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Improved experimental-analytical approach to compute speed-varying tool-tip FRF
Accepted Manuscript Title: Improved experimental-analytical approach to compute speed-varying tool-tip FRF Author: N. Grossi L. Sallese A. Scippa G. Campatelli PII: DOI: Reference:
S0141-6359(16)30375-0 http://dx.doi.org/doi:10.1016/j.precisioneng.2016.11.011 PRE 6487
To appear in:
Precision Engineering
Received date: Revised date: Accepted date:
18-5-2016 25-8-2016 17-11-2016
Please cite this article as: Grossi N, Sallese L, Scippa A, Campatelli G.Improved experimental-analytical approach to compute speed-varying tool-tip FRF.Precision Engineering http://dx.doi.org/10.1016/j.precisioneng.2016.11.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Improved experimental‐analytical approach to compute speed‐varying tool‐tip FRF N. Grossia,*, L. Sallesea, A. Scippaa, G. Campatellia aDepartment of Industrial Engineering, University of Firenze, via di Santa Marta 3, 50139, Firenze, Italy. *corresponding author. Tel. +390552758726. E‐mail: [email protected] HIGHLIGTHS An efficient method to identify the speed‐varying tool‐tip FRFs is proposed A quick cutting test is combined with the inversion of chatter analytical solution Experimental chatter limits are obtained by a dedicated test, called SSR By fitting experimental and predicted conditions, speed‐varying FRFs are computed Speed‐varying SLD reconstructed with the computed FRFs is experimentally validated Abstract Chatter stability prediction is crucial to improve the performances of modern milling process, and it gets even more important at high speeds, for which very productive cutting parameters can be achieved if the suitable spindle speed is selected. Unfortunately, the available chatter predictive models suffer from reduced accuracy at high speed due to inaccuracies in the input data, especially the machine tool dynamics that is acquired in stationary configurations but could sensibly change with spindle speed. In this paper, an efficient method to identify the speed‐varying Frequency Response Functions (FRFs) under operational conditions is presented. The proposed approach is based on the definition of some experimental chatter limits (i.e., chatter frequency and related depth of cut), obtained by a dedicated test, called Spindle Speed Ramp‐up. The experimental results are then combined with the analytical stability solution. By minimizing the differences between the experimental and predicted chatter conditions, a dedicated algorithm computes the speed‐varying FRFs. Few tests and simple equipment (i.e., microphone) are enough to calculate the FRFs in a wide range of spindle speeds. The proposed technique was validated in real machining applications, the identified tool‐tip FRFs are in accordance with expected trend reported in scientific literature. Speed‐varying stability lobe diagram reconstructed with the computed FRFs is proven to be accurate in predicting stable cutting parameters. Keywords: Chatter, Milling, Dynamics, Spindle, In‐process dynamics identification 1. INTRODUCTION The milling process could be significantly affected by detrimental vibrations, that represent one of the main limitations to further increase the process performance. The unstable vibration, known as chatter, is the most dangerous phenomenon because it could grow uncontrollably, causing poor surface finish, excessive tool wear and possible tool breakage, hence affecting productivity [1]. In the last decades many chatter prediction methods have been developed [2]; the main result of these approaches is a chart showing chatter‐free combinations of cutting parameters, known as Stability Lobe Diagram (SLD). These diagrams are very useful, especially at high speeds where lobes are spaced and higher depth of cut can be safely exploited if the appropriate spindle speed is selected. However, at high spindle speeds every predictive model shows reduced accuracy, generally not caused by the approximation introduced in the
analytical chatter model, but due to the inputs inaccuracies instead [3]. Indeed, the input data, such as cutting force coefficients and machine tool dynamics, may sensibly vary with spindle speed, even though commonly obtained at low speed, or in idle condition, and used in the entire spindle speed range [4]. In particular, the system dynamics (i.e., tool‐tip Frequency Response Function (FRF)), essential to forecast the SLD, is generally acquired via experimental modal analysis in the stationary condition (i.e., without spindle rotating). This is a critical issue, since the tool‐tip FRFs are influenced by the spindle speed due to gyroscopic moments, centrifugal forces and temperature, affecting the spindle bearings stiffness and the spindle‐holder‐tool dynamic behavior [5]. Using the stationary FRFs for predicting the SLD at high speed could return significant errors: the discrepancies in natural frequency of dominant modes will reflect in an inaccurate prediction of lobes positioning, the differences in modal damping will instead affect depth of cut values [3]. In the last years, extensive investigations have been carried out to identify the speed‐ dependent machine tool dynamics, both through models and experimental techniques. Gagnol et al. [6] developed an analytical dynamic model of a high‐speed spindle, Cao et al. [7] presented an alternative approach based on FE model of the spindle; both models take into account gyroscopic moments and centrifugal forces, and highlight the bearing stiffness dependence on spindle speed [8]. Li et al. [9] implemented an integrated model considering both thermal and mechanical effects; they found out that the bearing stiffness generally decreases at high spindle speeds. Although the predictive approaches provide promising results, a complex set of inputs is required (e.g., bearing preload, stiffness), limiting their application. In addition, these variables are subjected to uncertainty and their experimental identification could be challenging. For these reasons, several experimental techniques have been developed. Spindle dynamics was measured during rotation by impact hammer and Laser Doppler Vibrometer (LDV) [7] or capacitance probe [3]. Bediz et al. [10] developed a custom‐made impact excitation system and employed two fiber optic LDVs to automatically acquire speed‐varying tool‐tip FRFs in micro‐machining. This approach has some safety issues due to the impact on a rotating spindle, moreover the acquired measures are affected by relevant noise. Other authors [11– 13] developed excitation systems based on electromagnetic actuators. Application of such approaches is limited to research laboratories, due to the expensive equipment required and the restrictions on usable tooling geometries. In order to have a holistic view of the process dynamics, including both the machine tool and the process interaction, experimental techniques based on analysis during cutting operations have been developed. Some authors [14,15] applied the Operational Modal Analysis (OMA) principles to the cutting process, but the harmonic nature of the cutting forces still represents a challenge for its implementation. To overcome this issue, measurement techniques based on machining dedicated workpieces with specific shapes were investigated [16,17]. However, for these approaches, tool‐tip FRFs identification is limited to low spindle speeds due to force sensors bandwidth limitations and noisy results. Recently, Özşahin et al. [18] proposed an alternative approach, based on a similar idea already introduced by Suzuki et al. [19]: the tool‐tip FRFs are identified by matching chatter experimental results with analytical predicted ones. Natural frequency and damping of the dominant mode is determined by inverting the analytical chatter solution [20], starting from the experimental values of chatter frequency and depth of cut limit. This approach allows to indirectly reconstruct the FRFs at different spindle speeds without the need of expensive equipment or complex signal processing procedures. However, for each spindle speed several cutting tests are required to assess the chatter stability limits. As a result, the procedure could become time‐consuming when speed dependency of tool‐tip FRFs needs to be investigated, given that a large number of tests must be performed.
In this paper a test, the Spindle Speed Ramp‐up (SSR) [21], is implemented to efficiently extract chatter limits (i.e., chatter frequency and depth of cut limit), and a FRF identification technique is proposed based on the test results. The SSR has been chosen because it is able to return several chatter conditions at a specific depth of cut quickly and using simple and low‐ cost sensor (e.g., microphone). Then, tool‐tip FRFs are calculated in the entire spindle speed range by comparing predicted chatter limits and experimental results. The previous approach [18] relies on single chatter conditions to assess speed‐dependent FRFs at the specific spindle speed, implying high sensitivity to noise and errors of experimental values acquired, as discussed in the paper. A method to identify the speed‐varying tool‐tip FRFs in a more robust way is proposed here, based on the outcomes of the implemented test. 2. PROPOSED TEST The theory behind SSR is presented in detail in [21]. Basically, in the test the spindle speed is continuously increased, matching the feed rate in order to keep the feed per tooth constant. The main idea of the test is to investigate the SLD horizontally (i.e., fixed depth of cut, varying spindle speed) by changing the spindle speed in the entire range of interest during a single test. The sensor signals are acquired and analyzed in the frequency domain in order to investigate the presence of the characteristic frequency of the phenomenon (i.e., chatter frequency). This allows to distinguish between stable and unstable spindle speeds at the specific depth of cut. The time‐frequency domain analysis required by the method is achieved by means of the Order Analysis technique. The main output of this analysis is a waterfall plot or a colormap plot, that shows the frequency content of the signal changing with the spindle speed. Focusing on the chatter frequency detected in these spectra, stable and unstable cutting parameters are extracted. An example of this procedure is shown in Figure 1. Repeating the test a few times at different depths of cut allows to experimentally reconstruct the SLD in the entire spindle speed range. The test is simple, can be programmed in NC code and is very fast (e.g., a SSR test on aluminum investigating from 2,000 to 28,000 rpm lasted less than 3 seconds [21]). In this paper, the SSR test outcomes (i.e., the chatter limits) are used to calculate the speed‐varying FRF, as presented in the following sections. 3. PROPOSED IDENTIFICATION METHOD In this study, the tool‐tip FRFs under operational condition are calculated using: Experimental values of axial depth of cut and chatter frequency obtained by the SSR tests; Analytical formulations of axial depth of cut and chatter frequency. According to Budak and Altintas [20], these two parameters can be predicted analytically as in Eqs (1) and (2): 2 Λ 1 (1) 2
2 60
where: Λ Λ
(2)
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Λ
Λ
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Φ Φ Φ
Φ
4
(4) (5)
2
2
2
2
2
2
2
2
2
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2
2
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where N is the number of flutes, Kt is the tangential cutting force coefficient, Kr is the ratio of radial and tangential cutting force coefficients, st and ex are the start and exit angle of the cutting respectively (using coordinate system as in [20]), n is the spindle speed, k is the integer number of full vibration waves (i.e., lobes), and xx and yy are the tool‐tip FRFs that can be expressed as a function of modal parameters. In this paper, complex eigenvalues generic formulation, adopted also in [22], is used: ∗ 〈 〉 〈 〉 Φ , ∗ (7) ,
∗
1
where n is the number of modes; H is complex conjugate transpose of a matrix; {i} are the mode shapes; are the modal participation factors; [LR] is the lower residual; [UR] is the upper residual; i is the damping ratio, i the natural frequency of ith mode. According to the theory presented, alim and c can be used to identify the modal parameters of the tool‐tip FRF (i, i). At a given spindle speed, these parameters are calculated by matching the analytical axial depth of cut and the chatter frequency with the experimental ones. As in [18], some assumptions are considered: A specific chatter condition is caused by a single dominant mode, therefore the proposed method is applied to identify dominant modes causing chatter. Mode shapes {i} and participation factors could be considered to be independent over spindle speed, so they are assumed as constants (identified from the stationary condition). According to these considerations, only the damping ratio and the natural frequencies of the dominant mode are the unknowns. In [18], these unknowns are solved by the equivalence between the analytical and experimental axial depths of cut and the chatter frequencies of two close spindle speeds. This would allow to extract an exact solution. However, considering that experimental values are influenced by noise and chatter detection errors, the accuracy of this method could be reduced. In this work, starting from the SSR results, a high number of chatter limiting conditions can be extracted in the entire spindle speed range, so a more robust approach can be pursued. In particular, the identification of the damping ratio and the natural frequency of the dominant mode is calculated on several chatter limit conditions. A Genetic Algorithm (GA), as proposed in [4], is used for the purpose. Modal parameters are calculated by minimizing the objective function fo, computed as:
‖ ‖
(8)
‖ ‖
where f1 and f2 are error vectors on the depth of cut limit and the chatter frequency respectively, j is the jth chatter limiting condition considered, PRE stands for analytical predicted value, EXP for experimental obtained value, || || is the 2‐norm of the vector. The identification procedure can be summarized in the following steps: The tool‐tip FRFs in the idle condition is identified via experimental modal analysis, and the modes affecting the stability are extracted. The natural frequency and damping ratio of the defined modes are considered as unknowns. The chatter frequencies and the depths of cut limit are experimentally identified with few SSR tests in the spindle speed range of interest. The differences between the analytical and experimental chatter frequencies and axial depths of cut are minimized by the optimization algorithm (GA) for the limiting conditions considered, to compute the modal parameters. By repeating this procedure for a series of limiting conditions in different spindle speed ranges, the speed‐varying modal parameters are extracted. The speed‐varying tool‐tip FRF is reconstructed and the SLD recalculated to check the accuracy of the proposed approach. 4. EXPERIMENTAL VALIDATION In order to validate the accuracy of the proposed method, some experimental tests were performed using a NMV 1500 DCG Mori Seiki 5 axis milling machine. A series of SSR tests at different depths of cut in slotting operation (i.e., full radial immersion) was performed. An aluminum 6082‐T4 alloy test case was machined with a two flutes end mill (8 mm diameter) Garant 201770. The machine was equipped with a microphone (Bruel & Kjaer type 4165) installed inside the cutting chamber close to the cutting zone. The signals were acquired by LMS Scadas III and elaborated in LMS Test.Lab software. 4.1. Stationary FRFs Before performing the SSR tests, impact tests (Brüel & Kjaer Type 8202 impulse hammer and a PCB 352C22 accelerometer) were carried out to compute the tool‐tip FRF in stationary condition and extract the modal parameters. In Figure 2 the experimental tool‐tip FRFs in x and y directions are presented and, for the proposed machine setup, these could be considered symmetric. Henceforth, the tool‐tip FRFs in x and y directions will be considered equal. Once the tool‐tip FRFs are determined, the modal parameters and the mode shapes were identified with the Polymax [22] estimator, adopted also in [16,23]. The modal parameters are presented in Table 1. Modal extraction was performed based on the FRFs formulation using complex eigenvalues (Eq. 7), therefore the multiplications of mode shapes and modal participation factors are reported in the table, instead of modal stiffness. The computed FRFs are presented in Figure 2 to show the identification accuracy in reconstructing the experimental FRFs. 4.2 Spindle Speed Ramp‐up
The proposed test was carried out to identify the chatter limit conditions to be used in the FRFs identification procedure. In particular, six slotting tests were performed at different depths of cut (1.0, 1.5, 2.0, 2.5, 3.0, 3.5 mm) from 13,000 rpm to 30,000 rpm and 0.03 mm/tooth feed. Microphone signals were acquired and analyzed by means of Order Analysis. The results of the analysis are presented in Figure 3. The chatter frequency around the 5,300 Hz dominant mode was identified, and used to extract stable and unstable zones (green and red zones respectively). By performing only six tests (less than 30 seconds cutting) 34 limiting conditions were calculated (Table 2). In addition, in Figure 3, the experimental results are compared with the analytical SLD calculated via the stationary FRFs and the cutting force coefficients identified in [24]. As shown in the figure, the SLD calculated via the idle FRFs is accurate only at the lower speeds investigated (around 14,000 rpm), then increased discrepancies are detected both in the lobe positioning and depth of cut values, these differences are due to the FRFs variation with spindle speed. 5. RESULTS AND DISCUSSION In this section, the proposed FRF identification procedure is applied to the experimental results obtained by the SSR. The dominant mode (i.e., around 5,300 Hz), responsible of chatter, is investigated. Since a high number of limiting conditions are obtained by the experimental procedure (Figure 3), a robust identification based on multiple conditions is proposed. In particular, the 34 limits were divided into seven groups, selected on the basis of lobes distributions. Indeed, in the spindle speed range considered (13,000‐30,000) seven lobes are identified, for each of them limiting conditions are clustered. The identification algorithm is then performed for the different groups: seven tool‐tip FRFs are hence computed. For each lobe, the GA is used to find the best modal parameters able to minimize the objective function previously presented (Eq. 8). A population of 600 individuals and 10 generations were set for the algorithm. 5.1. Speed‐varying tool‐tip FRFs The speed‐varying tool‐tip FRFs, obtained by the proposed method, based on multiple conditions being part of the different lobes, are shown in Figure 4. The tool‐tip FRF deviates from the idle condition; in particular, increasing spindle speed, the natural frequency decreases and the damping increases. This behavior is in accordance with the experimental tests carried out to acquire the rotating tool‐tip FRFs, as presented in several works [6,7,18]. This trend is due to the gyroscopic moments and centrifugal forces that cause a decrease in bearing stiffness and an increase of damping [25]. Therefore, the computed speed‐varying FRF is in line with the physical model of spindle rotordynamics. In order to check the accuracy of tool‐tip identification on chatter prediction, the SLD is calculated based on the speed‐varying FRFs and compared with the limiting conditions extracted by the SSR tests, as shown in Figure 5.
The results show a good agreement between the reconstructed SLD and the limiting conditions both in terms of depths of cut and chatter frequencies values: the errors between analytical and experimental results, calculated as fo in Eq. 8, are less than 0.2% for all the lobes, as reported in Table 3. In the same table and in Figure 6, the calculated modal parameters changing with spindle speed are presented. Both the natural frequency and the damping ratio are changing linearly with the spindle speed. Only the modal parameters of the second lobe (15,353 rpm) are diverging from this trend. This is probably due to the low number of conditions used to identify the parameters for the lobe (only 2). The natural frequency decreases slowly from the idle condition of 5,375 Hz to about 5,178 Hz at the highest spindle speed tested, reaching a 4% reduction. This small decrease is enough to return a significant difference in the lobe positioning, as shown in Figure 5. On the other hand, the damping ratio is more sensible to spindle speed, growing from around 1.2% till nearly 4% with a 212% increase at 28,065 rpm. The proposed method based on multiple conditions provides results in accordance with simulative models [6], and experimental tests [7]. Furthermore, the proposed method was compared to the state of the art approach [18]. As discussed earlier, Özşahin et al. approach [18] relies on single chatter conditions to extract the exact solution for the tool‐tip FRFs causing the experimental chatter frequency and depth of cut values. Therefore, the proposed method was performed on the single chatter conditions (34 in total), and the results were compared with the ones obtained implementing the technique proposed by Özşahin et al. In Table 4, Table 5 and Figure 7 natural frequencies and damping values trends are presented. The results obtained by the two methods are very similar: the natural frequency and damping ratio show the same trend, changing with spindle speed. The differences in the natural frequency values are very small (less than 0.5%) while the differences in damping ratio reach higher values. This validation proves the equivalence between the results achieved with the proposed approach and the state of the art techniques, on the single conditions. Looking at the trends, relying only on the single conditions, as proposed by Özşahin et al., returns noisier results compared with proposed multi‐conditions approach. Although both the single and multi‐conditions methods result in similar overall trends of modal parameters, the multiple approach is more in line with the linear behavior of the physical models. The noisy results are due to the unavoidable inaccuracies in detecting chatter. Obviously, as stated in [18], these inaccuracies could be reduced if the chatter tests are repeated multiple times, excluding the ones for which the pattern is not in line with stability theory. However, this would imply an increased number of tests required and would need operator adjustments, making this approach less attractive. The SSR test is able to efficiently produce a high number of chatter limit conditions (the 34 conditions in the test case were identified in less than 30 seconds cutting), allowing the use of a more robust multi‐conditions identification technique, as the one here proposed. 5.2. Speed‐varying SLD experimental validation In order to further assess the accuracy of the proposed approach, the computed speed‐ varying SLD (Figure 5) is experimentally verified by chatter tests with constant parameters (single points in the diagram). Five different spindle speeds were investigated over two lobes (21,200, 21,800, 22,400, 23,000, 24,000 rpm). For each spindle speed the depth of cut was scanned every 0.5 mm till identifying chatter with at least two tests. The occurrence of chatter vibrations was assessed based on the frequency signals of the microphone and checking
chatter marks on the surface. The results achieved with constant parameters are compared with both speed‐varying and idle SLDs in Figure 8. Figure 8 shows how the SLD based on the speed‐varying tool‐tip FRFs identified by the proposed method is able to predict chatter occurrence in line with experimental tests. The maximum achieved experimental depth of cut was found at 22,400 rpm with chatter occurring at 5 mm, the maximum peak for the predicted lobes is at 22,380 rpm and 4.9 mm depth of cut limit, resulting in an error lower than 2%. On the contrary, the SLD based on idle FRFs fails in this prediction both in terms of lobe positioning and depth of cut values (at 22,400 the predicted depth of cut is about 2 mm against the 5 mm experimental value). This validation further proves the need of the identification of the tool‐tip FRF under operational conditions to accurately predict chatter stability at high speed. Among the dedicated identification methods presented in scientific literature, the proposed technique stands out for its simplicity, efficiency and low‐cost implementation. 6. CONCLUSION In this paper, a novel method to identify the tool‐tip FRFs under operational conditions is presented. A fast and efficient chatter identification test is combined with analytical stability solution to forecast the in‐process FRFs. The Spindle Speed Ramp‐up test is able to return a high number of experimental chatter frequencies and axial depths of cut with few cutting operations. By minimizing the differences between the experimental and analytically predicted limit conditions, the machine tool dynamics changing with spindle speed is calculated via an approach relying on multiple conditions. The proposed method was validated in milling operations, the tool‐tip FRFs varying with spindle speed are calculated without the need of expensive equipment, using a simple and low‐cost sensor and very few cutting tests (testing time less than 30 seconds). The proposed approach outcomes were proven to be in line with the expected linear trends found in the scientific literature. The speed‐varying SLD reconstructed by the proposed method was proven to be accurate in predicting chatter‐free cutting parameters. The technique still requires stationary tool‐tip FRF acquired via experimental modal analysis, however this could be replaced with FRF predicted by hybrid technique, such as Receptance Coupling Substructuring Analysis (RCSA) [26][27], even considering that high accuracy in the idle FRF is not required. The speed‐varying FRF computed by the proposed method would allow an accurate SLD prediction. In addition, it could be useful to investigate the influence of spindle speed on spindle dynamics and validate models. Moreover, this approach could be combined with RCSA to predict speed‐varying FRF of different tools, starting from few cutting tests with a specific tool. ACKNOWLEDGMENT The authors would like to thank the DMG Mori Seiki Co. and the Machine Tool Technology Research Foundation (MTTRF) for the loaned machine tool (Mori Seiki NMV1500DCG). REFERENCES [1] [2] [3] [4] [5]
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Figure 1 SSR colormap and stable and unstable zones. 1.6
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Figure 3 SSR results compared to analytical SLD
Figure 4 Speed‐varying FRF calculated by proposed method
Figure 5 SLD reconstructed with speed‐varying FRF
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0 20000 20500 21000 21500 22000 22500 23000 23500 24000 24500 25000
Spindle Speed [rpm] Figure 8 Experimental validation of speed‐varying SLD
Table 1 Modal parameters for stationary conditions Mode Natural frequency i [Hz] Damping ratio i [%] Mode shapes X participation factor 〈 〉[mm]
1st
2nd
3rd
4th
5th
6th
7th
1078.6
2103.3
2392.3
3604.6
5374.7
7064.9
7422.1
4.9
6.1
2.7
2.8
1.2
3.1
2.3
0.011 ‐ 0.072i
0.077 ‐ 0.265i
0.025 ‐ 0.037i
‐0.022 ‐ 0.076i
‐0.047 ‐ 0.627i
‐0.048 ‐ 0.025i
0.043 ‐ 0.247i
Table 2 Limiting conditions extracted via SSR
1
Spindle speed (rpm)
11
3
4
5
6
7
8
9
13,540 13,601 13,951 14,336 14,964 15,532 16,582 17,145 17,335
Limiting depth of cut (mm) Chatter frequency (Hz) 10
2
12
13
1.5
1.0
1.0
1.5
1.5
1.5
1.5
1.5
2.0
5,270
5,260
5,270
5,260
5,280
5,310
5,280
5,300
5,240
14
15
16
17
18
19
20
21
22
17,550 17,837 18,153 19,236 19,349 19,555 19,566 19,859 19,991 20,371 21,785 21,966 21,960 2.5
2.5
2.0
2.0
2.5
3.0
3.5
3.5
3.0
2.5
2.5
3.0
3.5
5,270
5,200
5,240
5,270
5,220
5,250
5,230
5,160
5,170
5,200
5,220
5,250
5,270
23
24
25
26
27
28
29
30
31
32
33
34
22,918 23,001 23,607 24,768 25,025 25,170 27,032 27,442 27,572 28,366 28,717 29,100
3.5
3.0
2.5
2.5
3.0
3.5
3.5
3.0
2.5
2.5
3.0
3.5
5,140
5,160
5,190
5,250
5,220
5,250
5,120
5,140
5,120
5,200
5,170
5,200
Table 3 Modal parameters and experimental values fitting for multiple conditions
Spindle speed Natural frequency i [Hz] Damping ratio i [%] Fitting error fo [%]
Idle condition
1st lobe 2nd lobe 3rd lobe 13,825 15,353 16,875 rpm rpm rpm
4th lobe 18,703 rpm
5th lobe 21,077 rpm
6th lobe 24,271 rpm
7th lobe 28,065 rpm
5374.7
5331.2
5340.8
5325. 4
5277.9
5256.7
5242.3
5178.1
1.2
1.3
1.8
1.8
2.2
2.8
3.3
3.9
‐
0.12
0.02
0.12
0.17
0.11
0.08
0.07
Table 4 Natural frequency computed for single conditions Natural frequency i [Hz] Proposed method
Spindle speed [rpm] 13,540 13,601 13,951 14,336 14,964 15,532 16,582 17,145 17,335 5300.4 5268.8 5248.7 5220.5 5298.8 5255. 3 5284.1 5257.0 5248.7
Ozahin et al. method
5315.3 5278.3 5236.6 5204.0 5301.7 5257.9 5290.8 5248.0 5244.0 0.28 0.18 ‐0.23 ‐0.32 0.056 0.049 0.13 ‐0.17 ‐0.09
Difference [%]
17,550 17,837 18,153 19,236 19,349 19,555 19,566 19,859 19,991 20,371 21,785 21,966 21,960 5268.8 5284.1 5284.1 5215.6 5188.9 5232.3 5234.8 5286.5 5277.9 5262.8 5138.7 5148.9 5149.9 5268.0 5289.5 5293.1 5203.0 5172.2 5215.1 5218.0 5289.1 5280.6 5266.3 5139.4 5157.6 5160.9 ‐0.01 0.10 0.17 ‐0.24 ‐0.32 ‐0.33 ‐0.32 0.05 0.05 0.07 0.01 0.17 0.21 22,918 23,001 23,607 24,768 25,025 25,170 27,032 27,442 27,572 28,366 28,717 29,100 5256.7 5264.6 5209.1 5174.8 5113.3 5120.4 5208.6 5178.1 5123.6 5178.1 5091.3 5087.4
5258.8 5265.5 5220.4 5169.8 5112.7 5126.8 5215.5 5187.8 5135.6 5174.8 5085.0 5083.9 0.04
0.02
0.22
‐0.10
‐0.01
0.12
0.13
0.19
0.23
‐0.06
‐0.13
‐0.07
Table 5 Damping ratio computed for single conditions Damping ratio i [%] Proposed method
Spindle speed [rpm] 13,540 13,601 13,951 14,336 14,964 15,532 16,582 17,145 17,335
Ozahin et al. method
1.85
1.36
1.13
0.96
1.99
1.12
2.02
1.29
1.03
1.63
1.30
0.98
0.92
1.95
0.98
2.01
1.08
0.96
‐4.56 ‐15.45
‐4.31
‐0.31 ‐18.81
‐7.15
‐13.41
Difference [%]
‐1.85 ‐14.47
17,550 17,837 18,153 19,236 19,349 19,555 19,566 19,859 19,991 20,371 21,785 21,966 21,960 0.88 2.35 2.46 1.39 0.96 1.12 1.68 2.58 2.64 3.08 1.17 0.95 1.52 0.89
2.04
0.88 ‐14.93
2.31
1.00
0.93
‐6.38 ‐38.98
‐3.72
1.02
1.40
2.15
2.36
3.04
1.08
1.02
1.46
‐9.83 ‐19.69 ‐19.78 ‐11.48
‐1.21
‐7.83
6.58
‐3.95
22,918 23,001 23,607 24,768 25,025 25,170 27,032 27,442 27,572 28,366 28,717 29,100 3.65 3.01 3.47 2.69 2.07 2.21 4.42 4.18 3.61 3.51 3.60 3.62
3.47
2.90
3.52
2.65
1.99
2.15
4.20
4.15
3.59
3.54
3.56
3.55
‐5.28
‐3.81
1.38
‐1.30
‐4.02
‐2.91
‐5.23
‐0.66
‐0.38
0.704
‐1.30
‐2.13
S0141-6359(16)30375-0 http://dx.doi.org/doi:10.1016/j.precisioneng.2016.11.011 PRE 6487
To appear in:
Precision Engineering
Received date: Revised date: Accepted date:
18-5-2016 25-8-2016 17-11-2016
Please cite this article as: Grossi N, Sallese L, Scippa A, Campatelli G.Improved experimental-analytical approach to compute speed-varying tool-tip FRF.Precision Engineering http://dx.doi.org/10.1016/j.precisioneng.2016.11.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Improved experimental‐analytical approach to compute speed‐varying tool‐tip FRF N. Grossia,*, L. Sallesea, A. Scippaa, G. Campatellia aDepartment of Industrial Engineering, University of Firenze, via di Santa Marta 3, 50139, Firenze, Italy. *corresponding author. Tel. +390552758726. E‐mail: [email protected] HIGHLIGTHS An efficient method to identify the speed‐varying tool‐tip FRFs is proposed A quick cutting test is combined with the inversion of chatter analytical solution Experimental chatter limits are obtained by a dedicated test, called SSR By fitting experimental and predicted conditions, speed‐varying FRFs are computed Speed‐varying SLD reconstructed with the computed FRFs is experimentally validated Abstract Chatter stability prediction is crucial to improve the performances of modern milling process, and it gets even more important at high speeds, for which very productive cutting parameters can be achieved if the suitable spindle speed is selected. Unfortunately, the available chatter predictive models suffer from reduced accuracy at high speed due to inaccuracies in the input data, especially the machine tool dynamics that is acquired in stationary configurations but could sensibly change with spindle speed. In this paper, an efficient method to identify the speed‐varying Frequency Response Functions (FRFs) under operational conditions is presented. The proposed approach is based on the definition of some experimental chatter limits (i.e., chatter frequency and related depth of cut), obtained by a dedicated test, called Spindle Speed Ramp‐up. The experimental results are then combined with the analytical stability solution. By minimizing the differences between the experimental and predicted chatter conditions, a dedicated algorithm computes the speed‐varying FRFs. Few tests and simple equipment (i.e., microphone) are enough to calculate the FRFs in a wide range of spindle speeds. The proposed technique was validated in real machining applications, the identified tool‐tip FRFs are in accordance with expected trend reported in scientific literature. Speed‐varying stability lobe diagram reconstructed with the computed FRFs is proven to be accurate in predicting stable cutting parameters. Keywords: Chatter, Milling, Dynamics, Spindle, In‐process dynamics identification 1. INTRODUCTION The milling process could be significantly affected by detrimental vibrations, that represent one of the main limitations to further increase the process performance. The unstable vibration, known as chatter, is the most dangerous phenomenon because it could grow uncontrollably, causing poor surface finish, excessive tool wear and possible tool breakage, hence affecting productivity [1]. In the last decades many chatter prediction methods have been developed [2]; the main result of these approaches is a chart showing chatter‐free combinations of cutting parameters, known as Stability Lobe Diagram (SLD). These diagrams are very useful, especially at high speeds where lobes are spaced and higher depth of cut can be safely exploited if the appropriate spindle speed is selected. However, at high spindle speeds every predictive model shows reduced accuracy, generally not caused by the approximation introduced in the
analytical chatter model, but due to the inputs inaccuracies instead [3]. Indeed, the input data, such as cutting force coefficients and machine tool dynamics, may sensibly vary with spindle speed, even though commonly obtained at low speed, or in idle condition, and used in the entire spindle speed range [4]. In particular, the system dynamics (i.e., tool‐tip Frequency Response Function (FRF)), essential to forecast the SLD, is generally acquired via experimental modal analysis in the stationary condition (i.e., without spindle rotating). This is a critical issue, since the tool‐tip FRFs are influenced by the spindle speed due to gyroscopic moments, centrifugal forces and temperature, affecting the spindle bearings stiffness and the spindle‐holder‐tool dynamic behavior [5]. Using the stationary FRFs for predicting the SLD at high speed could return significant errors: the discrepancies in natural frequency of dominant modes will reflect in an inaccurate prediction of lobes positioning, the differences in modal damping will instead affect depth of cut values [3]. In the last years, extensive investigations have been carried out to identify the speed‐ dependent machine tool dynamics, both through models and experimental techniques. Gagnol et al. [6] developed an analytical dynamic model of a high‐speed spindle, Cao et al. [7] presented an alternative approach based on FE model of the spindle; both models take into account gyroscopic moments and centrifugal forces, and highlight the bearing stiffness dependence on spindle speed [8]. Li et al. [9] implemented an integrated model considering both thermal and mechanical effects; they found out that the bearing stiffness generally decreases at high spindle speeds. Although the predictive approaches provide promising results, a complex set of inputs is required (e.g., bearing preload, stiffness), limiting their application. In addition, these variables are subjected to uncertainty and their experimental identification could be challenging. For these reasons, several experimental techniques have been developed. Spindle dynamics was measured during rotation by impact hammer and Laser Doppler Vibrometer (LDV) [7] or capacitance probe [3]. Bediz et al. [10] developed a custom‐made impact excitation system and employed two fiber optic LDVs to automatically acquire speed‐varying tool‐tip FRFs in micro‐machining. This approach has some safety issues due to the impact on a rotating spindle, moreover the acquired measures are affected by relevant noise. Other authors [11– 13] developed excitation systems based on electromagnetic actuators. Application of such approaches is limited to research laboratories, due to the expensive equipment required and the restrictions on usable tooling geometries. In order to have a holistic view of the process dynamics, including both the machine tool and the process interaction, experimental techniques based on analysis during cutting operations have been developed. Some authors [14,15] applied the Operational Modal Analysis (OMA) principles to the cutting process, but the harmonic nature of the cutting forces still represents a challenge for its implementation. To overcome this issue, measurement techniques based on machining dedicated workpieces with specific shapes were investigated [16,17]. However, for these approaches, tool‐tip FRFs identification is limited to low spindle speeds due to force sensors bandwidth limitations and noisy results. Recently, Özşahin et al. [18] proposed an alternative approach, based on a similar idea already introduced by Suzuki et al. [19]: the tool‐tip FRFs are identified by matching chatter experimental results with analytical predicted ones. Natural frequency and damping of the dominant mode is determined by inverting the analytical chatter solution [20], starting from the experimental values of chatter frequency and depth of cut limit. This approach allows to indirectly reconstruct the FRFs at different spindle speeds without the need of expensive equipment or complex signal processing procedures. However, for each spindle speed several cutting tests are required to assess the chatter stability limits. As a result, the procedure could become time‐consuming when speed dependency of tool‐tip FRFs needs to be investigated, given that a large number of tests must be performed.
In this paper a test, the Spindle Speed Ramp‐up (SSR) [21], is implemented to efficiently extract chatter limits (i.e., chatter frequency and depth of cut limit), and a FRF identification technique is proposed based on the test results. The SSR has been chosen because it is able to return several chatter conditions at a specific depth of cut quickly and using simple and low‐ cost sensor (e.g., microphone). Then, tool‐tip FRFs are calculated in the entire spindle speed range by comparing predicted chatter limits and experimental results. The previous approach [18] relies on single chatter conditions to assess speed‐dependent FRFs at the specific spindle speed, implying high sensitivity to noise and errors of experimental values acquired, as discussed in the paper. A method to identify the speed‐varying tool‐tip FRFs in a more robust way is proposed here, based on the outcomes of the implemented test. 2. PROPOSED TEST The theory behind SSR is presented in detail in [21]. Basically, in the test the spindle speed is continuously increased, matching the feed rate in order to keep the feed per tooth constant. The main idea of the test is to investigate the SLD horizontally (i.e., fixed depth of cut, varying spindle speed) by changing the spindle speed in the entire range of interest during a single test. The sensor signals are acquired and analyzed in the frequency domain in order to investigate the presence of the characteristic frequency of the phenomenon (i.e., chatter frequency). This allows to distinguish between stable and unstable spindle speeds at the specific depth of cut. The time‐frequency domain analysis required by the method is achieved by means of the Order Analysis technique. The main output of this analysis is a waterfall plot or a colormap plot, that shows the frequency content of the signal changing with the spindle speed. Focusing on the chatter frequency detected in these spectra, stable and unstable cutting parameters are extracted. An example of this procedure is shown in Figure 1. Repeating the test a few times at different depths of cut allows to experimentally reconstruct the SLD in the entire spindle speed range. The test is simple, can be programmed in NC code and is very fast (e.g., a SSR test on aluminum investigating from 2,000 to 28,000 rpm lasted less than 3 seconds [21]). In this paper, the SSR test outcomes (i.e., the chatter limits) are used to calculate the speed‐varying FRF, as presented in the following sections. 3. PROPOSED IDENTIFICATION METHOD In this study, the tool‐tip FRFs under operational condition are calculated using: Experimental values of axial depth of cut and chatter frequency obtained by the SSR tests; Analytical formulations of axial depth of cut and chatter frequency. According to Budak and Altintas [20], these two parameters can be predicted analytically as in Eqs (1) and (2): 2 Λ 1 (1) 2
2 60
where: Λ Λ
(2)
(3)
Λ
Λ
1 2
Λ
Φ Φ Φ
Φ
4
(4) (5)
2
2
2
2
2
2
2
2
2
2
2
2
(6)
where N is the number of flutes, Kt is the tangential cutting force coefficient, Kr is the ratio of radial and tangential cutting force coefficients, st and ex are the start and exit angle of the cutting respectively (using coordinate system as in [20]), n is the spindle speed, k is the integer number of full vibration waves (i.e., lobes), and xx and yy are the tool‐tip FRFs that can be expressed as a function of modal parameters. In this paper, complex eigenvalues generic formulation, adopted also in [22], is used: ∗ 〈 〉 〈 〉 Φ , ∗ (7) ,
∗
1
where n is the number of modes; H is complex conjugate transpose of a matrix; {i} are the mode shapes;
‖ ‖
(8)
‖ ‖
where f1 and f2 are error vectors on the depth of cut limit and the chatter frequency respectively, j is the jth chatter limiting condition considered, PRE stands for analytical predicted value, EXP for experimental obtained value, || || is the 2‐norm of the vector. The identification procedure can be summarized in the following steps: The tool‐tip FRFs in the idle condition is identified via experimental modal analysis, and the modes affecting the stability are extracted. The natural frequency and damping ratio of the defined modes are considered as unknowns. The chatter frequencies and the depths of cut limit are experimentally identified with few SSR tests in the spindle speed range of interest. The differences between the analytical and experimental chatter frequencies and axial depths of cut are minimized by the optimization algorithm (GA) for the limiting conditions considered, to compute the modal parameters. By repeating this procedure for a series of limiting conditions in different spindle speed ranges, the speed‐varying modal parameters are extracted. The speed‐varying tool‐tip FRF is reconstructed and the SLD recalculated to check the accuracy of the proposed approach. 4. EXPERIMENTAL VALIDATION In order to validate the accuracy of the proposed method, some experimental tests were performed using a NMV 1500 DCG Mori Seiki 5 axis milling machine. A series of SSR tests at different depths of cut in slotting operation (i.e., full radial immersion) was performed. An aluminum 6082‐T4 alloy test case was machined with a two flutes end mill (8 mm diameter) Garant 201770. The machine was equipped with a microphone (Bruel & Kjaer type 4165) installed inside the cutting chamber close to the cutting zone. The signals were acquired by LMS Scadas III and elaborated in LMS Test.Lab software. 4.1. Stationary FRFs Before performing the SSR tests, impact tests (Brüel & Kjaer Type 8202 impulse hammer and a PCB 352C22 accelerometer) were carried out to compute the tool‐tip FRF in stationary condition and extract the modal parameters. In Figure 2 the experimental tool‐tip FRFs in x and y directions are presented and, for the proposed machine setup, these could be considered symmetric. Henceforth, the tool‐tip FRFs in x and y directions will be considered equal. Once the tool‐tip FRFs are determined, the modal parameters and the mode shapes were identified with the Polymax [22] estimator, adopted also in [16,23]. The modal parameters are presented in Table 1. Modal extraction was performed based on the FRFs formulation using complex eigenvalues (Eq. 7), therefore the multiplications of mode shapes and modal participation factors are reported in the table, instead of modal stiffness. The computed FRFs are presented in Figure 2 to show the identification accuracy in reconstructing the experimental FRFs. 4.2 Spindle Speed Ramp‐up
The proposed test was carried out to identify the chatter limit conditions to be used in the FRFs identification procedure. In particular, six slotting tests were performed at different depths of cut (1.0, 1.5, 2.0, 2.5, 3.0, 3.5 mm) from 13,000 rpm to 30,000 rpm and 0.03 mm/tooth feed. Microphone signals were acquired and analyzed by means of Order Analysis. The results of the analysis are presented in Figure 3. The chatter frequency around the 5,300 Hz dominant mode was identified, and used to extract stable and unstable zones (green and red zones respectively). By performing only six tests (less than 30 seconds cutting) 34 limiting conditions were calculated (Table 2). In addition, in Figure 3, the experimental results are compared with the analytical SLD calculated via the stationary FRFs and the cutting force coefficients identified in [24]. As shown in the figure, the SLD calculated via the idle FRFs is accurate only at the lower speeds investigated (around 14,000 rpm), then increased discrepancies are detected both in the lobe positioning and depth of cut values, these differences are due to the FRFs variation with spindle speed. 5. RESULTS AND DISCUSSION In this section, the proposed FRF identification procedure is applied to the experimental results obtained by the SSR. The dominant mode (i.e., around 5,300 Hz), responsible of chatter, is investigated. Since a high number of limiting conditions are obtained by the experimental procedure (Figure 3), a robust identification based on multiple conditions is proposed. In particular, the 34 limits were divided into seven groups, selected on the basis of lobes distributions. Indeed, in the spindle speed range considered (13,000‐30,000) seven lobes are identified, for each of them limiting conditions are clustered. The identification algorithm is then performed for the different groups: seven tool‐tip FRFs are hence computed. For each lobe, the GA is used to find the best modal parameters able to minimize the objective function previously presented (Eq. 8). A population of 600 individuals and 10 generations were set for the algorithm. 5.1. Speed‐varying tool‐tip FRFs The speed‐varying tool‐tip FRFs, obtained by the proposed method, based on multiple conditions being part of the different lobes, are shown in Figure 4. The tool‐tip FRF deviates from the idle condition; in particular, increasing spindle speed, the natural frequency decreases and the damping increases. This behavior is in accordance with the experimental tests carried out to acquire the rotating tool‐tip FRFs, as presented in several works [6,7,18]. This trend is due to the gyroscopic moments and centrifugal forces that cause a decrease in bearing stiffness and an increase of damping [25]. Therefore, the computed speed‐varying FRF is in line with the physical model of spindle rotordynamics. In order to check the accuracy of tool‐tip identification on chatter prediction, the SLD is calculated based on the speed‐varying FRFs and compared with the limiting conditions extracted by the SSR tests, as shown in Figure 5.
The results show a good agreement between the reconstructed SLD and the limiting conditions both in terms of depths of cut and chatter frequencies values: the errors between analytical and experimental results, calculated as fo in Eq. 8, are less than 0.2% for all the lobes, as reported in Table 3. In the same table and in Figure 6, the calculated modal parameters changing with spindle speed are presented. Both the natural frequency and the damping ratio are changing linearly with the spindle speed. Only the modal parameters of the second lobe (15,353 rpm) are diverging from this trend. This is probably due to the low number of conditions used to identify the parameters for the lobe (only 2). The natural frequency decreases slowly from the idle condition of 5,375 Hz to about 5,178 Hz at the highest spindle speed tested, reaching a 4% reduction. This small decrease is enough to return a significant difference in the lobe positioning, as shown in Figure 5. On the other hand, the damping ratio is more sensible to spindle speed, growing from around 1.2% till nearly 4% with a 212% increase at 28,065 rpm. The proposed method based on multiple conditions provides results in accordance with simulative models [6], and experimental tests [7]. Furthermore, the proposed method was compared to the state of the art approach [18]. As discussed earlier, Özşahin et al. approach [18] relies on single chatter conditions to extract the exact solution for the tool‐tip FRFs causing the experimental chatter frequency and depth of cut values. Therefore, the proposed method was performed on the single chatter conditions (34 in total), and the results were compared with the ones obtained implementing the technique proposed by Özşahin et al. In Table 4, Table 5 and Figure 7 natural frequencies and damping values trends are presented. The results obtained by the two methods are very similar: the natural frequency and damping ratio show the same trend, changing with spindle speed. The differences in the natural frequency values are very small (less than 0.5%) while the differences in damping ratio reach higher values. This validation proves the equivalence between the results achieved with the proposed approach and the state of the art techniques, on the single conditions. Looking at the trends, relying only on the single conditions, as proposed by Özşahin et al., returns noisier results compared with proposed multi‐conditions approach. Although both the single and multi‐conditions methods result in similar overall trends of modal parameters, the multiple approach is more in line with the linear behavior of the physical models. The noisy results are due to the unavoidable inaccuracies in detecting chatter. Obviously, as stated in [18], these inaccuracies could be reduced if the chatter tests are repeated multiple times, excluding the ones for which the pattern is not in line with stability theory. However, this would imply an increased number of tests required and would need operator adjustments, making this approach less attractive. The SSR test is able to efficiently produce a high number of chatter limit conditions (the 34 conditions in the test case were identified in less than 30 seconds cutting), allowing the use of a more robust multi‐conditions identification technique, as the one here proposed. 5.2. Speed‐varying SLD experimental validation In order to further assess the accuracy of the proposed approach, the computed speed‐ varying SLD (Figure 5) is experimentally verified by chatter tests with constant parameters (single points in the diagram). Five different spindle speeds were investigated over two lobes (21,200, 21,800, 22,400, 23,000, 24,000 rpm). For each spindle speed the depth of cut was scanned every 0.5 mm till identifying chatter with at least two tests. The occurrence of chatter vibrations was assessed based on the frequency signals of the microphone and checking
chatter marks on the surface. The results achieved with constant parameters are compared with both speed‐varying and idle SLDs in Figure 8. Figure 8 shows how the SLD based on the speed‐varying tool‐tip FRFs identified by the proposed method is able to predict chatter occurrence in line with experimental tests. The maximum achieved experimental depth of cut was found at 22,400 rpm with chatter occurring at 5 mm, the maximum peak for the predicted lobes is at 22,380 rpm and 4.9 mm depth of cut limit, resulting in an error lower than 2%. On the contrary, the SLD based on idle FRFs fails in this prediction both in terms of lobe positioning and depth of cut values (at 22,400 the predicted depth of cut is about 2 mm against the 5 mm experimental value). This validation further proves the need of the identification of the tool‐tip FRF under operational conditions to accurately predict chatter stability at high speed. Among the dedicated identification methods presented in scientific literature, the proposed technique stands out for its simplicity, efficiency and low‐cost implementation. 6. CONCLUSION In this paper, a novel method to identify the tool‐tip FRFs under operational conditions is presented. A fast and efficient chatter identification test is combined with analytical stability solution to forecast the in‐process FRFs. The Spindle Speed Ramp‐up test is able to return a high number of experimental chatter frequencies and axial depths of cut with few cutting operations. By minimizing the differences between the experimental and analytically predicted limit conditions, the machine tool dynamics changing with spindle speed is calculated via an approach relying on multiple conditions. The proposed method was validated in milling operations, the tool‐tip FRFs varying with spindle speed are calculated without the need of expensive equipment, using a simple and low‐cost sensor and very few cutting tests (testing time less than 30 seconds). The proposed approach outcomes were proven to be in line with the expected linear trends found in the scientific literature. The speed‐varying SLD reconstructed by the proposed method was proven to be accurate in predicting chatter‐free cutting parameters. The technique still requires stationary tool‐tip FRF acquired via experimental modal analysis, however this could be replaced with FRF predicted by hybrid technique, such as Receptance Coupling Substructuring Analysis (RCSA) [26][27], even considering that high accuracy in the idle FRF is not required. The speed‐varying FRF computed by the proposed method would allow an accurate SLD prediction. In addition, it could be useful to investigate the influence of spindle speed on spindle dynamics and validate models. Moreover, this approach could be combined with RCSA to predict speed‐varying FRF of different tools, starting from few cutting tests with a specific tool. ACKNOWLEDGMENT The authors would like to thank the DMG Mori Seiki Co. and the Machine Tool Technology Research Foundation (MTTRF) for the loaned machine tool (Mori Seiki NMV1500DCG). REFERENCES [1] [2] [3] [4] [5]
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Figure 1 SSR colormap and stable and unstable zones. 1.6
# 10 -3
Amplitude [mm/N]
1.4 1.2 1 0.8 0.6 0.4 0.2 0
1000
2000
3000
4000
5000
6000
7000
8000
Frequency [Hz] Exp. tool-tip FRF - x direction Exp. tool-tip FRF - y direction Calculated tool-tip FRF using modal parameters Modes natural frequencies Figure 2 Tool‐tip FRFs at idle state
Figure 3 SSR results compared to analytical SLD
Figure 4 Speed‐varying FRF calculated by proposed method
Figure 5 SLD reconstructed with speed‐varying FRF
4
5400
Proposed method Idle condition 3.5
Damping Ratio [%]
Natural Frequency [Hz]
5350
5300
5250
3
2.5
2
5200 1.5 Proposed method idle condition
5150
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
Spindle Speed [rpm]
a)
1
3 # 10
4
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
Spindle Speed [rpm]
b)
3 # 10
4
Figure 6 Modal parameters varying with spindle speed extracted via proposed method compared to stationary condition a) natural frequency b) damping ratio
5350
4.5
Proposed method multiple Proposed method single Ozsahin et al.
3.5
Damping Ratio [%]
Natural Frequency [Hz]
5300
Proposed method multiple Proposed method single Ozshain et al.
4
5250
5200
5150
3
2.5
2
1.5
5100 1
5050
a)
0.5
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
Spindle Speed [rpm]
3 # 10
4
b)
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
Spindle Speed [rpm]
3 # 10 4
Figure 7 Modal parameters varying with spindle speed extracted via proposed method and state of the art approach on multiple and single conditions a) natural frequency b) damping ratio 7
Depth of cut [mm]
6 5
stationary SLD speed-varying SLD stable chatter
4 3 2 1
0 20000 20500 21000 21500 22000 22500 23000 23500 24000 24500 25000
Spindle Speed [rpm] Figure 8 Experimental validation of speed‐varying SLD
Table 1 Modal parameters for stationary conditions Mode Natural frequency i [Hz] Damping ratio i [%] Mode shapes X participation factor 〈 〉[mm]
1st
2nd
3rd
4th
5th
6th
7th
1078.6
2103.3
2392.3
3604.6
5374.7
7064.9
7422.1
4.9
6.1
2.7
2.8
1.2
3.1
2.3
0.011 ‐ 0.072i
0.077 ‐ 0.265i
0.025 ‐ 0.037i
‐0.022 ‐ 0.076i
‐0.047 ‐ 0.627i
‐0.048 ‐ 0.025i
0.043 ‐ 0.247i
Table 2 Limiting conditions extracted via SSR
1
Spindle speed (rpm)
11
3
4
5
6
7
8
9
13,540 13,601 13,951 14,336 14,964 15,532 16,582 17,145 17,335
Limiting depth of cut (mm) Chatter frequency (Hz) 10
2
12
13
1.5
1.0
1.0
1.5
1.5
1.5
1.5
1.5
2.0
5,270
5,260
5,270
5,260
5,280
5,310
5,280
5,300
5,240
14
15
16
17
18
19
20
21
22
17,550 17,837 18,153 19,236 19,349 19,555 19,566 19,859 19,991 20,371 21,785 21,966 21,960 2.5
2.5
2.0
2.0
2.5
3.0
3.5
3.5
3.0
2.5
2.5
3.0
3.5
5,270
5,200
5,240
5,270
5,220
5,250
5,230
5,160
5,170
5,200
5,220
5,250
5,270
23
24
25
26
27
28
29
30
31
32
33
34
22,918 23,001 23,607 24,768 25,025 25,170 27,032 27,442 27,572 28,366 28,717 29,100
3.5
3.0
2.5
2.5
3.0
3.5
3.5
3.0
2.5
2.5
3.0
3.5
5,140
5,160
5,190
5,250
5,220
5,250
5,120
5,140
5,120
5,200
5,170
5,200
Table 3 Modal parameters and experimental values fitting for multiple conditions
Spindle speed Natural frequency i [Hz] Damping ratio i [%] Fitting error fo [%]
Idle condition
1st lobe 2nd lobe 3rd lobe 13,825 15,353 16,875 rpm rpm rpm
4th lobe 18,703 rpm
5th lobe 21,077 rpm
6th lobe 24,271 rpm
7th lobe 28,065 rpm
5374.7
5331.2
5340.8
5325. 4
5277.9
5256.7
5242.3
5178.1
1.2
1.3
1.8
1.8
2.2
2.8
3.3
3.9
‐
0.12
0.02
0.12
0.17
0.11
0.08
0.07
Table 4 Natural frequency computed for single conditions Natural frequency i [Hz] Proposed method
Spindle speed [rpm] 13,540 13,601 13,951 14,336 14,964 15,532 16,582 17,145 17,335 5300.4 5268.8 5248.7 5220.5 5298.8 5255. 3 5284.1 5257.0 5248.7
Ozahin et al. method
5315.3 5278.3 5236.6 5204.0 5301.7 5257.9 5290.8 5248.0 5244.0 0.28 0.18 ‐0.23 ‐0.32 0.056 0.049 0.13 ‐0.17 ‐0.09
Difference [%]
17,550 17,837 18,153 19,236 19,349 19,555 19,566 19,859 19,991 20,371 21,785 21,966 21,960 5268.8 5284.1 5284.1 5215.6 5188.9 5232.3 5234.8 5286.5 5277.9 5262.8 5138.7 5148.9 5149.9 5268.0 5289.5 5293.1 5203.0 5172.2 5215.1 5218.0 5289.1 5280.6 5266.3 5139.4 5157.6 5160.9 ‐0.01 0.10 0.17 ‐0.24 ‐0.32 ‐0.33 ‐0.32 0.05 0.05 0.07 0.01 0.17 0.21 22,918 23,001 23,607 24,768 25,025 25,170 27,032 27,442 27,572 28,366 28,717 29,100 5256.7 5264.6 5209.1 5174.8 5113.3 5120.4 5208.6 5178.1 5123.6 5178.1 5091.3 5087.4
5258.8 5265.5 5220.4 5169.8 5112.7 5126.8 5215.5 5187.8 5135.6 5174.8 5085.0 5083.9 0.04
0.02
0.22
‐0.10
‐0.01
0.12
0.13
0.19
0.23
‐0.06
‐0.13
‐0.07
Table 5 Damping ratio computed for single conditions Damping ratio i [%] Proposed method
Spindle speed [rpm] 13,540 13,601 13,951 14,336 14,964 15,532 16,582 17,145 17,335
Ozahin et al. method
1.85
1.36
1.13
0.96
1.99
1.12
2.02
1.29
1.03
1.63
1.30
0.98
0.92
1.95
0.98
2.01
1.08
0.96
‐4.56 ‐15.45
‐4.31
‐0.31 ‐18.81
‐7.15
‐13.41
Difference [%]
‐1.85 ‐14.47
17,550 17,837 18,153 19,236 19,349 19,555 19,566 19,859 19,991 20,371 21,785 21,966 21,960 0.88 2.35 2.46 1.39 0.96 1.12 1.68 2.58 2.64 3.08 1.17 0.95 1.52 0.89
2.04
0.88 ‐14.93
2.31
1.00
0.93
‐6.38 ‐38.98
‐3.72
1.02
1.40
2.15
2.36
3.04
1.08
1.02
1.46
‐9.83 ‐19.69 ‐19.78 ‐11.48
‐1.21
‐7.83
6.58
‐3.95
22,918 23,001 23,607 24,768 25,025 25,170 27,032 27,442 27,572 28,366 28,717 29,100 3.65 3.01 3.47 2.69 2.07 2.21 4.42 4.18 3.61 3.51 3.60 3.62
3.47
2.90
3.52
2.65
1.99
2.15
4.20
4.15
3.59
3.54
3.56
3.55
‐5.28
‐3.81
1.38
‐1.30
‐4.02
‐2.91
‐5.23
‐0.66
‐0.38
0.704
‐1.30
‐2.13