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TECHNICAL PAPERS

Surface Friction Guiding for Reduced High-Frequency Lateral Vibration of Moving Media

[+] Author and Article Information
V. Kartik

IBM Research Laboratory, CH-8803 Rueschlikon, Zurich, Switzerland

J. A. Wickert1

Department of Mechanical Engineering,  Carnegie Mellon University, Pittsburgh, PA 15213wickert@cmu.edu

Unless noted otherwise, the model parameters are as follows: axial friction coefficient, $μx=0.1$; lateral friction coefficient, $μz=0.1$; tension at $x*=0$, $T0*=8.65$; transport speed, $v*=0.15$; central position of guide 1, $x*=0.3$; central position of guide 2, $x*=0.7$; guide wrap angles, $ϕ(i)=π∕4$; and guide radii, $Rg*(i)=0.159$.

1

Corresponding author.

J. Vib. Acoust 129(3), 371-379 (Oct 25, 2006) (9 pages) doi:10.1115/1.2732354 History: Received May 03, 2006; Revised October 25, 2006

Abstract

The free and forced vibration of a moving medium is examined in an application where distributed friction guiding is used to control lateral position passively. Subambient pressure features formed in the guides intentionally modify the naturally occurring self-pressurized air bearing and increase the contact force between the medium and the guide’s surface. These features increase friction to a level beyond that achievable based on the nominal wrap pressure. The moving medium is modeled as a beam that is transported over frictional regions and subjected to prescribed boundary disturbances arising from runout of a supply or take-up roll. For axial transport at a speed that is high compared to the velocity of lateral vibration, Coulomb friction between the guides and the moving medium can be well approximated by a derived expression for equivalent viscous damping. The equation of motion is developed for the cases of a single cylindrical guide and of a multiplicity of guides having arbitrary placement. The level of equivalent damping for each mode decreases with transport speed, and critical speeds exist where each vibration mode transitions between the overdamped and underdamped regimes. Parameter studies in the contact pressure, transport speed, and guide geometry identify preferred design configurations for maximizing dissipation in particular modes and for attenuating high-frequency response.

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Figures

Figure 1

(a) Magnetic tape is transported over a cylindrical surface friction guide that incorporates a subambient pressure feature. (b) The pressure relief channel vents the self-generated air bearing.

Figure 2

(a) Tape transport test stand for friction guiding measurements, (b) the path comprises three flanged posts and one test guide

Figure 3

Measured lateral vibration at the test guide’s location (a) with a conventional air bearing guide and (b) with a friction guide that incorporates subambient pressure features

Figure 4

Measured spectra for lateral vibration at the test guide’s location (a) with a conventional air bearing guide and (b) with a friction guide that incorporates subambient pressure features

Figure 5

Schematic diagram of a beam translating over two distributed contact friction guides and excited by prescribed displacements at its boundaries

Figure 6

Detail of the contact region on a friction guide

Figure 7

(a) Real and (b) imaginary eigenvalue component loci with increasing transport speed; μz=0.1 and α=0.2. The first two modes have branches labeled (i) and (ii) that bifurcate at critical speeds for overdamping. The frequency loci for an unguided beam are shown by the lighter line.

Figure 8

Evolution of the eigenvalue loci with transport speed for three choices of guide placement: (a) full friction guiding over x*∊(0,1), (b) partial guiding over x*∊[x0*(1),xe*(2)], and (c) two narrow guides with placement as in Fig. 5

Figure 9

(a) Real and (b) imaginary eigenvalue component loci with increasing subambient pressure, μz=0.1 and v*=0.4. The fundamental mode bifurcates into branches 1(i) and 1(ii) above the critical pressure coefficient for overdamping. The frequency loci for an unguided beam are shown by the lighter line.

Figure 10

First five mode shapes; α=0.02. Real components are shown by the solid line, and imaginary components, the dashed line

Figure 11

Hatched region denotes v*−α parameter combinations where the fundamental vibration mode is overdamped

Figure 12

Hatched region denotes v*−ϕ parameter combinations where the fundamental vibration mode is overdamped

Figure 13

Deflection envelope over one cycle of excitation with (a) ambient contact pressure (α=0) and (b) subambient contact pressure (α=0.2); Ω1*=20. The dashed line in (a) denotes the envelope in the absence of friction.

Figure 14

Frequency response function at x*=0.45 for different levels of friction guiding

Figure 15

Analogous single degree-of-freedom oscillator that undergoes lateral vibration in a frame that slides at speed v in the orthogonal direction over a surface with Coulomb friction. Force N acts orthogonal to the plane.

Figure 16

Transient response of the analogous single-degree-of-freedom oscillator and frame system moving over a surface with Coulomb friction. The displacement envelope based on the approximation 2 (dark line) is superposed on the numerical solution to Eq. 1 (lighter line).

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