Mechanical Domain Parametric Amplification

Proceedings of the ASME 2007 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2007 September 4-7, 2007, Las Vegas, Nevada, USA

DETC2007-35426

MECHANICAL DOMAIN PARAMETRIC AMPLIFICATION

Jeffrey F. Rhoads∗ Nicholas J. Miller Department of Mechanical Engineering Department of Mechanical Engineering Michigan State University Michigan State University East Lansing, Michigan 48824 East Lansing, Michigan 48824 Email: [email protected] Email: [email protected]

Steven W. Shaw Brian F. Feeny Department of Mechanical Engineering Department of Mechanical Engineering Michigan State University Michigan State University East Lansing, Michigan 48824 East Lansing, Michigan 48824 Email: [email protected] Email: [email protected]

ABSTRACT plications of parametric amplification in macroscale mechanical Though utilized for more than fifty years in a variety of systems. power and communication systems, parametric amplification, the process of amplifying a harmonic signal with a parametric pump, has received very little attention in the mechanical engi- INTRODUCTION neering community. In fact, only within the past fifteen to twenty years has the technique been implemented in micromechanical Parametric amplification, the process of amplifying an ex- systems as a means of amplifying the output of resonant micro- ternal, harmonic signal with a parametric pump, is a well es- transducers. While the vast potential of parametric amplifica- tablished concept in the field of electrical engineering, where tion has been demonstrated, to date, in a number of micro- and it has been employed for more than fifty years in applications nano-mechanical systems (as well as a number electrical sys- ranging from power and communication systems to Josephson tems), few, if any, macroscale mechanical amplifiers have been junctions [1, 2]. Though the technique has been widely imple- reported. Given that these amplifiers are easily realizable using mented in electrical systems, parametric amplification has re- larger-scale mechanical systems, the present work seeks to ad- ceived comparatively little attention in mechanical engineering dress this void by examining a simple, representative example: circles. In fact, only within the past fifteen to twenty years has a cantilevered beam with longitudinal and transverse base ex- the technique been implemented in micromechanical systems as citations. The work begins with the systematic formulation of a a means of amplifying the output of resonant micro-transducers. representative system model, which is used to derive a number The classical investigation of mechanical domain parametric am- of pertinent metrics. A series of experimental results, which val- plification, completed by Rugar and Grutter¨ in 1991, consid- idate the work’s analytical findings, are subsequently examined, ered resonant amplification in a microcantilever actuated by both and the work concludes with a brief look at some plausible ap- piezoelectric and electrostatic elements [3]. Subsequent studies have built upon this seminal work by examining both degenerate (where the pump is locked at twice the frequency of the ex-

∗Please address all correspondence to this author. 1 Copyright c 2007 by ASME ternal signal) and non-degenerate (where the pump is locked at a frequency distinct from that of the external signal) manifes- tations of parametric amplification in a variety of resonant mi- crosystems, including torsional microresonators [4, 5], electric force microscopes [6], optically-excited micromechanical oscillators [7], micro ring gyroscopes [8], MEMS diaphragms [9], coupled microresonators [10], and microcantilevers [11–13]. While the literature detailed above has demonstrated the vast potential of parametric amplification at the micro- and nano- scales, few, if any, macroscale mechanical amplifiers have been reported. Given that these amplifiers are easily realizable at larger scales, the present work seeks to address this void by examining a simple, macroscale parametric amplifier. Specifically, the work details a joint analytical and experimental investigation of mechanical domain parametric amplification in a cantilevered beam with longitudinal and transverse base excitations. The pa- per begins in the following section with the development of a Figure 1. SCHEMATIC OF A REPRESENTATIVE MECHANICAL PARA- consistent lumped-mass model for the macroscale amplifier that METRIC AMPLIFIER: A BASE-EXCITED CANTILEVERED BEAM. is amenable to analysis. Following this, analytical results, obtained using the method of averaging, are detailed. Results ac- principle, given by quired during the course of experimentation are presented in the subsequent section, and the work concludes with a brief discus- t l Z 2 Z 1 h 2 i sion of the potential macroscale applications of parametric am- δH = δ L¯ + λ 1 − 1 + u0 − (v0)2 dsdt t 0 2 plification. 1 (2) Z t2 Z l + (Quδu + Qvδv)dsdt, t1 0 MODELING Though parametric amplification can be easily realized in a where l represents the beam’s undeformed length, λ denotes the large number of macroscale systems, the present work, as noted Lagrange multiplier used to maintain the inextensibility con- above, limits itself to a simple, representative example: a base- straint, and Qu and Qv represent the generalized forces not ac- excited cantilever, such as that depicted schematically in Fig. 1. counted for in the specific Lagrangian, L¯, in the u and v direc- Given that the equation of motion governing the transverse dy- tions, respectively. Assuming that damping has an appreciable namics of this system can be recovered using the energy-based impact on only the local, transverse motion of the beam, these methods previously presented in Refs. [14, 15], only an outline forces can be approximated by of the procedure is detailed here. Assuming that the cantilever beam is uniform and has negligible rotational inertia, the specific Lagrangian associated with Qu = −ρAg, Qv = −cv˙, (3) the system can be approximated by where g represents the beam’s acceleration due to gravity and 1 h i 1 c represents a specific viscous damping coefficient. Substitut- L¯ = ρA (u˙ + u˙ )2 + (v˙+ v˙ )2 − EI(ψ0)2, (1) 2 p p 2 ing each of these forces, as well as the specific Lagrangian, into Eq. (2), integrating by parts successively, and introducing the kinematic constraint relating the angle ψ to the planar displace- ˙ 0 where u, v, and ψ are defined as in Fig. 1, (•) and (•) repre- ments u and v, sent temporal and spatial derivatives (determined with respect to time t and arc length variable s), up and vp specify the imposed v0 base motion in the longitudinal and transverse directions, and ρ, tanψ = , (4) A, E, and I represent the beam’s mass density, cross-sectional 1 + u0 area, modulus of elasticity, and cross-sectional moment of inertia, respectively. Noting this, and further assuming that the neu- results in a pair of coupled governing equations. The equation tral axis of the beam is inextensible, results in a governing varia- of motion governing transverse dynamics can be recovered from tional equation for the system, derived from extended Hamilton’s this set, by first solving the longitudinal equation to obtain the

2 Copyright c 2007 by ASME Lagrange multiplier and then substituting the obtained function Though the beam’s behavior could potentially be recovered into the second equation. Truncating the result such that only from the distributed-parameter model presented in Eq. (10), a linear terms remain (only linear amplifier operation is deemed lumped-mass model proves sufficient for the present analysis. pertinent and practical here) yields a distributed-parameter sys- Accordingly, the dynamic variablev ˆ is decomposed into spatial tem model given by and temporal components using the cantilever’s first mode shape Φ(sˆ), according to iv 00 0 ρAv¨+ cv˙+ EIv − v ρAu¨p(s − l) − v ρAu¨p 00 0 (5) vˆ = w(tˆ)Φ(sˆ). (12) − v ρAg(s − l) − v ρAg = −ρav¨p.

The result is then projected back onto the first mode shape using To reduce the number of free parameters in the system de- an inner product operator yielding a final lumped-mass model, tailed above, it proves convenient to rescale Eq. (5). Accord- given by ingly, the arc length variable and beam displacements are scaled by the beam’s undeformed length l and an additional characteris- 00 0 2 2 z + 2εζz + z + ελ1Ω cos(Ωτ + φ) + ελ2Ω cos(2Ωτ) z tic length v0 (e.g. the width or thickness of the beam), according (13) 2 2 to = εη1Ω cos(Ωτ + φ) + εη2Ω cos(2Ωτ), s sˆ = , (6) with nondimensional parameters defined as in Table 1. Note that l the imposed harmonic base motions included here are assumed to result from a uni-directional base excitationx ˆp, given by

v up vp ˆ ˆ vˆ = , uˆp = , vˆp = , (7) xˆp = Acos(ωt + φ) + Bcos(2ωt) v0 v0 v0 (14) = Aˆ cos(ωˆ tˆ+ φ) + Bˆ cos(2ωˆ tˆ), and time is scaled by a characteristic period of the system, according to and are deﬁned according to

t uˆp = xˆp sinα, vˆp = xˆp cosα. (15) tˆ = , (8) T The phase-dependent nature of this excitation is requisite in the where examination of degenerate parametric ampliﬁcation. An examination of non-degenerate ampliﬁcation, which is phase indepen- r dent and does not require the strict 2:1 frequency ratio utilized ρAl4 T = . (9) above, is left for subsequent studies. EI

This renders a final distributed-parameter system model given by ANALYSIS Though the equation of motion detailed in Eq. (13) is linear, v0 v0 its time-varying stiffness coefficient prevents the derivation of a vˆ¨+ cˆvˆ˙+ vîv − uˆ¨ (sˆ− 1)vˆ00 − uˆ¨ vˆ0 l p l p tractable closed-form solution. Accordingly, the method of aver- (10) ρAgl3 ρAgl3 aging is exploited here. To facilitate this approach, a constrained − (sˆ− 1)vˆ00 − vˆ0 = −vˆ¨ , EI EI p coordinate change is first introduced into Eq. (13), namely, where z(τ) = X(τ)cos(Ωτ) +Y(τ)sin(Ωτ), (16) cT z0(τ) = −X(τ)Ωsin(Ωτ) +Y(τ)Ωcos(Ωτ). cˆ = . (11) ρA Additionally, since near-resonant behavior is of particular inter- Note that the derivative operators have been redefined here in est, a detuning parameter σ, defined by terms of the new time and arc length variables, tˆ ands ˆ, respectively. Ω = 1 + εσ, (17)

3 Copyright c 2007 by ASME Table 1. NONDIMENSIONAL PARAMETER DEFINITIONS. NOTE 0.6 THAT ε REPRESENTS A ‘SMALL’ PARAMETER INTRODUCED FOR

ANALYTICAL PURPOSES AND ω REPRESENTS THE SYSTEM’S 0.5 ‘PHYSICAL’ BASE EXCITATION FREQUENCY. 0.4

2 0.3

z = w λ d(•) τ = ω tˆ, (•)0 = 0 dτ 0.2 ζ = 0 ωˆ 0.1 ˆ = T, = ζ = 0.05 ω ω Ω ζ = 0.1 ω0 0 Z 1 3 Z 1 Z 1 -0.1 -0.05 0 0.05 0.1 2 iv ρAl g 00 0 ω0 = ΦΦ dsˆ− ΦΦ (sˆ− 1)dsˆ+ ΦΦ dsˆ σ 0 EI 0 0

cˆ Figure 2. WEDGES OF INSTABILITY NEAR σ = 0. NOTE THAT SUC- εζ = 2ω0 CESSFUL PARAMETRIC AMPLIFICATION REQUIRES THAT THE SYS- TEM OPERATE BELOW ITS CORRESPONDING WEDGE, AS PARA- ˆ Z 1 Z 1 Av0 sinα 00 0 METRIC RESONANCE OCCURS WITHIN THE INSTABILITY REGION. ελ1 = ΦΦ (sˆ− 1)dsˆ+ ΦΦ dsˆ l 0 0 ˆ Z 1 Z 1 4Bv0 sinα 00 0 Utilizing this form of the oscillator’s amplitude, the gain associ- ελ2 = ΦΦ (sˆ− 1)dsˆ+ ΦΦ dsˆ l 0 0 ated with the ampliﬁer can be deﬁned as

ˆ ˆ εη1 = Acosα, εη2 = 4Bcosα a¯ G = pump on , (21) a¯pump o f f is utilized. Separating the constraint equation, as well as that which at σ = 0 yields which results from substitution, in terms of X0 and Y 0 and averaging the result over one period of the oscillator’s response (2 / ) s π Ω λ2 + 16ζ2 − 8λ ζsin2φ results in the system’s averaged equations, which are given by G(σ = 0) = 4ζ 2 2 . (22) 2 2 2 (λ2 − 16ζ )

1 X0 = − ε(λ Y + 4σY + 4ζX − 2η sinφ), Using Eqs. (19) and (22) the pertinent performance metrics of 4 2 1 (18) the mechanical ampliﬁer can be readily identiﬁed. 1 Y 0 = − ε(λ X − 4σX + 4ζY − 2η cosφ). Generally speaking, the resonator described herein can op- 4 2 1 erate at any point within the λ2-σ parameter space. However, due to the presence of parametric excitation in Eq. (13), the oscillator Using these averaged equations the steady-state behavior of the has a classical ‘wedge of instability’ (Arnold tongue) structure system can be recovered by setting (X0,Y 0) = (0,0). This reveals associated with it (see Fig. 2) [16, 17]. As such, parametric- that the system has a steady state solution given, in terms of am- resonance-induced oscillations, bounded by only by the sys- plitude and phase, by tem’s mechanical nonlinearities, are possible for pump amplitudes greater than v u 2 2 2 2 uη1 λ2 + 16(ζ + σ ) + 8λ2 (σcos2φ − ζsin2φ) q a¯ = 2t , (19) 2 2 2 2 2 2 λ2,crit = 4 σ + ζ . (23) λ2 − 16(σ + ζ )

As these motions are incompatible with linear signal ampliﬁca- tion, the present study limits itself to operation below the (λ2 − 4σ)sinφ − 4ζcosφ ψ¯ = arctan . (20) 2 2 2 (λ2 + 4σ)cosφ − 4ζsinφ λ2 − 16(σ + ζ ) = 0 (24)

4 Copyright c 2007 by ASME 40 1.6

λλ22 == 00 35 1.4 λλ22 == 0.0550.055 λ = 0.088 30 1.2 λ22 = 0.088

25 1

20 0.8 Gain 15 0.6

10 0.4 Normalized Amplitude 5 0.2 0 0 0.2 0.4 0.6 0.8 1 0 -1 -0.5 0 0.5 1 Normalized Pump Amplitude σ

Figure 3. THEORETICAL AMPLIFIER GAIN, G(σ = 0), PLOTTED Figure 4. THEORETICAL FREQUENCY RESPONSE CURVES, VERSUS PUMP AMPLITUDE, λ2/λ2,crit FOR AN OSCILLATOR OPER- a¯/max{a¯(λ2 = 0)} VS. σ, CORRESPONDING TO THREE DIF- ATING AT σ = 0 WITH φ = −π/4. NOTE THAT HERE AND IN FIG. 8 FERENT VALUES OF λ2 (PUMP AMPLITUDES). NOTE THAT THE PUMP AMPLITUDE HAS BEEN NORMALIZED SUCH THAT THE THE RESPONSES HAVE BEEN NORMALIZED SUCH THAT THE PARAMETRIC INSTABILITY OCCURS AT UNITY. ALSO NOTE THAT UNPUMPED SYSTEM HAS A RESONANT AMPLITUDE OF UNITY. HERE AND IN FIGS. 4-5 ζ = 0.07.

1.4 threshold, which corresponds to the principal parametric reso- 1.2 nance’s ‘wedge of instability’. 1 With the limitation on pump amplitude (λ2), detailed above, in place, the gain associated with the macroscale parametric 0.8 amplifier can be readily characterized. Figure 3, for exam- 0.6 ple, details the amplifier’s gain as a function of pump amplitude. As expected, the gain monotonically increases with in- 0.4 Normalized Amplitude creasing pump amplitude, eventually approaching infinity as λ2 0.2 approaches λ2,crit (as the system transitions into parametric resonance). This is in direct contrast to the effect of increasing 0 -45 0 45 90 135 the amplitude of the amplifier’s input signal (direct excitation), φ (deg) which has a negligible effect on the system’s gain, as detailed in Eq. (22). Also evident from Fig. 3 is the potential for relatively Figure 5. THEORETICAL AMPLITUDE, a¯/a¯(φ = 0), VERSUS high amplifier gains. As these gains require the ability to operate PHASE, φ. NOTE THAT HERE AND IN FIG. 9 THE RESPONSE AM- near, but not above, λ ,crit , however, extremely large gains (i.e. 2 PLITUDE HAS BEEN NORMALIZED SUCH THAT THE RESPONSE AT multiple orders of magnitude) may be quite difficult to realize in φ = 0 IS EQUAL TO UNITY. the face of system uncertainty. To further reinforce the amplification seen in Fig. 3, frequency response curves for the mechanical amplifier operating (and every 180◦ multiple), where sin2φ is minimum. Likewise, under three distinct pump conditions are included in Fig. 4. Once minimum gains are realizable at φ = 45◦ (and every 180◦ mul- again the effect of the parametric amplifier is evident, as gains of tiple), where sin2φ is maximum. The latter operating point re- approximately 1.25 and 1.45 are realized with pump amplitudes veals a potential application for parametric amplifiers which has well below λ2,crit . received little attention and is targeted for future study: vibration Though Figs. 3 and 4 detail the effect of the pump’s ampli- suppression. tude on the amplifier’s performance, as detailed in Eq. (22), the Before proceeding with an experimental investigation of the system gain also depends on the relative phase φ incorporated macroscale parametric amplifier, the noise characteristics of the into the amplifier’s input signal. Figure 5 details the effect of device should be briefly noted. Given that the amplifier described varying this phase in a representative amplifier. As evident, max- herein is designed to operate in a degenerate, phase sensitive imum gains are realized for a relative phase angle of φ = −45◦ mode, it, in theory, should be noise free down to a quantum

5 Copyright c 2007 by ASME 1.6 Power Amplifier 1.4 No Pump Summing 46.6% Pump Signal Generator Overtravel Vibration Op-Amp 1.2 Protector Exciter 74.3% Pump 1

0.8 Accelerometer 0.6

0.4

Oscilloscope Normalized Amplitude Signal Strain Gauges Cantilevered Conditioner Beam 0.2

0 10.5 11 11.5 12 12.5 f (Hz) Figure 6. BLOCK DIAGRAM OF THE EXPERIMENTAL SETUP USED TO OBTAIN THE RESULTS INCLUDED IN FIGS. 7-9. Figure 7. EXPERIMENTAL FREQUENCY RESPONSE CURVES, a¯/max{a¯(λ = 0)} VS. f , OBTAINED FOR THREE DIFFERENT mechanical level [3, 11, 18]. In practice, minimal amounts of 2 PUMP AMPLITUDES. noise can be expected, but these noise contributions should be appreciably smaller than the 3 dB of noise typically attributed to phase-insensitive, non-degenerate amplifiers [11]. logarithmic decrement methods and the beam’s response was recorded at discrete operating points over a wide range of forcing parameters (i.e. values of φ, Aˆ, Bˆ, etc.). Pertinent results recov- EXPERIMENTAL RESULTS ered during experimentation are detailed below. Though the preceding analytical investigation appears to Figure 7, recovered by holding the vibration exciter’s res- verify the feasibility of a macroscale parametric amplifier, ex- onant direct excitation amplitude (Aˆ) fixed and systematically perimental results were deemed necessary to validate the analyt- varying both the excitation frequency (ω) and pump ampli- ical results detailed herein. Accordingly, the experimental setup tude (Bˆ), depicts the cantilevered beam’s frequency response depicted schematically in Fig. 6 was assembled. The system’s for normalized pump values of 0%, 46.6%, and 74.3%, respec- signal flow is outlined below. tively (note that these values are normalized with respect to the To begin, a signal generator (Wavetek 2 MHz Variable Phase experimentally-determined parametric threshold – i.e. λ2,crit ). Synthesizer, model 650) is used to produce the two required input As evident, the system exhibits a linear resonance at approxi- signals, one at the driving frequency ω and another at 2ω, with a mately 11.5 Hz, which varies in normalized amplitude from 1 to fixed relative phase φ. These two signals are added together using approximately 1.5 with varying pump amplitude. This is in close a summing op-amp circuit, and the result is given as a command accordance with the predicted system behavior shown in Fig. 4, signal to the electromagnetic vibration exciter (MB Dynamics in terms of both Q and amplitude, and verifies that non-negligible model PM-500). This exciter, in turn, provides base excitation amplifier gains are experimentally realizable in a macroscale to a cantilevered spring steel beam (190 mm × 19 mm × 0.5 parametric amplifier. This conclusion is further reinforced by ◦ mm, f1 ≈ 11.5 Hz, f2 ≈ 73.3 Hz), which is orientated at α = 80 Fig. 8, which depicts the amplifier’s gain as a function of nor- to induce both direct and parametric excitations. To ensure that malized pump amplitude. Here, as in Fig. 3, the amplifier’s gain the shaker output matches that desired by the operator, the base is shown to increase with increasing pump amplitude (Bˆ). How- excitation is measured using a three-axis accelerometer (Analog ever, unlike Fig. 3, which predicts multiple-order-of-magnitude Devices ADXL105EM-3) attached directly to the exciter table. gains, the experimentally-acquired amplifier gains appear to only Beam deflections are measured using two strain gauges (Mea- reach approximately 1.6 before the onset of parametric resonance surements Group Inc. Micro Measurements Division, EA-13- (as verified by turning off the direct excitation signal – i.e. setting 120LZ-120) mounted in a half-bridge configuration. The strain Aˆ=0). Though this difference could be due to geometric or iner- signal from these gauges is balanced and low-pass filtered (100 tial nonlinearities, the symmetric, non-hysteretic nature of the Hz cut-off frequency) by a signal conditioning amplifier (Mea- frequency response near this operating condition seems to indi- surements Group Inc. Instruments Division, Model 2210). For cate the presence of another limiting factor. Accordingly, the au- measurement purposes, the command, accelerometer, and strain thor’s are currently exploring other reasons for this experimental signals are each recorded using an Agilent 54624A oscilloscope. gain limitation, including the effect that the non-resonant direct With the experimental test-rig depicted in Fig. 6 assembled, excitation has on the system’s behavior near the onset of para- the beam’s damping ratio (εζ ≈ 0.007) was determined using metric resonance.

6 Copyright c 2007 by ASME 1.6 plifiers have been reported. The present work fills this apparent void by demonstrating that parameter amplification can be easily 1.5 realized in even the simplest of macroscale mechanical systems, including a base-excited cantilever with longitudinal and trans- 1.4 verse excitations. Furthermore, the work demonstrates that non- trivial amplifier gains on the order of 1.4 to 1.6 are relatively easy 1.3

Gain to realize in practice and that appreciable gains may be recover-

1.2 able with further study. Accordingly, mechanical amplifiers are believed to be of practical use in a wide-variety of macroscale ap- 1.1 plications, including the amplification of output signals in some acoustic systems, such as cavity resonators, vibration test equip- 1 ment, etc. Additionally, in certain applications, parametric am- 0 0.2 0.4 0.6 0.8 1 plification, with careful phase selection, may facilitate vibration Normalized Pump Amplitude suppression and thus offers a potential alternative to classical vibration absorbers, which typically require the implementation of Figure 8. EXPERIMENTALLY-OBTAINED GAIN, G(σ = 0), PLOTTED additional hardware. On-going work is aimed at extending the AS A FUNCTION OF PUMP AMPLITUDE, λ /λ . 2 2,crit results described herein to the aforementioned applications, as well as identifying the factor(s) limiting the mechanical ampli- 1.4 fier’s gain.

1.2

1 ACKNOWLEDGMENT This work was supported by the National Science Founda- 0.8 tion under grant NSF-ECS-0428916. The authors would also like

0.6 to extend their gratitude to Dr. Alan Haddow and Mr. Umar Fa- rooq for their assistance in preparing the experimental setup. 0.4 Normalized Amplitude

0.2 REFERENCES 0 [1] Mumford, W. W., 1960. “Some notes on the history of -45 -25 -5 15 35 55 75 95 115 135 parametric transducers”. Proceedings of the IRE, 48(5), φ (deg) pp. 848–853. [2] Louisell, W. H., 1960. Coupled Mode and Parametric Elec- Figure 9. EXPERIMENTAL AMPLITUDE, a¯/a¯(φ = 0), VERSUS tronics. John Wiley & Sons, New York. PHASE, φ. [3] Rugar, D., and Grutter,¨ P., 1991. “Mechanical parametric amplification and thermomechanical noise squeezing”. To validate the phase-dependent nature of the degenerate Physical Review Letters, 67(6), pp. 699–702. amplifier, system behaviors akin to those illustrated in Fig. 5 [4] Carr, D. W., Evoy, S., Sekaric, L., Craighead, H. G., and were also examined experimentally . The results of this study are Parpia, J. M., 2000. “Parametric amplification in a tor- summarized in Fig. 9. As evident, the system exhibits a phase- sional microresonator”. Applied Physics Letters, 77(10), depend amplitude, which varies between approximately 0.5 and pp. 1545–1547. 1.1 with varying φ. Maximum and minimum response ampli- [5] Baskaran, R. and Turner, K. L., 2003. “Mechanical do- tudes, as previously predicted, occur near -45◦ and 45◦, respec- main coupled mode parametric resonance and amplification tively, and the phase relationship is repeated on 180◦ intervals. in a torsional mode micro electro mechanical oscillator”. This is in close accordance with the results detailed in Fig. 5. Journal of Micromechanics and Microengineering, 13(5), pp. 701–707. [6] Ouisse, T., Stark, M., Rodrigues-Martins, F., Bercu, B., CONCLUSION Huant, S., and Chevrier, J., 2005. “Theory of electric force As noted in the introduction, parametric amplification has microscopy in the parametric amplification regime”. Phys- been implemented, to date, in a wide variety of electrical and ical Review B, 71, 205404. micro- and nano-mechanical systems. However, to the best of [7] Zalalutdinov, M., Olkhovets, A., Zehnder, A., Ilic, B., the authors’ knowledge, few, if any, macroscale mechanical am- Czaplewski, D., Craighead, H. G., and Parpia, J. M., 2001.

7 Copyright c 2007 by ASME “Optically pumped parametric amplification for micromechanical oscillators”. Applied Physics Letters, 78(20), pp. 3142–3144. [8] Gallacher, B. J., Burdess, J. S., Harris, A. J., and Harish, K. M., 2005. “Active damping control in MEMS using parametric pumping”. In Proceedings of Nanotech 2005: The 2005 NSTI Nanotechnology Conference and Trade Show, Vol. 7, pp. 383–386. [9] Raskin, J.-P., Brown, A. R., Khuri-Yakub, B. T., and Re- beiz, G. M., 2000. “A novel parametric-effect MEMS amplifier”. Journal of Microelectromechanical Systems, 9(4), pp. 528–537. [10] Olkhovets, A., Carr, D. W., Parpia, J. M., and Craighead, H. G., 2001. “Non-degenerate nanomechanical parametric amplifier”. In Proceedings of MEMS 2001: The 14th IEEE International Conference on Micro Electro Mechani- cal Systems, pp. 298–300. [11] Dana, A., Ho, F., and Yamamoto, Y., 1998. “Mechan- ical parametric amplification in piezoresistive gallium ar- senide microcantilevers”. Applied Physics Letters, 72(10), pp. 1152–1154. [12] Roukes, M. L., Ekinci, K. L., Yang, Y. T., Huang, X. M. H., Tang, H. X., Harrington, D. A., Casey, J., and Artlett, J. L., 2004. “An apparatus and method for two-dimensional elec- tron gas actuation and transduction for gas NEMS”. Inter- national Patent, WO/2004/041998 A2. [13] Ono, T., Wakamatsu, H., and Esashi, M., 2005. “Para- metrically amplified thermal resonant sensor with pseudo- cooling effect”. Journal of Micromechanics and Microengi- neering, 15(12), pp. 2282–2288. [14] Crespo da Silva, M. R. M., and Glynn, C. C., 1978. “Non- linear flexural-flexural-torsional dynamics of inextensional beams. I: Equations of motion”. Journal of Structural Me- chanics, 6(4), pp. 437–448. [15] Malatkar, P., 2003. “Nonlinear vibrations of cantilever beams and plates”. Ph.D. Thesis, Virginia Polytechnic In- stitute and State University. [16] Nayfeh, A. H., and Mook, D. T., 1979. Nonlinear Oscilla- tions. Wiley-Interscience. [17] Stoker, J. J., 1950. Nonlinear Vibrations in Mechanical and Electrical Systems. John Wiley & Sons, New York. [18] Caves, C. M., 1982. “Quantum limits on noise in linear amplifiers”. Physical Review D, 26(8), pp. 1817–1839.