Framework to study the Nonlinear Vibrations of Geometrically-Exact Mechanical Systems

• Applicant: Valentin Sonneville, Ph.D.
• Subject Area: Mechanics ; Applied Mechanics, Statics and Dynamics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 536955101

In the pursuit of innovation and efficiency, the structural components of mechanical systems are becoming increasingly slenderer. Indeed, slenderer components are lighter, which reduces the amount of material needed to manufacture them and the amount of power to operate them. Furthermore, they can deform and vibrate substantially and this ability is the operating principle of several advanced systems. Nevertheless, structural integrity and optimal performance require a careful design procedure because the substantial effects of deformation and vibration incurred by the slenderness may lead to complex nonlinear dynamical behaviors. On the one hand, nonlinear dynamics is a growing field that develops methods for analyzing nonlinear dynamical phenomena. Nevertheless, most developments in this area have been carried out on approximated mathematical descriptions and have been limited to isolated components or simple systems of components. On the other hand, analytical and numerical tools to process nonlinear dynamics without such approximations, often referred to as geometrically-exact theories, have been developed in nonlinear structural mechanics. These developments were made possible by the use of advanced mathematical tools, including differential geometry and group theory, and have led to efficient modeling and solution techniques. This framework is commonly employed in structural analysis and has been used successfully to model large, complex mechanical systems but does not appear to be standard practice in nonlinear dynamics. This detrimental omission stems from the lack of appropriate tools to analyze the nonlinear dynamics of systems modeled with geometrically-exact theories. In light of these observations, the objective of this proposal is to develop a framework to study the nonlinear vibrations of large, complex mechanical systems modeled with geometrically-exact theories. To that end, state-of-the-art methods of nonlinear dynamics will be developed and formulated in a mathematical language that allows the use of geometrically-exact models and associated state-of-the-art solution methods. The work program includes: • Numerical methods for the computation of periodic solutions via shooting (WP 1) and discretization (WP 2), • Numerical computation of nonlinear normal modes by continuation of periodic solutions (WP 3), • Development of an appropriate software environment (WP 4), • Validation against benchmarks and case studies (WP 5), • Evaluation of possible extensions of the framework (WP 6).

• DFG Programme: Research Grants