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Research Papers

Intrinsic Finite Element Modeling of Nonlinear Dynamic Response in Helical Springs

[+] Author and Article Information
Michael J. Leamy

 George W. Woodruff School of Mechanical Engineering,Georgia Institute of Technology, Atlanta, GA 30332-0405michael.leamy@me.gatech.edu

Unlike convected basis vectors, the basis vectors employed always remain orthonormal. This primarily results from defining B 1 perpendicular to B 2 and B 3 and not oriented along the centerline tangent.

The cross-section is assumed to be planar in this configuration, which even for small initial curvatures and strains may account for some lack of symmetry in the derived constitutive relationships.

J. Comput. Nonlinear Dynam 7(3), 031007 (Mar 26, 2012) (9 pages) doi:10.1115/1.4005820 History: Received June 10, 2011; Revised December 01, 2011; Published March 26, 2012

This paper presents an efficient intrinsic finite element approach for modeling and analyzing the forced dynamic response of helical springs. The finite element treatment employs intrinsic curvature (and strain) interpolation and vice rotation (and displacement) interpolation and, thus, can accurately and efficiently represent initially curved and twisted beams with a sparse number of elements. The governing equations of motion contain nonlinearities necessary for large curvatures. In addition, a constitutive model is developed, which captures coupling due to nonzero initial curvature and strain. The method is employed to efficiently study dynamically-loaded helical springs. Convergence studies demonstrate that a sparse number of elements accurately capture spring dynamic response, with more elements required to resolve higher frequency content, as expected. Presented results also document rich, amplitude-dependent frequency response. In particular, moderate loading amplitudes lead to the presence of secondary resonances (not captured by linearized models), while large loading amplitudes lead to complex dynamics and transverse buckling.

Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 11

FFT of K̂2 at node 30 following application of moderate loading–example fixed-free spring discretized using 48 elements (97 nodes)

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Figure 12

Time histories of curvature K̂2 at node 30 following application of harmonic loading–example fixed-free spring discretized using 48 elements (97 nodes). Two loading cases are depicted: 52.43 Hz, which is a potential secondary resonance, and 60.0 Hz where no resonances are expected.

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Figure 13

Time histories of curvature K̂2 at node 30 following application of harmonic loading–example fixed-free spring discretized using 48 elements (97 nodes). Three loading cases are depicted: 52.43 Hz, 60.0 Hz, and 63.1 Hz.

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Figure 14

Initial spring configurations resulting from application of large-magnitude impulsive transverse loading

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Figure 15

FFT of K̂2 at node 30 following application of large-amplitude loading

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Figure 1

Configurations employed in developing the helical spring model. Material points along a straight reference configuration Ωref are mapped to an initial configuration Ω0 via initial curvature and strain K0,γ0. Similar mappings hold for a deformed configuration Ωf. The latter two configurations are related by mappings, which invoke net curvature and strain, K̂ and γ̂.

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Figure 2

Position and basis vectors used in the development. The position and basis vectors shown can be specialized to either the initial or deformed configuration using subscripts 0 and f.

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Figure 3

Depiction of transverse shear in a planar rectangular beam – a second component of transverse shear (γ13 ) is not shown for sake of clarity

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Figure 4

For formulations without shear, the normal n and bi-normal b directions are related to the basis vectors B2 and B3 via an angle ϕ as depicted

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Figure 5

Configurations for the example fixed-fixed spring illustrating the application of loading starting from the initial time until the maximum spring deflection is achieved (last configuration shown). The yellow line tracks the small deformation of a center coil.

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Figure 6

FFT of K̂2 at node 30–example fixed-fixed spring discretized using 48 elements (97 nodes)

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Figure 7

Modal convergence with number of intrinsic finite elements used for first example spring–longitudinal modes

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Figure 8

Modal convergence with number of intrinsic finite elements used for first example spring–transverse modes

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Figure 9

FFT of K̂2 at node 30–example fixed-free spring discretized using 48 elements (97 nodes)

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Figure 10

Initial spring configurations resulting from application of moderate-magnitude impulsive longitudinal loading. Ensuing motion is on the same order of magnitude as the configurations depicted.

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