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Research Papers

# Dynamic Stability of a Class of Second-Order Distributed Structural Systems With Sinusoidally Varying Velocities

[+] Author and Article Information
W. D. Zhu

Professor
Fellow ASME

K. Wu

Department of Mechanical Engineering,
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250

1Corresponding author.

Manuscript received August 31, 2012; final manuscript received December 4, 2012; accepted manuscript posted February 12, 2013; published online August 19, 2013. Assoc. Editor: Wei-Chau Xie.

J. Appl. Mech 80(6), 061008 (Aug 19, 2013) (15 pages) Paper No: JAM-12-1429; doi: 10.1115/1.4023638 History: Received August 31, 2012; Revised December 04, 2012; Accepted February 12, 2013

## Abstract

Parametric instability in a system is caused by periodically varying coefficients in its governing differential equations. While parametric excitation of lumped-parameter systems has been extensively studied, that of distributed-parameter systems has been traditionally analyzed by applying Floquet theory to their spatially discretized equations. In this work, parametric instability regions of a second-order nondispersive distributed structural system, which consists of a translating string with a constant tension and a sinusoidally varying velocity, and two boundaries that axially move with a sinusoidal velocity relative to the string, are obtained using the wave solution and the fixed point theory without spatially discretizing the governing partial differential equation. There are five nontrivial cases that involve different combinations of string and boundary motions: (I) a translating string with a sinusoidally varying velocity and two stationary boundaries; (II) a translating string with a sinusoidally varying velocity, a sinusoidally moving boundary, and a stationary boundary; (III) a translating string with a sinusoidally varying velocity and two sinusoidally moving boundaries; (IV) a stationary string with a sinusoidally moving boundary and a stationary boundary; and (V) a stationary string with two sinusoidally moving boundaries. Unlike parametric instability regions of lumped-parameter systems that are classified as principal, secondary, and combination instability regions, the parametric instability regions of the class of distributed structural systems considered here are classified as period-1 and period-$i$ ($i>1$) instability regions. Period-1 parametric instability regions are analytically obtained; an equivalent total velocity vector is introduced to express them for all the cases considered. While period-$i$ ($i>1$) parametric instability regions can be numerically calculated using bifurcation diagrams, it is shown that only period-1 parametric instability regions exist in case IV, and no period-$i$ ($i>1$) parametric instability regions can be numerically found in case V. Unlike parametric instability in a lumped-parameter system that is characterized by an unbounded displacement, the parametric instability phenomenon discovered here is characterized by a bounded displacement and an unbounded vibratory energy due to formation of infinitely compressed shock-like waves. There are seven independent parameters in the governing equation and boundary conditions, and the parametric instability regions in the seven-dimensional parameter space can be projected to a two-dimensional parameter plane if five parameters are specified. Period-1 parametric instability occurs in certain excitation frequency bands centered at the averaged natural frequencies of the systems in all the cases. If the parameters are chosen to be in the period-$i$ ($i≥1$) parametric instability region corresponding to an integer $k$, an initial smooth wave will be infinitely compressed to $k$ shock-like waves as time approaches infinity. The stable and unstable responses of the linear model in case I are compared with those of a corresponding nonlinear model that considers the coupled transverse and longitudinal vibrations of the translating string and an intermediate linear model that includes the effect of the tension change due to axial acceleration of the string on its transverse vibration. The parametric instability in the original linear model can exist in the nonlinear and intermediate linear models.

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## Figures

Fig. 1

Schematic of a translating string with a sinusoidally varying velocity and a sinusoidally varying domain

Fig. 2

Domains of the (a) original and (b) transformed equations and their boundary curves; waves from a point (ξ,η) in the region L1 in (b) propagate to the point (ξ˜,η˜) after one reflection from the two boundary curves

Fig. 3

Compression of an infinitesimal wave after reflection from the left boundary x = xl(t)

Fig. 4

Sum of three velocity vectors ν→eq1, ν→eqr, and ν→eql in the polar coordinates

Fig. 7

Wave pattern of the translating string in case II with (ω,ν1) = (3.5,0.3), which is in the period-2 parametric instability region with k = 3, at t = 96

Fig. 8

Wave pattern of the translating string in case II with (ω,ν1) = (3.15,0.3), which is in the period-3 parametric instability region with k = 4, at t = 312

Fig. 11

Wave pattern of the translating string in case III with (ω,ν1) = (3.4,0.3), which is in the period-2 parametric instability region with k = 3, at t = 48

Fig. 5

Period-1 through period-3 parametric instability regions in case II with ν0 = 0.5, ν21 = 0.1414, ν22 = 0, φ1 = 0, and φ2 = 0

Fig. 6

Wave pattern of the translating string in case II with (ω,ν1) = (2.3562,0.3), which is in the period-1 parametric instability region with k = 1, at t = 24

Fig. 9

Period-1 through period-3 parametric instability regions in case III with ν0 = 0.5, ν21 = 0.1414, ν22 = 0.1414, φ1 = 0.5π, and φ2 = 1.5π

Fig. 10

Wave pattern of the translating string in case III with (ω,ν1) = (2.3562,0.3), which is in the period-1 parametric instability region with k = 1, at t = 24

Fig. 12

Wave pattern of the translating string in case III with (ω,ν1) = (3.055,0.3), which is in the period-3 parametric instability region with k = 4, at t = 72

Fig. 13

Vibratory energy growth in case III: (a) (ω,ν1) = (2.3562,0.3), which is in the period-1 parametric instability region; (b) (ω,ν1) = (3.4,0.3), which is in the period-2 parametric instability region; and (c) (ω,ν1) = (3.055,0.3), which is in the period-3 parametric instability region

Fig. 15

Wave patterns of the stationary string in case IV with period-1 parametric instability at t = 24: (a) (ω,ν21) = (π,0.3) corresponding to k = 1, (b) (ω,ν21) = (2π,0.3) corresponding to k = 2, (c) (ω,ν21) = (3π,0.3) corresponding to k = 3, and (d) (ω,ν21) = (4π,0.3) corresponding to k = 4

Fig. 16

Period-1 parametric instability regions in case V with ν0 = 0, ν1 = 0, ν22 = 0.1414, φ1 = 0.5π, and φ2 = 1.5π

Fig. 17

Wave patterns of the stationary string in case V with period-1 parametric instability at t = 24: (a) (ω,ν21) = (π,0.3,) corresponding to k = 1, (b) (ω,ν21) = (2π,0.3) corresponding to k = 2, (c) (ω,ν21) = (3π,0.3) corresponding to k = 3, and (d) (ω,ν21) = (4π,0.3) corresponding to k = 4

Fig. 18

Schematic of a translating string with a constant length and an arbitrary translation velocity V(T)

Fig. 14

Period-1 parametric instability regions in case IV with ν0 = 0, ν1 = 0, ν22 = 0, φ1 = 0, and φ2 = 0

Fig. 19

Transverse displacement of the translating string at X = 5 m calculated using Galerkin’s method from (a, b) the original linear model, (c, d) the nonlinear model, and (e, f) the intermediate linear model for (a, c, e) V(T) = 15.7720 + 3.1544 sin(6.3088T) m/s, which corresponds to stable solutions, and (b, d, f) V(T) = 15.7720 + 7.4696 sin(7.7844T) m/s, which corresponds to unstable solutions; 50 terms are retained in (a) and (e), 1000 terms are retained in (b) and (f), 10 terms are retained for U(X,T) and W(X,T) in (c), and 40 terms are retained for U(X,T) and W(X,T) in (d)

Fig. 20

Transverse vibratory energy of the translating string from (a, b) the original linear model, (c, d) the nonlinear model, and (e, f) the intermediate linear model for (a, c, e) V(T) = 15.7720 + 3.1544 sin(6.3088T) m/s, which corresponds to stable solutions, and (b, d, f) V(T) = 15.7720 + 7.4696 sin(7.7844T) m/s, which corresponds to unstable solutions

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