求解 Rayleigh 阻尼系数的加权最小二乘法

摘要

在地震反应分析过程中，提出了一种优化方法以解决 Rayleigh 阻尼系数计算时选择两阶合理参考频率的难题。该方法是以反应谱理论为基础，以结构位移峰值误差最小为目标函数。将位移反应谱用一阶 Taylor 级数近似计算，从而将目标函数简化为加权最小二乘法的方程。随后以框架结构为例，讨论了模态个数和阻尼比模型对 Rayleigh 阻尼系数计算的影响，并与传统方法、最小二乘法及基于多参考振型的加权最小二乘法进行比较。计算结果表明，最小二乘法由于忽略了模态贡献的影响，不是计算 Rayleigh 阻尼系数的合理方法。当模态个数所包含的累积振型参与质量达90%以上，本文方法所得 Rayleigh 阻尼系数计算结果稳定，结构动力反应的计算精度高。%During the seismic response analysis,an optimization method is developed to overcome the problem on selection of two reasonable reference frequencies for the calculation of Rayleigh damping co-efficients.In the method,based on response spectral analysis,an objective function is established by mini-mizing an error term of the peak displacement of a structure.After approximating the displacement response spectrum to its first-order Taylor’s series,the objective function is simplified as a weighted least squares equation.Then,the effects of both mode numbers and exact damping model on the evalua-tion of Rayleigh damping coefficients are investigated with a frame structure,and are compared with traditional method,least squares method,weighted least squares method based on multi-reference modes. Numerical results show that the least squares method is not a reasonable method to evaluate Rayleigh damping coefficients due to neglecting the effect of mode contribution.The proposed method would obtain stability Rayleigh damping coefficients when the mode numbers cover accumulated modal partici-pating mass more than 90%,and make dynamic response results accuracy.