Vibration of an oscillator with random damping: Analytical expression for the probability density function

This study deals with the impact of probabilistic viscous and hysteretic damping upon the dynamic response of a linear single degree-of-freedom oscillator. Assuming that damping is governed by an unspecified probability density function (PDF), an analytical expression of the output PDF of the oscillator's transfer function is provided in terms of the input damping PDF. The instance of a uniform PDF is thoroughly treated for both viscous and hysteretic damping cases. The methodology also yields to analytical expressions of all moments of the output law of probability. A definition of a new concept of envelopes is then introduced and their analytical expression is derived. Exact analytical results are compared with Monte Carlo simulations for an academic case study. An identification procedure of a predefined input damping PDF is proposed allowing one to obtain probabilistic parameters from experimental frequency response data.