Computation of nonlinear normal modes through continuation methods

In mechanical engineering, performance enhancement usually results in lighter and more flexible structures and pushes the limits of the system operating envelope. Nonlinearity is therefore becoming a frequent occurrence and linear design tools show their limitations. To overcome these issues, nonlinear normal modes (NNMs) were introduced in structural dynamics as a direct extension of linear normal modes to nonlinear systems. Our contribution reviews the history and the new trends for the computation of NNMs in mechanical engineering.
Specifically, algorithms for the continuation of periodic solutions were first developed. Such algorithms are now well-established and applicable to large-scale systems such as real-life aerospace structures. To further extend the concept of NNMs to nonconservative systems, the definition of an NNM as an invariant manifold in the system’s phase space was introduced. Again, continuation techniques are particularly well suited for computing these invariant manifolds. The geodesic level set method developed by Krauskopf and Osinga [1] as well as the “PDE formulation” method of Guckenheimer and Vladimirsky [2] are both considered.

[1] Krauskopf, B. and H. Osinga (2003). "Computing Geodesic Level Sets on Global (Un)stable Manifolds of Vector Fields." SIAM Journal on Applied Dynamical Systems 2(4): 546-569.

[2] Guckenheimer, J. and A. Vladimirsky (2004). "A Fast Method for Approximating Invariant Manifolds." SIAM Journal on Applied Dynamical Systems 3(3): 232-260.