An Optimal Control of Transverse Vibrations of an Euler-Bernoulli Beam-String Complex System

In this paper, an optimal control of transverse vibrations of an Euler-Bernoulli beam and string complex system is studied. The dynamic response of the system is measured by a performance index that consists of a modified energy functional of two coupled structures at the terminal time and the expenditure of the control forces as a penalty term. The minimization of the performance index over these forces is subject to the equation of motion governing the structural vibrations, the imposed initial condition as well as the boundary conditions. The optimal control of distributed parameter systems is transformed into the optimal control of a linear timeinvariant lumped-parameter systems by employing Galerkin method in space. Fourier series approximation with unknown Fourier coefficients and frequencies is used to parameterize the control function. The applicability and effectiveness of the proposed method is demonstrated by a numerical example.