时变动力学Legendre级数解的Duhamel积分改进算法  

Improved Duhamel integral algorithm for Legendre series solutions to time-varying dynamic system

作　　者：金鹏飞 史治宇[1] 彭徐俊 JIN Pengfei;SHI Zhiyu;PENG Xujun(State Key Lab of Mechanics and Control for Aerospace Structures,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China)

机构地区：[1]南京航空航天大学航空航天结构力学及控制全国重点实验室,南京210016

出　　处：《振动与冲击》2024年第17期260-268,共9页Journal of Vibration and Shock

基　　金：国家自然科学基金(12272172);南京航空航天大学研究生科研与实践创新计划项目(xcxjh20230102)。

摘　　要：时变系统受迫振动的响应计算理论仍不完善,数值解法不能准确反映振动响应的解析性。利用Legendre多项式逼近法(Legendre polynomial approximation method, LPAM)可以得到时变系统响应的连续近似函数解,但该方法的计算效率较低。将时不变系统的时域响应求解理论移植到时变系统,基于线性系统的叠加原理提出基于Duhamel积分的改进Legendre级数算法,利用Dirac函数的Legendre多项式近似和Duhamel积分求解时变系统的响应。通过一个一阶非齐次变系数常微分方程组说明改进算法的有效性。设计刚度阻尼均线性变化和非线性变化的单自由度系统仿真算例,对比利用该方法得到的瞬态激励和简谐激励下系统位移响应和四阶Runge-Kutta数值法计算结果,说明了改进算法敛散性问题和计算速度的提升。The calculation theory of forced vibration response in time-varying systems is still incomplete,and numerical solutions can’t correctly reflect analytical nature of vibration response.Legendre polynomial approximation method(LPAM)can be used to obtain continuous approximate function solutions to time-varying system responses,but its computational efficiency is relatively low.Here,the time-domain response solving theory of time-invariant systems was transplanted to time-varying systems.On the basis of the superposition principle of linear systems,improved Legendre series algorithm based on Duhamel integration was proposed.Legendre polynomial approximation of Dirac function and Duhamel integration were used to solve response of time-varying system.The effectiveness of the improved algorithm was demonstrated through a set of first-order non-homogeneous variable coefficient ordinary differential equations.Simulation examples of a single-DOF system with both stiffness and damping linear changes and nonlinear changes were designed,and the system’s displacement responses under transient excitation and simple harmonic excitation obtained with this method were compared to the calculation results of the system responses with the 4th order Runge-Kutta numerical integration method to show convergence and divergence problems of the improved algorithm and improvement of calculation speed.

关 键 词：时变系统 Legendre多项式逼近 线性叠加 时域响应 

分 类 号：O322[理学—一般力学与力学基础]