- Non-Smooth Dynamical Systems
- Differential and Integral Equations
- Spectrum of Lyapunov exponents of non-smooth dynamical systems of integrate-and-fire type.
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If the dynamical system is linear in all subspaces then an implicit global analytical solution can be given, as the times when non-smoothness occurs have to be determined first. This leads to the necessity of solving a set of nonlinear algebraic equations. To illustrate the non-smooth dynamical systems and the methodology of solving them, three mechanical engineering problems have been studied. Firstly the vibro-impact system in a form of moling device was modelled and analysed to understand how the progression rates can be maximised.

Periodic trajectories can be reconstructed as they go through three linear subspaces no contact, contact with progression and contact without progression. In the second application frictional chatter occurring during metal cutting has been examined via numerical simulation method. The analysis has shown that the bifurcation analysis can be very useful to make an appropriate choice of the system parameters to avoid chatter.

The last problem comes from rotordynamics, where nonlinear interaction between the rotor and the snubber ring are studied. The results obtained from the developed mathematical model confronted with the experiment have shown a good degree of correlation. Keywords: Nonlinear dynamics, non-smooth systems, mechanical vibrations. Most of real systems are nonlinear and their nonlinearities can be manifested in many different forms. One of the most common in mechanics is the non-smoothness. One may think of the noise of a squeaking chalk on a blackboard, or more pleasantly of a violin concert.

Mechanical engineering examples include noise generation in railway brakes, impact print hammers, percussion drilling machines or chattering of machine tools. These effects are due to the non-smooth characteristics such as clearances, impacts, intermittent contacts, dry friction, or combinations of these effects. Non-smooth dynamical systems have been extensively studied for nearly three decades showing a huge complexity of dynamical responses even for a simple impact oscillator or Chua's circuit.

The theory of discontinuous and non-smooth dynamical systems has been rapidly developing and now we are in much better position to understand those complexities occurring in the non-smooth vector fields and caused by generally discontinuous bifurcations.

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There are numerous practical applications, where the theoretical findings on nonlinear dynamics of non-smooth systems have been applied in order to verify the theory and optimize the engineering performance. However from a mathematical point of view, problems with non-smooth characteristics are not easy to handle as the resulting models are dynamical systems whose right-hand sides are discontinuous, and therefore they require a special mathematical treatment and robust numerical algorithms to produce reliable solutions.

Practically, a combination of numerical, analytical and semi-analytical methods is used to solve and analyse such systems and this particular aspect will be explored here. The main aim of the paper is to outline a general methodology for solving of non-smooth dynamical systems, and to apply it to practical problems. The methodology will be illustrated and examined through three case studies. Firstly periodic responses of a drifting vibro-impact system with drift will be investigated through a novel semi-analytical method, developed by Pavlovskaia and Wiercigroch a , which allows to determine the favourable operating conditions.

The model accounts for visco-elastic impacts and is capable to mimic dynamics of a bounded progressive motion a drift. Then the frictional chatter in orthogonal metal cutting will be modelled and analysed using numerical and analytical methods see Wiercigroch and Krivtsov, In this paper an extensive nonlinear dynamic analysis has been performed giving some new light on the frictional chatter occurrence, i. Finally, the dynamic responses of a Jeffcott rotor system with bearing clearances will be examined see Karpenko et al.

Non-Smooth Dynamical Systems. In many engineering applications, characteristics of the system can be either discontinuous or non-smooth. It includes modelling of non-smooth systems by discontinuous functions and modelling of discontinuities by smooth functions. In the latter case extra care is required as smoothing discontinuities can produce a ghost solution Karpenko et al. The first approach considers first a dynamical system, which is continuous in global hyperspace W , and in autonomous form can be described as. Then we assume that the dynamical system 1 is continuous but only in N subspaces X i of the global hyperspace W see Fig.

Vibro-Impact Systems.

## Non-Smooth Dynamical Systems

Vibro-impact systems are inherently nonlinear and have been widely used in civil and mechanical engineering applications. One may give examples of ground moling machines, percussive drilling, ultrasonic machining and mechanical processing cold and hot forging. In the past all these machines and processes have been designed based on linear dynamic analysis. Imagine for example, a vibro-impact system driving a pile into the ground.

During its operation the driving module moves downwards, and its motion can be viewed as a sum of a progressive motion and bounded oscillations. The simplest physical model exhibiting such behaviour is comprised of a mass loaded by a force having static and harmonic components, and a dry friction slider. Despite its simple structure, a very complex dynamics was revealed. The main result from that work was a finding that the best progression occurs when the system responds periodically.

A more realistic model including visco-elastic properties of the ground and its optimal periodic regimes were studied in Pavlovskaia et al.

Modelling of Vibro-Impact Moling. As a first approximation a vibro-impact moling system may be represented as an oscillating mass with a frictional visco-elastic slider as shown in Fig. The frictional visco-elastic slider models well the hysteretic soil resistance depicted in Fig.

This model allows mimicking the separation between the mole head and the front face of the hole. The weightless slider has a linear visco-elastic pair of stiffness k and damping c. As has been reported in Pavlovskaia et al. Similarly to the stick-slip phenomenon, the progressive motion of the mass occurs when the force acting on the slider exceeds the threshold of the dry friction force d , x , z , v represent the absolute displacements of the mass, slider top and slider bottom, respectively. It is assumed that the model operates in a horizontal plane, or the gravitational force is compensated.

The considered system operates at the time in one of the following modes: No contact , Contact without progression , and Contact with progression. A detailed consideration of these modes and dimensional form of the equations of motion can be found in Pavlovskaia et al. The equations of motion covering all modes can be written using Heaviside step functions H i in the following form:. The basic function of the investigated system is to penetrate through soil. Despite the fact that the considered model has only two degrees-of-freedom, its dynamics is very complex.

Since displacements of the system elements are moving from the origin, the mass velocity has been used to view the structural changes in the system responses due to the fact that it is bounded.

### Differential and Integral Equations

The control parameter in form of static force, b proved to be very useful for determining the regions of the best progression. The construction of the bifurcation diagrams has brought some practical insight regarding progression rates. Since the system drifts towards larger displacements, v , one way to monitor progression rate is to calculate displacement in a finite time, which in our computations was equal to 50 periods of external loading.

The method constructs a periodic response assuming the global solution is comprised of a sequence of distinct phases for which local analytical solutions are known explicitly. A solution may consist of the following sequential phases see Fig. Progressions per period were calculated from the numerical simulation of the system dynamics and then compared with the results from the devised semi-analytical method thick solid line in Fig.

As can be seen from Fig.

Vibrations in Metal Cutting. Despite the continuing effort in the field, and generation of new theories, there is no consistent explanation for the existence of chatter. We then show that SNAs do exist in such nonsmooth dynamical system with quasiperiodic force. The dynamical behavior of the nonsmooth system is analyzed as a parameter is varied. The dynamics is analyzed through phase diagrams and bifurcation diagrams, Lyapunov exponents, singular continuous power spectrum, phase sensitivity of time series and rational approximations.

Strange nonchaotic attractors in a nonsmooth dynamical system. N2 - Strange nonchaotic attractors SNAs have fractal geometric structure, but are nonchaotic in the dynamical sense. AB - Strange nonchaotic attractors SNAs have fractal geometric structure, but are nonchaotic in the dynamical sense.

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## Spectrum of Lyapunov exponents of non-smooth dynamical systems of integrate-and-fire type.

Abstract Strange nonchaotic attractors SNAs have fractal geometric structure, but are nonchaotic in the dynamical sense. Fingerprint Strange attractor. Dynamical systems. Dynamical system. Piecewise Linear Systems. The book is divided into four parts, the rst three parts being sketched in Fig. The aim of the rst part is to present the main tools from mechanics and applied mathematics which are necessary to understand how nonsmooth dynamical systems may be numerically simulated in a reliable way.

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The second and third parts are dedicated to a detailed presentation of the numerical schemes. A fourth part is devoted to the presentation of the software platform Siconos. This book is not a textbook on - merical analysis of nonsmooth systems, in the sense that despite the main results of numerical analysis convergence, order of consistency, etc. Neem contact met mij op over Events Sprekers Incompany.

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