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Research Papers

Coupling FEM With Parameter Continuation for Analysis of Bifurcations of Periodic Responses in Nonlinear Structures

[+] Author and Article Information
Giovanni Formica

Department of Studies on Structures,
Roma Tre University,
Rome00146, Italy

Walter Lacarbonara

Department of Structural and Geotechnical Engineering,
Sapienza University of Rome,
Rome00184, Italy

Harry Dankowicz

Department of Mechanical Science and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received Decemeber 20, 2011; final manuscript received July 29, 2012; published online August 31, 2012. Assoc. Editor: Henryk Flashner.

J. Comput. Nonlinear Dynam 8(2), 021013 (Aug 31, 2012) (8 pages) Paper No: CND-11-1247; doi: 10.1115/1.4007315 History: Received December 20, 2011; Revised July 29, 2012

A computational framework is proposed to perform parameter continuation of periodic solutions of nonlinear, distributed-parameter systems represented by partial differential equations with time-dependent coefficients and excitations. The path-following procedure, encoded in the general-purpose Matlab-based computational continuation core (referred to below as coco), employs only the evaluation of the vector field of an appropriate spatial discretization; for example as formulated through an explicit finite-element discretization or through reliance on a black-box discretization. An original contribution of this paper is a systematic treatment of the coupling of coco with Comsolmultiphysics, demonstrating the great flexibility afforded by this computational framework. Comsolmultiphysics provides embedded discretization algorithms capable of accommodating a great variety of mechanical/physical assumptions and multiphysics interactions. Within this framework, it is shown that a concurrent bifurcation analysis may be carried out together with parameter continuation of the corresponding monodromy matrices. As a case study, we consider a nonlinear beam, subject to a harmonic, transverse direct excitation for two different sets of boundary conditions and demonstrate how the proposed approach may be able to generate results for a variety of structural models with great ease. The numerical results include primary-resonance, frequency-response curves together with their stability and two-parameter analysis of multistability regions bounded by the loci of fold bifurcations that occur along the resonance curves. In addition, the results of comsol are validated for the Mettler model of slender beams against an in-house constructed finite-element discretization scheme, the convergence of which is assessed for increasing number of finite elements.

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Figures

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Fig. 1

Uniform beam subject to the time-varying transverse load λ cos ωt per unit reference length

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Fig. 2

Frequency-response curves, obtained using 3 FEs, for different load amplitudes: λ=(0.25,0.5,1,3,10)λ0. Solid and dashed lines refer to stable and unstable branches, respectively.

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Fig. 3

Frequency-response curves, obtained using 3 FEs, for different slenderness ratios: α=(1,2,3)α0. Solid and dashed lines refer to stable and unstable branches, respectively.

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Fig. 4

Sensitivity to number of finite elements of frequency-response curves for different load amplitudes λ=(10,50)λ0 when the beam is subject to a primary resonance of the second symmetric mode

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Fig. 5

Sensitivity to number of finite elements of frequency-response curves for different load amplitudes λ=(10,50)λ0 when the beam is subject to a primary resonance of the third symmetric mode

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Fig. 6

Sensitivity to number of finite elements of frequency-response curves for different load amplitudes λ=(10,50)λ0 when the beam is subject to a primary resonance of the fourth symmetric mode

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Fig. 7

Instability/multistability regions obtained by continuation of the fold points found in one-parameter continuation using the 3 finite elements, discretization scheme for different values of the nonlinearity coefficient α. The circled numbers mean α=(0.5,1,2,3)α0.

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Fig. 8

Instability/multistability regions obtained by continuation of the fold points found in one-parameter continuation using the 3 finite elements, discretization scheme for different damping ratios: L is very lightly damped (ζ=0.01), W is weakly damped (ζ=0.02) and M is moderately damped (ζ=0.04)

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Fig. 9

Frequency-response curves obtained for the Mettler beam model, comparing the in-house finite-element discretization and the Comsol implementation, using 3 finite elements in both cases, for different load amplitudes: λ=(1,3)λ0

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Fig. 10

Frequency-response curves, obtained using the in-house finite-element discretization of the unshearable Mettler beam model, and the Comsol implementation of the field equations originating in the Euler-Bernoulli (unshearable) beam theory and the shearable special Cosserat theory. Here, we assumed α=300 and λ=(0.1875,0.375). Note that the frequency of the lowest flexural mode exhibited by the unshearable beam models equals π2.

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Fig. 11

Frequency-response curves, obtained using the Comsol implementation of the FE Mettler model, the Cosserat theory and the Euler-Bernoulli model when α=1.2·104 and λ=(0.075,0.225).

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Fig. 12

Frequency-response curves of a hinged-hinged Cosserat beam (hardening) contrasted with those of a simply supported Cosserat beam with a lumped mass, γ=8.1, on a roller support (softening) when λ=(0.0375,0.075,0.15). The figure highlights the change of response from hardening to softening depending on the boundary conditions. Solid and dashed curves refer to stable and unstable branches, respectively.

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