Flowchart of setting Chatter-free Cutting Condition

Transcription

1 Flowchart of setting Chatter-free Cutting Condition 1

2 Procedures for setting Chatter-free Cutting Condition -Illustration by case study- Hoshi Technical Research, 12 March 2017 revise 30 March Part 1 Chatter-free Rough Cutting Conditions 1.1 Foreword Chatter-free cutting conditions are necessary to be prepared without repeating try and error search on the shop floor for right conditions to work with. Cutting conditions for chatter-free machining can be obtained by prediction of stability pockets calculated using CutPRO software and recent results of studies on Process Damping. 1.2 Theories used Stability pocket Regenerative chatter theory found during 1950`s by late Professor Tlusty reached wide understanding during 1960`s which predicted that wide chatter-free regions termed as Stability Pockets exist at very high range of spindle speed. Research continued by Professor Tlusty and his students made the stability pocket concept successfully practiced in about year 2000, first in high speed milling of aluminum alloy aircraft structural parts. Chatter-free cutting conditions set by the stability pocket are characterized by relatively high spindle rotation. For using stability pocket, a set of software called CutPRO is used supplied by Professor Y. Altintas of MAL Company of the British Colombia University in Canada. The software is used, first for experimentally assessing the structural dynamics FRF of the tip of the cutting tool mounted on the machine spindle, and second for computing stability border diagram for setting the spindle speed to use. Using an impulse-hammer attached with a force sensor that detects dynamic force (this is the input signal), end of the tool is lightly impacted, whose response vibration (this is 2

3 the output signal) is detected by an accelerometer attached at the end of the tool. The test is called Impulse Hammer Test. FRF stands for Frequency Response Function that describe according to the frequency, two variables (usually in the control theory, the Gain that is the amplitude ratio of input and output signals, and Phase, the time difference between the two signals; and in the chatter theory, after the detected accelerometer signal is transformed into displacement amplitude, two variables termed as real and imaginary parts). The theory of chatter developed by Professor Tlusty is giving two characteristic points of practical importance. One is that the maximum negative value of the real part, termed as Maximum Real Negative Part, Gmin is indicative of the likelihood of chattering, and the second, that chatter happens at the frequency giving the Maximum Real Negative Part. Workpiece after machining may be of thin-walled geometry, hence highly likely to chatter. The Maximum Real Negative Part may be measured of the workpiece and by comparison with that of the end of the tool, it will be known which of the workpiece or the tool is responsible for the chatter to occur Process Damping Generation of dynamic cutting force that suppresses onset of chatter at low cutting speed is attributable to Process Damping that may consist of two different generation mechanisms. The mechanism illustrated in Fig.1.1 is more widely understood in which the edge radius and the flank wear width interfere with the cut surface, generating the dynamic force that suppress vibration of the cutting edge. CutPRO software has been successful in including this mechanism in computation of the stability border diagram as illustrated in Fig.1.2. The result is effective for use in selecting the finish cutting conditions because in the finish cutting, edge radius rt and flank wear width vb shown in Fig.1.1 are of the comparable order of magnitude as the radial depth of cut shown in Fig

4 Fig. 1.1 Illustration of interference of the edge radius and the flank wear. Fig.1.2 Process Damping computation by CutPRO software 4

5 1.2.3 Process Damping associated with directional deviation of the principal cutting force. Principal cutting force is a force component working in the direction of cutting speed. When vibration happens in the direction normal to cut surface, the direction of the cutting speed deviates so that a new dynamic force is additionally generated in the direction normal to the cut surface that tends to suppress the vibration. Das Tobias proposed this hypothesis in followed by cutting tests conducted later by independent authors that proved that such vibration suppressive dynamic force is generated according to the Das Tobias ( )Model. [Reference 1] Inducted from this experimental proof, an equation is used to calculate Asymptotic Cutting Spindle Speed, Sas below which chatter does not occur in taking any large depths of cut. For calculating Sas, the important variable is Frequency of Chatter Vibration to Happen that can be identified by conducting Impulse testing described in Section 1.1. Chatter-free cutting conditions can be set as long as the spindle speed is selected lower than Sas, which tends to be in a relatively low speed range and this condition is capable of covering rough cutting Chatter avoidance at the sharp corners. Chatter is highly like to occur when the cut profile includes a concave sharp corner that necessitates some special precautions for conducting both rough and finish cutting. In a situation as illustrated on the left side of Fig.1.3, finishing along a straight profile, chatter is hard to occur because the radial depth of cut is small. But when reaching the sharp corner as illustrated on the right side of the figure, chatter becomes highly prone to occur because of suddenly increased angle of immersion. While the situation was quite less likely to chatter as it was the finish cutting with a small radial depth of cut, entering the sharp corner enforces the situation to change all of a sudden into that of rough cutting. Although this situation lasts only for a very short period of time, chatter becomes highly likely to occur analogous to the rough machining. The difficulty can be avoided by taking either one of the following two precautions. When taking a relatively high spindle speed selected by the stability pocket calculation of CutPRO software, the radial depth of cut has to be set so large that chatter does Footnote: APPENDIX: [Reference 1] Prediction of Low Speed Stability by Das Tobias ( ) 5

6 Model Fig.1.3 Difference in angles of immersion in finish cutting between straight profile and profile of corner radius. not happen even when the angle of immersion becomes very large. It is safe to set the radial depth of cut equal to the diameter of the tool so as to prepare for the case angle of immersion is its maximum value 180 degrees. When taking a relatively low spindle speed selected below the Asymptotic Cutting Spindle Speed Sas, chatter is guaranteed not to occur no matter how large depths of cut is assumed, and when angle of immersion becomes large, the situation is still inside the chatter-free condition Vibration mark residual at the sharp corner. Even though precautions described in the previous section are practiced, very slight vibration mark may be visible on the sharp corner. By analysis of the frequency of vibration during cutting, it is confirmed that the vibration mark is not due to chatter, but it is the mark left by natural vibration of the spindle in torsion. Although such torsional natural vibration is happening at very high frequency, it is possible to be avoided by chucking the tool using a sleeve or a collet made of vibration absorbing metal as illustrated in Fig

7 Fig.1.4 Chucking sleeves and collets of vibration absorbing material 1.3 Case Study 1 Aluminum alloy -1- Work material 5056 Aluminum alloy Machine Tool Okuma MU400VA BT40 Max Spindle Speed 15,000rpm Stroke: X762mm, Y460mm, Z460mm End Mill: Diameter 20mm, number of flutes 4 Fig.1.5 Measured result of Tool tip dynamics FRF 7

8 1.3.1 High speed range chatter-free rough cutting condition. [Procedure 1] Measuring tool tip dynamics FRF Cutting tool is mounted on the spindle of the machine to use. Impulse test is conducted using MalTF module of CutPRO software. As illustrated in Fig.1.5, frequency of chatter to occur is identified at a relatively high value of 2265Hz. Maximum Real Negative Part has been identified at 0.73μm/N, a relatively small value, hence the tool can cut pretty well. [Procedure 2] Calculate stability border diagram using the tool dynamics FRF above. Fig.1.6 Computed stability border diagram. From the computed result Fig.1.6, using the stability pocket #3, spindle speed rpm has been selected closest to and below the Maximum spindle speed 15,000rpm of the machine. As the corresponding axial depth of cut is 1.3mm, a smaller value 1.0mm is selected for use. [Procedure 3] Confirm selected cutting condition by milling process simulation in view of chatter stability and spindle power requirement Fig.1.7 Milling process simulation confirms that chatter will not be occurring. 8

9 Fig.1.8 Confirm spindle power consumption by milling process simulation. It has been calculated that 2.1kW spindle power is requird at 1.0mm axial depth of cut that is within the capacity of the machine Low speed range chatter-free rough cutting condition. [Procedure 1] From the measured result Fig.1.5, since the frequency of chatter to occur is 2265Hz, Asymptotic Cutting Spindle Speed Sas is computed as shown in Fig.1.9. Fig.1.9 Calculation of Asymptotic Cutting Spindle Speed by Das Tobias ( ) model 9

10 In preparation for the sharp corner where the angle of immersion suddenly increases, it is decided to use 3,000rpm spindle speed by which chatter is avoided no matter how large depths of cut are selected. Other conditions are feed f 0.2mm/tooth, F600mm/min, radial depth of cut 20mm (slot cutting) and axial depth of cut 10mm. [Procedure 2] Confirm by milling process simulation that the spindle power requirement is within machine capacity. Fig.1.10 Confirmation by milling process simulation of the spindle power requirement. 6.3kW power requirement of the spindle drive is confirmed to be inside the capability of the machine. 10

11 1.3.3 Comparison among chatter-free slot rough machining conditions. Fig.1.11 Comparisons among chatter-free slot rough cutting conditions. In Low Speed Range, axial depth of cut set at 10mm quoted in Section may be further increased without onset of chatter for attaining higher machining rate, but subject to limitation by the spindle drive power capability and fracture risk of the end mill. In High Speed Range using the stability pocket #3 10,755rpm quoted in Section may be applied to higher spindle speeds, as high as the spindle speed capability is possible with the machine in use for attaining higher machining rate without much increase of the amount of cutting force because the axial depth of cut stays no greater than 2.3mm. 11

12 Part 2 Chatter-free Finish Cutting Conditions Finishing side of a straight high wall and also at the sharp corner where direction of the tool advance suddenly changes by 90deree are rather difficult situations for chatter-free cutting. Chatter-free condition will utilize Process Damping computations possible with the CutPRO software as well as Das Tobias ( ) model. As long as finish cutting is concerned, it is recommended to take a cutter having only one flute as illustrated for example in Fig.2.1. There are two reasons why single flute tool is recommended: (1) Chatter is less likely to occur when number of flutes of the cutter is smaller. Therefore single fluted tool is least likely to chatter. (2) Even in using a tool having multiple flutes, the surface geometry is generated by a cutting edge having largest runout. One flute, thus one cutting edge has minimum amount of the runout, therefore giving the best surface quality. Fig.2.1 Example single flute end mill 12

13 2.1 High Speed Range chatter-free finish cutting conditions Case Study 2 Aluminum alloy -1- Work material 5056 Aluminum alloy] Machine Tool Okuma MU400VA BT40 Max Spindle Speed 15,000rpm Stroke: X762mm, Y460mm, Z460mm End Mill: Diameter 16mm, number of flutes 1 [Procedure 1] Measuring tool tip dynamics FRF Fig.2.2 Measured result of Tool tip dynamics FRF Cutting tool is mounted on the spindle of the machine to use. Impulse test is conducted using MalTF module of CutPRO software. As illustrated in Fig.2.2, frequency of chatter to occur has been identified at 1,545Hz. Maximum Real Negative Part, at 2.0μm/N. 13

14 [Procedure 2] Calculate stability border diagram using the tool dynamics FRF above. Fig.2.3 Deciding cutting conditions by the calcltaed stability border diagram Radial depth of cut has been set at large 11mm in consideration of finishing the sharp corner where chatter is most likely to happen. As the maximum spindle speed of the machine is 15,000rpm, stability pocket #6 (14,685rpm) is used for the spindle speed. Axial depth of cut is possible up to 1.9mm by computation, therefore set at 1.5mm which is less than the calculated value. Fig.2.4 Situation of final finish 14

15 Fig.2.5 Case study 2, Work piece surface after final finish Good surface has been obtained both along the straight and harp corner parts of the profile. 2.2 Low Speed Range chatter-free finish cutting conditions Case Study 3 Aluminum alloy -2- Work material 5056 Aluminum alloy Machine Tool Mitsubishi M-V58 BT40 Max Spindle Speed 8,000rpm Stroke: X800mm, Y510mm, Z460mm End Mill: Diameter 16mm, # of flutes 1 Height of wall to finish 50mm 15

16 [Procedure 1] Measuring tool tip dynamics FRF Fig.2.6 Measured result of Tool tip dynamics FRF [Procedure 2] Calculate stability border diagram using the tool dynamics FRF above. Fig.2.7 Stability border diagram computed by CutPRO Stability pocket down to #5 is possible to use but the spindle speed below cannot be used with certainty. Since maximum spindle speed is limited to 8000rpm, chatter-free 16

17 condition is not possible in high speed range. Only low speed range is possible below asymptotic cutting spindle speed Sas that can be calculated for the frequency of chatter to happen at 1,483Hz as noted below: Fig.2.8 Cutting condition by asymptotic cutting spindle speed Fig.2.9 Calculate axial depth of cut by Process Damping computation by CutPRO 17

18 For 2,800rpm spindle speed calculated by Sas from Das Tobias (1960^64) model process damping, determined is axial depth of cut possible for finishing the straight part of the profile using Process Damping computation of CutPRO. From Fig.2.9 above, 9.4mm axial depth of cut is known possible at deepest. It has been decided to use 8mm axial depth of cut in finishing the profile. But by miss-instruction passed to NC program generation, actual finish cut was conducted with 25mm axial depth of cut resulting in finish surface with chatter mark. Fig.2.10 Situation of final finish Fig.2.11 Work piece surface after the final finish 18

19 Although computation was indicating 8mm axial depth of cut, because actual machining took place with 25mm axial depth of cut due to miss-instruction to the down stream data handling, chatter mark is visible on the finished surface. Part 3 High Performance Machining of Hard-to-Cut Materials 3.1 Vibration problem specific to hard-to-cut materials In cutting hard-to-machine materials including hardened steel, stainless steel, high manganese steel, titanium alloy and nickel base alloys, a hypothesis is proposed assuming that vibration happens due to formation of shear-type chip at extremely high frequencies besides possibility of conventionally known regenerative chatter. Vibration specific to hard-to-cut materials are associated with the shear-type chip formation as shown in Fig.3.1. Fig.3.1 Two examples of shear-type chip formation. 19

20 Every time a slice of the shear-type chip is formed, the cutting edge is displaced as depicted as a pulse in Fig.3.2 in the cutting speed direction (Ydirection). Width of the pulse is a short time less than 0.1 mill second. As the slice formation repeats, the pulsed displacement of the cutting edge repeats forming a pulse train. Time period L of the slice formation is in the order of mill seconds. Fourier s transform of the pulsed train forms a densely spaced frequency components distributed across a wide frequency range of over 10kHz as illustrated in the lower figure. If the tool has a natural frequency within the frequency range, a group of surrounding frequency components will be simultaneously excited causing the chipping of the CBN cutting edge. Fig.3.2 Shear-type chip formation causing (upper figure) pulse train-like Y-direction displacement of the tool tip and (lower figure) densely populated frequency components across a wide frequency range. 3.2 Characteristics of the cutting vibration caused by periodic chip formation Direction of the vibration 20

21 Vibration may be happening either at the tool tip or the work piece, but in either case, cutting vibration caused by the periodic chip formation is happening in the direction of the cutting speed (Y-direction) which is in contrast to the well- known regenerative chatter that is associated with vibration in the direction normal to cut surface (X-direction) Frequency of vibration Problems are associated with high frequency up to a few thousand Hz Frequency spectra As illustrated in Fig.3.2, vibration happens in Y- direction (direction of cutting speed) of the tool tip at a group of surrounding frequencies among the densely populated frequency components Method to control The vibration is known to be prevented by taking small enough thickness of the cut. Also it is effective to devise a method to absorb high frequency vibration in the direction of cutting speed (Y-direction). Specifically, either use of vibration absorbing material or use of friction damper Problems encountered due to shear-type chip formation. 1. Chipping of CBN cutting tool edge in hard turning. Chipping of carbide tool in milling and turning of hard-to-machine materials. 2. High frequency vibration caused by the resonance of the spindle in torsion. 3. Vibration mark left on the surface finished by hard turning. 4. Induction of regenerative chatter Effect of inducing regenerative chatter Fig.3.3 Effect of inducing regenerative chatter 21

22 Figure in the above compares strong chatter occurring in rough milling of titanium alloy TiAl4V at a feed rate of 0.08mm/tooth disappearing by reduction of feed rate to 0.07mm/tooth. This change is interpreted to have occurred by shear-type chip formation disappearing when the feed rate was reduced. Due to the observation, it is understood that machining titanium alloy should be planned with thickness of cut equal to or smaller than 0.07mm by which the situation becomes analogous to regular materials and fracture of tool edge will be prevented Specifics of cutting conditions for Titanium alloy. (1) Reduced thickness of cut For avoiding induction of regenerative chatter due to shear-type chip formation, and for preventing fracture of tool edge, thickness of cut needs to be kept equal to or smaller than 0.07mm (or smaller than 0.1mm). Variations by different situations are illustrated in Fig.3.4. Fig. 3.4 Thickness of cut variations by different situations (2) Lower cutting speed Because of faster tool wear due to the hard-to-machine material, cutting speed is usually selected relatively low among 60 to 200 m/min. 22

23 Case Study 4 Face cutting of TiAl4V Cutting tool used Fig.3.5 Outline of tool used for face cutting Cutting conditions for face cutting [Procedure 1] Measuring tool tip dynamics FRF Fig.3.6 Dynamic FRF of the tool tip measured by the impulse test. 23

24 [Procedure 2] Calculate stability border diagram using the tool dynamics FRF above. Fig.3.7 Stability border diagram Computed for impulse test result shown in Fig.3.6, stability border diagram Fig.3.7 indicates that the stability pocket #5 at S650rpm (V102m/min) may be useful but up to only mm axial depth of cut that may be acceptable for finishing condition. Stability pocket at lower speed is not possible to use because the speed width becomes even narrower. [Procedure 3] Confirm face finish milling condition by Milling Process Simulation. Fig.3.8 Milling Process Simulation of face finish cutting at 0.06mm axial depth of cut. 24

25 It has been confirmed as shown in Fig.3.8 by the milling process simulation that the #5 stability pocket is free from chatter when axial depth of cut is selected as low as 0.06mm Cutting test at face finish condition obtained from stability pocket #5. Fig.3.9 Result of cutting test at face finish condition obtained from stability pocket #5. Chatter-free cutting has been experimentally confirmed Face rough cutting condition by application of process damping calculation. Fg.3.10 Selecting face rough cutting condition by use of process damping calculation 25

26 As the natural frequency responsible for chatter is about 400Hz as known from Fig. 3.6, asymptotic cutting spindle speed Sas is calculated as above Cutting test for face roughing condition selected by process damping calculation. Fig.3.11 Result of cutting test at face rough condition selected by process damping calculation. It has been confirmed that face roughing is conducted without chatter Face rough cutting by the condition traditional in the factory. Fig.3.12 Face rough cutting test at the condition traditional in the factory 26

27 As compared to spindle speed S160rpm (V25m/min) and axial depth of cut 2.5mm in Fig.3.11, traditional condition of the factory Fig.3.12 has been successful at a faster spindle speed S255 (V40m/min) and a deeper axial depth of cut 2.7mm Comparison of face cutting experiments Fig.3.13 Results of comparison among face cutting tests Because the dynamics FRF of the end of the tool has been found rather week, stability pocket could define only finish cutting condition while conditions for rough cutting had to be defined by using Das Tobias ( ) model process damping calculation. Case Study 5 Rough slot cutting of TiAl4V Work material Titanium alloy Ti6Al4V Machine Tool Okuma 5 axes vertical machining center BT50 End Mill: Diameter 12mm, 4 fluted, under chuck length L70mm 5.1 Setting chatter-free condition of slot rough milling [Procedure 1] Measuring tool tip dynamics FRF Cutting tool is mounted on the spindle of the machine to use. Impulse test is conducted using MalTF module of CutPRO software. 27

28 Test Fig.3.14 Impulse test result of the end of the tool mounted on the machine spindle Test result shows that 3 natural frequencies are found as listed from the highest one; Natural frequency A 2,333Hz Natural frequency B 1,682/1692Hz Natural frequency C 398Hz [Procedure 2] Calculate stability border diagram using the tool dynamics FRF above. Fig.3.15 Stability border diagram computed for T33 end mill for rough slotting 28

29 #1 to #5 stability pocket exist at far high spindle speeds so that they can not be utilized. On the other hand, unconditional stability limit Alim is found as small as 0.031mm. Only possible way is to use low speed range conditions using process damping calculation. As there are three natural frequencies, Sas is going to be calculated for each 0f the natural frequencies. [Procedure 3] Calculating Sas for process damping application 29

30 Fig.3.16 Tuning spindle speed for rough slot cutting by use of process damping [Procedure 4] Checking on spindle power requirement and amount of cutting force applied on the tool edge. 30

31 Fig.3.17 Spindle power requirement and maximum cutting force calculated by the Milling Process Simulation. Fig.3.18 Outline of sot cutting tests 31

32 5.3 Slot rough cutting test Fig.3.19 Result of vibration analysis during slot rough cutting Although spindle speed has been set at 2,000rpm where chatter is not supposed to occur at any large depths of cut, chatter of 2379Hz has been identified both at the start of the cut and at the steady cutting. The amplitude, however, is very small (0.088μm) so that it actually did not cause a problem. 6.1 Selecting cutting condition for slot medium finish Fig.3.20 Stability border diagram by CutPRO enabling process damping calculation for medium finish condition. 32

33 Consideringg the sharp corner, spindle speed has been selected at 2,000rpm by Sas calculation in Section 5.1, axial depth of cut is calculated by CutPRO including Process Daming. Although the result of calculation is indicating that axial depth of cut is possible up to 26.7mm, axial depth of cut has been set at 30mm. 6.2 Medium finish cutting test of the profile including a sharp corner. Fig.3.21 Results of medium finish cutting test of a profile including a sharp corner 33

34 Very slight chatter is occurring at 1666Hz and 2599Hz during the steady cutting. Vibration is occurring at 2395Hz during both steady cutting and sharp corner cutting which is not at the frequency of chatter expected (2333Hz by Fig.3.14) but is the forced vibration. Fig.3.22 Situation of profile medium finish Chatter Fig.3.23 Case study 6, work piece surface after medium finish 34

35 Chatter mark is not visible on the straight side of the profile that has been finished by steady cutting. Slight vibration mark is visible on the sharp corner of the profile, which should be prevented. Slight over-cut is noticed on the sharp corner that must have happened by elastic deformation of the tool during steady cutting presumably by about 0.3mm because of the large radial depth of cut 1mm set for the medium finish. The surface had better be final finished once more by setting the radial depth of cut 0.1mm. If such final finish may be planned in the future, cutting condition may refer to a staility border diagram predicted by CutPRO using process damping computation as follows. Fig.3.24 Result of Stability border diagram computation using process damping calculation of CutPRO suggesting condition for final finish of the profile including sharp corner possible up to 92mm axial depth of cut. 7. Conclusions. 7.1 Objective of setting chatter-free cutting conditions. As already noted in Section 1.1 Foreword, chatter-free cutting conditions are wanted to be prepared without repeating try and error search at the factory floor for doing the job right. By powerful use of recent technologies, typically CutPRO software and process damping calculations, production engineers would be able to quickly and easily write up list of chatter-free cutting conditions for a new job to prepare. 35

36 7.2 Basic methodologies. As stability pockets are calculated at relatively high speed conditions, chatter-free conditions primarily refer to high speed range. If spindle speed capability of the machine does not allow high speed rotation, or if high speed rotation is not acceptable from tool life point of view that may often be the case in cutting hard-to-cut materials, lower speed range can be alternatively selected by conducting process damping calculation of asymptotic cutting spindle speed Sas by Das & Tobias ( ) model. 7.3 Finish cut For finish cutting, Process Damping computation embedded in CutPRO software is useful because that by Das Tobias ( ) model is valid only for roughing cut. 7.4 Flowchart Procedure for setting chatter-free cutting conditions discussed in the text is summarized by a flowchart as illustrated in Fig.3.25 shown on the next page. All situations covering roughing/finishing, include/not include sharp corner, high/low range conditions, and hard-to-machine materials can be processed by simply following the flowchart. 7.5 Machining rate comparisons By using the procedure and the flowchart, chatter-free cutting conditions are obtained without difficulty and ambiguity. Machining rate will be comparable to that normally practiced conventionally. When using the stability pocket and using high speed range conditions, machining rate will be definitely greater than conventional practice, Especially when the long tooling is used, which could be possible conventionally only by reducing the cutting speed, use of high sped range cutting condition will remarkably increase the machining rate. 7.6 Future tasks Procedure for setting chatter-free cutting may be rather complicated and not simple to follow as summarized in this note. One may need certain accumulation of experiences, which the author sincerely wishes to support. 36

37 Fig.3.25 Flowchart of setting chatter-free cutting conditions. 37

38 [APPENDIX] Prediction of Low Speed Stability by Process Damping Theory Tetsutaro Hoshi, 11 March 2008, last edit 24, January Introduction It is well known among machinists that taking low enough cutting speed increases dynamic stability so that chatter is prevented. Fig. 1 Experimental result of stability border diagram measured in turning test. (Lines represent results of computational simulation) 38

39 Principal direction of natural mode of vibration of tool supporting structure: depth of cut direction, natural frequency: 150 Hz. Feed rate: mm/rev, flank wear width: Vb = 0-50 μm, work piece material S35C plain carbon steel. [T. Takemura, Research on Avoidance of Chatter in Turning, Doctor Dissertation submitted to Kyoto University, )] When two parameters, cutting speed and depth of cut are mapped in a Cartesian coordinate as illustrated in Fig. 1 and amplitudes of vibration are marked as measured in cutting at selected combinations of two parameters, there can be defined unconditional stability limit Amin of the depth of cut below which system is stable so that chatter does not occur irrespective of cutting speed. It is noticed that as cutting speed is selected lower, greater depth of cut is possible without onset of chatter. This phenomenon, called Low Speed Stability, is understood to be caused by Process Damping which induces increased amount of stabilizing force on cutting tool at lower cutting speed. In academic circle, historical study of Das and Tobias 2-4) concluded that the effect is due to instant oscillation of the orientation of total cutting force acting on the cutting edge according to the slope of the inner modulation on which vibrating cutting tool traces. The concept described in the hypothesis of Das and Tobias has been later confirmed experimentally, based on which present study tries to formulate a method of predicting the upward expansion of stability limit at lower cutting speed range. 39

40 Ff F Fp Y Position in cutting direction X: Displacement in direction of uncut chip thickness λ: Wave Length Ft Inner Modulation Fig. 2 Hypothesis of Process Damping after Das and Tobias. 2. Mathematical Model of Process Damping. 2.1 Hypothesis by Das and Tobias Their conclusion was that the damping force that suppresses chatter is generated by instantaneous oscillation of the orientation of total cutting force according to the slope of inner modulation. As illustrated in Fig. 2, hypothesis noted in the above, sometimes referred to as Imaginary Part Effect of Inner Modulation can be modeled as follows: Generation of Fp is in proportion to the width of cut b, modeled by: 40

41 F p dx dt bf t dy dt bft ( j2 ( Frequency _ of _ Vibrationin Hz) X ) ( CuttingSpeed _ in _ mm/sec) Since wavelength is: (1) ( Cutting Speed in mm / sec) ( Frequency of Vibration in Hz) (2) Fp in the above equation (1) is reduced to: F p bft ( j 2 ) X (3) The Stiffness Frequency Response Function Tpx is then, (4) T px bfp j 2 bft X Cutting force components are obtained by generic two-dimensional cutting tests and related to uncut chip thickness h and width of 41

42 cut b as illustrated below. 2.2 Experimental Verification of Process Damping Model Ft in equation (4) is modeled as F t K oy h Tpx is then:, Koy : Static tangential force coefficient T px j 2 K oy b h (7) Experimental proof for the generation model of process damping as described in the above equation (7) is presented in a result of measurement, Fig.3 conducted in turning tests using tool support system having variable natural frequencies (146 to 1167Hz) at variable feed rates f (0.05 to 0.2 mm/rev) and depths of cut d (0.1 to 0.6mm). 42

43 Fig.3 Experimental value of imaginary part effect of inner modulation measured in turning test. Koy=Ft/bh, f=h, d=b [Hoshi 5), 6)] 3. Coordinating Process Damping in Stability Border 3.1 Stability Border without Process Damping Referring to conventional stability border model illustrated in Fig. 4, the so called Maximum Real Negative Part of structure dynamics Gmin defines Unconditional Stability Limit Amin by following equation: A min 2 K 1 G fc min (8) 43

44 Cutting dynamics is represented in the figure by a vertical line marked red. Equation (8) is valid only when overlap factor is unity 1 (Machining situations such as in parting or grooving of width b, or end milling of axial depth of cut b). In Machining situations where the overlap factor is less than 1, the cutting dynamics line (vertical line colored in red in next figure 4) is reduced to a curve inscribed on the left-hand side of the vertical red line, the stability border is reached by taking width of cut b greater than calculated by equation (8). Fig. 4 Conventional stability border model in compliance vector diagram. 44

45 3.2 Stability Border considering Process Damping In a stiffness vector diagram Fig.5, the cutting dynamics without considering process damping is represented by a smaller red circle at right upper corner. Stiffness FRF F/X of machine structure is represented by a horizontal straight line tangential to the small circle. It used to be represented by a circle in the compliance vector diagram in Fig.4, but in the stiffness vector diagram Fig.5 represented by a straight line. The distance of the straight line from the origin O of the coordinate is the minimum stiffness of the structure and it is the inverse of the maximum compliance 2Gmin shown in Fig.4. Center of the red circle represents inner modulation, while the circle itself corresponds to the outer modulation having variable phase angle to the inner modulation. When considering Process Damping, the circle center is shifted downward by the amount of Tpx, therefore stability border is reached only by taking greater width of cut b which is the Stability Limit with Process Damping Alim. 45

46 Fig. 5 Stability border model in stiffness vector diagram. The distance between the down shifted circle center and the horizontal straight line at the top representing the structure stiffness is the width of cut b= Alim at the stability border multiplied by Kfc. (10) Using equation (10) and replacing b by Alim, Tpx in equation (4) is (11) Noting in the stiffness diagram Fig.6, an equality KfcAlim= KfcAmin+Tpx is holding and replacing b with Alim, and dividing both sides of the equation by Kfc and introducing a variable that stands for tangential to feed dynamic cutting force ratio, 46

47 K K tc fc condition for the outer modulation reaching stability border is represented by: (12) 1 2 Since wavelength is ( Diameter) ( rpm) 60( Frequency) Finally, stability border including process damping: 1 120,,, (13) There occurs an asymptotic spindle speed Sas to which Alim approaches infinity when denominator of (13) reduces to zero, namely: 120, /, (14) The parameter Cs in the above needs to be 1 when applied to turning and boring operations where uncut chip thickness h is equal to feed rate f, and number of tool engaged is always one. For milling application; the equation needs to be adjusted as noted 47

48 below. Referring to Fig.6, ratio of average uncut chip thickness to h is represented by a new parameter Cs: C s DA EA (cos ) d DA d EA 180(sin DA sin EA) ( DA EA) (15) Feed rate h (mm/tooth) in equation (14) has to be replaced with average uncut chip thickness h x Cs. When disengage angle DE is fixed at 90deg, and calculating Cs for engage angle EA=-90 to +90deg, Cs is found as illustrated in Fig.8 to be maximum at engage angle EA=40 deg. Fig. 6 Average values for milling. 48

49 Fig.7 Variation of Cs for engage angle EA=-90 to +90deg when disengage angle DA is fixed at 90deg In rough milling applications where engage angle EA tends to be large negative value (right side of Fig.7), Cs values tend to be high that may cause Sas to be high. Process damping by this Das, Tobias ( ) model seems to be effective in rough machining situations. On the other hand, for finishing applications, EA is going to be a small angle (left side of the figure) Cs and hence Sas assumes very small value making the effect expected to be negligible small. As cutting speed gradually reduces and approaches the recommended cutting speed defined by equation (16), unconditional stability border Alim increases as illustrated in Fig.8. 49

50 Fig Numerical Example Profile of low speed stability. Work material: AL6061-T6 Tool: Diameter 20, = Ktc / Kfc = /698.8 = Cutting conditions: f = h = 0.05mm/tooth Frequency: 1,000Hz, Engage Angle (EA)=30deg, Disengage Angle (DA)=90deg Cs=180(1-Sin30deg)/(90-30)π= , /, = 120 x 1000 x x 0.478x0.05/20 = 323rpm 5. Concluding Remarks Referring to hypothesis of process damping mechanism proposed by Das and Tobias during 1964 to 1967 and later experimentally confirmed in 1972, a method has been investigated for predicting 50

51 increased stability at lower cutting speed that prevents onset of regenerative chatter. According to the mathematical model constructed for the hypothesis, effect of process damping increases in proportion to the magnitude of total cutting force in tangential direction, frequency of vibration and in inverse proportion to cutting speed. As cutting speed is reduced, effect of process damping increases up to a point beyond which chatter can no longer occur at any large widths of cut. This lower limit of speed is represented by asymptotic spindle speed equation (14) prepared by the study. S as whose value can be calculated using Also the profile of stability border can be calculated as shown in Fig.8 by which the unconditional stability limit A min defined by regenerative chatter theory is shifted upward in lower speed range. REFERENCES 1) T. Takemura, Research on Avoidance of Chatter in Turning, Doctor Dissertation submitted to Kyoto University, ) M. K. Das and S. A. Tobias: The Basis of a Universal Machinability Index, Proc. 5th Int. MTDR Conf., (1964) ) M. K. Das and S. A. Tobias: Statistical Basis of a Universal Machinability Chart, Proc. 6th MTDR Conf., (1965) ) M. K. Das and S. A. Tobias: The Relation between the Static and the Dynamic Cutting of Metals, Int. J. MTDR, 7 (1967)

52 5) T. Hoshi and K. Okushima, Cutting dynamics associated with vibration normal to cut surface, Annals CIRP 21/1, 1972, 101 6) T. Hoshi and T. Takemura, Cutting Dynamics Associated with Vibration Normal to Cut Surface, Memoirs of Faculty of Engineering, Kyoto University, Vol. XXXIV, Part 4 (OCTOBER 1972) 373. End of text 52