Math @ Duke

Publications [#244232] of Thomas P. Witelski
search www.ams.org.Papers Published
 Santillan, ST; Plaut, RH; Witelski, TP; Virgin, LN, Large oscillations of beams and columns including selfweight,
International Journal of NonLinear Mechanics, vol. 43 no. 8
(2008),
pp. 761771, ISSN 00207462 [007], [doi]
(last updated on 2018/06/21)
Abstract: Largeamplitude, inplane beam vibration is investigated using numerical simulations and a perturbation analysis applied to the dynamic elastica model. The governing nonlinear boundary value problem is described in terms of the arclength, and the beam is treated as inextensible. The selfweight of the beam is included in the equations. First, a finite difference numerical method is introduced. The system is discretized along the arclength, and secondorderaccurate finite difference formulas are used to generate time series of largeamplitude motion of an upright cantilever. Secondly, a perturbation method (the method of multiple scales) is applied to obtain approximate solutions. An analytical backbone curve is generated, and the results are compared with those in the literature for various boundary conditions where the selfweight of the beam is neglected. The method is also used to characterize largeamplitude firstmode vibration of a cantilever with nonzero selfweight. The perturbation and finite difference results are compared for these cases and are seen to agree for a large range of vibration amplitudes. Finally, largeamplitude motion of a postbuckled, clampedclamped beam is simulated for varying degrees of buckling and selfweight using the finite difference method, and backbone curves are obtained. © 2008 Elsevier Ltd.


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