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Research Papers

# Parametric Instability of a Traveling Plate Partially Supported by a Laterally Moving Elastic Foundation

[+] Author and Article Information
V. Kartik1

IBM Zurich Research Laboratory, CH 8803 Rüschlikon, Switzerlandkve@zurich.ibm.com

J. A. Wickert

Department of Mechanical Engineering, Iowa State University, Ames, IA 50011

The model is applied for the parameter values $h∗=1.88×10−4$, $δ∗=0.02$, $α=4$, $ν=0.3$, $T∗=5.58×103$, $v∗=90$, $kf∗=7×106$, $Lf∗=0.25$, and $Bf∗=0.35$, which are representative of the motivating application, unless noted otherwise.

A natural frequency here refers to the magnitude of the imaginary part of the corresponding eigenvalue $γ∗$, i.e., $ω∗=|Im(γ∗)|$.

1

Corresponding author.

J. Vib. Acoust 130(5), 051006 (Aug 13, 2008) (8 pages) doi:10.1115/1.2948394 History: Received May 04, 2007; Revised January 31, 2008; Published August 13, 2008

## Abstract

The parametric excitation of an axially moving plate is examined in an application where a partial foundation moves in the plane of the plate and in a direction orthogonal to the plate’s transport. The stability of the plate’s out-of-plane vibration is of interest in a magnetic tape data storage application where the read/write head is substantially narrower than the tape’s width and is repositioned during track-following maneuvers. In this case, the model’s equation of motion has time-dependent coefficients, and vibration is excited both parametrically and by direct forcing. The parametric instability of out-of-plane vibration is analyzed by using the Floquet theory for finite values of the foundation’s range of motion. For a relatively soft foundation, vibration is excited preferentially at the primary resonance of the plate’s fundamental torsional mode. As the foundation’s stiffness increases, multiple primary and combination resonances occur, and they dominate the plate’s stability; small islands, however, do exist within unstable zones of the frequency-amplitude parameter space for which vibration is marginally stable. The plate’s and foundation’s geometry, the foundation’s stiffness, and the excitation’s amplitude and frequency can be selected in order to reduce undesirable vibration that occurs along the plate’s free edge.

###### FIGURES IN THIS ARTICLE
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Copyright © 2008 by American Society of Mechanical Engineers
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## Figures

Figure 3

Schematic of a traveling plate in contact with a partial elastic foundation (shaded) that moves in the direction y orthogonal to the plate’s transport

Figure 4

Dependence of the plate’s natural frequencies on the transport speed: kf∗=7×106, (xf∗,yf∗)=(0.5,0.5), and (M,N)=(5,7)

Figure 1

A conventional read/write assembly of width substantially greater than the tape’s

Figure 2

Measured lateral tape vibration: (a) time record and (b) frequency spectrum

Figure 5

Dependence of the plate’s natural frequencies on the foundation’s stiffness: v∗=90; (xf∗,yf∗)=(0.5,0.5) (dot line type) and (xf∗,yf∗)=(0.5,0.35) (solid line type); (M,N)=(5,7). Loci for the bending modes (0, 0) and (1, 0) at kf∗=0 are also shown for comparison (dot-dash line type).

Figure 6

(a) Response of the point (0.3, 1.0) on the plate’s free edge to transient repositioning of the foundation (t0∗=2×10−3) and (b) variation of the peak overshoot with the repositioning time; A0∗=0.15 and (M,N)=(3,4)

Figure 7

Transient profiles of the plate during repositioning of the elastic foundation: (a) lateral cross section at x∗=0.3 and (b) free edge at y∗=1; t0∗=2×10−3, A0∗=0.15, and (M,N)=(3,4). The shape prior to repositioning is shown in the dashed line type, and the steady-state shape (t∗=∞) is shown in the bold line type.

Figure 8

Contact pressure distributions between the plate and the elastic foundation during and after a transient repositioning maneuver: t0∗=2×10−3, A0∗=0.15, and (M,N)=(3,4)

Figure 9

Stable and unstable (shaded) regions of the plate’s homogeneous response in the foundation’s frequency-amplitude parameter plane: kf∗=7×106 and (M,N)=(1,4)

Figure 11

Stable and unstable (shaded) regions of the plate’s homogeneous response in the foundation’s frequency-amplitude parameter plane: kf∗=7×108 and (M,N)=(1,4)

Figure 12

Representative response of the plate at position (0.3,1.0) for excitation amplitude and frequency combinations corresponding to the marginally stable parameters indicated in Fig. 1: (a) Ω0∗=6800, A0∗=0.12 and (b) Ω0∗=7600, A0∗=0.07; (M,N)=(1,4)

Figure 13

Frequency response at position (0.3,1.0) on the plate’s edge under harmonic foundation motion: A0∗=8×10−3, kf∗=7×106, and (M,N)=(3,4)

Figure 14

Frequency response of the plate-foundation contact pressure at position (0.5,yf∗(t∗)−B0∗/2) on the foundation’s edge under harmonic foundation motion: A0∗=8×10−3, kf∗=7×106, and (M,N)=(3,4)

Figure 10

Representative responses of the plate at position (0.3,1.0) for excitation amplitude and frequency combinations corresponding to the (a) stable (Ω0∗=4000, A0∗=0.08) and (b) unstable (Ω0∗=5200, A0∗=0.08) regions indicated in Fig. 9; (M,N)=(1,4)

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