非线性度对修正双步长显式法及常用逐步积分法的影响

Effects of nonlinearity on a corrected explicit method of double time steps and common step-by-step integration methods

为了掌握非线性度对逐步积分法的影响,研究了几种积分算法在不同非线性度振动系统中的响应。通过3个典型非线性算例,对修正双步长显式法、蛙跳式中心差分法、Newmark法、广义α法和精细积分法的计算精度和稳定性能等进行了比较。结果表明:非线性度对广义α法、精细积分法和Newmark法的稳定性有影响;高非线性度对Newmark法的计算稳定性影响最大;时间步长越小,算法精度和计算量越高;相同小步长情况下,精细积分法的精度最高,而修正双步长显式法的计算量最小;在时间步长较大时,低非线性度会引起精细积分法不稳定,修正双步长显式法的精度最高,修正双步长显式法在非线性系统中具有很强的鲁棒性。

In order to comprehend the influence of the degree of nonlinearity on step-by-step integration methods,the responses of several integration methods in the vibration systems with different degrees of nonlinearity are studied.Three representative nonlinear examples are examined withαcorrected explicit method of double time steps,the leapfrog central difference method,the Newmark method,the generalizedαmethod and the precise integration method for comparison purposes.The results manifest that the degree of nonlinearity has an influence on the stability of the generalizedαmethod,the precise integration method and the Newmark method.High degree of nonlinearity has significant effect on the computational stability of the Newmark method.The smaller the time step is,the higher the algorithm accuracy and the computation effort are.If the time step is smaller,the precise integration method possesses the highest accuracy.Meanwhile,the computation cost of the corrected explicit method of double time steps is the least.The corrected explicit method of double time steps is endowed with the highest accuracy if the time step is larger.However,the precise integration method is unstable due to low degree of nonlinearity and larger time steps.The corrected explicit method of double time steps is strongly robust for nonlinear systems.