MS12.7: Nonlinear Dynamics for Engineering Design

The Hopf calculations allow us to estimate the amplitude of the limit cycle emerging close to the bifurcation point analytically. According to the Hopf theorem, the radius of the limit cycle grows as a square root function of the bifurcation parameter. Many mechanical models consider exactly third degree nonlinearities in the Taylor expansion, see for instance the Duffing oscillator. However, second degree terms break the symmetry of the nonlinearities and that of the emerging limit cycle branches. This way, the usual analytic prediction of the limit cycles might be inaccurate for these asymmetric cases, that is, an estimation with the calculated radius is not satisfactory. This work presents the possibilities of correction, and some oscillatory systems where these corrections are relevant and necessary.

We present experiments on a new hydraulically amplified self-healing electrostatic (HASEL) actuator and propose a method for modeling its forced response on data-driven spectral submanifolds (SSMs). First, we find that the fast periodic dynamics is captured by a two-dimensional forced SSM. Second, over longer timescales and slower excitation, we discover hysteretic behavior and creep, and propose a new method for modeling this behavior on hyperbolic invariant manifolds. Our method, based on geometric singular perturbation theory, demonstrates for the first time how to account for memory effects in SSM-based modeling and provides a low-dimensional model suitable for control.

This study addresses the dynamic response variability in friction-damped structures, with a focus on the uncertainty stemming from the non-uniqueness of residual tractions. Utilizing a recently developed Nonlinear-Mode based method, we first perform uncertainty quantification on amplitude-dependent modal parameters through a surrogate model-based optimization, and then obtain frequency response bounds with an interval analysis. The benchmark consists of realistic turbine blades coupled with an underplatform damper. Results reveal significant variability in the dynamic behavior of the structure, and the effectiveness of the method is demonstrated with accurately-captured response bounds.

We propose a new application of the basin stability tool which allows to update the information on the system properties under uncertainties. The concept is presented using classical mechanical setup of coupled pendula, exchanging the energy via the supporting structure. Depending on the support parameters, the model can exhibit different types of co-existing synchronous patterns, as well as remain desynchronized. We calculate basin stability maps of particular behaviours and combine them with prior parameters distributions using Bayesian inference. The obtained posterior results, based on the attractors occurrence, update our knowledge on the system properties in the terms of probabilities.