Due to the existence of gap or backlash, many mechanical systems can be simplified into piecewise linear models. The dynamic study on mechanical system should be based on reliable mathematical model, so that it is very important to determine the contact point and separation point between the primary system and the auxiliary spring system (ASS) in the piecewise linear systems. In most existing literatures on piecewise linear systems, the contact point and the separation point of the mathematical model are fixed at the gap. But in this paper, it is found that the contact point and separation point actually change with the system parameters when the ASS contains damper, which implies the most existing mathematical model are incorrect. It is firstly demonstrated through numerical solution that the primary system will prematurely separate from the ASS before reaching the gap under harmonic excitation, which shows the incorrectness of the classical mathematical model. Then, based on the mechanical model and engineering practice, two corrected mathematical models are proposed. And the motions of the primary system and ASS after prematurely separation in the corrected models are studied. Finally, through comparison of the contact point, separation point, amplitude-frequency curves and motion state between the corrected models and the classical mathematical model, it can be concluded that the corrected models are more reasonable. And the comparisons with the experimental data implies that the corrected models can better reflect the engineering practice. These results will be helpful to the study and design of the piecewise linear system.