Abstract: This paper is concerned with vibration problems of linear System with viscous damping.A complex mode theory is presented. The characteristics of free vibration of a damped system, such as complex conjugate, decaying vibration, decoupling and orthogonality, are analyzed byusing the sta -te space method. The basic concepts "complex frequency" and "complex mode shape" are discussed.The impulse response matrix and transfer function matrix of a damped system are also presented in this paper, and their response characteristics are analyzed.The initial value of the modal component of impulse response matrix[hi（0）]indicates the non-proportional component of the viscous damping or, simply, the"complex" of a damped system.The modal velocity frequency response function matrix of a damped system approximately provides the admittance circle property in the phase plane under the condition of rather small modal damping ratio. The concept of resonance is tentatively treated.