MS15.4 - Nonlinear Dynamics and Dynamic Stability

In order to design effective energy-absorbing shields made of modern composite materials (e.g., fibrous composite panels), it is necessary to use deformable materials whose important feature is the largest possible dissipation of impact energy. Rheological properties of these materials differ significantly from dissipative and elastic properties described by the Hooke and Kelvin linear models. Moreover, the range of material deformation at the place of impact is so wide that the above-mentioned models are not sufficient to properly describe the process of energy dissipation. The value of dissipated energy is also important for the determination of the lifespan of dissipative and elastic elements subjected to cyclic dynamic loading and made of modern structural materials (fatigue phenomena). The authors proposed a dynamic model in which the rheological properties of the material are described by two spring parameters c, c0 and one parameter of viscous damping k0 in the configuration used in the so-called rheological standard model (Zener model). In addition, the model introduces the element of dry friction described by the appropriate function h(x). Furthermore, it was assumed that the non-linear element present in this model describes the effect of the occurrence of dry friction hSign(dx/dt) in the tested material, the value of which h is not constant, but depends on the extent at which the level of material deformation (variable x) is described by the relevant function. In this way, a non-linear third-order equation was obtained describing the motion of the so-called degenerate system. The developed method was tested on a computer system.

Increasing the damping properties of a structure while limiting the addition of mass is a challenging task. Originally developed for their acoustic absorption capabilities, microperforated plates (MPP) are simple devices that can feature additional viscous damping at low frequencies. Dissipation arises in the boundary layers of the microperforations through viscous friction mechanisms induced by the relative motion between the solid and the fluid in which the MPP is immersed. This leads to sound absorption in an acoustic context and to an added viscous damping in a solid dynamics context. The added viscous damping exhibited by the MPP can be captured by considering adapted fluid-solid interaction terms in the governing equations. A model was experimentally validated in a linear context. However, when the fluid velocity in the microperforations becomes sufficiently large, the inertial effects occurring in the microperforations can also have a significant impact. These are captured by Forchheimer's correction. Introducing this correction into the MPP linear vibration model leads to a nonlinear system with an additional antisymmetric quadratic damping term. The resulting equations of motion are solved numerically for a large amplitude forcing and quantitative analyses are performed. It is shown that the influence of antisymmetric quadratic nonlinear damping on the added viscous damping at the MPP depends on the magnitude of the external mechanical force. The model suggests the existence of maximum added viscous damping as a function of the external mechanical force magnitude.

An approach for the identification of discontinuous and nonsmooth nonlinear forces, as those generated by frictional contacts, in mechanical systems that can be approximated by a single-degree-of-freedom model is presented. To handle the sharp variations and multiple motion regimes introduced by these nonlinearities in the dynamic response, the partially-known physics-based model and noisy measurements of the system's response to a known input force are combined within a switching Gaussian process latent force model (GPLFM). In this grey-box framework, multiple Gaussian processes are used to model the unknown nonlinear force across different motion regimes and a resetting model enables the generation of discontinuities. The states of the system, nonlinear force and regime transitions are inferred by using filtering and smoothing techniques for switching linear dynamical systems. The proposed switching GPLFM is applied to a simulated dry friction oscillator and an experimental setup consisting in a single-storey frame with a brass-to-steel contact. Excellent results are obtained in terms of the identified nonlinear and discontinuous friction force for varying: (i) normal load amplitudes in the contact; (ii) measurement noise levels and (iii) number of samples in the datasets.