Sparse identification of the dynamics of a nonlinear multistable oscillator

Abstract

AbstractRecently, data‐driven modeling approaches are getting increasingly examined regarding their applicability for nonlinear mechanical or mechatronic systems. With a high data availability and often insufficiently accurate descriptions of complex behavior of real systems using established physical models, statistical models provide promising alternatives. Alongside machine learning techniques like deep neural networks, sparse regression is increasingly used to obtain models from measurement data. With sparse regression, governing equations are estimated from a given function space so that the data are explained with as few terms as possible while maintaining a low model error. This method is implemented in a framework called sparse identification of nonlinear dynamics. This paper demonstrates the application of this method on free and forced vibrations of a two degree of freedom nonlinear rotary oscillator with two stable equilibrium positions. The setup and data acquisition as well as the application of sparse identification are described. The selected function space containing the candidate functions is essential for an accurate representation of the system at hand. Monomials form a commonly used function space because they can approximate a wide variety of nonlinear characteristics. However, in this work, it is shown that monomials alone are insufficient when dry friction appears. Therefore, to account for Coulomb friction, a sign‐function is added as a candidate to the function space of monomials up to the fifth order. Adaptions of the common optimization algorithms turned out to be necessary for the inclusion of Coulomb friction. As a result, it is found that the addition of the sign‐function for Coulomb friction increases the model quality significantly.