Regulating the vibratory motion of beams using shape optimization

In this paper, shape optimization is used to regulate the vibrations of an Euler–Bernoulli beam having constant material volume. This is achieved by varying appropriately the beam cross-section and thus its stiffness and mass properties along its length, so that the beam vibrates with its minimum, maximum or a prescribed eigenfrequency as well as with the minimum or maximum difference between two successive eigenfrequencies. The problem is reduced to a nonlinear optimization problem under equality and inequality constraints as well as specified lower and upper bounds. The evaluation of the objective function requires the solution of the free vibration problem of a beam with variable mass and stiffness properties. This problem is solved using the analog equation method (AEM) for hyperbolic differential equations with variable coefficients. Besides its accuracy, this method overcomes the shortcoming of a FEM solution, which would require resizing of the elements and re-computation of their stiffness and mass properties during the optimization process. Certain example problems are presented, which illustrate the method and demonstrate its efficiency.