《结构动力学》中多频激励多自由度系统稳态解的新方法

A NEW METHOD FOR STEADY-STATE RESPONSE OF MULTIDOF SYSTEM WITH MULTI-FREQUENCY HARMONIC EXCITATIONS IN STRUCTURAL DYNAMICS

本文建立了一种多频激励下多自由度系统的特征值和稳态解的解析求解新方法.基于叠加原理,首先将稳态响应根据激励频率数量展开成多个简谐响应的叠加;其次,根据简谐平衡原理,将弹簧力、激励荷载以及惯性力分解成同样个数的对应荷载叠加;再次,根据达朗贝尔原理建立弹簧和质点的动态平衡方程;最后,根据传递矩阵法进行求解.通过4自由度系统的算例表明该方法求解结果和振型叠加法完全一致.研究表明该方法可同时求解特征值问题和系统的稳态解,在求解稳态解时不需先求特征值问题.

A new method is developed to obtain the eigen frequency and steady state response of multi-DOF system with multi-frequency harmonic excitation.Based on the superposition principle,the steady state response is divided into sev⁃eral components with different excitation frequencies.Correspondingly,the spring force,external excitation force and in⁃ertia force of the system are also divided into a number of components similar to that of the response.The quasi-static equilibrium among the spring force,external excitation force and inertia force of mass based on D'Alembert's principle is imposed between the corresponding components of these forces.Then,the transfer matrix method is employed to attain the eigen frequency and steady state response of multi-DOF system without resolving the eigenvalue problem in advance.A 4-DOF system is used to demonstrate the proposed method,suggesting that application of the new method can easily obtain analytical solutions,which is verified by the modal superposition method and shows that the proposed method is also effective for obtaining the eigen frequency and mode shape of multi-DOF system without resolving the eigenvalue problem in advance.