求强非线性动力系统周期解的MH法和EMH法

The MH and EMH method for strong nonlinear dynamic systems

提出了改进的谐波平衡法即MH法和EMH法,用于求强非线性系统周期解。MH法直接应用于求解大参数参变系统:x¨+a(t) x+b(t)x=c(t)。应用谐波平衡法求解强非线性系统,当谐波项取得较少时,求解结果精度低,为此引入最小二乘原理对谐波平衡法加以改进,计算结果精度高。EMH法用于求强非线性自治与非自治系统:x¨+F(x, x)=0和x¨+F(x, x,Ωt)=0的周期解,由于F呈非线性,不能直接应用MH法,为此,先由能量原理得出一次近似解,再引入牛顿迭代原理,得到关于修正量的周期系数方程;用MH法求此修正量,得到的结果精度较高。

A modified harmonic balance method (MH method) and an energy-modified harmonic balance method (EMH method) are proposed. MH method is applied directly for periodic solution of parametric oscillation systems with large parameters: x ¨ +a(t) +b(t)x=c(t). The parametric oscillation systems can be solved by means of the method of harmonic balance. But the accuracy would be low when the number of items of the series is small. Therefore, harmonic balance method is modified with least square technique principle. EMH method is applied to solve the strong nonlinear autonomous and non-autonomous systems: x ¨ +F(x, )=0 and x ¨ +F(x, ,Ωt)=0. Because of MH method can not be used directly, the first order of approximate expression of solution is gotten by using energy principle. Then the differential equation with periodic coefficient about the modified value is gotten which is higher order solution then lower order solution. MH method is applied to solve the modified value. The comparison of results between this method and Rung-Kuta method of example shows that this method is effective and its accuracy is high.