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Research Papers

# Mechanical Domain Parametric Amplification

[+] Author and Article Information

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907jfrhoads@purdue.edu

Nicholas J. Miller

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824mille820@msu.edu

Steven W. Shaw

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824shawsw@egr.msu.edu

Brian F. Feeny

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824feeny@egr.msu.edu

J. Vib. Acoust 130(6), 061006 (Oct 15, 2008) (7 pages) doi:10.1115/1.2980382 History: Received August 21, 2007; Revised May 16, 2008; Published October 15, 2008

## Abstract

Though utilized for more than $50years$ in a variety of 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 engineering community. In fact, only within the past $15–20years$ has the technique been implemented in micromechanical 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 number of micro- and nanomechanical systems (as well as a number electrical systems), few, if any, macroscale mechanical amplifiers have been reported. Given that these amplifiers are easily realizable using larger-scale mechanical systems, the present work seeks to address this void by examining a simple representative example: a cantilevered beam with longitudinal and transverse base excitations. The work begins with the systematic formulation of a representative system model, which is used to derive a number of pertinent metrics. A series of experimental results, which validate the work’s analytical findings, are subsequently examined, and the work concludes with a brief look at some plausible applications of parametric amplification in macroscale mechanical systems.

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## Figures

Figure 1

Schematic of a representative mechanical parametric amplifier: a base-excited cantilevered beam

Figure 2

Wedges of instability, recovered for various levels of damping, near σ=0. Note that successful linear parametric amplification requires that the system operates below its corresponding wedge, as parametric resonance occurs within the instability region.

Figure 3

Amplifier gain G(σ=0) plotted versus normalized pump amplitude λ2∕λ2,crit for an oscillator operating at σ=0 with ϕ=−π∕4. Note that the pump amplitude has been normalized such that the parametric instability (be it analytically or experimentally determined) occurs at unity. Also note that in this figure lines are used to designate analytical results and data points are used to designate experimental results. Finally, note that here, and in Figs.  45, ζ=0.065.

Figure 4

Frequency response curves, a¯∕max{a¯(λ2=0)} versus σ, corresponding to three different values of λ2 (pump amplitudes). Note that the responses have been normalized such that the unpumped system has a resonant amplitude of unity. Also note that in this figure, lines are used to designate analytical results and data points are used to designate experimental results.

Figure 5

Gain G(σ=0) versus phase ϕ. Note that in this figure, lines are used to designate analytical results and data points are used to designate experimental results.

Figure 6

The experimental setup used to obtain the results included in Figs.  345

Figure 7

Block diagram of the experimental setup used to obtain the results included in Figs.  345

Figure 8

The cantilever driven near resonance (a) without and (b) with a strong parametric pump

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