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TECHNICAL PAPERS

Effect of Boundary Conditions on Nonlinear Vibrations of Circular Cylindrical Panels

[+] Author and Article Information
M. Amabili

Dipartimento di Ingegneria Industriale, Università di Parma, Parco Area delle Scienze 181/A, Parma 43100, Italymarco.amabili@unipr.it

J. Appl. Mech 74(4), 645-657 (Feb 06, 2006) (13 pages) doi:10.1115/1.2424474 History: Received August 31, 2005; Revised February 06, 2006

Geometrically nonlinear vibrations of circular cylindrical panels with different boundary conditions and subjected to harmonic excitation are numerically investigated. The Donnell’s nonlinear strain–displacement relationships are used to describe geometric nonlinearity; in-plane inertia is taken into account. Different boundary conditions are studied and the results are compared; for all of them zero normal displacements at the edges are assumed. In particular, three models are considered in order to investigate the effect of different boundary conditions: Model A for free in-plane displacement orthogonal to the edges, elastic distributed springs tangential to the edges and free rotation; Model B for classical simply supported edges; and Model C for fixed edges and distributed rotational springs at the edges. Clamped edges are obtained with Model C for the very high value of the stiffness of rotational springs. The nonlinear equations of motion are obtained by the Lagrange multimode approach, and are studied by using the code AUTO based on the pseudo-arclength continuation method. Convergence of the solution with the number of generalized coordinates is numerically verified. Complex nonlinear dynamics is also investigated by using bifurcation diagrams from direct time integration and calculation of the Lyapunov exponents and the Lyapunov dimension. Interesting phenomena such as (i) subharmonic response; (ii) period doubling bifurcations; (iii) chaotic behavior; and (iv) hyper-chaos with four positive Lyapunov exponents have been observed.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 2

Natural frequency of mode (1,1) of supported panel computed with model A versus k; 19 DOF

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Figure 1

Geometry of the panel, coordinate system and symbols

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Figure 3

Natural frequency of mode (1,1) of panel with fixed edges computed with model C versus c; 39 DOF

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Figure 4

Nondimensional response of the panel for different boundary conditions versus non-dimensional excitation frequency; mode (1,1), f=0.021, ζ1,1=0.004. (– –) classical simply supported panel (model B), 9 DOF; (— - —) model A with k=4×109N∕m2, 19 DOF; (—) model A with k=0 (in-plane free edges), 19 DOF. (a) Maximum of generalized coordinate w1,1; and (b) minimum of w1,1.

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Figure 5

Nondimensional response of the panel for different boundary conditions versus nondimensional excitation frequency; mode (1,1), f=0.021, ζ1,1=0.004. (– –) classical simply supported panel (model B), 9 DOF; (— - —) model C with c=5×104N∕rad (practically clamped), 39 DOF; (—) model C with c=0 (fixed edges), 39 DOF. (a) Maximum transverse displacement at the center of the panel; and (b) minimum transverse displacement at the center of the panel.

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Figure 6

Convergence of model C (c=5×104N∕rad) for nonlinear forced vibrations of the clamped panel; nondimensional response of generalized coordinate w1,1 versus non-dimensional excitation frequency; fundamental mode (1,1), f=0.021, ζ1,1=0.004. (— - —) 24 DOF; (– –) 27 DOF; (—) 39 DOF.

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Figure 7

Response of panel computed with model A for k=4×109N∕m2; fundamental mode (1,1), f=0.021, ζ1,1=0.004, 19 DOF. (—) stable periodic response; (– –) unstable periodic response. (a) Maximum of the generalized coordinate w1,1; (b) maximum of the generalized coordinate w1,3; (c) maximum of the generalized coordinate w3,1; (d) maximum of the generalized coordinate w3,3; (e) maximum of the generalized coordinate u1,0; and (f) maximum of the generalized coordinate v0,1.

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Figure 8

Response of panel with fixed edges computed with model C for c=0; fundamental mode (1,1), f=0.021, ζ1,1=0.004, 39 DOF. (—) stable periodic response; (– –) unstable periodic response. (a) Maximum of the generalized coordinate w1,1; (b) maximum of the generalized coordinate w3,1; (c) maximum of the generalized coordinate w1,3; and (d) maximum of the generalized coordinate w3,3.

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Figure 10

Computed time response of the panel with fixed edges computed with model C for c=5×104N∕rad (practically clamped) for excitation frequency ω=1.06ω1,1; fundamental mode (1,1), f=0.021, ζ1,1=0.004, 39 DOF. (a) Force excitation; (b) Generalized coordinate w1,1; (c) generalized coordinate w3,1; (d) generalized coordinate w1,3; (e) generalized coordinate w3,3; and (f) generalized coordinate u2,1.

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Figure 11

Frequency spectrum of the response of the panel with fixed edges computed with model C for c=5×104N∕rad (practically clamped) for excitation frequency ω=1.06ω1,1; fundamental mode (1,1), f=0.021, ζ1,1=0.004, 39 DOF. (a) Generalized coordinate w1,1; (b) generalized coordinate w3,1; (c) generalized coordinate w1,3; (d) generalized coordinate w3,3; and (e) generalized coordinate u2,1.

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Figure 12

Bifurcation diagram of Poincaré maps and maximum Lyapunov exponent for the panel with in-plane free edges (model A with k=0) under decreasing harmonic load f̃ with frequency ω=ω1,1 (linear resonance condition); ζ1,1=0.004; 19 DOF model. (a) Bifurcation diagram: generalized coordinate w1,1; T=response period equal to excitation period; 2T=periodic response with two times the excitation period; 9T=periodic response with nine times the excitation period; (PD) period-doubling bifurcation; (M) amplitude modulations; (C) chaos; (b) maximum Lyapunov exponent; (c) bifurcation diagram: generalized coordinate w1,3; (d) bifurcation diagram: generalized coordinate w1,1, enlarged scale; and (e) 3D representation of the bifurcation diagram: generalized coordinate w1,1.

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Figure 13

All the 38 Lyapunov exponents for the panel with in-plane free edges (model A with k=0); excitation frequency ω=ω1,1 (linear resonance condition); f̃=5952.4N; ζ1,1=0.004, 19 DOF model.

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Figure 9

Response of panel with fixed edges computed with model C for c=5×104N∕rad (practically clamped); fundamental mode (1,1), f=0.021, ζ1,1=0.004, 39 DOF. (—) stable periodic response; (– –) unstable periodic response. (a) Maximum of the generalized coordinate w1,1; (b) maximum of the generalized coordinate w3,1; (c) maximum of the generalized coordinate w1,3; (d) maximum of the generalized coordinate w3,3; and (e) maximum of the generalized coordinate w1,5.

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