Effects of uncertainties on the dynamic response of a system with non-regular non-linearities
Résumé
During the past decades, numerous studies have been done to understand and to model the nonlinear phenomena in structural dynamics. Most of these models deal with deterministic parameters. But in these systems, geometrical, materials and non linear parameters are often uncertain due to the manufacturing process for example. The effects of uncertainties on the nonlinear dynamic responses remain misunderstood and most of the classical stochastic methods used in the linear case fail to solve a non linear problem. The Multi- Harmonic Balance Method is one of the most classical mathematical approaches to determine the nonlinear stationary response of mechanical systems with regular or non-regular non-linearities. To take into account uncertainties into non linear models, few methods have been developped e.g. the Monte Carlo simulations or the perturbation methods. However, these methods in the non linear case are either not enough accurate or difficult to implement or too costly. For example, the most general technique Monte Carlo Simulations, adapted for linear or non-linear problems, generates samples of the random input parameters and solve the determinist problem for each one. This method has then a high computational cost since a high number of samples is necessary to obtain the convergence of this method : it is then not realistic when the deterministic problems are already high-dimensional problems. Then, here, we choose another method that belongs to the parametric methods : the Polynomial Chaos Expansion that allows to reduce the computational cost. This method has proved its robustness and efficiency on linear dynamic problems. Here, it is extended on one non linear problem indeed by being coupled with the Multi- Harmonic Balance Method. Besides, the Multi- Harmonic Balance Method has to be used with an alternating time-frequency approach (AFT) in order to evaluate the non linear forces. Here again, to integrate the uncertainties into the stochastic model, we will use either an Alternating Frequency Time method with Probabilistic Collocation that is a theoretical extension of AFT for nonlinear mechanical systems with presence of uncertainties. To demonstrate the robustness and the efficiency of this new mixed method, the non linear dynamic response of the mechanical system in case of presence of various uncertainties will be investigated for mono or multiple excitation frequencies. The effects of the following three kinds of nonlinearities will be examined: cubic stiffness, contact/no contact, systems with frictional interface. Finally, the case of a non linear rotor system will be treated. Besides, a comparison of the results will be done with those obtained from the Monte Carlo Simulations.