The reduced space method for calculating the periodic solution of nonlinear systems

A hybrid method combined the reduced Sequential Quadratic Programming (SQP) method with the harmonic balance method has been developed to analyze the characteristics of mode localization and internal resonance of nonlinear bladed disks. With the aid of harmonic balance method, the nonlinear equality constraints for the constrained optimization problem are constructed. The reduced SQP method is then utilized to deal with the original constrained optimization problem. Applying the null space decomposition technique to the harmonic balance algebraic equations results in the vanishing of the nonlinear equality constraints and a simple optimization problem involving only upper and lower bound constraints on the optimization variables is formed and solved. Finally, numerical results are given for several test examples to validity the proposed method. The efficiency of the solution method to trace the family of energy dependent nonlinear modes is illustrated. The localization nonlinear normal modes of bladed disks related to various types of internal resonances are explored.