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Mechanical Domain Parametric Amplification

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Mechanical Domain Parametric Amplification Mechanical Domain Parametric Jeffrey F. Rhoads School of Mechanical Engineering, Purdue University, Amplification West Lafayette, IN 47907 e-mail: [email protected] Though utilized for more than 50 years in a variety of power and communication sys- tems, parametric amplification, the process of amplifying a harmonic signal with a para- metric pump, has received very little attention in the mechanical engineering community. Nicholas J. Miller In fact, only within the past 15–20 years has the technique been implemented in micro- e-mail: [email protected] mechanical systems as a means of amplifying the output of resonant microtransducers. While the vast potential of parametric amplification has been demonstrated, to date, in a Steven W. Shaw number of micro- and nanomechanical systems (as well as a number electrical systems), e-mail: [email protected] few, if any, macroscale mechanical amplifiers have been reported. Given that these am- plifiers are easily realizable using larger-scale mechanical systems, the present work Brian F. Feeny seeks to address this void by examining a simple representative example: a cantilevered e-mail: [email protected] beam with longitudinal and transverse base excitations. The work begins with the sys- tematic formulation of a representative system model, which is used to derive a number of Department of Mechanical Engineering, pertinent metrics. A series of experimental results, which validate the work’s analytical Michigan State University, findings, are subsequently examined, and the work concludes with a brief look at some East Lansing, MI 48824 plausible applications of parametric amplification in macroscale mechanical systems. ͓DOI: 10.1115/1.2980382͔ 1 Introduction following section with the development of a consistent lumped- mass model for the macroscale amplifier that is amenable to Parametric amplification, the process of amplifying an external analysis. Following this, analytical results, obtained using the harmonic signal with a parametric pump, is a well established method of averaging, are detailed. The results acquired during the concept in the field of electrical engineering, where it has been course of experimentation are presented in the subsequent section, employed for more than 50 years in applications ranging from ͓ ͔ and the work concludes with a brief discussion of the potential power and communication systems to Josephson junctions 1,2 . macroscale applications of parametric amplification. Though the technique has been widely implemented in electrical systems, parametric amplification has received comparatively little attention in mechanical engineering circles. In fact, only within the past 15–20 years has the technique been implemented 2 Modeling in micromechanical systems as a means of amplifying, with mini- Though parametric amplification can be easily realized in a mal noise, the output of resonant microtransducers. The classical large number of macroscale systems, the present work, as noted investigation of mechanical domain parametric amplification, above, limits itself to a simple representative example: a base- completed by Rugar and Grütter in 1991, considered low-noise excited cantilever, such as that depicted schematically in Fig. 1. resonant amplification and noise squeezing in a microcantilever Given that the equation of motion governing the transverse dy- actuated by both piezoelectric and electrostatic elements ͓3͔. Sub- namics of this system can be recovered using the energy-based sequent studies have built upon this seminal work by examining methods previously presented in Refs. ͓14,15͔, only an outline of the feasibility of both degenerate ͑where the pump is locked at the procedure is detailed here. twice the frequency of the external signal͒ and nondegenerate Assuming that the cantilever beam is uniform and has negli- ͑where the pump is locked at a frequency distinct from twice that gible rotational inertia, the specific Lagrangian associated with the of the external signal͒ manifestations of parametric low-noise sig- system can be approximated by nal amplification in a variety of resonant microsystems, including ¯ 1 ␳ ͓͑˙ ˙ ͒2 ͑˙ ˙ ͒2͔ 1 ͑␺Ј͒2 ͑ ͒ torsional microresonators ͓4,5͔, electric force microscopes ͓6͔, op- L = 2 A u + up + v + vp − 2 EI 1 tically excited micromechanical oscillators ͓7͔, microring gyro- where u, v, and ␺ are defined as in Fig. 1, ͑•˙͒ and ͑•͒Ј represent scopes ͓8͔, microelectromechanical system ͑MEMS͒ diaphragms the temporal and spatial derivatives ͑determined with respect to ͓9͔, coupled microresonators ͓10͔, and microcantilevers ͓11–13͔. time t and arc length variable s͒, u and v specify the imposed While the literature detailed above has demonstrated the vast p p base motion in the longitudinal and transverse directions, and ␳, potential of parametric amplification at the micro- and nanoscales, A E I few, if any, macroscale mechanical amplifiers have been reported. , , and represent the beam’s mass density, cross-sectional area, modulus of elasticity, and cross-sectional moment of inertia, Given that these amplifiers are easily realizable at larger scales, respectively. Noting this and further assuming that the neutral axis the present work seeks to address this void by examining a simple of the beam is inextensible result in a governing variational equa- macroscale parametric amplifier. Specifically, the work details a tion for the system, derived from extended Hamilton’s principle, joint analytical and experimental investigation of mechanical do- given by main parametric amplification in a cantilevered beam with longi- t l tudinal and transverse base excitations. This paper begins in the 2 1 ␦H = ␦͵ ͵ ͭ¯L + ␭͓1−͑1+uЈ͒2 − ͑vЈ͒2͔ͮdsdt 2 t1 0 Contributed by the Technical Committee on Vibration and Sound of ASME for t2 l publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August ͵ ͵ ͑ ␦ ␦ ͒ ͑ ͒ 21, 2007; final manuscript received May 16, 2008; published online October 15, + Qu u + Qv v dsdt =0 2 2008. Assoc. Editor: I. Y. ͑Steve͒ Shen. Paper presented at the ASME 2007 Design t1 0 Engineering Technical Conferences and Computers and Information in Engineering Conference ͑DETC2007͒, Las Vegas, NV, September 4–7, 2007. where l represents the beam’s undeformed length, ␭ denotes the Journal of Vibration and Acoustics Copyright © 2008 by ASME DECEMBER 2008, Vol. 130 / 061006-1 Downloaded 15 Oct 2008 to 128.46.184.236. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm v u v p p ͑ ͒ vˆ = , uˆ p = , vˆ p = 8 v0 v0 v0 and time is scaled by a characteristic period of the system, accord- ing to t ˆt = ͑9͒ T where ␳Al4 T = ͱ ͑10͒ EI This renders a final distributed-parameter system model given by v v ␳Agl3 ␳Agl3 vˆ¨ + cˆvˆ˙ + vˆ iv − 0 uˆ¨ ͑sˆ −1͒vˆЉ − 0 uˆ¨ vˆЈ − ͑sˆ −1͒vˆЉ − vˆЈ l p l p EI EI ¨ ͑ ͒ =−vˆ p 11 where cT cˆ = ͑12͒ ␳A Note that the derivative operators have been redefined here in Fig. 1 Schematic of a representative mechanical parametric terms of the new time and arc length variables, ˆt and sˆ, respec- amplifier: a base-excited cantilevered beam tively. Though the beam’s behavior could potentially be recovered from the distributed-parameter model presented in Eq. ͑11͒,a lumped-mass model proves sufficient for the present analysis. Ac- Lagrange multiplier used to maintain the inextensibility con- cordingly, the dynamic variable vˆ is decomposed into spatial and straint, temporal components using the cantilever’s first mode shape ⌽͑sˆ͒, ͑1+uЈ͒2 + ͑vЈ͒2 =1 ͑3͒ according to ͑ ͒⌽͑ ͒͑͒ and Qu and Qv represent the generalized forces not accounted for vˆ = w ˆt sˆ 13 in the specific Lagrangian, ¯L,intheu and v directions, respec- The result is then projected back onto the first mode shape using tively. Assuming that damping has an appreciable impact on only an inner product operator yielding a final lumped-mass model the local transverse motion of the beam, these forces can be ap- given by proximated by ␧␨ ͓␧␭ ⍀2 ͑⍀␶ ␾͒ ␧␭ ⍀2 ͑ ⍀␶͔͒ zЉ +2 zЈ + z + 1 cos + + 2 cos 2 z ␳ ˙ ͑ ͒ Qu =− Ag, Qv =−cv 4 ␧␩ ⍀2 ͑⍀␶ ␾͒ ␧␩ ⍀2 ͑ ⍀␶͒͑͒ = 1 cos + + 2 cos 2 14 where g represents the beam’s acceleration due to gravity and c with nondimensional parameters and operators defined as in Table represents a specific viscous damping coefficient. Substituting 1. Note that the imposed harmonic base motions included here are each of these forces, as well as the specific Lagrangian, into Eq. ˆ ͑2͒, integrating by parts successively, and introducing the kine- assumed to result from a unidirectional base excitation xp given matic constraint relating the angle ␺ to the planar displacements u by and v, ˆ ͑␻ ␾͒ ˆ ͑ ␻ ͒ ˆ ͑␻ ␾͒ ˆ ͑ ␻ ͒ xˆp = A cos t + + B cos 2 t = A cos ˆ ˆt + + B cos 2 ˆ ˆt vЈ ͑ ͒ ␺ ͑ ͒ 15 tan = Ј 5 1+u and are defined according to results in a pair of coupled governing equations. The equation of uˆ = xˆ sin ␣, vˆ = xˆ cos ␣ ͑16͒ motion governing transverse dynamics can be recovered from this p p p p set, by first solving the longitudinal equation to obtain the Excitations of frequency ␻ are used to provide direct excitation to Lagrange multiplier and then substituting the obtained function the system, while excitations of frequency 2␻ are used for para- into the second equation. Truncating the result such that only lin- metric pumping. Also note that the phase-dependent nature of this ear terms remain ͑only linear amplifier operation is deemed perti- excitation is requisite in the examination of degenerate parametric nent and practical here͒ yields a distributed-parameter system amplification. An examination of nondegenerate amplification, model given by which is phase independent and does not require the strict 2:1 frequency ratio utilized above, is left for subsequent studies. ␳ iv ␳ ͑ ͒ ␳ ␳ ͑ ͒ Av¨ + cv˙ + EIv − vЉ Au¨ p s − l − vЈ Au¨ p − vЉ Ag s − l ␳ ␳ ͑ ͒ − vЈ Ag =− Av¨ p 6 To reduce the number of free parameters in the system detailed 3 Analysis ͑ ͒ above, it proves convenient to rescale Eq.
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