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A model for the cutting force system in the gear broaching process
~ Pergamon
Int, I. Mich. Tools MlumfJg:t. Vol. 37, No. 10, pp. 1409--1421, 1997 1997 Publi~ed by Elsevier ,Science Lid. All rights tts,c~ed Printed in Great Britain 0~J0..6955/97$17.00 + .00
PII: S0890-6955(97)00014-X
A M O D E L F O R T H E C U T I ' I N G F O R C E S Y S T E M IN T H E G E A R BROACHING PROCESS J.W. SUTHERLAND,t E.J. SALISBURY~t§ and F.W. HOGEI[ (Received 5 May 1995; in.final form 19 December 1996)
A~--A mechanistic model for the cutting force system in a broaching process is developed. The broach geometry considered in the analysis is for a production operation used to produce an internal helical ring gear. The model for the force system is based on a description of the instantaneous chip load geometry and a relationship betwcen the chip load and the three-dimensional cutting force system. The chip load geometry is based on the process kinematics and the involute-shaped cutting teeth of the broach. Shaping tests are performed to develop empirical equations for the cutting and thrust pressures. The model is validated through a comparison with actual forces generated during the production process. © 1997 Published by Elsevier Science Ltd
i. INTRODUCTION
The broaching process may be used to generate irregular internal and external part features, and therefore has many potential industrial applications. One application in which the broaching process has been used is the production of internal helical (ring) gears. For this application, one of the advantages of the broaching process, over other competitive processes such as bobbing, is its higher productivity. However, since the material removal rates are relatively high for broaching, the cutting forces are also high. Because large deflections are generally associated with large cutting forces, a broaching process may produce high surface errors (dimensional inaccuracies). For this reason, careful attention must be directed towards the design of the broach geometry and the selection of process conditions. Historically, broaches have been designed based on experience or through trial and error. As more emphasis is placed on part accuracy and precision, it becomes less likely that a satisfactory broach geometry can be designed based solely on experience. Additionally, as manufacturers struggle to reduce costs and nonproductive time, it becomes clear that trial and error approaches to tooling design will not be satisfactory. To design a broach early in the life cycle of a product, a model for how a broach will perform during the operation would be extremely advantageous. Such a model should be capable of predicting the finished part dimensional characteristics, given the raw part and broach geometries. As a first step towards the development of a comprehensive broach performance model, a sub-model for the cutting force system in the broaching process must be developed. Many mechanistic models for cutting force systems in machining processes are described in the literature. The overall structure of many of these models, and their similarities/differences are described by Smith and Tlusty [1] and Sutherland et al. [2] Models have been developed for single point tool processes such as turning [3] and boring [4, 5]. Models have also been developed for multi-point tool processes such as end milling [6-9], face milling [10], and more recently thread chasing [11]. These models all have as fundamental components sub-models for the chip load and chip load-force relationship. This paper describes a mechanistic model for the cutting force system in the gear broach-
tDopartment of Mechanical Engineering, Engineering Mechanics, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931, U.S.A. ~:Department of Industrial and Manufacturing Systems Engineering, Iowa State University, 205 Engineering Annex, Ames, IA 50011, U.S.A. §To whom all correslx:mdence should be addressed. IIAdvancedTransmission, Ford Motor Company, Livonia, MI, U.S.A. 1409
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ing process. The geometry of the specific broaching operation of interest will be described. Attention will then be focused on the model for the cutting force system. Like other mechanistic models, this force model is dependent on sub-models for the chip load and chip load-cutting force relationship. The sub-models developed for the broaching operation of interest will be described. Model conditioning will then be accomplished through a small set of tests performed using a shaping operation. The cutting force system model will then be validated through force data obtained from a production broaching operation. Finally, some conclusions and directions for future work will be presented. 2. MODELDEVELOPMENT The development of a mechanistic force model for a gear broaching operation, structurally similar to those described in the literature, requires that sub-models be developed for the following components: • The tool-work contact area (chip load). This sub-model should describe the chip load as a function of tool, work and process geometry. • The chip load-cutting force relationship. This sub-model should describe how the cutting forces change as a function of the chip load. Typically this model is empirical. The details concerning these two sub-models will be presented following a discussion of some of the geometric/kinematic details concerning the specific broaching operation under consideration.
2.1. Broach, part and process geometries The specific operation considered in this paper is the broaching of an internal helical gear. The process geometry and kinematics for such an operation are shown in Fig. 1. As is evident from the figure, the broach is pulled vertically through the uncut workpiece to generate the gear teeth. Additionally, to generate a helical gear, the broach is also rotated as it is moved through the part. Also apparent from the figure is the fact that the broach consists of a number of rows of teeth. The part and process geometries for the actual product and process of interest are given in Table 1. The shape of the teeth on a internal helical gear are involute helicoids [12]. The procedure by which this gear geometry is generated involves the use of three types of broach teeth (see Fig. 2): • Roughing teeth. These broach teeth remove the majority of the material located between the gear teeth. The geometry of these broach teeth are involutes. • Gear crest finishing teeth. These broach teeth are responsible for generating the final shape of the crests of the gear teeth. For this reason, they are also responsible for generating the inside diameter of the finished gear.
|
h
Rotational motion of the broac~
[I
l
Axial motion
of the broach
r"
Initial radius
~ [
Rows of parallel teeth
I
I
j
Workpieoe
I I
I
T
r|se pet
I
Fad View of Workpiec¢ Fig. I. Gear broaching process geometry and kinematics.
A model for the cutting force system in the gear broaching process
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Table I. Broach and work geometry Broach geometry
Number of broach sections: 7 Rp = Pitch radius (transverse): 2.41481 in. a = Rake angle (roughing leeth): 15° Number of teeth per row: 94 = Pressure angle (transverse): 19.1508° = Rake angle (finishing teeth): 18° Initial part diameter:. 4.7750 in = Helix angle: 22° Pan height 1.7323 in. V = Cutting speed: 30 fl/min
Work geometry and proce~ conditions
Roughii$ Tooth
-Bue Circle
f
~
Pitch Circle
Flank Finishing Tooth age trmn nnmmc~
~..arteothprofit,
/
Dc~i\rl[/len~d~for "" BaseCircle finished gear
Fig. 2. Broach tooth geomeUies.
• Finishing teeth. These broach teeth are responsible for generating the final shape of the flanks of the gear teeth. The shape of these teeth are also based on an involute geometry. The roughing teeth all have the same involute geometry (pressure angle, pitch radius, teeth per row, etc.). The cutting edge for the roughing teeth is at the tooth radius, Ri. For the roughing teeth, the difference between the radius of a given tooth of interest and the tooth that generated the surface being cut (Ri - R i - ,) is termed the rise per tooth or uncut chip thickness. As is evident from Fig. 2, the crest finishing gear teeth can be described by a single parameter, the tooth radius, R~. As with the roughing teeth, the cutting edge is at the radius of the tooth, and the difference between the radius of the current tooth and the tooth that generated the surface being cut defines the uncut chip thickness. The broach flank finishing teeth are responsible for generating the final shape of the flanks of the gear teeth. Consequently, the geomelrj of these broach teeth is based on the geometry of the gear teeth, for example the pressure angle, pitch radius, etc. In this case, the radii of the roughing teeth become successively larger along the axis of the broach and therefore the tip of the roughing tooth is responsible for performing the cutting process.
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For the finishing teeth, however, the tooth radius is constant. Thus the finishing teeth remove material by using teeth with successively larger tooth thicknesses. The thickness of a given finishing tooth is then based on the geometry of the gear teeth to be generated and some offset of the broach tooth from the final gear tooth profile. To further complicate the geometry of the finishing teeth, only one side of the gear tooth profile is generated at a time. In other words, several rows of finishing teeth are used to generate one side of a gear tooth, and then several rows of finishing teeth are used to generate the opposing side. To avoid interference of the noncutting edge of a finishing tooth, an offset may be employed for the noncutting edge. As noted previously, the geometry of the crest finishing broach teeth may be described simply with a radius. The shape of both the roughing and finishing broach teeth may be described using an involute geometry. The arc tooth thickness of the roughing and finishing teeth is therefore given by [12] t = 2r(2-~p + (tan 4) - 4 ) ) - (tan ( p - (~)) -- ml -- mr
(l)
where: tp is the arc tooth thickness at the pitch circle; Rp is the radius of the pitch circle; r is the radius at which the tooth thickness is to be determined; 4) is the pressure angle corresponding to pitch radius, Rp; q~ is the pressure angle corresponding to radius, r, and is equal to cos - '(RJr); Rb is the radius of the base circle, and is equal to (Rp cos4)); Z~ is the offset of the cutting edge on the left side of a tooth from the involute profile; A is the offset of the cutting edge on the right side of a tooth from the involute profile; and t is the arc tooth thickness corresponding to a radius, r. The broach geometry parameters as listed above are defined in the transverse direction (as opposed to the normal direction). Therefore, Eqn (1) will yield the transverse tooth thickness. It may also be noted that the offsets, AI and A, are utilized only for the finishing teeth, i.e. they are zero for the roughing teeth. The roughing, crest finishing, and flank finishing teeth are typically organized on a broach in a fashion similar to that displayed in Table 2 (broach geometry for an actual production machining operation). Typically included among the broach's cutting teeth are a number of float teeth which are used to lengthen a broach's useful life. These float teeth are extra teeth interspersed among the cutting teeth and do not participate in the cutting operation until the radii of previous teeth have been reduced due to tool wear and resharpening. The float teeth ensure the dimensions of the manufactured part remain accurate, so care must be taken when sharpening broaches to avoid oversharpening the float teeth. Table 2. Broach sections and broach teeth geometry Section
I 2 3 4 5 6 7
Tooth type
Number of rows in section
Starting diameter
Approximate rise per tooth
Roughing Roughing Roughing Roughing Crest finishing Flank finishing Flank finishing
4 53 21 8 7 8 8
4.7750 4.7930 5.0040 5.0540 4.7725 5.0450 5.0450
0.00225 0.00200 0.00125 0.00025 0.00088 0.00055* 0.00055*
*Tooth offset ,~, or A~.
Starting axial Axial spacing position
0.0000 2.3750 28.875 39.375 39.953 50.000 54.000
0.5938 0.5000 0.5000 0.9688 0.0688 0.5000 0.5000
A model for the cutting force system in the gear broaching process
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2.2. Chip load model In the preceding section, the geometry and kinematics of the internal helical gear broaching operation were described. Additionally, models for the geometry of the broach teeth were presented. Attention now turns to the development of a model for the chip load, or in other words the time-varying tool-work contact area. The chip loads must be defined in a plane perpendicular to the cutting direction, i.e. the normal direction of the helical gear. However, since the geometry of the broach teeth is provided in the transverse direction, the transverse chip load, At, for each type of broach teeth will first be determined and then later be converted to normal chip loads, AN. As was previously mentioned, for the roughing teeth the tip of the tooth does the cutting. The transverse chip load for the roughing teeth is indicated by the shaded area in Fig. 3(a). This area, ArR (the second subscript, "R", indicates that a roughing tooth is considered), can be approximated by Eqn (2): Ri
A~ = Ri
f ~
R~
2r{ 2 ~ + (tan(4~) - 4~) - (tan(~0) - ~0)}dr
tdr= I
Ri -
(2)
I
which is the integral of the tooth thickness [Eqn (1)] from the previous tooth radius, Ri- i, to the current tooth radius, Ri. The result of this integration, after substituting q~ = cos-I(Rdr) is shown in Eqn (3). ATR =
{r2~r -~+r2(tanck-
-
R~(r 2 -
6)-
2 3 ~b (r2-R~)
/ (3)
R~)~ ---
|
The transverse chip load for a crest finishing tooth is obtained in a similar fashion as the chip load for the roughing tooth. The crest finishing teeth cut the inner diameter of the workpiece. The transverse chip load for a crest finishing tooth, Arc, is indicated by the shaded area in Fig. 3(b), and is given by the calculation shown in Eqn (4):
//ft'•orkpiece-•• Roughing Tooth (a)
Rfm
Crest Finishing Tooth (b)
Rfnfia Fhmk FinishingTooth (¢)
Fig. 3. Chip load a~as (transverse sections).
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m~
rr(R/2 - R/z- i) N
et al.
{ ra'n" -
~
2 + r~(~n 6 - 4,) - ~ ,
3 (r ~ - R~,)
(4) Ri- I
In Eqn (4), the first term is an annulus with inner radius Ri - ~ and outer radius Ri divided by the number of teeth. This gives an annulus arc area encompassing one tooth on the gear and a neighboring gap. The second term in Eqn (4) is the area of the annulus over one gap (the involute shape of the last roughing tooth). The second term is subtracted from the first term to give the transverse chip load, Arc. The transverse chip load for a flank finishing tooth is indicated by the shaded area in Fig. 3(c). The calculated chip load, Arv, for the flank finishing tooth is given by Eqn (5): A x v = (Rfm~
(5)
- Rfmin)A
The shaded area has a uniform width, A, with a height of Rf,~ - Rfm~,, where Rfmax is the maximum radius of the finishing tooth, Rf,,i. is the radius of the last crest finishing tooth and A is the rise per tooth. It can be noted that the cutting edge on a flank finishing tooth is curved and not symmetric about the vertical axis as is the case with both the roughing and crest finishing teeth. This results in a nonzero average cutting edge orientation angle, C., (analogous to the lead angle for a single point tool). This angle is equal to the average of ~0 - / 3 over the portion of the flank finishing tooth that is involved in cutting (see Fig. 4). The calculation is as follows:
I
(~
Cs= ~ - ~ - "
-
fl)dr
(6)
r2 -- rl
where
r!
Rb
Fig. 4. Diagram of cutting edge orientation angle.
A model for the cutting force system in the gear broaching process
tp + t a n t ~ - ~ q~= a cos ( ~ ) and/3 = 2rp
1415
tanq~+~
After substitution and integration,
c,-
-
to
N2
~-~
2rp
r2
-
NI r)
(7a)
where I
1
Nl=~
~--
1
r , - R bIn
N,=2
~-
I
r2-Rbln
I
+
- 1
(7b)
I
(7c)
and ~
+ ~-
Equations (3)-(5) represent the chip loads in the transverse direction; however, the relations to be developed in the next section assume the chip load is defined in a plane perpendicular to the cutting velocity. Since the velocity direction is defined by the helix angle, $, the chip loads in the normal direction are needed. The normal chip loads, As, are calculated by multiplying the transverse chip loads by the cos(S) as shown in Eqn (8): AN = AT COS $
(8)
With the relations for the chip loads established, the cutting force relationships can now be developed. 2.3. Cutting force model To understand the cutting force system in the broaching process consider the oblique cutting process displayed in Fig. 5. In Fig. 5, V is the cutting velocity, Vc is the chip flow velocity, i is the inclination angle, tic is the chip flow angle, a. is the normal rake angle and ae is the effective rake angle. For such a process, the chip load is the product of the undeformed chip thickness and the width of cut. The mechanistic modeling approach considers a plane containing the cutting velocity and chip velocity vectors as the "orthogonal cutting plane" [13]. This plane may be determined given an assumed chip flow angle (in
Fig. 5. Diagram of oblique cutting (adapted from ref. [13]).
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et aL
this case estimated using Stabler's chip flow rule, r/¢ = i [14]) and knowledge of the tool geometry. For the orthogonal cutting plane, the cutting and thrust forces are related to the chip load by the following relations: F c = KcAN, F-rI.IR =
KTAN
(9)
where Kc and Kr are the cutting and thrust pressures, respectively, and AN is the chip area. In practice the cutting and thrust pressures are dependent on such variables as the undeformed chip thickness, cutting speed and rake angle [15]. It is the use of empirically based relations for Kc and Kx that distinguish this mechanistic modeling approach from approaches that attempt to describe cutting processes at an even more fundamental level, e.g. shear angle relationships [16--18] and finite element models [19] for the cutting process. In this empirical approach, which will be described in Section 2.4, the effective rake angle (the angle between the V¢ and the line OB which is the projection of the Z-axis onto the V-V, plane as shown in Fig. 5) is required to determine the cutting and thrust pressures. For the broaching process, the effective rake angle can be evaluated by or, = sin*-*ffsin2i + cos2i sin an)
(10)
where an is the normal rake angle, and i is the inclination (and assumed chip flow) angle [13]. Based on ASA cutting tool definitions [13]: i = a m [ t a n Orbcos Cs - tan as sin Cs]
(ll)
a, = atan {[tan ax cos Cs + tan ab sin Clcos i}
(12)
where ab is the back rake angle, as is the side rake angle and Cs is the average cutting edge orientation angle as defined in Eqn (7a). The force components of Eqn (9) can be resolved into the (X,Y,Z) coordinate system shown in Fig. 5 by Fx = KcAN, Fr = Ka,AN sin(0), Fz = KTAN cos(0)
(13)
where Fz, Fr and Fx are the forces acting on the workpiece, and 0 is the angle between the thrust force direction OB and the Z-axis in Fig. 5 and is given by
0 = atan
sin r/~ cos i - cos r/c sin t~. sin i] cos r?~c-os ot~
J
(14)
Typically, these forces are then resolved into three force components in an external coordinate system using the process and cutting geometries. The external coordinate system is chosen to coincide with the coordinate system that defines the force components directly measured during cutting operations. These measured forces are subsequently used to validate the force model. The forces defined by the external coordinate system in the gear broaching process include the forces acting on the workpiece in the radial (F~At>), axial (FAx) and tangential directions (Fr^N), as shown in Fig. 6. To determine bRAD, FAX and FTAr~, the forces shown in Eqn (13) must be resolved into the external coordinate system. However, due to different cutting geometries, the forces for each tooth type (roughing, crest finishing, flank finishing) are resolved into the external coordinate system in a different manner. Equations that have been previously established [Eqns (10)-(14)] are based on singlepoint tool geometry variables such as as, ab, etc. The broach teeth geometry needs to be mapped into that form for each set of broach teeth. The roughing and crest finishing teeth in the broaching process both have the same cutting geometry coordinate system. The
A model for the cutting force system in the gear broaching process
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FZ FX
Normal Section
a) Roughing and Crest FinishingTeeth
J
Normal Section b) Right Rank Finishing Tooth
J
Normal Section
c) Left Flank Finishing Tooth Fig. 6. Force components for broach teeth.
back rake angle (Orb) is equal to the helix angle (~) and the side rake angle (a0 is equal to the tool rake angle (at) that is given in Table 2 and shown in Fig. 1. The cutting forces of Eqn (13) are shown in Fig. 6(a), and are resolved into the axial, radial and tangential components by Eqn (15): FrAN = Fx sin ~ + Fr cos qJ, FAx = Fx cos ~b - Fr sin ~, FRAD = F z
(15)
For the roughing and crest finishing teeth the undeformed chip thickness, a,, is equal to (Ri - Ri _ 1) and the effective rake angle can be determined from Eqn (10). The forces due to the right flank finishing teeth are shown in Fig. 6(b). In this case the back rake angle is equal to the tool rake angle and the side rake angle is equal to the helix angle. The cutting forces of Eqn (13) can be resolved into the axial, radial and tangential components by Eqn (16): F'rAr~ = Fx sin $ + Fz cos q/, FAx = Fx cos ~ - Fz sin ~, FRAD = f v
(16)
The forces due to the left flank finishing teeth are shown in Fig. 6(c). The back rake angle is equal to the tool rake angle and the side rake angle is equal to the negative of the helix angle. The cutting forces of Eqn (13) can be resolved into the axial, radial and tangential components by Eqn (17): FTAr~ = Fxsin ~ - Fzcos $, FAx = Fxcos q/+ Fzsin Ik, FRAD= Fr
(17)
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For the right and left flank finishing teeth the undeformed chip thicknesses are equal to A~cos$ and Atcos~b, respectively and the effective rake angle can be determined by Eqn (10). 2.4. Model conditioning In order to accurately predict the forces in the broaching process, models for the cutting pressures Kc and Kr need to be developed. Suthedand et al. [15] demonstrated that models depending on tool geometry (rake angle), cutting speed and chip thickness adequately characterize the cutting and thrust pressures. To obtain the necessary data for the development of pressure models, a set of 12 orthogonal cutting experiments were conducted using a shaper. Tests were run at three rake angles, two cutting speeds and two chip thicknesses while using the same work material as used in the broaching operation. For each of the 12 tests, the cutting and thrust forces were measured and from knowledge of the chip load, Ac, the cutting and thrust pressures Kc and Kr were calculated. The results of the 12 tests are shown in Table 3. Using the data from the 12 tests, models for Kc and Kr, in the form given by Sutherland et al. [15] can be determined. However, due to the difficulty in accurately modeling cutting pressures over a wide range of rake angles, two different models were developed for each pressure, one for the positive and one for the negative rake angles. The models for neutral to positive rake angle are given in Eqn (18), with the models for neutral to negative rake given by Eqn (19). ln(Kc) = 10.80844 - 0.51153 In(at) - 0.10801 In(V) - 1.64854ote, ln(Kr) = 10.66259 - 0.49354 In(at) - 0.10313 In(V) - 7.43166ae
(18)
ln(Kc) = 12.08827 - 0.24925 In(at) - 0.11320 In(V) + 0.174641a e, ln(Kr) = 11.30114 - 0.23638 In(at) - 0.00350 In(V) - 0.82360t~
(19)
It may be noted that Eqns (18) and (19) provide slightly different values for the cutting and thrust pressures for neutral rake angles at the chip thicknesses employed in the broaching process; this is because the model conditioning experiments were performed at chip thicknesses higher than those used in the broaching process. Smaller chip thicknesses could not be examined in the model conditioning experiments due to difficulties with system deflections. 3. MODEL VERIFICATION
To validate the model for the broaching process it is necessary to compare actual process forces with model predicted forces. The actual force signal was obtained by instrumenting Table 3. Kc and KT calibration data Test
I 2 3 4 5 6 7 8 9 l0 II 12
Rake (°)
Speed (ft/min)
Chip thickness (in.)
A, (in 2)
20 20 20 20 3 3 3 3 - 20 - 20 - 20 - 20
16.7 62.5 16.7 62.5 16.7 62.5 16.7 62.5 16.7 62.5 16.7 62.5
0.004 0.004 0.008 0.008 0.004 0.004 0.008 0.008 0.004 0.004 0.008 0.008
0.001 0.001 0.002 0.002 0.001 0.001 0.002 0.002 0.001 0.001 0.002 0.002
Cutting Thrust force (lbf) force (Ibf)
350 340 480 370 500 480 890 700 450 425 875 675
37 35 51 41 310 275 500 430 340 425 680 700
Kc (Ibf/in2)
KT (Ibf/in2)
350 000 340 000 240 000 185 000 500000 480 000 445 000 350 000 450 000 425 000 437 500 337 500
37 000 35 000 25 500 20 500 310000 275 000 250 000 215 000 340 000 425 000 340 000 350 000
A model for the cutting force system in the gear broaching process
1419
the production broaching operation with a strain-gage sensor and then digitally recording this signal. The sensor measured the axial force transmitted through the workpiece and applied to the fixturing. The forces measured during the production of a typical ring gear are shown in Fig. 7(a). The model for the cutting force system in the broaching process that has been described was used to establish a computer-based simulation model for the broaching process. The simulation model permitted a user to manipulate the work, process and tool geometries, and then explore how such changes would impact the time-varying forces applied to the workpiece. For the conditions specified previously, the simulation model was used to predict the axial force applied to the workpiece by the broach. This force signature is depicted in Fig. 7(b). A comparison of the actual forces of Fig. 7(a) and the predicted forces of Fig. 7(b) shows good agreement. This indicates that the model adequately models the process behavior. Examining Fig. 7, there appears to be six distinct regions within the force signature. The first region (0--12 in. of broach movement) is associated with the entry of the broach into the part. Some difference between the actual and model predicted forces may be attributed to the fact that the part deforms elastically (breathes) under the application of the cutting forces. The high forces in this entry region may also produce increased wear levels that significantly change the broach teeth geometry. In the second region (12-30 in.), also associated with the roughing section of the broach, the model prediction matches the actual force very well. The third region evident in the force signatures (30--40 in.) is associated with the broach teeth that have the largest radius. Differences between the actual and model predicted forces suggest that an improvement is required in describing the geometry of these teeth. The fourth (40-50 in.), fifth (50-55 in.) and sixth (55--60 in.) regions of the force signal are associated with the crest finishing, and fight and left flank finishing teeth, respectively. Once again the forces in these three regions match very well. 4. SUMMARY AND CONCLUSIONS
A model for the cutting force system in the broaching process has been developed. This model is based on sub-models that describe the instantaneous area of contact (chip load) 104
A 3
1 L
A
A
I
A
10
20
30
40
50
60
4o
5o
60
~
M~o~nt
a) Experimental Results
4
x 104
3
8.2
£
1
0 0
t
&
i
10
2o
3o 84r~J~ Movon~nt
b) Simulation Results Fig. 7. Axial forces.
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J.W. Sutherland et aL
between the broach and the workpiece and the dependence of the force system on the chip load, cutting edge orientation and cutting speed. The chip load sub-model utilized the process geometry and kinematics to characterize the amount of engagement of each broach tooth with the workpiece. Three types of broach teeth were considered: (i) roughing teeth that remove the majority of the material located between the gear teeth; (ii) crest finishing teeth that are responsible for generating the final shape of the crests of the gear teeth (i.e. responsible for generating the inside diameter of the finished gear); and (iii) finishing teeth that are responsible for generating the final shape of the flanks of the gear teeth. The geometry for both the roughing and finishing teeth were characterized by an involute geometry. Given knowledge of the instantaneous chip load as a function of time, force relationships as a function of chip load were established. These relationships consider an orthogonal cutting plane and relate cutting and thrust forces to the chip load via coefficients for cutting and thrust pressures. After coordinate system transformation, the cutting and thrust forces may be mapped into values for the axial, radial and tangential forces on the workpiece. Empirical models for the cutting and thrust pressures as a function of undeformed chip thickness, effective rake angle and cutting speed were developed using test data obtained from a shaping process. A computer-based representation of the cutting force system was then used to predict the axial forces arising under actual production conditions. These predicted forces accurately matched the measured axial force signature for the same production conditions. The research effort described herein was driven by the need to establish a computerbased tool to predict the force behavior of the broaching process under different broach geometries, process settings and workpiece conditions. Specific research findings resulting from this activity include: • The chip load for an involute-shaped tooth can be described via the equations presented here. • From a theoretical standpoint, there is a significant difference between the left and right flank finishing teeth on the broach owing to the difference between their cutting geometries with respect to the velocity vector. This difference was also evident from the measured axial force signal. • Both the measured and model predicted forces indicate that the forces associated with the flank finishing are large and significant when compared to the forces associated with the roughing teeth. • The forces associate with the crest finishing of the gear teeth are relatively small. • Empirical relations between force and chip load developed for a shaping process can be applied to other nonshaping operations. • The entry-exit behavior of the broach results in an oscillatory behavior in the force signal. Such cyclical behavior may excite structural vibrational modes within the broaching machine, or be transmitted through the floor to other machines. These findings provide further support that the mechanistic modeling procedure is an effective technique for the characterization of machining operations. Several unresolved issues were identified during the course of this investigation and represent potential topics for continuing research study. As noted, it would be desirable to use the forces predicted by the broach force model together with a model for the structural response of the system to produce surface error/accuracy predictions for a broached part. Additionally, the broaching forces may be used to design fixturing for the operation. Small inaccuracies in the force signal for the region where the gear root is formed were identified, and it is believed that further study is needed to better understand the process in this region. Finally, the oscillatory nature of the broaching force signature suggests that modifications to the broach geometry may be investigated with the hope of "smoothing out" the force signal.
A model for the cutting force system in the gear broaching process
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Acknowledgements--The authors gratefully acknowledge the assistance of David Dodds and the financial support of the Ford Motor Company during the course of this research. The model conditioning experiments described herein were capably performed by Dr Robert Radalescu (now at Michelin) in the Chao-Trigger machining research laboratory at the University of Illinois at Urbana--Champalgn. Finally the authors would like to thank Professors R.E. DeVor and S.G. Kapoor of the University of Illinois for their helpful comments during this research investigation. REFERENCES [I] Smith, S. and Tlusty, J., J. Engng Ind. Trans. ASME, 1991, 113, 169-175. [2] Sutherland, J. W., DeVor, R. E., Kapoor, S. G. and Ferreira, P. M., SAE Trans., 1988, 97(5), 215-226. [3] Endres, W. J., Sutherland, J. W., DeVor, R. E. and Kapoor, S. G., ASME Bound VoI.-PED, 1990, 44, 193-212. [4] Subramani, G., Suvada, R., Kapoor, S. G., DeVor, R. E. and Meingast, W., Proc. N. Am. Manufact. Res. Conf., 1987, 15, 439--446. [5] Zhang, G. M. and Kapoor, S. G., J. Engng Ind. Trans. ASME, 1987, 109(3), 219-226. [6] Kline, W. A. and DeVor, R. E., Int. J. Mach. Tool Des. Res., 1983, 23(2), 123. [7] Tlusty, J., J. Engng Ind. Trans. ASME, 1986, 108, 59-67. [8] Sutherland, J. W., ASME Bound Vol.-PED, 1988, 33, 53-62. [9] Montgomery, D. and AItintas, Y., J. Engng Ind. Trans. ASME, 1991, 113, 160--168. [10] Fu, H. J., DeVor, R. E. and Kapoor, S. G., J. Engng Ind. Trans. ASME, 1984, 106(!), 81-88. Ill] Sutherland, J. W. and Zdeblick, W. J., SAE Trans., 1992, 101(5), 711-719. [12] J. E. Shigley and J. J. Uicker, Theory of Machines and Mechanisms. McGraw-Hill, New York (1980). [13]. N. H. Cook, Manufacturing Analysis. Addison-Wesley, Reading, MA 0966). [14] Stabler, G. V., Proc. Inst. Mech. Engrs, 1951, 165, 14-26. [15] Sutherland, J. W., Subramani, G., Kuhl, M. J., DeVor, R. E. and Kapoor, S. G., Trans. NAMRI/SME, 1988, 16, 264-272. [16] Usui, E., Hirota, A. and Masuko, M., J. Engng Ind. Trans. ASME, 1978, 100, 222-228. [17] Usui, E. and Hirola, A., Z Engng Ind. Trans. ASME, 1978, 100, 229-235. [18] Usui, E., Shirakashi, T. and Kitagawa, T., J. Engr. Ind. Trans. ASME, 1978, 100, 236-243. [19] Strenkowski, J. S. and Mitchum, G. L., Proc. N. Am. Manufact. Res. Conf., 1987, 15, 506-509.
Int, I. Mich. Tools MlumfJg:t. Vol. 37, No. 10, pp. 1409--1421, 1997 1997 Publi~ed by Elsevier ,Science Lid. All rights tts,c~ed Printed in Great Britain 0~J0..6955/97$17.00 + .00
PII: S0890-6955(97)00014-X
A M O D E L F O R T H E C U T I ' I N G F O R C E S Y S T E M IN T H E G E A R BROACHING PROCESS J.W. SUTHERLAND,t E.J. SALISBURY~t§ and F.W. HOGEI[ (Received 5 May 1995; in.final form 19 December 1996)
A~--A mechanistic model for the cutting force system in a broaching process is developed. The broach geometry considered in the analysis is for a production operation used to produce an internal helical ring gear. The model for the force system is based on a description of the instantaneous chip load geometry and a relationship betwcen the chip load and the three-dimensional cutting force system. The chip load geometry is based on the process kinematics and the involute-shaped cutting teeth of the broach. Shaping tests are performed to develop empirical equations for the cutting and thrust pressures. The model is validated through a comparison with actual forces generated during the production process. © 1997 Published by Elsevier Science Ltd
i. INTRODUCTION
The broaching process may be used to generate irregular internal and external part features, and therefore has many potential industrial applications. One application in which the broaching process has been used is the production of internal helical (ring) gears. For this application, one of the advantages of the broaching process, over other competitive processes such as bobbing, is its higher productivity. However, since the material removal rates are relatively high for broaching, the cutting forces are also high. Because large deflections are generally associated with large cutting forces, a broaching process may produce high surface errors (dimensional inaccuracies). For this reason, careful attention must be directed towards the design of the broach geometry and the selection of process conditions. Historically, broaches have been designed based on experience or through trial and error. As more emphasis is placed on part accuracy and precision, it becomes less likely that a satisfactory broach geometry can be designed based solely on experience. Additionally, as manufacturers struggle to reduce costs and nonproductive time, it becomes clear that trial and error approaches to tooling design will not be satisfactory. To design a broach early in the life cycle of a product, a model for how a broach will perform during the operation would be extremely advantageous. Such a model should be capable of predicting the finished part dimensional characteristics, given the raw part and broach geometries. As a first step towards the development of a comprehensive broach performance model, a sub-model for the cutting force system in the broaching process must be developed. Many mechanistic models for cutting force systems in machining processes are described in the literature. The overall structure of many of these models, and their similarities/differences are described by Smith and Tlusty [1] and Sutherland et al. [2] Models have been developed for single point tool processes such as turning [3] and boring [4, 5]. Models have also been developed for multi-point tool processes such as end milling [6-9], face milling [10], and more recently thread chasing [11]. These models all have as fundamental components sub-models for the chip load and chip load-force relationship. This paper describes a mechanistic model for the cutting force system in the gear broach-
tDopartment of Mechanical Engineering, Engineering Mechanics, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931, U.S.A. ~:Department of Industrial and Manufacturing Systems Engineering, Iowa State University, 205 Engineering Annex, Ames, IA 50011, U.S.A. §To whom all correslx:mdence should be addressed. IIAdvancedTransmission, Ford Motor Company, Livonia, MI, U.S.A. 1409
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et al.
ing process. The geometry of the specific broaching operation of interest will be described. Attention will then be focused on the model for the cutting force system. Like other mechanistic models, this force model is dependent on sub-models for the chip load and chip load-cutting force relationship. The sub-models developed for the broaching operation of interest will be described. Model conditioning will then be accomplished through a small set of tests performed using a shaping operation. The cutting force system model will then be validated through force data obtained from a production broaching operation. Finally, some conclusions and directions for future work will be presented. 2. MODELDEVELOPMENT The development of a mechanistic force model for a gear broaching operation, structurally similar to those described in the literature, requires that sub-models be developed for the following components: • The tool-work contact area (chip load). This sub-model should describe the chip load as a function of tool, work and process geometry. • The chip load-cutting force relationship. This sub-model should describe how the cutting forces change as a function of the chip load. Typically this model is empirical. The details concerning these two sub-models will be presented following a discussion of some of the geometric/kinematic details concerning the specific broaching operation under consideration.
2.1. Broach, part and process geometries The specific operation considered in this paper is the broaching of an internal helical gear. The process geometry and kinematics for such an operation are shown in Fig. 1. As is evident from the figure, the broach is pulled vertically through the uncut workpiece to generate the gear teeth. Additionally, to generate a helical gear, the broach is also rotated as it is moved through the part. Also apparent from the figure is the fact that the broach consists of a number of rows of teeth. The part and process geometries for the actual product and process of interest are given in Table 1. The shape of the teeth on a internal helical gear are involute helicoids [12]. The procedure by which this gear geometry is generated involves the use of three types of broach teeth (see Fig. 2): • Roughing teeth. These broach teeth remove the majority of the material located between the gear teeth. The geometry of these broach teeth are involutes. • Gear crest finishing teeth. These broach teeth are responsible for generating the final shape of the crests of the gear teeth. For this reason, they are also responsible for generating the inside diameter of the finished gear.
|
h
Rotational motion of the broac~
[I
l
Axial motion
of the broach
r"
Initial radius
~ [
Rows of parallel teeth
I
I
j
Workpieoe
I I
I
T
r|se pet
I
Fad View of Workpiec¢ Fig. I. Gear broaching process geometry and kinematics.
A model for the cutting force system in the gear broaching process
1411
Table I. Broach and work geometry Broach geometry
Number of broach sections: 7 Rp = Pitch radius (transverse): 2.41481 in. a = Rake angle (roughing leeth): 15° Number of teeth per row: 94 = Pressure angle (transverse): 19.1508° = Rake angle (finishing teeth): 18° Initial part diameter:. 4.7750 in = Helix angle: 22° Pan height 1.7323 in. V = Cutting speed: 30 fl/min
Work geometry and proce~ conditions
Roughii$ Tooth
-Bue Circle
f
~
Pitch Circle
Flank Finishing Tooth age trmn nnmmc~
~..arteothprofit,
/
Dc~i\rl[/len~d~for "" BaseCircle finished gear
Fig. 2. Broach tooth geomeUies.
• Finishing teeth. These broach teeth are responsible for generating the final shape of the flanks of the gear teeth. The shape of these teeth are also based on an involute geometry. The roughing teeth all have the same involute geometry (pressure angle, pitch radius, teeth per row, etc.). The cutting edge for the roughing teeth is at the tooth radius, Ri. For the roughing teeth, the difference between the radius of a given tooth of interest and the tooth that generated the surface being cut (Ri - R i - ,) is termed the rise per tooth or uncut chip thickness. As is evident from Fig. 2, the crest finishing gear teeth can be described by a single parameter, the tooth radius, R~. As with the roughing teeth, the cutting edge is at the radius of the tooth, and the difference between the radius of the current tooth and the tooth that generated the surface being cut defines the uncut chip thickness. The broach flank finishing teeth are responsible for generating the final shape of the flanks of the gear teeth. Consequently, the geomelrj of these broach teeth is based on the geometry of the gear teeth, for example the pressure angle, pitch radius, etc. In this case, the radii of the roughing teeth become successively larger along the axis of the broach and therefore the tip of the roughing tooth is responsible for performing the cutting process.
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For the finishing teeth, however, the tooth radius is constant. Thus the finishing teeth remove material by using teeth with successively larger tooth thicknesses. The thickness of a given finishing tooth is then based on the geometry of the gear teeth to be generated and some offset of the broach tooth from the final gear tooth profile. To further complicate the geometry of the finishing teeth, only one side of the gear tooth profile is generated at a time. In other words, several rows of finishing teeth are used to generate one side of a gear tooth, and then several rows of finishing teeth are used to generate the opposing side. To avoid interference of the noncutting edge of a finishing tooth, an offset may be employed for the noncutting edge. As noted previously, the geometry of the crest finishing broach teeth may be described simply with a radius. The shape of both the roughing and finishing broach teeth may be described using an involute geometry. The arc tooth thickness of the roughing and finishing teeth is therefore given by [12] t = 2r(2-~p + (tan 4) - 4 ) ) - (tan ( p - (~)) -- ml -- mr
(l)
where: tp is the arc tooth thickness at the pitch circle; Rp is the radius of the pitch circle; r is the radius at which the tooth thickness is to be determined; 4) is the pressure angle corresponding to pitch radius, Rp; q~ is the pressure angle corresponding to radius, r, and is equal to cos - '(RJr); Rb is the radius of the base circle, and is equal to (Rp cos4)); Z~ is the offset of the cutting edge on the left side of a tooth from the involute profile; A is the offset of the cutting edge on the right side of a tooth from the involute profile; and t is the arc tooth thickness corresponding to a radius, r. The broach geometry parameters as listed above are defined in the transverse direction (as opposed to the normal direction). Therefore, Eqn (1) will yield the transverse tooth thickness. It may also be noted that the offsets, AI and A, are utilized only for the finishing teeth, i.e. they are zero for the roughing teeth. The roughing, crest finishing, and flank finishing teeth are typically organized on a broach in a fashion similar to that displayed in Table 2 (broach geometry for an actual production machining operation). Typically included among the broach's cutting teeth are a number of float teeth which are used to lengthen a broach's useful life. These float teeth are extra teeth interspersed among the cutting teeth and do not participate in the cutting operation until the radii of previous teeth have been reduced due to tool wear and resharpening. The float teeth ensure the dimensions of the manufactured part remain accurate, so care must be taken when sharpening broaches to avoid oversharpening the float teeth. Table 2. Broach sections and broach teeth geometry Section
I 2 3 4 5 6 7
Tooth type
Number of rows in section
Starting diameter
Approximate rise per tooth
Roughing Roughing Roughing Roughing Crest finishing Flank finishing Flank finishing
4 53 21 8 7 8 8
4.7750 4.7930 5.0040 5.0540 4.7725 5.0450 5.0450
0.00225 0.00200 0.00125 0.00025 0.00088 0.00055* 0.00055*
*Tooth offset ,~, or A~.
Starting axial Axial spacing position
0.0000 2.3750 28.875 39.375 39.953 50.000 54.000
0.5938 0.5000 0.5000 0.9688 0.0688 0.5000 0.5000
A model for the cutting force system in the gear broaching process
1413
2.2. Chip load model In the preceding section, the geometry and kinematics of the internal helical gear broaching operation were described. Additionally, models for the geometry of the broach teeth were presented. Attention now turns to the development of a model for the chip load, or in other words the time-varying tool-work contact area. The chip loads must be defined in a plane perpendicular to the cutting direction, i.e. the normal direction of the helical gear. However, since the geometry of the broach teeth is provided in the transverse direction, the transverse chip load, At, for each type of broach teeth will first be determined and then later be converted to normal chip loads, AN. As was previously mentioned, for the roughing teeth the tip of the tooth does the cutting. The transverse chip load for the roughing teeth is indicated by the shaded area in Fig. 3(a). This area, ArR (the second subscript, "R", indicates that a roughing tooth is considered), can be approximated by Eqn (2): Ri
A~ = Ri
f ~
R~
2r{ 2 ~ + (tan(4~) - 4~) - (tan(~0) - ~0)}dr
tdr= I
Ri -
(2)
I
which is the integral of the tooth thickness [Eqn (1)] from the previous tooth radius, Ri- i, to the current tooth radius, Ri. The result of this integration, after substituting q~ = cos-I(Rdr) is shown in Eqn (3). ATR =
{r2~r -~+r2(tanck-
-
R~(r 2 -
6)-
2 3 ~b (r2-R~)
/ (3)
R~)~ ---
|
The transverse chip load for a crest finishing tooth is obtained in a similar fashion as the chip load for the roughing tooth. The crest finishing teeth cut the inner diameter of the workpiece. The transverse chip load for a crest finishing tooth, Arc, is indicated by the shaded area in Fig. 3(b), and is given by the calculation shown in Eqn (4):
//ft'•orkpiece-•• Roughing Tooth (a)
Rfm
Crest Finishing Tooth (b)
Rfnfia Fhmk FinishingTooth (¢)
Fig. 3. Chip load a~as (transverse sections).
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J.W. Sutherland
m~
rr(R/2 - R/z- i) N
et al.
{ ra'n" -
~
2 + r~(~n 6 - 4,) - ~ ,
3 (r ~ - R~,)
(4) Ri- I
In Eqn (4), the first term is an annulus with inner radius Ri - ~ and outer radius Ri divided by the number of teeth. This gives an annulus arc area encompassing one tooth on the gear and a neighboring gap. The second term in Eqn (4) is the area of the annulus over one gap (the involute shape of the last roughing tooth). The second term is subtracted from the first term to give the transverse chip load, Arc. The transverse chip load for a flank finishing tooth is indicated by the shaded area in Fig. 3(c). The calculated chip load, Arv, for the flank finishing tooth is given by Eqn (5): A x v = (Rfm~
(5)
- Rfmin)A
The shaded area has a uniform width, A, with a height of Rf,~ - Rfm~,, where Rfmax is the maximum radius of the finishing tooth, Rf,,i. is the radius of the last crest finishing tooth and A is the rise per tooth. It can be noted that the cutting edge on a flank finishing tooth is curved and not symmetric about the vertical axis as is the case with both the roughing and crest finishing teeth. This results in a nonzero average cutting edge orientation angle, C., (analogous to the lead angle for a single point tool). This angle is equal to the average of ~0 - / 3 over the portion of the flank finishing tooth that is involved in cutting (see Fig. 4). The calculation is as follows:
I
(~
Cs= ~ - ~ - "
-
fl)dr
(6)
r2 -- rl
where
r!
Rb
Fig. 4. Diagram of cutting edge orientation angle.
A model for the cutting force system in the gear broaching process
tp + t a n t ~ - ~ q~= a cos ( ~ ) and/3 = 2rp
1415
tanq~+~
After substitution and integration,
c,-
-
to
N2
~-~
2rp
r2
-
NI r)
(7a)
where I
1
Nl=~
~--
1
r , - R bIn
N,=2
~-
I
r2-Rbln
I
+
- 1
(7b)
I
(7c)
and ~
+ ~-
Equations (3)-(5) represent the chip loads in the transverse direction; however, the relations to be developed in the next section assume the chip load is defined in a plane perpendicular to the cutting velocity. Since the velocity direction is defined by the helix angle, $, the chip loads in the normal direction are needed. The normal chip loads, As, are calculated by multiplying the transverse chip loads by the cos(S) as shown in Eqn (8): AN = AT COS $
(8)
With the relations for the chip loads established, the cutting force relationships can now be developed. 2.3. Cutting force model To understand the cutting force system in the broaching process consider the oblique cutting process displayed in Fig. 5. In Fig. 5, V is the cutting velocity, Vc is the chip flow velocity, i is the inclination angle, tic is the chip flow angle, a. is the normal rake angle and ae is the effective rake angle. For such a process, the chip load is the product of the undeformed chip thickness and the width of cut. The mechanistic modeling approach considers a plane containing the cutting velocity and chip velocity vectors as the "orthogonal cutting plane" [13]. This plane may be determined given an assumed chip flow angle (in
Fig. 5. Diagram of oblique cutting (adapted from ref. [13]).
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J.W. Sutherland
et aL
this case estimated using Stabler's chip flow rule, r/¢ = i [14]) and knowledge of the tool geometry. For the orthogonal cutting plane, the cutting and thrust forces are related to the chip load by the following relations: F c = KcAN, F-rI.IR =
KTAN
(9)
where Kc and Kr are the cutting and thrust pressures, respectively, and AN is the chip area. In practice the cutting and thrust pressures are dependent on such variables as the undeformed chip thickness, cutting speed and rake angle [15]. It is the use of empirically based relations for Kc and Kx that distinguish this mechanistic modeling approach from approaches that attempt to describe cutting processes at an even more fundamental level, e.g. shear angle relationships [16--18] and finite element models [19] for the cutting process. In this empirical approach, which will be described in Section 2.4, the effective rake angle (the angle between the V¢ and the line OB which is the projection of the Z-axis onto the V-V, plane as shown in Fig. 5) is required to determine the cutting and thrust pressures. For the broaching process, the effective rake angle can be evaluated by or, = sin*-*ffsin2i + cos2i sin an)
(10)
where an is the normal rake angle, and i is the inclination (and assumed chip flow) angle [13]. Based on ASA cutting tool definitions [13]: i = a m [ t a n Orbcos Cs - tan as sin Cs]
(ll)
a, = atan {[tan ax cos Cs + tan ab sin Clcos i}
(12)
where ab is the back rake angle, as is the side rake angle and Cs is the average cutting edge orientation angle as defined in Eqn (7a). The force components of Eqn (9) can be resolved into the (X,Y,Z) coordinate system shown in Fig. 5 by Fx = KcAN, Fr = Ka,AN sin(0), Fz = KTAN cos(0)
(13)
where Fz, Fr and Fx are the forces acting on the workpiece, and 0 is the angle between the thrust force direction OB and the Z-axis in Fig. 5 and is given by
0 = atan
sin r/~ cos i - cos r/c sin t~. sin i] cos r?~c-os ot~
J
(14)
Typically, these forces are then resolved into three force components in an external coordinate system using the process and cutting geometries. The external coordinate system is chosen to coincide with the coordinate system that defines the force components directly measured during cutting operations. These measured forces are subsequently used to validate the force model. The forces defined by the external coordinate system in the gear broaching process include the forces acting on the workpiece in the radial (F~At>), axial (FAx) and tangential directions (Fr^N), as shown in Fig. 6. To determine bRAD, FAX and FTAr~, the forces shown in Eqn (13) must be resolved into the external coordinate system. However, due to different cutting geometries, the forces for each tooth type (roughing, crest finishing, flank finishing) are resolved into the external coordinate system in a different manner. Equations that have been previously established [Eqns (10)-(14)] are based on singlepoint tool geometry variables such as as, ab, etc. The broach teeth geometry needs to be mapped into that form for each set of broach teeth. The roughing and crest finishing teeth in the broaching process both have the same cutting geometry coordinate system. The
A model for the cutting force system in the gear broaching process
1417
FZ FX
Normal Section
a) Roughing and Crest FinishingTeeth
J
Normal Section b) Right Rank Finishing Tooth
J
Normal Section
c) Left Flank Finishing Tooth Fig. 6. Force components for broach teeth.
back rake angle (Orb) is equal to the helix angle (~) and the side rake angle (a0 is equal to the tool rake angle (at) that is given in Table 2 and shown in Fig. 1. The cutting forces of Eqn (13) are shown in Fig. 6(a), and are resolved into the axial, radial and tangential components by Eqn (15): FrAN = Fx sin ~ + Fr cos qJ, FAx = Fx cos ~b - Fr sin ~, FRAD = F z
(15)
For the roughing and crest finishing teeth the undeformed chip thickness, a,, is equal to (Ri - Ri _ 1) and the effective rake angle can be determined from Eqn (10). The forces due to the right flank finishing teeth are shown in Fig. 6(b). In this case the back rake angle is equal to the tool rake angle and the side rake angle is equal to the helix angle. The cutting forces of Eqn (13) can be resolved into the axial, radial and tangential components by Eqn (16): F'rAr~ = Fx sin $ + Fz cos q/, FAx = Fx cos ~ - Fz sin ~, FRAD = f v
(16)
The forces due to the left flank finishing teeth are shown in Fig. 6(c). The back rake angle is equal to the tool rake angle and the side rake angle is equal to the negative of the helix angle. The cutting forces of Eqn (13) can be resolved into the axial, radial and tangential components by Eqn (17): FTAr~ = Fxsin ~ - Fzcos $, FAx = Fxcos q/+ Fzsin Ik, FRAD= Fr
(17)
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For the right and left flank finishing teeth the undeformed chip thicknesses are equal to A~cos$ and Atcos~b, respectively and the effective rake angle can be determined by Eqn (10). 2.4. Model conditioning In order to accurately predict the forces in the broaching process, models for the cutting pressures Kc and Kr need to be developed. Suthedand et al. [15] demonstrated that models depending on tool geometry (rake angle), cutting speed and chip thickness adequately characterize the cutting and thrust pressures. To obtain the necessary data for the development of pressure models, a set of 12 orthogonal cutting experiments were conducted using a shaper. Tests were run at three rake angles, two cutting speeds and two chip thicknesses while using the same work material as used in the broaching operation. For each of the 12 tests, the cutting and thrust forces were measured and from knowledge of the chip load, Ac, the cutting and thrust pressures Kc and Kr were calculated. The results of the 12 tests are shown in Table 3. Using the data from the 12 tests, models for Kc and Kr, in the form given by Sutherland et al. [15] can be determined. However, due to the difficulty in accurately modeling cutting pressures over a wide range of rake angles, two different models were developed for each pressure, one for the positive and one for the negative rake angles. The models for neutral to positive rake angle are given in Eqn (18), with the models for neutral to negative rake given by Eqn (19). ln(Kc) = 10.80844 - 0.51153 In(at) - 0.10801 In(V) - 1.64854ote, ln(Kr) = 10.66259 - 0.49354 In(at) - 0.10313 In(V) - 7.43166ae
(18)
ln(Kc) = 12.08827 - 0.24925 In(at) - 0.11320 In(V) + 0.174641a e, ln(Kr) = 11.30114 - 0.23638 In(at) - 0.00350 In(V) - 0.82360t~
(19)
It may be noted that Eqns (18) and (19) provide slightly different values for the cutting and thrust pressures for neutral rake angles at the chip thicknesses employed in the broaching process; this is because the model conditioning experiments were performed at chip thicknesses higher than those used in the broaching process. Smaller chip thicknesses could not be examined in the model conditioning experiments due to difficulties with system deflections. 3. MODEL VERIFICATION
To validate the model for the broaching process it is necessary to compare actual process forces with model predicted forces. The actual force signal was obtained by instrumenting Table 3. Kc and KT calibration data Test
I 2 3 4 5 6 7 8 9 l0 II 12
Rake (°)
Speed (ft/min)
Chip thickness (in.)
A, (in 2)
20 20 20 20 3 3 3 3 - 20 - 20 - 20 - 20
16.7 62.5 16.7 62.5 16.7 62.5 16.7 62.5 16.7 62.5 16.7 62.5
0.004 0.004 0.008 0.008 0.004 0.004 0.008 0.008 0.004 0.004 0.008 0.008
0.001 0.001 0.002 0.002 0.001 0.001 0.002 0.002 0.001 0.001 0.002 0.002
Cutting Thrust force (lbf) force (Ibf)
350 340 480 370 500 480 890 700 450 425 875 675
37 35 51 41 310 275 500 430 340 425 680 700
Kc (Ibf/in2)
KT (Ibf/in2)
350 000 340 000 240 000 185 000 500000 480 000 445 000 350 000 450 000 425 000 437 500 337 500
37 000 35 000 25 500 20 500 310000 275 000 250 000 215 000 340 000 425 000 340 000 350 000
A model for the cutting force system in the gear broaching process
1419
the production broaching operation with a strain-gage sensor and then digitally recording this signal. The sensor measured the axial force transmitted through the workpiece and applied to the fixturing. The forces measured during the production of a typical ring gear are shown in Fig. 7(a). The model for the cutting force system in the broaching process that has been described was used to establish a computer-based simulation model for the broaching process. The simulation model permitted a user to manipulate the work, process and tool geometries, and then explore how such changes would impact the time-varying forces applied to the workpiece. For the conditions specified previously, the simulation model was used to predict the axial force applied to the workpiece by the broach. This force signature is depicted in Fig. 7(b). A comparison of the actual forces of Fig. 7(a) and the predicted forces of Fig. 7(b) shows good agreement. This indicates that the model adequately models the process behavior. Examining Fig. 7, there appears to be six distinct regions within the force signature. The first region (0--12 in. of broach movement) is associated with the entry of the broach into the part. Some difference between the actual and model predicted forces may be attributed to the fact that the part deforms elastically (breathes) under the application of the cutting forces. The high forces in this entry region may also produce increased wear levels that significantly change the broach teeth geometry. In the second region (12-30 in.), also associated with the roughing section of the broach, the model prediction matches the actual force very well. The third region evident in the force signatures (30--40 in.) is associated with the broach teeth that have the largest radius. Differences between the actual and model predicted forces suggest that an improvement is required in describing the geometry of these teeth. The fourth (40-50 in.), fifth (50-55 in.) and sixth (55--60 in.) regions of the force signal are associated with the crest finishing, and fight and left flank finishing teeth, respectively. Once again the forces in these three regions match very well. 4. SUMMARY AND CONCLUSIONS
A model for the cutting force system in the broaching process has been developed. This model is based on sub-models that describe the instantaneous area of contact (chip load) 104
A 3
1 L
A
A
I
A
10
20
30
40
50
60
4o
5o
60
~
M~o~nt
a) Experimental Results
4
x 104
3
8.2
£
1
0 0
t
&
i
10
2o
3o 84r~J~ Movon~nt
b) Simulation Results Fig. 7. Axial forces.
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between the broach and the workpiece and the dependence of the force system on the chip load, cutting edge orientation and cutting speed. The chip load sub-model utilized the process geometry and kinematics to characterize the amount of engagement of each broach tooth with the workpiece. Three types of broach teeth were considered: (i) roughing teeth that remove the majority of the material located between the gear teeth; (ii) crest finishing teeth that are responsible for generating the final shape of the crests of the gear teeth (i.e. responsible for generating the inside diameter of the finished gear); and (iii) finishing teeth that are responsible for generating the final shape of the flanks of the gear teeth. The geometry for both the roughing and finishing teeth were characterized by an involute geometry. Given knowledge of the instantaneous chip load as a function of time, force relationships as a function of chip load were established. These relationships consider an orthogonal cutting plane and relate cutting and thrust forces to the chip load via coefficients for cutting and thrust pressures. After coordinate system transformation, the cutting and thrust forces may be mapped into values for the axial, radial and tangential forces on the workpiece. Empirical models for the cutting and thrust pressures as a function of undeformed chip thickness, effective rake angle and cutting speed were developed using test data obtained from a shaping process. A computer-based representation of the cutting force system was then used to predict the axial forces arising under actual production conditions. These predicted forces accurately matched the measured axial force signature for the same production conditions. The research effort described herein was driven by the need to establish a computerbased tool to predict the force behavior of the broaching process under different broach geometries, process settings and workpiece conditions. Specific research findings resulting from this activity include: • The chip load for an involute-shaped tooth can be described via the equations presented here. • From a theoretical standpoint, there is a significant difference between the left and right flank finishing teeth on the broach owing to the difference between their cutting geometries with respect to the velocity vector. This difference was also evident from the measured axial force signal. • Both the measured and model predicted forces indicate that the forces associated with the flank finishing are large and significant when compared to the forces associated with the roughing teeth. • The forces associate with the crest finishing of the gear teeth are relatively small. • Empirical relations between force and chip load developed for a shaping process can be applied to other nonshaping operations. • The entry-exit behavior of the broach results in an oscillatory behavior in the force signal. Such cyclical behavior may excite structural vibrational modes within the broaching machine, or be transmitted through the floor to other machines. These findings provide further support that the mechanistic modeling procedure is an effective technique for the characterization of machining operations. Several unresolved issues were identified during the course of this investigation and represent potential topics for continuing research study. As noted, it would be desirable to use the forces predicted by the broach force model together with a model for the structural response of the system to produce surface error/accuracy predictions for a broached part. Additionally, the broaching forces may be used to design fixturing for the operation. Small inaccuracies in the force signal for the region where the gear root is formed were identified, and it is believed that further study is needed to better understand the process in this region. Finally, the oscillatory nature of the broaching force signature suggests that modifications to the broach geometry may be investigated with the hope of "smoothing out" the force signal.
A model for the cutting force system in the gear broaching process
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Acknowledgements--The authors gratefully acknowledge the assistance of David Dodds and the financial support of the Ford Motor Company during the course of this research. The model conditioning experiments described herein were capably performed by Dr Robert Radalescu (now at Michelin) in the Chao-Trigger machining research laboratory at the University of Illinois at Urbana--Champalgn. Finally the authors would like to thank Professors R.E. DeVor and S.G. Kapoor of the University of Illinois for their helpful comments during this research investigation. REFERENCES [I] Smith, S. and Tlusty, J., J. Engng Ind. Trans. ASME, 1991, 113, 169-175. [2] Sutherland, J. W., DeVor, R. E., Kapoor, S. G. and Ferreira, P. M., SAE Trans., 1988, 97(5), 215-226. [3] Endres, W. J., Sutherland, J. W., DeVor, R. E. and Kapoor, S. G., ASME Bound VoI.-PED, 1990, 44, 193-212. [4] Subramani, G., Suvada, R., Kapoor, S. G., DeVor, R. E. and Meingast, W., Proc. N. Am. Manufact. Res. Conf., 1987, 15, 439--446. [5] Zhang, G. M. and Kapoor, S. G., J. Engng Ind. Trans. ASME, 1987, 109(3), 219-226. [6] Kline, W. A. and DeVor, R. E., Int. J. Mach. Tool Des. Res., 1983, 23(2), 123. [7] Tlusty, J., J. Engng Ind. Trans. ASME, 1986, 108, 59-67. [8] Sutherland, J. W., ASME Bound Vol.-PED, 1988, 33, 53-62. [9] Montgomery, D. and AItintas, Y., J. Engng Ind. Trans. ASME, 1991, 113, 160--168. [10] Fu, H. J., DeVor, R. E. and Kapoor, S. G., J. Engng Ind. Trans. ASME, 1984, 106(!), 81-88. Ill] Sutherland, J. W. and Zdeblick, W. J., SAE Trans., 1992, 101(5), 711-719. [12] J. E. Shigley and J. J. Uicker, Theory of Machines and Mechanisms. McGraw-Hill, New York (1980). [13]. N. H. Cook, Manufacturing Analysis. Addison-Wesley, Reading, MA 0966). [14] Stabler, G. V., Proc. Inst. Mech. Engrs, 1951, 165, 14-26. [15] Sutherland, J. W., Subramani, G., Kuhl, M. J., DeVor, R. E. and Kapoor, S. G., Trans. NAMRI/SME, 1988, 16, 264-272. [16] Usui, E., Hirota, A. and Masuko, M., J. Engng Ind. Trans. ASME, 1978, 100, 222-228. [17] Usui, E. and Hirola, A., Z Engng Ind. Trans. ASME, 1978, 100, 229-235. [18] Usui, E., Shirakashi, T. and Kitagawa, T., J. Engr. Ind. Trans. ASME, 1978, 100, 236-243. [19] Strenkowski, J. S. and Mitchum, G. L., Proc. N. Am. Manufact. Res. Conf., 1987, 15, 506-509.