Response analysis of the vibro-impact system under fractional-order joint random excitation

As a kind of good damping material, viscoelastic material is widely used in machinery, civil engineering, and other fields. In this paper, the viscoelasticity of the system is described by fractional differentiation. The dynamic response of a unilateral vibro-impact system with a viscoelastic oscillator under joint random excitation is studied, in which joint random excitation is composed of additive and multiplicative white noise. The fractional-order derivative was calculated based on Caputo’s definition, and the fractional derivative was equivalent to the corresponding linear damping force and linear restoring force. As a result, a new random system without fractional-order terms was obtained. A non-smooth transformation was introduced, which was equivalent to the original system to a new system without a velocity jump. The steady-state probability density functions of fractional-order vibro-impact systems under joint random excitation are solved by using the random average method and non-smooth transformation. In addition, the effects of parameters on the steady-state response of the system are analyzed.