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Research Papers

On the Static and Dynamic Contact Problem of an Inflated Spherical Viscoelastic Membrane

[+] Author and Article Information
Nirmal Kumar

Department of Mechanical Engineering and
Centre for Theoretical Studies,
Indian Institute of Technology Kharagpur,
Kharagpur 721302, India
e-mail: nirmal.mit@gmail.com

Anirvan DasGupta

Department of Mechanical Engineering and
Centre for Theoretical Studies,
Indian Institute of Technology Kharagpur,
Kharagpur 721302, India
e-mail: anir@mech.iitkgp.ernet.in

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received June 5, 2015; final manuscript received August 29, 2015; published online October 1, 2015. Assoc. Editor: Nick Aravas.

J. Appl. Mech 82(12), 121010 (Oct 01, 2015) (8 pages) Paper No: JAM-15-1296; doi: 10.1115/1.4031484 History: Received June 05, 2015; Revised August 29, 2015

Inflated membrane structures, useful in vibration/shock isolation devices, terrestrial and space structures, etc., rely on the internal dissipation in the membrane for vibration attenuation. In this work, using the Christensen viscoelastic material model, we study the contact mechanics, displacement-controlled relaxation response, force-controlled creep response, dynamic contact, and energy dissipation due to oscillatory contact in an inflated spherical nonlinear viscoelastic membrane. We consider an inflated spherical membrane squeezed between two large rigid, frictionless, parallel plates. The effective stiffness and damping in the membrane–plate assembly are determined, and a phenomenological model is developed. Under oscillatory contact condition, the energy dissipation per cycle is determined. Further, using the free-vibration test, the damped natural frequency of the membrane–plate system is calculated.

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Figures

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Fig. 1

Geometry of spherical membrane before and after inflation and contact with two large rigid plates

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Fig. 2

Free-body diagram of an elementary part of the membrane in noncontacting region

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Fig. 3

Relaxation response of the membrane for C = 1

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Fig. 4

Creep response of the membrane for C = 1

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Fig. 5

Variation of (a) amplitude and (b) phase of the contact force during oscillatory motion with frequency of oscillation ω for material constant C = 1. Note that the farthest curve from the list of initial stretches λs and mean equilibrium positions y0 correspond to the first initial stretch and mean equilibrium position values, respectively.

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Fig. 6

Hysteresis cycles for material constant C = 1, equilibrium force F0=6.81, amplitude of oscillation yA=0.05, and different initial stretches and mean equilibrium position

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Fig. 7

Phenomenological model consists of nonlinear springs and damper for viscoelastic membrane–plate assembly

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Fig. 8

Comparison of the numerically calculated E0 (shown in bullet points) with the fit given in Eq. (24) (shown in lines). Note that the farthest curve from the list of values of C corresponds to the first value of degree of viscoelasticity C.

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Fig. 9

Comparison of the numerically calculated Cd (shown in bullet points) with the fit given in Eq. (25) (shown in lines)

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Fig. 10

Comparison of the numerically calculated creep response (shown in broken lines) for λs=2 with the results obtained from phenomenological model (shown in solid lines)

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Fig. 11

Comparison of the numerically determined (shown in bullet points and joined by dotted lines): (a) amplitude of oscillation and (b) phase of contact force during frequency of oscillation ω with the amplitude and phase obtained from phenomenological model (shown in lines) for degree of viscoelasticity C = 1

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Fig. 12

(a) The variation of damped natural frequency with equilibrium position for material constant C = 1 and gravitational acceleration parameter g¯=1. (b) Free vibration response of the membrane–plate system for amplitude yA = 0.05 and equilibrium position y0 = 0.5.

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Fig. 13

The variation of vibration decay timescales ξ (mean position) and ζ (amplitude) with equilibrium position for material constant C = 1 and gravitational acceleration parameter g¯=1

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