Singular solutions of truss size optimization for considering fundamental frequency constraints

For engineering structures, the limits of internal stresses, nodal displacements and fundamental frequencies must be simultaneously considered. This had been paid attention in the theory of structural optimization. Actually, most examples of only considering static constraints or only considering dynamic constraints were presented for simultaneously considering static and dynamic constraints. A few examples of considering both static and dynamic constraints were presented, but the advantage could not be presented. It is the reason that the singularity of structural optimization for considering dynamic constraints has not been discussed. To discover the singularity, an optimization model to simultaneously consider static and dynamic constraints is used for the truss size optimization. And according to the extremum conditions of the optimization problem, Ratio-Extremum method is proposed to solve the optimization problems of considering both static and dynamic constraints and only considering dynamic constraints, in which a new searching direction of design variables is to be discussed. Particularly, the step-size factors can be determined by formulas to iteratively solve Lagrangian multipliers and design variables. Numerical examples of 15-bar planar and 72-bar spatial trusses are used to show the singular solutions. On the convergent points, the optimization weights of only considering dynamic constraints are about 66.17% and 71.14% more than the weights of considering both static and dynamic constraints, respectively. The convergent solutions of only considering dynamic constraints are not the best results. However, additional static constraints can be helpful to obtain better results for considering dynamic constraints.