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TECHNICAL PAPERS

Optimal Shape of a Rotating Rod With Unsymmetrical Boundary Conditions

[+] Author and Article Information
Teodor M. Atanackovic

Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovica 6, 21000 Novi Sad, Serbia

Since the pair (λ1*,λ2*) is given we shall not differentiate it with respect to a(t).

J. Appl. Mech 74(6), 1234-1238 (Feb 23, 2007) (5 pages) doi:10.1115/1.2744041 History: Received December 02, 2005; Revised February 23, 2007

Governing equations of a compressed rotating rod with clamped–elastically clamped (hinged with a torsional spring) boundary conditions is derived. It is shown that the multiplicity of an eigenvalue of this system can be at most equal to two. The optimality conditions, via Pontryagin’s maximum principle, are derived in the case of bimodal optimization. When these conditions are used the problem of determining the optimal cross-sectional area function is reduced to the solution of a nonlinear boundary value problem. The problem treated here generalizes our earlier results presented in Atanackovic, 1997, Stability Theory of Elastic Rods, World Scientific, River Edge, NJ. The optimal shape of a rod is determined by numerical integration for several values of parameters.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

Coordinate system and loading configuration

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Figure 2

Interaction curves for: (a) constant cross section; and (b) optimal cross section

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Figure 3

Buckling modes corresponding to optimal cross-sectional area for λ1=0, λ2=27.22, k=0.1

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Figure 4

Optimal cross-sectional area for λ1=0, λ2=27.22, k=0.1

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Figure 5

Buckling modes corresponding to optimal cross-sectional area for λ1=30, λ2=37.5, k=4

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Figure 6

Optimal cross-sectional area for λ1=30, λ2=37.5, k=4

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Figure 7

Buckling modes corresponding to optimal cross-sectional area for λ1=40, λ2=40, k=100

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Figure 8

Optimal cross-sectional area for λ1=40, λ2=40, k=100

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