MS15.6 - Nonlinear Dynamics and Dynamic Stability

The dynamic behaviour of a freestanding block rocking on a rigid base when subjected to a horizontal ground excitation is a classical problem of nonlinear dynamics that has drawn considerable attention. Despite the apparent structural simplicity, rocking motion is characterised by various nonlinear and nonsmooth dynamics phenomena, which compose a complex and often chaotic behaviour. Chattering is a feature of nonlinear dynamics that might appear during the low amplitude forced oscillations of a rocking block. Chattering can be complete or incomplete. Complete chattering occurs when the block undergoes a theoretically infinite sequence of impacts in finite time, that eventually bring the block to the state of persistent (continuous) contact. On the contrary, incomplete chattering does not bring the block to rest after the theoretically infinite number of impacts. This paper analytically investigates the conditions under which a rigid rocking block undergoes complete chattering when subjected to low amplitude mathematical pulses. Specifically, the analysis proves the existence of a ground acceleration amplitude threshold below which the rocking block terminates its motion. Importantly, this work shows that the acceleration threshold mainly depends on the coefficient of restitution while it remains almost independent of the frequency of the ground excitation. Furthermore, the paper approximates the duration needed for the block to come to rest, namely chattering time. To this end, it adopts perturbation theory and proposes an iterative algorithm that accurately approximates chattering time. Finally, the analysis reveals how the acceleration amplitude of the ground excitation affects chattering time.

The one of the authors proposed the causal hysteresis damping and the extended Rayleigh damping as damping models with low frequency dependence for damping ratio. The causal hysteretic model is based on the causality theory and past displacements are used in the calculation. The extended Rayleigh damping uses the causal hysteresis damping instead of the stiffness proportional part while both stiffness proportional part and mass proportional part are used in ordinal Rayleigh damping. However, only linear analysis has been studied well, and non-linear analysis has not been sufficiently studied for these damping properties. In this report, nonlinear time history response analyses are examined in addition to linear analyses for following models. The 20-story (60 m height) building is studied by a lumped mass model and its structure is either reinforced concrete (RC) or steel. For nonlinear analyses, the degrading-trilinear hysteretic curve is used for RC structure while the normal-trilinear hysteresis curve is used for steel structure. For linear analyses, Rayleigh damping, causal damping, and extended Ray damping are studied by comparing with modal damping. For nonlinear analyses, the same damping models as above of initial and tangent types are studied by comparing with nonlinear modal damping. The study is carried out by expanding the frequency range where damping ratio should be kept constant defined by 2 frequencies (F1 and F2). In the expansion analyses, F2 is changed from the horizontal second eigen frequency to the vertical first eigen frequency while F1 remains the horizontal first order frequency. This expansion is needed to consider from horizontal input only analysis to both horizontal and vertical simultaneous inputs analysis. Following results were obtained. 1) It can be confirmed that causal damping and extended Rayleigh damping are more accurate than Rayleigh damping in both linear and nonlinear analyses. 2) However, the greater the nonlinearity, the smaller the difference between these results and they were almost similar, finally. 3) For steel structures using the normal-trilinear hysteretic curve, both initial and tangent types have almost the same accuracy. However, for RC structures using the degrading-trilinear hysteretic curve, tangent type is better than initial type.

Applying an oscillatory load is one of the most efficient ways to alter friction forces. Several theoretical and experimental studies on the influence of oscillatory loads on friction have been conducted, investigating the effect of both in-and out-of-plane oscillations for different tribological pairings and ranges of oscillation amplitudes, frequencies, and sliding velocities. However, the effect of external load on the frictional property has been studied with an emphasis on dynamic loads characterized by a high-frequency content, while a clear statement as to what is considered high-frequency is still missing. The common method of analysis for high-frequency is the method of direct separation of motion (MDMS). However, when studying the effect of a general sinusoidal excitation on friction by means of the MDMS, the analytical solutions become cumbersome or even impossible to obtain. Therefore, this study aims to show that a general relation, regardless of the frequency range, accounting for the induced effect of excitation on friction, can be easily obtained by utilizing the mobility transfer function of the linear dynamic system. Besides the study of linear systems, this work also presents the effect of excitation on friction for some nonlinear systems. To solve the case of a nonlinear system, the harmonic balance method will be used. The proposed method will be applied to a classical mass-spring-dashpot system on a moving belt, and Amontons-Coulomb and Stribeck laws will be considered.