Z. C. Feng

Dynamics of a Mechanical System with a Shape Memory Alloy Bar

We study the dynamics of a simple mechanical system containing a mass and a bar made of shape memory alloys. Our objective is to understand the effect of stress induced phase transtormation on the system dynamics. Specifically, the effectiveness of an idealized vibration isolator made of shape memory alloys is examined both numerically and experimentally. We find that the stress induced phase transformation within the material has two effects: the resonance frequency of the system is lower and the peak response near the resonance is heavily suppressed. Furthermore, both of these effects depend on the forcing amplitude and they are more pronounced at large forcing amplitudes. We also find that nonlinearity associated with the phase transformation gives rise to complicated dynamics. In particular, period-three responses are found for some forcing parameters. It the case of a biased load, period-doubling cascade and chaotic motion can occur. However, this complicated dynamics disappears at large forcing amplitudes, making the shape memory alloy vibration isolator an effective device at large forcing amplitudes.

Instability of a liquid jet emerging from a droplet upon collision with a solid surface

A linear perturbation theory is developed to investigate the interface instabilities of a radially-expanding, liquid jet in cylindrical geometries. The theory is applied to rapidly spreading droplets upon collision with solid surfaces as the fundamental mechanism behind splashing. The analysis is based on the observation that the instability of the liquid sheet, i.e., the formation of the fingers at the spreading front, develops in the extremely early stages of droplet impact. The shape evolution of the interface in the very early stages of spreading is numerically simulated based on the axisymmetric solutions obtained by a theoretical model. The effects that factors such as the transient profile of an interface radius, the perturbation onset time, and the Weber number have on the analysis results are examined. This study shows that a large impact inertia, associated with a high Weber number, promotes interface instability, and prefers high wave number for maximum instability. The numbers of fingers at the spreading front of droplets predicted by the model agree well with those experimentally observed.

Global Bifurcations in the Motion of Parametrically Excited Thin Plates

In this paper we investigate global bifurcations in the motion of parametrically excited, damped thin plates. Using new mathematical results by Kovačič and Wiggins in finding homoclinic and heteroclinic orbits to fixed points that are created in a resonance resulting from perturbation, we are able to obtain explicit conditions under which Silnikov homoclinic orbits occur. Furthermore, we confirm our theoretical predictions by numerical simulations.

Global bifurcation and chaos in parametrically forced systems with one-one resonance

Global bifurcations of a fourth-order Hamiltonian system with Z2 ⨭ Z2 symmetry are studied. The system represents normal-form equations that arise in a variety of problems which have one-one internal resonance and which are forced sinusoidally at the natural frequencies. Four qualitatively different types of global behaviours are shown to occur. Using a generalization of the Melnikov method, three different heteroclinic cycles are shown to break, generating Smale horseshoes and resulting in chaotic phenomena. The theoretical results are verified by numerical simulations. The main conclusion of the analysis is that chaotic phenomena are very common in this class of system

Rubbing phenomena in rotor–stator contact

Abstract This paper discusses the vibration phenomena of a rotor rubbing with a stator caused by an initial perturbation. The analytical model consists of a simple disc-shaft rotor and a fixed stator. The perturbation is an instantaneous change of the radial velocity when the rotor is rotating in its normal steady state. It is found that under certain conditions, the rotor will remain rubbing with the stator, even if the initial perturbation no longer exists. In the case of no friction on the contact surface between the rotor and the stator, the full rubbing behaves as forward whirling. When friction is present, the full rubbing behaves as backward whirling.

Global Bifurcations in Parametrically Excited Systems with Zero-to-One Internal Resonance

In this work we study the existence of Silnikov homoclinicorbits in the averaged equations representing the modal interactionsbetween two modes with zero-to-one internal resonance. The fast mode isparametrically excited near its resonance frequency by a periodicforcing. The slow mode is coupled to the fast mode when the amplitude ofthe fast mode reaches a critical value so that the equilibrium of theslow mode loses stability. Using the analytical solutions of anunperturbed integrable Hamiltonian system, we evaluate a generalizedMelnikov function which measures the separation of the stable and theunstable manifolds of an annulus containing the resonance band of thefast mode. This Melnikov function is used together with the informationof the resonances of the fast mode to predict the region of physicalparameters for the existence of Silnikov homoclinic orbits.

Translational instability of a bubble undergoing shape oscillations

This paper studies the translational instability of an oscillating bubble. It is shown that when a spherical bubble undergoing volume oscillation becomes unstable, giving rise to shape oscillations of two neighboring modes, the translational mode is intimately coupled with the two shape modes and this results in translational instability of the bubble. The main contribution is twofold. First, the integral relations for motions of bubbles in an infinite perfect liquid are not relied on, hence result is applicable to liquids with weak viscous effect. Second, the method of deriving the amplitude equations, which is similar to that of normal form calculations for ordinary differential equations, has not been applied to partial differential equations before.

Bifurcation and chaos in friction-induced vibration

Abstract Friction-induced vibration is a phenomenon that has received extensive study by the dynamics community. This is because of the important industrial relevance and the ever-evolving development of new friction models. In this paper, we report the result of bifurcation study of a single-degree-of-freedom mechanical oscillator sliding over a surface. The friction model we use is that developed by Canudas de Wit et al., a model that is receiving increasing acceptance from the mechanics community. Using this model, we find a stable limit cycle at intermediate sliding speed for a single-degree-of-freedom mechanical oscillator. Moreover, the mechanical oscillator can exhibit chaotic motions. For certain parameters, numerical simulation suggests the existence of a Silnikov homoclinic orbit. This is not expected in a single-degree-of-freedom system. The occurrence of chaos becomes possible because the friction model contains one internal variable. This demonstrates a unique characteristic of the friction model. Unlike most friction models, the present model is capable of simultaneously modeling self-excitation and predicting stick–slip at very low sliding speed as well.

Vibration Characteristics of Conical Shell Panels With Three-Dimensional Flexibility

A first known investigation on the three-dimensional vibration characteristics of conical shell panels is reported. A linear frequency equation is derived based on an exact three-dimensional, small-strain, linearly elastic theory. Sets of one and two-dimensional polynomial series are employed to approximate the spatial displacements of the conical shell panels in three dimension. The perturbation of frequency responses due to the variations of relative thickness L/h, slanted length L/S, vertex angle γ v , and subtended angle γ o is investigated. First known frequency parameters and three-dimensional deformed mode shapes of the conical shell panels are presented in vivid graphical forms. The new results may serve as benchmark references for validating the new refined shell theories and new computational techniques.

On energy transfer in resonant bubble oscillations

In this paper, energy transfer mechanism due to resonant interactions of bubble shape and volume deformation modes is studied. In the process, a number of recent investigations that have focused on specific sets of initial conditions are generalized. Using phase‐space analysis and a reduction of the mode dynamics equations to an integrable Hamiltonian, two types of resonant interactions are considered. The first is the so‐called one–two resonance, when the natural frequency of the breathing mode is twice (or approximately twice) that of a shape mode. In this case, it is found that there is nearly always a continuous, periodic exchange of energy between the shape and volume modes. However, there exists a particular class of initial conditions, that was the subject of most earlier studies, for which the energy transfer is one way, from the shape mode to the volume mode. The second type of resonant interaction occurs when the natural frequency of the breathing mode is approximately equal to that of the shape...