摘要
应用牛顿谐波平衡法求解一个具有有理式恢复力的非线性振子的近似频率和近似周期解.这种方法先用牛顿法将非线性方程线性化再用谐波平衡法求解,这样避免直接使用谐波平衡法时需要求解非常复杂的非线性代数方程组.用这种方法可以容易得到高阶近似角频率和近似周期解的显式表达式,这些近似解对小振幅和大振幅的非线性振动问题都有效.当振幅很大时,一阶近似角频率与精确角频率的百分比误差为7.845%,而二阶近似角频率与精确角频率的百分比误差为2.636%.与数值方法给出的"精确"周期解比较,二阶近似解析周期解比一阶近似解析周期解要精确的多.
The Newton-harmonic balancing approach is applied to a strong nonlinear oscillator with the restoring force having a rational form.The approach is characterized by that the linearization of governing differential equations by Newton's method is conducted prior to harmonic balancing.It doesn't need to solve a set of very complicated nonlinear algebraic equation.With this approach we give explicit approximate formulas for the exact angular frequency and periodic solution.These approximate solutions are valid for small as well as large amplitudes of oscillation.When the amplitudes are large,the percentage error of the first-order approximate period in relation to the exact one is 7.845%,and the percentage error of the second-order approximate period is 2.636%.A comparison of the first and second analytical approximate periodic solutions with the numerically exact solutions shows that the second analytical approximate periodic solution is much more accurate than the first one.
出处
《湖南科技大学学报（自然科学版）》
CAS
北大核心
2010年第3期121-123,共3页
Journal of Hunan University of Science And Technology：Natural Science Edition
基金
国家自然科学基金(50775071)
湖南科技大学研究生创新基金(S090105)资助项目
