Analysis of the dynamics of bilinear systems is critical for a variety of civil, mechanical and aerospace structures that contain gaps or prestress that are caused by cracks, delamination, joints or interfaces amongst components. Recently, a method was developed called bilinear amplitude approximation (BAA) to estimate the response of bilinear systems without gaps or prestress. This method was developed on the idea that the bilinear system can be separated into two time intervals both of which the system behaves as a distinct linear system: (1) the open state and (2) the closed or sliding state. In order to couple the linear vibrational response for each time interval, both geometric and momentum constraints are applied as transition conditions between the states. This paper expands the previous BAA method for the case where there are either gaps or prestress in the system. The new method requires the forcing magnitude to be known so that it can accurately determine when the system transitions between the two states, and the new equilibrium positions for each state for a given forcing magnitude. The new method also finds the bilinear frequency of the system, which cannot be computed using the bilinear frequency approximation (BFA) method previously developed since that method is only accurate for the zero gap and no prestress case. The new BAA and BFA methods are demonstrated on single degree of freedom and three degree of freedom systems for a variety of forcing conditions.

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