Available online at www.sciencedirect.com
ScienceDirect Procedia CIRP 67 (2018) 274 – 277
11th CIRP Conference on Intelligent Computation in Manufacturing Engineering, CIRP ICME ‘17
Relation between power and linear model of dynamic cutting coefficients Dominika Śniegulska-Grądzkaa, Mirosław Nejmana, Krzysztof Jemielniaka,* a
Dept. of Automation, Machine Tools and Metal Cutting, Warsaw University of Technology, Narbutta 86, 02-524 Warsaw, Poland
* Corresponding author. Tel.: +48-22-234-86-14; fax: +48-22-849-02-85. E-mail address:
[email protected]
Abstract Fundamental importance for the stability analysis results has assumed model of the dynamic cutting coefficients. In most cases it is based on the model of cutting forces in steady state conditions. The models most often used are based on Kienzle power function and linear, proposed by Altintas. The paper presents interdependence of these models and the way of calculation one from another –linear from power and power from linear. The in-depth analysis, how the linear model of Altintas and Kienzle power function model can be potentially applied for modelling dynamic cutting characteristics is presented in the paper.
© 2017 The Authors. Published by Elsevier B.V.
© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the scientific committee of the 11th CIRP Conference on Intelligent (http://creativecommons.org/licenses/by-nc-nd/4.0/). Manufacturing Engineering. under responsibility of the scientific committee of the 11th CIRP Conference on Intelligent Computation in Manufacturing Engineering Peer-review
Computation in
Keywords: Cutting process dynamic characteristics; Modelling of cutting forces; Self-excited vibrations
1. Introduction The self-exciting vibrations are one of the most significant factors having the negative impact on the cutting process as well as the machining results. Therefore, there is a need for creating models for calculating the stability limit based on the cutting process characteristics and parameters of the massspring-damper (MSD) system. The dynamic characteristics of the cutting process, that is the dependence of the dynamic component of the cutting force on dynamic changes of cutting conditions, have been a subject of intensive research activities for a few decades [1, 4, 5]. The most important factor influencing changes of the cutting force is a variation of uncut chip thickness h influenced by current relative displacement between the workpiece and the tool r(t), called inner modulation of h, and waviness on the workpiece surface created in the previous tool pass r(t-T), called outer modulation of h, see Fig. 1: ݄ ൌ ݄ ݄ௗ ൌ ݄ െ ݎሺݐሻ ݎሺ ݐെ ܶሻ
r(t) – relative displacement between the workpiece and the tool (inner modulation of uncut chip thickness) , ݎሺ ݐെ ܶሻ – relative displacement between the workpiece and the tool in previous tool pass (outer modulation of uncut chip thickness), T – time between subsequent passes of the tool.
(1)
where: h, h0, hd – instantaneous, static (steady state) and dynamic uncut chip thickness,
Fig. 1. Variation of instantaneous uncut chip thickness h caused by inner and outer modulation.
2212-8271 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 11th CIRP Conference on Intelligent Computation in Manufacturing Engineering
doi:10.1016/j.procir.2017.12.212
Dominika niegulska-Gr dzka et al. / Procedia CIRP 67 (2018) 274 – 277
275
The formula (5) is a dynamic characteristic of the cutting process, which is base for stability limit calculation. E.g. for single-point machining operations stability lobes can be as described as follows [3]: ܾ ൌ
ିଵ
(7)
ଶೝ ୖୣሺீሻ
where: Re(G) – real part of the MSD system transfer function Thus crucial role in the determination of the dynamic cutting stiffness krd plays model of the cutting force dependence on uncut chip thickness (3). The purpose of this paper has been to compare the two most often used models – Altintas and Kienzle, determine the consequences of assuming one of them, and to deduce the formula allowing calculation one from another.
Fig. 2. Instantaneous cutting force Fr and its static Fr0 and dynamic Frd components.
The static cutting force ܨ is a function of steady state (static) uncut chip thickness, while the instantaneous force under vibratory cutting Fr is a function of instantaneous uncut chip thickness h:
2. Modeling of cutting force The most often used model of cutting force in stationary conditions is a power function proposed by Kienzle [6,7]: ݕ
ܨ ൌ ݂ሺ݄ ሻǢ
(2)
ܨ ൌ ݂ሺ݄ሻ
(3)
where Fr0, Fr – static and instantaneous radial cutting force. Vibration velocity r’(t) is much smaller than cutting speed, and slope of the tool path practically does not exceed some 2°÷3° – it is much smaller than in Fig. 1. Therefore, it can be assumed that relations (2) and (3) can be described by the same function – see Fig. 2 [5]. Consequently the instantaneous cutting force can be considered as a sum of its static and dynamic component: ܨ ൌ ܨ ܨௗ ൌ ݂ሺ݄ ሻ ݂ሺ݄ௗ ሻ
(4)
What matters in the stability analysis is not the whole instantaneous cutting force Fr but only its dynamic component Frd, because the static component Fr0 results in static deflection of the MSD system, with no influence on dynamic behavior of the system. Moreover, it is noteworthy that at the early stage of chatter development, interesting for stability analysis, vibration amplitude is small. Both the uncut chip thickness changes from hmin to hmax and instantaneous cutting force changes from Frmin to Frmax are small (see Fig. 2). Therefore function (3) can be linearized with a high degree of approximation. Thus dynamic component of cutting force can be quite exactly described as:
ଵ డிೝ ሺሻ డ
ቚ
where kr1.1 – unit specific cutting force - the cutting force required to cut a chip of cross section b·h=1mm·1mm. Substituting Eq. (8) into Eq. (6) dynamic cutting stiffness krd_K derived from Kienzle model can be expressed as follows: ሺ ݕെͳሻ
݇ௗ̴ ൌ ݇ଵǤଵ ݕ ݄Ͳ ݎ
(9)
The Kienzle model predicts both nonlinear dependence of cutting force on uncut chip thickness h due to changes of the shear angle, and growing stability limit accompanying increase of h (or feed). As the model is nonlinear, it is inconvenient, especially in milling where uncut chip thickness is inherently variable, sometimes from zero to the maximum, so there is not steady state uncut chip thickness. Nowadays the most often used in machine tool dynamics is Altintas model [2,3]. In accordance with this model the cutting force consists of two components. The first one is the edge component Fre related to the friction and ploughing on the cutting edge. This component is proportional to the length of the cutting edge b but is not dependent on uncut chip thickness h. The second one is cutting component Frc related to material shearing and it is proportional to undeformed chip cross section bh0. (10)
(5)
where krd – dynamic cutting stiffness – derivative of Fr(h) function at the steady state point (h=h0, Fr=Fr0) related to 1 mm of width of cut b:
(8)
ܨ ൌ ܨ ܨ ൌ ݇ ܾ ݇ ܾ݄
ܨௗ ൌ ݇ௗ ܾ݄ௗ
݇ௗ ൌ
ܨ ൌ ݇ଵǤଵ ܾ݄Ͳݎ
ୀబ
where b – width of cut.
(6)
where kre – edge coefficient, krc – cutting coefficient. Only the second component of the radial force Fr is influenced by changes of uncut chip thickness, consequently in accordance with this model the cutting coefficient is directly equal to the dynamic cutting stiffness krd_A in equation (5) and (7): ݇ௗ̴ ൌ ݇
(11)
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Fig. 3. Measurement data of average radial force Fr [1] and its modeling using Kienzle and Altintas models.
The Altintas model is linear, thus it is simple and very convenient. However, it does not take into account well known influence of the chip thickness on the friction and shear angles, and the yield shear stress, making cutting coefficient not constant, not linear [3]. To compare both models, results of cutting force measurement for orthogonal turning were used [1]. Cutting conditions were as follows: workpiece material AL 6061-T6, cutting speed vc=250m/min, width of cut b=2.3 rake angle Jo=12°, clearance angle Do=12°. Results of experiments and both model estimations are presented in Fig. 3. Kienzle model: kr1.1 = 227.5 N/mm2, mr = 0.436, yr = 0.564 ܨ ൌ ʹʹǤͷܾ݄Ǥହସ
(12)
Fig. 4. Comparison of dynamic cutting stiffness based on Kienzle and Altintas models.
In order to present the impact of dynamic cutting stiffness modelling on the on the stability limit calculation, the following example of 1DOF MSD system was used: mass m=100kg, dumping c = 2 Ns/mm, stiffness k = 50 kN/mm, natural frequency Z0=707.1 rad/s. For a such system minimum of stability limit can be described as: ܾ୪୧୫̴ ൌ
ఠబ
(16)
ೝ
Substituting dynamic cutting stiffness values and MDS system parameters to Eq. (16), the dependence between stability limit and uncut chip thickness (or feed) according to the Kienzle model is described by: ܾ୪୧୫̴ ൌ ͳͳǤͲ͵݄Ǥସଷ
(17)
Altintas model:
while in the Altintas model it is constant:
kre =31.1N/mm, krc = 301.6 N/mm2
ܾ୪୧୫̴ ൌ ͶǤ
ܨ ൌ ͵ͳǤͳܾ ͵ͲͳǤܾ݄
(13)
(18)
Both are compared in Fig. 5.
Correlation coefficient r2 is higher for the Kienzle model, which is understandable, but the difference is not very high.
4. Conversion of Kienzle model into Altintas model and vice-versa
3. Comparison of stability analysis based on Kienzle and Altintas model
It is quite common need in the research activities of the cutting process to use data from previous publications and experiments. Hence, it would be useful to have tools for simple and efficient transformation of parameters between those two models. The first transformation – from Kienzle to Altintas model consist in the linearization of entire instantaneous cutting force in working point (h=h0, Fr =Fr0). Altintas cutting coefficient – krc is equal to the dynamic cutting stiffness krd_K from Eq. (9). Substitution Eq. (9) into Eq. (5) results in:
Much more important are the consequences of the application of both models in stability analysis. Accordingly to Eq. (9) and results presented above, dynamic cutting stiffness for Kienzle model is: ݇ௗ̴ ൌ ͳʹͺǤ͵݄െͲǤͶ͵ Ͳ
(14)
while in the Altintas model it is constant:
௬ ିଵ
௬
ܨௗ ൌ ݇ଵǤଵ ܾݕ ݄ ೝ ݄ௗ ൌ ൫݇ଵǤଵ ܾ݄ ೝ ൯ݕ ݇ௗ̴ ൌ ͵ͲͳǤ Both are compared in Fig. 4.
(15)
బ
ൌ ܨ ݕ
బ
ሺͳͻሻ
Dominika niegulska-Gr dzka et al. / Procedia CIRP 67 (2018) 274 – 277
ݕ ൌ ೖೝ బ ೖೝ
ାబ
277
(27)
The unit specific cutting force kr1.1 can be derived from Eq. (8): ݇ଵǤଵ ൌ
ೝ ݕെͳ
௬ೝ ݄Ͳݎ
(28)
Again the values of the Kienzle model parameters depend on the assumed working point. It is reasonable to take the value h0 from the middle of the interesting range. Substitution h0=0.14 mm and Altintas model coefficients from Eq. (13) into (27) and (28) results in yr=0.576 and kr1.1=222.2 N/mm2 which is not far from the values determined directly from the experiments Fig. 5. Comparison minimum of stability limits based on Kienzle and Altintas models.
Entire instantaneous cutting force is the sum of static and dynamic component. Substitution of Eq. (19) into (4) results in: ܨ ൌ ܨ ܨ ݕ
బ
ൌ ܨ ቀͳ ݕ
బ
ቁ
(20)
The force on the cutting edge according to the Altintas Fre model can be determined by substituting: ݄ௗ ൌ െ݄ ,
(21)
into Eq. 20 and using Eq. (8), which results in: ௬
ܨ ൌ ܾ݇ ൌ ܨ ሺͳ െ ݕ ሻ ൌ ݇ଵǤଵ ܾሺͳ െ ݕ ሻ݄ ೝ ௬
௬
݇ଵǤଵ ܾ݄Ͳ ೝ ൌ ܾ݇ ݇ଵǤଵ ܾݕ ݄Ͳ ೝ
ሺʹʹሻ (23)
௬
(24)
Of course values of the Altintas model coefficients obtained from Kienzle model depend on assumed working point (steady state uncut chip thickness h0). For example taking h0=0.14 mm which is the average value from the experiments presented in Fig. 3 results in kre =32.7N/mm, krc = 302.3 N/mm2 which is quite close to the values determined directly from the experiments. For the purpose of calculating Kienzle model parameters from the Altintas model, the edge coefficient (Eq. 24) should be divided by cutting coefficient equal to dynamic stiffness krd_K from formula (9): ೝ ೝ
ൌ
బ ೝ ሺଵି௬ೝ ሻ ሺ ݎݕെͳሻ
௬ೝ ݄Ͳ
ൌ
ሺଵି௬ೝ ሻబ ௬ೝ
(25)
Hence Kienzle exponent is: ݕ ൌ
ೝ బ ೝ ାೝ బ
Finally:
The paper present that the two most commonly used dynamic cutting forces models, liner proposed by Altintas and based on Kienzle power function. It has been proved that those models are not equal, and using the simplified Altintas model has a significant impact on the obtained stability limits. On the other hand, developed methodology of conversion of one model into another allows for comparison and exploitation of data between them. Acknowledgements Financial support of Operational Programme Smart Growth - Project "Expert system design of machining process elements", Nr POIR.01.02.00-00-0013/15 is gratefully acknowledged. References
Hence edge coefficient in Altintas model is: ݇ ൌ ݇ଵǤଵ ݄ ೝ ሺͳ െ ݕ ሻ
5. Summary
(26)
[1] Altintas Y. Machining process Modeling, Machine Tap testing and Chatter Vibration Avoidance. MAL Inc. User Manual for CutPro.exe. Vancouver: Manufacturing Automation Laboratory; 2002. [2] Altintas Y, Kilic Z. M. Generalized dynamic model of metal cutting operations. CIRP Annals-Manufacturing Technology. 2013;62:47-50. [3] Altintas Y. Manufacturing Automation. Second Edition. New York: Cambridge University Press; 2012. [4] Das M. K, Tobias S. A. The Relation Between the Static and the Dynamic Cutting of Metals. International Journal of Machine Tool Design and Research. 1967;7:63-89. [5] Jemielniak K. Modelling of Dynamic Cutting Coefficients in Three Dimensional Cutting. International Journal of Machine Tools and Manufacture. 1992;32/4:509-519. [6] Kienzle O. Die Bestimmung von Kräften und Leistungen an spanenden Werkzeugen und Werkzeugmaschinen. VDI-Z. 1952;94:299-305. [7] Klocke F. Manufacturing Processes 1. Cutting. RWTH Edition. Berlin Heilderberg: Springer-Verlag; 2011.
