Recent advances in nonlinear passive vibration isolators

Modal analysis is used extensively for understanding the dynamic behavior of structures. However, a major concern for structural dynamicists is that its validity is limited to linear structures. New developments have been proposed in order to examine nonlinear systems, among which the theory based on nonlinear normal modes is indubitably the most appealing. In this paper, a different approach is adopted, and proper orthogonal decomposition is considered. The modes extracted from the decomposition may serve two purposes, namely order reduction by projecting high-dimensional data into a lower-dimensional space and feature extraction by revealing relevant but unexpected structure hidden in the data. The utility of the method for dynamic characterization and order reduction of linear and nonlinear mechanical systems is demonstrated in this study.

The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview

Proper orthogonal decomposition and its applications—part i: theory

Control of chaos: Methods and applications in engineering

A study of a nonlinear vibration isolator with a quasi-zero stiffness characteristic

Performance of Non-linear Vibration Isolators Under Harmonic Excitation

Nonlinear dynamics of one-mode approximation of an axially moving continuum such as a moving magnetic tape is studied. The system is modeled as a beam moving with varying speed, and the transverse vibration of the beam is considered. The cubic stiffness term, arising out of finite stretching of the neutral axis during vibration, is included in the analysis while deriving the equations of motion by Hamilton's principle. One-mode approximation of the governing equation is obtained by the Galerkin's method, as the objective in this work is to examine the low-dimensional chaotic response. The velocity of the beam is assumed to have sinusoidal fluctuations superposed on a mean value. This approximation leads to a parametrically excited Duffing's oscillator. It exhibits a symmetric pitchfork bifurcation as the axial velocity of the beam is varied beyond a critical value. In the supercritical regime, the system is described by a parametrically excited double-well potential oscillator. It is shown by numerical simulation that the oscillator has both period-doubling and intermittent routes to chaos. Melnikov's criterion is employed to find out the parameter regime in which chaos occurs. Further, it is shown that in the linear case, when the operating speed is supercritical, the oscillator considered is isomorphic to the case of an inverted pendulum with an oscillating support. It is also shown that supercritical motion can be stabilised by imposing a suitable velocity variation.

Low-dimensional chaotic response of axially accelerating continuum in the supercritical regime

A general framework for local control of nonlinearity in nonautonomous systems using feedback strategies is considered in this work. In particular, it is shown that a system exhibiting chaos can be driven to a desired periodic motion by designing a combination of feedforward controller and a time-varying controller. The design of the time-varying controller is achieved through an application of Lyapunov–Floquet transformation which guarantees the local stability of the desired periodic orbit. If it is desired that the chaotic motion be driven to a fixed point, then the time-varying controller can be replaced by a constant gain controller which can be designed using classical techniques, viz. pole placement, etc. A sinusoidally driven Duffing's oscillator and the well-known Rossler system are chosen as illustrative examples to demonstrate the application.

A general approach in the design of active controllers for nonlinear systems exhibiting chaos

The effect of a strictly dissipative force (velocity to the pth power model) on the response and bifurcations of driven, soft Duffing oscillators is considered. The method of harmonic balance is used to obtain the steady state harmonic response. An anomalous jump in the harmonic response (signifying a break in the resonance curve), obtained in the case of linearly damped, soft Duffing oscillators, is shown to persist even in the presence of nonlinear damping. It is shown that the bifurcation structure and the structure of the chaotic attractor are quite insensitive to the damping exponent p. However, the threshold values of the parameters, at which bifurcations occur, depend both on the damping index and the damping coefficient. The Melinkov criterion and an analytical criterion for the period-doubling bifurcation have been obtained in the presence of combined linear and cubic damping.

Role of nonlinear dissipation in soft Duffing oscillators.

The steady-state, harmonic response of a vibration isolation system with a cubic, hard non-linear restoring force and combined Coulomb and viscous damping is presented. The results have been obtained by using the method of harmonic balance. It has been assumed that the motion is continuous without any stop. An anomalous jump in the response, similar to the one obtained in earlier studies on certain soft systems, is observed when the isolation system is subjected to a base excitation. Linear stability analysis is carried out to determine the status of the anomalous jump. The effect of the damping parameters on the jump in the response is investigated. Transmissibility curves are plotted for various parameter values to study the performance characteristics. The obtained results extend the previous works of Den Hartog and Ruzicka who considered a linear restoring element.

B. Ravindra

Papers

Performance of Non-linear Vibration Isolators Under Harmonic Excitation

Low-dimensional chaotic response of axially accelerating continuum in the supercritical regime

A general approach in the design of active controllers for nonlinear systems exhibiting chaos

Role of nonlinear dissipation in soft Duffing oscillators.

Hard duffing-type vibration isolator with combined Coulomb and viscous damping