Spiralling in BTA deep-hole drilling How to model varying frequencies

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1 Spiralling in BTA deep-hole drilling How to model varying frequencies Nils Raabe, Oliver Webber and Winfried Theis September 1, 2004 Abstract One serious problem in deep-hole drilling is the formation of a dynamic disturbance called spiralling which causes holes with several lobes. Since such lobes are a severe impairment of the bore hole the formation of spiralling has to be prevented. Gessesse et al. (1994) explain spiralling by the coincidence of bending modes and multiples of the rotary frequency. This they derive from an elaborate finite elements model of the process. In online measurements we detected slowly changing frequency patterns similar to those calculated by Gessesse et al. We therefore propose a method to estimate a slow change of frequencies over time from spectrogram data. This makes it possible to significantly simplify the usage of the explanation of spiralling in practice because the finite elements model has to be correctly modified for each machine and tool assembly while the statistical method uses observable measurements. Estimating the variation of the frequencies as good as possible opens up the opportunity to prevent spiralling by e.g. changing the rotary frequency. 1 Introduction The work presented in this paper has been carried out as part of a project aimed at modelling the BTA deep hole drilling process, with special emphasis on dynamic aspects. The longterm goal is online-prediction of dynamic disturbances which in future may be used as a basis for intelligent control of the process. Deep hole drilling methods are used for producing holes with a high length-to-diameter ratio, good surface finish and straightness. For drilling 1

2 holes with a diameter of 20 mm and above, the BTA (Boring and Trepanning Association) deep hole machining principle is usually employed (VDI, 1974). The working principle is illustrated in Figure 1. Figure 1: BTA deep hole drilling, working principle (VDI, 1974). For obtaining a low deviation of the bore hole center axis from the ideal straight line, which is an important objective for machining holes with a high length to diameter ratio, deep hole drilling tools use the bore hole wall section machined in the immediate past as a guiding surface. This is achieved by an asymmetric cutting edge arrangement in combination with guiding pads on the circumference of the tool. The high surface finish of the bore hole wall is a side effect of the guiding action. When drilling with standard twist drills, chip removal becomes more and more unreliable with increasing drilling depth. This sooner or later leads to process failure. To solve this problem, deep hole drilling tools feature forced chip removal through high cooling lubricant flow rates via low restriction passages. In the case of BTA deep hole drilling, oil is supplied around the outside of the boring bar and the chips are transported away through the internal volume of the tube. Machining of bore holes with a high length to diameter ratio necessitates slender tool-boring bar assemblies. These components therefore have low dynamic stiffness properties which in turn can be the cause of dynamic disturbances such as chatter vibration and spiralling. Whereas chatter mainly leads to increased tool wear along with marks on the generally discarded bottom of the bore hole, spiralling causes a multi-lobe shaped deviation of the cross section of the hole from absolute roundness often constituting a 2

3 significant impairment of the workpiece. The effects of these disturbances on the workpiece can be seen in Fig. 2. Figure 2: Radial chatter marks on the bottom of the bore hole (left) and effects of spiralling on the bore hole wall (right). As the deep drilling process is often used during the last production phases of expensive workpieces, process reliability is of prime importance. To achieve an optimal process design with the aim of reducing the risk of workpiece damage, a detailed analysis of the process dynamics is necessary. In this paper we focus on spiralling which can be observed to occur either reproducibly at a certain drilling depth and fixed machining parameters or at random drilling depths. Gessesse et al. (1994) have modelled the process with finite elements and derived from this model that a reason for the reproducible occurence of spiralling is the intersection of changing bending modes and uneven multiples of the rotational frequency. They have shown in some experiments that this actually was a good prediction of spiralling. We observed the movement of the bending modes in on-line measurements of the bending moment of the boring bar and in measurements of the lateral acceleration of the boring bar. Here we propose a method to detect a region of moving frequencies and to estimate the functional form of this movement. 2 Simulations on Varying Frequencies First we checked by some simple simulations if it is indeed possible to reconstruct slow movements of frequencies from spectrogram data. Therefore, we used the following procedure: 1. Select a fixed number K of frequencies f k which shall change within frequency-bands (in the most simple case well separated). 2. For each frequency f k select a function f k (t) for the change within the frequency band. (And perhaps select also a function A k (t) for the amplitude on f k ) 3

4 3. Calculate where ε t N (0, σ 2 ). K x t = A k (t) cos(fk (t)lt + ϕ) + ε t (1) k=1 4. Calculate spectra on time-segments of a fixed length, possibly overlapping, to form the spectrogram matrix. 5. Check if the peaks in the spectrogram change similar to functions of the frequencies. Figure 3 shows two typical results from such a simulation. On the left hand side the change in the frequencies is very slow (.0004 per time segment) making the functions f 1 (t) nearly constant on the sections used for the construction of the spectrogram matrix. On the right hand side a case is presented where the change is too fast (more than.05 per time segment) which leads to completely misleading peaks or no peaks at all. These simulations have been produced with the following parameters: 1. Amplitudes on the changing frequency were fixed at 20 for both experiments observations were generated from the models (l = 2000 per second). 3. The additive error was N (0,.25) distributed. 4. The functions of the changing frequencies were (a) f a (t) = c 0 + c 1 t, c 0 = 0.2, c 1 = and (b) f b (t) = exp( t t 0 d ) + 0.2, t 0 = 35, d = The length of the segments in the spectrograms was set equally to 2000 with an overlap of 1800 observations, to make sure to have a sufficiently high resolution. 4

5 a) b) Figure 3: Left hand panel: slowly changing frequencies; right hand panel: fast changing frequencies. Checking Figure 3 a) more closely it transpires that the frequency with the maximum amplitude fa max (t) at each time t is not identical to fa (t) (depicted by the dashed white line). In fact the following holds: fa max (t) = d(f a (t)lt) d(lt) = c 0 + 2c 1 t. (2) This latter function is depicted by the dotted white line, which actually follows the dark area of significant frequencies. In Figure 3 b) the function f b (t) represented by the dashed white line obviously also matches the maximum amplitudes except of the area around t = 35, where the frequency rapidly decreases, and all frequencies are equally apparent. The corresponding function of the maximum amplitude (dotted white line) is given by f max b (t) = 0.1 [1 + exp( t t 0 )(1 t )] d d [1 + exp( t t (3) 0 d )]2 5

6 a) b) Figure 4: a) Simulation with similar settings as in Figure 3 b) but d = 10. b) Simulation with a million observations, turning point t 0 at 50 seconds, d = 12.5 and segments of length with an overlap of The graphs in Figure 4 show cases of slow changing frequencies following f b (t). Again this function is depicted by dashed white lines and the theoretical frequency-function by dotted ones. Figure 4 b) shows that spectrograms with higher resolutions lead to narrower confidence bands. Finally Figure 5 a) displays a case where the frequency crosses the Nyquistfrequency 0.5. Note that higher frequencies appear as their principal aliases in the spectrogram. Figure 5 b) displays the effect of a lower resolution for the spectrogram data which is not as bad as could be expected. a) b) Figure 5: a) Linear change in amplitude with slope c 1 = where the frequency crosses 0.5. b) Data from Figure 3 a) with segments of length 200 and without overlap. 6

7 To close this discussion we checked, whether more than one frequency can be distinguished from the spectrogram data. In Figure 6 the result of a simulation with two changing frequencies which intersect is shown. Obviously they do not influence each other and therefore could be estimated independently. Figure 6: Result of a simulation with two varying frequencies with intersection. These simulations show that it is possible to use spectrogram data to estimate changes in frequencies. Based on this data the parameters of given functions fk (t) resp. fk max (t) can be estimated as well as the following section will show. Application We now apply the idea of time-varying frequencies to a pair of experiments whereby spiralling occurred during the second. Note that in this case the reason for the spiralling was not the intersection of a bending mode and a multiple of the rotary frequency. That frequency lies much below the lowest apparent frequency. In fact the spiralling resulted from worn guiding 7

8 pads. Both experiments were driven with feed mm/rev, cutting speed 60 m/min and oil flow rate 300 l/min. The rotary frequency was 5.31 Hz. We use the lateral acceleration data for our investigations. Figure 7 shows the spectrograms of the two experiments, which are restricted to frequencies below 400 Hz. a) b) Figure 7: Left hand panel: Experiment without spiralling; Right hand panel: Experiment with spiralling. Both pictures in Figure 7 suggest three bands, in which frequencies vary. The margins for these band lie around 100, 160 and 300 Hz. Furthermore Figure 7 a) and b) look very similar, so the spiralling seems not to effect the spectrogram. To keep calculations simple we from now on only consider the entries of the spectrogram which are significant on a level of Next the remaining observations are clustered bivariate (frequency and amplitude) by a k-means method. The results of these clusterings are displayed in Figure 8. 8

9 a) b) Figure 8: Clustering results, left hand panel: without spiralling; right hand panel: with spiralling. Obviously the clustering results in both cases coincide with the first impression of three frequency bands. Note that even though the amplitudes are used for clustering as well, the bands are perfectly divided by the frequencies. This shows that in fact the amplitudes of the frequencies within on band are similar as well. This circumstance confirms the assumption of varying frequencies within frequency bands. Because of the visibly U-shaped variation we decide to fit a model of the following form for each frequency band k: f max k (t) = c k0 + c k1 t + c k2 t 2, k = 1,..., 3. (4) Then the functions within the underlying harmonic process with varying frequencies as defined in 1 compute as follows: fk (t) = c k0 + c k 1 2 t + c k 2 3 t2, k = 1,..., 3. (5) Figure 9 shows the resulting curves by estimating c ki for the experiment without spiralling. 9

10 Figure 9: OLS-Estimation of varying frequencies. The regression curves in figure 9 reproduce the dynamics of the frequencies quite well. Similar results can be obtained for a series of 21 experiments, which were driven with different settings of the factors feed and cutting speed. Conlusion This paper shows that it is possible to reconstruct the dynamic behavior of frequencies within a harmonic process from spectrogram data. This result may be used as a basis for online-intervention strategies for the prevention of spiralling. For example in cases where a bending mode runs into danger of meeting an uneven multiple of the rotary frequency statistical methods like control cards can be used. Also combinations of models with varying frequencies and amplitudes are possible. Models with varying amplitudes of relevant frequencies were used in (Theis, 2004) to predict chatter. However, equivalent models did not perform well in the prediction of spiralling. So it may be interesting if taking into account time-dependence of both frequencies and amplitudes could improve the results of single models. This theory is supported by the fact, that in 10

11 our experiments amplitudes are similar within frequency bands. So changes of the process conditions may also show in changes of amplitudes. Another topic of future work may be the connection between process parameters and the dynamic of the frequencies. Acknowledgements This work has been supported by the Collaborative Research Centre Reduction of Complexity in Multivariate Data Structures (SFB 475) of the German Research Foundation (DFG). References Y.B. Gessesse, V.N. Latinovic, and M.O.M. Osman. On the problem of spiralling in bta deep-hole machining. Journal of Engineering for Industry, 116: , Winfried Theis. Modelling Varying Amplitudes. PhD thesis, University of Dortmund, VDI. Tiefbohrverfahren. VDI Düsseldorf,