An optimization of a Mindlin‐Timoshenko beam with a dynamic contact on the boundary

Abstract

We deal with the optimal control problem governed by a hyperbolic variational inequality describing the perpendicular vibrations of a Mindlin-Timoshenko beam clamped on the left end with a rigid obstacle at the right end. A variable thickness of a beam plays the role of a control parameter.

Regularized Optimal Design Problem for a Viscoelastic Plate Vibrating Against a Rigid Obstacle

“…The dynamic contact for a viscoelastic bridge in a contact with a fixed road has been solved in [3]. The similar optimal control problems for the beams in a boundary contact are investigated in [1] and [4] respectively.…”

“…Solving the state problem we apply the Galerkin method in the same way as in [1], where the rigid obstacle acting against a beam is considered or in [2] where the problem for a viscoelastic von Kármán plate vibrating against a rigid obstacle has been solved. The compactness method will be used in solving the minimum problem for a cost functional.…”

Regularized optimal control problem for a beam vibrating against an elastic foundation

“…We have considered the dynamic state problem in [5]. There the equation for the deflections has the form e(x)u tt + d e 3 Further we assume the distance of the middle line of the beam and the foundation to be 1 2 e max > 0. Due to the variable thickness the equation e(x)u tt + d e 3 is more suitable.…”

“…Solving the state problem we apply the Galerkin method in the same way as in [1], where the rigid obstacle acting against a beam is considered or in [2], where the problem for an elastic plate vibrating against a rigid obstacle has been solved. The compactness method will be used in solving the minimum problem for a cost functional.…”