Applications of incremental harmonic balance method combined with equivalent piecewise linearization on vibrations of nonlinear stiffness systems

An improved method is proposed to combine the incremental harmonic balance method (IHBM) with the equivalent piecewise linearization for solving the dynamics of complicated nonlinear stiffness systems. Compared with the traditional IHBM, the proposed method has a wider application range and can provide a standard and unified procedure to deal with various forms of nonlinear stiffness systems. Firstly, the equivalent piecewise linearization is adopted to approximate the nonlinear restoring force of the system to the piecewise-linear restoring force and the linearized error is put forward to evaluate the computation error of this approximation. The procedure is suitable for any form of nonlinear stiffness expression that can be linearized into piecewise-linear stiffness expression. Then the IHBM procedure for the general piecewise-linear stiffness system is derived considering the sub-harmonic responses. The Duffing oscillator is presented to validate the proposed method. Finally, three cases of nonlinear stiffness systems are carried out. The results of these cases show that the proposed method is capable of analyzing the dynamics of these complicated nonlinear stiffness systems.