Modeling and analysis of a system of two self-balancing pendulums is presented in this paper. Such systems are commonly used as elements of automotive door latch mechanisms that can be subjected to oscillatory excitation or vibratory inertia forces occurring during crash events. In order to avoid an unwanted behavior such as opening of the door, the considered mechanism should be properly designed and its dynamical response well understood and predictable. One pendulum of the double-pendulum system, playing the role of a counterweight (CW), is used to reduce the second (or main) pendulum motion under inertia loading. The interaction force between the pendulums is defined as the reaction of a holonomic constraint linking the rotations of both pendulums. Another reaction force acts between one of the pendulums and the support, reinforced by the action of a preloaded spring. An important aspect of the model is its discontinuous nature due to the presence of a gap in the interface area. This may result in impacts between both pendulums and between one of the pendulums and the support. High-frequency/high-acceleration amplitude vibratory motion of the base part provides inertia input to the system. Classical multibody dynamics approach is adopted first to solve the equations of motion. It is shown that the considered system under certain conditions responds with a high-amplitude irregular motion. A special methodology is used in order to study the regions of chaotic motion, with the goal to gain more understanding of the considered system dynamics. Bifurcation diagrams are presented together with quantitative and qualitative analysis of the motion. The sensitivity of solutions to variation of system parameters and input characteristics is also analyzed in the paper.