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Plate insertion as a means to improve the damping capacity of a cutting tool system
International Journal of Machine Tools & Manufacture 38 (1998) 1209–1220
Plate insertion as a means to improve the damping capacity of a cutting tool system Etsuo Maruia,*, Satoshi Emab, Masatoshi Hashimotoc, Yasunori Wakasawac a
Faculty of Engineering, Gifu University, 1-1 Yanagido, Gifu-shi 501-11, Japan b Faculty of Education, Gifu University, 1-1 Yanagido, Gifu-shi 501-11, Japan c Department of Mechanical Engineering, Toyota National College of Technology, 2-1 Eisei-cho, Toyota-shi 471, Japan Received 27 October 1997
Abstract Effective chatter prevention during cutting operations is achieved by increasing the damping capacity of a cutting tool system. It is well known that damping capacity is generated through (i) micro-slip at the interface between the tool shank and tool post, (ii) slip at the grain boundary within a vibrating body (that is, internal friction), and (iii) friction between the surface of the vibrating body and the surrounding air. Among these three causes of damping capacity, micro-slip at the interface between the tool shank and tool post is the greatest factor affecting the damping capacity of the cutting tool system. In the research investigation, it is shown that the damping capacity of a cutting tool system is improved by friction acting between the inner wall of a rectangular hole made at the overhanging shank of the cutting tool system and the surface of a plate inserted into this rectangular hole. The damping capacity improvement proposed in this paper is realized by a mechanism similar to the inner friction mechanism. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Damping capacity; Cutting tool system; Improvement; Friction; Plate insertion
1. Introduction Chatter vibration occurring in cutting operations is prevented by providing additional structural damping from an electro-dynamic active damper [1]. Although this method is quite effective,
*Corresponding author. 0890-6955/98/$19.00 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 8 ) 0 0 0 0 1 - 7
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special equipment is required and the cost is considerable. The application of a simple Lanchester damper [2] and impact damper [3,4], which do not have such inconvenience, was proposed. The effectiveness of these dampers depends strongly on the characteristics of the main vibratory system. The problem is, the damping characteristics must be adjusted for each and every machine tool system. On the other hand, if the damping capacity improvement of the cutting tool system is realized by some simple methods, machine tools can operate at full efficiency [5]. The damping capacity of the cutting tool system is generated through micro-slip at the interface between cutting tool and tool post, slip at the grain boundary within the vibrating body (that is, inner friction), and friction between the vibrating body and the surrounding air. The micro-slip between cutting tool and tool post has the greatest effect on the damping capacity generation. The damping effect of the friction by surrounding air cannot be expected. Much research has been reported on the effect of micro-slip at the tool post. The damping capacity of the vibration modes parallel to [6,7] and normal to [8] the contact surface has been measured. Surface topography influences the generation of the damping capacity by micro-slip [9]. The present authors investigated the mechanism of the damping capacity generation of the cutting tool system induced by micro-slip and the effect of clamping condition [10,11]. In a previous study, it was clarified experimentally by the present authors [12] that a remarkable improvement in damping capacity of a circular plate system can be expected by surface topography optimization of the clamping jig and the insertion of adhesive tape into the clamping interface. In the present study, to improve the damping capacity of the cutting tool system by a mechanism similar to the inner friction, a plate is introduced into the cutting tool shank. A rectangular hole is cut in the overhanging tool shank. Then a plate slightly thicker than the height h of the hole, is inserted into this rectangular hole to increase the damping capacity of the system by friction during vibration acting between the inner wall of the hole and the inserted plate surface. The effectiveness of this device in improving the damping capacity is reported herewith. 2. Experimental apparatus and experimental method 2.1. Experimental apparatus The experimental apparatus for an indicial response experiment is shown in Fig. 1. A vibratory body made from carbon steel, which is equivalent to a cutting tool system, is clamped by a vise via two load cells. The clamped portion of the vibratory body is made thick so as to have a high rigidity in order to decrease the influence of the clamping portion on the damping capacity. The distance between the two load cells is 55 mm. The displacement of the vibratory body is measured at the front end by an eddy current displacement sensor of non-contact type. The configuration of the vibratory body, which simulates a cutting tool system of cantilever type, is shown in Fig. 2. This vibratory body is called a cutting tool system, hereinafter. An elongated column with 15 mm-square cross section and 105 mm-overhang vibrates. A rectangular hole h mm high and 30 mm long is cut in the cutting tool system at a point 20 mm from the front end, as shown in Fig. 2(a). A thin plate is inserted into this rectangular hole, and the damping capacity improvement is expected to result from the friction generated between the inner wall of
E. Marui et al. / International Journal of Machine Tools & Manufacture 38 (1998) 1209–1220
Fig. 1. Experimental apparatus for indicial response experiment.
Fig. 2. Configuration of cutting tool system.
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the hole and the plate surface. In Fig. 2(b), the hole is cut at a point 20 mm from the rear end. A thin plate of 15 mm wide and 30 mm long is inserted into this rectangular hole. The plate is slightly thicker than the hole height h. Five cutting tool systems (TS, TSF5, TSR5, TSR6, TSR7) having a configuration shown in Fig. 2 are prepared, including a cutting tool system without the rectangular hole. The specifications of the cutting tool systems are given in Table 1. The letter “F” indicates that a rectangular hole is cut at the position 20 mm from the front end, and the letter “R” at the position 20 mm from the rear end of the cutting tool system. A cutting tool system with a plate of small interference inserted is shown by the letter “S” and of large interference by the letter “L”. In Table 1, the height of the rectangular hole, plate thickness, interference, spring constant and fundamental natural frequency of each cutting tool system are shown. The spring constant is obtained from a separate static load test, and the fundamental natural frequency is obtained from the below-mentioned indicial response experiment. Some of the experiments are carried out on the cutting tool system of straight shank type, in which the cross-sectional dimension of the overhang portion is the same as that of the clamped portion. 2.2. Experimental method The clamping load acting on the vise may be varied by eight settings from 6 kN to 20 kN. An impulse input is added in the direction vertical to the clamping surface of the vise by a plastic hammer, and the indicial response of the cutting tool system is then measured. The output displacement of the front end of the cutting tool system is measured by an eddy current type displacement sensor, and is analyzed by an FFT analyzer. A fundamental natural frequency is obtained. The damping capacity is estimated by the ratio between two successive amplitudes of damping vibration and the above-mentioned fundamental frequency. Hence, the damping capacity discussed in the following is the fundamental mode of vibration of the system. Table 1 Specification and characteristics of cutting tool system Tool system
Height of square hole (mm)
Plate thickness (mm)
Interference (m)
Spring constant (kN/m)
Natural frequency (Hz)
TS TSF5 TSF5S TSF5L TSR5 TSR5S TSR5L TSR6 TSR6S TSR6L TSR7 TSR7S TSR7L
– 4.89 4.89 4.89 4.88 4.88 4.88 5.87 5.87 5.87 6.90 6.90 6.90
– – 4.98 5.01 – 4.99 5.01 – 5.98 6.00 – 6.99 7.05
– – 90 120 – 110 130 – 110 130 – 90 150
2200 2240 2200 2250 2200 2200 2250 2020 2040 2070 1930 1970 2000
1033 1050 1000 1025 933 938 1023 933 938 1013 875 926 1013
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Damping capacity is indicated on the basis of the magnitude of the damping ratio, which is a ratio of the measured damping coefficient and the critical damping coefficient. 3. Damping capacity improvement by plate insertion 3.1. Effect of clamping portion on damping capacity The damping capacity of system TS, which has a thick clamped portion and no rectangular hole, is compared with that of the cutting tool system of straight shank type, whose clamped portion is the same size as the overhang portion. Using this result, the effect of the clamped portion on the damping capacity is examined. The result is shown in Fig. 3, in which the relation between the damping ratio of two separate cutting tool systems and the clamping load acting on one load cell is plotted. The damping capacity of the cutting tool system TS having a large clamped portion is little affected by the clamping load. On the contrary, in the cutting tool system of straight shank type, the following tendencies are recognized. The damping ratio within the small clamping load range is somewhat large, and decreases with the increase in clamping load. Any damping ratio beyond the clamping load of 14 kN is almost constant. This tendency in the damping ratio indicates that some vibratory displacement occurs at the clamped portion of the system corresponding to the vibration displacement at the overhang portion, and the damping capacity of this system is mainly generated owing to the vibration energy consumption due to the friction between the vise and the contact surface. The displacement at the clamped portion decreases with the increase of clamping load, and the vibration energy consumption becomes small. As a result, the damping ratio of the system TS shown in Fig. 3 is obtained. The displacement at the clamped portion tends to occur less readily as the clamped portion’s cross-section is large and the rigidity there is higher. Thus, the energy consumption due to friction at the clamped portion becomes small. This is why the damping ratio of the cutting tool system
Fig. 3.
Measurement example of damping ratio (effect of configuration at clamping portion).
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TS having a very rigid clamped portion is considerably lower than that of the straight shank type cutting tool system. 3.2. Effect of position of rectangular hole or plate insertion on damping capacity Some experimental results on the effect of the rectangular hole position on the damping capacity are given in Figs 4–6. Figure 4 shows the damping ratios for three cutting tool systems: TS having no rectangular hole. TSF5 having a rectangular hole 5 mm high at the front end of the system, and TSR5 having a rectangular hole 5 mm high at the rear end of the system. A plate is not inserted here. In the system TSR5, the damping ratio at the small clamping load range is somewhat large, however, it becomes constant beyond the clamping load of 14 kN. Although the damping ratio of system TSR5 is somewhat larger as the clamping load increases, there is not much difference among them. The damping ratio in the case in which a plate of small interference is inserted into the system of TSF5 and TSR5 is shown in Fig. 5. In this figure, the damping ratio of the system TS having no rectangular hole is also plotted. A weak clamping load dependency of the damping ratio is recognized in the cutting tool systems with a plate inserted, when the clamping load is small. The damping ratio becomes almost constant once the clamping load is larger than 14 kN. The magnitude of the damping ratio increases in the order of the cutting tool system with no hole, a rectangular hole at the front end, and a rectangular hole at the rear end near the clamped portion. When regarding the cutting tool system as a cantilever arrangement, the inclination angle of the cantilever becomes large going away from the clamped portion. So, the damping ratio is expected to be large when the plate is inserted in the hole near the front end. However, the reverse experimental result is obtained. This result is obtained because the deflection curve of the cutting tool system (that is, the vibration mode) is changed by the insertion of the plate into the rectangular hole and the interference between the inner wall of the hole and the inserted plate is different in each cutting tool system. Figure 6 shows the result for the same cutting tool systems as Fig. 5 with the insertion of a
Fig. 4. Measurement example of damping ratio (effect of position of rectangular hole, without plate insertion).
E. Marui et al. / International Journal of Machine Tools & Manufacture 38 (1998) 1209–1220
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Fig. 5. Measurement example of damping ratio (effect of position of rectangular hole, with plate insertion of small interference).
Fig. 6. Measurement example of damping ratio (effect of position of rectangular hole, with plate insertion of large interference).
plate having a large interference. The damping ratio becomes small with the increase of the interference, for both rectangular hole positions at the front or rear end of the cutting tool system. Owing to the increase in interference, the slip between the inner wall of the hole and the plate surface tends to occur less easily and the energy consumption by friction becomes small. 3.3. Effect of interference between rectangular hole and inserted plate The effect of the interference between the rectangular hole and the inserted plate on the damping ratio is given in Figs 7–9. These results are obtained for the nominal plate thicknesses of 5 mm, 6 mm and 7 mm. In the figures, the damping ratios obtained for the cutting tool systems TS
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Fig. 7. Measurement example of damping ratio (effect of interference, plate thickness of 5 mm).
Fig. 8. Measurement example of damping ratio (effect of interference, plate thickness of 6 mm).
(without rectangular hole), TSR5, TSR6 and TSR7 (with rectangular hole but no inserted plate) are plotted for reference. In what follows, we discuss the damping ratio at the clamping load beyond 14 kN, where the clamping load dependency of the damping ratio is not recognized. The damping ratio becomes large by the insertion of a plate of any thickness. Small interference is desirable for a plate 5 mm thick, and large interference is desirable for the plates 6 mm and 7 mm thick to improve the damping capacity. Thus, the difference in damping ratio in terms of the amount of interference may be explained by a combined effect of the lower slippage and greater friction resulting between the hole wall and the plate due to greater interference. In any case, it is confirmed that the damping ratio becomes large by inserting the plate. This shows that intentionally amplified friction acting within a vibrating portion is effective for damping capacity improvement. It also indicates specifically that the inner friction within vibrating materials must be used for the damping capacity improvement.
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Fig. 9. Measurement example of damping ratio (effect of interference, plate thickness of 7 mm).
4. Evaluation of damping capacity improvement 4.1. Evaluation method In the previous section, it is shown that the damping capacity can be improved and useful chatter prevention may be expected by inserting a plate of various nominal thicknesses and interference at various positions in a cutting tool system. In this section, the extent of damping capacity improvement is evaluated from the viewpoint of chatter prevention in cutting operations. The following discussion is based on the regenerative chatter vibration theory developed by Merritt[5]. A cutting tool system is approximated as a lumped parameter system of one degree of freedom, whose mass is m, the damping coefficient c and the spring constant k. Setting the tool displacement as x and the actual cutting depth of the tool as u, the equation of motion of this cutting tool system is written as follows: m
dx d 2x + kx = kcu +c 2 dt dt
(1)
where kc is a cutting stiffness. By Laplace transformation of Eq. (1), ms2X + csX + kX = kcU
(2)
In Eq. (2), s = j* (*: chatter frequency), then kc X = U (k ⫺ m*2) + jc* Real and imaginary parts of Eq. (3) are written as follows:
(3)
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Re =
kc(k ⫺ m*2) (k ⫺ m*2)2 + (c*)2
(4)
Im =
⫺ ckc* (k ⫺ m*2)2 + (c*)2
(5)
Eq. (3) is plotted on a complex number plane, where chatter frequency * is set as a parameter. Real part Re is plotted on the abscissa and the imaginary part on the ordinate. This curve is called the characteristic curve of the cutting tool system. Referring to the regenerative chatter vibration theory of Merritt, a vibratory system is stable when a whole characteristic curve exists in the right side zone of the vertical straight line Re = ⫺ 0.5. Differentiating Eq. (4) with respect to *, the minimum value of the real part Re is obtained as follows: (Re)min =
⫺ kc 4k(␦0 + ␦ 20)
(6)
where ␦0 is a damping ratio and is written as ␦0 = c/√mk. From Eq. (6), it is clear that the following condition is necessary to improve the stability of the cutting operations. Cutting rigidity kc must be kept small. Both the spring constant k and the damping ratio ␦0 of the characteristics of the vibratory system are as large as possible. Hence, the following evaluation factor for chatter prevention is defined. This factor corresponds to the denominator of Eq. (6). Now we evaluate the damping capacity improvement of all cutting tool systems in the present paper.
= k(␦0 + ␦ 20)
(7)
4.2. Evaluation result of damping capacity improvement Evaluation factor is calculated from the experimentally obtained damping ratio and spring constant for all cutting tool systems. The evaluated result is shown in Fig. 10, as a function of the spring constant. In the figure, experimental results for a different cutting tool system group are classified by the three different types of lines plotted. The chain line indicates the results of the evaluation of either a cutting tool system with the rectangular hole or a system with no hole and no plate insertion. The broken line is for evaluation of the system with the rectangular hole in the front end and a plate inserted. A solid line, on the other hand, is for a system with the hole in back and a plate inserted. Most of the experimental results are thus indicated by the solid line. The curves indicated by the broken and chain lines are for cases in which there were few experimental conditions, and their changes were expected. It is clear from the evaluation that there is a spring constant which maximizes the evaluation factor in cutting tool systems with an inserted plate.
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Fig. 10. Evaluation of damping capacity improvement.
The evaluation factor of the cutting tool systems with a plate inserted into the rectangular hole at the rear end is the largest among the three indicated cases. This agrees with the above-mentioned description regarding the damping ratio only. Thus, the maximum damping ratio or the maximum stability for the chatter vibration can be effectively raised by cutting a rectangular hole at the rear end. The extent of stability improvement is dependent on the interference between the inner wall of the hole and the plate surface.
5. Conclusion In this paper, a means is proposed to raise the damping capacity of the cutting tool system. The proposed means is the insertion of a thin plate into a rectangular hole cut in the cutting tool system so that the consumption of vibratory energy by friction acting between the plate surface and the inner wall of the hole is effectively used. The extent of damping capacity improvement is affected by the position of the hole and the interference between the rectangular hole and the inserted plate. An evaluation factor is introduced to estimate the degree of stability against the chatter vibration of the cutting tool system. By this factor, the damping capacity improvement due to the proposed means can be estimated. The stability of the cutting tool system against the chatter vibration is enhanced by the plate insertion. It is indicated that there is an optimum spring constant to maximize the stability increase. The results of this investigation suggest that the effective use of inner friction within the cutting tool system is also important for the damping capacity improvement.
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References [1] A. Cowley, A. Boyle, Active damper for machine tools, Annals of the CIRP 18 (1970) 213. [2] S. Kato, E. Marui, H. Kurita, Some considerations on prevention of chatter vibration in boring operations, ASME Journal of Engineering for Industry 91 (1969) 717. [3] M.M. Sadek, B. Mills, Effect of gravity on the performance of an impact damper: Part 1: Steady-state motion, Journal of Mechanical Engineering Science 12 (1970) 268. [4] S. Ema, E. Marui, Damping characteristics of an impact damper and its application, International Journal of Machine Tools and Manufacture 36 (1996) 293. [5] H.E. Merritt, Theory of self-excited machine-tool chatter: Contribution to machine-tool chatter research—1, ASME Journal of Engineering for Industry 87 (1965) 447. [6] M. Masuko, Y. Ito, Horizontal stiffness and micro-slip on a bolted joint subjected to repeated tangential static load, Bulletin of the JSME 17 (1974) 1494. [7] P.E. Rogers, G. Boothroyd, Damping at metallic interfaces subjected to oscillating tangential loads, ASME Journal of Engineering for Industry 97 (1975) 1087. [8] R.H. Thornley, F. Koenigsberger, Damping characteristics of machined joints loaded and excited normal to the joint face, Annals of the CIRP 19 (1971) 459. [9] A.S.R. Murty, K.K. Padmanabhan, Effect of surface topography on damping in machine joints, Precision Engineering 4 (1982) 185. [10] E. Marui, M. Hashimoto, S. Kato, Damping capacity of turning tools, Part 1: Effect of clamping conditions and optimum clamping load, ASME Journal of Engineering for Industry 115 (1993) 362. [11] M. Hashimoto, E. Marui, S. Kato, Damping capacity of turning tools, Part 2: Mechanism initiating the damping capacity, ASME Journal of Engineering for Industry 115 (1993) 366. [12] E. Marui, S. Ema, R. Miyachi, An experimental investigation of circular saw vibration via a thin plate model, International Journal of Machine Tools and Manufacture 34 (1994) 893.
Plate insertion as a means to improve the damping capacity of a cutting tool system Etsuo Maruia,*, Satoshi Emab, Masatoshi Hashimotoc, Yasunori Wakasawac a
Faculty of Engineering, Gifu University, 1-1 Yanagido, Gifu-shi 501-11, Japan b Faculty of Education, Gifu University, 1-1 Yanagido, Gifu-shi 501-11, Japan c Department of Mechanical Engineering, Toyota National College of Technology, 2-1 Eisei-cho, Toyota-shi 471, Japan Received 27 October 1997
Abstract Effective chatter prevention during cutting operations is achieved by increasing the damping capacity of a cutting tool system. It is well known that damping capacity is generated through (i) micro-slip at the interface between the tool shank and tool post, (ii) slip at the grain boundary within a vibrating body (that is, internal friction), and (iii) friction between the surface of the vibrating body and the surrounding air. Among these three causes of damping capacity, micro-slip at the interface between the tool shank and tool post is the greatest factor affecting the damping capacity of the cutting tool system. In the research investigation, it is shown that the damping capacity of a cutting tool system is improved by friction acting between the inner wall of a rectangular hole made at the overhanging shank of the cutting tool system and the surface of a plate inserted into this rectangular hole. The damping capacity improvement proposed in this paper is realized by a mechanism similar to the inner friction mechanism. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Damping capacity; Cutting tool system; Improvement; Friction; Plate insertion
1. Introduction Chatter vibration occurring in cutting operations is prevented by providing additional structural damping from an electro-dynamic active damper [1]. Although this method is quite effective,
*Corresponding author. 0890-6955/98/$19.00 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 8 ) 0 0 0 0 1 - 7
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special equipment is required and the cost is considerable. The application of a simple Lanchester damper [2] and impact damper [3,4], which do not have such inconvenience, was proposed. The effectiveness of these dampers depends strongly on the characteristics of the main vibratory system. The problem is, the damping characteristics must be adjusted for each and every machine tool system. On the other hand, if the damping capacity improvement of the cutting tool system is realized by some simple methods, machine tools can operate at full efficiency [5]. The damping capacity of the cutting tool system is generated through micro-slip at the interface between cutting tool and tool post, slip at the grain boundary within the vibrating body (that is, inner friction), and friction between the vibrating body and the surrounding air. The micro-slip between cutting tool and tool post has the greatest effect on the damping capacity generation. The damping effect of the friction by surrounding air cannot be expected. Much research has been reported on the effect of micro-slip at the tool post. The damping capacity of the vibration modes parallel to [6,7] and normal to [8] the contact surface has been measured. Surface topography influences the generation of the damping capacity by micro-slip [9]. The present authors investigated the mechanism of the damping capacity generation of the cutting tool system induced by micro-slip and the effect of clamping condition [10,11]. In a previous study, it was clarified experimentally by the present authors [12] that a remarkable improvement in damping capacity of a circular plate system can be expected by surface topography optimization of the clamping jig and the insertion of adhesive tape into the clamping interface. In the present study, to improve the damping capacity of the cutting tool system by a mechanism similar to the inner friction, a plate is introduced into the cutting tool shank. A rectangular hole is cut in the overhanging tool shank. Then a plate slightly thicker than the height h of the hole, is inserted into this rectangular hole to increase the damping capacity of the system by friction during vibration acting between the inner wall of the hole and the inserted plate surface. The effectiveness of this device in improving the damping capacity is reported herewith. 2. Experimental apparatus and experimental method 2.1. Experimental apparatus The experimental apparatus for an indicial response experiment is shown in Fig. 1. A vibratory body made from carbon steel, which is equivalent to a cutting tool system, is clamped by a vise via two load cells. The clamped portion of the vibratory body is made thick so as to have a high rigidity in order to decrease the influence of the clamping portion on the damping capacity. The distance between the two load cells is 55 mm. The displacement of the vibratory body is measured at the front end by an eddy current displacement sensor of non-contact type. The configuration of the vibratory body, which simulates a cutting tool system of cantilever type, is shown in Fig. 2. This vibratory body is called a cutting tool system, hereinafter. An elongated column with 15 mm-square cross section and 105 mm-overhang vibrates. A rectangular hole h mm high and 30 mm long is cut in the cutting tool system at a point 20 mm from the front end, as shown in Fig. 2(a). A thin plate is inserted into this rectangular hole, and the damping capacity improvement is expected to result from the friction generated between the inner wall of
E. Marui et al. / International Journal of Machine Tools & Manufacture 38 (1998) 1209–1220
Fig. 1. Experimental apparatus for indicial response experiment.
Fig. 2. Configuration of cutting tool system.
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the hole and the plate surface. In Fig. 2(b), the hole is cut at a point 20 mm from the rear end. A thin plate of 15 mm wide and 30 mm long is inserted into this rectangular hole. The plate is slightly thicker than the hole height h. Five cutting tool systems (TS, TSF5, TSR5, TSR6, TSR7) having a configuration shown in Fig. 2 are prepared, including a cutting tool system without the rectangular hole. The specifications of the cutting tool systems are given in Table 1. The letter “F” indicates that a rectangular hole is cut at the position 20 mm from the front end, and the letter “R” at the position 20 mm from the rear end of the cutting tool system. A cutting tool system with a plate of small interference inserted is shown by the letter “S” and of large interference by the letter “L”. In Table 1, the height of the rectangular hole, plate thickness, interference, spring constant and fundamental natural frequency of each cutting tool system are shown. The spring constant is obtained from a separate static load test, and the fundamental natural frequency is obtained from the below-mentioned indicial response experiment. Some of the experiments are carried out on the cutting tool system of straight shank type, in which the cross-sectional dimension of the overhang portion is the same as that of the clamped portion. 2.2. Experimental method The clamping load acting on the vise may be varied by eight settings from 6 kN to 20 kN. An impulse input is added in the direction vertical to the clamping surface of the vise by a plastic hammer, and the indicial response of the cutting tool system is then measured. The output displacement of the front end of the cutting tool system is measured by an eddy current type displacement sensor, and is analyzed by an FFT analyzer. A fundamental natural frequency is obtained. The damping capacity is estimated by the ratio between two successive amplitudes of damping vibration and the above-mentioned fundamental frequency. Hence, the damping capacity discussed in the following is the fundamental mode of vibration of the system. Table 1 Specification and characteristics of cutting tool system Tool system
Height of square hole (mm)
Plate thickness (mm)
Interference (m)
Spring constant (kN/m)
Natural frequency (Hz)
TS TSF5 TSF5S TSF5L TSR5 TSR5S TSR5L TSR6 TSR6S TSR6L TSR7 TSR7S TSR7L
– 4.89 4.89 4.89 4.88 4.88 4.88 5.87 5.87 5.87 6.90 6.90 6.90
– – 4.98 5.01 – 4.99 5.01 – 5.98 6.00 – 6.99 7.05
– – 90 120 – 110 130 – 110 130 – 90 150
2200 2240 2200 2250 2200 2200 2250 2020 2040 2070 1930 1970 2000
1033 1050 1000 1025 933 938 1023 933 938 1013 875 926 1013
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Damping capacity is indicated on the basis of the magnitude of the damping ratio, which is a ratio of the measured damping coefficient and the critical damping coefficient. 3. Damping capacity improvement by plate insertion 3.1. Effect of clamping portion on damping capacity The damping capacity of system TS, which has a thick clamped portion and no rectangular hole, is compared with that of the cutting tool system of straight shank type, whose clamped portion is the same size as the overhang portion. Using this result, the effect of the clamped portion on the damping capacity is examined. The result is shown in Fig. 3, in which the relation between the damping ratio of two separate cutting tool systems and the clamping load acting on one load cell is plotted. The damping capacity of the cutting tool system TS having a large clamped portion is little affected by the clamping load. On the contrary, in the cutting tool system of straight shank type, the following tendencies are recognized. The damping ratio within the small clamping load range is somewhat large, and decreases with the increase in clamping load. Any damping ratio beyond the clamping load of 14 kN is almost constant. This tendency in the damping ratio indicates that some vibratory displacement occurs at the clamped portion of the system corresponding to the vibration displacement at the overhang portion, and the damping capacity of this system is mainly generated owing to the vibration energy consumption due to the friction between the vise and the contact surface. The displacement at the clamped portion decreases with the increase of clamping load, and the vibration energy consumption becomes small. As a result, the damping ratio of the system TS shown in Fig. 3 is obtained. The displacement at the clamped portion tends to occur less readily as the clamped portion’s cross-section is large and the rigidity there is higher. Thus, the energy consumption due to friction at the clamped portion becomes small. This is why the damping ratio of the cutting tool system
Fig. 3.
Measurement example of damping ratio (effect of configuration at clamping portion).
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TS having a very rigid clamped portion is considerably lower than that of the straight shank type cutting tool system. 3.2. Effect of position of rectangular hole or plate insertion on damping capacity Some experimental results on the effect of the rectangular hole position on the damping capacity are given in Figs 4–6. Figure 4 shows the damping ratios for three cutting tool systems: TS having no rectangular hole. TSF5 having a rectangular hole 5 mm high at the front end of the system, and TSR5 having a rectangular hole 5 mm high at the rear end of the system. A plate is not inserted here. In the system TSR5, the damping ratio at the small clamping load range is somewhat large, however, it becomes constant beyond the clamping load of 14 kN. Although the damping ratio of system TSR5 is somewhat larger as the clamping load increases, there is not much difference among them. The damping ratio in the case in which a plate of small interference is inserted into the system of TSF5 and TSR5 is shown in Fig. 5. In this figure, the damping ratio of the system TS having no rectangular hole is also plotted. A weak clamping load dependency of the damping ratio is recognized in the cutting tool systems with a plate inserted, when the clamping load is small. The damping ratio becomes almost constant once the clamping load is larger than 14 kN. The magnitude of the damping ratio increases in the order of the cutting tool system with no hole, a rectangular hole at the front end, and a rectangular hole at the rear end near the clamped portion. When regarding the cutting tool system as a cantilever arrangement, the inclination angle of the cantilever becomes large going away from the clamped portion. So, the damping ratio is expected to be large when the plate is inserted in the hole near the front end. However, the reverse experimental result is obtained. This result is obtained because the deflection curve of the cutting tool system (that is, the vibration mode) is changed by the insertion of the plate into the rectangular hole and the interference between the inner wall of the hole and the inserted plate is different in each cutting tool system. Figure 6 shows the result for the same cutting tool systems as Fig. 5 with the insertion of a
Fig. 4. Measurement example of damping ratio (effect of position of rectangular hole, without plate insertion).
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Fig. 5. Measurement example of damping ratio (effect of position of rectangular hole, with plate insertion of small interference).
Fig. 6. Measurement example of damping ratio (effect of position of rectangular hole, with plate insertion of large interference).
plate having a large interference. The damping ratio becomes small with the increase of the interference, for both rectangular hole positions at the front or rear end of the cutting tool system. Owing to the increase in interference, the slip between the inner wall of the hole and the plate surface tends to occur less easily and the energy consumption by friction becomes small. 3.3. Effect of interference between rectangular hole and inserted plate The effect of the interference between the rectangular hole and the inserted plate on the damping ratio is given in Figs 7–9. These results are obtained for the nominal plate thicknesses of 5 mm, 6 mm and 7 mm. In the figures, the damping ratios obtained for the cutting tool systems TS
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Fig. 7. Measurement example of damping ratio (effect of interference, plate thickness of 5 mm).
Fig. 8. Measurement example of damping ratio (effect of interference, plate thickness of 6 mm).
(without rectangular hole), TSR5, TSR6 and TSR7 (with rectangular hole but no inserted plate) are plotted for reference. In what follows, we discuss the damping ratio at the clamping load beyond 14 kN, where the clamping load dependency of the damping ratio is not recognized. The damping ratio becomes large by the insertion of a plate of any thickness. Small interference is desirable for a plate 5 mm thick, and large interference is desirable for the plates 6 mm and 7 mm thick to improve the damping capacity. Thus, the difference in damping ratio in terms of the amount of interference may be explained by a combined effect of the lower slippage and greater friction resulting between the hole wall and the plate due to greater interference. In any case, it is confirmed that the damping ratio becomes large by inserting the plate. This shows that intentionally amplified friction acting within a vibrating portion is effective for damping capacity improvement. It also indicates specifically that the inner friction within vibrating materials must be used for the damping capacity improvement.
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Fig. 9. Measurement example of damping ratio (effect of interference, plate thickness of 7 mm).
4. Evaluation of damping capacity improvement 4.1. Evaluation method In the previous section, it is shown that the damping capacity can be improved and useful chatter prevention may be expected by inserting a plate of various nominal thicknesses and interference at various positions in a cutting tool system. In this section, the extent of damping capacity improvement is evaluated from the viewpoint of chatter prevention in cutting operations. The following discussion is based on the regenerative chatter vibration theory developed by Merritt[5]. A cutting tool system is approximated as a lumped parameter system of one degree of freedom, whose mass is m, the damping coefficient c and the spring constant k. Setting the tool displacement as x and the actual cutting depth of the tool as u, the equation of motion of this cutting tool system is written as follows: m
dx d 2x + kx = kcu +c 2 dt dt
(1)
where kc is a cutting stiffness. By Laplace transformation of Eq. (1), ms2X + csX + kX = kcU
(2)
In Eq. (2), s = j* (*: chatter frequency), then kc X = U (k ⫺ m*2) + jc* Real and imaginary parts of Eq. (3) are written as follows:
(3)
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Re =
kc(k ⫺ m*2) (k ⫺ m*2)2 + (c*)2
(4)
Im =
⫺ ckc* (k ⫺ m*2)2 + (c*)2
(5)
Eq. (3) is plotted on a complex number plane, where chatter frequency * is set as a parameter. Real part Re is plotted on the abscissa and the imaginary part on the ordinate. This curve is called the characteristic curve of the cutting tool system. Referring to the regenerative chatter vibration theory of Merritt, a vibratory system is stable when a whole characteristic curve exists in the right side zone of the vertical straight line Re = ⫺ 0.5. Differentiating Eq. (4) with respect to *, the minimum value of the real part Re is obtained as follows: (Re)min =
⫺ kc 4k(␦0 + ␦ 20)
(6)
where ␦0 is a damping ratio and is written as ␦0 = c/√mk. From Eq. (6), it is clear that the following condition is necessary to improve the stability of the cutting operations. Cutting rigidity kc must be kept small. Both the spring constant k and the damping ratio ␦0 of the characteristics of the vibratory system are as large as possible. Hence, the following evaluation factor for chatter prevention is defined. This factor corresponds to the denominator of Eq. (6). Now we evaluate the damping capacity improvement of all cutting tool systems in the present paper.
= k(␦0 + ␦ 20)
(7)
4.2. Evaluation result of damping capacity improvement Evaluation factor is calculated from the experimentally obtained damping ratio and spring constant for all cutting tool systems. The evaluated result is shown in Fig. 10, as a function of the spring constant. In the figure, experimental results for a different cutting tool system group are classified by the three different types of lines plotted. The chain line indicates the results of the evaluation of either a cutting tool system with the rectangular hole or a system with no hole and no plate insertion. The broken line is for evaluation of the system with the rectangular hole in the front end and a plate inserted. A solid line, on the other hand, is for a system with the hole in back and a plate inserted. Most of the experimental results are thus indicated by the solid line. The curves indicated by the broken and chain lines are for cases in which there were few experimental conditions, and their changes were expected. It is clear from the evaluation that there is a spring constant which maximizes the evaluation factor in cutting tool systems with an inserted plate.
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Fig. 10. Evaluation of damping capacity improvement.
The evaluation factor of the cutting tool systems with a plate inserted into the rectangular hole at the rear end is the largest among the three indicated cases. This agrees with the above-mentioned description regarding the damping ratio only. Thus, the maximum damping ratio or the maximum stability for the chatter vibration can be effectively raised by cutting a rectangular hole at the rear end. The extent of stability improvement is dependent on the interference between the inner wall of the hole and the plate surface.
5. Conclusion In this paper, a means is proposed to raise the damping capacity of the cutting tool system. The proposed means is the insertion of a thin plate into a rectangular hole cut in the cutting tool system so that the consumption of vibratory energy by friction acting between the plate surface and the inner wall of the hole is effectively used. The extent of damping capacity improvement is affected by the position of the hole and the interference between the rectangular hole and the inserted plate. An evaluation factor is introduced to estimate the degree of stability against the chatter vibration of the cutting tool system. By this factor, the damping capacity improvement due to the proposed means can be estimated. The stability of the cutting tool system against the chatter vibration is enhanced by the plate insertion. It is indicated that there is an optimum spring constant to maximize the stability increase. The results of this investigation suggest that the effective use of inner friction within the cutting tool system is also important for the damping capacity improvement.
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