【摘要】 本文是《多自由度复模态理论的摄动方法(一)一阶摄动》[1]的继续,讨论重特征值及高阶摄动修正问题,对于有重特征值的实模态摄动修正已有论述,本文将论述复特征值的修正。一般而言,一阶摄动已有足够精度,但当参数变化范围稍大时,需要二阶或更高阶的摄动修正,Meirovitch等人讨论了无阻尼,非陀螺系统的二阶摄动修正,并用于响应计算。当阻尼系数增大时,复特征值的误差将随之增大。本文将给出二阶摄动修正及任意阶摄动修正,从而得到二阶及二阶以上的复特征值及复特征矢量的近似公式。Aubrun采用Jacobin公式讨论了有阻尼系统的摄动解,给出了一阶及二阶的阻尼,频率修正公式及一阶复模态,但是由于非按照正规的摄动方法来求解,其一阶阻尼系数与本文虽一致,但对频率则无修正,阻尼对复模态的修正也只有虚部而无实部。为了改善收敛速度,本文提出了将阻尼阵中可对角化部分作为与质量,刚度阵同量级列入方程,而不可对角化部分列入一阶摄动量。这种改进的摄动法以复特征值及实振型为零阶近似,从而可以提高精度改善收敛速度,使对阻尼阵作为一阶小量的限制放宽。作为复模态理论摄动法的应用,讨论了陀螺特征值问题。文末并给出了简单的算例。更多还原

【Abstract】 In the paper of "A perturbation Method For The Complex mode Theory Of Multi-DOF Linear Systems (I) First-order perturbation",the Perturbation equations of any order have been derived and the results of first-order perturbation in the case without multiple-eigenvalue have been studied in detail. In dealing with practical structures such as those with certain kind of symmetry,there appears so called multiple-eigenvalue problem to which the perturbation formulas presented there are no longer available. In this paper, a process is developed to tackle this special situation. Although, as a general rule,the results of first-order p erturbation may have satisfactory accuray in many cases of practice,the second-or higher-order Perturbations must be taken into account in order to guarantee the validity of the approximations when the Perturbation parameters vary over a rather wide range. Meirovitch etal have developed a second-order perturbation theory for the response of nongyroscopic dynamic systems without damping. It is known that as the coefficients of damping get large so will the errors of complex modes and eigenvalues given by the first-order perturbation equations. In this Paper,the results of complex mode and eigenvalues of second-and higher-order perturbations are presented. Aubrun has got the perturbation formulas of damping coefficients and frequencies of first-and second-order and the formulas of first-order complex modes,but the results except damping coefficients are different from those obtained here because the pertnrbation techniques used in two papers arc not quite the same.In order to improve the rate of convergence of perturbation method,this paper suggests an approach that the damping matrix is divided into two parts with the first being the part that can be diagonalized by real mode transformation,the second part being the remaider and only the second part of damping matrix is treated as small perturbated parameters. In this way, accuracy and convergence rate of the perturbation method are then improved without complicating the process and the previous restriction that the damping must be small on the whole is now modified.As an application of the Perturbation method of the complex mode theory, cigensolution problem of gyroscopic dynamical systems is studied to some extent.Two numerical examples are given at the end of this Paper.更多还原