The mechanism of energy dissipation in mechanical systems is often nonlinear. Even though there may be other forms of nonlinearity in the dynamics, nonlinear damping is the dominant source of nonlinearity in a number of practical systems. The analysis of such systems is simplified by the fact that they show no jump or bifurcation behaviour, and indeed can often be well represented by an equivalent linear system, whose damping parameters depend on the form and amplitude of the excitation, in a ‘quasi-linear’ model. The diverse sources of nonlinear damping are first reviewed in this paper, before some example systems are analysed, initially for sinusoidal and then for random excitation. For simplicity, it is assumed that the system is stable and that the nonlinear damping force depends on the
n
th power of the velocity. For sinusoidal excitation, it is shown that the response is often also almost sinusoidal, and methods for calculating the amplitude are described based on the harmonic balance method, which is closely related to the describing function method used in control engineering. For random excitation, several methods of analysis are shown to be equivalent. In general, iterative methods need to be used to calculate the equivalent linear damper, since its value depends on the system’s response, which itself depends on the value of the equivalent linear damper. The power dissipation of the equivalent linear damper, for both sinusoidal and random cases, matches that dissipated by the nonlinear damper, providing both a firm theoretical basis for this modelling approach and clear physical insight. Finally, practical examples of nonlinear damping are discussed: in microspeakers, vibration isolation, energy harvesting and the mechanical response of the cochlea.
a b s t r a c tWith the advent of wireless sensors, there has been an increasing amount of research in the area of energy harvesting, particularly from vibration, to power these devices. An interesting application is the possibility of harvesting energy from the track-side vibration due to a passing train, as this energy could be used to power remote sensors mounted on the track for strutural health monitoring, for example. This paper describes a fundamental study to determine how much energy could be harvested from a passing train. Using a time history of vertical vibration measured on a sleeper, the optimum mechanical parameters of a linear energy harvesting device are determined. Numerical and analytical investigations are both carried out. It is found that the optimum amount of energy harvested per unit mass is proportional to the product of the square of the input acceleration amplitude and the square of the input duration. For the specific case studied, it was found that the maximum energy that could be harvested per unit mass of the oscillator is about 0.25 J/kg at a frequency of about 17 Hz. The damping ratio for the optimum harvester was found to be about 0.0045, and the corresponding amplitude of the relative displacement of the mass is approximately 5 mm.
Abstract. Many structures or machines interact with some internal nonconservative forces and present asymmetric systems in which the stiffness and damping matrices are asymmetric. Examples include friction-induced vibration and aeroelastic flutter. Asymmetric systems are prone to flutter instability as a result of the real parts of some poles becoming positive when certain system parameters vary.This paper presents a receptance-based inverse method for assigning a number of complex poles of second-order damped asymmetric systems while keeping other unassigned poles unchanged. It uses state-feedback (active damping and active stiffness) to shift the poles to desired locations where all poles have negative real parts. Receptances at only a small, limited number of degrees-of-freedom of the underlying symmetric system are required. Simulated numerical examples indicate that this is an effective method and is capable of assigning negative real parts to unstable poles to stabilise an otherwise unstable second-order dynamic system.
This paper presents active vibration control to reduce the stick-slip oscillations in drill-strings. A simplified two degree-offreedom drill-string torsional model is considered. The nonlinear interaction between the rock and the bit is included in the model, where its parameters are fitted with field data from a 5km drill-string system. Different PD-control strategies are employed and compared, including the one that takes into account the weight-on-bit (axial force) and the bit speed. Optimization problems are proposed to obtain the values of the gain coefficients, and a torsional stability map for different weight-on-bit values and top-drive speeds is constructed. It is noted that the information of the dynamics at the bottom increases the performance of the PD-controller significantly in terms of the torsional vibration suppression, for the system analyzed.
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