CISM-AIMETA Advanced School on "Dynamic Stability and Bifurcation in Nonconservative Mechanics"
April 10, 2017 — April 14, 2017
- Oleg Kirillov (Russian Academy of Sciences - Steklov Mathematical Institute, Moscow, Russia)
- Davide Bigoni (Università di Trento, Trento, Italy)
Nonconservative mechanical systems have been examined since the end of the XIX-th century when Greenhill posed a problem on the buckling of a screw-shaft of a steamer subject to both an end thrust and a torsion with the torque vector being tangential to the deformed axis of the shaft (the follower torque). Such follower loads represent non-potential positional forces, also referred to as circulatory or curl forces, producing non-zero work on a closed contour (e.g. optical tweezers can generate a circulatory force field). At the same time Kelvin and Tait, studying problems of planet formation, discovered the destruction of gyroscopic stabilization in rotating ellipsoids filled with liquid by dissipation. This was the first example of dissipation-induced instabilities of negative energy modes. Nowadays dissipative and circulatory forces are recognized as the two fundamental nonconservative forces in the growing number of scientific and engineering disciplines including physics, fluid and solids mechanics, fluid-structure interactions, and modern multidisciplinary research areas such as biomechanics, micro- and nanomechanics, optomechanics, robotics, and material science.
From the very beginning, nonconservative systems demonstrated unusual and counter-intuitive dynamics and stability properties. Efficient prediction of flutter and divergence instabilities which, in non-conservative systems can be both desirable, as in energy harvesting and harmful as in aircraft structures, is mathematically challenging. This is due to the non-self-adjoint (non-Hermitian) character of the governing equations that, as a rule, depend on multiple parameters. However, traditional university curricula do not offer a coherent collection of modern mathematical tools for the analysis of multiparameter families of non-self-adjoint differential equations combined with a first-hand demonstration of how they actually work in practical applications.
The present course fills this gap and offers a unified view on classical results and recent advances in the dynamics of nonconservative systems. The theoretical fundamentals are presented systematically and include: Lyapunov stability theory, Hamiltonian and reversible systems, negative energy modes, anomalous Doppler effect, non-holonomic mechanics, sensitivity analysis of non-self-adjoint operators, dissipation-induced instabilities, and absolute and convective instabilities. They are applied to engineering situations that include the coupled mode flutter of wings, flags and pipes, flutter in granular materials, piezoelectric mechanical metamaterials, wave dynamics of infinitely long structures, stability of high-speed trains, experimental realization of follower forces, soft-robot locomotion, wave energy converters, friction-induced instabilities, brake squeal, non-holonomic sailing, and stability of bicycles.
The course is targeted at young researchers, doctoral students and engineers working in fields associated with the dynamics of structures and materials. The course will help to get a comprehensive and systematic knowledge on the stability, bifurcations and dynamics of nonconservative systems and establish links between approaches and methods developed in different areas of mechanics and physics and modern applied mathematics.