Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities

The following sections are included:

• Preface

• Contents

The following sections are included:

• Introduction

• Scope of the Book
  • Outlining the Basis and Methods
  • Impacting Systems
  • Systems with Dry Friction
  • Complex Mechanical Systems

• Bibliography

The chapter outlines two models for describing and solving dynamical systems with motion dependent discontinuities such as clearances, impacts, dry friction, or combination of these phenomena. The first approach assumes any dynamic system can be considered as continuous in a finite number of continuous subspaces, which together form so-called global hyperspace. Global solution is obtained by “gluing” together local solutions obtained by solving the problem in the continuous subspaces. An efficient numerical algorithm is presented, and then used to solve dynamics of a piecewise oscillator, which has been also verified experimentally. The second approach considers that in reality the system parameters do not change in an abrupt manner. Therefore a contiunuous function is used to model transitions between the subspaces. The sigmoid function is employed allowing to control the degree of abruptness. An unsymmetrical, piecewise linear oscillator has been used to examine this method providing ultimately recommendations regarding validity of this approach.

The two different methods using the idea of non-smooth transformation of variables are discussed. The first method deals with the positional variables (coordinates) and can be applied to a class of systems with absolutely rigid constraints. The constraints are eliminated by means of a special non-smooth transformation of the coordinates. Another method can be qualified as a non-smooth transformation of arguments. A role of the arguments may play either time when considering the oscillating processes or spatial independent variables of the periodic elastic structures.

In this study, modifications and extensions of cell mapping (CM) methods are discussed. CM methods are tools for the global investigation of the long term behaviour of nonlinear dynamic systems. By means of CM, periodic as well as chaotic solutions of the equations of motion can be determined. Additionally, application of CM enables the determination of the basins of attraction of the stable solutions.

In this study we will mainly focus on the ‘simple’ cell mapping (SCM) method. This method is based on a discretization of the state space in cells, followed by a determination—by means of numerical integration—of corresponding image cells. Then groups of periodic cells represent the system's long term behaviour. Some modifications for systems with discontinuities will be discussed.

Additionally, two important extensions of cell mapping methods are presented. The first extension contains a parameter variation technique, suited for the sensitivity-analysis of CM results with respect to system parameters. With this technique, the evolution of the basin boundaries due to a parameter variation can be obtained in relatively little CPU-time. In this way, global bifurcations can easily be predicted. The introduced concept has been applied for the analysis of a modified Duffing equation.

The second extension is a new CM method, termed ‘multi-DOF’ cell mapping (MDCM), which can be applied to systems of many degrees of freedom. Since the number of cells—and hence the CPU-time and storage requirements—grows exponentially with the state space dimension, application of conventional CM methods to these systems is very impractical. Under MDCM, the CPU-time grows only linearly with the system dimension while the order of the storage requirements remains constant. To illustrate the power of the technique a 4-DOF beam with nonlinear support will be investigated.

It can be concluded that the presented modifications and extensions have merit. Further, the additional value of CM methods is emphasized with respect to more established methods of investigation, such as periodic solvers and regular numerical integration.

This chapter presents results of theoretical and simulation analysis of the motion of a typical impacting system - the impact oscillator. It is assumed, that impacts between a moving body and the rigid stop are centric, direct and negligibly short and the elementary Newton theory with the coefficient of restitution can be used (the motion of mechanical systems with impacts is strongly nonlinear). Many different types of periodic and chaotic impact motions exist even for simple systems with external periodic excitation forces. The group of fundamental periodic motions is characterized by the different number of impacts in one motion period, which equals the excitation force period. Every motion has a region in the space of system parameters in which the solution can exist and is stable. There exist transition regions, so-named hysteresis regions and beat motion regions, which lie between the zones of neighbouring fundamental impact motions. Transition regions are determined by the boundaries of existence which correspond to grazing bifurcations and by the boundaries of stability corresponding to the period-doubling and saddle-node bifurcations. The transition between neighbouring periodic impact motions is never continuous, with the exception of singular points, where the existence boundaries and stability boundaries intersect. Jump phenomena appear on the hysteresis region boundaries, and subharmonic and chaotic motions exist in the beat motion region.

This chapter presents a review of previous analytical work on the dynamics of periodically excited piecewise linear oscillators. The common characteristic of these oscillators is that their stiffness and damping properties may change abruptly at specific displacement values. First, an analytical method is presented for determining periodic steady-state response of these oscillators. This analysis takes advantage of the fact that the exact solution form for any solution piece, included within any time interval where the stiffness and damping properties of the oscillator remain constant, is known. Then, an appropriate methodology is also presented, revealing the stability properties of the located periodic motions. This methodology is based on the derivation of a matrix relation which determines how an arbitrary but small perturbation at the beginning of a periodic solution propagates to the end of a response period. Then, some useful results obtained by bifurcation analysis of the periodic solutions are also derived. Finally, typical numerical results are presented, by considering example mechanical models.

This study deals with the quenching problem of self-excited vibrations by using an impact damper composed of a ball and impact walls. First, the quenching problem of a single-degree-of-freedom self-excited system was treated. The Runge-Kutta-Gill method with variable time division is applied to the numerical analysis of a Rayleigh's type self-excited system. The solutions obtained from this method are divided into two main categories; periodic solutions and chaos. Further, two types of chaos are found, namely, chaos after period doubling bifurcation and that of the intermittent type. An optimum approach for quenching the self-excited vibration was discussed. As a result, it was clarified that for the optimum design of the impact damper there existed a certain relation between the coefficient of restitution and clearance. It was found that the condition for vibration quenching was located at the edge of chaos. This quenching of self-excited vibration was a unique example of chaos utilization in the problem of vibration quenching. Experiments were performed to quench vortex-induced vibration by using the impact damper. The experimental results agreed well, qualitatively, with those from the numerical analysis. Secondly, since a wide quenching range of the impact damper was confirmed in the above-mentioned system, the problem of quenching vortex-induced vibration of a two-degree-of-freedom system using this range was treated. From the experiment and numerical analysis, it turns out that the vortex-induced vibrations of the first and the second modes are quenched by a single impact damper. It was confirmed that there existed optimum parameters for the quenching vibration, and this quenching of vortex-induced vibration employed chaos utilization.

In the present paper, different models of gear transmission systems were reviewed. The conditions of periodic and chaotic vibration types occurrence of the system with a backlash, governed by non-linear differential equations with a periodically changing stiffness, were discussed with details. In particular, for specified parameters, using Lyapunov exponent, conditions leading to chaotic motion were determined. The influence of system parameters on dynamic loads of gears was investigated. Due to the backlash, the stiffness in the system was assumed to have a piecewise linear characteristics, however the influence of additional cubic nonlinearities was also discussed in the present paper. Couplings with other parts and teeth errors effects were simulated by an additional external random force in the system.

It is the aim of the present section to first explain some mathematically rigorous methods that can be used to analyze dry friction problems, or non-smooth systems in general. The first concept, Lyapunov exponents, is well-accepted for smooth systems to predict the long-term behaviour, and the existence of at least one positive Lyapunov exponent is often used as the definition of “chaos”. Then in section 9.2 we will turn to a different issue, namely to verify that some system has undergone a bifurcation when some external parameter has been varied. Often a good indicator for bifurcations appearing is that some “index” changes when the parameter crosses a critical value. We will introduce the so-called Conley index and explain how it applies to detect such bifurcations. Since usually the dynamical behaviour will be quite complicated, often only numerical analysis will yield satisfactory results. In section 9.3 we will describe some phenomena observed in a single-mass friction oscillator which are due to the fact that trajectories can collide with discontinuity surfaces. Finally, Section 9-4 contains some conclusions. Most of the material is taken from [13].

This study deals with a forced self-excited vibrating system accompanied by dry friction as an example of a nonlinear system with discontinuities. The numerical method used here is the direct numerical integral method as we have presented previously, which is a shooting method able to obtain highly accurate periodic solutions of systems with discontinuities. The resonance curves of entrained harmonic, higher-harmonic, and subharmonic vibrations are obtained. Chaos is also found. The influences of amplitude and frequency of an external force on the resonance curves and the stability of solutions are discussed. It is also found that bifurcations are realized from the discontinuous characteristics of fractional force. Due to this type of bifurcation, periodic solution changes its stability abruptly and chaos occurs suddenly halfway on the period doubling route in the system. This characteristic of periodic solution is verified using characteristic multipliers, and that of chaos is verified using Lyapunov exponents, bifurcation diagrams, and Poincaré maps. As another system with discontinuity, preloaded compliance system is dealt with, and a similar discontinuity of stability and route to chaos are briefly shown.

Nonsmooth processes such as stick–slip may introduce problems with phase-space reconstructions. We examine chaotic single-degree-of-freedom stick–slip friction models and use the method of delays to reconstruct the phase space. We illustrate that this reconstruction process can cause pseudo trajectories to collapse in a way that is unlike, yet related to, the dimensional collapse in the original phase-space. As a result, the reconstructed attractor is not topologically similar to the real attractor. Standard dimensioning tools are applied in effort to recognize this situation. The use of additional observables is examined as a possible remedy for the problem.

In engineering practice, dry friction often causes undesirable side effects such as the generation of self-sustained oscillations also called stick—slip vibrations.

Torsional stick—slip vibrations for example have been observed in drill strings, that are used in rotary drilling for oil and gas. In rotary drilling, deep wells are drilled with a rock-crushing tool, called a drill bit. The drill bit is driven by an electric motor at the surface, whose torque is transmitted by the (long and flexible) drill string. Between the drill bit and parts of the drill string on the one hand, and the rock on the other hand, dry friction occurs. When a drill string is undergoing a torsional stick—slip vibration, the top is rotating at a constant speed, whereas the speed at the drill bit varies between zero (the stick phase) and a speed, much higher than the speed at the top (the slip phase), which can lead to damage of drilling components.

In literature, different dry friction models can be found, of which Coulomb's friction law is the most familiar. Because of the discontinuous nature of dry friction, highly nonlinear differential equations are found. In this study, periodic solutions of systems, experiencing dry friction are determined, using a module STRDYN of the finite element code DIANA. This module offers several numerical algorithms for investigating multi-degree-of-freedom finite element models with local nonlinearities. To avoid numerical complications a smooth approximation of the discontinuous friction model will be used. The techniques will be applied to a 2-masses-on-belt model with dry friction, taken from literature. The friction model is approximated, using an arctan-function. Very good correspondence is found between the DIANA results and results which could also be found in literature but also some new solutions will be discussed.

The dynamics of rotating disks involves a multiplicity of fractionally induced and parametric vibration phenomena, which, in turn, can have a considerable effect on machine and installation performance. This article is intended to review and highlight the most significant phenomena, and therefore to indicate the level of understanding of disk dynamics that has now been reached. Sections 13.1 to 13.3 review the fundamentals of plate and disk vibrations and the dynamics of rotating membranes and disks, whilst section 13.4 gives an overview of the effects of imperfections in the form of asymmetries. Section 13.5 introduces the additional complexity of external spring-mass-damper loads on a disk, initially for the case of a stationary disk and a moving load, then vice versa, with follower force models given in sections 13.6 and 13.7. The role of the negative (-velocity characteristic in conjunction with parametric excitation due to a follower force is also examined in section 13.7.

The motion with impacts and dry friction forces appears in some mechanical systems as mechanisms with clearances, (e.g., in gearings, pins, slots, guides, valve gears etc.), impact dampers, relays, forming and mailing machines, power pics etc. Such mechanisms include one or more pairs of impacting bodies, which introduce the strong nonlinearity into the system motion. The motion of the general pair of bodies with the both-sides impacts and dry friction forces is assumed (Figure 14.1). It can be the part of a more complex chain of masses in the mechanical system. Dead zones in the relative motion of bodies can be caused by assumed nonlinearities. The mathematical conditions controlling the numerical simulations or analytical solution of the motion are introduced. The application of this method is explained by the study of the influence of dry friction force on amplitude-frequency characteristics of four types of dynamical and impact dampers with optimised parameters.

An in depth study on the nonlinear dynamic interactions occuring during metal cutting process is given. A general mathematical model of the machine tool-cutting process is established, and then a high accuracy numerical algorithm for a multibody dynamic system excited by a cutting process is developed. Next, a simple model of orthogonal metal cutting, where all nonlinearities have been accounted for in the cutting process has been investigated in the first instance. Then stochastic properties of the material being cut were introduced to reflect variations in the workpiece properties, in particular in the cutting resistance. Nonlinear dynamics techniques such as constructing bifurcation diagrams and Poincaré maps are employed to ascertain dynamics responses for both the deterministic and the stochastic model. Untypical routes to chaos and unusual topology of Poincaré cross-sections were observed. The analysis conducted has led to formulation of some practical design recommendations. Finally, occurrence of chatter was investigated experimentally using a specially designed rig.

Most research in ultrasonic drilling has been concentrated on experimental aspects of the process, and this has led to a substantial gap in a depth of treatment of experimental and theoretical investigations. The basic aim of this chapter is to review and develop appropriate mathematical models, which can be used to predict the main features of the ultrasonic drilling of hard materials. Among them, an impact oscillator and dry friction models are used to provide a valuable insight into this complex process involving dynamic fracture of brittle materials. It is postulated that the main mechanism of the enhancement of material removal rate (MRR) in ultrasonic machining is associated with high amplitudes forces generated by impacts, which act on the workpiece and help to develop micro-cracks in the cutting zone. Novel procedures for calculating the MRR are proposed, which for the first time explain the experimentally observed fall in MRR at higher static forces. These theoretical studies are verified by the experimental investigations carried out in our laboratory.