Volterra System Identification Using Wigner-Ville Representations

The assumption of linearity in the characterisation of the dynamical behaviour of structures, usually valid under standard operational conditions, must be carefully re-evaluated when addressing more complex scenarios. Indeed, even simple dynamical systems may exhibit nonlinear behaviour, making it fundamental to capture these nonlinearities for accurate prediction, understanding, and monitoring of their response. This becomes evident in the case of forced dynamic tests, where the nonlinearities are often more pronounced, and thus readily detectable. The presence of nonlinearities adds complexity even to the dynamic response of Single-Degree-Of-Freedom (SDOF) systems, which in some cases serve as a fundamental representation in simple structural schemes. To effectively address this issue, selecting appropriate models and identification approaches is critical. When models are constructed with suitable variables and relationships, they can achieve high efficiency with minimised computational effort. A powerful framework is given by the Volterra series representation of nonlinear systems, which allows for gradual description of the nonlinearities by modelling the system as an infinite series of functionals. However, obtaining a sufficiently large set of experimental measurements for estimating all statistical moments in Volterra systems is typically unfeasible, especially in situ, but this issue can be partially mitigated by localizing the frequency components of the signals in time. Among the time-frequency transforms, the Wigner-Ville distribution (WVD), belonging to Cohen’s class, satisfies several desirable properties. This work focuses on developing a reliable and easily scalable tool for characterising the dynamic behaviour of structural typologies that can be approximated by SDOF systems with polynomial nonlinearities. In particular, this is done by employing the Wigner-Ville distribution in time-frequency framework combined with a Volterra series representation, especially in scenarios where polynomial nonlinearities present significant challenges to the accurate prediction of dynamic responses.