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TECHNICAL PAPERS

# Optimal Shape of a Rotating Rod With Unsymmetrical Boundary Conditions

[+] Author and Article Information
Teodor M. Atanackovic

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

## Abstract

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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## Figures

Figure 1

Figure 2

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

Figure 3

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

Figure 4

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

Figure 5

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

Figure 6

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

Figure 7

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

Figure 8

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

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