The vibration response of a damped linear system is always calculated based on the mode superposition method. However, the construction of the damping matrix is difficult for the conventional mode superposition methods based on the viscous damping model, and this problem is much more serious for nonproportionally damped linear systems. The damping matrix based on the hysteretic damping model is easy to construct and unique, which is determined only by the structural stiffness and material loss factor. The time-domain motion equation of a multi-degree-of-freedom nonproportionally damped linear system is easily constructed based on the hysteretic damping model. According to the characteristics of external excitation, the general solution of the corresponding homogeneous equation and special solution of the corresponding nonhomogeneous equation can be solved. By the aid of the easiness of the damping matrix, a complex mode superposition method based on the hysteretic damping model is proposed for the nonproportionally damped linear system. Based on the proposed method, a user subroutine in ANSYS and MATLAB is developed to calculate vibration responses in time-domain dynamic analyses. A shaking table test of a cantilever plate composed of host and damping layers is conducted to validate the proposed method. The method proposed in this article is unconditionally convergent, and its convergence is independent of the time step of time-domain analyses. Compared with the common complex mode superposition method based on the viscous damping model, the simulation results of the proposed method are closer to the test results, and its accuracy and efficiency are higher. In addition, the calculation results of the proposed method are unique, which is irrelevant to the choice of vibration modes.

阻尼线性系统的振动响应始终基于模式叠加法来计算。然而，对于基于粘滞阻尼模型的常规模式叠加方法，阻尼矩阵的构建是困难的，并且对于非比例阻尼线性系统，该问题更为严重。基于滞后阻尼模型的阻尼矩阵易于构造且唯一，它仅由结构刚度和材料损耗因子决定。基于滞后阻尼模型，可以轻松构建多自由度非比例阻尼线性系统的时域运动方程。根据外部激励的特点，可以求解相应的齐次方程的一般解和相应的非齐次方程的特殊解。借助于阻尼矩阵的易用性，针对非比例阻尼线性系统，提出了一种基于滞后阻尼模型的复模叠加方法。基于所提出的方法，在ANSYS和MATLAB中开发了一个用户子程序来计算时域动态分析中的振动响应。进行了由主体和阻尼层组成的悬臂板的振动台试验，以验证所提出的方法。本文提出的方法是无条件收敛的，其收敛与时域分析的时间步长无关。与基于粘性阻尼模型的普通复模叠加方法相比，所提方法的仿真结果更接近测试结果，其准确性和效率更高。另外，该方法的计算结果是唯一的，与振动模式的选择无关。