Research Article
Motion characteristic between die and
workpiece in spline rolling process with
round dies
Advances in Mechanical Engineering
2016, Vol. 8(7) 1–12
Ó The Author(s) 2016
DOI: 10.1177/1687814016655961
aime.sagepub.com
Da-Wei Zhang1,2,3, Sheng-Dun Zhao1 and Hengan Ou2
Abstract
In the spline rolling process with round dies, additional kinematic compensation is an essential mechanism for improving
the division of teeth and pitch accuracy as well as surface quality. The motion characteristic between the die and workpiece under varied center distance in the spline rolling process was investigated. Mathematical models of the instantaneous center of rotation, transmission ratio, and centrodes in the rolling process were established. The models were
used to analyze the rolling process of the involute spline with circular dedendum, and the results indicated that (1) with
the reduction in the center distance, the instantaneous center moves toward workpiece, and the transmission ratio
increases at first and then decreases; (2) the variations in the instantaneous center and transmission ratio are discontinuous, presenting an interruption when the involute flank begins to be formed; (3) the change in transmission ratio at the
forming stage of the workpiece with the involute flank can be negligible; and (4) the centrode of the workpiece is an
Archimedes line whose polar radius reduces, and the centrode of the rolling die is similar to Archimedes line when the
workpiece is with the involute flank.
Keywords
External spline, cold rolling, plane meshing, center distance variation, instantaneous center, transmission ratio
Date received: 7 August 2015; accepted: 3 May 2016
Academic Editor: David R Salgado
Introduction
The rolling process has been widely used to manufacture spline or short-toothed gears due to its proven
advantage of efficient utilization of material, short process time, high strength, and wear resistance of rolled
parts.1–3 However, the motion in the rolling process of
spline is very complex, where the workpiece is driven by
the rolling die, and the rolling die has a rotation combining feed-in motion, and the center distance between
the die and workpiece changes continuously in the process. The forming qualities, such as division of teeth,
pitch accuracy, and surface quality, are prone to error
because the motion between the workpiece and dies is
not coordinated. Thus, it is necessary to explore the
motion characteristic between the die and workpiece in
the rolling process.
Much research is focused on mechanics analysis,3
processing experiment,4 parts performance,5 and numerical simulation.6 Taking into consideration the frictional
moment, Zhang et al.7 modeled the rotatory condition
at the initial stage to guarantee precise division of teeth.
The rotatory condition reflects the relationship among
1
School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, P.R.
China
2
Department of Mechanical, Materials and Manufacturing Engineering,
University of Nottingham, Nottingham, UK
3
Suzhou Academy, Xi’an Jiaotong University, Suzhou, P.R. China
Corresponding author:
Da-Wei Zhang, School of Mechanical Engineering, Xi’an Jiaotong
University, No. 28, Xianning West Road, Xi’an, Shaanxi 710049, P.R.
China.
Email: [email protected]; [email protected]
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License
(http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without
further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/
open-access-at-sage).
2
the diameter of the billet before rolling, the outside diameter of die, decrement (depending on feed-in and rotating speeds of die), and coefficient of friction, but the
change of the center distance was not considered.
Neugebauer et al.1,2 presented that the rolling circle
changes in the rolling process, and the workpiece and
die should be forced for synchronizated rotation in order
to roll higher tooth profiles. The synchronization is realized by speed-control of the workpiece (an additional
kinematic compensation), and the revolution of the
workpiece in the rolling process is determined by the
rolling circle diameter, while the revolution of die1,8 or
the revolution of the workpiece multiplied by rolling circle diameter is a constant.2 However, the determination
of the rolling circle diameter in the rolling process was
not mentioned in the relevant literatures.
For gear mesh, a mathematical model was developed to modify the tooth profile of gear pairs under a
certain shaft displacement or deflection by Dooner and
Santana.9 Tsai et al.10 studied the influence of center
distance error on the stress of flank in planetary gear
mechanism. The effects of center distance variation on
the meshing stiffness of a gear pair also have been
researched.11,12 Zhao et al.13 investigated the center distance modification for the machining of worm drives.
Considering certain assembly errors, Wang et al.14
studied the tooth profile and meshing for double-roller
enveloping hourglass worm. However, it appears that
there are not sufficient reports in the literature on
motion characteristic under the condition of continuously changed center distance.
In this study, the motion characteristic between the
die and workpiece in the spline rolling process was investigated, where the center distance changes continuously.
The tooth profile ( fw) of the spline/workpiece is the conjugate curve of the tooth profile ( fd) of the rolling die.
In the final rolling position, where the die does not have
feed-in motion, the conjugate curve of fd is the desired
tooth shape of the spline/workpiece, and thus the
expected tooth profile can be determined by the tooth
profile of the die. According to two known tooth profiles, mathematical models of the instantaneous center
of rotation, transmission ratio, and the centrodes of
workpiece and die in the spline rolling process with varying center distance have been established. Thus, the
motion characteristic in the rolling process can be
obtained, and this provides a basis for process active
control of the spline rolling process to ensure rolling successfully and to improve forming quality. The relative
calculation procedure has been compiled by MATLAB.
Description of spline rolling process
The spline rolling process with round dies is based on
the principle of cross rolling, which can be presented as
Advances in Mechanical Engineering
Figure 1. Spline rolling process with round dies.
Figure 2. Forming stages divided by diffident rules (two round
dies).
follows: two spline rolling dies with the same tooth profiles synchronously rotate in the same direction, and
the workpiece rotates in the opposite direction, and one
rolling die or two rolling dies feed-in under constant
speed or constant force, as shown in Figure 1.
The rolling process can be divided into four stages
according to decrement or be divided into two stages by
taking into consideration of forming tooth profile,15 as
shown in Figure 2. The rolling process with two round
dies was taken as example in Figure 2. The decrement is
the difference between the root radius of the workpiece
before the spline rolling die contact with the workpiece
and the root radius of the workpiece after the rolling
die’s separation from the workpiece, which depends on
feed-in and rotating speeds of the rolling die.
According to decrement, the spline rolling process
can be divided into four forming stages as described in
Figure 2. The decrement increases from 0 to a constant
at the first rolling stage, and is the constant at the second rolling stage, and decreases from the constant to 0
at the third rolling stage, and is zero at the fourth rolling stage. The center distance between the die and
Zhang et al.
workpiece changes continuously at the first and second
stages. According to the tooth profile, the rolling process can be divided into two forming stages, such as the
divide-tooth stage and form-tooth stage. The number
of teeth is determined at the divide-tooth stage, and the
flank of workpiece has been formed at the form-tooth
stage. It can be seen from Figure 2 that the forming
time (or rotation angle of workpiece) of the dividetooth stage is the same as that of the first rolling stage,
and the form-tooth stage corresponds to the second,
third, and fourth stages.
In the rolling process of the involute spline, if the relative slip between the workpiece and the rolling die is
neglected, the angular velocity of the workpiece or transmission ratio is stable at the form-tooth stage as reported
from previous research.15 However, in the rolling process
of the involute spline with circular dedendum, it was
found in this study that this only occurs in cases of the
involute flank of the workpiece being formed. At the initial forming stage, without the involute flank of the
workpiece being formed, the workpiece only has circular
dedendum; after a certain amount of feed-in of the die,
the involute flank of the workpiece is formed.3 The involute flank of the workpiece may be formed at the stage
of tooth formation as shown in Figure 2, but it may be
formed at the divide-tooth stage if the feed-in speed of
the die increases. The workpiece with different curves
has an influence on the motion characteristic. The forming stages considering the formation of the involute
flank were used in this study.
Zhang et al.7 presented that at the divide-tooth
stage, the rotation of the workpiece is mainly driven by
frictional moment; at the form-tooth stage, flank of the
workpiece has been formed, so the motion between the
workpiece and rolling die can be regarded as meshing
motion between two tooth profiles. Thus, friction effect
on motion is considered at the divide-tooth stage and
friction effect on deformation is considered at the
form-tooth stage. An additional kinematic compensation is added in the rolling process only taking meshing
motion into consideration in the study.1,2,8 So the friction effect is also not considered in the analysis of
motion characteristic in this study.
3
Figure 3. Coordinate systems and tooth curves in the spline
rolling process.
which rotates and moves with the rotation and feed-in
of the rolling die.
fd is a known tooth profile of the rolling die and fw is
an unknown tooth profile of the spline/workpiece. Two
tooth profiles contact and are tangent to point M, and
line MP is the common normal line at point M. The
common normal line MP intersects line Ow Od at point
P, where Ow is the center of the billet/workpiece and Od
is the center of the rolling die. The rolling die rotates
clockwise.
Tooth profile fw in final rolling position
The desired tooth profile fw is the conjugate curve of
the tooth profile fd in the final rolling position where
the rolling die stops the feed-in and only rotates. In the
final rolling position, the center distance is certain and
there is no variation, and the transmission ratio i also
becomes constant. The tooth profile of the spline/workpiece in the final rolling position can be obtained using
theories of conjugate curves and envelope curve. Based
on the theories, the tooth profile of the workpiece in the
spline rolling process under the certain center distance
and transmission ratio was modeled in Zhang et al.16
In the coordinate system Od xd yd , fd can be expressed
as follows
fd :
xd = xd (h)
yd = yd (h)
ð1Þ
Mathematical models of motion
characteristic
where h is the parameter of parametric equation.
In this study, the standard homogeneous coordinates
have been adopted in order to easily synthesize the rotation transformation and translation transformation.
The family of curves fdu (h, u) in the coordinate system
Ow xw yw can be expressed as follows
Coordinate system
0
Three rectangular coordinate systems, that is, Oxy,
Ow xw yw , and Od xd yd , are established, as shown in
Figure 3. Where Oxy is rigidly connected to the frame
which is fixed; Ow xw yw is rigidly connected to the workpiece, which rotates with the rotation of the workpiece;
and the Od xd yd is rigidly connected to the rolling die,
B C B
@ yw A = @ sin if u cos if u
1
0
0
0
10 1
cos u sin u 0
xd
B
CB C
y
sin
u
cos
u
0
@
A@ d A
1
0
0
1
xw
1
0
cos if u
sin if u
0
10
1
0
CB
0 A@ 0
1
1
0
0
af
1
C
0A
1
ð2Þ
4
Advances in Mechanical Engineering
0 1 0
1
x
@yA=@0
0
1
Table 1. Basic parameters for involute.
Parameter
Symbol
Unit
Value
Module
Number of spline/workpiece teeth
Number of rolling die teeth
Pressure angle of reference circle
Addendum coefficient of spline
Dedendum coefficient of spline
m
Zw
Zd
ar
ha
hf
mm
–
–
°
–
–
1
20
200
37.5
0.45
0.7
where af is the center distance between the die and the
workpiece at the final rolling position; if is the transmission ratio at the final rolling position
if =
Zd
Zw
ð3Þ
where Zd is the number of rolling die teeth; Zw is the
number of spline teeth.
The tooth profile fw consists of equations (2) and (4)
∂yw ∂xw ∂yw ∂xw
=0
∂h ∂u
∂u ∂h
a=
a0 vt
af
v . 0
v=0
ð8Þ
where v is the feed-in speed of rolling die, a0 is the initial
center distance, and t is the rolling time.
The coordinate system Ow xw yw is concentric with the
coordinate system Oxy, and then the tooth profile ( fw )
of the spline/workpiece is expressed by equations (2)
and (6). However, the coordinate transformation for fw
may be needed at center distance a, which is determined
according to the specific curve.
The tooth profile fd is tangent to fw at contact point
M. The coordinate of the point is (x(hM ), y(hM )), which
in fd and fw can be expressed as (x(hMd ), y(hMd )) and
(x(hMw ), y(hMw )), respectively, and then equation (9)
would be obtained
ð4Þ
8
x(hMd ) = x(hMw )
>
>
< y(hM ) = y(hM )
w d
dfw > dfd >
=
: dx dx h = hMw
h = hM
ð9Þ
d
ð5Þ
Substituting the expressions of fd and fw into equation (9), then the coordinate of point M would be
solved.
According to the meshing equation (equation (10))
in plane meshing, the common normal line of the tooth
profiles at point M should pass the instantaneous center point P at this time17
v(12) n = 0
Substituting equation (5) and the partial derivatives
of xw and yw into equation (4), equation (6) can be
obtained
(if + 1)(xd sin g + yd cos g)
g
u = arcsin
if af
ð7Þ
where
Letting
8
dxd
>
>
>
>
>
dh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
sin g = s
>
2 2ffi
>
>
dxd
dyd
>
>
>
+
>
>
dh
dh
>
>
>
>
dy
>
d
<
dh
cos g = sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2ffi
>
dxd
dyd
>
>
>
+
>
>
dh
dh
>
>
>
>
dx
>
d
>
>
>
>
>
tan g = dh
>
>
dyd
>
:
dh
10 1
xd
0 a
1 0 A @ yd A
0 1
1
where v(12) is the relative speed at point M; n is the normal vector.
In the coordinate system Oxy, the common normal
line at point M can be expressed as follows
ð6Þ
The tooth profile of the spline/workpiece in the final
rolling position is composed of equations (2) and (6).
Transmission ratio, instantaneous center, and
centrodes
Under center distance a, the tooth profile ( fd ) of the
rolling die can be expressed as follows in the coordinate
system Oxy
ð10Þ
dx dy (x xM ) +
(y yM ) = 0 ð11Þ
dh h = hM
dh h = hM
In the coordinate system Oxy, the line Ow Od passes
the center (O, Ow ) of the billet/workpiece and the center
(Od ) of the rolling die, that is, y = 0. Thus, the intersection P can be solved from equation (11), such as follows
8
dy >
>
<
dh h = h
M
xP = xM + yM dx
ð12Þ
>
dh h = hM
>
:
yP = 0
Zhang et al.
5
The intersection P is the instantaneous center of
rotation, and thus the transmission ratio i under center
distance a can be expressed as follows17,18
ia =
Od P
Ow P
ð13Þ
The angular velocity of the rolling die is vd , which
often is a constant in the rolling process, and thus the
centrode of the rolling die can be expressed as follows
0 1 0
1
x
B C B
=
y
@ A @0
0
1
1
B
@0
0
0
1
10
Ð
a
cos vd dt
Ð
CB
0 A@ sin vd dt
0
0 1
10 1
0 a
xP
CB C
1 0 A@ y P A
0
1
0
Ð
sin vd dt
Ð
cos vd dt
0
0
1
ð14Þ
Then, the centrode of the workpiece can be expressed
as follows
Ð
sinÐ ivd dt
cos ivd dt
0
10
8
< r = rbd sec a
u = inv a inv ard
:
a 2 ½afd , aT C
0A
1
1
0 1 0
Ð
x
cos Ð ivd dt
@ y A = @ sin ivd dt
1
0
die, and ard is the pressure angle of reference circle of
the rolling die.
In the final rolling position, taking Od shown in
Figure 3 as polar point and taking negative xd -axis
shown in Figure 3 as polar axis, a polar coordinate system is established. The involute flank section fwinv of the
tooth profile ( fw ) of the workpiece is tangent to fdinv of
fd at point M. If point M is in line with line Ow Od , then
parametric equation of fdinv in the polar coordinate system can be expressed as follows
where rbd is the base circle radius of the rolling die; afd
is the pressure angle in dedendum circle of the rolling
die when engagement has no headspace.
According to equation (17), the fdinv in the coordinate
system Od xd yd can be expressed as follows
8
< xd = r cos u = xd (a)
yd = r sin u = yd (a)
:
a 2 ½afd , aT 1
xP
0
0 A@ yP A ð15Þ
1
1
Application for rolling process of involute
spline with circular dedendum
In general, the spline cold rolling process has been used
to form circular dedendum spline, where one knuckle
curve connects two involute flanks to dedendum circle.
Correspondingly, a knuckle curve connects two flanks
to addendum circle of the rolling die. The discussion in
this section is based on these profiles. The spur-involute
spline is discussed in this section, and the basic parameters used in this section are listed in Table 1.
Tooth profiles
In the rolling process for the involute spline with circular dedendum, the tooth profile (fd ) of the rolling die is
composed of involute flank section fdinv and knuckle of
addendum section fdcir . The knuckle is tangent to two
involute flanks and addendum circle. The tangent point
between the knuckle and involute flank is point T, and
the pressure angle and polar radius at point T are aT
and rT , respectively. Then, equation (16) can be
obtained according to triangular cosine theorem
re2 + (rad re )2 2re (rad re )
p
p
+ tan aT cos
inv ard r2T = 0
2
2Zd
ð16Þ
where re is the knuckle radius for addendum of the rolling die, rad is the addendum circle radius of the rolling
ð17Þ
ð18Þ
Substituting the differentiation of equation (18) into
equation (5), then the following expression would be
obtained
tan g = cot(tan a inv ard )
ð19Þ
that is
g=
p
tan a + inv ard
2
ð20Þ
Then, the involute flank fwinv of the workpiece can be
obtained according to equations (2), (6), (18), and (20).
In the final rolling position, the coordinate of center
(Oc ) of knuckle for addendum of the rolling die in the
coordinate system Od xd yd can be expressed as follows
p
p
, (rad re ) sin
ð21Þ
(rad re ) cos
2Zd
2Zd
Then, fdcir in the coordinate system Od xd yd can be
expressed as follows
8
p
>
=
r
cos
b
(r
r
)
cos
= xd (b)
x
>
d
e
a
e
d
>
d
<
2Z
yd = re sin b (rad re ) sin 2Zpd = yd (b)
>
h
i
>
>
: b 2 p + tan a inv a , p + p
T
r
d
2
2Zd
ð22Þ
Substituting the differentiation of equation (22) into
equation (5), then the following expression would be
obtained
tan g = tan b
ð23Þ
6
Advances in Mechanical Engineering
Under center distance a, dedendum circle radius of
workpiece is rfa , which can be expressed as follows
rfa = rZ + a a0
Figure 4. Tooth curves of the die and workpiece in the final
rolling position.
that is
g=p b
ð24Þ
Then, the circular dedendum fwcir of the workpiece
can be obtained according to equations (2), (6), (22),
and (24).
The above calculations were carried out under
MATLAB software environment, and Figure 4 illustrates
the tooth profiles of the die and workpiece in the final
rolling position, where coordinate system Oxy is adopted.
Models for rolling process of circular dedendum
spline
At the initial forming stage, the involute flank of the
workpiece is not formed; after a certain amount of
feed-in of the rolling die, the involute flank of the
workpiece is formed.3 At the forming stage where the
workpiece only has circular dedendum, a circular arc
meshing motion occurs; with the decrease in center distance, the involute flank of the workpiece has been
formed, and then involute meshing motion occurs. The
different meshing motions are based on different
expressions of tooth curves, and thus the models of
motion characteristics are also different.
With the increase in the amount of feed-in of the
rolling die, the center distance between the die and the
workpiece decreases, and the tooth height of the workpiece increases. Thus, there is a critical center distance
acrit , where if a acrit , then the workpiece only has circular dedendum, and the involute flank is not formed;
if a\acrit , the workpiece with the involute flank and
tooth profile consists of circular dedendum and involute flank. The acrit can be determined according to volume constancy principle in the rolling process (see
Appendix 2).
ð25Þ
where rZ is the initial radius of the billet before rolling.
The involute flank section of the workpiece is similar
to the modified gear, and thus the expressions of fwinv in
section ‘‘Tooth profiles’’ are also used in here.
However, the circular dedendum starts from root rfa ;
and the results in Zhang et al.15,16 indicated that the
contact between circular dedendum of the workpiece
and knuckle of the rolling die appears around x-axis,
so fwinv would rotate by p=(2Zw ) in order to find the
contact point. Thus, under center distance a, fwcir of the
workpiece in the coordinate system Oxy should be
expressed by equation (26)
1
0 1 0
x
1 0 rf a rf w
C
B C B
0 A
@yA=@0 1
0 0
1
1
0
1
p
p
sin
0 0x 1
cos
2Zw
2Zw
B
C w
B
CB C
p
B sin p
@ yw A
cos
0C
@
A
2Zw
2Zw
1
0
0
1
ð26Þ
where rfw is the dedendum circle radius of the spline.
Under center distance a, the tooth profile of the rolling die contacts with and is tangent to tooth profile fw
after rotating by f. Then, tooth profile fd in the coordinate system Oxy should be expressed by equation (27)
0 1 0
x
1
@yA=@0
1
0
10
0 a
cos f
1 0 A@ sin f
0 1
0
sin f
cos f
0
10 1
0
xd
0 A@ yd A
1
1
ð27Þ
Workpiece only with circular dedendum. At the initial forming stage where the workpiece only has circular dedendum, the coordinate of contact point is (x(bM ), y(bM )),
which in fdcir and fwcir can be expressed as
(x(bMd ), y(bMd )) and (x(bMw ), y(bMw )), respectively; the
following expression can be obtained according to
equation (9)
8
x(bMd ) = x(bMw )
>
>
< y(bM ) = y(bM )
w d
dfd dfw >
>
=
: dx dx b = bM
b = bM
d
ð28Þ
w
where there are three unknowns with three equations,
so the unknowns including bMw , bMd , and f can be
solved. Substituting equations (2), (6), (22), (24), (26),
Zhang et al.
7
and (27) into equation (28), the following expression
can be obtained
the coordinate of contact point is (x(aM ), y(aM )), which
in fdinv and fwinv can be expressed as (x(aMd ), y(aMd )) and
8
>
re cos (bMd f) (rad re ) cos 2Zpd f + a
>
>
>
>
>
>
>
= re cos bMw (if + 1)u + 2Zpw (rad re ) cos 2Zpd (if + 1)u + 2Zpw + af cos if u 2Zpw + (rfa rfw )
>
>
>
>
> r sin (b f) (r r ) sin p f
>
>
e
a
e
Md
d
>
<
2Zd
p
= re sin bMw (if + 1)u + 2Zw (rad re ) sin 2Zpd (if + 1)u + 2Zpw af sin if u 2Zpw
>
>
>
dy >
>
db
>
cot
(b
f)
=
>
dx Md
>
db
>
b = bM w
>
>
>
>
>
p
(i
+
1)(r
r
>
f
ad
e ) sin bM 2Z
>
w
d
: u = arcsin
+ bMw p
if af
ð29Þ
where
fwcir :
8
dx
p
>
>
= re sin b (if + 1)u +
>
>
db
2Zw
>
>
>
>
>
du
p
p
p
p
>
>
(if + 1)re sin b (if + 1)u +
(if + 1)(rad re ) sin
if af sin if u +
(if + 1)u +
>
>
>
db
2Zw
2Zd
2Zw
2Zw
>
>
>
>
p
> dy
>
<
= re cos b (if + 1)u +
db
2Zw
>
>
du
p
p
p
p
>
>
+ (if + 1)(rad re ) cos
if af cos if u +
(if + 1)re cos b (if + 1)u +
(if + 1)u +
>
>
>
db
2Zw
2Zd
2Zw
2Zw
>
>
>
>
p
>
>
cos
b
> du
2Zd
>
>
= 1 + rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
>
2
ffi
>
db
>
i
a
2
f
f
p
:
sin
b
(if + 1)(ra re )
2Zd
ð30Þ
d
When the workpiece only has circular dedendum,
the instantaneous center P in the coordinate system
Oxy can be expressed as follows
xP = xM yM cot (bMd f)
yP = 0
ð31Þ
Then, at center distance a, the transmission ratio ia
can be determined by equation (13), and the centrodes
of the rolling die and the workpiece can be determined
by equations (14) and (15).
Workpiece with involute flank. At the forming stage where
the workpiece is with involute and circular dedendum,
(x(aMw ), y(aMw )), respectively; equation (32) can be
obtained according to equation (9)
8
x(aMd ) = x(aMw )
>
<
y(a Md ) = y(aMw)
>
: dfdxd = dfdxw a = aMd
ð32Þ
a = aM w
where there are three unknowns with three equations,
so the unknowns including aMw , aMd , and f can be
solved. Substituting equations (2), (6), (18), (20), and
(27) into equation (32), the following expression can be
obtained
8
rbd sec aMd cos(invaMd inv ard f) + a
>
>
>
rbd (if + 1)
>
>
sec
a
sin
a
+
arcsin
+
i
u
+ af cos if u
=
r
b
M
M
f
>
d
w
w
if af
>
>
>
>
sec
a
sin(inv
a
inv
a
f)
r
bd
Md
rd
>
Md
>
>
rbd (if + 1)
>
>
sec
a
cos
a
+
arcsin
+
iu
af sin if u
=
r
bd
Mw
Mw
>
if af
>
>
>
< tan (tan aMd inv ard f)
rbd (if + 1)
rbd (if + 1)
>
r
sin
arcsin
+
iu
+
i
r
sec
a
sin
a
+
arcsin
+
i
u
if af
bd
f bd
Mw
Mw
f
>
>
if af
if a f
>
>
=
>
>
rbd (if + 1)
rb (if + 1)
>
>
>
rbd cos arcsin
+ iu if rbd sec aMw cos aMw + arcsin d
+ iu if af
>
>
a
i
if af
>
f f
>
>
r (i + 1)
>
>
u = arcsin bd iffaf g
>
>
:
g = p2 tan aMw + inv ard
ð33Þ
cos if u
sin if u
8
Advances in Mechanical Engineering
Figure 6. x Axial coordinate of instantaneous center in rolling
process.
Figure 5. Contact point and instantaneous center at different
stages: (a) workpiece only with circular dedendum and (b)
workpiece with involute flank.
Figure 7. Transmission ratio in rolling process.
When the workpiece is formed with the involute
flank, the instantaneous center P in the coordinate system Oxy can be expressed as follows
xP = xM + yM tan(tan aMd inv ard f)
yP = 0
ð34Þ
Then, the transmission ratio ia can be determined by
equation (13), and the centrodes of the rolling die and
the workpiece can be determined by equations (14) and
(15).
Results and discussion
Under MATLAB software environment, a calculation
program was developed to obtain the relevant data.
Figure 5 illustrates the contact point and instantaneous
center at different forming stages where the workpiece
has different curves.
When the workpiece is formed only with circular
dedendum, the contact point is far from the
instantaneous center, and the instantaneous center is
generally outside of addendum circle of the workpiece.
Conversely, when the workpiece with the involute
flank, the contact point is close to the instantaneous
center, and the instantaneous center is generally inside
of addendum circle of the workpiece. Because of this,
the change in the instantaneous center at former stage
is larger than that at latter stage, as shown in Figure 6.
In the forming process, the changes in the instantaneous center and transmission ratio are shown in
Figures 6 and 7, respectively. There is a notable difference for two forming stages. With the decrease in center distance, the instantaneous center moves along
negative x-axis and the position of the instantaneous
center remains the same after the rolling die stops feedin in the radial direction.
At the stage where the workpiece is only with circular dedendum, the changes of the instantaneous center
and transmission ratio are notable, and the x axial
Zhang et al.
9
Figure 8. Centrodes in the spline rolling process: (a) centrodes of rolling die and workpiece, (b) enlarging area at initial stage, and
(c) enlarging area for workpiece.
coordinate of the instantaneous center reduces and the
transmission increases about 1.3701%.
However, at the stage where the workpiece is formed
with the involute flank, the instantaneous center varies
slowly, and the transmission ratio has a slight decrease
which is about 20.0007%. The change of transmission
ratio at this stage is consistent with the transmission
principle of involute gearing.
The different curve meshing is the main reason for a
remarkable difference of the change in the instantaneous center or transmission ratio at the two forming
stages. Under the parameters used in Figure 7, the difference of the transmission ratio is about 3.0211%
when the workpiece only with circular dedendum turns
into the workpiece with the involute flank.
Figure 8 illustrates the centrodes for the rolling die
and the workpiece in the rolling process, and it can be
found from the figure that the centrodes are not closed
in the changing process of the center distance. Before
rolling die stops feed-in in the radial direction, the centrode of the workpiece is an Archimedes line where the
polar radius reduces. The centrode is determined by the
instantaneous center. The position of the instantaneous
center has a notable change when the workpiece only
with circular dedendum turns into the workpiece with
the involute flank, so the centrode also has a notable
change at this moment, as shown in Figure 8(c).
Figure 9. Polar radius of centrode and rotary center for rolling
die.
The polar radius of the centrode for the rolling die
almost remains unchanged at the stage where the workpiece is only with circular dedendum, and the centrode
for the rolling die is similar to Archimedes line at the
forming stage of the workpiece with the involute flank,
where the polar radius reduces with the decrease in center distance, as shown in Figure 9. The rotary center of
the rolling die moves along negative x-axis with the
decrease in center distance. Thus, the later centrode for
the rolling die may be outside of the initial centrode, as
10
Advances in Mechanical Engineering
3.
Figure 10. Kinematic compensation for workpiece in the
rolling process.
shown in Figure 8(b). When the workpiece only with
circular dedendum turns into the workpiece with the
involute flank, the polar radius of the centrode of the
rolling die increases. However, rotary center of the rolling die moves toward workpiece and the radius of the
rolling die is much larger than the radius of the workpiece, and then the change of centrode at this moment
is not as sharp as that for the workpiece.
In order to keep the synchronization rotation
between the workpiece and die, an additional kinematic
compensation is implemented for the workpiece over
the process. The angular velocity of the workpiece over
the whole rolling process was determined by the changing transmission ratio and the angular velocity of the
rolling die, as shown in Figure 10.
Conclusion
Based on the principle of plane meshing and the processing characteristic of spline rolling, mathematical
models of instantaneous center, transmission ratio, and
centrodes in the spline rolling process under continuously varied center distance have been developed.
Application of the developed models to investigate the
rolling process of the involute spline with circular
dedendum has been carried out, and the following conclusions can be drawn:
1.
2.
With the feed-in of the rolling die in the radial
direction, the center distance between the die
and workpiece decreases, and the instantaneous
center moves along negative x-axis. The position of the instantaneous center varies sharply
at the stage where the workpiece is only with
circular dedendum.
The transmission ratio increases at first and
then decreases with the decrease in center
distance. The transmission ratio increases when
the workpiece is only with circular dedendum,
and the decrease in the transmission ratio when
the workpiece is with the involute flank can be
neglected, and there is an interruption between
two stages.
The centrode of the workpiece is an Archimedes
line, where the polar radius is reducing. The
polar radius of the centrode of the rolling die
almost remains unchanged at the stage where
the workpiece is only with circular dedendum,
and the polar radius is reducing with the
decrease in center distance after that stage.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this
article: The authors would like to gratefully acknowledge the
supports of the National Natural Science Foundation of
China (grant no. 51305334), National Natural Science
Foundation of China for key program (grant no. 51335009),
China Postdoctoral Science Foundation (grant no.
2014T70913), Shaanxi Province Natural Science Foundation
of China (grant no. 2014JQ7273), and Science and
Technology Planning Project of Suzhou, China (grant no.
SYG201447).
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Appendix 1
fd , fdcir , fdinv , fw ,
fwcir , fwinv
ha
hf
h, b
i, ia , if
m
M, P, T
n
Od
Ow
Oxy, Od xd yd ,
O w xw yw
ra0
racrit
ra d
rb d
rb w
rC
re
rf a
rfcrit
rf w
rZ
s0a , sfa , sfw
Scrit
t
v
v(12)
Zd
Zw
a, a0a , aB , aD ,
afd , ar , ard ,
arw , aT
g, u
d
u, uA , uB , uD
p
Notation
a, acrit , af ,a0
Oc
center distance between die and
workpiece
curves
addendum coefficient of spline
dedendum coefficient of spline
parameters of parametric
equations
transmission ratio
module of spline
points
normal vector
r, rT
f
uw
vd
center of the knuckle radius for
addendum of rolling die
center of rolling die
center of billet/workpiece
rectangular coordinate systems
radius of (operating) pitch circle
addendum circle radius of
workpiece under center distance
acrit
addendum circle radius of rolling
die
base circle radius of rolling die
base circle radius of spline
radius of workpiece at the
tangent point between dedendum
circle and flank
knuckle radius
dedendum circle radius of
workpiece under center distance a
dedendum circle radius of
workpiece under center distance
acrit
dedendum circle radius of spline
initial radius of billet
tooth thickness
area outside dedendum circle for
one tooth of workpiece under
center distance acrit
rolling time
infeeding speed of rolling die
relative speed
number of rolling die teeth
number of spline teeth
pressure angle
parameters of envelop curve
half of angle of knuckle curve at
spline root
polar angle
ratio of a circle’s circumference to
its diameter
polar radius
rotation angle
rotation angle of workpiece in
rolling process
angular velocity of rolling die
Appendix 2
Critical center distance
The addendum circle radius of the workpiece is racrit
under the critical center distance acrit , and the full
12
Advances in Mechanical Engineering
prZ2 = prf2crit + Zw Scrit
ð37Þ
where Scrit is the area outside dedendum circle for one
tooth of the workpiece under center distance acrit .
According to the study in Zhang,19 Scrit can be
expressed as follows
rb2w
(tan3 aB tan3 aD )
3
+ 2re2 ( tan d d) rf2crit (uA uD )
Scrit = ra2crit (uA uB ) +
Figure 11. Convex at root of rolled spline:19 (a) finite element
analysis and (b) experiment.
circular dedendum is just formed at this moment, and
thus the following expression can be obtained
racrit rfcrit = rC rfw
ð35Þ
where rfcrit is the dedendum circle radius of the workpiece at center distance acrit , which is expressed by
equation (36); rC is the radius of the workpiece at the
tangent point between dedendum circle and flank
rfcrit = rZ + acrit a0
ð36Þ
In the spine rolling process, there is no deformation
in the axial direction. The experiment and finite element
analysis indicated that19 there only exists a slight convex at root on free end cross section of the workpiece,
as shown in Figure 11, which affects the small range
from the end cross section. This corner filling problem
will be prevented by adding a restriction (such as bar)
on the free end cross section, as shown in Figure 11(b).
Thus, the deformation in the axial direction can be
neglected, and then equation (37) can be obtained
according to volume constancy principle
ð38Þ
where
8
0
< uA = 2rs 0aa + inva0 a
u = invaB
: B
uD = invaD
8
m cos arw (Zw + Zd )
>
>
a0 a = arc cos
>
>
2(ra2 + rfcrit )
>
>
<
rbw
aB = arc cos
>
racrit
>
>
>
r
>
>
: aD = arc cos bw
rfcrit
8
mZw cos arw
>
0
>
>
> ra = 2 cos a0 a
>
>
>
<
r0 a
s0a = sfa
2r0 a (inva0 a invaD )
>
r
f
crit
>
>
>
>
2prfcrit 2prfw
>
>
: sf a =
+ sf w
Zw
Zw
ð39Þ
ð40Þ
ð41Þ
where d is the half of angle of knuckle curve at spline
root, arw is the pressure angle of reference circle of
spline, and sfw is the tooth thickness at reference circle
of spline.
The critical center distance acrit can be solved by
equations (35)–(41).