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Existence and approximation results for shape optimization problems in rotordynamics. (English) Zbl 1207.74108

Summary: We consider a shape optimization problem in rotordynamics where the mass of a rotor is minimized subject to constraints on natural frequencies. Our analysis is based on a class of rotors described by a Rayleigh beam model including effects of rotary inertia and gyroscopic moments. The solution of the equation of motion leads to a generalized eigenvalue problem. The governing operators are non-symmetric due to the gyroscopic terms. We prove the existence of solutions for the optimization problem by using the theory of compact operators. For the numerical treatment of the problem, a finite element discretization based on a variational formulation is considered. Applying results on spectral approximation of linear operators, we prove that the solution of the discretized optimization problem converges towards the solution of the continuous problem if the discretization parameter tends to zero. Finally, a priori estimates for the convergence order of the eigenvalues are presented and illustrated by a numerical example.

MSC:

74P05 Compliance or weight optimization in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H45 Vibrations in dynamical problems in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
74H20 Existence of solutions of dynamical problems in solid mechanics
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