## 1 Introduction  
Damping is one of critical factors affecting a structure's responses under dynamic excitations. In practical engineering structures, damped systems are commonly characterized non-proportional damping, such as the soil-structure system [1; 2; 3], the steel-concrete structures [4; 5; 6], and structures with supplemental dampers [7; 8]. Even though the dynamic response of a non-proportionally damped system can be solved by the direct integration method, the mode superposition method is frequently used. One reason is that the natural vibration properties can help to realize the dominant frequency region. Another, is that few modes usually provide sufficiently accurate results, making the mode superposition method highly efficient. The important key contribution of mode superposition method is decoupling the coupled governing equations of motion in physical coordinates into a series of generalized single degree-of-freedom (DOF) equations of motion by mode shape coordinates. It is well known that traditional mode shapes based on the modal analysis of undamped systems can decouple undamped or proportional damped systems [9]. For the non-proportionally damped system, the traditional mode shapes cannot decouple the equations of motion, and an additional technique must be adopted [10]. [PERSON] [11] proposed a state-space method to transform the equation of a damped linear dynamic system to an uncoupled set of complex modal system, but the increased computational cost hindered the wider application of this damped modal analysis. The equation of motion of the state-space method is double in dimensions compared to the traditional method, and it is necessary to solve the eigenvalue problem with complex-valued operation. Consequently, calculating the complex-value mode shapes in state space requires approximately eight times the numerical effort as the calculation of undamped real mode shapes [12]. [PERSON] [13] forced diagonalization of the transformed damping matrix by ignoring the off-diagonal terms. However, these solutions would cause unpredictable errors [14; 15]. Furthermore, the equivalent uniform damping ratio method is a simple and approximate procedure for irregularly damped cases [16; 17] such as the modal strain energy (MSE) method [18; 19], but that method could significantly overestimate damping ratios for certain systems [20]. [PERSON] and [PERSON] [12], [PERSON] and [PERSON] [21], [PERSON] [22] proposed iterative solution methods which account for the effect of off-diagonal terms, and acceptable accuracy and efficiency have been observed. The non-proportionally damped system can be viewed as having been modified from a damped system, so the dynamic characteristics and responses can be determined by the perturbation method [23; 24; 25; 26]. [PERSON] [27] used the modes of an undamped system as a basis to find the first-order perturbation solution for an arbitrary but weakly non-proportionally damped system. [PERSON] and [PERSON] [28; 29] proposed a direct modal perturbation method (MPM) that treated the new system as a minor change based on the original system. [PERSON] [30], using the first-order perturbation method to evaluate the complex eigensolution of a linearly proportional system supplemented with a viscous damper. [PERSON] and [PERSON] [31] proposed a perturbation method which can deal with the undamped system with repeated eigenvalues. Those perturbation methods are usually suitable to analyze the weakly non-proportionally damped system [27; 31; 32; 33], and need the complete eigensolution set of the unperturbed system [27; 30]. In this paper, a modal perturbation method for non-proportional damping problems applied to the vibrations of a highly non-proportionally damped system was proposed. In the method, an equivalent proportionally damped system, which is taken as the unperturbed system, is constructed from the real mode shapes of a corresponding undamped system and a non-proportional damping matrix.