Phd in THEORETICAL AND APPLIED MECHANICS

Thesis title: Nonlinear Dynamic Substructuring of Systems with Localized Nonlinearities

The dynamic analysis of complex engineering structures has always represented a challenging task, since the reliability of the results significantly depends on the accuracy of the model. The dynamic behavior of engineering systems can be often described by using a linearized set of algebraic or differential equations. However, since the use of linearized models might not provide reliable results, an enhanced description of the system based on a set of nonlinear equations can be necessary. Different techniques have been developed over the years to analyze the dynamics of mechanical systems. Dynamic Substructuring represents a powerful tool that deals with the subsystems that make up the assembly, allowing their characterization with the simultaneous use of different types of data (experimental, numerical or analytical). However, this approach is developed in a predominant way under the assumption that the analyzed systems can be described using linear equations. The objective of this research is to use dynamic substructuring to predict the dynamic behavior of systems described using nonlinear equations, thus addressed as nonlinear. In particular, a specific and yet wide class of cases is considered, namely those in which two or more substructures are jointed together through connections that introduce significant nonlinear effects, such as bolted joints and wire rope isolators to name a few. In these cases the connecting elements play such a relevant role in the nonlinear behavior of the assembled system that they can be considered as nonlinear substructures while the connected subsystems as linear. This research aims at developing an energy based modal coupling procedure to analyze the dynamics of these systems. It is performed by evaluating how the energy is distributed among the substructures at a given energy level, which also leads to a reduced computational burden. The theory of Nonlinear Normal Modes is used to account for the presence of the nonlinear connections in the coupled assembly. Lumped parameter models are first analyzed to evaluate the effectiveness of the procedure and the accuracy of the results by comparing them with those obtained through well established techniques, such as the Harmonic Balance. These models are suitable to account for the presence of multiple connections between the coupled substructures, allowing also to easily evaluate the effects of different types of nonlinear laws, either hardening and softening. The substructuring procedure successfully retraces the dynamics of these systems, thus the analysis is extended to discretized continuous subsystems connected through nonlinear connections with hardening behavior. The results are coherent with the ones achieved through the HB method, thus confirming the effectiveness of the approach. Also, the effects of modal truncation on the dynamics of the system in terms of resonance frequencies and mode shapes are analyzed. Once the effectiveness of the nonlinear coupling procedure is assessed, a real case is considered. A nonlinear connecting element, specifically designed and tested to have a nearly cubic behavior, is used to connect two linear subsystems and the nonlinear response of the assembly is measured. The numerical substructuring results are compared to the experimental ones in order to evaluate the accuracy of the procedure. Eventually, the substructuring procedure is modified to account for the presence of damping in one of the subsystems or in the connection. It is applied to analyze the dynamics of a non-conservative lumped parameter model, providing accurate results. Overall, it is possible to affirm that the procedure developed so far is effective in analyzing complex systems with localized nonlinearities.