

sin(

Ωτ

2+ ψ(τ)

). (12)

Note that this conversion requires that a certain constraint equation involvinga′ and ψ′ be satisfied, just as in the method of variation of parameters; see, forexample, [19]. In order to capture the dynamics near the principal parametricresonance, an excitation frequency detuning parameter σ is introduced, definedby,

Ω = 2 + εσ, (13)

where σ measures the closeness of the excitation frequency to the principalparametric resonance condition. Averaging the resulting equations, now interms of amplitude and phase coordinates, over the period 4π/Ω in the τ -domain produces the system’s averaged equations [1, 5, 6]. These are given by

a′ =1

8aε[−8ζ +

(2λ1 + a2λ3

)sin 2ψ

]+O

(ε2), (14)

ψ′ =1

8ε[3a2γ3 + 4ν1 − 4σ + 2

(λ1 + a2λ3

)cos 2ψ

]+O

(ε2). (15)

Note that the presence of parametric excitation in the cubic term, capturedby the parameter λ3, introduces additional coupling in these equations that isnot present in the case of purely linear parametric excitation. Consequently,the nontrivial steady-state solutions of the system take on a more complicatedform and the system has much more interesting response characteristics.

7

4 Steady-State Responses, Their Stability, and Their Effective Non-linearities

Since the characteristic form of the system nonlinearity (i.e., hardening orsoftening) is a critical feature of the response for current purposes, and thisis largely unaffected by damping, zero damping (ζ = 0) is assumed to sim-plify the analysis. We will consider the effects of small damping subsequently.Steady-state responses are captured by setting (a′, ψ′) = (0, 0). Examining theresulting equations reveals that the system has a trivial solution and three dis-tinct sets of nontrivial solution branches. The first two nontrivial sets appearin pairs with amplitudes and associated phases given by

a1 = ±√

4σ + 2λ1 − 4ν1

3γ3 − 2λ3

, (16)

ψ1 =π

2, (17)

and

a2 = ±√

4σ − 2λ1 − 4ν1

3γ3 + 2λ3

, (18)

ψ2 = 0. (19)

Note that each ± pair represents the same physical response; they appear thisway due to the phase relation of this subharmonic response relative to theexcitation. Likewise, all responses with magnitude π phase shifts represent thesame physical response, and thus are not included. For each of these solutions,the signs of the terms under the square root determine the frequency rangeover which these branches are real valued, and thus physically meaningful.The numerators of these terms are dictated by linear frequency terms, andthe denominators are set by nonlinear terms. Of direct interest here is therole played by the nonlinearities, which differ for each of these frequency-dependent branches. To describe these effects, effective nonlinearities η1 andη2 are defined that dictate the behavior of these branches. These nonlinearitiesare taken from the denominator of the expressions above and are given by [1, 5]

η1 = 3γ3 − 2λ3 (20)

and

8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

1.5

γ3

λ 3I

Vb

IV

IIb

x 10-1

x 10-1

IIa

III

Va

VI

η 1< 0

η2 > 0

η 1> 0

η2 < 0 η 1

> 0

η 1< 0

η2 > 0

η2 < 0

Fig. 2. The γ3 - λ3 parameter space [1]. Those regions designated I and VI ex-hibit hardening (or quasi-hardening) nonlinear characteristics, those designated IIIand IV exhibit softening (or quasi-softening) nonlinear characteristics, and thosedesignated IIa, IIb, Va, and Vb exhibit mixed nonlinear characteristics.

η2 = 3γ3 + 2λ3. (21)

The case η1 < 0 and η2 < 0 results in the usual softening nonlinearity. Likewise,the usual hardening nonlinearity occurs for η1 > 0 and η2 > 0. However, dueto the nature of this equation of motion, two mixed cases also exist, namely,η1 > 0 and η2 < 0, and η1 < 0 and η2 > 0, which correspond to the twobranches delineated in Eqs. (16) and (18) bending toward or away from oneanother, as determined by λ1 and ν1. This result is summarized in Fig. 2,which delineates the various response regions within the γ3 - λ3 parameterspace. Note that a transition between response types occurs when one orboth of the effective nonlinearities equals zero, in which case the bifurcationassociated with crossing the boundary of the parametric instability (given byan Arnold tongue) becomes degenerate. Accordingly, phenomena near thesedegeneracies cannot be captured without the inclusion of high-order nonlinearterms, 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, by

a3 = ±√−2λ1

λ3

, (22)

9

ψ3 = ±1

2arccos

(−3γ3λ1 + 2λ3ν1 − 2λ3σ

λ1λ3

). (23)

Note that there will be four such solutions, only two of which will be physi-cally distinct (as evident in the next section), corresponding to each branchof the respective inverse cosine function. Accordingly, these solutions will ap-pear in either the upper or lower half-plane of the parameter space delineatedin Fig. 2, as dictated by the sign of λ1. As such, the system undergoes abifurcation across the λ3 = 0 axis in Fig. 2, which corresponds to the cre-ation or annihilation of the constant amplitude solution. It should be notedhowever, that this creation (or annihilation) is a “smooth” process, as thesolution appears from positive or negative infinity due to the presence of λ3

in the solution’s denominator. As will be shown shortly, the presence of theseadditional solution branches significantly complicates the system’s dynamicswithin certain parameter regimes, especially when viewed from a frequencyresponse perspective. Before turning to the generation of frequency responsecurves, the stability of the various response branches is first determined.

Though the local stability of steady-state responses is generally addressedby considering the local linear behavior of the averaged equations near thesteady-state responses in the (a, ψ) polar coordinate space, singularities at theorigin require that a coordinate change be introduced. As such, the followingcoordinate change is evoked, which converts the polar coordinates (a, ψ) intoCartesian coordinates (x, y):

x(τ) = a(τ) cos[ψ(τ)], (24)

y(τ) = a(τ) sin[ψ(τ)]. (25)

This coordinate change results in averaged equations of the form

x′ =1

8ε[(2λ1 − 4ν1 + 4σ) y − 3γ3x

2y + (−3γ3 + 2λ3) y3 − 8ζx

]+O

(ε2), (26)

y′ =1

8ε[(2λ1 + 4ν1 − 4σ)x+ 3γ3xy

2 + (3γ3 + 2λ3)x3 − 8ζy

]+O

(ε2). (27)

Using these equations, with ζ = 0, local stability results are inferred by con-sidering the response of the system when linearized about the steady states.

10

Letting the state vector be

X(τ) =

x(τ)y(τ)

(28)

and the steady-state values be given by

X∗ =

x∗y∗

, (29)

the local linearized equation of motion for the system can be written as

Y ′(τ) = J |X∗Y (τ), (30)

where

Y (τ) = X(τ)−X∗ (31)

and J is the Jacobian matrix of the averaged equations, Eqs. (14) and (15),evaluated at steady state, X∗. The stability of the steady-state response isgoverned by the eigenvalues of the corresponding Jacobian.

To simplify the analysis, we will consider the stability in terms of the traceand determinant of the 2× 2 Jacobian [20]. In particular, the eigenvalues canbe expressed in terms of the trace, T , and the determinant, ∆, as

Λ1,2 =1

2

(T ±

√T 2 − 4∆

). (32)

For the undamped case under consideration here, the trace of the Jacobian foreach of the steady-state responses is zero, thus only two generic equilibriumtypes are possible for nontrivial eigenvalues: saddles (unstable) and centers(marginally stable). Which of these equilibrium types exists depends solelyon the sign of the determinant. In particular, when ∆ > 0 the equilibrium isa center, and when ∆ < 0 the equilibrium is a saddle. The remaining case,∆ = 0, corresponds to the degenerate case of two identically zero eigenvalues,and as such is used here only to identify where stability changes occur.

The trivial solution features a determinant of the form

11

∆0 = − 1

16ε2[λ1

2 − 4 (ν1 − σ)2]. (33)

By finding the roots of ∆0 = 0, the critical frequency values (detuning values)at which stability changes occur can be shown to be

σ1 = ν1 −λ1

2(34)

and

σ2 = ν1 +λ1

2, (35)

respectively. The stability in each frequency and detuning region is then deter-mined by evaluating the determinant at a single point within the domain or bycalculating the derivative of the determinant at the critical frequency value.This shows that the trivial solution appears as a center for σ < min(σ1, σ2)and σ > max(σ1, σ2), and as a saddle for min(σ1, σ2) < σ < max(σ1, σ2).These results are, of course, unaffected by the nonlinear excitation term andare closely related to the results obtained for a standard Mathieu equation.The distinct difference here, however, is that “wedge of instability” whichappears in the λ1 - Ω parameter space is generally distorted away from thetypical configuration seen in [21] and [22], for example, due to the presence ofthe frequency-shifting ν1 term (this is considered extensively in [1]).

Using similar techniques, the constant amplitude solutions of Eq. (22) canbe shown to be saddles for all frequency values where the solutions exist,as determined by considering parameter values where their phases are real-valued. It is found that these solutions exist between the detuning values

σ =−3λ1γ3 − λ1λ3 + 2λ3ν1

2λ3

(36)

and

σ =−3λ1γ3 + λ1λ3 + 2λ3ν1

2λ3

. (37)

Note that each of these values may serve as an upper or lower bound, depend-ing on the sign of λ1.

The remaining two branches are slightly more complicated in that their sta-bility is heavily dependent on which region of the γ3 - λ3 parameter space

12

(see Fig. 2) the system falls within. Regardless, the stability for each of thebranches can be found by evoking the fact that the response branch presentedin Eq. (16) has a determinant given by

∆ =ε2 (λ1 − 2ν1 + 2σ) (3λ1γ3 − λ1λ3 − 2λ3ν1 + 2λ3σ)

12γ3 − 8λ3

, (38)

which features critical frequency values of

σ = ν1 −λ1

2, (39)

corresponding to the bifurcation at which this branch emanates from the ori-gin, and

σ =−3λ1γ3 + λ1λ3 + 2λ3ν1

2λ3

, (40)

corresponding to the bifurcation which connects this branch to the constantamplitude branches. The response branch presented in Eq. (18) has a deter-minant given by

∆ =ε2 (λ1 + 2ν1 − 2σ) (3λ1γ3 + λ1λ3 − 2λ3ν1 + 2λ3σ)

12γ3 + 8λ3

, (41)

which features critical frequency values of

σ = ν1 +λ1

2, (42)

corresponding to the bifurcation at which this branch emanates from the ori-gin, and

σ =−3λ1γ3 − λ1λ3 + 2λ3ν1

2λ3

, (43)

corresponding to the bifurcation which connects this branch to the constantamplitude branches.

Combining these results with the steady-state solutions derived above yields acomplete picture of the oscillator’s possible frequency responses, as describedin detail in Section 5.

13

Response Amplitude vs. Detuning in Region I (γ3 = 0.005, λ3 = 0.005)

0

10

20

30

40

50

60

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

σ

aA B C

Fig. 3. Representative frequency response plot showing response amplitude versusdetuning frequency for Region I in Fig. 2. Note that here, as well as in Figs. 4-14,solid lines indicate stable response branches and dashed lines unstable responsebranches.

Response Amplitude vs. Detuning in Region IIa (γ3 = 0.005, λ3 = 0.010)

0

5

10

15

20

25

30

35

40

45

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

σ

a

D E F

Fig. 4. Representative frequency response plot showing response amplitude versusdetuning frequency for Region IIa in Fig. 2.

14

Response Amplitude vs. Detuning in Region IIb (γ3 = -0.005, λ3 = 0.010)

0

5

10

15

20

25

30

35

40

45

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

σ

a

G H I

Fig. 5. Representative frequency response plot showing response amplitude versusdetuning frequency for Region IIb in Fig. 2.

Response Amplitude vs. Detuning in Region III (γ3 = -0.005, λ3 = 0.005)

0

10

20

30

40

50

60

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

σ

a

J K L

Fig. 6. Representative frequency response plot showing response amplitude versusdetuning frequency for Region III in Fig. 2.

15

Response Amplitude vs. Detuning in Region IV (γ3 = -0.005, λ3 = -0.005)

0

5

10

15

20

25

30

35

40

45

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

σ

a

M K LC J

Fig. 7. Representative frequency response plot showing response amplitude versusdetuning frequency for Region IV in Fig. 2.

Phase Angle vs. Detuning in Region IV (γ3 = -0.005, λ3 = -0.005)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5

σ

ψ (r

ad)

M K LC J

Fig. 8. Representative frequency response plot showing phase angle versus detuningfrequency for Region IV in Fig. 2.

16

Response Amplitude vs. Detuning in Region Va (γ3 = -0.005, λ3 = -0.010)

0

10

20

30

40

50

60

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

σ

a

F Q DN

OP

Fig. 9. Representative frequency response plot showing response amplitude versusdetuning frequency for Region Va in Fig. 2.

Phase Angle vs. Detuning in Region Va (γ3 = -0.005, λ3 = -0.010)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5

σ

ψ (r

ad)

F Q DNO P

Fig. 10. Representative frequency response plot showing phase angle versus detuningfrequency for Region Va in Fig. 2.

17

Response Amplitude vs. Detuning in Region Vb (γ3 = 0.005, λ3 = -0.010)

0

10

20

30

40

50

60

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

σ

a

I R GU

ST

Fig. 11. Representative frequency response plot showing response amplitude versusdetuning frequency for Region Vb in Fig. 2.

Phase Angle vs. Detuning in Region Vb (γ3 = 0.005, λ3 = -0.010)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5

σ

ψ (r

ad)

I R GUS T

Fig. 12. Representative frequency response plot showing phase angle versus detuningfrequency for Region Vb in Fig. 2.

18

Response Amplitude vs. Detuning in Region VI (γ3 = 0.005, λ3 = -0.005)

0

5

10

15

20

25

30

35

40

45

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

σ

a

A B JVC

Fig. 13. Representative frequency response plot showing response amplitude versusdetuning frequency for Region VI in Fig. 2.

Phase Angle vs. Detuning in Region VI (γ3 = 0.005, λ3 = -0.005)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5

σ

ψ (r

ad)

A B JVC

Fig. 14. Representative frequency response plot showing phase angle versus detuningfrequency for Region VI in Fig. 2.

19

5 System Frequency Responses

To facilitate understanding of the analysis presented in Section 4, consider arepresentative single degree of freedom dynamical system modeled by Eq. (10).In particular, consider an undamped version (ζ = 0) with ν1 = λ1 = 1 (Theresults can easily be extended for other values of ν1 and λ1, both positive andnegative). By applying the results of the aforementioned section to this typicalcase, frequency response plots, such as those shown in Figs. 3-14, are easilygenerated for each of the regions depicted in Fig. 2. A brief explanation ofeach of these plots follows.

To begin, consider the amplitude-vs.-frequency response curves depicted inFigs. 3 and 6, corresponding to Regions I and III in Fig. 2, respectively. As aquick examination reveals, these regions appear consistent with normal soften-ing and hardening nonlinear behavior. That is, the nontrivial solutions branchoff in two distinct pitchfork bifurcations, one subcritical and the other su-percritical, and all solutions remain globally bounded (Note that the termglobally bounded is used here to designate the fact that all initial conditionsresult in bounded responses). Though not shown, the phase response will beequally mundane in that it remains constant across the branches, as dictatedby Eqs. (17) and (19). The frequency responses shown in Figs. 4 and 5, corre-sponding to the topologically equivalent regions labeled IIa and IIb in Fig. 2,are slightly more interesting 1 . Here the nontrivial solutions also branch off,but in each instance a subcritical bifurcation occurs. The net result is thatthe two response curves actually bend away from one another and some solu-tions are globally unbounded (specifically, all non-zero initial conditions leadto unbounded motions over the frequency range where the trivial solution isunstable, and where the trivial solution is stable, initial conditions with suf-ficiently large energies lead to unbounded responses). The phase responses ofthe solutions within these regions, much like those considered above, remaininvariant in frequency and thus graphic representations are omitted.

In the lower half-plane of the parameter space depicted in Fig. 2, the exis-tence of the additional nontrivial, constant amplitude, solutions complicatesmatters. First, consider Figs. 7 and 13, corresponding to Regions IV and VIin Fig. 2, respectively. Here the near zero amplitude responses resemble thoseof Figs. 3 and 6, respectively. However, as the frequency is swept further awayfrom zero detuning (σ = 0), the system undergoes two additional local bifur-cations, corresponding to the creation and annihilation of the constant ampli-

1 Note that these regions are given different designations only due to the fact thatthe nontrivial response branches to the left and right of the resonance have differentamplitudes, which are reversed in these two regions. This is not physically important,since both branches are unstable. However, in other regions, this distinction will haveconsequences in terms of the observed frequency response behavior.

20

tude response, which connects the other two nontrivial branches. Though thisstill results in a globally bounded system with quasi- softening or hardeningcharacteristics, it is distinct from the usual hardening and softening responsecurves of the upper half-plane, due to the fact that the stability in the two non-constant branches switches via the constant amplitude branch. This switchingcan be further reconciled by considering the phase response plots shown inFigs. 8 and 14, which show the transition from one phase to another as theconstant amplitude branch is traversed.

The responses shown in Figs. 9 and 11, corresponding to the regions labeledVa and Vb in Fig. 2, are also quite unique. Again, the response features fourdistinct local bifurcations, two of the pitchfork variety where the nontrivialsolutions branch off from zero, and two corresponding to the creation and an-nihilation of the constant amplitude solution, again connecting the other twonontrivial branches. The net result is a globally unbounded system (that is,sufficiently large initial energies will lead to unbounded responses) with non-trivial response branches which bend towards one another and are both ini-tially stable. Both of these branches lose stability via bifurcations where theyconnect with the constant amplitude branch. It should also be noted that aglobal bifurcation, in particular, a saddle connection, occurs within these tworegimes as well. This can be verified geometrically by considering the phaseportraits presented in Fig. 19 below, which clearly show the connection of themanifolds originating from the saddle at the origin with the manifolds orig-inating from the saddles corresponding to the constant amplitude solutions.This can also be confirmed analytically by performing a simple invariant man-ifold calculation. In particular, it is observed that when the saddle connectionsexist, they are linear manifolds, and the attendant x and y dynamics of thesystem at the specified bifurcation point can be related according to

y = αx (44)

and

y′ = αx′, (45)

where α is a to-be-determined constant that represents the slopes of the saddle-connection invariant manifolds. Substituting these relationships into the un-damped versions of Eqs. (26) and (27) and equating like powers of x (or y)yields both a constraint on the magnitude of α and a frequency, or more ac-curately, a detuning condition, at which the bifurcation occurs. In particular,it can be shown that the global bifurcation occurs for

21

σ =−3γ3λ1 + 4λ3ν1

4λ3

, (46)

which interestingly enough also corresponds to the frequency value at whichthe two stable nontrivial branches “intersect” in the specified amplitude-vs.-frequency response plots (for Regions Va and Vb). It should be noted that thephase dynamics within these regions follow the general trends outlined abovefor Regions IV and VI.

To provide additional insight into the bifurcations described above, represen-tative phase planes corresponding to each of the frequency response regimesdelineated in Figs. 3-14 are presented. These plots, corresponding to the con-servative system explored in this section, are shown in Figs. 15-19. Note thatthe capital letters designate topologically distinct phase portraits, and thatthese are associated with the similarly-labeled frequency ranges in the fre-quency response plots shown in Figs. 3-14.

While the results presented above are for a special case of the equation ofmotion in question, it should be noted, as previously mentioned, that theresults are easily extendible for arbitrary ν1 and λ1. In particular, other valuesof these parameters typically results in two notable differences. First, selectingdifferent values for ν1 and λ1 results in a distortion of the Arnold tongueassociated with the stability of the solution’s trivial solution. This is easilyreconciled through consideration of Eqs. (34) and (35). In addition, specifyingthe value of λ1 to be negative results in a symmetric reflection of the γ3 - λ3

parameter space about the λ3 = 0 axis. This can be reconciled by noting thatthe existence of a3 depends on the sign of the ratio λ1/λ3, and thus for λ1 < 0the constant amplitude solutions will appear in the upper half-plane of theparameter space instead of the lower half-plane.

It should also be noted that, in addition to the phenomena described above,the system exhibits a number of additional interesting features. First, notethat in the absence of damping the phase portraits show a distinct symmetry.The breaking of this symmetry is considered in the following section, wherethe effects of small system damping on the system response are considered.Also, and related to the symmetry, over certain frequency (detuning) ranges,an invariant circle exists, given by

x2 + y2 =−2λ1

λ3

. (47)

Note that the radius of this circle corresponds to the magnitude of the con-stant amplitude branches and, therefore, these solutions lie on this invariantcircle. Furthermore, these solutions are connected to one another via invariant

22

manifolds consisting of circular arcs.

These analytical results, and the graphical representations expressed in termsof frequency response curves and phase portraits, give a complete picture ofthe response of the system for zero damping. Small damping will destroy thesymmetry that permits such a thorough analysis, but the resulting responsescan be inferred by straightforward extrapolations of the undamped results.Steps in this direction, including results from numerical analysis of the dampedequations of motion, are considered in the next section.

23

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

yAA

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

BB

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

CC

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

yDD

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

EE

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

FF

Fig. 15. Representative phase planes corresponding to the regions labeled A-F inFigs. 3-14.

24

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

yGG

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

HH

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

II

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

yJJ

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

KK

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

LL

Fig. 16. Representative phase planes corresponding to the regions labeled G-L inFigs. 3-14.

25

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

yMM

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

NN

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

OO

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

yPP

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

QQ

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

RR

Fig. 17. Representative phase planes corresponding to the regions labeled M-R inFigs. 3-14.

26

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

SS

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

TT

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

UU

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

VV

Fig. 18. Representative phase planes corresponding to the regions labeled S-V inFigs. 3-14.

27

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40

x

y

Fig. 19. Representative phase planes corresponding to the frequency (detuning)boundaries OP (left) and ST (right) where the saddle connection global bifurcationsoccur.

28

6 The Effects of Damping

Thus far, our analysis of Eq. (10) has considered the case of zero damping(ζ = 0). To gain a better understanding of the full oscillator dynamics, we nowpresent numerical bifurcation results for the system with nonzero damping andwith the detuning (σ) treated as a bifurcation parameter. These results are ob-tained using AUTO [23], which requires that we rewrite the non-autonomousEq. (10) as a system of autonomous first order equations. Rewriting this asfirst order equations is accomplished by defining v = z′ and considering equa-tions for z′ and v′. We make this system autonomous by augmenting Eq. (10)with equations for a nonlinear oscillator with a stable periodic orbit with onevariable equal to cos Ωτ , and replacing the appropriate terms of Eq. (10) withthis variable. Most of our attention will be on Region IV from Fig. 2, as thisdisplays the most interesting bifurcation behavior. We will also present typicalresults for Regions IIb, III, and Va; Regions I, IIa, Vb, and VI are related tothe presented regions by symmetry.

As a typical point in Region IV, we take λ1 = ν1 = 1 and γ3 = λ3 = −0.005,as in Fig. 7. Throughout this section, we take ε = 0.1. The no-motion state(z = z′ = 0) exists for all values of ζ and σ, and undergoes bifurcations givingrise to branches of periodic orbits which may be followed with AUTO. Theseperiodic orbits display response at half the driving frequency Ω, and have thesymmetry property

(z, z′)(τ + T/2) = (−z,−z′)(τ), (48)

where T = 2π/Ω is the period of the response. Already for small dampingwe find the interesting result that the constant amplitude branch shown inFig. 7 splits such that the branches with the distinct phases now have slightlydifferent amplitudes; see Fig. 20 for results with ζ = 0.001. Here the stableperiodic orbit branch which bifurcates from the trivial solution at σ ≈ 1.5undergoes a saddle-node bifurcation at σ ≈ −1 before returning to the trivialstate in the bifurcation at σ ≈ 0.5. The branch containing the stable periodicorbit for, e.g., σ = −1 undergoes a saddle-node bifurcation at σ ≈ 0, and isdisconnected from the no-motion state. Periodic orbits on this disconnectedbranch also display the symmetry given in Eq. (48).

For larger damping, we find additional bifurcations on the disconnected branch;see Fig. 21 for ζ = 0.1 (a very large value for a MEM resonator). A periodicorbit with the symmetry of Eq. (48) is stable from −0.570 < σ < −3.739;a typical orbit is shown in Fig. 22(a). At the upper limit it undergoes asaddle-node bifurcation, while at the lower limit it undergoes a symmetry-breaking pitchfork bifurcation. The symmetry-breaking bifurcation leads to

29

-2 -1 0 1 2 3 4

-10

0

10

20

30

40

50

-1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25

18.0

18.5

19.0

19.5

20.0

20.5

21.0

z max

σ

-0.75 -0.25 0.25

21.0

20.0

19.0

18.0-1.25

Fig. 20. Bifurcation diagram showing the maximum value of the displacement (zmax)for ζ = 0.001 and other parameters as described in the text. In this and other bifur-cation diagrams, diamonds, squares, and triangles indicate crossing the boundary ofan Arnold tongue, period-doubling, and symmetry-breaking pitchfork bifurcations,respectively. Furthermore, solid and dashed lines represent stable and unstable so-lutions, respectively.

two symmetry-related asymmetric periodic orbits, which lack the symmetrygiven in Eq. (48); see Fig. 22(b) for an example of one of these (the other is ob-tained by letting (z, z′) → (−z,−z′)). Each of these symmetry-related asym-metric periodic orbits undergoes a period-doubling bifurcation at σ = −3.884;one of the period-doubled orbits is shown in Fig. 22(c). This is the beginningof a period-doubling cascade to chaos as σ decreases, with chaotic attractoras shown in Fig. 22(d). This is one of two symmetry-related chaotic attractors(the other is obtained by letting (z, z′) → (−z,−z′)). We note that the basinof attraction for this chaotic attractor is quite small, making it somewhatdifficult to locate (and likely unobservable in physical experiments). We alsonote that from [24], it is necessary for a symmetric periodic orbit such as thatshown in Fig. 22(a) to first undergo a symmetry-breaking bifurcation beforea period-doubling cascade can occur.

Figure 23 shows results from a two-parameter (σ and ζ) bifurcation study forthe Region IV parameters used above. The no-motion state is unstable insidethe dashed curve. The width of this instability region decreases as the damp-ing ζ increases, consistent with the effect of damping for the related Math-ieu equation [21]. The dot-dashed lines show bifurcation sets for the saddle-node bifurcations of the symmetric periodic orbits. The crossing of these at(σ, ζ) ≈ (−0.3, 0.05) does not correspond to a codimension-two bifurcation;the two saddle-node bifurcations are just (independently) occurring at the

30

-5 -4 -3 -2 -1 0 1 2 3

-25

0

25

50

75

100

z max

σ

Fig. 21. Bifurcation diagram for ζ = 0.1 and other parameters corresponding toRegion IV, as described in the text.

same parameter values. The dashed and one dash-dotted curve (correspond-ing to the saddle-node bifurcation of the periodic orbit branch which bifurcatesfrom the no-motion branch) intersect at (σ, ζ) ≈ (0.796, 0.222). Here the pe-riodic orbit branch emerging from the no-motion branch bifurcates vertically.For smaller ζ values, there is an unstable periodic orbit branch bifurcating tosmaller σ values (which then turns around in a saddle-node bifurcation), whilefor larger ζ values there is a stable branch which bifurcates to larger σ values.

We now consider the point in Region Va with λ1 = ν1 = 1 and γ3 =−0.005, λ3 = −0.01, as in Fig. 9. For small ζ we similarly find that the constantamplitude solution branch shown in Fig. 9 splits, giving a periodic orbit branchwhich is disconnected from the no-motion state; see Fig. 24 for ζ = 0.001. Thisdisconnected branch persists for larger ζ; see Fig. 25 for ζ = 0.1, for which wedo not find any further bifurcations. Note that we obtained unstable periodicorbits on the disconnected branch using a shooting method, and these werethen followed with AUTO.

A similar period-doubling cascade to chaos, as found above for Region IV,is present for ζ = 0.1 for the point in Region III with λ1 = ν1 = 1 andγ3 = −0.005, λ = 0.005, as in Fig. 6; see Fig. 26. In this region, there are noconstant amplitude solutions for ζ = 0, as there were for Region IV.

We close this section by noting that no interesting bifurcation behavior isfound for ζ = 0.1 for the point in Region IIb with λ1 = ν1 = 1 and γ3 =−0.005, λ3 = 0.01, as in Fig. 5.

31

-80

-40

0

40

80

0 1 2 3 4 5 6 7

-60

-30

0

30

60

-60 -30 0 30 60-80

-40

0

40

80

0 2 4 6 8 10 12 14

-60

-30

0

30

60

-60 -30 0 30 60

-80

-40

0

40

80

0 1 2 3 4 5 6

-4

-2

0

2

4

-56 -55 -54 -53

-60

-30

0

30

60

-60 -30 0 30 60

-60

-30

0

30

60

-60 -30 0 30 60

z

z

z

z z

z′

z′

z′

z

z

z

z′z′

z′z′ z′

(a)

(b)

(c)

(d)

τ

τ

τ

Fig. 22. Stable solutions for parameters as in Fig. 21 and with forcing ∼ cos Ωτ : (a)symmetric periodic orbit for σ = −1.5; (b) asymmetric periodic orbit for σ = −3.85;(c) period doubled orbit for σ = −3.895; (d) chaotic solution for σ − 3.90235, withright panel zoomed in. The timeseries in (a), (b), and (c) show one period.

32

−1.5 −1 −0.5 0 0.5 1 1.50

0.05

0.1

0.15

0.2

0.25

ζ

σ

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 showbifurcation sets for the saddle-node bifurcations of the symmetric periodic orbits.

-2 -1 0 1 2 3 4 5 6

-5

0

5

10

15

20

15.0

14.0

13.0

16.0

−1.0 −0.5 0.0 0.5 1.0

17.0

z max

σ

Fig. 24. Bifurcation diagram for ζ = 0.001 and other parameters corresponding toRegion Va, as described in the text.

33

-2 -1 0 1 2 3

-10

0

10

20

30

40

50

σ

z max

Fig. 25. Bifurcation diagram for ζ = 0.1 and other parameters corresponding toRegion Va, as described in the text.

-5 -4 -3 -2 -1 0 1 2 3

-25

0

25

50

75

100

σ

z max

Fig. 26. Bifurcation diagram for ζ = 0.1 and other parameters corresponding toRegion III, as described in the text.

34

7 Application to MEM Resonators

The results developed in the previous sections are directly applicable to theequations of motion for parametrically-excited MEM oscillators of the typedeveloped in Section 2. However, in this case, the coefficients in the equationof motion that are related to the electrostatic forces depend on the amplitudeof the AC voltage, VA. Accordingly, the qualitative nature of both the linearand nonlinear behavior of the oscillator depend on VA.

The effect of varying VA on the linear behavior of the system is easily notedby considering the linear stability chart for an undamped MEM oscillator aspresented in Fig. 27. As clearly shown, as VA is increased, the width (in termsof detuning) of the linear instability zone increases. As noted in Section 4, thisis largely consistent with the behavior of a typical Mathieu system, but withtwo notable exceptions. First, due to the nonlinear relationship between λ1 andVA the boundaries of the instability feature a distinct curvature. In addition,due the presence of the frequency shift caused by the electrostatic stiffnessterm (λ1), the instability region is distorted from its nominally symmetricconfiguration. Each of these differences are examined extensively in [1]. Froma practical point of view, it is also important to note that in the presence ofdamping, the bottom of the instability wedge in Fig. 27 pulls up from zero

Linear Stability Chart in the VA - f Parameter Space (Undamped)

0

5

10

15

20

25

30

51000 52000 53000 54000 55000 56000 57000 58000 59000

f (Hz)

V A (V

)

Stable StableUnstable

Fig. 27. Linear stability chart in the VA - f (excitation frequency) parameter spacefor an undamped MEM oscillator with system parameters as specified in Table 2.Experimentally derived instability points are also shown. Note that in each case thenoted stability corresponds to the trivial solution.

35

53900 54100 54300 54500

0.0

1.6

3.2

4.8

Driving Frequency (Hz)

Vel

ocity

Am

plitu

de

(125

mm

/s)

Fig. 28. Numerical frequency response curves for VA = 7.6 V. Note the hardeningbehavior exhibited in the system’s frequency response.

voltage and becomes rounded off, as shown numerically in both Fig. 35 and[1] and experimentally in both Fig. 27 and [15].

The effect of VA on the nonlinear behavior of the system can be realized bynoting that as the excitation amplitude is varied the system can transitionfrom one response regime to another (in Fig. 2). This can be confirmed byexamining the form of the nonlinearity in the γ3 - λ3 parameter space. Tak-ing the definitions of λ3 and γ3 from Table 1 it is seen that at zero voltagethe system starts at (γ3, λ3) = (χ, 0) (the point corresponding to the purelymechanical hardening nonlinearity), which lies on the boundary between Re-gions I and VI, and moves along a straight line in the γ3 - λ3 parameter spacegiven by

λ3 = γ3 − χ (49)

as VA is increased [1, 5]. To validate this, consider the oscillator design de-scribed by the parameters in Table 2, which were obtained, as subsequentlydescribed, from a device similar to that shown in Fig. 1. As Fig. 31 shows,the nonlinearity for this device will move through regions VI, Vb, Va, and IVas the amplitude of the AC voltage is varied. Thus, the system will transi-tion from a quasi-hardening nonlinearity, to a mixed nonlinearity, and finallyto a quasi-softening nonlinearity as the oscillator’s input voltage increases.Figs. 28-30 show response curves for the experimental parameter values forthree different values of VA. Note that the response indeed transitions fromhardening, to mixed, to softening, as predicted.

With the aforementioned behavior in mind, a designer can account for thesetransitions and estimate whether or not they will occur in a given device’s

36

54000 54500 55000 55500

0.0

1.6

3.2

4.8

6.4

Vel

ocity

Am

plitu

de

(125

mm

/s)

Driving Frequency (Hz)

Fig. 29. Numerical frequency response curves for VA = 16.6 V. Note the mixednonlinear behavior exhibited in the system’s frequency response.

53000 55000 57000 59000

0.0

1.6

3.2

4.8

6.4

Vel

ocity

Am

plitu

de

(125

mm

/s)

Driving Frequency (Hz)

Fig. 30. Numerical frequency response curves for VA = 33.0 V. Note the softening(quasi-softening) behavior exhibited in the system’s frequency response.

operating voltage range. The following relationships, derived by setting η1

and η2 equal to zero, provide expressions for the transition voltages, that is,the critical voltage values corresponding to qualitative changes in the system’snonlinearity. For a given oscillator,

VA,C1 =

√−3k3

5r3A

(50)

represents the voltage at which η2 = 0, and

VA,C2 =

√−3k3

r3A

, (51)

37

Table 2Design Parameters for a Representative MEM Oscillator

Parameter Value

r1A 8× 10−4 µN

V 2µm

r3A −1.2× 10−4 µN

V 2µm3

k1 7.15µN

µm

k3 0.042µN

µm3

c 2.11× 10−8 kg

s

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05

γ3

λ 3

IIII

VIIV

IIaIIb

VbVa

VA = 0 V

VA = 14.49 VVA = 32.40 V

Fig. 31. Sample parameter space showing how the system nonlinearity transitionsbetween response regimes under a varying AC voltage amplitude for the devicepresented in Table 2.

represents the voltage at which η1 = 0. For the device under consideration,these voltages are predicted to be 14.49 V and 32.40 V, respectively.

In order to confirm the general behavior predicted by this analysis, some pre-viously obtained experimental results from a MEM oscillator are included inFigs. 32-34. These figures depict frequency response curves of velocity am-plitude versus AC voltage frequency for the device in question at AC voltage

38

Velocity Amplitude vs. Driving Frequency - VA = 7.6 V

0

0.2

0.4

0.6

0.8

1

1.2

53950 54000 54050 54100 54150 54200 54250 54300 54350 54400 54450

Driving Frequency (Hz)

Velo

city

Am

plitu

de (1

25 m

m/s

)

Up SweepDown Sweep

Fig. 32. Experimental frequency response curves for VA = 7.6 V. Note the hardeningbehavior exhibited in the system’s frequency response.

Veloicty Amplitude vs. Driving Frequency - VA = 16.6 V

0

0.5

1

1.5

2

2.5

54000 54200 54400 54600 54800 55000 55200 55400 55600

Driving Frequency (Hz)

Velo

city

Am

plitu

de (1

25 m

m/s

)

Up SweepDown Sweep

Fig. 33. Experimental frequency response curves for VA = 16.6 V. Note the mixednonlinear behavior exhibited in the system’s frequency response.

excitation amplitudes (VA) of 7.6 V, 16.6 V, and 33.0 V, respectively (the samevalues as used for the simulated results). As predicted, the oscillator’s nonlin-ear characteristics qualitatively change as the excitation voltage is increased.Specifically, the oscillator exhibits hardening characteristics at 7.6 V, mixednonlinear characteristics at 16.6 V, and softening characteristics at 33.0 V, incomplete qualitative agreement with the simulation results.

It is important to note that the quantitative compatibility of the simula-

39

Velocity Amplitude vs. Driving Frequency - VA = 33.0 V

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

53000 54000 55000 56000 57000 58000 59000

Driving Frequency (Hz)

Velo

city

Am

plitu

de (1

25 m

m/s

)

Up SweepDown Sweep

Fig. 34. Experimental frequency response curves for VA = 33.0 V. Note the softeningbehavior exhibited in the system’s frequency response.

tion and experimental results presented is based on the fact that the set ofexperiments that generated Figs. 32-34 were used for parameter identifica-tion. Specifically, the experimentally identified bifurcation points associatedwith crossing the Arnold tongue were compared with numerical simulations todetermine the linear system parameters (r1A, k1, and c) and the experimen-tally identified saddle-node bifurcations associated with the periodic orbitswere compared to numerical simulations and the analytically-derived tran-sition voltages to determine the nonlinear system parameters (r3A and k3).Though this identification process was largely done in an ad-hoc manner (sys-tematically, but visually), the results are quite promising. In fact, a quitecomplete comparison of the predicted and measured bifurcation sets is givenin Fig. 35. It is seen that the Arnold tongue and the higher frequency saddle-node bifurcations are well predicted, but that the model does not accuratelycapture the frequencies at which the lower-frequency saddle-node occurs. How-ever, it should be noted that the model does a very good job at predicting theranges of VA over which hardening, mixed, and softening nonlinear behaviorsare observed.

While the experimental results presented are promising indications of the va-lidity of the analysis presented, some issues remain to be resolved. For example,the absence of effects associated with the additional constant amplitude so-lution is slightly disconcerting. This, however, is largely reconcilable with thefact that the authors were not aware of such phenomena previous to and dur-ing the experimentation which produced the included results, and thus thisphenomena was not taken into account during frequency sweeps and devicecharacterization. In addition, the parameter identification technique used to

40

52000 54000 56000 58000

0

10

20

30

Dri

ving

Vol

tage

Am

plitu

de

Driving Frequency (Hz)

AT

AT

SN

SN

Fig. 35. The voltage amplitude vs. excitation frequency parameter space which wasused to recover the system parameters presented in Table 2. The solid lines desig-nated AT and SN correspond to the numerically predicted location of the Arnoldtongue and saddle-node bifurcations, respectively. The various data points, whichwere determined experimentally, correspond to the Arnold tongue (diamonds), leftsaddle-node bifurcation (crosses), and right saddle-node bifurcation (triangles). Notethe strong correlation produced by the parameter identification process.

determine the parameters presented in Table 2 could undergo refinement. Forexample, robust nonlinear curve fitting techniques could be used in place ofthe visual approach employed here. Each of these issues and others, such asthe discrepancy between numerically predicted and experimentally determinedresponse amplitudes, will be addressed in forthcoming works.

8 Conclusion

As shown in this work, parametrically-excited MEM oscillators and other sys-tems with generalized forms of parametric resonance can exhibit a wide arrayof interesting dynamical behavior, most of which can be attributed to theexistence of nonlinear parametric excitation in their respective equations ofmotion. One interesting result of this nonlinear parametric excitation is thatsuch systems fail to exhibit a single effective nonlinearity that is capable ofcharacterizing their nonlinear behavior (in terms of frequency response). In-stead, the systems feature multiple effective nonlinearities, which only whencollectively analyzed reveal their nonlinear nature. Accordingly, such systemsare capable of displaying not only typical hardening or softening nonlinearcharacteristics, but also mixed nonlinear characteristics wherein the principalresponse branches bend toward or away from one another near resonance. Infact, for certain devices, perhaps most notably parametrically-excited MEM

41

oscillators, such behavior can be shown to be explicitly dependent on theexcitation amplitude of the system. That is, such systems can be shown totransition between various qualitatively different nonlinear regimes as the ex-citation amplitude is varied.

Experimentation has yielded results that are in good agreement with theanalytical predictions developed. In particular, experimentally produced fre-quency response and bifurcation diagrams confirm the assertion that the qual-itative nature of the systems’ nonlinear response can change with the systems’excitation amplitude.

Despite the general agreement between the analytical, numerical, and experi-mental results presented in this work, some issues remain unresolved. As high-lighted in the previous section, these issues include the experimental identifica-tion of effects associated with the constant amplitude solutions, which shouldgive rise to isolate response branches, and improvement in parameter estima-tion techniques. Both of these issues are currently being addressed through anexperimental campaign designed to systematically characterize such systemsin light of the analytical and numerical results presented in this work.

In terms of device design, it is often desirable to have a system that is eitherpurely hardening, or purely softening. It is relatively simple to achieve overallhardening behavior, since virtually all mechanical suspensions lead to hard-ening nonlinearities. Thus, one can design comb drives that will maintain thesystem in Region I or VI as VA is varied. The case of softening nonlinearity ismore complicated, since no known mechanical suspension leads to softeningbehavior. In this case, the suspension should be made as linear as possible,and the comb drive designed so that the system transitions from Region VIthrough Regions Va and Vb at very low voltages, resulting in a system thatis softening beyond a low threshold. This is not problematic, however, sincethe applied voltage must be above a certain threshold in order to activate theparametric resonance (due to the presence of damping).

The results presented are directly applicable to other microscale systems, suchas electrostatically-actuated microbeams and torsional oscillators, and are alsoextendable to a variety of systems on the macroscale. It is hoped that thegeneral response features and predictive tools described herein are beneficialin the design and evaluation of these systems.

42

Acknowledgements

This work was supported by the Air Force Office of Scientific Research (AFOSR)under contract F49620-02-1-0069.

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