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 degener- ate (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 oscil- lators [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 ex- amining 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, ob- tained 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 neg- ligible 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 iner- tia, 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, ac- cording 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 exami- nation 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˙+ vˆiv − 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).