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Research Papers

Singularities in Differential-Algebraic Boundary-Value Problems Governing the Excitation Response of Beam Structures

[+] Author and Article Information
Mehdi Saghafi

Department of Mechanical
Science and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: msagha2@illinois.edu

Harry Dankowicz

Department of Mechanical
Science and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: danko@illinois.edu

1Corresponding author.

2Elements of this paper have appeared in Saghafi & Dankowicz, “Nondegenerate Continuation Problems for the Excitation Response of Nonlinear Beam Structures,” Proceedings of ASME 2013 IDETC/CIE, Paper #13115, 2013.

Manuscript received September 10, 2013; final manuscript received March 14, 2014; published online October 14, 2014. Assoc. Editor: Arend L. Schwab.

J. Comput. Nonlinear Dynam 10(1), 011017 (Oct 14, 2014) (10 pages) Paper No: CND-13-1218; doi: 10.1115/1.4027208 History: Received September 10, 2013; Revised March 14, 2014

The objective of this paper2 is to identify and, where possible, resolve singularities that may arise in the discretization of spatiotemporal boundary-value problems governing the steady-state behavior of nonlinear beam structures. Of particular interest is the formulation of nondegenerate continuation problems of a geometrically-nonlinear model of a slender beam, subject to a uniform harmonic excitation, which may be analyzed numerically in order to explore the parameter-dependence of the excitation response. In the instances of degeneracy investigated here, the source is either found (i) directly in a differential-algebraic system of equations obtained from a finite-element-based spatial discretization of the governing partial differential boundary-value problem(s) together with constraints on the trial functions or (ii) in the further collocation-based discretization of the time-periodic boundary-value problem. It is shown that several candidate spatial finite-element discretizations of a mixed weak formulation of the governing boundary-value problem either result in (i) spatial group symmetries corresponding to equivariant vector fields and one-parameter families of periodic orbits along the group symmetry orbit or (ii) temporal group symmetries corresponding to ghost solutions and indeterminacy in a subset of the field variables. The paper demonstrates several methods for breaking the spatial equivariance, including projection onto a symmetry-reduced state space or the introduction of an artificial continuation parameter. Similarly, the temporal indeterminacy is resolved by an asymmetric discretization of the governing differential-algebraic equations. Finally, in the absence of theoretical bounds, computation is used to estimate convergence rates of the different discretization schemes, in the case of numerical calibration experiments performed on equilibrium and periodic responses for a linear beam, as well as for the full nonlinear models.

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References

Figures

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Fig. 1

Frequency-response curves for different values of the nonlinearity coefficient α (reprinted from Ref. [2])

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Fig. 2

Schematic of the beam with clamped-clamped boundary condition

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Fig. 3

Illustration of the effect of oddness and evenness of m (4 in left panels and 3 in right panels) and N (2 in upper panels and 3 in lower panels) on the satisfaction of periodicity condition of the ghost solution δχ(t)

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Fig. 4

Equivariance results in a symmetry group orbit of periodic solutions shifted along v∧. A unique solution is obtained by appending a boundary condition at t = 0.

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Fig. 5

Frequency-response curves obtained using the index-2 discretization (dashed) for M = 2, 4, and 6 and p = 4 compared to that obtained using the discretized scalar model (solid) for M = 8 and p = 4

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