This paper considers a small oscillatory motion of a pendulum suspended on a single-degree-of-freedom system subjected to a narrow band excitation and impacts due to a barrier located at a certain distance from the system׳s equilibrium position. The influence of a impacting motion of the single-degree-of-freedom system onto the instability boundaries of the pendulum is investigated. It is demonstrated that the impacting motion significantly changes the shape of the instability domain compared to the traditional one inherent to the Mathieu equation.
Alevras, P., Bobryk, R., & Yurchenko, D. (2014). Stability of an autoparametric pendulum system with impacts. Journal of Sound and Vibration, 333(26), 7233–7247. https://doi.org/10.1016/j.jsv.2014.08.003