Parameter estimation of a two regime stochastic differential model for a bilinear oscillator subjected to random loads

Abstract

Many real-life mechanical and structural systems respond dynamically to random loads. The need to understand the response of the structure and problems involving different behaviors provide useful information for assessment of structural damage and failure, usually leads to stochastic inference problems. In this work we consider a simple structure modeled by a stochastic differential equation assuming two regimes both corresponding to the equations of linear single degree of freedom harmonic oscillators forced by a stochastic external load. We assume that the structure switches from one regime to the other at a given time. The regimes differ in the value of the stiffness coefficient that characterizes the model. This includes the scenario of a structure that possibly suffers from a loss of stiffness, of unknown amount, after a certain time. The stochastic forces acting on the system are assumed to be driven by a Wiener process. The state space model, describing displacements and velocities over time, is assumed to be completely observed and observations are assumed to be taken in discrete times. We describe how to compute the maximum likelihood estimates of the parameters of the stochastic model from the observations, as well as the natural vibration frequencies of the structure in both regimes, before and after the transition between regimes occurs, using all available data and from the data collected concerning displacements and velocities along the time we obtain the maximum likelihood estimates of the previous and actual values of the stiffness coefficient as well as the associated natural vibration frequencies. We present a simulation study for a particular example that illustrates the procedure and the associated accuracy. The Yuima software is used to simulate the stochastic response of the system.