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Research Papers

Frequency Response of Cylindrical Resonators in a Viscous Fluid

[+] Author and Article Information
Michael J. Martin1

Naval Research Laboratory, Washington, DC 20375-5320martinm2@asme.org

Brian H. Houston

Naval Research Laboratory, Washington, DC 20375-5320

1

Corresponding author. Present address: Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA 70809.

J. Vib. Acoust 133(3), 031009 (Mar 29, 2011) (6 pages) doi:10.1115/1.4003203 History: Received February 07, 2010; Revised August 30, 2010; Published March 29, 2011; Online March 29, 2011

Abstract

The frequency response of a cylinder in a viscously damped fluid is a problem of fundamental engineering interest, with applications ranging from microsystems to offshore structures. The analytic solution for the drag in a vibrating cylinder in the laminar flow regime is combined with the equations of motion for forced vibration of a cylinder attached to a spring. The resulting model gives an analytic solution for the dynamic response of the system, including the gain, frequency lag, resonant frequency, quality factor, and stability of the system. The results show that the response of the system is nonlinear, with the phase of the system differing from the phase predicted by linear models. The gain, quality factor, resonant frequency, and crossover frequency all increase with the nondimensional natural frequency $β$ and decrease with the ratio of the fluid density to the resonator density.

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Figures

Figure 1

Cylinder and spring geometry

Figure 2

Lumped-parameter model of cylindrical resonator

Figure 3

Bode plots and Nyquist diagram for m′=0.01: (a) gain versus s, (b) phase versus s, and (c) Im[G(s)] versus Re[G(s)]

Figure 4

Bode plots and Nyquist diagram for m′=0.1: (a) gain versus s, (b) phase versus s, and (c) Im[G(s)] vs Re[G(s)]

Figure 5

Bode plots and Nyquist diagram for m′=1.0: (a) gain versus s, (b) phase versus s, and (c) Im[G(s)] versus Re[G(s)]

Figure 6

Bode plots and Nyquist diagram for m′=10.0: (a) gain versus s, (b) phase versus s, and (c) Im[G(s)] versus Re[G(s)]

Figure 7

ωres/ωn versus βn

Figure 8

Gain at sn versus βn

Figure 9

ωc/ωn versus βn

Figure 10

Phase margin

Figure 11

Q versus βn

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