Updating of a Nonlinear Finite Element Model Using Discrete-Time Volterra Series

Abstract

In this study, the discrete-time Volterra series are used to update parameters in a nonlinear finite element model. The main idea of the Volterra series is to describe the discrete-time output of a nonlinear system using multidimensional convolutions between the Volterra kernels represented in a Kautz orthogonal basis and the excitations. A metric based on the residue between the experimental and the numerical Volterra kernels is used to identify the parameters of the numerical model. First, the identification of the linear parameters is performed using a metric based only on the first order Volterra kernels. Then the nonlinear parameters are identified through a metric based on the higher-order kernels. The originality of this nonlinear updating method stems from the decoupling of linear and nonlinear parameters and the use of global nonlinear model. In order to put in light the applicability of this technique, this work focus on the identification of the parameters in a nonlinear finite element model of a beam that was preloaded by compression mechanism. This work shows that the updated numerical model was able to represent the behaviour observed in the experimental measurements.Keywords Discrete-time Volterra series, Kautz filters, model updating, nonlinear finite element model, nonlinear identification, sensitivity. Updating of a Nonlinear Finite Element Model Using Discrete-Time Volterra Series 1 INTRODUCTIONIn many cases, the dynamical behaviour is needed to design a mechanical system. Moreover, the knowledge of the dynamical behaviour of a structure is important to control its integrity and to predict its service life. In the practical systems, nonlinear effects due to large displacements, gaps, jumps, discontinuities, etc. are very common (e.g. Worden and Tomlinson (2001)). The nonlinear finite element method is a very powerful tool to study the nonlinear vibrations. Nevertheless, it is not easy to identify the parameters of the nonlinear model due to the uncertainties in the modelling of the experimental system (e.g. the mechanical properties, the boundary conditions). Despite of the important number of investigations, the nonlinear model updating techniques are not mature as the classical linear ones.A bibliography review on the state of the art nonlinear system identification techniques in structure dynamics is presented by Noël and Kerschen (2017). Furthermore, a bibliography review of the nonlinear updating methods is exposed by Bussetta et al. (2017). Generally, the updating methods are based on the minimisation of an objective function or a metric that represents the difference between the numerical results and the experimental data. This objective function can use data in the frequency domain or in the time domain. The Volterra series can be used to defined this objective function (e.g. Wu and Kareem (2014); Guo et al. (2013)). In the paper of Shiki et al. (2012) the authors made a numerical comparison of a nonlinear model updating technique based on Volterra series and proper orthogonal ...

Nonlinear updating method: a review

“…Nonlinear finite element model has also been updated by Ebrahimian et al [31] using a batch Bayesian estimation approach. Many other examples can be found into the literature, as well as the finite element model with localized nonlinearity studied by da Silva et al [95] and the nonlinear finite element model treated by Bussetta et al [12]. Both examples use the Volterra series into the updating process.…”

“…This updating method was used by Shiki et al [90,92] for damage detection in nonlinear structures. Another metrics based on first three Volterra kernels was used by Bussetta et al [12] to update large nonlinear FE model. Third-order Volterra model was used to simulate nonlinear bridge aerodynamics by Wu and Kareem [107,108].…”

Finite element model updating using crow search algorithm with Levy flight

“…Alimouri and Khademi‐Zahedi recognized the natural frequencies of physical structure by stochastic subspace identification method and used bees optimization algorithm to update parameters of its FE model. Bussetta used the discrete‐time Volterra series to update parameters in a nonlinear FE model, which stems from the decoupling of linear and nonlinear parameters and the use of global nonlinear model.…”

A Model Updating Method for Plate Elements Using Particle Swarm Optimization (PSO), Modeling the Boundary Flexibility, Including Uncertainties on Material and Dimensional Properties

“…A problem arises because of the non-linear relationship between the vibration data and the physical parameters, making the solution more complex (Bussetta et al, 2017). To overcome such problem, the application of the Particle Swarm Optimization (PSO) has been growing in recent years.…”