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Research Papers

Dynamic Behavior of Elastic Bars and Beams Impinging on Ideal Springs

[+] Author and Article Information
Song Wang, Zhilong Huang

Department of Engineering Mechanics,
Zhejiang University,
Hangzhou 310027, China

Yong Wang

Department of Engineering Mechanics,
Zhejiang University,
Hangzhou 310027, China
e-mail: ypwang@zju.edu.cn

T. X. Yu

Department of Mechanical
and Aerospace Engineering,
Hong Kong University of Science
and Technology,
Kowloon 999077, Hong Kong

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 23, 2015; final manuscript received November 15, 2015; published online December 10, 2015. Editor: Yonggang Huang.

J. Appl. Mech 83(3), 031002 (Dec 10, 2015) (10 pages) Paper No: JAM-15-1447; doi: 10.1115/1.4032048 History: Received August 23, 2015; Revised November 15, 2015

The impacting and rebounding behaviors of straight elastic components are investigated and a unified approach is proposed to analytically predict the whole process of the collision and rebounding of straight elastic bars and beams after each of them impinges on ideal (massless) elastic spring(s). The mathematical problems with definitive solution are formulated, respectively, for both the constrained-motion and free-motion stages, and the method of mode superposition, which is concise and straightforward especially for long-time interaction and multiple collision cases, is successfully utilized by repeatedly altering boundary and initial conditions for these successive stages. These two stages happen alternatively and the collision process terminates when the constrained motion no longer occurs. In particular, three examples are investigated in detail; they are: a straight bar impinges on an ideal elastic spring along its axis, a straight beam vertically impinges on an ideal elastic spring at the beam's midpoint, and a straight beam vertically impinges on two ideal springs with the same stiffness at the beam's two ends. Numerical results show that the coefficient of restitution (COR) and the nondimensional rebounding time (NRT) only depend on the stiffness ratio between the ideal spring(s) and the elastic bar/beam. Collision happens only once for the straight bar impinging on spring, while multiple collisions occur for the straight beam impinging on springs in the cases with large stiffness ratio. Once multiple collisions occur, COR undergoes complicated fluctuation with the increase of stiffness ratio. Approximate analytical solutions (AASs) for COR and NRT under the cases of small stiffness ratio are all derived. Finally, to validate the proposed approach in practical collision problems, the influence of the springs' mass on the collision behavior is demonstrated through numerical simulation.

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Figures

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Fig. 1

A straight elastic bar impinges on an ideal spring along its axis

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Fig. 2

Displacement history at the contact point for a straight bar impinging on an ideal spring

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Fig. 3

Dependences of the COR (solid line) and the NRT (chain line) on the stiffness ratio. Discrete symbol: results from finite-element analysis (FEA); dashed line: AAS.

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Fig. 4

A straight elastic beam impinges on an ideal spring at its midpoint

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Fig. 5

Displacement history at the contact point for a straight beam impinging on an ideal spring at its midpoint

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Fig. 6

Dependences of the COR (solid line), the NRT (chain line), and the collision number on the stiffness ratio. (a) COR and NRT and (b) collision number. Discrete symbol: results from FEA; dashed line: AAS.

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Fig. 7

A straight elastic beam impinges on two ideal springs at its end-points

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Fig. 8

Displacement history at the contact point for a straight beam impinging on two ideal springs at its end-points

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Fig. 9

Dependences of the COR (solid line), the NRT (chain line), and the collision number on the stiffness ratio. (a) COR and NRT and (b) collision number. Discrete symbol: results from FEA; dashed line: AAS.

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Fig. 10

COR varying with the density ratio ρ¯

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