Igor Grabec

Chaotic dynamics of the cutting process

Abstract In the article a manufacturing system is treated as a simple two dimensional elastic structure in which the cutting force is generated by material flow against the tool. Generalized empirical relations corresponding to orthogonal cutting are applied to describe the non-linear dependence of a cutting force on chip velocity and thickness. The formulated mechanical model of a two dimensional coupled oscillator thus represents a typical example of a non-linear dissipative system, the dynamics of which require description in a four dimensional phase space. The numerical solutions of the governing dynamical equations reveal chaotic oscillations if the characteristic cutting parameter is selected in a region corresponding to intensive cutting. The properties of chaotic oscillations are illustrated by the time dependence of tool displacement, acceleration, cutting force, and power dissipated for material deformation or exchanged with an oscillating tool. The manufactured surface profile is found to resemble a wavy water surface. The phase portraits, Lissajoux figures and spectral densities of calculated signals indicate a similarity with quasiperiodic movement. The information dimension of a corresponding strange attractor, estimated by the correlation exponent, is approximately 3 for a typical example of cutting chaos. The tool displacement X in the direction of input velocity appears to be the most characteristic variable. Its return map is constructed by successive observations of maximal displacement values. The corresponding approximate analytical expression X n+1 =5 X n (1.1- X n )-0.70 is similar to the prototypical map which is frequently applied in the study of chaotic phenomena.

Chaos generated by the cutting process

Abstract Due to nonlinearity and coupling forces chaotic oscillations take place in an elastic manufacturing machine. A transition from quasiperiodic via chaotic to synchronised anharmonic oscillations with increasing cutting intensity is demonstrated by spectral distributions and the correlation exponent.

Analysis of acoustic emission signals and monitoring of machining processes

Monitoring of a machining process on the basis of sensor signals requires a selection of informative inputs in order to reliably characterize and model the process. In this article, a system for selection of informative characteristics from signals of multiple sensors is presented. For signal analysis, methods of spectral analysis and methods of nonlinear time series analysis are used. With the aim of modeling relationships between signal characteristics and the corresponding process state, an adaptive empirical modeler is applied. The application of the system is demonstrated by characterization of different parameters defining the states of a turning machining process, such as: chip form, tool wear, and onset of chatter vibration. The results show that, in spite of the complexity of the turning process, the state of the process can be well characterized by just a few proper characteristics extracted from a representative sensor signal. The process characterization can be further improved by joining characteristics from multiple sensors and by application of chaotic characteristics.

Automatic chatter detection in grinding

Two methods for automatic chatter detection in outer diameter plunge feed grinding are proposed. The methods employ entropy and coarse-grained information rate (CIR) as indicators of chatter. Entropy is calculated from a power spectrum, while CIR is calculated directly from fluctuations of a recorded signal. The methods are verified using signals of the normal grinding force and RMS acoustic emission. The results show that entropy and CIR perform equally well as chatter indicators. Based on the normal grinding force, they detect chatter in its early stage, while only cases of strong chatter are detected based on RMS acoustic emission.

Self-organization of neurons described by the maximum-entropy principle

In the article the maximum-entropy principle and Parzen windows are applied to derive an optimal mapping of a continuous into a descrete random variable. The mapping can be performed by a network of self-organizing information processing units similar to biological neurons. Each neuron is selectively sensitized to one prototype from the sample space of the discrete random variable. The continuous random variable is applied as the input signal exciting the neurons. The response of the network is described by the excitation vector which represents the encoded input signal. Due to the interaction between neurons adaptive changes of prototypes are caused by the excitations. The derived mathematical model explains this interaction in detail; a simplified self-organization rule derived from it corresponds to that of Kohonen. One and two-dimensional examples of self-organization simulated on a computer are shown in the article.

Nonlinear model predictive control of a cutting process

Abstract Nonlinear model predictive control (MPC) of a simulated chaotic cutting process is presented. The nonlinear MPC combines a neural-network model and a genetic-algorithm-based optimizer. The control scheme comprises a process, a model, an optimizer, a controller and a corrector. Neural networks are used to build a nonlinear experimental model of the process which is applied to recursive prediction in MPC. A robust genetic-algorithm-based optimizer is used for the optimization of control trajectories. A neural-network-based controller is included in the control scheme for enhanced optimizer initialization and for autonomous control after the learning period. The nonlinear MPC is applied to control the simulated chaotic cutting process. The dynamics of a cutting process are very complex due to the nonlinear effects of high order involved. The control objective is to construct an on-line control system capable of improving the quality of the manufactured surface by preventing tool oscillations which result in the rough surface of the workpiece. A feedforward network is applied as an experimental model of the cutting process, and MPC strategy with tool support manipulation as a control variable is investigated. The results show considerable improvement of the manufacturing quality obtained by the proposed nonlinear model predictive control.

A New Method for Chatter Detection in Grinding

Abstract A new method for automatic chatter detection in outer-diameter grinding is proposed which exploits significant changes in grinding dynamics caused by the onset of chatter. The method is based on monitoring of a non-linear statistic called the coarse-grained entropy rate. The entropy rate is calculated from the fluctuations of the normal grinding force. Values of the entropy rate close to zero are typical of chatter, whereas larger values are typical of chatter-free grinding. If the entropy rate is normalized, a threshold value can be set which enables automatic distinction between chatter-free grinding and chatter.

Recent developments in quantitative ultrasonic NDE of composites

Abstract This paper summarizes several recent developments which are facilitating new approaches for both active and passive quantitative ultrasonic measurements in composite materials. These include the development of point sources and point receivers, a theory for analysing the propagation of transient elastic waves through a bounded, dispersive and attenuative medium and the development and implementation of appropriate signal processing algorithms by which the detected ultrasonic signals can be processed to recover the elastic properties of an anisotropic material, the characteristics of wave propagation, or the characteristics of the source. An alternative to these deterministic approaches is a processing scheme based on a simulated intelligent system which processes ultrasonic signals like a neutral network. Examples of applications of these ideas to the ultrasonic NDE of composite materials are given.

A chaotic cutting process and determining optimal cutting parameter values using neural networks

A model of an orthogonal cutting system is described as an elastic structure deformable in two directions. In the system, a cutting force is generated by material flow against the tool. Nonlinear dependency of the cutting force on the cutting velocity can cause chaotic vibrations of the cutting tool which influence the quality of a manufactured surface. The intensity and the characteristics of vibrations are determined by the values of the cutting parameters. The influence of cutting depth on system dynamics is described by bifurcation diagrams. The properties of oscillations are illustrated by the time dependence of tool displacement, the corresponding frequency spectra and phase portraits. The corresponding strange attractors are characterized by correlation dimension. The vibrations are characterized by the maximum Lyapunov exponent. The manufactured surface at the first cut is taken as the incoming surface in the second cut, thus incorporating the influence of the rough surface in the model. Again, bifurcation diagrams, the correlation dimension and the maximum Lyapunov exponent are employed to describe the effects of parametrical excitation on the cutting dynamics. A cost function is defined which describes the dependence of the cutting performance on cutting depth. The cost function is empirically modeled using a self-organizing neural network. A conditional average estimator is applied to determine the optimal value of the cutting depth applicable as a control variable of the cutting process.

STATISTICAL APPROACH TO MODELING OF SPATIOTEMPORAL DYNAMICS

Often in the analysis of spatially extended dynamic systems, we do not know an analytical model of the system dynamics, but we can provide spatiotemporal records of the characteristic state variable. The question then arises of how to extract a model of the system dynamics from the corresponding data. As a quite general solution of this problem, we propose a nonparametric statistical method of local modeling. The performance of the proposed method is demonstrated by predicting typical examples of spatiotemporal chaotic data. The results of modeling indicate that the statistical method can be applied to modeling the deterministic properties of spatiotemporal dynamics in terms of recorded data.