Minimization of the vibration energy of thin-plate structure

An optimization method is proposed to reduce the vibration of thin-plate structures. The method is based on a finite-element shell analysis, a modal analysis, and a structural optimization method. In the finite-element analysis, a triangular shell element with 18 degrees of freedom is used. In the optimization, the overall vibration energy of the structure is adopted as the objective function, and it is minimized at the given exciting frequency by varying the thickness of the elements. The technique of modal analysis is used to derive the sensitivity of the vibration energy with respect to the design variables. The sensitivity is represented by the sensitivities of both eigenvalues and eigenvectors. The optimum value is computed by the gradient projection method and a unidimensional search procedure under the constraint condition of constant weight. A computer code, based on the proposed method, is developed and is applied to design problems using a beam and a plate as test cases. It is confirmed that the vibration energy is reduced at the given exciting frequency. For the beam excited by a frequency slightly less than the fundamental natural frequency, the optimized shape is close to the beam of uniform strength. For the plate, the optimum shape is obtained such that the changes in thickness have the effect of adding a stiffener or a mass.