Intermittency and multiscale dynamics in milling of fiber reinforced composites
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Abstract
We have analyzed the variations in cutting force during milling of a fiber-reinforced composite material. In particular, we have investigated the multiscale dynamics of the cutting force measured at different spindle speeds using multifractals and wavelets. The multifractal analysis revealed the changes in complexity with varying spindle speeds. The wavelet analysis identified the coexistence of important periodicities related to the natural frequency of the system and its multiple harmonics. Their nonlinear superposition leads to the specific intermittent behavior. The workpiece used in the experiment was prepared from an epoxy-polymer matrix composite reinforced by carbon fibers.
Keywords
Milling Wavelets Multifractals Nonlinear vibration Mechanics of machines1 Introduction
Milling is one of the most common machining operations performed in the manufacturing industry. It is often used as a process for material removal, edge finishing and other functions. The cutting force in a milling operation evolves on multiple timescales and exhibits complex dynamics. Over the past few years, new technological development in milling provided a reliable high-speed cutting procedure. However, in spite of recent progress in the understanding of the nonlinear mechanisms leading to vibrations [1, 2, 3, 4, 5], the strategies for controlling conditions for stable machining are not clearly understood. Consequently, it would of great interest to gain a deeper understanding of the complex dynamics of the milling process. Some progress has been made in this direction using the adaptive control concept, based on identification of relatively short time series [6, 7].
Fiber-reinforced composites, in view of their high specific strength and stiffness, are now widely used in various industrial applications. However, due to material discontinuity, non-homogeneity and anisotropy, machining of composites is more challenging than machining of simple metals and their alloys. Because of various possible damage mechanisms such as fiber pullout, fiber fragmentation and delamination, matrix burning, and matrix cracking, poor surface quality can occur [8, 9]. Workpiece-tool vibrations appearing during machining increase considerably the temperature of contact, and this effect cannot be minimized by cooling fluid as the material can easily absorb it. Recently, Rusinek [10] and Litak et al. [11] investigated the cutting dynamics in milling of a fiber-reinforced composite material using nonlinear time series analysis techniques. In this study, we analyze the dynamics of the cutting force variations in milling of fiber-reinforced composites using wavelets and multifractals.
The present paper is composed of five sections. Following this section which provides an introduction to the main topics of the paper, we describe the experimental set up and the measurement procedure in Sect. 2. Section 3 is devoted to the multifractal approach, whereas Sect. 4 presents the wavelet analysis. Finally in Sect. 5, a few concluding remarks are given.
2 Experimental set up and force measurement
3 Multifractal analysis
A characteristic feature of the complex dynamics of the milling process is that they occur on multiple time scales. A convenient way to describe the dynamics of such multiscale processes is to use a multifractal formalism. The multifractal approach is based on the spectrum of Hölder exponents which can be used as a measure of complexity [12, 13, 14]. A fractal (or a monofractal) process is self-similar in the sense that its dynamics can be described in terms of a single power-law scaling exponent such as the Hurst exponent [15], and may be considered a homogeneous process. Accordingly, its complexity can be described by means of a single fractal dimension. In contrast, a multifractal process is heterogeneous and evolves on different time scales with different scaling exponents. It is therefore necessary to use several scaling exponents or fractal dimensions to describe the multiscale features of a multifractal process. This can be done by calculating the singularity spectrum in terms of the so-called Hölder exponent. For an infinitely long monofractal process, the singularity spectrum reduces to a single point. On the other hand, if the singularity spectrum does not reduce to a single point, it is indicative of multifractal behavior. A multifractal process may be considered to be locally self-similar and the Hölder exponent may be treated as a local Hurst exponent. The broadness of the singularity spectrum is a measure of complexity of the multifractal process. Multifractal processes are known to occur in a wide variety of applications (see, for example, [16, 17, 18]).
We use the following two attributes of a multifractal spectrum: (i) the value of the Hölder exponent, α=α_{0}, corresponding to the spectral peak, and (ii) the broadness, Δα, which is the distance between the (extrapolated) points of intersection of the spectral curve with the α-axis. The parameter, α_{0}, represents the most dominant fractal exponent, and it reflects the degree of persistence or correlation in a time series. In particular, the value α_{0}=0 corresponds to Gaussian white noise, α_{0}=0.5 to Brownian walk, both of which indicate an uncorrelated process. On the other hand, the values of α_{0}<0.5, and α_{0}>0.5 indicate anti-persistent and persistent walks, implying positive and negative correlations, respectively, between the events in the time series [16, 17, 18]. The broadness of a singularity spectrum describes the range of possible fractal exponents and thus gives a measure of multifractality or complexity of the time series. A large value of broadness describes a richer multifractal structure whereas a small value approaches a monofractal limit.
Summary of statistical and multifractal parameters of the force F_{x}
Speed [rpm] | Average value [N] | Standard deviation SD [N] | α_{0} | Δα |
---|---|---|---|---|
2000 | 7.32 | 22.07 | 0.294 | 0.804 |
3500 | 11.70 | 24.45 | 0.125 | 0.624 |
5000 | 12.34 | 21.65 | 0.190 | 0.632 |
6500 | 8.02 | 26.90 | 0.080 | 0.528 |
8000 | 9.68 | 19.20 | 0.047 | 0.253 |
4 Wavelet analysis
Wavelets have been used for time series analysis in a wide variety of applications. Wavelet analysis provides a spectral-temporal approach to identify the dominant modes of variability in a time series and to delineate how these modes vary over time. It is particularly useful for analyzing transient and intermittent phenomena. A wavelet-based approach has advantages over the more traditional methods such as the Fourier transform or the windowed Fourier transform. The Fourier transform is a purely frequency domain technique which seeks to determine the periodicities in a signal through spectral peaks, but it cannot delineate the time spans over which the periodicities may persist. A windowed Fourier transform also known as a short-time Fourier transform (STFT) circumvents this limitation by applying the Fourier transform on a short segment of the signal using a window of fixed size and then sliding the window in time. The temporal variations of the periodicities, if any, can thus be determined. However, because a fixed-size window is used in STFT, the frequency resolution as well as the time resolution is fixed. As a consequence, for a given signal either the frequency resolution may be poor or the time localization may be less precise, depending on the size of the chosen window. In contrast, using variable-size windows, wavelet analysis provides an elegant way of adjusting the time and frequency resolutions in an adaptive fashion. A wavelet transform uses a window that narrows when focusing on small-scale or high-frequency features of the signal, and widens on large-scale or low-frequency features, analogous to a zoom lens [19]. Recently we have used wavelet analysis in our studies of pressure fluctuations in internal combustion engines [16], and other applications [20, 21, 22] including the turning process [17]. We present below a brief description of the wavelet analysis methodology and then apply it to the cutting force time series shown in Fig. 2.
5 Concluding remarks
We have examined the dynamics of cutting force variations in milling of a fiber-reinforced composite material by analyzing the experimental time series of the largest force component (F_{x}) using wavelets, and multifractals. The wavelet analysis revealed that the short or high-frequency periodicities of 2 and 10 sampling periods are intermittent. Interestingly, the largest detected intermittencies coincided with the maximum of fluctuations measured by the standard deviation. The above conclusions have been confirmed also by the multifractal measures which showed that there is a noticeable deviation of the changing trend in the correlation α_{0} and complexity measure Δα at the particular spindle speed (ω=3500 rpm). This peculiar behavior was formerly investigated by the multiscaled entropy showing its nonhomogeneous behavior against the scaling and similarity factors which confirmed the main changes in short period (intermittencies) as well as long-period (modulation) changes [11].
Note, in the standard approach to milling one can follow the uniform material which could fit the mathematical model and the corresponding regions of stability [3, 26, 27, 28, 29, 30]. On the other hand, the results obtained here, for a composite material, can be used further to study the stability of the milling process and to develop new methods of control. To get further insight, it would be of interest to analyze the cutting force variations during milling of different types of composite materials. Results of such analysis will be reported in a future publication.
Acknowledgements
This research has been partially supported by the European Union within the framework of Integrated Regional Development Operational Program under the project POIG.0101.02-00-015/08, and the European Union Seventh Framework Programme (FP7/2007–2013), FP7-REGPOT-2009-1, under grant agreement no. 245479.
Open Access
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