MS8.2 - Dynamic Stability, Multistability and Buckling-induced Smart Applications

In this work, Koiter’s theory is applied to study the nonlinear large amplitude vibrations of simply supported shallow spherical panels. The nonlinear equations of motion are determined by using the Rayleigh-Ritz method in an energetic approach. The displacement fields are approximated using an expansion of trigonometric functions that satisfy the boundary conditions. Therefore, the backbone curves are determined using multiple shooting method and an Euler-Newtonian predictor-corrector continuation algorithm and the Floquet theory is applied to determine the stability of nonlinear solutions. The obtained backbone curves show multiple internal resonances due to the coupling between nonlinear normal modes. Time responses of some points of the backbone curves are depicted to analyse the internal resonances, which can represent loss of stability and sudden changes in the structural behaviour of shells submitted by external forces.

We investigate the steady configuration of a visco-elastic belt drive and its loss of stability under varying driving speed. The belt is considered as a linearly visco-elastic slender beam with small bending stiffness. Due to the presence of the small damping parameter and the small bending stiffness, the equations of motion are severly singularly perturbed. By variation of system parameters, like the driving speed, the damping coefficient, the tension force and the radius of the drums, we calculate the steady configuration and the stability limit of the belt. Preliminary calculations show, that the viscous damping, the distributed load and the strong bending at the drums can decrease the critical driving speed significantly and lead to flutter oscillations. Drive belts are frequently used tools for power transmission and their stable behaviour is important for the proper operation of the facility. For slowly moving belts it is usually sufficient to consider the equilibrium states, but if the belt speed approaches the wave speed in the belt, the influence of the drive motion has to be taken into account. In this talk we focus on the calculation of the steady configuration for a circular belt, which is in frictional contact with two drums, and on the calculation of its stability and the possible onset of flutter. Assuming that the driving pulley rotates with constant angular velocity and a constant angular momentum acts on the driven pulley, we obtain two sets of nonlinear partial differential equations for both free spans, which are connected by the boundary conditions at the contact points. Since we assume, that due to large friction coefficients between the belt and the drums the sliding zone on the pulleys is negligible small and the tension in the belt is transferred between the run-up and run-off points, also small state-dependent time delays have to be taken into account.

Non-linear dynamical systems of different areas of engineering and technology are subject to parameters that should most of the times be realistically modelled as random variables, although they are usually considered to be deterministic. This paper addresses in a simple, almost naïve fashion, the reliability analysis of those systems, starting from a deterministic analysis, but then bringing into it the statistical properties of input variables. Although the concept of dynamical integrity has greatly contributed to establishing safe thresholds in dynamical systems, requiring that the basins of attraction should be robust, a reliability measure is still missing in that respect. In fact, supposing that the erosion curve of a dynamic integrity measure I (for instance, the integrity factor) has been obtained in terms of a system parameter A (for instance, a load amplitude, a load frequency, or still an imperfection parameter) using a deterministic approach, one can estimate the output statistical properties for I, in terms of those of the input parameter A, now considered as a random variable in its own right. Hence, once reference values for the integrity measure I_ref and the system parameter A_ref have been established, and assuming that increase of A beyond A_ref may prove to be dangerous, the probability that I≥I_ref, provided that A≤A_ref, would give a sound reliability assessment. A very simple approach towards this aim is proposed and applied herewith to an archetypal model of a rigid column asymmetrically constrained by a linear spring, subject to a conservative axial compression and a small dynamical transversal load, playing the role of a random dynamic imperfection. It should be said that this work is a continuation of another one, with the same archetypal model, yet taking into account only the effect of a statical imperfection. A versatile in-house code is used to obtain the basins of attraction and the erosion curves that give support to the methodology proposed. The reliability assessment is carried out in two different scenarios, namely varying either the dynamic imperfection amplitude or its frequency, and identifying the influence of nearness to either buckling, or to external resonance or even parametric resonance. Since the proposed methodology is simple and easy to be applied, it is hoped that it can be absorbed in engineering design practice without its traditional resistance to incorporate new trends.