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Research Papers

Parametric Instability in a Taut String With a Periodically Moving Boundary

[+] Author and Article Information
K. Wu

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

W. D. Zhu

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

1Corresponding author.

Manuscript received August 15, 2013; final manuscript received November 25, 2013; accepted manuscript posted December 9, 2013; published online January 30, 2014. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 81(6), 061002 (Jan 30, 2014) (23 pages) Paper No: JAM-13-1345; doi: 10.1115/1.4026181 History: Received August 15, 2013; Revised November 25, 2013; Accepted December 09, 2013

Parametric instability in a taut string with a periodically moving boundary, which is governed by a one-dimensional wave equation with a periodically varying domain, is investigated. Parametric instability usually occurs when coefficients in governing differential equations of a system periodically vary, and the system is said to be parametrically excited. Since the governing partial differential equation of the string with a periodically moving boundary can be transformed to one with a fixed domain and periodically varying coefficients, the string is parametrically excited and instability caused by the periodically moving boundary is classified as parametric instability. The free linear vibration of a taut string with a constant tension, a fixed boundary, and a periodically moving boundary is studied first. The exact response of the linear model is obtained using the wave or d'Alembert solution. The parametric instability in the string features a bounded displacement and an unbounded vibratory energy, and parametric instability regions in the parameter plane are classified as period-i (i1) parametric instability regions, where period-1 parametric instability regions are analytically obtained using the wave solution and the fixed point theory, and period-i (i>1) parametric instability regions are numerically calculated using bifurcation diagrams. If the periodic boundary movement profile of the string satisfies certain condition, only period-1 parametric instability regions exist. However, parametric instability regions with higher period numbers can exist for a general periodic boundary movement profile. Three corresponding nonlinear models that consider coupled transverse and longitudinal vibrations of the string, only the transverse vibration, and coupled transverse and axial vibrations are introduced next. Responses and vibratory energies of the linear and nonlinear models are calculated for both stable and unstable cases using three numerical methods: Galerkin's method, the explicit finite difference method, and the implicit finite difference method; advantages and disadvantages of each method are discussed. Numerical results for the linear model can be verified using the exact wave solution, and those for the nonlinear models are compared with each other since there are no exact solutions for them. It is shown that for parameters in the parametric instability regions of the linear model, the responses and vibratory energies of the nonlinear models are close to those of the linear model, which indicates that the parametric instability in the linear model can also exist in the nonlinear models. The mechanism of the parametric instability is explained in the linear model and through axial strains in the third nonlinear model.

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Figures

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Fig. 1

Schematic of a taut string with a periodically moving boundary with indications of transverse, longitudinal, and axial vibrations at a point on the string

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Fig. 2

Domain of the governing equation in Eq. (1) with four different forms of the wave solution

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Fig. 3

Illustration of reflections of an infinitesimal wave from two boundaries of the string

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Fig. 4

(a) Different boundary movement profiles: sinusoidal function (solid line), piecewise linear function (short dashed line), piecewise quadratic function (long dashed line), and piecewise cubic function (dash-dotted line); (b) corresponding period-1 parametric instability regions, and (c) a bifurcation diagram for the sinusoidal boundary movement profile

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Fig. 5

(a) Boundary movement profile l(t)=1 - A0[cos(ωt)+cos(2ωt)]/1.5625+0.28A0, (b) period-1 through period-3 parametric instability regions, and (c) a bifurcation diagram for the boundary movement profile

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Fig. 6

(a) Piecewise linear boundary movement profile given by Eq. (54), (b) period-1 through period-3 parametric instability regions, and (c) a bifurcation diagram for the boundary movement profile

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Fig. 7

(a) Transverse displacements, (b) vibratory energies, and (c) the vibratory energy with a longer time duration of the linear string model with (ω, A0)=(2.5 rad/s, 0.06 m), which is in a stable region in Fig. 5(b): solid lines, the exact wave solution; short dashed lines, Galerkin's method; long dashed lines, the explicit finite difference method; and dash-dotted lines, the implicit finite difference method

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Fig. 8

(a) Transverse displacements and (b) vibratory energies of the linear string model with (ω, A0)=(π rad/s, 0.06 m), which is in a period-1 parametric instability region in Fig. 5(b): solid lines, the exact wave solution; short dashed lines, Galerkin's method; long dashed lines, the explicit finite difference method; and dash-dotted lines, the implicit finite difference method

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Fig. 9

(a) Transverse displacements and (b) vibratory energies of the linear string model with (ω,A0)=(1.5π rad/s,0.06 m), which is in a period-2 parametric instability region in Fig. 5(b): solid lines, the exact wave solution; short dashed lines, Galerkin's method; long dashed lines, the explicit finite difference method; and dash-dotted lines, the implicit finite difference method

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Fig. 10

(a) Transverse displacements, (b) longitudinal displacements, (c) vibratory energies, (d) the relative axial displacement, and (e) the vibratory energy with a longer time duration of the first nonlinear string model with (ω,A0)=(2.5 rad/s,0.06 m), which is in a stable region in Fig. 5(b): short dashed lines, Galerkin's method; long dashed lines, the explicit finite difference method; and dash-dotted lines, the implicit finite difference method

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Fig. 11

(a) Transverse displacements, (b) longitudinal displacements, (c) vibratory energies, and (d) the relative axial displacement of the first nonlinear string model with (ω,A0)=(π rad/s,0.06 m), which is in a period-1 parametric instability region in Fig. 5(b): short dashed lines, Galerkin's method; long dashed lines, the explicit finite difference method; and dash-dotted lines, the implicit finite difference method

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Fig. 12

(a) Transverse displacements, (b) longitudinal displacements, (c) vibratory energies, and (d) the relative axial displacement of the first nonlinear string model with (ω,A0)=(1.5π rad/s,0.06 m), which is in a period-2 parametric instability region in Fig. 5(b): short dashed lines, Galerkin's method; long dashed lines, the explicit finite difference method; and dash-dotted lines, the implicit finite difference method

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Fig. 13

(a) Transverse displacements and (b) vibratory energies of the second nonlinear string model with (ω,A0)=(2.5 rad/s,0.06 m), which is in a stable region in Fig. 5(b): short dashed lines, Galerkin's method; long dashed lines, the explicit finite difference method; and dash-dotted lines, the implicit finite difference method

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Fig. 14

(a) Transverse displacements and (b) vibratory energies of the second nonlinear string model with (ω,A0)=(π rad/s,0.06 m), which is in a period-1 parametric instability region in Fig. 5(b): short dashed lines, Galerkin's method; long dashed lines, the explicit finite difference method; and dash-dotted lines, the implicit finite difference method

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Fig. 15

(a) Transverse displacements and (b) vibratory energies of the second nonlinear string model with (ω,A0)=(1.5π rad/s,0.06 m), which is in a period-2 parametric instability region in Fig. 5(b): short dashed lines, Galerkin's method; long dashed lines, the explicit finite difference method; and dash-dotted lines, the implicit finite difference method

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Fig. 16

(a) The transverse displacement, (b) the relative axial displacement, (c) the axial strain, and (d) the vibratory energy of the third nonlinear string model with (ω,A0)=(2.5 rad/s,0.06 m), which is in a stable region in Fig. 5(b)

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Fig. 17

(a) The transverse displacement, (b) the relative axial displacement, (c) the axial strain, and (d) the vibratory energy of the third nonlinear string model with (ω,A0)=(π rad/s,0.06 m), which is in a period-1 parametric instability region in Fig. 5(b)

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Fig. 18

(a) The transverse displacement, (b) the relative axial displacement, (c) the axial strain, and (d) the vibratory energy of the third nonlinear string model with (ω,A0)=(1.5π rad/s,0.06 m), which is in a period-2 parametric instability region in Fig. 5(b)

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Fig. 19

Transverse displacements of the string at t=5 s calculated from (a) the explicit finite difference method, and (b) the exact wave solution and the implicit finite difference method, with (ω,A0)=(π rad/s,0.05 m), which is in a period-1 parametric instability region in Fig. 5(b): long dashed line, the explicit finite difference method; solid line, the exact wave solution; and dash-dotted line, the implicit finite difference method

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Fig. 20

Longitudinal displacements of the string at t=5 s calculated from (a) the explicit finite difference method and (b) Galerkin's and implicit finite difference methods, and transverse displacements at t=5 s calculated from (c) Galerkin's method, the explicit finite difference method, and the implicit finite difference method, with (ω,A0)=(π rad/s,0.06 m), which is in a period-1 parametric instability region in Fig. 5(b): long dashed lines, the explicit finite difference method; short dashed lines, Galerkins method; and dash-dotted lines, the implicit finite difference method

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Fig. 21

(a) Transverse displacements and (b) vibratory energies of the linear string model with (ω,A0)=(π rad/s,0.2 m), which is in a period-1 parametric instability region in Fig. 5(b): solid lines, the exact wave solution; short dashed lines, the implicit finite difference method with (M,N)=(501,5001); dash-dotted lines, the implicit finite difference method with (M,N)=(101,5001); and long dashed lines, the implicit finite difference method with (M,N)=(101,5001) and numerical damping ς=0.004

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