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Non-uniform flank rolling measurement for shaped noncircular gears
Accepted Manuscript Non-uniform flank rolling measurement for shaped noncircular gears Fangyan Zheng, Lin Hua, Xinghui Han, Bo Li PII: DOI: Reference:
S0263-2241(17)30487-6 http://dx.doi.org/10.1016/j.measurement.2017.07.048 MEASUR 4888
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
16 April 2017 20 June 2017 24 July 2017
Please cite this article as: F. Zheng, L. Hua, X. Han, B. Li, Non-uniform flank rolling measurement for shaped noncircular gears, Measurement (2017), doi: http://dx.doi.org/10.1016/j.measurement.2017.07.048
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Non-uniform flank rolling measurement for shaped noncircular gears Fangyan Zhenga, Lin Hua*a, Xinghui Han, Bo Lib a School of Automotive Engineering, Hubei Key Laboratory of Advanced Technology of Automotive components, Wuhan University of Technology, Wuhan, 430070, China b School of Logistics Engineering, Wuhan University of Technology, Wuhan, 430063 Abstract: Noncircular gears can be applied to realize non-uniform transmission ratio for various mechanical systems. However, due to the complex geometry, the manufacture and measurement constitute great hindrance to the application. Recently, a practical shaping method has been applied to the manufacture of noncircular gear. Yet along with this, still no investigation has been conducted into the measurement method or device for noncircular gears. In this regard, the paper aims for a quick measurement of shaped noncircular gears. The manufacture of noncircular gears is discussed first, a mathematical model for non-uniform flank rolling measurement is established then, and finally, the measurement processing is introduced, including the developed measurement device, the computerized measurement system and a measurement experiment. Keywords: Noncircular gear, flank rolling measurement
1
1. Introduction Following the rapid development of the automotive and robot industry, the demand for speed variation mechanical systems has greatly increased. Noncircular gears, regarded as the ideal solution with accurate high speed and compact structure, have hence been paid more attention to than ever. A variety of noncircular gear applications have been researched, such as function generator [1][2], wearable robot [3], torque balancing [4], geared linkage [5], steering mechanism [6], planetary gear train [7], gear pump [8], indexing device [9], and so on. However, as the geometry of noncircular gears is far more complicated than that of traditional circular gears, traditional methods consequently fail to realize the manufacture and measurement for noncircular gears. Countering this, a noncircular gear shaping method, applicable to all kinds of cylindrical noncircular gears, has been developed [10]. And a number of subsequent works have been focused on the mathematical model and craft processing of the gear shaping [11] [12], which in turn have found application in the gear products for gear pump and indexing device [13][14]. [13] proposed a basic linkage model and the corresponding cutting processing for noncircular external straight gear. Further, [14] discussed a synthesis model, applicable to noncircular external or internal gear type, with straight or helix gear lengthwise. As for gear measurement, a number of methods have been proposed till now: Traditional instruments, such as Vernier caliper gauges, gear pitch micrometers and profile meters, are most commonly used in practice. While with them, the measurement becomes convenient, quick and of low cost, defects of low measurement accuracy and susceptibility to the operator are obvious and irresolvable. Another method is flank rolling testing (FRT), in which a gear product is matched with a standard gear or the paired gear. Applications of FRT include the single or double flank rolling testing for cylindrical gear [15], the load rolling experiment for spiral bevel gear [16][17], FRT for hypoid gear [18], for face gear [19] and for worm gear drive [20]. With FRT, the results are reflected by the fluctuation of the center distance, the rotating angle of driven gear, the contact path, the noise and vibration. The integrated errors and properties of gearing, best reflecting the usability of real gear product, are quickly obtainable, proving in turn the adaptability of FRT to mass production. The defect, however, lies again in the integrated errors, which can hardly be analyzed as influenced by a number of factors. The most precise method, coordinate measurement method, is realized through probing into a coordinate measurement machine (CMM) to acquire tooth geometric information [21][21][22]. With this method, tooth topology of the product can be detected, each erroneous item can be calculated, and the real tooth geometry can even be numerically reconstituted [23]. Some relevant works are focused on the principle and mathematic model for the measurement of cylindrical gear [24], straight bevel gear [25], spiral bevel gear [26], hypoid gear [29][28][29] and worm gear[30], while others on the reference design and the evaluating method [31][32] for CMM accuracy 2
[33]. Still there are more methods: the computer vision method [34], the double-flank rack probe (DFRP) method [35] and the rotary profiling method [36]. However, due to the nonlinear geometry, the methods and devices discussed above cannot be directly applied to noncircular gear measurement in practice. Anyway, with the proposed mathematic model and algorithm, CMM method has been applied to the pitch deviation measurement of curve-face gear pair [37]. Yet, as discussed above, due to its low efficiency, the CMM method is still not readily applicable for mass production. To tackle this problem, the paper introduces non-uniform flank rolling testing (NFRT), in which a noncircular gear is matched with a standard circular gear. And different from that of FRT, the center distance between the measured gear and the standard gear constantly varies during the mesh. Thus, a non-uniform rolling tester is developed in place of the traditional rolling tester. Additionally, an automotive computerized measurement system is programmed for quick measurement. In all, this paper elaborates the non-uniform flank rolling measurement for shaped noncircular gears following the steps below: first, introduce the theory and practice of noncircular gear shaping; second, establish the mathematical model for NFRT; third, describe the development of the non-uniform flank rolling tester and the measurement system; and finally, perform the measurement experiment for a 3-lobe noncircular gear and briefly analyze the errors.
2 Manufacture of noncircular gear 2.1 Pitch curve of noncircular gear Based on the velocity relations, the pitch curve of noncircular gear can be defined by a polar coordinate. Here, the polar angle is defined as and the radius as
rp ( ) . In a Cartesian coordinate, the pitch curve can be represented as follows: Ap ( ) rp ( ) cos( ) rp ( )sin( ) 0
T
(1)
Thus, the tangent vector:
T( )
dAp ( ) d
rp '( ) cos( ) rp ( )sin( ) rp '( )sin( ) rp ( ) cos( ) 0
(2)
Where, r '( ) dr ( ) / d , and its corresponding unit velocity is t( ) T( )/ | T( ) |
Further, the unit normal vector is 3
(3)
0 n( ) t ( ) 0 1
(4)
2.2 Kinematics of noncircular gear shaping
y0
Pitch circle O1 (O2 )
Centers line of generator
rg
y1
g t
P
y2
x2
Pitch curve
O0
x1
x0
Fig. 1 Coordinate system for noncircular gear shaping
The principle of noncircular gear generation is to ensure a pure rolling relation between the pitch curve of the noncircular gear and the pitch circle of the cutter. As shown in Fig. 1, coordinate system S0 (O0 -x 0 y0 ) is fixed on the ground and attached to the noncircular gear. P represents the contact point between the pitch curve of the gear and the pitch circle of the cutter. Coordinate system S1 (O1 -x1y1 ) is set at the center of the shaper cutter, of which x1 -axis and y1 -axis are respectively along vector n and t . The distance between O1 and P is the pitch radius of the cutter, defined as O1P rg . The center of the cutter, located at the pitch curve normal offset, can be calculated as follows:
Ac ( ) Ap ( ) n( )rg
(5)
Thus, the coordinate transformation from S1 to S0 can be yielded through the vector method of transformation [8]:
4
n( )0 n( ) 1 M 01 ( ) 0 0
t ( )0 t ( )1 0 0
0 Ac ( )0 0 A c ( )0 1 0 0 1
(6)
where, subscript 0 and 1 refer respectively to the first and second vector component. Reference frame S2 (O2 -x 2 y2 ) is attached on the shaper cutter, and its angle relative to S1 is the rotation angle of the shaper cutter, which can be calculated in terms of the pure rolling relations [13] as follows
g
Sr ( ) rg
[r '( )]2 r ( ) 2 d
0
(7)
rg
where, Sr ( ) refers to the rolling arc-length. Hence, the coordinate transformation from coordinate system S2 to S1 : cos( g ) sin( g ) sin( ) cos( ) g g M12 ( ) 0 0 0 0
0 0 1 0
0 0 0 1
(8)
Finally, the coordinate transformation from the cutter to the gear, namely, from coordinate system S2 to S0 , can also be obtained: M02 ( ) M01 ( )M12 ( )
(9)
2.3 Shaping processing With the kinematics in a CNC shaping machine discussed above, the tooth profile of noncircular gear can be obtained. The design data is listed in Table 1, along with the cutter parameters in Table 2. Based on the discussion in [13][14], the interpolation, feeding and blank designs are shown in Table 3. The shaping processing and the final tooth profile of generated noncircular gear are illustrated respectively in in Fig. 2(a) and Fig. 2(b). Table 1 Design data of noncircular gear Tooth number
42
Tooth addendum (mm)
2
Tooth dedendum (mm)
2.5
5
Pitch curve function
r ( )
82.605/[2.019 0.2sin(3 )]
Normal module (mm)
2
Pressure angle (deg)
20 Table 2 Cutter Parameters
Example
1
Tooth number
13
Tooth addendum (mm)
2.5
Tooth dedendum (mm)
2.5
Normal module (mm)
2
Pressure angle (deg)
20
Pitch radius
16.25 Table 3 Interpolation, feeding and blank designs
Example
1
Interpolation method
Equal-B-axis
Interpolation cycle
0.01
Interpolation steps
9×104
Cutting cycle
4
Depth in each cycle (mm)
3; 2; 0.5; 0.2
Initial angle of blank
8.509
(a)
(b)
Fig. 2 Processing (a) and product (b) of shaping noncircular gear
3 The mathematical model for measurement 3.1 Measurement principle The basic idea of rolling testing is to make the noncircular gear product mesh 6
with a high-accuracy standard circular gear. As a noncircular gear can engage with a circular gear under varying center distances based on the discussion of variable center distance gears [8], the center distance between the measured noncircular gear and the standard gear surely varies in meshing. Thus, the integrated errors are reflected by the translation and rotation of the standard gear. And because of the varying center distance, the mathematical model for measurement of shaped noncircular gear is much more complicated than that for a traditional circular gear. 3.2 Applied coordinate system
yF
yS yG
xS
Q OF
G G
Noncircular gear
xG
rS
S
xF
OS
Standard gear
Fig. 3 Coordinate systems for measurement of noncircular gear
As shown in Fig. 3, three coordinate systems are established to describe the relations in noncircular gear measurement. Coordinate frame SF (OF -x F yF ) is fixed on the device, with its origin located at the center of the measured gear and its x F -axis along the direction of the slider. Coordinate system SG (OG -x G yG ) is attached to the measured gear. The rotating angle of the measured gear G is the angle between reference frame SF and SG . The coordinate transformation matrix from SF to SG is as follows:
7
cos(G ) sin(G ) sin( ) cos( ) G G M GF (G ) 0 0 0 0
0 0 1 0
0 0 0 1
(10)
Coordinate system SS (OS -xS yS ) is connected to the standard gear. Its angle relative to SF refers to the rotating angle of standard gear S , and its position relative to SF refers to the center distance ES . The coordinate transformation matrix from SF to SS is as follows: cos( S ) sin( S ) ES sin( ) cos( ) 0 S S M SF ( S ) 0 0 0 0 0 0
0 0 0 1
(11)
Suppose the angular velocity of the measured gear is G dG / dt , the screw of the measured gear in the reference frame SF is [38]:
Si [si ; ςi ] [0,0, G ,0,0,0]T
(12)
Where, s i and ς i are correspondingly the primary part (main vector) and the secondary part (dual vector) of the input screw. And the output screw (screw of output gear) in the fixed reference frame S0 is
So [so ; ςo ]
(13)
where, with the angular velocity of the standard gear defined as S dS / dt , the main vector so is identified to be determined by the gear ratio function
so 0 0 S
T
(14)
And the dual vector ς o (1 ) can be calculated as follows [38]:
vE ES vE ς o 0 0 so ES S 0 0 0 where, vE is velocity of the slide, calculated as vE dES / dt 8
(15)
In term of the theorem of three axes [39], the instantaneous screw between drive and driven gears can be calculated as
Sis So Si [sis ; ςis ] [0,0, S G ; vE , ESG ,0]T
(16)
where, the main vector s is is found to be parallel to z0 -axis through further calculation of its direction vector (the unit vector of main vector) as follows:
suis
sis T 0 0 1 | sis |
(17)
The position vector of the instantaneous screw axis, namely, the instant center of the gear pair Q , can be obtained as [38] T
E s ς vE rQ (G , S , ES ) is is S S 0 (18) sis sis S G S G where, the instant center is shown to be not on the center line of the gear pair, complicating the error calculation. Still, the pitch angle of measured gear can be obtained:
G G a tan
vE ES S
(19)
Additionally, the pitch radius of the measured gear is
rM (G , S , ES ) | rQ (G , S , ES ) |
ES 2S 2 vE 2 | S G |
(20)
Furthermore, the pitch radius of the standard gear is
ES ES 2G 2 vE 2 rS (G , S , ES ) rQ (G , S , ES ) 0 S G 0
(21)
3.3 Error calculation For a circular cylindrical gear, with a flank gear rolling tester applied, two types of errors can be measured. One type is the radial integrated error, decided by fluctuation of the center distance, and the other type is the tangential integrated error, decided by wave of the measured angular displacement relative to the theoretical angular displacement for the standard gear. Likewise, for a noncircular gear, with this tester applied, the above two types of errors can also be obtained. However, the calculation of these errors will not be easy, as previously discussed that the instant center is not on the center line of the gear pair. In order to calculate the errors, it should be understood first that the above instant center is a vital to the calculation of errors . And in practice, in term of Eq.(21), the 9
radius of the standard gear cannot be constant if S is the measured angle of the standard gear. S is then defined as the theoretical angle of the standard gear, and the measured angle of the standard gear is represented as m . Thus, with the pitch radius of the standard gear set as rs , the following equations must be satisfied:
ES 2G 2 vE 2 rS (G , S , ES ) rs | S G |
(22)
Where, the angle of the measured gear G , the distance of the slider ES and the velocity of the slider are obtained through measurement. S can in turn be solved:
ES 2G 2 vE 2 S G rs
(23)
So can the angle of standard gear t
S S (t )dt
(24)
0
Substitute Eq.(23) and Eq.(24) into Eq. (20), the measured pitch curve of noncircular gear can be obtained as follows:
rm rM (G , S , ES )
(25)
Thus, the radial and the tangential integrated errors can be resolved as follows:
r ( G ) rm (G ) rp ( G ) t ( G ) [ m (G ) S ( G )]rs
(26)
And for a clear demonstration of the distribution of errors , the errors are magnified 100 times. The enlarged error pitch radius is correspondingly calculated: rpE ( G ) rp ( G ) 100r ( G )
(27)
Thus, the enlarged error pitch curve of noncircular gear is ApE ( ) rpE ( G ) cos( G ) rpE ( G )sin( G ) 0
4 Measurement of errors
10
T
(28)
4.1 Measurement device
Measured gear Standard gear Weight gear slider Angle encoder
Linear rail
Grating sensor Drive motor Fig. 4 The configuration of the non-uniform flank rolling tester
Fig. 5 The developed flank rolling device
In order to measure the flank rolling errors of a shaped noncircular gear, a nonuniform flank rolling tester is needed. As shown in Fig. 4, similar to the case of a traditional tester for circular gear, in the case of a non- uniform flank rolling tester for a shaped noncircular gear, the two rotating axes and a linear axis are configured. The measured gear and the standard gear are attached to the two rotating axes. The standard gear axis is attached to the gear slider (linear axis) and moves along with the slider’s translation. A drive motion, used to provide the rotating angle of the measured gear, is attached to the measured gear through a worm gear driving. And an angle encoder, used to obtain the rotating angle of the standard gear, is rigidly connected to the standard gear. Still, there are some differences. Unlike the case for traditional tester, as the center distance of the gear pair changes, a grating sensor, is set between the slider and the linear rail, along with an additional weight configured to ensure the contact between the measured and the standard gear, to measure the standard gear’s translation. Based on the above discussion of configuration, a non-uniform flank rolling tester is developed as shown in Fig. 5. With this device applied, the rotating angle of the 11
noncircular gear is acquired by the motion, the angular displacement of the standard gear is obtained by the angle encoder, and the center distance between the noncircular and the circular gear is captured by the grating sensor. 4.2 Measurement presetting
yF
t
P
OF
0
rp 0
rg
rs
Oc
xF
OS
Noncircular gear
Standard gear
Fig. 6 Presetting of the measured noncircular gear
As discussed in shaping processing [16][17], a circumferential datum A is set in the gear blank perpendicular to the center line between cutter and gear, which has already been applied to cutter presetting in manufacture. And here, this datum is also taken as a reference in the measurement. Yet, as the pitch radius of the standard gear is inconsistent with that of the cutter in general, datum A will be not perpendicular to the linear rail in measurement and an angle of datum A must be set. As shown in Fig. 6, rp 0 is the initial pitch radius of the noncircular gear and can be calculated as rp 0 rp (0)
(29)
Based on the discussion above, the angle between the tangential vector and the pitch radius is
( ) drp ( ) / d Thus, the center distance between the cutter and the gear is 3 | OF Oc | rp 0 2 rg 2 2rp 0 rg cos[ (0)] 2 Likewise, the center distance between the cutter and the standard gear is 3 | OF OS | rp 0 2 rs 2 2rp 0 rs cos[ (0)] 2
(30)
(31)
(32)
And the angle of datum A relative to y F , namely, the presetting angle of the gear in measurement, can be obtained as 12
0 a cos
| OF OS |2 | OF OS |2 (rs rg ) 2 2 | OF OS || OF OS |
4.3 Computerized measurement systems
Fig. 7 The interface of the computerized measurement system
Fig. 8 Flow chart of Matlab and C++ mixed programing 13
(33)
As shown in Fig. 7, the computerized measurement system is developed to communicate with the non-uniform flank rolling tester, as well as analyze and calculate the errors. While the communication with the tester and the interface design of the system can be realized in the C++ environment, an algorithm programming for data analysis and error calculation can hardly be realized. Matlab, meanwhile, inclusive of numerous mathematical functions, can be an ideal tool for numerical calculation. In this regard, the developed system is designed to be a programming mixed by these two environments. And the flow chart for mixed programming is shown in Fig. 8. With the measured and the standard gears installed, and the computerized measurement system developed, a flank rolling testing can be realized automatically following the flow chart in Fig. 9. In processing, some measurement parameters, such as speed of the motor and sampling frequency should be input manually, while the operations, including communication port initialization, data collection, conversion, filter, error calculation and display, are all carried out by the system automatically.
Fig. 9 The flow chart of the measurement
4.4 Experiment and discussion With the measurement device developed and the computerized measurement designed, the shaped gear, as shown in Fig. 2, is applied to measurement experiment. A standard gear, with identical module and a pitch radius rs 46(mm) , is used to match with the noncircular gear. The processing of measurement is shown in Fig. 10. 14
The radial and tangential integrated errors for the measured gear can ultimately be obtained. Fig. 11 shows the radial integrated errors of the measured noncircular gear, the maximum error being 27 um and the minimum being -26 um. It is not hard to find that the cycle of the errors corresponds approximately with that of the noncircular gear. Fig. 12 shows the enlarged error pitch curve of the measured gear. In the figure, the distribution of the errors in the theoretical pitch curve is clearly illustrated. And it can be observed that the error pitch curve rotates an angle anticlockwise or so relative to the theoretical pitch curve. This could be attributed to the circumference location in the machine cutter presetting before the cutting, as is discussed in [17], the circumference location of blank brings the major difference as well as difficulty for noncircular gear shaping. Besides, this phenomenon could also be explained by the non-accurate position of the cutter center relative to the cutting gear, for which again the nonlinear interpolation method for noncircular gear could account, as discussed in [14]. Fig. 13 shows the tangential integrated errors of the measured noncircular gear, the maximum error being 15 um while the minimum error -13 um. Overall, the trend of the error curve is similar to that of radial integrated error curve, indicating the two errors to be internally associated. This is actually true since both errors are measured through the tooth profile of the gear. In careful comparison, however, the tangential errors are found to smaller, which can be explained by the fact that tangential errors are greatly influenced by rotation of the cutter in cutting processing and the rotation of the cutter is linear with equal-arc-length cutting method adopted[17]. Thus, for noncircular gear, no universal measurement standard can be used to evaluate the accuracy. And in reference to ISO system of accuracy for cylindrical gear, only with both radial and tangential integrated errors taken into consideration, will the shaped noncircular gear reach grade 8 [40].
Fig. 10 The non-uniform gear rolling measurement
15
Fig. 11 The radial integrated errors of noncircular gear
Fig. 12 The theoretical and enlarged error pitch curve
Fig. 13 The tangential integrated errors of noncircular gear
16
5. Conclusions This paper aims for a quick measurement of noncircular gears for mass production. The mathematic model for measurement, the non-uniform rolling tester and the computerized measurement system are systematically described. And based on the discussions above, several conclusions are drawn: 1) With a variable center distance, a noncircular gear can be meshed with a standard cylindrical gear, and thus the accuracy of the product noncircular gear can be reflected. 2) Different from the uniform rolling testing for circular gear, the errors of noncircular gear cannot be directly obtained, but calculated through a complicated mathematical model. 3) Based on the configuration of the developed measurement device, both radial and tangential integrated errors of shaped noncircular gear can be obtained. 4) Since the pitch radius of the standard gear is not always consistent with that of the shaper cutter, the presetting datum of measurement is different from that in practical manufacture. 5) With the developed non-uniform rolling tester and the computerized measurement system, the integrated errors of shaped noncircular gear are quickly obtained, demonstrating their correctness and versatility.
Nomenclature rp ( ) = pitch radius of noncircular gear
Ap ( ) = pitch curve of noncircular gear
t ( ) , n( ) = tangent and normal vector of the pitch curve rg = pitch radius of the shaper cutter
Ac ( ) = normal offset of pitch curve
Sr ( ) = rolling arc-length of the shaper cutter g = rotation angle of the shaper cutter
G = rotating angle of measured gear
17
ES = centers distance between the measured and standard gear
S = rotating angle of standard gear
G = angular velocity of the measured gear S i = screw of the measured gear s i , ς i = primary and secondary part of the measured gear screw
S o = screw of standard gear so , ς o = primary and secondary part of the standard gear screw
S = angular velocity of the standard gear vE = translate velocity of the standard gear S is = instant screw of the measured and standard gear
sis , ς is = primary and secondary part of the instant screw suis , rQ = direction and position vector of the instant screw
G = pitch angle of measured gear m = measured angle of the standard gear rM (G , S , ES ) = calculated pitch radius of the measured gear rS (G , S , ES ) = calculated pitch radius of the standard gear rs = given pitch radius of the standard gear
rm = measured pitch curve of noncircular gear r ( G ), t ( G ) = radial and the tangential integrated errors of noncircular gear rpE ( G ) = enlarged errors pitch curves of noncircular gear ApE ( ) = enlarged errors pitch curves of noncircular gear rp 0 = initial pitch radius of the noncircular gear 18
( ) = the angle between the tangential vector and the pitch radius of noncircular gear
0 = measurement presetting angle of the gear
Acknowledgments The authors would like to thank the Natural Science Foundation of China (No. 51575416), Innovative Research Team Development Program of Ministry of Education of China (No. IRT13087), High-End Talent Leading Program of Hubei Province (No. 2012-86), Science and Technology Support Program of Hubei Province (No. 2015BAA039) and Natural Science Foundation of Hubei Province (No. 2014CFB876) for the support given to this research.
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[27] A. Guenther, Interpretation of bevel gear topography measurements, CIRP Ann. – Manuf. Technol. 60 (2011) 551–554. [28] Anke Guenther, Karin Kniel, Frank Härtig, Ingo Lindner, Introduction of a new bevel gear measurement standard, CIRP Ann. – Manuf. Technol. 62 (2013) 515–518. [29] José Sánchez, Gerardo Ruiz, Sergio Padilla, Benjamín Valera, Geometrical principles for analysing hypoidal gears in Coordinate Measuring Machines: Involutometry, protocols, data interpretation, Measurement 45 (2012) 2368–2375 [30] Zhaoyao Shi, Bo Yu, Fayang He, Precision measurement of planar double-enveloping hourglass worms, Measurement 91 (2016) 177–185 [31] Masaharu Komori, Fumi Takeoka, Koshi Kondo, Yohan Kondo, Toshiyuki Takatsuji, Sonko Osawa, Design method of double ball artifact for use in evaluating the accuracy of a gear-measuring instrument, ASME J. Mech. Des. 132(2010) p071010 [32] Maria Pia Sammartini, Leonardo De Chiffre, Development and validation of a new reference cylindrical gear for pitch measurement, Precision Eng. 24 (2000) 302–309 [33] Kondo Y, Osawa S, Sato O, Komori M, Takatsuji T. Evaluation of instruments for pitch measurement using a sphere artifact. Precis. Eng. 2012;36(4):604–11. [34] E.S. Gadelmawla, Computer vision algorithms for measurement and inspection of spur gears, Measurement 44 (2011) 1669–1678 [35] Jie Tang, Jianqiao Jia, Zhiqiang Fang, Zhaoyao Shi, Development of a gear measuring device using DFRP method, Precision Engineering 45 (2016) 153–159 [36] Bin Xu , Yuki Shimizu, So Ito, Wei Gao, Pitch deviation measurement of an involute spur gear by a rotary profiling system, Precision Engineering 39 (2015) 152–160 [37] Chao Lin, Xi-Jun Cao, Yu Fan, Dong Zeng, Pitch deviation measurement and analysis of curve-face gear pair, Measurement 81 (2016) 95–101
[38] Dai, J. S. Geometrical Foundations and Screw Algebra for Mechanisms and Robotics, Higher Education Press, Beijing, 2014,50-60 [39] Phillips, J. R. and Hunt, K. H., on the theorem of three axes in the spatial motion of three bodies, Australian Journal of Applied Science, 15(1964), 267-287 [40] ISO 1328-2:1997, Cylindrical gears – ISO system of accuracy – Part 2: Definitions and allowable values of deviations relevant to radial composite deviations and run out information.
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(1) Establish a mathematical model for non-uniform flank rolling measurement. (2) Develop a non-uniform rolling tester and computerized measurement system. (3) Investigate the radial and tangential integrated error for shaped noncircular gear.
22
S0263-2241(17)30487-6 http://dx.doi.org/10.1016/j.measurement.2017.07.048 MEASUR 4888
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
16 April 2017 20 June 2017 24 July 2017
Please cite this article as: F. Zheng, L. Hua, X. Han, B. Li, Non-uniform flank rolling measurement for shaped noncircular gears, Measurement (2017), doi: http://dx.doi.org/10.1016/j.measurement.2017.07.048
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Non-uniform flank rolling measurement for shaped noncircular gears Fangyan Zhenga, Lin Hua*a, Xinghui Han, Bo Lib a School of Automotive Engineering, Hubei Key Laboratory of Advanced Technology of Automotive components, Wuhan University of Technology, Wuhan, 430070, China b School of Logistics Engineering, Wuhan University of Technology, Wuhan, 430063 Abstract: Noncircular gears can be applied to realize non-uniform transmission ratio for various mechanical systems. However, due to the complex geometry, the manufacture and measurement constitute great hindrance to the application. Recently, a practical shaping method has been applied to the manufacture of noncircular gear. Yet along with this, still no investigation has been conducted into the measurement method or device for noncircular gears. In this regard, the paper aims for a quick measurement of shaped noncircular gears. The manufacture of noncircular gears is discussed first, a mathematical model for non-uniform flank rolling measurement is established then, and finally, the measurement processing is introduced, including the developed measurement device, the computerized measurement system and a measurement experiment. Keywords: Noncircular gear, flank rolling measurement
1
1. Introduction Following the rapid development of the automotive and robot industry, the demand for speed variation mechanical systems has greatly increased. Noncircular gears, regarded as the ideal solution with accurate high speed and compact structure, have hence been paid more attention to than ever. A variety of noncircular gear applications have been researched, such as function generator [1][2], wearable robot [3], torque balancing [4], geared linkage [5], steering mechanism [6], planetary gear train [7], gear pump [8], indexing device [9], and so on. However, as the geometry of noncircular gears is far more complicated than that of traditional circular gears, traditional methods consequently fail to realize the manufacture and measurement for noncircular gears. Countering this, a noncircular gear shaping method, applicable to all kinds of cylindrical noncircular gears, has been developed [10]. And a number of subsequent works have been focused on the mathematical model and craft processing of the gear shaping [11] [12], which in turn have found application in the gear products for gear pump and indexing device [13][14]. [13] proposed a basic linkage model and the corresponding cutting processing for noncircular external straight gear. Further, [14] discussed a synthesis model, applicable to noncircular external or internal gear type, with straight or helix gear lengthwise. As for gear measurement, a number of methods have been proposed till now: Traditional instruments, such as Vernier caliper gauges, gear pitch micrometers and profile meters, are most commonly used in practice. While with them, the measurement becomes convenient, quick and of low cost, defects of low measurement accuracy and susceptibility to the operator are obvious and irresolvable. Another method is flank rolling testing (FRT), in which a gear product is matched with a standard gear or the paired gear. Applications of FRT include the single or double flank rolling testing for cylindrical gear [15], the load rolling experiment for spiral bevel gear [16][17], FRT for hypoid gear [18], for face gear [19] and for worm gear drive [20]. With FRT, the results are reflected by the fluctuation of the center distance, the rotating angle of driven gear, the contact path, the noise and vibration. The integrated errors and properties of gearing, best reflecting the usability of real gear product, are quickly obtainable, proving in turn the adaptability of FRT to mass production. The defect, however, lies again in the integrated errors, which can hardly be analyzed as influenced by a number of factors. The most precise method, coordinate measurement method, is realized through probing into a coordinate measurement machine (CMM) to acquire tooth geometric information [21][21][22]. With this method, tooth topology of the product can be detected, each erroneous item can be calculated, and the real tooth geometry can even be numerically reconstituted [23]. Some relevant works are focused on the principle and mathematic model for the measurement of cylindrical gear [24], straight bevel gear [25], spiral bevel gear [26], hypoid gear [29][28][29] and worm gear[30], while others on the reference design and the evaluating method [31][32] for CMM accuracy 2
[33]. Still there are more methods: the computer vision method [34], the double-flank rack probe (DFRP) method [35] and the rotary profiling method [36]. However, due to the nonlinear geometry, the methods and devices discussed above cannot be directly applied to noncircular gear measurement in practice. Anyway, with the proposed mathematic model and algorithm, CMM method has been applied to the pitch deviation measurement of curve-face gear pair [37]. Yet, as discussed above, due to its low efficiency, the CMM method is still not readily applicable for mass production. To tackle this problem, the paper introduces non-uniform flank rolling testing (NFRT), in which a noncircular gear is matched with a standard circular gear. And different from that of FRT, the center distance between the measured gear and the standard gear constantly varies during the mesh. Thus, a non-uniform rolling tester is developed in place of the traditional rolling tester. Additionally, an automotive computerized measurement system is programmed for quick measurement. In all, this paper elaborates the non-uniform flank rolling measurement for shaped noncircular gears following the steps below: first, introduce the theory and practice of noncircular gear shaping; second, establish the mathematical model for NFRT; third, describe the development of the non-uniform flank rolling tester and the measurement system; and finally, perform the measurement experiment for a 3-lobe noncircular gear and briefly analyze the errors.
2 Manufacture of noncircular gear 2.1 Pitch curve of noncircular gear Based on the velocity relations, the pitch curve of noncircular gear can be defined by a polar coordinate. Here, the polar angle is defined as and the radius as
rp ( ) . In a Cartesian coordinate, the pitch curve can be represented as follows: Ap ( ) rp ( ) cos( ) rp ( )sin( ) 0
T
(1)
Thus, the tangent vector:
T( )
dAp ( ) d
rp '( ) cos( ) rp ( )sin( ) rp '( )sin( ) rp ( ) cos( ) 0
(2)
Where, r '( ) dr ( ) / d , and its corresponding unit velocity is t( ) T( )/ | T( ) |
Further, the unit normal vector is 3
(3)
0 n( ) t ( ) 0 1
(4)
2.2 Kinematics of noncircular gear shaping
y0
Pitch circle O1 (O2 )
Centers line of generator
rg
y1
g t
P
y2
x2
Pitch curve
O0
x1
x0
Fig. 1 Coordinate system for noncircular gear shaping
The principle of noncircular gear generation is to ensure a pure rolling relation between the pitch curve of the noncircular gear and the pitch circle of the cutter. As shown in Fig. 1, coordinate system S0 (O0 -x 0 y0 ) is fixed on the ground and attached to the noncircular gear. P represents the contact point between the pitch curve of the gear and the pitch circle of the cutter. Coordinate system S1 (O1 -x1y1 ) is set at the center of the shaper cutter, of which x1 -axis and y1 -axis are respectively along vector n and t . The distance between O1 and P is the pitch radius of the cutter, defined as O1P rg . The center of the cutter, located at the pitch curve normal offset, can be calculated as follows:
Ac ( ) Ap ( ) n( )rg
(5)
Thus, the coordinate transformation from S1 to S0 can be yielded through the vector method of transformation [8]:
4
n( )0 n( ) 1 M 01 ( ) 0 0
t ( )0 t ( )1 0 0
0 Ac ( )0 0 A c ( )0 1 0 0 1
(6)
where, subscript 0 and 1 refer respectively to the first and second vector component. Reference frame S2 (O2 -x 2 y2 ) is attached on the shaper cutter, and its angle relative to S1 is the rotation angle of the shaper cutter, which can be calculated in terms of the pure rolling relations [13] as follows
g
Sr ( ) rg
[r '( )]2 r ( ) 2 d
0
(7)
rg
where, Sr ( ) refers to the rolling arc-length. Hence, the coordinate transformation from coordinate system S2 to S1 : cos( g ) sin( g ) sin( ) cos( ) g g M12 ( ) 0 0 0 0
0 0 1 0
0 0 0 1
(8)
Finally, the coordinate transformation from the cutter to the gear, namely, from coordinate system S2 to S0 , can also be obtained: M02 ( ) M01 ( )M12 ( )
(9)
2.3 Shaping processing With the kinematics in a CNC shaping machine discussed above, the tooth profile of noncircular gear can be obtained. The design data is listed in Table 1, along with the cutter parameters in Table 2. Based on the discussion in [13][14], the interpolation, feeding and blank designs are shown in Table 3. The shaping processing and the final tooth profile of generated noncircular gear are illustrated respectively in in Fig. 2(a) and Fig. 2(b). Table 1 Design data of noncircular gear Tooth number
42
Tooth addendum (mm)
2
Tooth dedendum (mm)
2.5
5
Pitch curve function
r ( )
82.605/[2.019 0.2sin(3 )]
Normal module (mm)
2
Pressure angle (deg)
20 Table 2 Cutter Parameters
Example
1
Tooth number
13
Tooth addendum (mm)
2.5
Tooth dedendum (mm)
2.5
Normal module (mm)
2
Pressure angle (deg)
20
Pitch radius
16.25 Table 3 Interpolation, feeding and blank designs
Example
1
Interpolation method
Equal-B-axis
Interpolation cycle
0.01
Interpolation steps
9×104
Cutting cycle
4
Depth in each cycle (mm)
3; 2; 0.5; 0.2
Initial angle of blank
8.509
(a)
(b)
Fig. 2 Processing (a) and product (b) of shaping noncircular gear
3 The mathematical model for measurement 3.1 Measurement principle The basic idea of rolling testing is to make the noncircular gear product mesh 6
with a high-accuracy standard circular gear. As a noncircular gear can engage with a circular gear under varying center distances based on the discussion of variable center distance gears [8], the center distance between the measured noncircular gear and the standard gear surely varies in meshing. Thus, the integrated errors are reflected by the translation and rotation of the standard gear. And because of the varying center distance, the mathematical model for measurement of shaped noncircular gear is much more complicated than that for a traditional circular gear. 3.2 Applied coordinate system
yF
yS yG
xS
Q OF
G G
Noncircular gear
xG
rS
S
xF
OS
Standard gear
Fig. 3 Coordinate systems for measurement of noncircular gear
As shown in Fig. 3, three coordinate systems are established to describe the relations in noncircular gear measurement. Coordinate frame SF (OF -x F yF ) is fixed on the device, with its origin located at the center of the measured gear and its x F -axis along the direction of the slider. Coordinate system SG (OG -x G yG ) is attached to the measured gear. The rotating angle of the measured gear G is the angle between reference frame SF and SG . The coordinate transformation matrix from SF to SG is as follows:
7
cos(G ) sin(G ) sin( ) cos( ) G G M GF (G ) 0 0 0 0
0 0 1 0
0 0 0 1
(10)
Coordinate system SS (OS -xS yS ) is connected to the standard gear. Its angle relative to SF refers to the rotating angle of standard gear S , and its position relative to SF refers to the center distance ES . The coordinate transformation matrix from SF to SS is as follows: cos( S ) sin( S ) ES sin( ) cos( ) 0 S S M SF ( S ) 0 0 0 0 0 0
0 0 0 1
(11)
Suppose the angular velocity of the measured gear is G dG / dt , the screw of the measured gear in the reference frame SF is [38]:
Si [si ; ςi ] [0,0, G ,0,0,0]T
(12)
Where, s i and ς i are correspondingly the primary part (main vector) and the secondary part (dual vector) of the input screw. And the output screw (screw of output gear) in the fixed reference frame S0 is
So [so ; ςo ]
(13)
where, with the angular velocity of the standard gear defined as S dS / dt , the main vector so is identified to be determined by the gear ratio function
so 0 0 S
T
(14)
And the dual vector ς o (1 ) can be calculated as follows [38]:
vE ES vE ς o 0 0 so ES S 0 0 0 where, vE is velocity of the slide, calculated as vE dES / dt 8
(15)
In term of the theorem of three axes [39], the instantaneous screw between drive and driven gears can be calculated as
Sis So Si [sis ; ςis ] [0,0, S G ; vE , ESG ,0]T
(16)
where, the main vector s is is found to be parallel to z0 -axis through further calculation of its direction vector (the unit vector of main vector) as follows:
suis
sis T 0 0 1 | sis |
(17)
The position vector of the instantaneous screw axis, namely, the instant center of the gear pair Q , can be obtained as [38] T
E s ς vE rQ (G , S , ES ) is is S S 0 (18) sis sis S G S G where, the instant center is shown to be not on the center line of the gear pair, complicating the error calculation. Still, the pitch angle of measured gear can be obtained:
G G a tan
vE ES S
(19)
Additionally, the pitch radius of the measured gear is
rM (G , S , ES ) | rQ (G , S , ES ) |
ES 2S 2 vE 2 | S G |
(20)
Furthermore, the pitch radius of the standard gear is
ES ES 2G 2 vE 2 rS (G , S , ES ) rQ (G , S , ES ) 0 S G 0
(21)
3.3 Error calculation For a circular cylindrical gear, with a flank gear rolling tester applied, two types of errors can be measured. One type is the radial integrated error, decided by fluctuation of the center distance, and the other type is the tangential integrated error, decided by wave of the measured angular displacement relative to the theoretical angular displacement for the standard gear. Likewise, for a noncircular gear, with this tester applied, the above two types of errors can also be obtained. However, the calculation of these errors will not be easy, as previously discussed that the instant center is not on the center line of the gear pair. In order to calculate the errors, it should be understood first that the above instant center is a vital to the calculation of errors . And in practice, in term of Eq.(21), the 9
radius of the standard gear cannot be constant if S is the measured angle of the standard gear. S is then defined as the theoretical angle of the standard gear, and the measured angle of the standard gear is represented as m . Thus, with the pitch radius of the standard gear set as rs , the following equations must be satisfied:
ES 2G 2 vE 2 rS (G , S , ES ) rs | S G |
(22)
Where, the angle of the measured gear G , the distance of the slider ES and the velocity of the slider are obtained through measurement. S can in turn be solved:
ES 2G 2 vE 2 S G rs
(23)
So can the angle of standard gear t
S S (t )dt
(24)
0
Substitute Eq.(23) and Eq.(24) into Eq. (20), the measured pitch curve of noncircular gear can be obtained as follows:
rm rM (G , S , ES )
(25)
Thus, the radial and the tangential integrated errors can be resolved as follows:
r ( G ) rm (G ) rp ( G ) t ( G ) [ m (G ) S ( G )]rs
(26)
And for a clear demonstration of the distribution of errors , the errors are magnified 100 times. The enlarged error pitch radius is correspondingly calculated: rpE ( G ) rp ( G ) 100r ( G )
(27)
Thus, the enlarged error pitch curve of noncircular gear is ApE ( ) rpE ( G ) cos( G ) rpE ( G )sin( G ) 0
4 Measurement of errors
10
T
(28)
4.1 Measurement device
Measured gear Standard gear Weight gear slider Angle encoder
Linear rail
Grating sensor Drive motor Fig. 4 The configuration of the non-uniform flank rolling tester
Fig. 5 The developed flank rolling device
In order to measure the flank rolling errors of a shaped noncircular gear, a nonuniform flank rolling tester is needed. As shown in Fig. 4, similar to the case of a traditional tester for circular gear, in the case of a non- uniform flank rolling tester for a shaped noncircular gear, the two rotating axes and a linear axis are configured. The measured gear and the standard gear are attached to the two rotating axes. The standard gear axis is attached to the gear slider (linear axis) and moves along with the slider’s translation. A drive motion, used to provide the rotating angle of the measured gear, is attached to the measured gear through a worm gear driving. And an angle encoder, used to obtain the rotating angle of the standard gear, is rigidly connected to the standard gear. Still, there are some differences. Unlike the case for traditional tester, as the center distance of the gear pair changes, a grating sensor, is set between the slider and the linear rail, along with an additional weight configured to ensure the contact between the measured and the standard gear, to measure the standard gear’s translation. Based on the above discussion of configuration, a non-uniform flank rolling tester is developed as shown in Fig. 5. With this device applied, the rotating angle of the 11
noncircular gear is acquired by the motion, the angular displacement of the standard gear is obtained by the angle encoder, and the center distance between the noncircular and the circular gear is captured by the grating sensor. 4.2 Measurement presetting
yF
t
P
OF
0
rp 0
rg
rs
Oc
xF
OS
Noncircular gear
Standard gear
Fig. 6 Presetting of the measured noncircular gear
As discussed in shaping processing [16][17], a circumferential datum A is set in the gear blank perpendicular to the center line between cutter and gear, which has already been applied to cutter presetting in manufacture. And here, this datum is also taken as a reference in the measurement. Yet, as the pitch radius of the standard gear is inconsistent with that of the cutter in general, datum A will be not perpendicular to the linear rail in measurement and an angle of datum A must be set. As shown in Fig. 6, rp 0 is the initial pitch radius of the noncircular gear and can be calculated as rp 0 rp (0)
(29)
Based on the discussion above, the angle between the tangential vector and the pitch radius is
( ) drp ( ) / d Thus, the center distance between the cutter and the gear is 3 | OF Oc | rp 0 2 rg 2 2rp 0 rg cos[ (0)] 2 Likewise, the center distance between the cutter and the standard gear is 3 | OF OS | rp 0 2 rs 2 2rp 0 rs cos[ (0)] 2
(30)
(31)
(32)
And the angle of datum A relative to y F , namely, the presetting angle of the gear in measurement, can be obtained as 12
0 a cos
| OF OS |2 | OF OS |2 (rs rg ) 2 2 | OF OS || OF OS |
4.3 Computerized measurement systems
Fig. 7 The interface of the computerized measurement system
Fig. 8 Flow chart of Matlab and C++ mixed programing 13
(33)
As shown in Fig. 7, the computerized measurement system is developed to communicate with the non-uniform flank rolling tester, as well as analyze and calculate the errors. While the communication with the tester and the interface design of the system can be realized in the C++ environment, an algorithm programming for data analysis and error calculation can hardly be realized. Matlab, meanwhile, inclusive of numerous mathematical functions, can be an ideal tool for numerical calculation. In this regard, the developed system is designed to be a programming mixed by these two environments. And the flow chart for mixed programming is shown in Fig. 8. With the measured and the standard gears installed, and the computerized measurement system developed, a flank rolling testing can be realized automatically following the flow chart in Fig. 9. In processing, some measurement parameters, such as speed of the motor and sampling frequency should be input manually, while the operations, including communication port initialization, data collection, conversion, filter, error calculation and display, are all carried out by the system automatically.
Fig. 9 The flow chart of the measurement
4.4 Experiment and discussion With the measurement device developed and the computerized measurement designed, the shaped gear, as shown in Fig. 2, is applied to measurement experiment. A standard gear, with identical module and a pitch radius rs 46(mm) , is used to match with the noncircular gear. The processing of measurement is shown in Fig. 10. 14
The radial and tangential integrated errors for the measured gear can ultimately be obtained. Fig. 11 shows the radial integrated errors of the measured noncircular gear, the maximum error being 27 um and the minimum being -26 um. It is not hard to find that the cycle of the errors corresponds approximately with that of the noncircular gear. Fig. 12 shows the enlarged error pitch curve of the measured gear. In the figure, the distribution of the errors in the theoretical pitch curve is clearly illustrated. And it can be observed that the error pitch curve rotates an angle anticlockwise or so relative to the theoretical pitch curve. This could be attributed to the circumference location in the machine cutter presetting before the cutting, as is discussed in [17], the circumference location of blank brings the major difference as well as difficulty for noncircular gear shaping. Besides, this phenomenon could also be explained by the non-accurate position of the cutter center relative to the cutting gear, for which again the nonlinear interpolation method for noncircular gear could account, as discussed in [14]. Fig. 13 shows the tangential integrated errors of the measured noncircular gear, the maximum error being 15 um while the minimum error -13 um. Overall, the trend of the error curve is similar to that of radial integrated error curve, indicating the two errors to be internally associated. This is actually true since both errors are measured through the tooth profile of the gear. In careful comparison, however, the tangential errors are found to smaller, which can be explained by the fact that tangential errors are greatly influenced by rotation of the cutter in cutting processing and the rotation of the cutter is linear with equal-arc-length cutting method adopted[17]. Thus, for noncircular gear, no universal measurement standard can be used to evaluate the accuracy. And in reference to ISO system of accuracy for cylindrical gear, only with both radial and tangential integrated errors taken into consideration, will the shaped noncircular gear reach grade 8 [40].
Fig. 10 The non-uniform gear rolling measurement
15
Fig. 11 The radial integrated errors of noncircular gear
Fig. 12 The theoretical and enlarged error pitch curve
Fig. 13 The tangential integrated errors of noncircular gear
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5. Conclusions This paper aims for a quick measurement of noncircular gears for mass production. The mathematic model for measurement, the non-uniform rolling tester and the computerized measurement system are systematically described. And based on the discussions above, several conclusions are drawn: 1) With a variable center distance, a noncircular gear can be meshed with a standard cylindrical gear, and thus the accuracy of the product noncircular gear can be reflected. 2) Different from the uniform rolling testing for circular gear, the errors of noncircular gear cannot be directly obtained, but calculated through a complicated mathematical model. 3) Based on the configuration of the developed measurement device, both radial and tangential integrated errors of shaped noncircular gear can be obtained. 4) Since the pitch radius of the standard gear is not always consistent with that of the shaper cutter, the presetting datum of measurement is different from that in practical manufacture. 5) With the developed non-uniform rolling tester and the computerized measurement system, the integrated errors of shaped noncircular gear are quickly obtained, demonstrating their correctness and versatility.
Nomenclature rp ( ) = pitch radius of noncircular gear
Ap ( ) = pitch curve of noncircular gear
t ( ) , n( ) = tangent and normal vector of the pitch curve rg = pitch radius of the shaper cutter
Ac ( ) = normal offset of pitch curve
Sr ( ) = rolling arc-length of the shaper cutter g = rotation angle of the shaper cutter
G = rotating angle of measured gear
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ES = centers distance between the measured and standard gear
S = rotating angle of standard gear
G = angular velocity of the measured gear S i = screw of the measured gear s i , ς i = primary and secondary part of the measured gear screw
S o = screw of standard gear so , ς o = primary and secondary part of the standard gear screw
S = angular velocity of the standard gear vE = translate velocity of the standard gear S is = instant screw of the measured and standard gear
sis , ς is = primary and secondary part of the instant screw suis , rQ = direction and position vector of the instant screw
G = pitch angle of measured gear m = measured angle of the standard gear rM (G , S , ES ) = calculated pitch radius of the measured gear rS (G , S , ES ) = calculated pitch radius of the standard gear rs = given pitch radius of the standard gear
rm = measured pitch curve of noncircular gear r ( G ), t ( G ) = radial and the tangential integrated errors of noncircular gear rpE ( G ) = enlarged errors pitch curves of noncircular gear ApE ( ) = enlarged errors pitch curves of noncircular gear rp 0 = initial pitch radius of the noncircular gear 18
( ) = the angle between the tangential vector and the pitch radius of noncircular gear
0 = measurement presetting angle of the gear
Acknowledgments The authors would like to thank the Natural Science Foundation of China (No. 51575416), Innovative Research Team Development Program of Ministry of Education of China (No. IRT13087), High-End Talent Leading Program of Hubei Province (No. 2012-86), Science and Technology Support Program of Hubei Province (No. 2015BAA039) and Natural Science Foundation of Hubei Province (No. 2014CFB876) for the support given to this research.
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(1) Establish a mathematical model for non-uniform flank rolling measurement. (2) Develop a non-uniform rolling tester and computerized measurement system. (3) Investigate the radial and tangential integrated error for shaped noncircular gear.
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