Experimental Investigation of Dynamics and Bifurcations of an Impacting Spherical Pendulum

Since Newton first considered the motion of a spherical pendulum over 200 years ago, many researchers have studied its dynamic response under a variety of conditions. The characteristic of the problem that has invited so much investigation was that a spherical pendulum paradigms much more complex phenomena. Understanding the response of a paradigm gives an almost multiplicative effect in the understanding of other phenomena that can be modeled as a variant of the paradigm. The spherical pendulum has been used to damp irregular motion in helicopters and on space stations as well as for many other applications. In this study an inverted impacting spherical pendulum with large deflection was investigated. The model was designed to approximate an ideal pendulum, with the pendulum bob contributing the vast majority of the mass moment of inertia of the system. Two types of bearing mechanisms and tracking devices were designed for the system, one of which had low damping coefficient and the other with a relatively high damping coefficient. An experimental investigation was performed to determine the dynamics of an inverted, impacting spherical pendulum with large deflection and vertical parametric forcing. The pendulum system was studied with nine different bobs and two different base configurations. During the experiments, the frequency of the excitation remained between 24.6 and 24.9 Hz. It was found that sustained conical motions did not naturally occur. The spherical pendulum system was analyzed to determine under what conditions the onset of Type I response (a repetitive motion in which the pendulum bob does not traverse through the apex. The bob strikes the same general area of the restraint without striking the opposite side of the restraint.), sustainable Type II response (this is the repetitive motion in which the pendulum bob traverses through the apex. The bob strikes opposite sides of the restraint.), and mixed mode response (motion in which the pendulum bob randomly strikes either the same area of the restrain or the opposite side of the restraint) occurred.     

Dynamics of a plastic-impact system with oscillatory and progressive motions

A mathematical model is developed to describe oscillatory and progressive motions in dynamics of a plastic impact oscillator with a frictional slider. Dynamics of the impact oscillator is analyzed by a five-dimensional map, which describes free flight and sticking solutions of two masses of the system, between impacts, supplemented by transition conditions at the instants of impacts. Piecewise property and singularity are found to exist in the Poincaré map. The piecewise property is caused by the transitions of free flight and sticking motions of impacting masses immediately after the impact, and the singularity of the map is generated via the grazing contact of impacting masses immediately before the impact. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. The influence of piecewise property, grazing singularities and various parameters on dynamics of the vibro-impact system is analyzed. The global bifurcation diagrams for before-impact velocity versus forcing frequency are plotted to predict much of the qualitative behavior of the system. The global bifurcations of period-n single-impact motions of the plastic-impact oscillator are found to exhibit extensive and systematic characteristics.     

Dynamics of two delay coupled van der Pol oscillators

In this paper, the dynamics of a system of two van der Pol oscillators with delayed position and velocity coupling is studied by the method of averaging together with truncation of Taylor expansions. According to the slow-flow equations, the dynamics of 1:1 internal resonance is more complex than that of non-1:1 internal resonance. For 1:1 internal resonance, the stability and the number of periodic solutions vary with different time delay for given coupling coefficients. The condition necessary for saddle-node and Hopf bifurcations for symmetric modes, namely in-phase and out-of-phase modes, are determined. The numerical results, obtained from direct integration of the original equation, are found to be in good agreement with analytical predictions.     

Complex and chaotic response of a non-linear oscillator with an isothermal gas spring

In this paper we study the dynamics of a non-linear one-degree-of-freedom system subjected to an external harmonic excitation, representing a simplified model for the synchronous hydraulic oscillations that can occur in the draft tube of Francis turbines at partial loads. The application of different typical numerical techniques has shown the existence of multiple coexisting periodic solutions, and the non-periodic bounded solutions which exhibit deterministic chaotic behaviour. The relevant strange attractor has been defined and the loss of memory associated with an exponential divergence in time of close initial conditions resulting in chaotic dynamics have been found and measured. A partial classification of qualitatively different dynamical behaviours for the system has been outlined in the control parameter space.

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Sommario In questo articolo viene studiata la dinamica di un sistema non-lineare ad un singolo grado di liberta' soggetto ad una forzante armonica esterna, rappresentante un modello semplificato per le oscillazioni idrauliche sincrone che hanno luogo nei diffusori delle turbine tipo Francis a carico parziale. Applicando differenti tecniche numeriche, viene mostrata l'esistenza di soluzioni periodiche multiple, oltre che soluzioni non-periodiche limitate con tipico comportamento caotico deterministico. L'attrattore strano corrispondente e' stato definito e caratterizzato: la perdita di memoria associata alla divergenza esponenziale di orbite inizialmente vicine, tipica della dinamica caotica, e' stata individuata e calcolata numericamente. Una prima parziale classificazione dei vari comportamenti dinamici per il sistema viene evidenziata attraverso la rappresentazione nello spazio parametrico.

     

Buoyancy-driven instability in a vertical cylinder: Binary fluids with Soret effect. Part II: Weakly non-linear solutions

The buoyancy-driven instability of a monocomponent or binary fluid that is completely contained in a vertical circular cylinder is investigated, including the influence of the Soret effect for the binary mixture. The Boussinesq approximation is used, and weakly-non-linear solutions are generated via Galerkin's technique using an expansion in the eigensolutions of the associated linear stability problem. Various types of fluid mixtures and cylindrical domains are considered. Flow structure and associated heat transfer are computed and experimental observations are cited when possible.     

A Mechanics Based Model for Study of Dynamics of Milling Operations

A unified mechanics based model with multiple degrees of freedom is developed and numerically simulated to study workpiece-tool interactions during milling of ductile workpieces with helical tools. A refined orthogonal cutting model is used at each section of the tool, and the milling forces are determined by using a spatial integration scheme along the axis of the tool. Both regenerative and loss of contact effects are considered in determining the cutting forces, which makes the model well suited for a wide range of milling operations. The model also allows for partial engagement of a tool with a workpiece, which is an important feature needed for milling operations with helical tools. Time domain simulations are carried out by using the developed model to predict the stability boundaries in the space of the tool spindle speed and the axial depth of cut. Poincaré sections are used to determine loss of stability from period-one motions to other motions such as two-period quasiperiodic motions, as a control parameter is varied.     

On Local Analysis of Oscillations of a Non-ideal and Non-linear Mechanical Model

It is of major importance to consider non-ideal energy sources in engineering problems. They act on an oscillating system and at the same time experience a reciprocal action from the system. Here, a non-ideal system is studied. In this system, the interaction between source energy and motion is accomplished through a special kind of friction. Results about the stability and instability of the equilibrium point of this system are obtained. Moreover, its bifurcation curves are determined. Hopf bifurcations are found in the set of parameters of the oscillating system.     

Stability and bifurcation of doubly curved shallow panels under quasi-static uniform load

The focus of this paper is on the investigation of the mathematical nature of buckling from the point of view of bifurcation theory. For the doubly curved orthotropic panels subjected to quasi-static uniform load and with hinged boundary conditions, the solution to the non-linear partial differential equation is partitioned into two parts and projected onto the complete space spanned by the eigenfunctions of the linear operator of the governing equation. Furthermore, the fundamental branch, from which a new solution will emanate, is approximated by the first single mode pair which is close to the real membrane state. Whereas the ensuing bifurcated branch is approximated by the other single mode pair, under the assumption that the coupling between modes can be neglected. The present analysis could give a deep insight into the mechanism of the instability of panel structures, and show that there exists a mode transition at the critical point and the snap-through, then results from saddle-node bifurcation on the bifurcated branch. As a conclusion, the buckling of the system studied can be stated as: a bifurcated branch emanates from the fundamental branch at a critical point, and a saddle-node bifurcation, behaving as jumping, then occurs on the ensuing bifurcated branch.     

Testing a fast dynamical indicator: The MEGNO

To investigate non-linear dynamical systems, like for instance artificial satellites, Solar System, exoplanets or galactic models, it is necessary to have at hand several tools, such as a reliable dynamical indicator.The aim of the present work is to test a relatively new fast indicator, the Mean Exponential Growth factor of Nearby Orbits (MEGNO), since it is becoming a widespread technique for the study of Hamiltonian systems, particularly in the field of dynamical astronomy and astrodynamics, as well as molecular dynamics.In order to perform this test we make a detailed numerical and statistical study of a sample of orbits in a triaxial galactic system, whose dynamics was investigated by means of the computation of the Finite Time Lyapunov Characteristic Numbers (FT-LCNs) by other authors.     

Multi-Field Coupled Dynamics for a Micro Beam

This paper proposes a multi-field coupled dynamics equation for a micro beam. The natural frequencies and the amplitude–frequency relationship of the micro beam in the coupled fields are investigated. Changes in the natural frequencies of the micro beam along with time, bias voltage, and dynamic viscosity of gas are discussed. The effects of the system parameters on the amplitude–frequency relationship are investigated. A number of useful results are obtained. These results are useful in the sensitivity design of resonant micro gas sensors excited by the electrostatic force.     

Modelling of Ground Moling Dynamics by an Impact Oscillator with a Frictional Slider

This paper describes current research into the mathematical modelling of a vibro-impact ground moling system. Due to the structural complexity of such systems, in the first instance the dynamic response of an idealised impact oscillator is investigated. The model is comprised of an harmonically excited mass simulating the penetrating part of the mole and a visco-elastic slider, which represents the soil resistance. The model has been mathematically formulated and the equations of motion have been developed. A typical nonlinear dynamic analysis reveals a complex behaviour ranging from periodic to chaotic motion. It was found out that the maximum progression coincides with the end of the periodic regime.     

平动弹性梁的刚-柔耦合动力学

本文建立了作大范围平动弹性梁的刚-柔耦合动力学控制方程。分析了大范围平动对弹性梁变形运动动力学性质的影响，发现了大范围平动与变形运动之间的耦合动力学与大范围转动与变形运动之间的耦合动力学存在显著的差异。大范围平动使弹性梁的刚度降低，同时使系统阻尼增加；而大范围转动使弹性梁的刚度增加，同时使系统产生了能量转换的陀螺效应。因此，柔性多体系统刚-柔耦合动力建模中必须包括大范围平动与柔性体变形运动之间的耦合动力学效应。     

Chaos in a coupled oscillators system with widely spaced frequencies and energy-preserving non-linearity

This paper is a sequel to Tuwankotta [Widely separated frequencies in coupled oscillators with energy-preserving nonlinearity, Physica D 182 (2003) 125-149.], where a system of coupled oscillators with widely separated frequencies and energy-preserving quadratic non-linearity is studied. We analyze the system for a different set of parameter values compared with those in Tuwankotta [Widely separated frequencies in coupled oscillators with energy-preserving nonlinearity, Physica D 182 (2003) 125-149.]. In this set of parameters, the manifold of equilibria are non-compact. This turns out to have an interesting consequence to the dynamics. Numerically, we found interesting bifurcations and dynamics such as torus (Neimark-Sacker) bifurcation, chaos and heteroclinic-like behavior. The heteroclinic-like behavior is of particular interest since it is related to the regime behavior of the atmospheric flow which motivates the analysis in Tuwankotta [Widely separated frequencies in coupled oscillators with energy-preserving nonlinearity, Physica D 182 (2003) 125-149.] and this paper.     

Experimental analysis of pre-compressed circular cylindrical shell under axial harmonic load

In this paper the nonlinear dynamics of circular cylindrical shells under axial static (compressive) and periodic resonant loads have been experimentally investigated, the goal is to study the dynamic scenario and to analyze nonlinear regimes. A special test rig has been developed for the experiment in order to apply a static axial load combined with a dynamic axial load. The setup allows for investigating the linear behavior under static preload by means of the usual modal testing techniques; moreover, it allows for analyzing the nonlinear response which occurs when the dynamic axial load is periodic and gives rise to complex resonances. The complex dynamics, arising when a periodic axial load excites the asymmetric (shell like) modes, are analyzed by means of amplitude frequency diagrams, waterfall spectrum diagrams, bifurcation diagrams of Poincaré maps; a deep analysis of time histories, spectra, phase portraits and Poincaré maps completes the study of the complex dynamic scenario.     

Comparative analysis of chaos control methods: A mechanical system case study

Chaos may be exploited in order to design dynamical systems that may quickly react to some new situation, changing conditions and their response. In this regard, the idea that chaotic behavior may be controlled by small perturbations allows this kind of behavior to be desirable in different applications. This paper presents an overview of chaos control methods classified as follows: OGY methods - include discrete and semi-continuous approaches; multiparameter methods - also include discrete and semi-continuous approaches; and time-delayed feedback methods that are continuous approaches. These methods are employed in order to stabilize some desired UPOs establishing a comparative analysis of all methods. Essentially, a control rule is of concern and each controller needs to follow this rule. Noisy time series is treated establishing a robustness analysis of control methods. The main goal is to present a comparative analysis of the capability of each chaos control method to stabilize a desired UPO.     

Chaos in a pendulum with feedback control

We study chaotic dynamics of a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small inductance, so that the feedback control system reduces to a periodic perturbation of a planar Hamiltonian system. This Hamiltonian system can possess multiple saddle points with non-transverse homoclinic and/or heteroclinic orbits. Using Melnikov's method, we obtain criteria for the existence of chaos in the pendulum motion. The computation of the Melnikov functions is performed by a numerical method. Several numerical examples are given and the theoretical predictions are compared with numerical simulation results for the behavior of invariant manifolds.     

Experimental and numerical investigation of chaotic regions in the triple physical pendulum

The experimental and numerical analysis of triple physical pendulum is performed. The experimental setup of the triple pendulum with the first body externally excited by the square function and the widely used LabView measure-programming system, which is designed especially for measure data processing and acquisition, are described. The mathematical model of the system is then introduced. The parameters of the model are estimated by minimization of the sum of squares of deviations between the signal from the simulation and the signal from the experiment. A good agreement between results from experiment and from simulation is shown in few examples, including periodic as well as chaotic solutions.     

Dynamics of Geometrically Nonlinear Rods: II. Numerical Methods and Computational Examples

The paper selects and develops appropriate numerical solutionmethods for initial boundary value problems of the equations of motionof geometrically nonlinear extensible Euler–Bernoulli-beams. A finiteelement method that uses first-order Hermitian polynomials as interpolationfunctions for the rod axis position vector is used as discretizationtechnique. An averaging method for the calculation of net forces andmoments is developed that achieves a better approximation than thedirect calculation from the strains. Time integration is done usingan energy and momentum conserving algorithm that is proposed in thispaper and Newmark type methods. The derived algorithms are used tosolve problems from space and marine engineering. The obtained simulationresults are compared with results which have been already publishedin the literature or were calculated by different methods.     

Dynamics of a Simple Damped Oscillator Undergoing Stick-Slip Vibrations

The dynamics of a simple dynamical system subjected to an elastic restoring force, viscous damping and dry friction forces is investigated. Self-sustained oscillations occur with non-standard attracting properties. Discontinuity of the governing equations leads to non-standard bifurcations, which are studied here, with analytical and numerical tools. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamical behaviors of nonlinear viscoelastic thick plates with damage

Based on the deformation hypothesis of Timoshenko's plates and the Boltzmann's superposition principles for linear viscoelastic materials, the nonlinear equations governing the dynamical behavior of Timoshenko's viscoelastic thick plates with damage are presented. The Galerkin method is applied to simplify the set of equations. The numerical methods in nonlinear dynamics are used to solve the simplified systems. It could be seen that there are plenty of dynamical properties for dynamical systems formed by this kind of viscoelastic thick plate with damage under a transverse harmonic load. The influences of load, geometry and material parameters on the dynamical behavior of the nonlinear system are investigated in detail. At the same time, the effect of damage on the dynamical behavior of plate is also discussed.