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Mathematical model for design and analysis of power skiving tool for involute gear cutting
Mechanism and Machine Theory 101 (2016) 195–208
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Mathematical model for design and analysis of power skiving tool for involute gear cutting Chung-Yu Tsai Department of Mechanical and Computer-Aided Engineering, National Formosa University, 64 Wunhua Road, Huwei, Yunlin 632, Taiwan
a r t i c l e
i n f o
Article history: Received 20 June 2015 Received in revised form 24 October 2015 Accepted 25 March 2016 Available online xxxx Keywords: Power skiving Involute gear Tool design Full-field angle analysis
a b s t r a c t Power skiving is an efficient means of producing high accuracy gears; particularly internal gears. However, the literature lacks a systematic mathematical model for the design and analysis of such tools. Accordingly, the present study proposes a comprehensive yet straightforward methodology for the design of resharpening power skiving tools based on conjugate surface theory and full-field angle analysis. The proposed methodology has three important features. First, mathematical models of the virtual-conjugate-surface and rake surfaces are derived, and thus the cutting edges can be obtained. Second, the power skiving is modeled by transformation matrices, and the normal/tangent vectors of the relative surface can be explicitly obtained. As a result, the working rake and clearance angles can be investigated in accordance with recommended ISO-standards. Finally, a method is prescribed for determining the working rake angles, clearance angles and wedge angles of the cutter based on the ISO-defined reference planes. Utilizing this method, a simple technique is proposed for redesigning the power skiving tool so as to avoid negative working clearance angles. The results of the illustrative example are shown that the proposed approach provides a comprehensive, simple and versatile technique for modeling a wide range of power skiving tool design features. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Power skiving is a kind of gear machining method first proposed by Wilhelm von Pittler in 1910 [1]. The advantages of power skiving result from combining the traditional machining processes of hobbing and shaping. However, due to its continuous chip removal capability, power skiving is faster than shaping, and more flexible than broaching; particularly in the machining of internal gears. Power skiving has always presented a significant challenge to machines and tools due to their low stiffness and high wear characteristics, respectively. However, over the past few years, developments in manufacturing engineering have overcome these limitations, and power skiving technology now provides an efficient and flexible approach for the machining of gear components. Power skiving methods can be broadly classified into two types, namely a simple type, in which only a shaft angle Σ is applied to the tool in the machining process, and a general type, in which both a shaft angle Σ and a tilt angle δ are applied (see Fig. 1). It is noted that the simple type is just a special case of the general type in which the tool tilt angle δ is set to zero. The simple type of power skiving has attracted significant attention in the literature in recent years. For example, Spath and Huhsam [2] performed a numerical investigation into the tool design, cutting angles and cutting load in the skiving of periodic structures. Bouzakis et al. [3] investigated the efficiency of various types of gear cutting process for the manufacturing of cylindrical gears, including power skiving. Guo et al. [4] proposed a method for the design and analysis of skiving tools for internal gears. Chen
E-mail address: [email protected].
http://dx.doi.org/10.1016/j.mechmachtheory.2016.03.021 0094-114X/© 2016 Elsevier Ltd. All rights reserved.
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Nomenclature Sθ Cθ (xyz)t (xyz)g (xyz)w θg θt δ Σ st rg rga rta zg zt mn αn αnt βg wr wq dam ddm wn Σ0 rt. sw t r t n rVCS Rrake Rclearance E Ltp g v ωt ft. γrake αclearance μwedge g mre g mn g mse g mrake g mflank tchip ξt0 ξt
sin(θ) cos(θ) power skiving tool frame gear frame rack frame rotation angle of gear rotation angle of power skiving tool tilt angle of meshing gear shaft angle of meshing gear feed distance of tool along gear axis standard pitch radius of gear active pitch radius of gear active pitch radius of power skiving tool teeth number of gear teeth number of power skiving tool normal module of gear normal pressure angle of gear operating normal pressure angle of gear helix angle of gear spur rack surface generation curve of spur rack surface addendum of gear dedendum of gear unit normal vector of rack shaft angle of meshing spur rack pitch radius of power skiving tool motion distance of spur rack surface with respect to tool frame (xyz)t unit normal vector with respect to tool frame (xyz)t VCS of tool rake surface clearance surface cutting edge leading pitch of power skiving tool linear cutting velocity with respect to gear frame (xyz)g tool rotation speed tool feed speed working rake angle clearance angle wedge angle unit normal vector of working reference plane unit tangent vector along cutting edge unit normal vector of cutting edge plane normal vector of rake surface unit normal vector of flank surface chip thickness along tool feed direction profile shifting coefficient of cutting edge change rate of profile shifting coefficient
et al. [5] presented a design approach for realizing error-free spur slice cutters [6]. However, general power skiving has attracted little attention since its introduction by Stadtfeld [7] in 2014. Accordingly, this study presents a comprehensive method for the design of power skiving tools for involute gear cutting based on conjugate surface theory and full-field angle analysis. Section 2 derives a mathematical model of the power skiving system. Section 3 determines the virtual conjugate surface (VCS) of the power skiving tool by means of conjugate surface theory. Section 4 analyzes the linear cutting speed in power skiving. Section 5 defines the working rake angles, working clearance angles and working wedge angles of the power skiving tool in accordance with the normal vectors of the ISO recommended reference planes. Section 6 presents a methodology based on full-field angle analysis for redesigning the power skiving tool so as to avoid negative working clearance angles. Section 7 provides an
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illustrative example to demonstrate the validity of the proposed approach. Finally, Section 8 presents some brief concluding remarks. In the homogenous coordinate notation system used in the present study, a position vector axi + ayj + azk is written in the form of a column matrix ja = [ax ay az 1]T, where the pre-superscript ‘j’ of the leading symbol ja indicates that the vector is referred with respect to coordinate frame (xyz)j. Given a point ja, its transformation ka is represented by the matrix product k a ¼ k A j j a , where kAj is a 4 × 4 matrix defining the position and orientation (referred to as the pose hereafter) of frame (xyz)j with respect to another frame (xyz)k [8]. It is noted that the aforementioned notation rules are also applicable to a T unit directional vector j n ¼ ½ nx ny nz 0 .
2. Mathematical model of power skiving system In the power skiving system, the pose of frame (xyz)g built in the gear with respect to the power skiving tool frame (xyz)t can be modeled via the following homogeneous matrix manipulation:
g
At ¼ Rot z; −θg Transð0; 0; st ÞTrans r ga ; 0; 0 Rotðx; ΣÞRot y; −cg δ Trans cg r ta ; 0; 0 Rotðz; θt Þ 2 CδCθg Cθt þ Sθg CΣSθt −cg SδSΣCθt CΣSθg Cθt þ cg SδSΣSθg −CδCθg Sθt 6 6 CΣCθg Cθt þ cg SδSΣCθg þ CδSθg Sθt CΣCθg Sθt −Cθt cg SδSΣCθg þ CδSθg ; ¼6 6 4 SΣSθt þ cg SδCΣCθt SΣCθt −cg CΣSδSθt 0 0 3 −cg SδCθg −CδSΣSθg r ga Cθg þ cg r ta CδCθg −cg SδSΣSθg 7 7 cg SδSθg −CδSΣCθg −r ga Sθg −cg r ta CδSθg þ cg SδSΣCθg 7 7 5 CδCΣ r CΣSδ þ s ta
0
ð1Þ
t
1
where Trans and Rot are translation and rotation operators, respectively, and are defined in Appendix A. Furthermore, referring to Fig. 1, θg and θt are the rotation angles of the gear and power skiving tool, respectively; and δ is the tilt angle of the tool and is set in such a way as to achieve the required clearance. Σ is the shaft angle of the tool required to provide the necessary sliding velocity for gear cutting, and st is the feed distance of the tool along the gear axis direction (i.e., the z-axis of the gear frame, (xyz)g).
Fig. 1. Illustration of power skiving machining model.
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Finally, rga and rta are the active pitch radii of the gear and skiving tool, respectively, and are given as r ga ¼ Cβg Cα nt
mn zg rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1þ
ð2Þ
tanðα n Þ 2 Cβg
and r ta ¼
mn zt rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ; 2C Σ−β g Cα nt 1 þ tanCβðαn Þ
ð3Þ
g
where zg and zt are the teeth number of the gear and power skiving tool, respectively; and mn, αn, αnt and βg are the normal module, normal pressure angle, operating normal pressure angle, and helix angle of the gear, respectively. For power skiving machining, θg, θt and st must satisfy the following relation: tan βg zt ; θg ¼ −cg θt −ch st zg rg
ð4Þ
where rg = zgmn/(2Cβg) is the standard pitch radius of the gear, and cg and ch are gear coefficients. cg has a value of +1 for external gears and −1 for internal gears. Similarly, ch has a value of +1 for right-handed helical gears and −1 for left-handed helical gears. 3. Power skiving tool design To design a power skiving tool, it is sufficient only to consider the kinematical relation between the tool and a spur rack (see Fig. 2). One important feature of spur racks is that their working surfaces are surfaces of protrusion. Consequently, the spur rack surface, wr, with respect to the rack frame, (xyz)w, can be obtained by protruding the generation curve w q ¼ ½ xðu1 Þ yðu1 Þ 0 1 T in the xy-plane about the protrusion z-axis (see Fig. 3), that is w
w
r ¼ Transð0; 0; u2 Þ q ¼ ½ xðu1 Þ
yðu1 Þ
u2
T
1 :
Fig. 2. Kinematical relation between power skiving tool and spur rack.
ð5Þ
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Fig. 3. Profile and parameters of spur rack.
For the case of an involute profile (see Fig. 3), the generation curve of a relative rack allows various gear normal profile parameters to be determined, including the gear normal module mn, the normal pressure angle αn, the helix angle βg, the addendum dam, and the dedendum ddm. The surface unit normal vector wn of the rack along the working surface can be determined by w
n ¼ nx
ny
nz
0
T
w ∂ r ∂w r ∂w r ∂w r = ¼ ∂u1 ∂u2 ∂u1 ∂u2
2 ¼
y0 ðu1 Þ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 x ðu1 Þ2 þ y0 ðu1 Þ2
x0 ðu1 Þ − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 x ðu1 Þ2 þ y0 ðu1 Þ2
3T 0
05 :
ð6Þ
In order to determine the VCS of a power skiving tool in terms of the motions of the rack (see Fig. 2) using conjugate surface theory [9], it is first necessary to define a power skiving tool coordinate frame (xyz)t with which to describe its position and orientation. The pose of frame (xyz)w built in the rack with respect to frame (xyz)t built in the power skiving tool can be obtained from its generation motion as follows: t
Aw 3 2 ¼ Rotðz; −θt ÞTransðr t ; 0; 0ÞRotðy; −δÞRotðx; −Σ0 ÞTransð0; sw ; 0Þ Cθt Cδ Sθt CΣ0 þ Cθt SδSΣ0 −Cθt SδCΣ0 þ Sθt SΣ0 r t ½Cθt þ θt CΣ0 ðSθt CΣ0 þ Cθt SδSΣ0 Þ 6 −Sθ Cδ Cθ CΣ −Sθ SδSΣ Sθt SδCΣ0 þ Cθt SΣ0 r t ½−Sθt þ θt CΣ0 ðCθt CΣ0 −Sθt SδSΣ0 Þ 7 t t 0 t 0 7; ¼6 5 4 Sδ −CδSΣ0 CδCΣ0 −r t θt CδCΣ0 SΣ0 0 0 0 1
ð7Þ
where θt is the rotation angle of the power skiving tool, δ is the tilt angle, and Σ0 is the shaft angle. Additionally, rt = ztmn/(2CΣ0) is the pitch radius of the power skiving tool, where zt and mn are, respectively, the number of tool teeth and the normal module of the rack (equal to that of the gear). The motion distance of the spur rack as the power skiving tool rotates is then obtained as sw = rtθtCΣ0. In the present study, the VCS of the tool is generated utilizing conjugate surface theory, which states that contact surfaces have a common normal vector at the conjugate point. Accordingly, the conjugate points and complete conjugate surface can be determined from T d t r T ∂ t Aw w r dθ T ∂ t Aw w r t t t t ¼ n ¼ n n ωt ¼ 0; dt dt ∂θt ∂θt
ð8Þ
where tr and tn are the surface equation and unit outward normal vector relative to the tool frame (xyz)t, respectively, and are obtained from the coordinate transformations t r ¼ t Aw w r and t n ¼ t Aw w n. Eq. (8) states that for continuous contact to be maintained between two moving surfaces, the relative sliding velocity d(tr)/dt must be orthogonal to the common normal vector tn at the conjugate point. If this condition is not satisfied, then either the contact between the surfaces breaks or gouging occurs.
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Imposing Eq. (8) on Eqs. (5) and (6), the relation between parameters μ1 and μ2 (denoted as u1 and u2 , respectively) at the conjugate point is obtained as 0
0
x ðu1 Þ½xðu1 ÞCδCΣ0 −u2 Sδ þ y ðu1 ÞCδfu2 SΣ0 þ CΣ0 ½yðu1 Þ þ r t θt CΣ0 g ¼ 0:
ð9Þ
For a particular value of u1, which represents a particular point wq on the generating curve, the value of u2 can be determined from the above equation as u2 ¼
CδCΣ0 xðu1 Þx0 ðu1 Þ þ y0 ðu1 Þ½yðu1 Þ þ r t θt CΣ0 : 0 0 x ðu1 ÞSδ−y ðu1 ÞCδSΣ0
ð10Þ
The VCS trVCS of the tool generated by the spur rack can then be obtained by substituting Eq. (10) into Eq. (5), and transforming wr to frame (xyz)t via the relation t rVCS ¼ t Aw w r, that is 3 yðu1 ÞCΣ0 Sθt þ Cθt ½xðu1 ÞCδ þ yðu1 ÞSδSΣ0 þ u2 ½SΣ0 Sθt −SδCΣ0 Cθt þ r t fCθt þ θt CΣ0 ½SδSΣ0 Cθt þ CΣ0 Sθt g 6 −xðu1 ÞCδSθt þ SδSθt ½u2 CΣ0 −yðu1 ÞSΣ0 þ Cθt ½yðu1 ÞCΣ0 þ u2 SΣ0 −r t fSθt þ θt CΣ0 ½SδSΣ0 Sθt −CΣ0 Cθt g 7 t w 7: ¼ Aw r ¼ 6 5 4 xðu1 ÞSδ−yðu1 ÞCδSΣ0 þ CδCΣ0 ½u2 −r t θt SΣ0 1 2
t
rVCS
ð11Þ During power skiving, only the cutting edge of the tool meshes with the tooth surface of the rack. Hence, the cutting edge can be regarded as an arbitrary curve on the VCS. For example, a cone surface tRrake, which is axially symmetrical with the tool axis, can be arbitrarily designed as the rake surface of the tool. Therefore, the intersection curve of the VCS trVCS and rake surface t Rrake can be taken as the cutting edge. Notably, this type of cutting edge has the fundamental advantage of easier tool resharpening. To achieve this cutting edge, the rake surface tRrake can be mathematically obtained by rotating its generating curve t qR ¼ ½ u3 0 ðu3 −r out Þ tanðτÞ 1 T in the xz-plane about the symmetrical rotating zt-axis (see Fig. 2), i.e., t
t
Rrake ¼ Rotðz; u4 Þ qR ¼ ½ u3 Cu4
ðu3 −r out Þ tanðτÞ
u3 Su4
T
1 ;
ð12Þ
where rout is the outer radius of the tool and is determined by the dedendum of the rack. The angle τ controls the rake angle. However, it is noted that the actual working rake angle and clearance angle do not remain constant during the power skiving process. (The variation of the two angles during the cutting process is discussed later in Section 5.) Substituting Eqs. (11) and (12) into the relation trVCS = tRrake, parameters θt, u3 and u4 can all be solved numerically relative to ~ 3, and u ~ 4, respectively). Substituting the solutions of ~θt, u ~ 3, and u ~ 4 into Eq. (12), the cutting edge tE of the skiving u1 (denoted as ~ θt , u tool with respect to the tool frame (xyz)t can be finally determined. Having determined the cutting edge tE, the clearance surface t Rclearance can be constructed by helically protruding the cutting edge with a leading pitch Ltp, i.e., t
Rclearane ¼ Transð0; 0; uc ÞRot z;
2πuc Ltp
! t
ð13Þ
r;
where uc is one of the arguments of the clearance surface tRclearane, and the leading pitch Ltp is given by zt mn π : CδSΣ0
Ltp ¼
ð14Þ
After determining the rake surface tRrake and clearance surface tRclearane, the full profile of the power skiving tool is finally obtained. 4. Linear cutting speed analysis Achieving a linear cutting speed is essential in power skiving in order to minimize tool damage, reduce vibration and improve the machining efficiency. In practice, the optimal cutting speed depends on the material properties of the tool and workpiece. In power skiving, the conjugate point on the cutting edge gE can be expressed with respect to the gear frame (xyz)g as g
g
t
E ¼ At E:
ð15Þ
The linear cutting velocity gv with respect to the gear frame (xyz)g can then be derived as g
v¼
dg E ¼ dt
g d At t E dt
" ¼
! ∂g At ∂g At ∂θg ωt þ þ ∂θt ∂θg ∂θt
! # ∂g At ∂g At ∂θg t f t E; þ ∂st ∂θg ∂st
ð16Þ
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where ωt is the tool rotation speed and ft is the tool feed speed. Furthermore, CΣSθg Cθt þ cg SδSΣSθg −CδCθg Sθt 6 ∂g At 6 CΣCθg Cθt þ cg SδSΣCθg þ CδSθg Sθt ¼6 6 ∂θt 4 SΣCθ −c SδCΣSθ 2
t
g
t
0
−CδCθg Cθt þ Sθg cg SδSΣCθt −CΣSθt CδSθg Cθt þ Cθg cg SδSΣCθt −CΣSθt −SΣSθt −cg SδCΣCθt 0
0 0
0
3
7 7 0 7; 7 05
ð17Þ
0 0 1
CΣCθg Cθt þ cg SδSΣCθg þ CδSθg Sθt CΣCθg Sθt −Cθt cg SδSΣCθg þ CδSθg 6 ∂g At 6 CδCθg Sθt − cg SδSΣSθt þ CΣCθt Sθg −CδCθg Cθt þ Sθg cg SδSΣCθt −CΣSθt ¼6 6 ∂θg 4 0 0 2
0 cg SδSθg −CδSΣCθg cg SδCθg þ CδSΣSθg 0 0 ∂θg ∂θt
¼−
cg zt ; zg
2
0 ∂ At 6 0 ¼6 40 ∂st 0 g
0 3 −rga Sθg −cg r ta cg SδSΣCθg þ CδSθg 7 7 −Cθg r ga þ cg r ta Cδ þ r ta SδSΣSθg 7; 7 5 0 1
0 0 0 0
ð18Þ
ð19Þ
0 0 0 0
3 0 07 7; 15 1
ð20Þ
:
ð21Þ
and ∂θg ∂st
¼−
ch tan βg rg
It is noted that the cutting speed |gv| in power skiving machining is determined mainly by the tool spindle rotation speed ωt since this speed is much larger than the feed speed ft (which is used basically to control the chip thickness). In other words, the first term of Eq. (16) constitutes the main component of the cutting speed. 5. Rake angle and clearance angle analysis The geometry and nomenclature of power skiving tools are surprisingly complicated subjects. It is difficult to calculate or even discuss the appropriate planes in which the various angles of a power skiving tool should be measured due to confusion among
Fig. 4. Tool-in-use planes for major cutting edge.
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Table 1 Definition of power skiving tool angles. Tool angle
Angle between two vectors
Angle measured in plane
Rake angle γrake Clearance angle αclearance Wedge angle μwedge
t
Pn Pn Pn
t t
Rrake and Pre Rclearance and Pse Rrake and tRclearance
current definitions. The International Organization for Standardization (ISO) has established two systems of planes for defining the various angles of the face and flank surfaces of single-point tools [10], namely tool-in-use and tool-in-hand (see Fig. 4). The tool-in-use system is defined in relation to the resultant cutting direction and is used in a machining operation. By contrast, the tool-in-hand system is defined in relation to the tool base and is used for the purposes of grinding and sharpening the tool. In this paper, the tool-in-use system is used to define the required planes, namely the working reference plane Pre (the plane perpendicular to the resultant cutting direction), the working cutting edge plane Pse (the plane tangential to the cutting edge and perpendicular to the working reference plane), and the cutting edge normal plane Pn (the plane perpendicular to the cutting edge and perpendicular to the working reference plane) (see Fig. 4). Table 1 shows the ISO definitions of the working rake angle γrake, clearance angle αclearance, and wedge angle μwedge =90° − γrake − αclearance with respect to a selected point on the cutting edge. The primary motions gv of the conjugate points along the cutting edge of a power skiving tool have been determined in the previous section. The unit normal vectors of the working reference planes at these points are given as
g
g
mre ¼
v jg vj:
ð22Þ
The unit tangent vectors along the cutting edge can be obtained as
g
mn ¼
∂g E g ∂t E ¼ At : ∂u1 ∂u1
ð23Þ
Fig. 5. Determination of angle between plane Pa and plane Pb measured in plane Pc.
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Table 2 ISO tool angle definitions. Tool angle
Angle between two planes
Rake angle γrake Clearance angle αclearance Wedge angle μwedge
g
g
mrake and mre mflank and gmse mrake and gmflank
Angle measured in plane g
mn mn mn
g
g
g
g
From the definition of the cutting edge plane Pse, the unit normal vectors of Pse at the conjugate points on the cutting edge can be determined by g
g
mse ¼
mn g mre : j mn g mre j
ð24Þ
g
The unit normal vectors of the rake surface can be obtained as g
g
t
g
mrake ¼ At mrake ¼ At
! ∂t Rrake ∂t Rrake : ∂u3 ∂u4
ð25Þ
The unit normal vectors of the flank surface can be determined by g
g
t
g
mflank ¼ At mflank ¼ At
! t t ∂ Rclearane ∂ Rclearane : ∂u1 ∂uc
ð26Þ
Furthermore, from basic geometric principles, the angle ψ between plane Pa and plane Pb measured in plane Pc is uniquely determined by the respective unit normal vectors ma, mb and mc as (see Fig. 5) Cψ ¼
ðma mc Þ ðmc mb Þ : jma mc jjmc mb j
ð27Þ
The definitions given in Table 2 presents definitions for the working rake angle γrake, working clearance angle αclearance, and working wedge angle μwedge along the cutting edge in terms of the unit normal vectors of the various reference planes. When all of the cutting angles of all the cutting points on the full cutting edge have been calculated for all moments of contact between the cutting edge and the workpiece, a full-field angle analysis of the tool can be performed. 6. Tool redesign based on full-field analysis of clearance angles The cutting mechanism in general power skiving machining is more complicated than that in traditional gear cutting since the working rake angle γrake and working clearance angle αclearance not only change from point to point along the cutting edge E at any arbitrary cutting moment, but also change continuously as the cutting motion proceeds. To verify the practical feasibility of the working rake angles and working clearance angles, the cutting path should first be determined. Fig. 6 shows the cutting path Cc traversed by cutting point QE located on cutting edge E of the skiving tool. Note that Tc is the trajectory traced by cutting point QE in space, Sc is the finished cutting surface produced by the previous full cutting edge,
Fig. 6. Illustration of cutting path of arbitrary point on cutting edge of power skiving tool.
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Table 3 Gear, tool and machining parameters for illustrative example. Type
Parameter
Initial value
Modified value
Gear
Normal module Normal pressure angle Operating normal pressure angle Helix angle Teeth number Teeth number Addendum Dedendum Tilt angle Rake angle Shaft angle to mesh a spur rack Profile shifting coefficient of cutting edge Profile shifting coefficient rate of flank Leading pitch Shaft angle for machining gear Tool rotation speed Chip thickness Feed speed
mn = 1.25 αn = 20° αnt βg = 0° zg = 92 zt = 37 dam = 1.563 mm ddm = 1.5 mm δ = 6° τ = 16° Σ0 = 15° ξt0 = 0 ζt = 0 mm−1 Ltp = 564.483 mm Σ = 15° ωt = 2000 rmp tchip = 0.1 mm ft = 80.435 mm/min
– – – – – zt = 47 – – δ = 10° – – ξt0 = 1.0 ζt = 0.15 mm−1 Ltp = 724.119 mm – – – ft = 102.174 mm/min
Tool
Machining
and S0 is the boundary surface of the raw materials (e.g., the internal cylindrical surface of an internal gear). Hence, the cutting path Cc begins at the intersection point Qc,start of curve Tc and surface Sc and ends at the intersection point Qc,end of curve Tc and surface S0. The curve gTc with respect to the gear frame (xyz)g can be determined as g
g
t
Tc ¼ At Q E :
ð28Þ
Meanwhile, the surface gSc with respect to the gear frame (xyz)g is given by g
g t Sc ¼ Trans 0; 0; −t chip At E;
ð29Þ
where tchip is the chip thickness along the tool feed direction (i.e., the z-axis of the gear frame (xyz)g) and is determined as t chip ¼
f t zg : ω t zt
ð30Þ
After determining the cutting paths using the method described in Section 5, the working rake angles and working clearance angles can be finally determined for all of the points on all of the cutting paths. In general, large negative rake angles increase
Fig. 7. Clearance angle in initial design.
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both the deformation of the metal chips and the cutting force during the cutting operation. Furthermore, negative clearance angles result in interference between the material and the clearance surface of the tool, and are thus unacceptable for machining. For tools with a constant cross-sectional profile, the profile does not depend on the number of resharpenings. In other words, the profile remains unchanged over the course of the tool life. However, such a tool design generally leads to negative working clearance angles when applied to power skiving applications. Accordingly, the following discussions present an approach for solving this negative clearance problem whilst retaining the resharpening advantage. The mathematical derivations presented in the previous sections show that the working clearance angle αclearance increases with an increasing tool tilt angle δ, an increasing tool teeth number zt, and an increasing profile shifting coefficient ξt0 of the cutting edge. Furthermore, the tool flank surface can also be designed via the protrusion of a continuously changing cross-section in which the profile shifting coefficient ξt(z) reduces along the direction of protrusion. In such a case, the profile shifting coefficient ξt(z), which is set to change nearly linearly along the (−z)-axis of the rack frame (xyz)w, can be expressed as ξt ðzÞ ¼ ξt0 −ζ t z;
ð31Þ
Fig. 8. Clearance angle in final design, in which green regions correspond to new tool and yellow regions correspond to end of tool life after resharpening: (a) clearance angle; (b) rake angle. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
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where ζt is the profile shifting coefficient changing rate of the tool flank surface along the (−z)-axis direction of the rack frame (xyz)w. By adopting such an approach, the working clearance angle can be increased while simultaneously retaining the resharpening characteristic.
7. Illustrative example This section presents an illustrative example to demonstrate the design method described in the previous sections. The necessary parameters of related gear, tool and machining are shown in Table 3. Note that the leading pitch of the tool (Ltp = 564.483 mm) is determined from Eq. (14). Meanwhile, the tool feed speed ft along the z-axis direction of the gear is determined from Eq. (30) as ft = 80.435mm/min. In the initial tool design, the cross-sectional profile is unchanged along the teeth helical protrusion direction, and is thus amenable for resharpening. Fig. 7 shows the full-field analysis results for the working clearance angles of the four main regions of the cutting edge, i.e., the left, top, right and root regions. It is noted that the x-axis indicates the angular positions of the individual cutting points on the cutting edge, the y-axis indicates the rotation angles of the power skiving tool, and the z-axis indicates the working clearance angle. It is seen that the working clearance angle in some portions of the left region of the tool is negative, and is thus unacceptable. Thus, as shown in the right-most column of Table 3, modifications were made to the tool teeth number zt, tool tilt angle δ, profile shifting coefficient of the cutting edge ξt0, and profile shifting coefficient change rate of the flank ζt. The modified values of the tool leading pitch Ltp and machining feed speed ft were obtained as Ltp = 724.119 mm and ft = 102.174mm/min, respectively. Fig. 8(a) shows the full-field analysis results for the working clearance angles of the modified tool. Note that the green regions correspond to the new tool, while the yellow regions correspond to the end-of-tool-life re-sharpened tool. It is seen that when the profile shifting coefficient of the cutting edge reduces to ξt0 = 0.3 following resharpening (i.e., the yellow region of Fig. 8(a)), the working clearance angles of some portions of the left region of the cutting edge reduce almost to zero. As a consequence, the tool can no longer be used for machining. For the modified tool design, the maximum permissible resharpening length along the tool axis is thus around 6.05 mm. In the illustrative example described above, the rake surface is specified as a simple cone (i.e., the cutting angles are constrained) in order to facilitate tool resharpening using a similar process to that used for a standard shaper cutter. However, by adopting such a design, several undesirable side effects inevitably occur if negative clearances are to avoided at all points on the cutting edge, namely: (1) the cutting angles (see Fig. 8) vary significantly in different regions of the tool and are thus difficult to control; (2) the rake angle, τ, increases to a large value (16°), and hence limits the tool life; and (3) the clearance angles also increase, and hence reduce the tool stiffness. In theory, these problems can be overcome by adopting a step-sharpened cutter teeth design (in which the rake surface of each tooth is oriented under a helix angle). In this way, a more flexible approach can be taken in designing the cutting angles. However, the greater flexibility of the design process is obtained at the expense of a more complex analysis task and resharpening process. Fig. 8(b) shows the working rake angles of the modified cutting edge. It is observed that a very large negative rake angle exists over the full extent of the left region of the cutting edge and the initial portions of the right and root regions, respectively. The negative rake angle greatly increases the deformation of the metal chips and the cutting force during the power skiving operation. Fig. 9 shows the full-field cutting speed analysis results for the modified tool design. The maximum cutting speed of the tool is around 195 m/min, and occurs in the top region of the tool at the end cutting points. By contrast, the minimum cutting speed is approximately 99 m/min, and occurs in the root region at the start cutting points. Following resharpening, the maximum and minimum cutting speeds occur in the same tool locations, but are reduced to approximately 187 m/min and 96 m/min, respectively. In other words, the cutting speed experienced by the tool varies by around 91 m/min. Thus, the choice of a suitable tool material (e.g., tungsten carbide tools) is essential in avoiding excessive tool wear and maintaining an adequate cutting performance. Fig. 10 shows a schematic illustration of the final (modified) cutter design.
Fig. 9. Cutting speed analysis of power skiving tool, in which green regions correspond to new tool and yellow regions correspond to end of tool life after resharpening. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 10. Schematic illustration of designed power skiving tool.
8. Conclusion This paper has presented a simple yet comprehensive mathematical modeling approach for the design and analysis of power skiving tools. Importantly, the proposed approach is applicable to both simple power skiving, which involves only a tool shaft angle, and general power skiving, which involves both a tool shaft angle and a tool tilt angle. In the proposed model, mathematical expressions are derived for the virtual conjugate surface of the tool and the rake surfaces. Consequently, the cutting edges can be obtained directly by numerical calculating the intersections of these two surfaces. In addition, the skiving tool and internal gear are modeled by transformation matrices. As a result, the normal and tangent vectors of the relative surface can be obtained explicitly, and hence the working rake and clearance angles of the cutting edge can be investigated in accordance with the recommended ISO standards. A simple method has been proposed for determining the full-field working rake angles, clearance angles and wedge angles based on the unit normal vectors of the ISO-recommended reference planes. A formulation for computing the full-field cutting speed has also be proposed. Finally, a method has been presented for redesigning the power skiving tool in such a way as to avoid negative working clearance angles. The validity of the proposed approach has been demonstrated by means of computer modeling. The mathematical model presented in this study is simple, easily applied, and capable of describing a wide range of power skiving tool design features. However, it is noted that the geometry of the modified power skiving tool proposed in the illustrative example is too complicated for real-world tool manufacturing processes (particularly high-precision tool grinding processes). Therefore, in a future study, the model will be applied to search for the optimum generating curve for the rake surface of a power skiving tool. In addition, a more detailed investigation will be performed into the feasibility of the proposed method for the real-world design and analysis of power skiving cutters manufactured using a grinding process. Acknowledgments The authors gratefully acknowledge the financial support provided to this study by the Ministry of Science and Technology, Taiwan under Grant No. MOST 103-2221-E-150-010. Appendix A 3 2 1 0 0 tx 6 0 1 0 ty 7 7 Trans t x ; t y ; t z ¼ 6 4 0 0 1 t z 5; 0 0 0 1 2
1 60 6 Rot ðx; ωx Þ ¼ 4 0 0
0 Cωx Sωx 0
2 Cωy 6 0 Rot y; ωy ¼ 6 4 −Sωy 0 2
Cωz 6 Sωz Rot ðz; ωz Þ ¼ 6 4 0 0
0 −Sωx Cωx 0
0 1 0 0
−Sωz Cωz 0 0
Sωy 0 Cωy 0
3 0 07 7; 05 1 3 0 07 7; 05 1
3 0 0 0 07 7: 1 05 0 1
ðA1Þ
ðA2Þ
ðA3Þ
ðA4Þ
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Note that S and C denote sine and cosine, respectively.
References [1] W. Pittler von, Verfahren zum Schneiden von Zahnrädern mittels eines zahnrad-artigen, an den Stirnflächen der Zähne mit Schneidkanten versehen-en Schneidwerkzeuges, Patent Application, Germany, March, 1910. [2] D. Spath, A. Huhsam, Skiving for high-performance machining of periodic structures, CIRP Ann. Manuf. Technol. 51 (2002) 91–94. [3] K.D. Bouzakis, E. Lili, N. Michailidis, O. Friderikos, Manufacturing of cylindrical gears by generating cutting processes: a critical synthesis of analysis methods, CIRP Ann. Manuf. Technol. 57 (2008) 676–696. [4] E. Guo, R. Hong, X. Huang, C. Fang, Research on the design of skiving tool for machining involute gears, J. Mech. Sci. Technol. 28 (2014) 5107–5115. [5] X.C. Chen, J. Li, B.C. Lou, A study on the design of error-free spur slice cutter, Int. J. Adv. Manuf. Technol. 68 (2013) 727–738. [6] X.C. Cchen, J. Li, B.C. Lou, J. Shi, Q. Yang, Effect of the cutter parameters and machining parameters on the interference in gear slicing, Chin. J. Mech. Eng. 26 (2013) 1118–1126. [7] H. Stadtfeld, Power skiving of cylindrical gears on different machine platforms, Gear Technol. 31 (2014) 52–62. [8] R.P. Paul, Robot Manipulators-Mathematics, Programming and Control, MIT press, 1982. [9] J. Chakraborty, S.G. Dhande, Kinematics and Geometry of Planar and Spatial Cam Mechanisms, John Wiley & Sons, New York, 1977. [10] G. Boothroyd, Fundamentals of Metal Machining and Machine Tools, McGraw-Hill, Englewood Cliffs, NJ, 1975.
Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Mathematical model for design and analysis of power skiving tool for involute gear cutting Chung-Yu Tsai Department of Mechanical and Computer-Aided Engineering, National Formosa University, 64 Wunhua Road, Huwei, Yunlin 632, Taiwan
a r t i c l e
i n f o
Article history: Received 20 June 2015 Received in revised form 24 October 2015 Accepted 25 March 2016 Available online xxxx Keywords: Power skiving Involute gear Tool design Full-field angle analysis
a b s t r a c t Power skiving is an efficient means of producing high accuracy gears; particularly internal gears. However, the literature lacks a systematic mathematical model for the design and analysis of such tools. Accordingly, the present study proposes a comprehensive yet straightforward methodology for the design of resharpening power skiving tools based on conjugate surface theory and full-field angle analysis. The proposed methodology has three important features. First, mathematical models of the virtual-conjugate-surface and rake surfaces are derived, and thus the cutting edges can be obtained. Second, the power skiving is modeled by transformation matrices, and the normal/tangent vectors of the relative surface can be explicitly obtained. As a result, the working rake and clearance angles can be investigated in accordance with recommended ISO-standards. Finally, a method is prescribed for determining the working rake angles, clearance angles and wedge angles of the cutter based on the ISO-defined reference planes. Utilizing this method, a simple technique is proposed for redesigning the power skiving tool so as to avoid negative working clearance angles. The results of the illustrative example are shown that the proposed approach provides a comprehensive, simple and versatile technique for modeling a wide range of power skiving tool design features. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Power skiving is a kind of gear machining method first proposed by Wilhelm von Pittler in 1910 [1]. The advantages of power skiving result from combining the traditional machining processes of hobbing and shaping. However, due to its continuous chip removal capability, power skiving is faster than shaping, and more flexible than broaching; particularly in the machining of internal gears. Power skiving has always presented a significant challenge to machines and tools due to their low stiffness and high wear characteristics, respectively. However, over the past few years, developments in manufacturing engineering have overcome these limitations, and power skiving technology now provides an efficient and flexible approach for the machining of gear components. Power skiving methods can be broadly classified into two types, namely a simple type, in which only a shaft angle Σ is applied to the tool in the machining process, and a general type, in which both a shaft angle Σ and a tilt angle δ are applied (see Fig. 1). It is noted that the simple type is just a special case of the general type in which the tool tilt angle δ is set to zero. The simple type of power skiving has attracted significant attention in the literature in recent years. For example, Spath and Huhsam [2] performed a numerical investigation into the tool design, cutting angles and cutting load in the skiving of periodic structures. Bouzakis et al. [3] investigated the efficiency of various types of gear cutting process for the manufacturing of cylindrical gears, including power skiving. Guo et al. [4] proposed a method for the design and analysis of skiving tools for internal gears. Chen
E-mail address: [email protected].
http://dx.doi.org/10.1016/j.mechmachtheory.2016.03.021 0094-114X/© 2016 Elsevier Ltd. All rights reserved.
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Nomenclature Sθ Cθ (xyz)t (xyz)g (xyz)w θg θt δ Σ st rg rga rta zg zt mn αn αnt βg wr wq dam ddm wn Σ0 rt. sw t r t n rVCS Rrake Rclearance E Ltp g v ωt ft. γrake αclearance μwedge g mre g mn g mse g mrake g mflank tchip ξt0 ξt
sin(θ) cos(θ) power skiving tool frame gear frame rack frame rotation angle of gear rotation angle of power skiving tool tilt angle of meshing gear shaft angle of meshing gear feed distance of tool along gear axis standard pitch radius of gear active pitch radius of gear active pitch radius of power skiving tool teeth number of gear teeth number of power skiving tool normal module of gear normal pressure angle of gear operating normal pressure angle of gear helix angle of gear spur rack surface generation curve of spur rack surface addendum of gear dedendum of gear unit normal vector of rack shaft angle of meshing spur rack pitch radius of power skiving tool motion distance of spur rack surface with respect to tool frame (xyz)t unit normal vector with respect to tool frame (xyz)t VCS of tool rake surface clearance surface cutting edge leading pitch of power skiving tool linear cutting velocity with respect to gear frame (xyz)g tool rotation speed tool feed speed working rake angle clearance angle wedge angle unit normal vector of working reference plane unit tangent vector along cutting edge unit normal vector of cutting edge plane normal vector of rake surface unit normal vector of flank surface chip thickness along tool feed direction profile shifting coefficient of cutting edge change rate of profile shifting coefficient
et al. [5] presented a design approach for realizing error-free spur slice cutters [6]. However, general power skiving has attracted little attention since its introduction by Stadtfeld [7] in 2014. Accordingly, this study presents a comprehensive method for the design of power skiving tools for involute gear cutting based on conjugate surface theory and full-field angle analysis. Section 2 derives a mathematical model of the power skiving system. Section 3 determines the virtual conjugate surface (VCS) of the power skiving tool by means of conjugate surface theory. Section 4 analyzes the linear cutting speed in power skiving. Section 5 defines the working rake angles, working clearance angles and working wedge angles of the power skiving tool in accordance with the normal vectors of the ISO recommended reference planes. Section 6 presents a methodology based on full-field angle analysis for redesigning the power skiving tool so as to avoid negative working clearance angles. Section 7 provides an
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197
illustrative example to demonstrate the validity of the proposed approach. Finally, Section 8 presents some brief concluding remarks. In the homogenous coordinate notation system used in the present study, a position vector axi + ayj + azk is written in the form of a column matrix ja = [ax ay az 1]T, where the pre-superscript ‘j’ of the leading symbol ja indicates that the vector is referred with respect to coordinate frame (xyz)j. Given a point ja, its transformation ka is represented by the matrix product k a ¼ k A j j a , where kAj is a 4 × 4 matrix defining the position and orientation (referred to as the pose hereafter) of frame (xyz)j with respect to another frame (xyz)k [8]. It is noted that the aforementioned notation rules are also applicable to a T unit directional vector j n ¼ ½ nx ny nz 0 .
2. Mathematical model of power skiving system In the power skiving system, the pose of frame (xyz)g built in the gear with respect to the power skiving tool frame (xyz)t can be modeled via the following homogeneous matrix manipulation:
g
At ¼ Rot z; −θg Transð0; 0; st ÞTrans r ga ; 0; 0 Rotðx; ΣÞRot y; −cg δ Trans cg r ta ; 0; 0 Rotðz; θt Þ 2 CδCθg Cθt þ Sθg CΣSθt −cg SδSΣCθt CΣSθg Cθt þ cg SδSΣSθg −CδCθg Sθt 6 6 CΣCθg Cθt þ cg SδSΣCθg þ CδSθg Sθt CΣCθg Sθt −Cθt cg SδSΣCθg þ CδSθg ; ¼6 6 4 SΣSθt þ cg SδCΣCθt SΣCθt −cg CΣSδSθt 0 0 3 −cg SδCθg −CδSΣSθg r ga Cθg þ cg r ta CδCθg −cg SδSΣSθg 7 7 cg SδSθg −CδSΣCθg −r ga Sθg −cg r ta CδSθg þ cg SδSΣCθg 7 7 5 CδCΣ r CΣSδ þ s ta
0
ð1Þ
t
1
where Trans and Rot are translation and rotation operators, respectively, and are defined in Appendix A. Furthermore, referring to Fig. 1, θg and θt are the rotation angles of the gear and power skiving tool, respectively; and δ is the tilt angle of the tool and is set in such a way as to achieve the required clearance. Σ is the shaft angle of the tool required to provide the necessary sliding velocity for gear cutting, and st is the feed distance of the tool along the gear axis direction (i.e., the z-axis of the gear frame, (xyz)g).
Fig. 1. Illustration of power skiving machining model.
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Finally, rga and rta are the active pitch radii of the gear and skiving tool, respectively, and are given as r ga ¼ Cβg Cα nt
mn zg rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1þ
ð2Þ
tanðα n Þ 2 Cβg
and r ta ¼
mn zt rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ; 2C Σ−β g Cα nt 1 þ tanCβðαn Þ
ð3Þ
g
where zg and zt are the teeth number of the gear and power skiving tool, respectively; and mn, αn, αnt and βg are the normal module, normal pressure angle, operating normal pressure angle, and helix angle of the gear, respectively. For power skiving machining, θg, θt and st must satisfy the following relation: tan βg zt ; θg ¼ −cg θt −ch st zg rg
ð4Þ
where rg = zgmn/(2Cβg) is the standard pitch radius of the gear, and cg and ch are gear coefficients. cg has a value of +1 for external gears and −1 for internal gears. Similarly, ch has a value of +1 for right-handed helical gears and −1 for left-handed helical gears. 3. Power skiving tool design To design a power skiving tool, it is sufficient only to consider the kinematical relation between the tool and a spur rack (see Fig. 2). One important feature of spur racks is that their working surfaces are surfaces of protrusion. Consequently, the spur rack surface, wr, with respect to the rack frame, (xyz)w, can be obtained by protruding the generation curve w q ¼ ½ xðu1 Þ yðu1 Þ 0 1 T in the xy-plane about the protrusion z-axis (see Fig. 3), that is w
w
r ¼ Transð0; 0; u2 Þ q ¼ ½ xðu1 Þ
yðu1 Þ
u2
T
1 :
Fig. 2. Kinematical relation between power skiving tool and spur rack.
ð5Þ
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199
Fig. 3. Profile and parameters of spur rack.
For the case of an involute profile (see Fig. 3), the generation curve of a relative rack allows various gear normal profile parameters to be determined, including the gear normal module mn, the normal pressure angle αn, the helix angle βg, the addendum dam, and the dedendum ddm. The surface unit normal vector wn of the rack along the working surface can be determined by w
n ¼ nx
ny
nz
0
T
w ∂ r ∂w r ∂w r ∂w r = ¼ ∂u1 ∂u2 ∂u1 ∂u2
2 ¼
y0 ðu1 Þ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 x ðu1 Þ2 þ y0 ðu1 Þ2
x0 ðu1 Þ − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 x ðu1 Þ2 þ y0 ðu1 Þ2
3T 0
05 :
ð6Þ
In order to determine the VCS of a power skiving tool in terms of the motions of the rack (see Fig. 2) using conjugate surface theory [9], it is first necessary to define a power skiving tool coordinate frame (xyz)t with which to describe its position and orientation. The pose of frame (xyz)w built in the rack with respect to frame (xyz)t built in the power skiving tool can be obtained from its generation motion as follows: t
Aw 3 2 ¼ Rotðz; −θt ÞTransðr t ; 0; 0ÞRotðy; −δÞRotðx; −Σ0 ÞTransð0; sw ; 0Þ Cθt Cδ Sθt CΣ0 þ Cθt SδSΣ0 −Cθt SδCΣ0 þ Sθt SΣ0 r t ½Cθt þ θt CΣ0 ðSθt CΣ0 þ Cθt SδSΣ0 Þ 6 −Sθ Cδ Cθ CΣ −Sθ SδSΣ Sθt SδCΣ0 þ Cθt SΣ0 r t ½−Sθt þ θt CΣ0 ðCθt CΣ0 −Sθt SδSΣ0 Þ 7 t t 0 t 0 7; ¼6 5 4 Sδ −CδSΣ0 CδCΣ0 −r t θt CδCΣ0 SΣ0 0 0 0 1
ð7Þ
where θt is the rotation angle of the power skiving tool, δ is the tilt angle, and Σ0 is the shaft angle. Additionally, rt = ztmn/(2CΣ0) is the pitch radius of the power skiving tool, where zt and mn are, respectively, the number of tool teeth and the normal module of the rack (equal to that of the gear). The motion distance of the spur rack as the power skiving tool rotates is then obtained as sw = rtθtCΣ0. In the present study, the VCS of the tool is generated utilizing conjugate surface theory, which states that contact surfaces have a common normal vector at the conjugate point. Accordingly, the conjugate points and complete conjugate surface can be determined from T d t r T ∂ t Aw w r dθ T ∂ t Aw w r t t t t ¼ n ¼ n n ωt ¼ 0; dt dt ∂θt ∂θt
ð8Þ
where tr and tn are the surface equation and unit outward normal vector relative to the tool frame (xyz)t, respectively, and are obtained from the coordinate transformations t r ¼ t Aw w r and t n ¼ t Aw w n. Eq. (8) states that for continuous contact to be maintained between two moving surfaces, the relative sliding velocity d(tr)/dt must be orthogonal to the common normal vector tn at the conjugate point. If this condition is not satisfied, then either the contact between the surfaces breaks or gouging occurs.
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Imposing Eq. (8) on Eqs. (5) and (6), the relation between parameters μ1 and μ2 (denoted as u1 and u2 , respectively) at the conjugate point is obtained as 0
0
x ðu1 Þ½xðu1 ÞCδCΣ0 −u2 Sδ þ y ðu1 ÞCδfu2 SΣ0 þ CΣ0 ½yðu1 Þ þ r t θt CΣ0 g ¼ 0:
ð9Þ
For a particular value of u1, which represents a particular point wq on the generating curve, the value of u2 can be determined from the above equation as u2 ¼
CδCΣ0 xðu1 Þx0 ðu1 Þ þ y0 ðu1 Þ½yðu1 Þ þ r t θt CΣ0 : 0 0 x ðu1 ÞSδ−y ðu1 ÞCδSΣ0
ð10Þ
The VCS trVCS of the tool generated by the spur rack can then be obtained by substituting Eq. (10) into Eq. (5), and transforming wr to frame (xyz)t via the relation t rVCS ¼ t Aw w r, that is 3 yðu1 ÞCΣ0 Sθt þ Cθt ½xðu1 ÞCδ þ yðu1 ÞSδSΣ0 þ u2 ½SΣ0 Sθt −SδCΣ0 Cθt þ r t fCθt þ θt CΣ0 ½SδSΣ0 Cθt þ CΣ0 Sθt g 6 −xðu1 ÞCδSθt þ SδSθt ½u2 CΣ0 −yðu1 ÞSΣ0 þ Cθt ½yðu1 ÞCΣ0 þ u2 SΣ0 −r t fSθt þ θt CΣ0 ½SδSΣ0 Sθt −CΣ0 Cθt g 7 t w 7: ¼ Aw r ¼ 6 5 4 xðu1 ÞSδ−yðu1 ÞCδSΣ0 þ CδCΣ0 ½u2 −r t θt SΣ0 1 2
t
rVCS
ð11Þ During power skiving, only the cutting edge of the tool meshes with the tooth surface of the rack. Hence, the cutting edge can be regarded as an arbitrary curve on the VCS. For example, a cone surface tRrake, which is axially symmetrical with the tool axis, can be arbitrarily designed as the rake surface of the tool. Therefore, the intersection curve of the VCS trVCS and rake surface t Rrake can be taken as the cutting edge. Notably, this type of cutting edge has the fundamental advantage of easier tool resharpening. To achieve this cutting edge, the rake surface tRrake can be mathematically obtained by rotating its generating curve t qR ¼ ½ u3 0 ðu3 −r out Þ tanðτÞ 1 T in the xz-plane about the symmetrical rotating zt-axis (see Fig. 2), i.e., t
t
Rrake ¼ Rotðz; u4 Þ qR ¼ ½ u3 Cu4
ðu3 −r out Þ tanðτÞ
u3 Su4
T
1 ;
ð12Þ
where rout is the outer radius of the tool and is determined by the dedendum of the rack. The angle τ controls the rake angle. However, it is noted that the actual working rake angle and clearance angle do not remain constant during the power skiving process. (The variation of the two angles during the cutting process is discussed later in Section 5.) Substituting Eqs. (11) and (12) into the relation trVCS = tRrake, parameters θt, u3 and u4 can all be solved numerically relative to ~ 3, and u ~ 4, respectively). Substituting the solutions of ~θt, u ~ 3, and u ~ 4 into Eq. (12), the cutting edge tE of the skiving u1 (denoted as ~ θt , u tool with respect to the tool frame (xyz)t can be finally determined. Having determined the cutting edge tE, the clearance surface t Rclearance can be constructed by helically protruding the cutting edge with a leading pitch Ltp, i.e., t
Rclearane ¼ Transð0; 0; uc ÞRot z;
2πuc Ltp
! t
ð13Þ
r;
where uc is one of the arguments of the clearance surface tRclearane, and the leading pitch Ltp is given by zt mn π : CδSΣ0
Ltp ¼
ð14Þ
After determining the rake surface tRrake and clearance surface tRclearane, the full profile of the power skiving tool is finally obtained. 4. Linear cutting speed analysis Achieving a linear cutting speed is essential in power skiving in order to minimize tool damage, reduce vibration and improve the machining efficiency. In practice, the optimal cutting speed depends on the material properties of the tool and workpiece. In power skiving, the conjugate point on the cutting edge gE can be expressed with respect to the gear frame (xyz)g as g
g
t
E ¼ At E:
ð15Þ
The linear cutting velocity gv with respect to the gear frame (xyz)g can then be derived as g
v¼
dg E ¼ dt
g d At t E dt
" ¼
! ∂g At ∂g At ∂θg ωt þ þ ∂θt ∂θg ∂θt
! # ∂g At ∂g At ∂θg t f t E; þ ∂st ∂θg ∂st
ð16Þ
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201
where ωt is the tool rotation speed and ft is the tool feed speed. Furthermore, CΣSθg Cθt þ cg SδSΣSθg −CδCθg Sθt 6 ∂g At 6 CΣCθg Cθt þ cg SδSΣCθg þ CδSθg Sθt ¼6 6 ∂θt 4 SΣCθ −c SδCΣSθ 2
t
g
t
0
−CδCθg Cθt þ Sθg cg SδSΣCθt −CΣSθt CδSθg Cθt þ Cθg cg SδSΣCθt −CΣSθt −SΣSθt −cg SδCΣCθt 0
0 0
0
3
7 7 0 7; 7 05
ð17Þ
0 0 1
CΣCθg Cθt þ cg SδSΣCθg þ CδSθg Sθt CΣCθg Sθt −Cθt cg SδSΣCθg þ CδSθg 6 ∂g At 6 CδCθg Sθt − cg SδSΣSθt þ CΣCθt Sθg −CδCθg Cθt þ Sθg cg SδSΣCθt −CΣSθt ¼6 6 ∂θg 4 0 0 2
0 cg SδSθg −CδSΣCθg cg SδCθg þ CδSΣSθg 0 0 ∂θg ∂θt
¼−
cg zt ; zg
2
0 ∂ At 6 0 ¼6 40 ∂st 0 g
0 3 −rga Sθg −cg r ta cg SδSΣCθg þ CδSθg 7 7 −Cθg r ga þ cg r ta Cδ þ r ta SδSΣSθg 7; 7 5 0 1
0 0 0 0
ð18Þ
ð19Þ
0 0 0 0
3 0 07 7; 15 1
ð20Þ
:
ð21Þ
and ∂θg ∂st
¼−
ch tan βg rg
It is noted that the cutting speed |gv| in power skiving machining is determined mainly by the tool spindle rotation speed ωt since this speed is much larger than the feed speed ft (which is used basically to control the chip thickness). In other words, the first term of Eq. (16) constitutes the main component of the cutting speed. 5. Rake angle and clearance angle analysis The geometry and nomenclature of power skiving tools are surprisingly complicated subjects. It is difficult to calculate or even discuss the appropriate planes in which the various angles of a power skiving tool should be measured due to confusion among
Fig. 4. Tool-in-use planes for major cutting edge.
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Table 1 Definition of power skiving tool angles. Tool angle
Angle between two vectors
Angle measured in plane
Rake angle γrake Clearance angle αclearance Wedge angle μwedge
t
Pn Pn Pn
t t
Rrake and Pre Rclearance and Pse Rrake and tRclearance
current definitions. The International Organization for Standardization (ISO) has established two systems of planes for defining the various angles of the face and flank surfaces of single-point tools [10], namely tool-in-use and tool-in-hand (see Fig. 4). The tool-in-use system is defined in relation to the resultant cutting direction and is used in a machining operation. By contrast, the tool-in-hand system is defined in relation to the tool base and is used for the purposes of grinding and sharpening the tool. In this paper, the tool-in-use system is used to define the required planes, namely the working reference plane Pre (the plane perpendicular to the resultant cutting direction), the working cutting edge plane Pse (the plane tangential to the cutting edge and perpendicular to the working reference plane), and the cutting edge normal plane Pn (the plane perpendicular to the cutting edge and perpendicular to the working reference plane) (see Fig. 4). Table 1 shows the ISO definitions of the working rake angle γrake, clearance angle αclearance, and wedge angle μwedge =90° − γrake − αclearance with respect to a selected point on the cutting edge. The primary motions gv of the conjugate points along the cutting edge of a power skiving tool have been determined in the previous section. The unit normal vectors of the working reference planes at these points are given as
g
g
mre ¼
v jg vj:
ð22Þ
The unit tangent vectors along the cutting edge can be obtained as
g
mn ¼
∂g E g ∂t E ¼ At : ∂u1 ∂u1
ð23Þ
Fig. 5. Determination of angle between plane Pa and plane Pb measured in plane Pc.
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Table 2 ISO tool angle definitions. Tool angle
Angle between two planes
Rake angle γrake Clearance angle αclearance Wedge angle μwedge
g
g
mrake and mre mflank and gmse mrake and gmflank
Angle measured in plane g
mn mn mn
g
g
g
g
From the definition of the cutting edge plane Pse, the unit normal vectors of Pse at the conjugate points on the cutting edge can be determined by g
g
mse ¼
mn g mre : j mn g mre j
ð24Þ
g
The unit normal vectors of the rake surface can be obtained as g
g
t
g
mrake ¼ At mrake ¼ At
! ∂t Rrake ∂t Rrake : ∂u3 ∂u4
ð25Þ
The unit normal vectors of the flank surface can be determined by g
g
t
g
mflank ¼ At mflank ¼ At
! t t ∂ Rclearane ∂ Rclearane : ∂u1 ∂uc
ð26Þ
Furthermore, from basic geometric principles, the angle ψ between plane Pa and plane Pb measured in plane Pc is uniquely determined by the respective unit normal vectors ma, mb and mc as (see Fig. 5) Cψ ¼
ðma mc Þ ðmc mb Þ : jma mc jjmc mb j
ð27Þ
The definitions given in Table 2 presents definitions for the working rake angle γrake, working clearance angle αclearance, and working wedge angle μwedge along the cutting edge in terms of the unit normal vectors of the various reference planes. When all of the cutting angles of all the cutting points on the full cutting edge have been calculated for all moments of contact between the cutting edge and the workpiece, a full-field angle analysis of the tool can be performed. 6. Tool redesign based on full-field analysis of clearance angles The cutting mechanism in general power skiving machining is more complicated than that in traditional gear cutting since the working rake angle γrake and working clearance angle αclearance not only change from point to point along the cutting edge E at any arbitrary cutting moment, but also change continuously as the cutting motion proceeds. To verify the practical feasibility of the working rake angles and working clearance angles, the cutting path should first be determined. Fig. 6 shows the cutting path Cc traversed by cutting point QE located on cutting edge E of the skiving tool. Note that Tc is the trajectory traced by cutting point QE in space, Sc is the finished cutting surface produced by the previous full cutting edge,
Fig. 6. Illustration of cutting path of arbitrary point on cutting edge of power skiving tool.
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Table 3 Gear, tool and machining parameters for illustrative example. Type
Parameter
Initial value
Modified value
Gear
Normal module Normal pressure angle Operating normal pressure angle Helix angle Teeth number Teeth number Addendum Dedendum Tilt angle Rake angle Shaft angle to mesh a spur rack Profile shifting coefficient of cutting edge Profile shifting coefficient rate of flank Leading pitch Shaft angle for machining gear Tool rotation speed Chip thickness Feed speed
mn = 1.25 αn = 20° αnt βg = 0° zg = 92 zt = 37 dam = 1.563 mm ddm = 1.5 mm δ = 6° τ = 16° Σ0 = 15° ξt0 = 0 ζt = 0 mm−1 Ltp = 564.483 mm Σ = 15° ωt = 2000 rmp tchip = 0.1 mm ft = 80.435 mm/min
– – – – – zt = 47 – – δ = 10° – – ξt0 = 1.0 ζt = 0.15 mm−1 Ltp = 724.119 mm – – – ft = 102.174 mm/min
Tool
Machining
and S0 is the boundary surface of the raw materials (e.g., the internal cylindrical surface of an internal gear). Hence, the cutting path Cc begins at the intersection point Qc,start of curve Tc and surface Sc and ends at the intersection point Qc,end of curve Tc and surface S0. The curve gTc with respect to the gear frame (xyz)g can be determined as g
g
t
Tc ¼ At Q E :
ð28Þ
Meanwhile, the surface gSc with respect to the gear frame (xyz)g is given by g
g t Sc ¼ Trans 0; 0; −t chip At E;
ð29Þ
where tchip is the chip thickness along the tool feed direction (i.e., the z-axis of the gear frame (xyz)g) and is determined as t chip ¼
f t zg : ω t zt
ð30Þ
After determining the cutting paths using the method described in Section 5, the working rake angles and working clearance angles can be finally determined for all of the points on all of the cutting paths. In general, large negative rake angles increase
Fig. 7. Clearance angle in initial design.
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both the deformation of the metal chips and the cutting force during the cutting operation. Furthermore, negative clearance angles result in interference between the material and the clearance surface of the tool, and are thus unacceptable for machining. For tools with a constant cross-sectional profile, the profile does not depend on the number of resharpenings. In other words, the profile remains unchanged over the course of the tool life. However, such a tool design generally leads to negative working clearance angles when applied to power skiving applications. Accordingly, the following discussions present an approach for solving this negative clearance problem whilst retaining the resharpening advantage. The mathematical derivations presented in the previous sections show that the working clearance angle αclearance increases with an increasing tool tilt angle δ, an increasing tool teeth number zt, and an increasing profile shifting coefficient ξt0 of the cutting edge. Furthermore, the tool flank surface can also be designed via the protrusion of a continuously changing cross-section in which the profile shifting coefficient ξt(z) reduces along the direction of protrusion. In such a case, the profile shifting coefficient ξt(z), which is set to change nearly linearly along the (−z)-axis of the rack frame (xyz)w, can be expressed as ξt ðzÞ ¼ ξt0 −ζ t z;
ð31Þ
Fig. 8. Clearance angle in final design, in which green regions correspond to new tool and yellow regions correspond to end of tool life after resharpening: (a) clearance angle; (b) rake angle. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
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where ζt is the profile shifting coefficient changing rate of the tool flank surface along the (−z)-axis direction of the rack frame (xyz)w. By adopting such an approach, the working clearance angle can be increased while simultaneously retaining the resharpening characteristic.
7. Illustrative example This section presents an illustrative example to demonstrate the design method described in the previous sections. The necessary parameters of related gear, tool and machining are shown in Table 3. Note that the leading pitch of the tool (Ltp = 564.483 mm) is determined from Eq. (14). Meanwhile, the tool feed speed ft along the z-axis direction of the gear is determined from Eq. (30) as ft = 80.435mm/min. In the initial tool design, the cross-sectional profile is unchanged along the teeth helical protrusion direction, and is thus amenable for resharpening. Fig. 7 shows the full-field analysis results for the working clearance angles of the four main regions of the cutting edge, i.e., the left, top, right and root regions. It is noted that the x-axis indicates the angular positions of the individual cutting points on the cutting edge, the y-axis indicates the rotation angles of the power skiving tool, and the z-axis indicates the working clearance angle. It is seen that the working clearance angle in some portions of the left region of the tool is negative, and is thus unacceptable. Thus, as shown in the right-most column of Table 3, modifications were made to the tool teeth number zt, tool tilt angle δ, profile shifting coefficient of the cutting edge ξt0, and profile shifting coefficient change rate of the flank ζt. The modified values of the tool leading pitch Ltp and machining feed speed ft were obtained as Ltp = 724.119 mm and ft = 102.174mm/min, respectively. Fig. 8(a) shows the full-field analysis results for the working clearance angles of the modified tool. Note that the green regions correspond to the new tool, while the yellow regions correspond to the end-of-tool-life re-sharpened tool. It is seen that when the profile shifting coefficient of the cutting edge reduces to ξt0 = 0.3 following resharpening (i.e., the yellow region of Fig. 8(a)), the working clearance angles of some portions of the left region of the cutting edge reduce almost to zero. As a consequence, the tool can no longer be used for machining. For the modified tool design, the maximum permissible resharpening length along the tool axis is thus around 6.05 mm. In the illustrative example described above, the rake surface is specified as a simple cone (i.e., the cutting angles are constrained) in order to facilitate tool resharpening using a similar process to that used for a standard shaper cutter. However, by adopting such a design, several undesirable side effects inevitably occur if negative clearances are to avoided at all points on the cutting edge, namely: (1) the cutting angles (see Fig. 8) vary significantly in different regions of the tool and are thus difficult to control; (2) the rake angle, τ, increases to a large value (16°), and hence limits the tool life; and (3) the clearance angles also increase, and hence reduce the tool stiffness. In theory, these problems can be overcome by adopting a step-sharpened cutter teeth design (in which the rake surface of each tooth is oriented under a helix angle). In this way, a more flexible approach can be taken in designing the cutting angles. However, the greater flexibility of the design process is obtained at the expense of a more complex analysis task and resharpening process. Fig. 8(b) shows the working rake angles of the modified cutting edge. It is observed that a very large negative rake angle exists over the full extent of the left region of the cutting edge and the initial portions of the right and root regions, respectively. The negative rake angle greatly increases the deformation of the metal chips and the cutting force during the power skiving operation. Fig. 9 shows the full-field cutting speed analysis results for the modified tool design. The maximum cutting speed of the tool is around 195 m/min, and occurs in the top region of the tool at the end cutting points. By contrast, the minimum cutting speed is approximately 99 m/min, and occurs in the root region at the start cutting points. Following resharpening, the maximum and minimum cutting speeds occur in the same tool locations, but are reduced to approximately 187 m/min and 96 m/min, respectively. In other words, the cutting speed experienced by the tool varies by around 91 m/min. Thus, the choice of a suitable tool material (e.g., tungsten carbide tools) is essential in avoiding excessive tool wear and maintaining an adequate cutting performance. Fig. 10 shows a schematic illustration of the final (modified) cutter design.
Fig. 9. Cutting speed analysis of power skiving tool, in which green regions correspond to new tool and yellow regions correspond to end of tool life after resharpening. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 10. Schematic illustration of designed power skiving tool.
8. Conclusion This paper has presented a simple yet comprehensive mathematical modeling approach for the design and analysis of power skiving tools. Importantly, the proposed approach is applicable to both simple power skiving, which involves only a tool shaft angle, and general power skiving, which involves both a tool shaft angle and a tool tilt angle. In the proposed model, mathematical expressions are derived for the virtual conjugate surface of the tool and the rake surfaces. Consequently, the cutting edges can be obtained directly by numerical calculating the intersections of these two surfaces. In addition, the skiving tool and internal gear are modeled by transformation matrices. As a result, the normal and tangent vectors of the relative surface can be obtained explicitly, and hence the working rake and clearance angles of the cutting edge can be investigated in accordance with the recommended ISO standards. A simple method has been proposed for determining the full-field working rake angles, clearance angles and wedge angles based on the unit normal vectors of the ISO-recommended reference planes. A formulation for computing the full-field cutting speed has also be proposed. Finally, a method has been presented for redesigning the power skiving tool in such a way as to avoid negative working clearance angles. The validity of the proposed approach has been demonstrated by means of computer modeling. The mathematical model presented in this study is simple, easily applied, and capable of describing a wide range of power skiving tool design features. However, it is noted that the geometry of the modified power skiving tool proposed in the illustrative example is too complicated for real-world tool manufacturing processes (particularly high-precision tool grinding processes). Therefore, in a future study, the model will be applied to search for the optimum generating curve for the rake surface of a power skiving tool. In addition, a more detailed investigation will be performed into the feasibility of the proposed method for the real-world design and analysis of power skiving cutters manufactured using a grinding process. Acknowledgments The authors gratefully acknowledge the financial support provided to this study by the Ministry of Science and Technology, Taiwan under Grant No. MOST 103-2221-E-150-010. Appendix A 3 2 1 0 0 tx 6 0 1 0 ty 7 7 Trans t x ; t y ; t z ¼ 6 4 0 0 1 t z 5; 0 0 0 1 2
1 60 6 Rot ðx; ωx Þ ¼ 4 0 0
0 Cωx Sωx 0
2 Cωy 6 0 Rot y; ωy ¼ 6 4 −Sωy 0 2
Cωz 6 Sωz Rot ðz; ωz Þ ¼ 6 4 0 0
0 −Sωx Cωx 0
0 1 0 0
−Sωz Cωz 0 0
Sωy 0 Cωy 0
3 0 07 7; 05 1 3 0 07 7; 05 1
3 0 0 0 07 7: 1 05 0 1
ðA1Þ
ðA2Þ
ðA3Þ
ðA4Þ
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Note that S and C denote sine and cosine, respectively.
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