Research Papers

Impact Mechanics of Elastic Structures With Point Contact

[+] Author and Article Information
Róbert Szalai

Department of Engineering Mathematics,
University of Bristol,
Merchant Venturers Building,
Woodland Road,
Bristol BS8 1UB, UK
e-mail: r.szalai@bristol.ac.uk

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 3, 2013; final manuscript received March 17, 2014; published online April 18, 2014. Assoc. Editor: Philip Bayly.

J. Vib. Acoust 136(4), 041002 (Apr 18, 2014) (6 pages) Paper No: VIB-13-1229; doi: 10.1115/1.4027242 History: Received July 03, 2013; Revised March 17, 2014

This paper introduces a modeling framework that is suitable to resolve singularities of impact phenomena encountered in applications. The method involves an exact transformation that turns the continuum, often partial differential equation description of the contact problem into a delay differential equation. The new form of the physical model highlights the source of singularities and suggests a simple criterion for regularity. To contrast singular and regular behavior the impacting Euler–Bernoulli and Timoshenko beam models are compared.

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Fig. 1

Impacting cantilever beam

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Fig. 2

Graph of [L(τ)]2 for the Euler–Bernoulli and the Timoshenko beam models. The inset shows that the Timoshenko beam model converges to a function where [L+]2≠0, while the Euler–Bernoulli model is singular since its L(τ) is continuous. The parameters are β = 4800, γ = 1/4.

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Fig. 3

Vibrations of the impacting Timoshenko beam using two impact models. The rigid stop as illustrated in Fig. 1 is placed at y1 = −0.05, and the forcing is fe(t) = 30 cos (13t) through the second mode. Trajectories of the reduced model (4,6,7) are shown in dark and the solution of the CoR model (1,15) is represented by light lines for comparison. The time step used to solve the reduced model is ε = 3.5 × 10−5 and the number of collocation points used to solve the CoR model is N = 20. Panels (a) and (b) show that the solution converges to a periodic orbit. A single period of the solution is illustrated in panels (c) and (d).

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Fig. 4

A sequence of two impacts within a period of the periodic solution in Fig. 3. The solution of the CoR model (1,15) as shown by the light lines is highly oscillatory. The dark lines show that the reduced model (4,6,7) is similarly accurate and eliminates the high-frequency chatter of the CoR model (a) and (b). Insets (d) and (e) show the small-scale dynamics of the CoR model. The contact force is finite and continuous after the initial contact (c).

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Fig. 5

(a) Coefficients (A2) for δt = 10−5, Dk = 0, and ωk = (kπ-(π/2))2. (b) The sum of the infinite series of coefficients (A2) is shown by the continuous line. The dashed line is the estimate of N, in Eq. (A4) that overestimates the sum.




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