A parametrically excited pendulum with irrational nonlinearity

•We present a novel parametrically excited pendulum with irrational nonlinearity.•Dynamic response curve of small oscillations is discussed by using analytical method and numerical verification.•This parametrical vibration system exhibits multiple bistable and discontinuous characteristics.•Different kinds of periodic solution and chaotic motion is displayed.

In this paper, we propose a parametrically excited pendulum with irrational nonlinearity which comprises a simple pendulum linked by a linear spring under base excitation. This parametric vibration system exhibits bistable state and discontinuous characteristics due to the geometry configuration. For small oscillations, this system can be described by Mathieu equation coupled with SD (Smooth and Discontinuous) oscillator whose dynamic response is examined analytically by using the averaging method in both smooth and discontinuous case. Numerical simulations are carried out to demonstrate the complicated dynamic behavior of multiple periodic motions and different types of chaotic motions.