Dynamics of a Duffing oscillator with the stiffness modeled as a stochastic process

Abstract

The Duffing equation has been extensively studied in the last decades, especially in systems of vibrating micro-beams subjected to electro-mechanical fluctuating fields. However, mechanical systems are often subjected to uncertainties coming from the excitation and/or the design parameters. These uncertainties can then be considered as stationary or non-stationary stochastic processes. In many engineering applications, we consider stationary stochastic processes as they are easier to model and simplify our problems. However, some of them do not present stationarity properties and it is desirable to know what are the consequences of leaving the non-stationarity aside and consider a stationary case. For this purpose, two stochastic processes are applied to a Duffing oscillator and the results are analyzed when its stiffness modeling is based on (i) a stationary stochastic process generated with Langevin’s equation; and when it is based on (ii) a non-stationary stochastic process known as Brownian Bridge. A methodology is proposed to modify these stochastic processes in order to establish a lower limit for their support. In fact, this methodology can be applied to stationary and non-stationary stochastic processes. The stochastic results showed that the case with uncertainties in the stiffness coefficient of the non-linear term presented the highest variation on system’s response. In addition, the non-stationary case presented a result much closer to the deterministic one for the set of parameters chosen. The smooth variation of the second moment with time might explain it. There are cases in which the assumption of a stationary process might be not appropriate and non-stationarity must be assessed.