1. No category

Strojniški vestnik - Journal of Mechanical Engineering Volume(Year)No, StartPage-EndPage
UDC xxx.yyy.z
Paper received: 00.00.200x
Paper accepted: 00.00.200x
Numerical Calculation of Tooth Profile of Non-circular
Curved Face Gear
1
Chao Lin1,* - Dong Zeng1 - Xianglu Zhao1 - Xijun Cao1
The State Key Laboratory of Mechanical Transmission, Chongqing University, P R China
Based on the cylindrical coordinates and the space engagement theory, from the perspective of
cylindrical coordinate system, the arbitrary curve equation was obtained and a method of curve
expansion was established. Then any order of pitch curve of non-circular curved face gear and normal
equidistant curve parameter equation could be derived. The tooth profile points of non-circular curved
face gear can be solved using the numerical calculation method that the tooth profile intersect with pitch
curve and normal equidistant curves of non-circular curved face gear. Finally the numerical method to
solve the tooth profile of the non-circular curved face gear was proved correctly using the way which
analyzed the error between measured data and the data of theoretical calculation.
©20xx Journal of Mechanical Engineering. All rights reserved.
Keywords: non-circular curved face gear, cylindrical coordinates, pitch curve, numerical
calculation of tooth profile, error analysis
0 INTRODUCTION
The non-circular curved face gear is also
called the orthogonal variable transmission ratio
face gear. The column gear of the ordinary face
gear pair is replaced by the non-circular gear and
gets the conjugate meshing face gear which is
called the non-circular curved face gear[1]. The
transmission ratio is variable since the tooth of
the non-circular curved face gear distributes on
the cylindrical surface where the gear and noncircular gear meshing transmission. That’s the
most significant feature compared to the face
gear. So the gear pair has a great application
prospect in the field of engineering, textile,
agriculture etc.
Buckingham came up with the concept of
face gear in the paper of "Analytical Mechanics
of Gears" for the first time in 1940, which was
defined as rack of changing tooth pitch and
pressure angle[2]; Litvin and his team made great
contribution to the research about face gear on the
basis of predecessors' research. His book “Gear
Geometry and Applied Theory” expanded depth
research about the surface of face gear in the view
of the gear geometry and meshing principle[3];
Zhu Ru-peng et al, did a lot of research about face
gear in the field of tooth surface contact analysis,
strength, coincidence degree theory and so on[4];
Lin Chao et al, proposed the non-circular curved
face gear for the first time and explored its tooth
profile
analysis,
machining
simulation,
*Corr.
measurement, etc. His study obtained significant
achievements. But there is rarely any research
about the tooth profile of non-circular curved face
gear[5]and[6].
The error of the tooth profile will not only
impact on the instantaneous transmission
accuracy but also cause plastic deformation on
the tooth surface of intermeshing gear. There
exist some methods to use the numerical method
to solve the gear tooth profile, for instance, Xia
Ji-qiang et al, calculated the tooth profile of bevel
gear by the numerical method which was based
on the spherical coordinates[7]to[9]. In the field
of gear measurement, Anke Guenther et.al put
forward some new theories and methods to
measure and analyze gear [10]to[12]. But at
present, there is no practical way to use numerical
method to solve the tooth profile of non-circular
curved face gear. This paper presents a new
general numerical method which is used to
calculate the tooth profile of non-circular curved
face gear. This method will serve as an important
reference basis in the field of modeling,
processing, analyzing the error of measurement
and so forth.
1. Parameters of the curve in cylindrical
coordinate system
The non-circular curved face gear is a kind
of cylindrical gear. In order to describe and
calculate the parameters of the gear more
Author's Address: Chongqing University State Key Laboratory of mechanical transmission, Shapingba
District sand 174 Center Street, Chongqing, P R China, linchao@cqu.edu.cn
1
Strojniški vestnik - Journal of Mechanical Engineering Volume(Year)No, StartPage-EndPage
accurately, this paper ordered a study on the angle
and the expansion of the curve under cylindrical
coordinate system.
Z
K
h1  1
1


M
the expansion of the plane OMN . Solve the
Eq.(2), then
N
2
O
The line links all points on the curve to the
center of the circle in the bottom surface will
compose a conical surface. As shown in Fig.1,
there are countless tangent planes through the
point N . But there exists unique tangent plane
which is vertical to the Z axis. There exists one
point K on the line intersected by the tangent
plane and the cylindrical surface where
LMN  LNK . At the same time the plane OKN is
h2
Y
LNK  LMN  
2
1

Fig. 1 Parameters of the curve in cylindrical
coordinate system
As shown in Fig.1, O  XYZ is the
cylindrical coordinate system,  is one curve on
the cylindrical surface, M and N are any two
points on the curve.  is the rotation angle which
is from the positive X axis to the point. h is the
distance between the point and the bottom surface,
 is the central angle of the point on the curve,
R is the cylindrical radius.
R 2  h'2  d
(4)
Then the expansion angle  corresponding
to the curve MN is:
M'  N'
2
X
2
1
R h
2
2

  
R 2  h '2  d
1
(5)
2. Pitch curve of non-circular curved face gear
2.1 The equation of the pitch curve
According to the meshing relationship of
non-circular curved face gear pair, the coordinate
system can be established.
Z2
1.1 The basic description of the curve
Assume that the equation of any curve is:
Γ   R cos
R sin
h  
Y1
O1
(1)
The arc length between points M and N
is:
LMN  
2
1
R 2  h'2  d
(2)
'
Where h   is the derivative of h   .
The central angle of one point on the curve
can be defined as the angle between the line
which links the point to the center of the circle in
the bottom surface and the bottom surface.
According to the geometric relationship the
formula can be got:
  arctan
h  
R
1.2 The expansion of the curve
2
(3)

X1
R
E
O2
P1
X2
2
r  1 
P2
Z1
1
Y2
h 2 
Fig. 2 Meshing coordinate of non-circular curved
face gear
As shown in Fig. 2, axis of the noncircular gear is orthogonal to axis of the noncircular curved face gear. As shown in Fig. 2,
O1  X 1Y1 Z1 is the fixed coordinate system of
non-circular gear. O2  X 2Y2 Z 2 is the fixed
Author's Surname, N. - Co-author's Surname, N.
Strojniški vestnik - Journal of Mechanical Engineering Volume(Year)No, StartPage-EndPage
coordinate system of non-circular curved face
gear. The point P1 is on the pitch curve of noncircular gear, and the point P2 is on the pitch
curve of non-circular curved face gear. When P1
coincides with P2 , the non-circular gear turns
angle of  1 , and non-circular curved face gear
turns angle of  2 . The radius of pitch curve of the
non-circular curved face gear is R , radius vector
of the non-circular gear is r 1  . In the condition
of Fig. 2, the distance between the axis of noncircular gear and the bottom surface of face gear
is E .  is the pitch curve of non-circular curved
face gear.
According to the theory of non-circular
gear, the equation of pitch curve of non-circular
gear is:
r 1  
a 1  e2 
(6)
1  e cos  n11 
Where,  1 is polar angle of elliptic gear.
a is half of longer axis of the ellipse. e is the
eccentricity of the ellipse. n1 is the order of the
non-circular gear.
When the parameters in Eq. (6) were
chosen as follows: a  35mm , e  0.1 and n1
changes from 1 to 4. The pitch curve of noncircular gear can be shown in Fig. 3.
120
90
120
60
150
330
240 270
120 90
210
330
240
270 300
(8)
Where,
R
n2
2
2  
1
0
2
n1
0

r   d  ;
1
1 1
d    r   d  ; n1 is the
i12
R 0
order of the non-circular gear. n2 is the order of
non-circular curved face gear. i12 is the
transmission ratio of non-circular curved face
gear pair and i12  R r 1  .
By substituting the equations above into
Eq. (8), the pitch curve of non-circular curved
face gear can be obtained.
2

n2
n1
x

cos


2 0 r    d 
2


2

n
r2  2    y  2 sin  2  n1 r   d 
0
2

 z  h  2   E  r 1 


(9)
2.2 Tooth modulus angle of the pitch curve plane
60
30
180
0
x  R cos  2


r2  
y  R sin  2
 z  h    E  r  
2
1

300
150
30
point on pitch curve and the bottom surface when
non-circular face gear turns an angle  2 .
By substituting the Eq. (6) and Eq. (7) into
the Eq. (1), the pitch curve of non-circular curved
face gear can be obtained.
330
240 270
90
120
60
180
0
210
300
150
30
180
0
210
60
150
30
180
90
Where, h 2  is the distance between the
0
210
330
240
270
300
Fig. 3 Impact of n1 to the pitch curve of noncircular gear
In the meshing process of non-circular
gear and non-circular curved face gear, the center
distance is fixed. From the geometric relationship
shown in Fig. 2, at any time the Eq. (7) is correct.
E  h  2   r 1   a  ae
(7)
As shown in the Fig. 4, based on the
method of the expansion of the curve, tooth 1 and
tooth 2 represent any 2 tooth on the pitch curve of
non-circular curved face gear. Point M and point
N are the points on the right tooth profile of
tooth 1 and tooth 2. There is a complete tooth
profile between M and N . By expanding the
tooth profile on the tangential circle of pitch
curve, the point K on the tangential circle can be
got. When the arc length of MN is equal to KN ,
the plane OKN is the expansion plane of one
tooth profile.
Numerical Calculation of Tooth Profile of Non-circular Curved Face Gear
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  a  ha   m



 f   ha  c   m
Z
Tangential
circle
Pitch curve
2
1
M
N
K
 m
1
Y
2
X
Where, ha is the addendum coefficient,
c  is the dedendum coefficient.  m is the
modulus angle.
So the addendum and dedendum of noncircular curved face gear are:
 ha  R tan     a   h  2 
(11)

h f  R tan     f   h  2 
Where,  is the center angle, R is the
R
O
Fig. 4 Tooth modulus angle of non-circular
curved face gear
When the angle   m ,  m is the
modulus angle of the tooth. The tooth profile of
the non-circular curved face gear changes when
direction of tooth width changes, so does the
modulus angle.
2.3 Addendum angle and dedendum angle
radius of the pitch curve of non-circular curved
face gear,  a is the addendum angle of noncircular curved face gear,  f is the dedendum
angle of non-circular curved face gear.
3. The numerical method to solve tooth profile
The fundamental of numerical method to
solve the tooth profile of non-circular curved face
gear is derived from the generating processing of
non-circular curved face gear. The generating
processing is shown in Fig. 6.

Z
O
M
P
f
R
Z 2  Z 2' 
1
a
N
1
2
3
2
Xk X
k'
Zk  Zk ' 

3
Ok
2
Y
Yk
Yk '
Y2'
O2
Y2
X
1-Addendum line 2- Pitch curve
3- Dedendum line
Fig. 5 The addendum angle and the dedendum
angle
As shown in Fig. 5, addendum line, pitch
curve and dedendum line of the non-circular
curved face gear are on the cylindrical surface  .
The tangential plane through the center point O
of bottom surface intersects with addendum line,
pitch curve and dedendum line at the point M ,
P and N . MN is the tangent section on the
cylindrical surface. Angles MOP and PON are
defined as the addendum angle and the dedendum
angle. And,
4
(10)
X 2 X 2'
Fig. 6 Coordinate system of generating
processing
As shown in Fig. 6, Ok  X k Yk Z k is rigidly
connected to the frame of the cutting machine.
Ok '  X k 'Yk ' Zk ' is rigidly connected to the cutter.
Likewise O2  X 2Y2 Z 2 is rigidly connected to the
frame of the cutting machine. O2'  X 2'Y2' Z2' is
rigidly connected to the non-circular curved face
gear.
Author's Surname, N. - Co-author's Surname, N.
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According to the principle of gear
engagement, the transformation matrix from M kk '
 en1 sin  n11  

 1  en1 cos  n11  
to M 2'2 can be derived as:
  arctan 
M 2' k '  M 2'2 M 2 k M kk '
 sin sin  2
sin cos 
2

 cos

0

cos sin  2
cos cos  2
 sin
0
 cos  2
sin  2
0
0
R cos  2 
 R sin  2 
h  2  

1

(12)
Where,  is the rotating angle of cutter
when the non-circular curved face gear rotates an
angle of  2 .
The rotation angle of the cutter can be
obtained from the processing geometric
relationships.

 r 1  
 1  arctan 

2
 r ' 1  



1
0
r 2    r '2  d 
(13)
rk
'
Where r   is the derivative of r    .
rk is the radius of the pitch curve of cutter gear .
3.1 The equation of equidistant curve of pitch
curve
The equation of the normal equidistant
curve can be derived from the pitch curve
equation of non-circular curved face gear.
Z
1
O
Pa
P
Pf
ha
hf
(14)
Assume that there are n normal equidistant
curves between addendum line and dedendum
line. The parametric equations of n / 2
equidistant curves from the pitch curve to
addendum line and another n / 2 equidistant
curves from the dedendum line to the pitch curve
can be derived from Eq. (9).

2t1ha sin  


 x1  R cos   2 
nR




2t1ha sin  

 y1  R sin   2 

nR




2t1ha
cos
 z1  h  2  
n

(15)

2t2 h f sin  

 x2  R cos   2 

nR




2t2 h f sin  


 y2  R sin   2 

nR




2t h
 z2  h  2   2 f cos
n

(16)
Where, n is the number of the normal
equidistant curves. The t1 and t2 represent
respectively the t1  th equidistant curve from the
pitch curve to addendum line and the t2  th
equidistant curve from the pitch curve to
dedendum line respectively. When t1  n / 2 , it is
the addendum line. When t2  n / 2 , it is the
dedendum line.
2

through point P .  is the angle of normal and
vertical direction. So  can be expressed as:
3
Y
Fig. 7 Planimetric pitch curve of non-circular
curved face gear
In Fig. 7, The curve 1, 2, 3 stand for the
expansion of addendum line, pitch curve and
dedendum line of non-circular curved face gear
respectively. Pa is the intersection of the
addendum line and the normal line of pitch curve
through point P . Pf is the intersection of the
3.2 The numerical method to solve the tooth
profile
The tooth profile of the non-circular
curved face gear can be derived by building the
equation of normal equidistant curve of the noncircular curved face gear using the method above.
During the process, the intersection is the point if
it satisfies the engagement conditions when
cylindrical cutter tooth profile associates with
dedendum line and the normal line of pitch curve
Numerical Calculation of Tooth Profile of Non-circular Curved Face Gear
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equidistant curve of the pitch curve of noncircular curved face gear.
According to mechanical principles, the
tooth profile equation of involute cylindrical gear
cutter is[2]:
 xk   rb sin  ko   k    k cos  ko   k 

 yk  rb cos  ko   k    k sin  ko   k 

z k  uk

(17)
Where,  stands for left or right tooth
profile of the cutter gear. rb is the radius of base
circle.  k is the angle of any point on the involute.
 ko is the angle from the involute-starting point to
the center line of the gullet,  ko   / 2 z  inv k .
u k is the parameter in the direction of tooth width.
 k is the pressure angle of cutter gear.
When u k is determined, the tooth profile
of cutter gear can be calculated by the software of
Matlab. And according to the coordinate
transformation equation P2'  M 2' k ' Pk ' , the tooth
profile of cutter gear is transferred to the
following coordinates of non-circular curved face
gear. Then, the intersections are calculated though
the equations of the line of the points obtained
above and the equidistant curve equation of pitch
curve of non-circular curved face gear.
Tooth profile
of cutter
Pitch curve
Normal
equidistant
curve

Y1
2  
Tangent line
P2
Ok 2

O1

1
P1'
Xk2
Pitch curve of
non-circular gear
Yk 2
Yk 1
Ok 1
P1 X k 1
Pitch curve
of cutter
(b) Space location of the gear shaper cutter
Fig. 8 The solving process of tooth profile
As shown in Fig. 8(a), the left tooth profile
of the cutter exists four points which have the
equal distance. P12' , P22' , P32' , P42' are obtained by
transforming the points to the coordinates
O2'  X 2'Y2' Z2' . Three lines can be obtained by
linking the adjacent points. Intersections are
calculated though the equations of the lines and
equidistant curves.
The gear pair can be obtained through
generating processing. There also exist
intersection points where the tooth profile of the
cutter intersects with equidistant curves when the
gullet is processed. And those points of tooth
profile can be verified on the meshing condition.
According to the principle of gear engagement,
the points of tooth profile of non-circular curved
face gear must satisfy the meshing equation. So
the intersection points obtained above satisfy the
meshing equation [1]:
N  v 2k  rb  cos     r 2 1   rk2  2rk r 1  sin 
P11'
i21  R  uk  cos 
P22'
X1
(18)
0
P33'
P44'
(a) The intersection of tooth profile and normal
equidistant curves
Where, The N stands for the normal line
of cutter tooth surface. The v 2k stands for the
relative velocity of the cutter and the non-circular
curved face gear. These two parameters need to
be expressed in the same coordinate system. As
shown in Fig. 8(b), the rb  rk cos k ,
    1   k   ko .
Finally the points which satisfy the Eq. (18)
are on the tooth profile.
3.3 The examples of calculation
The 4-order non-circular curved face gear
is processed using the involute cylindrical gear
cutter which the parameters are the tooth number
6
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Strojniški vestnik - Journal of Mechanical Engineering Volume(Year)No, StartPage-EndPage
z  12 , the pressure angle  k  20 , the module
m  4 . The gear pair is shown in Fig. 9.
Non-circular gear
E
D
C
B
A
1 2
3
4
5
Non-circular curved face gear
Fig. 9 material object of non-circular curved face
gear
As shown in Fig. 10(a), according to the
principle of numerical method to solve the tooth
profile[12]to[14], the u k is divided in accordance
with the tooth width of 72.5mm, 74.5mm, 76.5mm,
78.5mm, 80.5mm. The 4 equidistant curves
distance the pitch curve 1.5mm and 3mm
respectively, which are used to divide the tooth
along the tooth depth direction.
Calculate the coordinate value of the left
tooth surface of tooth 1. The results of the mesh
generation are shown in Fig. 10(b). Parts of the
coordinate values are shown in Table 1.
Right tooth
Left tooth
surface
surface
uk
(b) the node number on tooth 1
Fig. 10 the division of measurement grids
Table.1 The coordinate value of theoretical points
on the left surface of tooth 1 (Units: mm )
1
3
5
-80.49168,
E -1.13698,
3
-76.48175,
-1.66847,
3
-72.46920,
-2.11100,
3
-80.43873,
C -3.14385,
0
-76.43156,
-3.23707,
0
-72.42771,
-3.23691,
0
-80.32150,
A -5.35782,
-3
-76.32929,
-5.10585,
-3
-72.34882,
-4.67954,3
3.4 The measurement of non-circular curved
surface gear
(a) The mesh generation of tooth depth direction
As shown in Fig. 11, the non-circular
curved face gear was measured by the contour
scanning software of the German Klingelnberg
P26 automatic CNC controlled gear measuring
center. Since each tooth profile in one cycle on
the gear is different, the method of measuring
should choose the contour scanning.
Numerical Calculation of Tooth Profile of Non-circular Curved Face Gear
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Strojniški vestnik - Journal of Mechanical Engineering Volume(Year)No, StartPage-EndPage
Where, R*m is the coordinate value of
measured points. R m is the coordinate value of
theoretical points. nm is the normal direction of
the tooth surface.
The deviation values are obtained by the
means of calculation of living example. The
results are shown in the Fig. 12.
units： m
Fig. 11 profile measurement of non-circular
curved face gear
According to the measurement and
theoretical calculation process, the radius of tooth
width which is the motion trajectory of the probe
is the same as the one which is calculated above.
The coordinate values of the grid nodes on the
left tooth surface of tooth 1 are obtained using the
measurement points from the neighborhood space
of the theoretical point from the measured data.
The specific values are shown in Table.2.
Table.2 The coordinate value of measured points
on the left surface of tooth 1 (Units: mm )
1
3
5
E
-80.477,
-1.138,
3
-76.484,
-1.661,
3
-72.459,
-2.105,
3
C
-80.441,
-3.132,
0
-76.434,
-3.224,
0
-72.419,
-3.227,
0
A
-80.323,
-5.369,
-3
-76.323,
-5.106,
-3
-72.349,
-4.667,
-3
3.5 The error analysis of tooth profile
Since there exist errors in actual
processing, measurement and the precision of the
numerical method, there also exist errors between
the measured values and the theoretical values.
Assume the normal distance between the
measured point and the theoretical point is the
normal error d , which is used to measure the
deviation between two points. According to the
following formula[3]:
d   R*m  Rm   nm
8
（19）
14.7
15.8
12.1
15.7
11.3
4.9
7.8
5.6 4.7
5.7
5.6
5.3 8.6
5.5
6.3
13.5
11.8
12.8 11.5
11.9 13.2
15.7 11.4
12.5 12.5
Inner of
the tooth
1
2
Measured value
Theoretical value
1-The grid of measured value
2-The grid of theoretical value
Fig. 12 The distribution of errors
The normal deviation value is in the range
of 5-16μm. The precision of deviation value has
satisfied the requirements of GB/T 10095.1-2001
precision class grade 6-7, and has met the
engineering demand. Therefore the numerical
method of the tooth profiles can be proved
accurate.
4. Conclusion
In the article, a new kind of gear was
generated. Its geometric and mathematical models
were established. A numerical method used to
calculate the tooth profile of non-circular curved
face gear was developed. And the correctness of
the method was examined by experiment. The
results of performed research allow the following
conclusions to be drawn:
(a) The relevant parameters of arbitrary
curves in cylindrical coordinate were calculated
in terms of the cylindrical coordinate system. The
parametric equations of the pitch curve of noncircular curved face gear and the normal
equidistant curve equation were obtained.
(b) The tooth profile of non-circular
curved face gear in the tooth width direction was
obtained by changing the point on the tooth
profile of cutter to the coordinate system of the
non-circular curved face gear through the method
Author's Surname, N. - Co-author's Surname, N.
Strojniški vestnik - Journal of Mechanical Engineering Volume(Year)No, StartPage-EndPage
of the coordinate conversion and intersecting with
its pitch curve and the normal equidistant curve.
(c) The error of the tooth profile between
the theoretical calculations and the actual
measurement was analyzed. The results showed
that the numerical method to solve the tooth
profile of non-circular curved face gear possess
the characteristics of high precision and good
commonality. The calculation method serves as a
significant reference in the field of the error
analysis of tooth surface, the accuracy assessment
of tooth surface and manufacture etc.
[6]
[7]
ACKNOWLEDGEMENTS
[8]
The authors would like to appreciate their
supports from the National Natural Science
Foundation of China (51275537).
[9]
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Numerical Calculation of Tooth Profile of Non-circular Curved Face Gear
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