Abstract: Due to the existence of gap, many mechanical systems can be simplified into piecewise linear models, where the auxiliary spring system (ASS) usually contains damping. In most classical mathematical models established in the literatures, the contact point and separation point of the primary system and ASS are generally fixed at the gap. In this paper, it is found that due to the different mechanical characteristics of the spring and damper in the ASS, the positions of the contact point and separation point actually change with the system parameters and motion state. If this case is ignored, the subsequent dynamical analysis including bifurcation and chaos may incur errors. In this paper, based on the classical mathematical model of piecewise linear system, it is firstly demonstrated through numerical solution that the primary system is prematurely separated from the ASS before returning to the gap under harmonic excitation, which explains the incorrectness of the classical model. Based on the classical mechanical model, the further study shows that there is not only the premature separation but also contact hysteresis in the primary system. Accordingly, a more reasonable mathematical model is proposed by correcting the contact and separation conditions. It is found that the contact point, separation point and the amplitude-frequency response of the corrected model differ greatly from the classical mathematical model, and the characteristic of complex dynamics may change after correction, which proves that the corrected model is more reasonable and can better reflect the engineering reality. Then, the integration interval of the averaging method is generalized so the analytical solution of amplitude-frequency response after correction is obtained. The correctness of analytical solution is verified through the Runge-Kutta method and the stability discrimination formula is obtained through the analytical solution. Finally, the influence of the parameters of the ASS on the amplitude-frequency response is explored.