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Precision measurement of planar double-enveloping hourglass worms
Accepted Manuscript Precision measurement of planar double-enveloping hourglass worms Zhaoyao Shi, Bo Yu, Fayang He PII: DOI: Reference:
S0263-2241(16)30167-1 http://dx.doi.org/10.1016/j.measurement.2016.05.021 MEASUR 4037
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
21 April 2015 6 May 2016 7 May 2016
Please cite this article as: Z. Shi, B. Yu, F. He, Precision measurement of planar double-enveloping hourglass worms, Measurement (2016), doi: http://dx.doi.org/10.1016/j.measurement.2016.05.021
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Precision measurement of planar double-enveloping hourglass worms Zhaoyao Shi*, Bo Yu, Fayang He Beijing Engineering Research Center of Precision Measurement Technology and Instruments, Beijing University of Technology, Pingleyuan 100, Beijing 100124, China * Corresponding author. E-mail: [email protected] (Zhaoyao Shi)
Abstract The measurement of planar double-enveloping hourglass worms has always been a technical problem in the field of gear metrology. To solve this problem, we propose a new method and develop a measuring machine to inspect each deviation of the worms. In a cylindrical-coordinate system, the measuring principle adopts electronic generative metrology. The mathematical model of the worms is established according to the forming process of the worms. Based on the definitions of error items to be measured, the model of each deviation will be determined. The model could be transformed to the control path that leads the probe by controlling the movements of X axis, Y axis and Θ axis. In this process, probe radius compensation is necessary to correct the control path. At last, the helix deviations, the tooth profile deviations and the topologic deviations of a worm are inspected. It is the first time that the topologic deviation of this kind of worm is measured. The analysis shows that the deviations of the worm could be a reference to adjust the processing of the worms. The measuring machine could inspect the hourglass hobs, the cylindrical worms and the hobs as well. Keywords: hourglass worm, worm measurement, measuring machine, topologic deviation
1. Introduction In 1865, Hindley first proposed the idea of the hourglass worm [1]. For over a hundred years, this kind of worm has been improved. In 1977, the Chinese engineers invented the planar double-enveloping hourglass worm gear (TP worm) [2]. It proved to be a worm with high performance theoretically. This worm has complicated geometrical flanks. Compared with the cylindrical worm, it has the advantages of more meshing teeth and the double-line contacting worm flank. All these qualities bring it high load capacity and enhance its significant role in lightweight gear equipment. However, it is difficult to manufacture the worm with high quality to guarantee its excellent performance. The measurement of the
complicated geometry is critical to solve the problem. Cylindrical worm measurement is relatively fully developed [3], while TP worm measuring technique is still in the early stage of development. So far, there are two ways to test the TP worm: individual measurement and composite measurement. The former is to check the individual deviations of the worm on Coordinate Measuring Machine (CMM) or Gear Measuring Center (GMC) [4, 5]. The latter could only test the transmission error on a special instrument [6]. The methods above pose some problems. First, as the error items for representing the TP worm are many, the above two methods couldn’t check all the errors. Second, the topologic deviation couldn’t be tested. Last, the low measuring speed makes them not suitable for in-process rapid test. The backward measuring technique hinders the development and application of TP worms. Therefore, we propose the TP worm measuring method which could also measure TP hobs.
2. Mathematic model
2.1 Model of the worm Worm
1
Plane grinding wheel
3 Basic circle
rb
2
Figure 1. Forming principle of TP worm The model of the worm is based on the forming principle of the worm, as shown in Figure 1. The plane in which the grinding wheel is placed is the generating plane. The grinding wheel rotates around the basic circle and the worm rotates around the axis of itself at the speeds of 2 and 1 , respectively. 1 / 2 is
constant. The whole process could be regarded as the contact line of the plane is copied to the flank of the worm after coordinate transformation [7].
2.1.1 Coordinate systems
Y3 Hourglass worm
Zn Z2
Z j Z1 Z3
Oj Y1
O1
Xj
1
X3
O3
Xn
X1
Yj
Worm gear
a
X2
2 Yn
O nO 2 rb
Y2
Figure 2. Coordinate of the TP worm and the worm gear The coordinate systems of the worm and the worm gear are placed as shown in Figure 2 [8, 9]. S j (O j X jY j Z j ) and Sn (On X nYn Zn ) are static coordinate systems. Z j axis and Z n axis are in
the directions of the axes of the worm and the worm gear respectively. S1 (O1 X1Y1Z1 ) and
S2 (O2 X 2Y2 Z2 ) are rotational coordinate systems fixed to the worm and the worm gear. Z1 axis and
Z 2 axis are also in the directions of the axes of the worm and the worm gear respectively. S3 (O3 X 3Y3 Z3 ) is the coordinate system fixed to the generating plane. X 3 axis is tangent to the basic circle at point O3 . The shaft angle between the worm and the worm gear is 90 and the center distance between them is a . The radius of the basic circle is rb and the slope angle of the generating plane is .
2.1.2 Model of the generating plane
Z2 Z3
Y2
O2 (1)
u
t
X2
N
X3 Figure 3. Generating plane (1) In Figure 3, the formula of the generating plane (1) could be established as following .
r3 [u, rb t sin , t cos ， 1]T r3 (u, t )
(1)
In Eq. (1), u and t are variables of the generating plane. The meanings of them are shown in Figure 3.
2.1.3 Model of the worm During the processing, i12 1 / 2 . Base on the meshing principle, the equation of the contact line on the generating plane could be determined as
r3 (u, t ) cos ) (rb sin 2 a)sin u (sin cos 2 i12 t sin 2
(2)
Eq. (2) is simplified as r2 (u, t , 2 ) . After transforming Eq. (2) into S1 , the contact line could be expressed as Eq. (3).
r1 (u, t , 2 ) M12 r2 (u, t , 2 )
M12
cos 1 cos 2 cos sin 1 2 sin 1 0
sin 1 cos 2 sin 1 sin 2 cos 1 0
sin 2 cos 2 0 0
(3)
a cos 2 a sin 2 0 1
In Eq. (3), 1 and 2 are the rotational angles of the worm and the generating plane respectively. M12 is transformation matrix from S2 to S1 . The position of the contact line varies with 2 . Combining all
the contact lines together, the flank of the worm will be determined. As u u( x1 , y1 , z1 ) , t t ( x1 , y1 , z1 ) and 2 2 ( x1 , y1 , z1 ) , Eq. (3) could be replaced by r1 f ( x1 , y1 , z1 ) .
2.2 Model of the helix The surface of the worm and a rotational surface around the axis of the worm intersect, and the line of intersection is the helix, as shown in Figure 4. Unlike the cylindrical worm, the lead of the TP worm changes along with the axial direction.
Flank of the worm
Z1
O1
Rotating surface X1 a
Ri
Z1 Y1
X1
O2
Helix
Figure 4. Helix of the TP worm
Figure 5. Rotational surface
In Figure 5, there is an arc with its radius Ri in X1O1Z1 plane. The equation of the rotational surface could be determined when the arc rotates around the Z1 axis.
a Combining Eq.(4) and
x12 y12
z 2
2 1
Ri 2
(4)
r1 f ( x1 , y1 , z1 ) , the model of the helix could be formed as
Fh ( x1 , y1 , z1 ) 0 .
2.3 Model of the axial tooth profile The tooth profile in the axial plane is the axial tooth profile [9, 10], as shown in Figure 6. The axial tooth profile changes when the axial plane is different. The study focuses on the symmetry plane because the profiles on the right and left flanks are symmetric around X1O1Y1 plane in this plane. The model of the axial tooth profile nearest to X1O1Y1 plane is discussed.
O1
Z1
X1
Flank of the worm
Z1 Y1
2
Axial section
X1
O2
Axial tooth profile
Figure 6. Axial tooth profile of the TP worm
Figure 7. Axial tooth profile in the symmetry plane
For any point in the same axial plane, the ratio of x1 to y1 is constant. To locate the axial tooth profile, it is necessary to know the rotational angle of the symmetry plane. When the axial plane becomes the symmetry plane, the generating plane is rotated 2 and the rotational angle of the symmetry plane is
1 =i12 2 . According to the Figure 7, 2 0.5
(5)
y1 x1 tan[i12 ( 0.5 )]
(6)
Therefore,
By combining Eq. (6) with r1 f ( x1 , y1 , z1 ) , the model of the helix could be formed as
Ff ( x1 , y1 , z1 ) 0 .
2.4 Probe radius compensation In the measurement, the radius of the probe can’t be neglected. It is essential to compensate the error brought in by the probe radius. The probe used in the worm measurement is a conical probe as shown in Figure 8. Point B is the contact point on the flank of the worm. Point A is the center of the bottom of the cone. Assume a sphere with the radius rs and the center Os contacts with the worm at point B. The bottom circle of the cone is on the sphere. rp is the radius of the bottom circle.
Flank of the worm
Virtual sphere in which the probe is located
Z1
O1
B
rs
rp
O1
N
RA
A
Os
Conical probe
Figure 8. Axial tooth profile of the TP worm As the sphere contacts with the worm at point B, BO s is the normal vector of the flank. With BOs (rs nx , rs ny , rs nz ) and O1Os ( x1 rs nx , y1 rs ny ,0) , the angle between them is clear.
cos O1Os B
( x1 rs nx )nx ( y1 rs n y )n y ( x1 rs nx )2 ( y1 rs n y )2
(7)
Meanwhile, there is another relationship about O1Os B inside the sphere.
cos O1Os B
rs 2 rp 2
(8)
rs
It is easy to get the following equation. ( x1 rs nx )nx ( y1 rs n y )n y ( x1 rs nx )2 ( y1 rs n y ) 2
rs 2 rp 2 rs
(9)
When point B is a known point, the normal vector at this point is determined. In Eq. (9), only rs is unknown. As point A and point Os is in the same radial direction, the coordinate of point A could be calculated.
RA x A O O xos 1 s RA yos yA O1Os z z os A
(10)
The models of the deviations with probe radius compensation will be expressed in a unified form
FA ( x1 , y1 , z1 ) 0 in the following sections.
3. Deviations
3.1 Helix deviation Based on the Chinese National Standard GB/T16445 Planar double-enveloping worm gearing accuracy, the helix deviation is the distance between two design helix traces which enclose the actual helix trace over the evaluation range [11], as shown in Figure 9. 80
Actualhelix helixtrace trace Actual
Worm face width
70
Designhelix helixtrace trace Design
60 50 40
f h
30 20 10 0 0
50 2
100
4 150
6200
Rotational angle of the worm Figure 9. Definition of the helix deviation
250
300
Unlike the deviation of cylindrical worms divided into different segments by module, the helix tolerance is divided by the center distance, as shown in Table 1.
Table 1. Helix tolerance Center distance (mm) 80 ~ 160
160 ~ 315
315 ~ 630
630 ~ 1250
Item Accuracy grade Helix tolerance ( m )
6
7
6
7
6
7
6
7
28
40
36
50
45
63
63
90
3.2 Axial tooth profile deviation The axial tooth profile deviation of the worm is the distance between two design profile traces which enclose the actual profile trace over the evaluation range. The profile on the toot flank changes with the position of the axial plane.
3.3 Index deviation Based on GB/T16445, the index deviation is the error of the equipartition property of the helixes on different teeth. It is expressed as the arc length along the pitch circle in the throat plane [11]. The index deviation is also a reason for the pitch deviation of the worm.
4. Measuring principle and the measuring machine
4.1 Measuring principle As the TP worm is a kind of revolution, the measurement could adopt cylindrical-coordinate system measuring principle [12]. The measuring machine uses electronic generative metrology to form curves which are consistent with the definitions of the deviations. It is necessary to transform the coordinates to the cylindrical-coordinate system.
x x2 y 2 1 1 z z 1 sin x / y 1 1
(11)
The equation FA ( x1 , y1 , z1 ) 0 could be changed to FA ( x, z, ) 0 . Then, A set of discrete points,
:{xi , zi , i | i 1, 2, 3,
, n} , which compose the control path are calculated by a numerical method. n
represents the number of the control points. The machine leads the probe to move along the control path. For example, the measurement of the
helix deviation is realized by the movement of X axis, Y axis and axis. As the control points include probe radius compensation, the measurement of the axial deviation is also realized by the movements of X axis, Y axis and axis. The index deviation could be inspected without the movement of Y axis. When the probe moves, the data acquisition system collects the signals of the encoders and the probe as the original data, :{z pk , xmk , zmk , mk | k 1, 2, 3,
, nm } . nm represents the number of the data. The
data at point k will be processed to get the theoretical value ( x1k , y1k , z1k ) . Therefore the error at point k is as following. k zmk z pk z1k
(12)
The deviation curve could be drawn with the errors of all the measured points.
4.2 Measuring machine The measuring machine consists of translations in X axis, Y axis and Z axis and a rotation in axis, as shown in Figure 10(a). X axis and Y axis use linear guide rails whose strokes are 150mm and 70mm respectively, and Z axis uses a V-plane rail with a stroke of 850mm. The radial runout of axis is 1 m and the face runout is 1.5 m . X axis, Z axis and axis driven by servo motors achieve the three-axis linkage with control system, and Y axis is manually controlled. The resolutions of linear encoders are 0.2
m and that of the rotary encoder with two readheads is 0.45 . The one-dimensional inductance probe is equipped on Y axis with the stroke 0.6mm of and resolution of 0.15 m , as shown in Figure 10(b).
Θ Y X
Z
(a)
(b)
Figure 10. Measuring machine (a) and measuring process (b)
4.3 Precision analysis The precision of the machine is the basis for guaranteeing the reliability of the measurement. The main
factors influencing the precision of the machine are mechanical error, precision of components and signal processing and software algorithm error [13]. If the measured worms are different, the influences of the factors will be changed. Using the worm with the diameter 70mm of the pitch circle in the throat plane as an example, the uncertainty of the machine is analyzed. The uncertainties caused by the main factor are listed in Table 2. The assumed distributions in the evaluation of each uncertainty of precision of components and signal processing and software algorithm errors are uniform distributions. Those of mechanical errors and repeatability of the helix deviation are normal distributions. Table 2. Uncertainty analysis of the machine Item
Value
Uncertainty (ui)
Up generatrix parallelism
2μm/700mm
0.03μm
Side generatrix parallelism
2μm/700mm
0.06μm
Perpendicularity
4μm/800mm
0.29μm
Misalignment
1μm
0.11μm
Rotary encoder
1＂
0.02μm
Precision of
Axial linear encoder
3μm/m
0.18μm
components
Radial linear encoder
1μm/m
0.01μm
Probe
0.7μm
0.20μm
Signal processing
Rotary encoder
0.015μm
0.01μm
and software
Linear encoder
0.2μm
0.10μm
algorithm errors
Software algorithm
0.005μm
0.00μm
1.75μm
1.75μm
Mechanical errors
Repeatability of the helix deviation
As the uncertainties in the table are mutually independent, they are combined based on the principle of uncertainty and the combined uncertainty is as following. uc
ui2
1.80 m
(13)
5. Measuring practice and discussion
Figure 11. TP155-2-15 type worm The machine could check the TP worm with a center distance from 50mm to 250mm. In the factory, the measuring practice proved that the machine could achieve the precision measurement of the TP worm. The following are the results of TP155-2-15 type worm (Figure 11) and its parameters are shown in Table 3. Table 3. Parameters of the measured gear Parameter Center distance(a/mm)
Value 155
Tooth( Z w )
2
Transmission ratio( i12 )
15
Diameter of the pitch circle in the throat plane( D1 /mm)
70
Diameter of the basic circle( Db /mm) Slope angle of the generating plane( / ) Radius of the probe( rp /mm)
97.28 15 1.035
5.1 Results of the helix deviation According to the measurement, the helix deviations of the worm are shown in Figure 12. The results of the left flanks of Tooth 1 and Tooth 2 are 51.81μm and 43.88μm, and those of the right flank are 37.07μm and 37.05μm [14]. Table 4 shows the repeatability of measurement results of helix deviation. The standard deviations of the results are less than 1μm. If the confidence level is 0.95, the confidence intervals are [48.07, 53.62] and [39.81, 45.36] for the left flanks of Tooth 1 and Tooth 2, and [34.16, 39.71] and [34.18, 39.73] for right flanks of Tooth 1 and Tooth 2.
f h1 / m
f h1 / m
20
20
0 20
2
0 20
/ rad
2
/ rad
2
/ rad
f h1 / m
f h1 / m
20
20
0 20
2
0 20
/ rad
(a)
(b)
Figure 12. Helix deviation curves of the left flank(a) and the right flank(b) Table 4. Repeatability of measurement results of helix deviation No.
Tooth 1
Tooth 2
Left
Right
Left
Right
1
51.81μm
37.07μm
43.88μm
37.05μm
2
51.21μm
36.68μm
42.31μm
36.53μm
3
50.22μm
37.26μm
42.36μm
36.71μm
4
50.33μm
36.87μm
42.13μm
37.53μm
5
50.66μm
36.78μm
42.24μm
36.97μm
Repeatability
1.59μm
0.58μm
1.75μm
1.00μm
Standard deviation
0.66μm
0.23μm
0.73μm
0.38μm
It is obvious that the curves shown in Figure 12(b) present the first order error. After processing the data with FFT, Figure 13 shows that there is a peak at the rotating frequency, which is caused by installation eccentricity during the processing. As the angle between the two teeth is 180 , the curves show the phase difference of 180 . Considering the above analysis, the radial runout of the worm clamped on the machine is checked and the value of the eccentricity is found to be 14.26μm. After adjusting the clamping of the worm to reduce the eccentricity, the helix deviations of the right flank are less than 22μm, as Figure 14.
X(f)
f:0.996 X:9.347
f (Hz)
Figure 13. FFT of the helix deviation on the right flank of Tooth 1 f h1 / m
10 0
2
2
/ rad
10
10 0
f h1 / m
/ rad
10
Figure 14. Helix deviation curves of the right flank after adjustment
5.2 Results of the axial tooth profile deviation According to the measurement, the axial tooth profile deviations of the worm are shown in Figure 15. The results of the left flanks of Tooth 1 and Tooth 2 are 7.02μm and 6.81μm, and those of the right flank are 68.28μm and 73.83μm [14]. Table 5 shows the repeatability of measurement results of axial profile deviation. The standard deviations of the results are also less than 1μm. If the confidence level is 0.95, the confidence intervals are [5.39, 10.94] and [4.32, 9.88] for the left flanks of Tooth 1 and Tooth 2, and [65.44, 70.99] and [70.79, 76.35] for right flanks of Tooth 1 and Tooth 2.
f f / m
f f / m 20 0 20
1
2
3
4
5
6
20 0 20
7 X 1 / mm
1
2
3
4
5
6
7 X 1 / mm
2
3
4
5
6
7 X 1 / mm
f f / m
f f / m 20 0 20
1
2
3
4
5
6
20 0 20
7 X 1 / mm
1
(a)
(b)
Figure 15. Axial profile deviation curves of the left flank(a) and the right flank(b) Table 5. Repeatability of measurement results of axial profile deviation No.
Tooth 1
Tooth 2
Left
Right
Left
Right
1
7.02μm
68.28μm
6.81μm
73.83μm
2
8.73μm
67.34μm
8.05μm
72.73μm
3
8.16μm
67.53μm
7.22μm
73.43μm
4
9.18μm
69.30μm
7.16μm
73.38μm
5
7.72μm
68.61μm
6.26μm
74.48μm
Repeatability
2.16μm
1.96μm
1.79μm
1.75μm
Standard deviation
0.85μm
0.80μm
0.65μm
0.64μm
The deviations of the left flanks are much smaller than those of the right flanks. This may be caused by the processing order. The left flanks were ground first with better processing parameters which changed when grinding the right flanks. The incline of the curves in Figure 15(b) is related to error of the slope angle of the generating plane, because when finishing the processing of the left flanks and staring that of the right ones, only the grinding wheel is turned around to a new angle, which makes the slope angle of the generating plane changed. In order to confirm the value of the error, the tooth profile deviation curve of different slope angles of the generating plane are simulated, as Figure 16. It is obvious that the curve with an error of ** degree fits the measured curve better. After adjusting the slope angles of the generating plane, the tooth profile deviations of the right flank are less than 10μm, as Figure 17.
f f / m =-0.5
=0
=0.3 =0.5
=0.55
X 1 / mm
Figure 16. Simulated tooth profile deviation curves of different slope angles f f / m 5 0 5
1
2
3
4
5
6
7
X 1 / mm
2
3
4
5
6
7
X 1 / mm
f f / m 5 0 5
1
Figure 17. Tooth profile deviation curves of the right flank after adjustment
5.3 Results of the topologic deviation The topologic deviation is got by measuring the helixes in different radial positions on the flank, as shown in Figure 18. These helixes distribute evenly with a distance one-tenth m on either side of the reference toroid of the worm. m represents module. ‘+’ means outside the reference toroid and ‘–’ means inside the reference toroid.
Helix
0.5m 0.4m 0.3m 0.2m 0.1m 0 -0.1m -0.2m -0.3m -0.4m -0.5m
Z1 Y1
X1
Flank of the worm
Figure 18. Distribution of the helixes to be measured The topologic graphs are shown in Figure 19. Deviations on the flanks are more intuitive. In Figure 19(c) and Figure 19(d), the first order error of the helix deviation exists on the whole flank. Meanwhile, the helix deviation curve moves up from the dedendum to the addendum on the right flank, which also conforms to the incline in Figure 15(b). In Figure 19(b), there is a break at the addendum because the addendum circle reduces and the probe falls out of the flank.
/ rad
(d)
1
X
/m m
f / m
1
X /m m
f / m
1
X
/m m
f / m
/ rad
(a)
/ rad
(b)
/ rad
(c)
1
X
/m m
f / m
Figure 19. Topologic deviations on the left flanks of Tooth 1 (a) and Tooth 2 (b) and on the right flanks of Tooth 1 (c) and Tooth 2 (d)
6. Conclusions (1) Based on cylindrical-coordinate system, the TP worm measuring method is proposed. The models with the probe radius compensation lead the machine to measure the deviations by the three-axis linkage. (2) According to this method, the TP worm measuring machine is developed to achieve the inspection of the deviations by the three-axis linkage with control system. It can check the TP worm with a center distance from 50mm to 250mm. (3) The helix deviation, the axial tooth profile deviation and the topological deviation of TP155-2-15 type worm are inspected. The measuring practice in the factory verifies the feasibility of the method. More work will be performed to analyze the results for adjusting the processing parameters in the future, which improve the quality of the worms. (4) The topologic deviation including the helix deviation and the tooth profile deviation could reflect the errors on the whole flank obviously.
Acknowledgements This work was funded by Specialized Research Fund for Doctoral Program of Higher Education (No.3B001013201301).
References [1] Liangyong Zhou. Hourglass Worm Modification Principle and Manufacturing Technology. Beijing: National University of Defense Technology Press, 2005, p. 1-2. [2] Yannian Li, Yunfang Shen. Theoretical research and design of the parameters for SG-71 type worm with small transmission ratio. Journal of Beijing University of Iron and Steel Technology, 1981, 4: 1-8. [3] Cedric Barber. Worm gear measurement. Gear Technology, 1997, 5: 18-23. [4] Datong Qin, Jia Yan, Guanghui Zhang. Diagnosis of machining errors and improvement of manufacture precision on plane-generated hourglass worm. Mechanical Transmission, 1996, 2: 34-37 [5] Yongjun Ji, Qicheng Lao. Study on measurement of hourglass worm with gear measuring center. Tool Engineering, 2008, 12: 102-5. [6] Huakun Xie, Xinmin Zhang, Chen Wang. The new tester developed for double enveloping worm
gearing. Tool Engineering, 1994, 28: 32-6. [7] Faydor L Litvin. Gear Geometry and Applied Theory. Shanghai: Shanghai Science and Technology Press, 2008, p. 105-32. [8] L V Mohan, M S Shunmugam Geometrical aspects of double enveloping worm gear drive Mechanism and Machine Theory, 2009, 44: 2053-65. [9] Illés Dudás. The Theory and Practice of Worm Gear Drives. London: Penton Press, 2000, p. 44-63. [10] Zhaoyao Shi, Xiaoqing Ji. Tooth profile measurement error for ZK type worm gears. Journal of Harbin Engineering University, 2009, 30: 1141-5. [11] GB/T16445-1996 Planar Double-enveloping Worm Gearing Accuracy, China. [12] Zhaoyao Shi, Yong Ye. Research on the generalized polar-coordinate method for measuring involute profile deviations. Chinese Journal of Scientific Instrument, 2001, 22: 140-2, 153. [13] Zhaoyao Shi. Research on the theory & technology of measurement for characterized curves of complex helicoidal surfaces. Hefei University of Technology, 2001, p. 94-99. [14] Illés Dudás, Sándor Bodzás. The analysis of cutting edges of face gear hob with analytical calculation and three coordinate measuring machine. The 11th International Symposium of Measurement Technology and Intelligent Instruments, Aachen and Braunschweig, Germany, 2013, p. 30-36.
Highlights: 1) Proposing a new method to measure planar double-enveloping hourglass worms. 2) Developing a measuring machine to inspect each deviation of the worms. 3) Presenting a method to compensate the probe radius in the measurement. 4) Analyzing how to adjust the processing of the worms by the measuring results.
S0263-2241(16)30167-1 http://dx.doi.org/10.1016/j.measurement.2016.05.021 MEASUR 4037
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
21 April 2015 6 May 2016 7 May 2016
Please cite this article as: Z. Shi, B. Yu, F. He, Precision measurement of planar double-enveloping hourglass worms, Measurement (2016), doi: http://dx.doi.org/10.1016/j.measurement.2016.05.021
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Precision measurement of planar double-enveloping hourglass worms Zhaoyao Shi*, Bo Yu, Fayang He Beijing Engineering Research Center of Precision Measurement Technology and Instruments, Beijing University of Technology, Pingleyuan 100, Beijing 100124, China * Corresponding author. E-mail: [email protected] (Zhaoyao Shi)
Abstract The measurement of planar double-enveloping hourglass worms has always been a technical problem in the field of gear metrology. To solve this problem, we propose a new method and develop a measuring machine to inspect each deviation of the worms. In a cylindrical-coordinate system, the measuring principle adopts electronic generative metrology. The mathematical model of the worms is established according to the forming process of the worms. Based on the definitions of error items to be measured, the model of each deviation will be determined. The model could be transformed to the control path that leads the probe by controlling the movements of X axis, Y axis and Θ axis. In this process, probe radius compensation is necessary to correct the control path. At last, the helix deviations, the tooth profile deviations and the topologic deviations of a worm are inspected. It is the first time that the topologic deviation of this kind of worm is measured. The analysis shows that the deviations of the worm could be a reference to adjust the processing of the worms. The measuring machine could inspect the hourglass hobs, the cylindrical worms and the hobs as well. Keywords: hourglass worm, worm measurement, measuring machine, topologic deviation
1. Introduction In 1865, Hindley first proposed the idea of the hourglass worm [1]. For over a hundred years, this kind of worm has been improved. In 1977, the Chinese engineers invented the planar double-enveloping hourglass worm gear (TP worm) [2]. It proved to be a worm with high performance theoretically. This worm has complicated geometrical flanks. Compared with the cylindrical worm, it has the advantages of more meshing teeth and the double-line contacting worm flank. All these qualities bring it high load capacity and enhance its significant role in lightweight gear equipment. However, it is difficult to manufacture the worm with high quality to guarantee its excellent performance. The measurement of the
complicated geometry is critical to solve the problem. Cylindrical worm measurement is relatively fully developed [3], while TP worm measuring technique is still in the early stage of development. So far, there are two ways to test the TP worm: individual measurement and composite measurement. The former is to check the individual deviations of the worm on Coordinate Measuring Machine (CMM) or Gear Measuring Center (GMC) [4, 5]. The latter could only test the transmission error on a special instrument [6]. The methods above pose some problems. First, as the error items for representing the TP worm are many, the above two methods couldn’t check all the errors. Second, the topologic deviation couldn’t be tested. Last, the low measuring speed makes them not suitable for in-process rapid test. The backward measuring technique hinders the development and application of TP worms. Therefore, we propose the TP worm measuring method which could also measure TP hobs.
2. Mathematic model
2.1 Model of the worm Worm
1
Plane grinding wheel
3 Basic circle
rb
2
Figure 1. Forming principle of TP worm The model of the worm is based on the forming principle of the worm, as shown in Figure 1. The plane in which the grinding wheel is placed is the generating plane. The grinding wheel rotates around the basic circle and the worm rotates around the axis of itself at the speeds of 2 and 1 , respectively. 1 / 2 is
constant. The whole process could be regarded as the contact line of the plane is copied to the flank of the worm after coordinate transformation [7].
2.1.1 Coordinate systems
Y3 Hourglass worm
Zn Z2
Z j Z1 Z3
Oj Y1
O1
Xj
1
X3
O3
Xn
X1
Yj
Worm gear
a
X2
2 Yn
O nO 2 rb
Y2
Figure 2. Coordinate of the TP worm and the worm gear The coordinate systems of the worm and the worm gear are placed as shown in Figure 2 [8, 9]. S j (O j X jY j Z j ) and Sn (On X nYn Zn ) are static coordinate systems. Z j axis and Z n axis are in
the directions of the axes of the worm and the worm gear respectively. S1 (O1 X1Y1Z1 ) and
S2 (O2 X 2Y2 Z2 ) are rotational coordinate systems fixed to the worm and the worm gear. Z1 axis and
Z 2 axis are also in the directions of the axes of the worm and the worm gear respectively. S3 (O3 X 3Y3 Z3 ) is the coordinate system fixed to the generating plane. X 3 axis is tangent to the basic circle at point O3 . The shaft angle between the worm and the worm gear is 90 and the center distance between them is a . The radius of the basic circle is rb and the slope angle of the generating plane is .
2.1.2 Model of the generating plane
Z2 Z3
Y2
O2 (1)
u
t
X2
N
X3 Figure 3. Generating plane (1) In Figure 3, the formula of the generating plane (1) could be established as following .
r3 [u, rb t sin , t cos ， 1]T r3 (u, t )
(1)
In Eq. (1), u and t are variables of the generating plane. The meanings of them are shown in Figure 3.
2.1.3 Model of the worm During the processing, i12 1 / 2 . Base on the meshing principle, the equation of the contact line on the generating plane could be determined as
r3 (u, t ) cos ) (rb sin 2 a)sin u (sin cos 2 i12 t sin 2
(2)
Eq. (2) is simplified as r2 (u, t , 2 ) . After transforming Eq. (2) into S1 , the contact line could be expressed as Eq. (3).
r1 (u, t , 2 ) M12 r2 (u, t , 2 )
M12
cos 1 cos 2 cos sin 1 2 sin 1 0
sin 1 cos 2 sin 1 sin 2 cos 1 0
sin 2 cos 2 0 0
(3)
a cos 2 a sin 2 0 1
In Eq. (3), 1 and 2 are the rotational angles of the worm and the generating plane respectively. M12 is transformation matrix from S2 to S1 . The position of the contact line varies with 2 . Combining all
the contact lines together, the flank of the worm will be determined. As u u( x1 , y1 , z1 ) , t t ( x1 , y1 , z1 ) and 2 2 ( x1 , y1 , z1 ) , Eq. (3) could be replaced by r1 f ( x1 , y1 , z1 ) .
2.2 Model of the helix The surface of the worm and a rotational surface around the axis of the worm intersect, and the line of intersection is the helix, as shown in Figure 4. Unlike the cylindrical worm, the lead of the TP worm changes along with the axial direction.
Flank of the worm
Z1
O1
Rotating surface X1 a
Ri
Z1 Y1
X1
O2
Helix
Figure 4. Helix of the TP worm
Figure 5. Rotational surface
In Figure 5, there is an arc with its radius Ri in X1O1Z1 plane. The equation of the rotational surface could be determined when the arc rotates around the Z1 axis.
a Combining Eq.(4) and
x12 y12
z 2
2 1
Ri 2
(4)
r1 f ( x1 , y1 , z1 ) , the model of the helix could be formed as
Fh ( x1 , y1 , z1 ) 0 .
2.3 Model of the axial tooth profile The tooth profile in the axial plane is the axial tooth profile [9, 10], as shown in Figure 6. The axial tooth profile changes when the axial plane is different. The study focuses on the symmetry plane because the profiles on the right and left flanks are symmetric around X1O1Y1 plane in this plane. The model of the axial tooth profile nearest to X1O1Y1 plane is discussed.
O1
Z1
X1
Flank of the worm
Z1 Y1
2
Axial section
X1
O2
Axial tooth profile
Figure 6. Axial tooth profile of the TP worm
Figure 7. Axial tooth profile in the symmetry plane
For any point in the same axial plane, the ratio of x1 to y1 is constant. To locate the axial tooth profile, it is necessary to know the rotational angle of the symmetry plane. When the axial plane becomes the symmetry plane, the generating plane is rotated 2 and the rotational angle of the symmetry plane is
1 =i12 2 . According to the Figure 7, 2 0.5
(5)
y1 x1 tan[i12 ( 0.5 )]
(6)
Therefore,
By combining Eq. (6) with r1 f ( x1 , y1 , z1 ) , the model of the helix could be formed as
Ff ( x1 , y1 , z1 ) 0 .
2.4 Probe radius compensation In the measurement, the radius of the probe can’t be neglected. It is essential to compensate the error brought in by the probe radius. The probe used in the worm measurement is a conical probe as shown in Figure 8. Point B is the contact point on the flank of the worm. Point A is the center of the bottom of the cone. Assume a sphere with the radius rs and the center Os contacts with the worm at point B. The bottom circle of the cone is on the sphere. rp is the radius of the bottom circle.
Flank of the worm
Virtual sphere in which the probe is located
Z1
O1
B
rs
rp
O1
N
RA
A
Os
Conical probe
Figure 8. Axial tooth profile of the TP worm As the sphere contacts with the worm at point B, BO s is the normal vector of the flank. With BOs (rs nx , rs ny , rs nz ) and O1Os ( x1 rs nx , y1 rs ny ,0) , the angle between them is clear.
cos O1Os B
( x1 rs nx )nx ( y1 rs n y )n y ( x1 rs nx )2 ( y1 rs n y )2
(7)
Meanwhile, there is another relationship about O1Os B inside the sphere.
cos O1Os B
rs 2 rp 2
(8)
rs
It is easy to get the following equation. ( x1 rs nx )nx ( y1 rs n y )n y ( x1 rs nx )2 ( y1 rs n y ) 2
rs 2 rp 2 rs
(9)
When point B is a known point, the normal vector at this point is determined. In Eq. (9), only rs is unknown. As point A and point Os is in the same radial direction, the coordinate of point A could be calculated.
RA x A O O xos 1 s RA yos yA O1Os z z os A
(10)
The models of the deviations with probe radius compensation will be expressed in a unified form
FA ( x1 , y1 , z1 ) 0 in the following sections.
3. Deviations
3.1 Helix deviation Based on the Chinese National Standard GB/T16445 Planar double-enveloping worm gearing accuracy, the helix deviation is the distance between two design helix traces which enclose the actual helix trace over the evaluation range [11], as shown in Figure 9. 80
Actualhelix helixtrace trace Actual
Worm face width
70
Designhelix helixtrace trace Design
60 50 40
f h
30 20 10 0 0
50 2
100
4 150
6200
Rotational angle of the worm Figure 9. Definition of the helix deviation
250
300
Unlike the deviation of cylindrical worms divided into different segments by module, the helix tolerance is divided by the center distance, as shown in Table 1.
Table 1. Helix tolerance Center distance (mm) 80 ~ 160
160 ~ 315
315 ~ 630
630 ~ 1250
Item Accuracy grade Helix tolerance ( m )
6
7
6
7
6
7
6
7
28
40
36
50
45
63
63
90
3.2 Axial tooth profile deviation The axial tooth profile deviation of the worm is the distance between two design profile traces which enclose the actual profile trace over the evaluation range. The profile on the toot flank changes with the position of the axial plane.
3.3 Index deviation Based on GB/T16445, the index deviation is the error of the equipartition property of the helixes on different teeth. It is expressed as the arc length along the pitch circle in the throat plane [11]. The index deviation is also a reason for the pitch deviation of the worm.
4. Measuring principle and the measuring machine
4.1 Measuring principle As the TP worm is a kind of revolution, the measurement could adopt cylindrical-coordinate system measuring principle [12]. The measuring machine uses electronic generative metrology to form curves which are consistent with the definitions of the deviations. It is necessary to transform the coordinates to the cylindrical-coordinate system.
x x2 y 2 1 1 z z 1 sin x / y 1 1
(11)
The equation FA ( x1 , y1 , z1 ) 0 could be changed to FA ( x, z, ) 0 . Then, A set of discrete points,
:{xi , zi , i | i 1, 2, 3,
, n} , which compose the control path are calculated by a numerical method. n
represents the number of the control points. The machine leads the probe to move along the control path. For example, the measurement of the
helix deviation is realized by the movement of X axis, Y axis and axis. As the control points include probe radius compensation, the measurement of the axial deviation is also realized by the movements of X axis, Y axis and axis. The index deviation could be inspected without the movement of Y axis. When the probe moves, the data acquisition system collects the signals of the encoders and the probe as the original data, :{z pk , xmk , zmk , mk | k 1, 2, 3,
, nm } . nm represents the number of the data. The
data at point k will be processed to get the theoretical value ( x1k , y1k , z1k ) . Therefore the error at point k is as following. k zmk z pk z1k
(12)
The deviation curve could be drawn with the errors of all the measured points.
4.2 Measuring machine The measuring machine consists of translations in X axis, Y axis and Z axis and a rotation in axis, as shown in Figure 10(a). X axis and Y axis use linear guide rails whose strokes are 150mm and 70mm respectively, and Z axis uses a V-plane rail with a stroke of 850mm. The radial runout of axis is 1 m and the face runout is 1.5 m . X axis, Z axis and axis driven by servo motors achieve the three-axis linkage with control system, and Y axis is manually controlled. The resolutions of linear encoders are 0.2
m and that of the rotary encoder with two readheads is 0.45 . The one-dimensional inductance probe is equipped on Y axis with the stroke 0.6mm of and resolution of 0.15 m , as shown in Figure 10(b).
Θ Y X
Z
(a)
(b)
Figure 10. Measuring machine (a) and measuring process (b)
4.3 Precision analysis The precision of the machine is the basis for guaranteeing the reliability of the measurement. The main
factors influencing the precision of the machine are mechanical error, precision of components and signal processing and software algorithm error [13]. If the measured worms are different, the influences of the factors will be changed. Using the worm with the diameter 70mm of the pitch circle in the throat plane as an example, the uncertainty of the machine is analyzed. The uncertainties caused by the main factor are listed in Table 2. The assumed distributions in the evaluation of each uncertainty of precision of components and signal processing and software algorithm errors are uniform distributions. Those of mechanical errors and repeatability of the helix deviation are normal distributions. Table 2. Uncertainty analysis of the machine Item
Value
Uncertainty (ui)
Up generatrix parallelism
2μm/700mm
0.03μm
Side generatrix parallelism
2μm/700mm
0.06μm
Perpendicularity
4μm/800mm
0.29μm
Misalignment
1μm
0.11μm
Rotary encoder
1＂
0.02μm
Precision of
Axial linear encoder
3μm/m
0.18μm
components
Radial linear encoder
1μm/m
0.01μm
Probe
0.7μm
0.20μm
Signal processing
Rotary encoder
0.015μm
0.01μm
and software
Linear encoder
0.2μm
0.10μm
algorithm errors
Software algorithm
0.005μm
0.00μm
1.75μm
1.75μm
Mechanical errors
Repeatability of the helix deviation
As the uncertainties in the table are mutually independent, they are combined based on the principle of uncertainty and the combined uncertainty is as following. uc
ui2
1.80 m
(13)
5. Measuring practice and discussion
Figure 11. TP155-2-15 type worm The machine could check the TP worm with a center distance from 50mm to 250mm. In the factory, the measuring practice proved that the machine could achieve the precision measurement of the TP worm. The following are the results of TP155-2-15 type worm (Figure 11) and its parameters are shown in Table 3. Table 3. Parameters of the measured gear Parameter Center distance(a/mm)
Value 155
Tooth( Z w )
2
Transmission ratio( i12 )
15
Diameter of the pitch circle in the throat plane( D1 /mm)
70
Diameter of the basic circle( Db /mm) Slope angle of the generating plane( / ) Radius of the probe( rp /mm)
97.28 15 1.035
5.1 Results of the helix deviation According to the measurement, the helix deviations of the worm are shown in Figure 12. The results of the left flanks of Tooth 1 and Tooth 2 are 51.81μm and 43.88μm, and those of the right flank are 37.07μm and 37.05μm [14]. Table 4 shows the repeatability of measurement results of helix deviation. The standard deviations of the results are less than 1μm. If the confidence level is 0.95, the confidence intervals are [48.07, 53.62] and [39.81, 45.36] for the left flanks of Tooth 1 and Tooth 2, and [34.16, 39.71] and [34.18, 39.73] for right flanks of Tooth 1 and Tooth 2.
f h1 / m
f h1 / m
20
20
0 20
2
0 20
/ rad
2
/ rad
2
/ rad
f h1 / m
f h1 / m
20
20
0 20
2
0 20
/ rad
(a)
(b)
Figure 12. Helix deviation curves of the left flank(a) and the right flank(b) Table 4. Repeatability of measurement results of helix deviation No.
Tooth 1
Tooth 2
Left
Right
Left
Right
1
51.81μm
37.07μm
43.88μm
37.05μm
2
51.21μm
36.68μm
42.31μm
36.53μm
3
50.22μm
37.26μm
42.36μm
36.71μm
4
50.33μm
36.87μm
42.13μm
37.53μm
5
50.66μm
36.78μm
42.24μm
36.97μm
Repeatability
1.59μm
0.58μm
1.75μm
1.00μm
Standard deviation
0.66μm
0.23μm
0.73μm
0.38μm
It is obvious that the curves shown in Figure 12(b) present the first order error. After processing the data with FFT, Figure 13 shows that there is a peak at the rotating frequency, which is caused by installation eccentricity during the processing. As the angle between the two teeth is 180 , the curves show the phase difference of 180 . Considering the above analysis, the radial runout of the worm clamped on the machine is checked and the value of the eccentricity is found to be 14.26μm. After adjusting the clamping of the worm to reduce the eccentricity, the helix deviations of the right flank are less than 22μm, as Figure 14.
X(f)
f:0.996 X:9.347
f (Hz)
Figure 13. FFT of the helix deviation on the right flank of Tooth 1 f h1 / m
10 0
2
2
/ rad
10
10 0
f h1 / m
/ rad
10
Figure 14. Helix deviation curves of the right flank after adjustment
5.2 Results of the axial tooth profile deviation According to the measurement, the axial tooth profile deviations of the worm are shown in Figure 15. The results of the left flanks of Tooth 1 and Tooth 2 are 7.02μm and 6.81μm, and those of the right flank are 68.28μm and 73.83μm [14]. Table 5 shows the repeatability of measurement results of axial profile deviation. The standard deviations of the results are also less than 1μm. If the confidence level is 0.95, the confidence intervals are [5.39, 10.94] and [4.32, 9.88] for the left flanks of Tooth 1 and Tooth 2, and [65.44, 70.99] and [70.79, 76.35] for right flanks of Tooth 1 and Tooth 2.
f f / m
f f / m 20 0 20
1
2
3
4
5
6
20 0 20
7 X 1 / mm
1
2
3
4
5
6
7 X 1 / mm
2
3
4
5
6
7 X 1 / mm
f f / m
f f / m 20 0 20
1
2
3
4
5
6
20 0 20
7 X 1 / mm
1
(a)
(b)
Figure 15. Axial profile deviation curves of the left flank(a) and the right flank(b) Table 5. Repeatability of measurement results of axial profile deviation No.
Tooth 1
Tooth 2
Left
Right
Left
Right
1
7.02μm
68.28μm
6.81μm
73.83μm
2
8.73μm
67.34μm
8.05μm
72.73μm
3
8.16μm
67.53μm
7.22μm
73.43μm
4
9.18μm
69.30μm
7.16μm
73.38μm
5
7.72μm
68.61μm
6.26μm
74.48μm
Repeatability
2.16μm
1.96μm
1.79μm
1.75μm
Standard deviation
0.85μm
0.80μm
0.65μm
0.64μm
The deviations of the left flanks are much smaller than those of the right flanks. This may be caused by the processing order. The left flanks were ground first with better processing parameters which changed when grinding the right flanks. The incline of the curves in Figure 15(b) is related to error of the slope angle of the generating plane, because when finishing the processing of the left flanks and staring that of the right ones, only the grinding wheel is turned around to a new angle, which makes the slope angle of the generating plane changed. In order to confirm the value of the error, the tooth profile deviation curve of different slope angles of the generating plane are simulated, as Figure 16. It is obvious that the curve with an error of ** degree fits the measured curve better. After adjusting the slope angles of the generating plane, the tooth profile deviations of the right flank are less than 10μm, as Figure 17.
f f / m =-0.5
=0
=0.3 =0.5
=0.55
X 1 / mm
Figure 16. Simulated tooth profile deviation curves of different slope angles f f / m 5 0 5
1
2
3
4
5
6
7
X 1 / mm
2
3
4
5
6
7
X 1 / mm
f f / m 5 0 5
1
Figure 17. Tooth profile deviation curves of the right flank after adjustment
5.3 Results of the topologic deviation The topologic deviation is got by measuring the helixes in different radial positions on the flank, as shown in Figure 18. These helixes distribute evenly with a distance one-tenth m on either side of the reference toroid of the worm. m represents module. ‘+’ means outside the reference toroid and ‘–’ means inside the reference toroid.
Helix
0.5m 0.4m 0.3m 0.2m 0.1m 0 -0.1m -0.2m -0.3m -0.4m -0.5m
Z1 Y1
X1
Flank of the worm
Figure 18. Distribution of the helixes to be measured The topologic graphs are shown in Figure 19. Deviations on the flanks are more intuitive. In Figure 19(c) and Figure 19(d), the first order error of the helix deviation exists on the whole flank. Meanwhile, the helix deviation curve moves up from the dedendum to the addendum on the right flank, which also conforms to the incline in Figure 15(b). In Figure 19(b), there is a break at the addendum because the addendum circle reduces and the probe falls out of the flank.
/ rad
(d)
1
X
/m m
f / m
1
X /m m
f / m
1
X
/m m
f / m
/ rad
(a)
/ rad
(b)
/ rad
(c)
1
X
/m m
f / m
Figure 19. Topologic deviations on the left flanks of Tooth 1 (a) and Tooth 2 (b) and on the right flanks of Tooth 1 (c) and Tooth 2 (d)
6. Conclusions (1) Based on cylindrical-coordinate system, the TP worm measuring method is proposed. The models with the probe radius compensation lead the machine to measure the deviations by the three-axis linkage. (2) According to this method, the TP worm measuring machine is developed to achieve the inspection of the deviations by the three-axis linkage with control system. It can check the TP worm with a center distance from 50mm to 250mm. (3) The helix deviation, the axial tooth profile deviation and the topological deviation of TP155-2-15 type worm are inspected. The measuring practice in the factory verifies the feasibility of the method. More work will be performed to analyze the results for adjusting the processing parameters in the future, which improve the quality of the worms. (4) The topologic deviation including the helix deviation and the tooth profile deviation could reflect the errors on the whole flank obviously.
Acknowledgements This work was funded by Specialized Research Fund for Doctoral Program of Higher Education (No.3B001013201301).
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Highlights: 1) Proposing a new method to measure planar double-enveloping hourglass worms. 2) Developing a measuring machine to inspect each deviation of the worms. 3) Presenting a method to compensate the probe radius in the measurement. 4) Analyzing how to adjust the processing of the worms by the measuring results.