Second order approximation of natural frequency constraints in structural synthesis

Structural optimization problems with constraints imposed on natural frequencies are studied, giving special attention to the inherent non‐linearity of natural frequency constraints. Assuming that the system mass and stifiness matrices are expressed as linear functions of the design variables, it is shown that there are two sources of non‐linearity for natural frequencies in such a design space. Furthermore, simple variable transformation does not alleviate these non‐linearities as it does in the static response case. Recognizing the inherent non‐linear characteristics of natural frequency constraints, a second order Taylor series approximation for each eigenvalue of the equations of motion is derived and applied successfully to improve stability and overall efficiency of the automated synthesis process.