中文摘要：

针对传统增量谐波平衡法求解多频激励局部非线性系统周期响应耗时太长的问题,提出了降维增量谐波平衡方法。首先通过谐波平衡理论分析了多频激励局部非线性系统响应中各自由度各次谐波的定量对比关系,并且根据该定量对比关系使系统的维数降至与非线性自由度个数相同;其次针对降维后的复数非线性系统推导了多频增量谐波平衡法,以及原系统各自由度各阶响应的还原方法;最后利用双频激励局部非线性悬臂梁系统进行了所提方法的精度和效率验证。结果表明：该方法的精度与传统方法一致,但是在局部非线性自由度较少时其效率远高于传统方法。

英文摘要：

The time-consuming problem of the traditional incremental harmonic balance method can be often encountered when determining the steady state response of local nonlinear system with multi-frequency excitation.The demension-reductive incremental harmonic balance method is proposed to increase the efficiency.For this method,the quantitative comparison relationship of every harmonic in each degree of freedom in the response of the local nonlinear system is analyzed by using the harmonic balance theory,and the dimension of system is reduced to be equal to the dimension of the nonlinear structure by using this relationship.Then the multi-frequency incremental harmonic balance method for the reduced complex system,as well as the response restoring method of every harmonic in each degree of freedom of the original system is deduced.Furthermore,the accuracy and efficiency of the proposed method is verified by using the dual-frequency excitation local nonlinear cantilever beam system.Studies show that the accuracy of the proposed method is in line with the traditional method but the efficiency is much higher with the system of less degree of freedom.