Optimal Shape and First Integrals for Inverted Compressed Column

Abstract

We study optimal shape of an inverted elastic column with concentrated force at the end and in the gravitational field. We generalize earlier results on this problem in two directions. First we prove a theorem on the bifurcation of nonlinear equilibrium equations for arbitrary cross-section column. Secondly we determine the cross-sectional area for the compressed column in the optimal way. Variational principle is constructed for the equations determining the optimal shape and two new first integrals are constructed that are used to check numerical integration. Next, we apply the Noether’s theorem and determine transformation groups that leave variational principle Gauge invariant. The classical Lagrange problem follows as a special case. Several numerical examples are presented.

On the generalized Clausen problem

“…We proved that optimally shaped rod will have supercritical pitchfork bifurcation at prescribed load. 2.The procedure used here is based on Pontryagin's principle and is used earlier for single and bi modal optimizations, see [18–20]. The use of first integral () 1 and optimality () 2 simplifies the numerical procedure. …”

“…2. The procedure used here is based on Pontryagin's principle and is used earlier for single and bi modal optimizations, see [18][19][20]. The use of first integral (16) 1 and optimality (16) 2 simplifies the numerical procedure.…”