Dynamics of thinwalled element milling expressed by recurrence analysis
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Abstract
This paper presents the results of experimental research on the stability of a milling process for producing a thinwalled part made of AL7075 aluminium alloy. The part was machined on a CNC milling machine with a decreasing wall thickness. The acceleration and cutting forces in the process were measured and analyzed to determine stability limit using classical stability diagrams as well as recurrence plots and recurrence quantification analysis.
Keywords
Milling stability Thinwalled element milling Recurrence plot1 Introduction
A relative motion between a cutting tool and a workpiece in cutting processes very often occurs as selfexcited vibrations, also called chatter. Chatter is an undesired phenomenon because it deteriorates the finished surface and promotes the wear of tools and components of cutting machines [19]. Moreover, chatter vibrations can cause the amplitude increase to a higher extent than in stable (vibrationfree) cutting, which is particularly dangerous in practice [9]. The description of physical mechanisms that cause chatter phenomena can be found in [39]. Usually, the regenerative chatter (also called secondary chatter) is one of the most common causes of instability in the cutting processes. However, Grabec [14] and Wiercigroch et al. [40] showed that frictional effect is also important, because it leads to the socalled frictional or primary chatter. Generally, the problem of regenerative and frictional chatter is also discussed in [18, 31, 40]. From a practical point of view the choice of proper cutting conditions is very important because it can prevent chatter vibrations. In order to define a set of parameters which ensure stable cutting conditions the so called stability lobes diagrams (SLD) should be employed. SLDs help determine the value of acceptable cutting width (or depth) as a function of spindle speed. This is particularly important in high speed machining (HSM). HSM is often used in practice in order to increase process efficiency and reduce production costs. An indepth overview of both selfexcited vibrations in cutting processes and methods for chatter prediction is given in [29].
Aluminium alloys are widely used in the aviation industry because of their low mass density. Although more and more thinwalled structural components are nowadays made of composite materials [35], some applications still require that these components be made of aluminum alloys which are typically used for relatively rigid thinwalled elements with complicated design. On the other hand, the machining of thinwalled components means removing even up to 90 % of their volume. In order to do this effectively, high cutting depths and feed rates should be applied, which leads to an increase of cutting forces. If the forces are too high they can cause stability loss of a thinwalled workpiece. The literature provides a great deal of analytical, numerical and experimental methods for stability prediction. For instance, Altintas and Budak describe an analytical method for predicting milling stability lobes based on the mean of the Fourier series of the dynamic milling coefficients [2]. This method is fast but cannot predict additional stability regions and period doubling bifurcation for a low radial depth of cut [10]. To overcome this problem, a multifrequency solution of chatter stability was developed by Budak and Altintas [6] and then extended by Merdol and Altintas [28].
One of the most popular numerical methods for chatter prediction is the Finite Element Method (FEM) [1, 13, 24, 37]. Bayly et al. [5] propose the use of a temporal finite element analysis for milling and interrupted turning [4]. Moreover, Voronov [37] and especially Adetoro et al. [1] propose an improved model of the classical milling process. The author of the work [1] takes into account the wellknown stability model and supplements it with considerations about the nonlinearity of the cutting force coefficients and axial immersion angle as a function of the axial depth of cut. Besides, the authors indicate that thinwalled workpiece dynamics is nonlinear. The proposed model is validated and the theoretical findings show very good agreement with the experimental results [1]. Although, experimental results are very often compared with FEM results [3, 17], the problem of experimental signal analysis remains an open question. Different methods of signal analysis are applied in order to recognize chatter vibrations in cutting operations, including multifractal and wavelet approaches [38], multiscale entropy [23] Hilbert–Huang transform [20], recurrence analysis [22], flickernoise spectroscopy [21] and audio signal analysis [32]. As it was mentioned before, the HSM method is also applied to thinwalled aluminium components for the aircraft industry to increase productivity [8]. Then, the flexibility of the tool and the workpiece, high spindle speeds and the inherent impact of nonlinearities in the milling process lead to complicated toolworkpiece interactions. In the paper [8] the method of delay space reconstruction is applied to show, that the workpiece motion is characterized by fractal geometry. The autobispectra suggests quadratic phase coupling among the spectral peaks associated with the cutter frequency. Finally, the authors propose a mechanicsbased model with an impact so as to explain the obtained results. Their predictions agree well with the experimental observations. Arnaud et al. [3] presents the problem of modelling of machining vibrations of thinwalled aluminium workpieces at high productivity rate. These authors employ numerical simulations, FEM analysis and experimental tests in order to determine optimal cutting conditions. As a results, chatter frequencies are obtained. Moreover their study draws attention to the fact that the prediction of amplitudes in case of unstable milling is a very complex problem. Therefore, the simulations of thinwalled components machining require the applications of developed cutting force models including contact modelling and ploughing due to the variations in clearance angle during machining. In the machining process of thinwalled structure, it can be observed that the dynamic behaviour of the tool workpiece system depends on the tool position in the workpiece. This phenomenon and the problem of the influence of several modes on stability lobes are discussed in [36]. On the other hand, modal interaction can affect surface roughness in milling thinwalled structures [33].
The models of cutting processes are described by delay differential equations which may pose some problems in numerical calculations. To accelerate the integration procedure the semi or full discretization methods for prediction of milling stability are proposed in several papers [10]. The semidiscretization method, developed by Insperger and Stepan is an efficient numerical technique for stability analysis of linear delayed systems. However, Ding et al. [10] present a fulldiscretization method for prediction of milling stability which is even more effective. Insperger [16] proposes the so called actandwait concept for continuoustime control systems with feedback delay. The problem of chatter control is also discussed in [34] as well.
This paper focuses on the problem of milling stability of thinwalled element made of aluminium alloy 7075. A similar problem is discussed in [7], where the authors present a stability prediction method for the workpiece dynamics based on a structural modification technique. They consider a continuous change in the inprocess workpiece structure during a machining cycle. With this method, the frequency response function of the workpiece is obtained by FEA only once, and it is continuously modified by depending on the workpiece volume removed during the cycle. The simulation results show good agreement with the experimental findings.
Here, in this paper an influence of wall thickness on process stability is valuated with an experiment. Next, the recurrence plot technique is applied to show various kinds of dynamical properties depending on wall thickness.
The primary objective of the paper is to determine milling stability expressed by means of the new index obtained from recurrence quantification analysis (RQA). Recurrence quantifiers are chosen for stability analysis because they enable finding a simple index for chatter recognition. Such index could be implemented later to a control system of chatter vibrations in any cutting processes.
2 Experimental methodology
3 Classical stability analysis
4 Theory of recurrence analysis
 Recurrence rate (RR) is the density of recurrence points in a recurrence plot. In physical terms, RR denotes the probability that the system will recur$$ {\text{RR}} = \frac{1}{{N^{2} }}\sum\limits_{i,j = 1}^{N} {M_{i,j} } $$(3)
 Determinism (DET) is the fraction of recurrence points forming diagonal lines$$ {\text{DET}} = \frac{{\sum\nolimits_{{l = l_{\hbox{min} } }}^{N} {lP\left( l \right)} }}{{\sum\nolimits_{i,j = 1}^{N} {M_{i,j} } }}, $$(4)
 Laminarity (LAM) is the fraction of recurrence points forming vertical lines. Vertical lines are typical for intermittency$$ {\text{LAM}} = \frac{{\sum\nolimits_{{v = v_{\hbox{min} } }}^{N} {vP(v)} }}{{\sum\nolimits_{v = 1}^{N} {vP(v)} }} $$(5)
 Laminarity to determinism ratio (LAM/DET)$$ {\text{LAM/DET}} = {{\frac{{\sum\nolimits_{{v = v_{\hbox{min} } }}^{N} {vP(v)} }}{{\sum\nolimits_{v = 1}^{N} {vP(v)} }}} \mathord{\left/ {\vphantom {{\frac{{\sum\nolimits_{{v = v_{\hbox{min} } }}^{N} {vP(v)} }}{{\sum\nolimits_{v = 1}^{N} {vP(v)} }}} {\frac{{\sum\nolimits_{{l = l_{\hbox{min} } }}^{N} {lP(l)} }}{{\sum\nolimits_{i,j = 1}^{N} {lP(l)} }}}}} \right. \kern0pt} {\frac{{\sum\nolimits_{{l = l_{\hbox{min} } }}^{N} {lP(l)} }}{{\sum\nolimits_{i,j = 1}^{N} {lP(l)} }}}} $$(6)
 Divergence (DIV) is an inverse of the longest diagonal length$$ {\text{DIV}} = \frac{1}{{\hbox{max} \left( {\{ l_{i} ;\;i = 1 \ldots N_{l} \} } \right)}} $$(7)
 Lentropy (L _{ ent }) is the Shannon’s entropy of the diagonal line segment distribution$$ L_{ent} =  \sum\limits_{l = l\hbox{min} }^{N} {P\left( l \right)\ln P\left( l \right)} $$(8)
 L _{max} is the length of the longest diagonal line$$ L_{\hbox{max} } = \hbox{max} (\{ l_{i} ;\;i = 1 \ldots N_{l} \} ) $$(9)
P(l) is the histogram of the lengths l of the diagonal lines, P(v) is the histogram of the lengths v of the vertical lines, N denotes the number of points on the phase space trajectory. The recurrence quantification analysis can provide useful information even for short and nonstationary data, where other methods fail. The RQA can be applied for various kinds of data to recognize dynamical behaviour [23]. Here the RQA is also applied to investigate milling process instability.
5 Recurrence plots and recurrence quantification analysis
The critical indexes of RQA should be estimated in an experiment, for every cutting operation individually. In the experiment reported in this paper, the critical value of indexes were: DET/RR_{ cr } = 2000, DET_{ cr } = 0.3, LAM/DET_{ cr } = 0.3, (L _{ ent })_{ cr } = 0.5, DIV_{ cr } = 0.2,(L _{max})_{ cr } = 10. If LAM/DET or DIV is smaller than the critical ones the milling process is stable. As for the other indexes, they should be set higher than critical ones to ensure that the process is stable.
6 Discussion and conclusions
This study investigated the problem of stability of the milling process for producing a thinwalled components. The stability limit estimated with the commercial software CutPro ver. 9 is not accurate either for the workpiece treated as a rigid body or even for the flexible workpiece described by the modal parameters taken from the impact test. The results of the experiment performed on the milling machine reveal that stability loss occurs much earlier than suggested by the CutPro9 module, despite the fact that the modal parameters of both the tool and the workpiece were measured after every tool pass and implemented in the programme for calculating stability diagrams. This means that the applied software is not suitable for investigating milling process for low mass elements (like thinwalled components) where the mass reduction is substantially high compared to the initial mass of the workpiece. However, undoubtedly, the main advantage of the CutPro software is the fact that it enables producing the SLD prior to running a cutting process (test). Stability loss is more visible in the cutting force signals and especially in the acceleration of vibrations. However, the acceleration signal indicates that a loss of stability is earlier (i.e. for a lower value of wall thickness) than the force signals suggest. This observation is in accordance with the experiment. The tests performed with the modal hammer let us determine the modal parameters of the workpiece which are changeable during the experiment. The decrease of the critical depth of cut with wall thickness is directly connected with a decrease in the damping ratio.
The recurrence diagrams and the method of phase space reconstruction method can determine stability loss of the milling process and provide a deeper insight into the system dynamics. From a practical point of view, however, these methods are not a convenient way of estimating stability limits. The use of recurrence quantification analysis is much better in this regard. When the process approaches instability, the stability index indicates that some technological parameters of the cutting process should be modified to prevent chatter. Therefore, we need the simplest numerical indexes. The reconstructed phase space analysis gives a deeper insight into process dynamics but the application of this analysis for the purpose of this paper is difficult. That is why, it was necessary to find a proper index which could be implemented later for chatter control system. The production of thinwalled parts, that are widely used in aircraft industry, is only one example of RQA applications. The selection of such RQA indexes as DET/RR, DET, LAM/DET, L _{ ent }, DIV and L _{max} was dedicated by the fact that they exhibit the exact moment when a thin walled element losses stability in milling. In the future the RQA indexes could be applied to control the stability of cutting processes online. RQA is an alternative method of finding stability limit. Sometimes, it is simpler to calculate statistical parameters of force or acceleration signal but, RP and RQA give deeper insight in system dynamics.

the variation of modal parameters in the cutting process for low mass (thin walled) elements is crucial for stability diagram prediction. Modal mass, stiffness and damping ratio are essentially important, too;

the proposed stability indexes based on recurrence quantification analysis can be used for determination of the stability limit if the critical (threshold) value is properly defined,

the critical value of stability index should be estimated individually for every cutting operations.
Notes
Acknowledgments
Financial support of Structural Funds in the Operational Programme—Innovative Economy (IE OP) financed from the European Regional Development Fund—Project “Modern material technologies in aerospace industry”, POIG.01.01.0200015/0800 is gratefully acknowledged.
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