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Research Papers: Hydrodynamic Lubrication

# On Squeeze Film Damping in Microsystems

[+] Author and Article Information
Victor Marrero, Diana-Andra Borca-Tasciuc

Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Instituteci, Troy, NY 12180-3590

John Tichy

Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Instituteci, Troy, NY 12180-3590tichyj@rpi.edu

J. Tribol 132(3), 031701 (Jun 04, 2010) (6 pages) doi:10.1115/1.4001620 History: Received May 06, 2009; Revised April 09, 2010; Published June 04, 2010; Online June 04, 2010

## Abstract

Classical hydrodynamic lubrication theory has been one of the most successful and widely used theories in all of engineering and applied science. This theory predicts that the force resisting the squeezing of a fluid between two parallel plates is inversely proportional to the cube of the fluid thickness. However, recent reports on liquid squeeze film damping in microsystems appear to indicate that experimentally measured damping force is proportional to the inverse of the fluid thickness to the first power—a large fundamental discrepancy from classical theory. This paper investigates potential limitations of lubrication theory in microsystems by theoretical and computational methods. The governing equations for a Newtonian incompressible fluid are solved subject to two-dimensional, parallel surface squeezing by an open-source computational fluid dynamics program called parallel hierarchic adaptive stabilized transient analysis (PHASTA ), and by a classical similarity solution technique. At low convective Reynolds numbers, the damping force is determined as a function of the ratio of a reference film thickness $H$ to a reference direction $B$ along the film. Good agreement with classical lubrication theory is found for aspect ratios $H/B$ as high as 1 despite the fact that lubrication theory requires that this ratio be “small.” A similarity analysis shows that when instantaneous convective Reynolds number is of order 10–100 (a range present in experiment), calculated damping deviates significantly from lubrication theory. This suggests that nonlinearity associated with high Reynolds numbers could explain the experimentally observed discrepancy in damping force. Dynamic analysis of beams undergoing small vibrations in the presence of a liquid medium further supports this finding.

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## Figures

Figure 1

Schematic of cantilever and squeeze film, side and end views

Figure 2

Squeeze force in arbitrary units, dots are computed points, and line is lubrication theory=const×Vr/(h/B)3

Figure 3

Velocity profiles and streamlines: h/B=0.1 and h/B=1.0

Figure 4

Reynolds number and squeeze force versus time, similarity solution n=1/2. The solid line denotes Ref=0.1 and the dashed line denotes Ref=0.5.

Figure 5

Dimensionless damping coefficient versus time, similarity solution n=1/2. The solid line denotes Ref=0.1 and the dashed line denotes Ref=0.5.

Figure 6

Beam deflection profile: local beam deflection divided by applied force per length. The solid line is 80 kHz and the dashed line is 40 kHz.

Figure 7

Response of cantilever: amplitude of beam end divided by applied force per length. The line with sharp peaks indicates structural damping only.

## Errata

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