单步无条件稳定结构动力求解隐式算法

A single-step unconditionally stable implicit algorithm for structural dynamic solution

基于量纲分析的思想,提出一种用于求解结构动力问题的单步无条件稳定隐式算法。该算法首先构造位移和速度关于质量、刚度、阻尼等变量的函数关系,然后对方程进行无量纲化处理,引入无量纲系数,最后化简得到位移和速度的递推格式。该算法在位移和速度递推格式中不包括加速度项,但能提供2阶精度加速度输出,从而实现真正的自启动。理论分析表明算法具有无条件稳定、2阶精度、可控数值阻尼和无超调特性。分别以单自由度结构和多自由度结构为对象,考虑不同的阻尼条件,对比分析各种算法的数值特性。研究结果表明:通过对算法中的自由参数进行设计,该算法可等价于Newmark平均加速度法,实现零振幅衰减和较低的周期延长,同时与Houbolt法和Wilson-θ法相比,具有更优的频谱特性,能得到更精确的计算结果;与Generalized-α法相比,该算法在有阻尼的情况下,位移、速度和加速度均有2阶精度,同时具有更低的计算误差。通过引入数值阻尼,该算法能更加有效地滤除高频响应分量,更大程度地保留低频响应的贡献,提供更好的计算精度。研究成果可以为结构动力问题的求解提供新方法,为算法的研究提供新思路。

Based on the idea of dimensional analysis,a single-step unconditionally stable implicit algorithm for solving structural dynamic problems was proposed.The algorithm first constructed a functional relationship between displacement and velocity with respect to variables such as mass,stiffness,and damping,then the equations were made dimensionless by introducing dimensionless coefficients and finally simplified to obtain a recursive format for displacement and velocity.The algorithm did not include acceleration terms in the displacement and velocity recursive formats,but provided second-order precision acceleration output,thus enabling true self-start.Theoretical analysis shows that the algorithm has unconditional stability,second-order accuracy,controlled numerical damping,and no overshoot characteristics.The numerical characteristics of various algorithms are compared and analyzed by considering different damping conditions for single degree of freedom and multi degree of freedom structures,respectively.The results show that by designing the free parameters in the algorithm,the algorithm can be equivalent to the Newmark average acceleration method,achieving zero amplitude decay and lower period elongation,while having better spectral characteristics and obtaining more accurate calculation results compared with the Houbolt method and Wilson-θmethod.Compared with the Generalized-αmethod,the algorithm has second-order accuracy in displacement,velocity and acceleration with damping,as well as lower computational error.By introducing numerical damping,the algorithm can filter out the high-frequency response components more effectively and retain the contribution of the low-frequency response to a greater extent,providing better computational accuracy.The research results can provide new methods for the solution of structural dynamic problems and new ideas for the research of the algorithms.