Research Papers

Exact Solutions for Free Vibrations and Buckling of Double Tapered Columns With Elastic Foundation and Tip Mass

[+] Author and Article Information
R. D. Firouz-Abadi

Assistant Professor
e-mail: Firouzabadi@sharif.edu

M. Rahmanian

e-mail: Rahmanian@ae.sharif.edu
Department of Aerospace Engineering,
Sharif University of Technology,
P.O. Box 11155-8639,
Tehran, Iran

M. Amabili

Department of Mechanical Engineering,
McGill University,
817 Sherbrooke Street West,
Montreal, Canada
e-mail: marco.amabili@mcgill.ca

Contributed by the Design Engineering Division of ASME for publication in the Journal of Vibration and Acoustics. Manuscript received February 13, 2012; final manuscript received January 26, 2013; published online June 18, 2013. Assoc. Editor: Massimo Ruzzene.

J. Vib. Acoust 135(5), 051017 (Jun 18, 2013) (10 pages) Paper No: VIB-12-1038; doi: 10.1115/1.4023991 History: Received February 13, 2012; Revised January 26, 2013

The present study aims at the free vibration analysis of double tapered columns. Foundation is assumed to be elastic and the effects of self-weight and tip mass with significant moment of inertia are considered. The governing equation of motion is obtained using the Hamilton principle, based on both the Euler–Bernoulli and Timoshenko beam models. Applying the power series method of Frobenius, the base solutions of the governing equations are obtained in the form of a power series via general recursive relations. Applying the boundary conditions, the natural frequencies of the beam/column are obtained using both models. The obtained results are compared with literature and a very good agreement is achieved. Subsequently, comprehensive studies are performed to provide an insight into the variation of the natural frequencies and instability conditions of the beam with respect to the tip mass, self-weight, taper ratio, slenderness, and foundation stiffness and eventually some general conclusions are drawn.

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Fig. 1

Schematic view of a doubly tapered column of rectangular cross section

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Fig. 2

Fundamental frequency variations versus slenderness ratio

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Fig. 3

Thickness taper ratio effect on the first three nondimensional frequencies

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Fig. 4

The fundamental frequency variations versus the tip and foundation parameters for r = 0.05, E/κG = 3.9 (a) κ2 = ∞,μ1 = μ2 = 0, (b) κ1 = ∞,μ1 = μ2 = 0, (c) μ2 = 1,κ1 = κ2 = ∞, (d) μ1 = 0.01,κ1 = κ2 = ∞

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Fig. 5

Tip-load/self-weight versus the fundamental frequency parameter (r = 0.05,E/κG = 3.9) (a) ν1 = 0, (b) ν2 = 0

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Fig. 6

Tip-load/self-weight versus the fundamental frequency parameter for (cb = ch = 0.5,E/κG = 3.9) (a) ν1 = 0, (b) ν2 = 0

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Fig. 7

Slenderness and tapering effects on the stability boundaries (a) cb = ch = 0.5,E/κG = 3.9, (b) r = 0.05,E/κG = 3.9




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