Research Article Advances in Mechanical Engineering 2016, Vol. 8(11) 1–11 Ó The Author(s) 2016 DOI: 10.1177/1687814016679311 aime.sagepub.com
Geometric error detection of curve-face gear pairs Chao Lin, Chao Huang, Xi-Jun Cao and Yu Fan
Abstract A new type of gear pair, the curve-face gear pair, composed of a curve-face gear and a non-circular gear with mutual engagement, was proposed to achieve variable ratio transmission of motion and power between the intersecting axes. Geometric error detection is required to evaluate the grade of the manufactured gears. Due to the complexity of the curve-face gear, direct detection has been conducted in a very limited way. In this article, a measurement method aimed at curve-face gear pair artifacts is presented based on the computer numerical control gear measuring center. By comparing the measured coordinate data of the surface points with the corresponding theoretical data, various errors such as tooth profile error and pitch deviation can be obtained, and accuracy grades are evaluated with reference to accuracy standards for cylindrical gear and bevel gear. The developed method is simple and robust without requiring a special measuring device; hence, it can be applied for the industrial practice as a means for measuring the tooth profile and pitch deviations which cannot be measured by conventional methods. Keywords Curve-face gear, gear measuring center, pitch deviation, profile error, geometric error detection
Date received: 12 April 2016; accepted: 20 October 2016 Academic Editor: Fakher Chaari
Introduction Gears are crucial components for modern precision machinery as a means for power transmission mechanism. Face gear drive is composed of a cylindrical gear and a bevel gear meshing with each other, which has unique advantage in heavy-duty and high-speed occasion such as helicopter transmission system.1 Curveface gear pair is proposed based on the face gear pair, generated by replacing cylindrical gear in face gear pair with a non-circular gear. It is composed of a noncircular gear and a curve-face gear in order to realize variable ratio transmission between the intersecting axes.2 The relevant researches of curve-face gear, especially the geometric error detections, mainly draw lessons from the existing cylindrical gear and face gear. For gear drives, the requirements are different in various machineries, but main requirements such as working accuracy, transmission efficiency, and load distribution uniformity are all related to gear accuracy.3,4
Pitch deviation and tooth profile error are important factors influencing gear accuracy. A lot of researches on measurement have been done, and two main methods are widely used to measure the errors, which are specialized measuring instruments, such as gear circular pitch measuring instrument to measure pitch deviation and involute tester to measure profile error, and universal measuring instruments, such as gear measuring center and coordinate measuring machine.5,6 Nowadays, the researches on geometric error detection mainly focused on the following. (1) The development of novel measuring device using new principle and structure.7–9
The State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing, P.R. China Corresponding author: Chao Lin, The State Key Laboratory of Mechanical Transmissions, Chongqing University, 400044 Chongqing, P.R. China. Email:
[email protected]
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).
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Advances in Mechanical Engineering forward based on the P26 automatic CNC-controlled gear measuring center from Klingelnberg of German, and related researches are carried out.
Design of curve-face gear pair The curve-face gear pair drive is composed of a curveface gear and a non-circular gear with mutual engagement. During the process of engagement, pitch curves of curve-face gear and non-circular gear do pure rolling with each other. As shown in Figure 1, at the initial moment, pitch curves of the curve-face gear pair contact at point P0 ; after a certain time, pitch curves of the curve-face gear pair contact at point P1 . During this time, non-circular gear turns u1 and curve-face gear dP1 and arc P0(2) dP1 are equal in turns u2 , where arc P0(1) length. Figure 1. Pitch curves of the curve-face gear pair. 1: non-circular gear; 2: curve-face gear.
Some novel measuring device are developed, such as a transmission error tester for face gear based on singleflank rolling principle, which can inspect several errors such as pitch deviation and transmission error.7 Tang et al.8 developed a gear measuring using double-flank rack probe (DFRP) method, which can be used to measure pitch and profile deviations on the left and right flanks simultaneously. (2) New measurement and evaluation methods of geometric error using universal measuring instruments.10–13 For example, Guenther10 proposed a method to evaluate runout deviation at bevel gears based on pitch measurements using coordinate measuring machines. Sa´nchez et al.11 introduced geometric principles for analyzing hypoid gears in coordinate measuring machines and obtained a method to trace the tooth shape of hypoid gears. Suh et al.12 measured spiral bevel gear tooth using a coordinate measuring machine and obtained profile error applying profile error measurement algorithm. For detection of bevel gear and cylindrical gear, there exist a variety of equipment and standards around the world,14–16 while there is not any specialized equipment or standard for the detection of curve-face gear. Thus, gear measuring center is used to measure the tooth flank, and a method to obtain and evaluate pith deviation and profile error using measured points is developed in this article. Curve-face gear is a new type of gear; in order to verify the practicability and correctness of the design, the curve-face gear pair was processed by three different ways, that is, five-axis computer numerical control (CNC) machining, additive manufacturing, and composite machining (combining additive manufacturing and five-axis CNC machining).17–18 Finally, a kind of pitch deviation, profile error detection, and evaluation method for curve-face gear pair artifacts are put
Pitch curves of the curve-face gear pair Non-circular gear is used to transmit inhomogeneous motion between two shafts, which has many advantages than other mechanism. The most used non-circular gears are ellipse gear, eccentric gear, and Pascal spiral gear. In this article, the pitch curve of non-circular gear is elliptic curve. According to the spatial meshing theory and coordinate relation, the complex pitch curve of curve-face gear can be deduced from the readily available noncircular gear.2 The equation of the pitch curve of curveface gear can be expressed as follows 2
2p
n2 6 2p cosðu2 Þ 6
2 3 x 6 4y5=6 6 6 z 4
Ðn1
3
rðu1 Þdu1 7 7 7 2p 7 n1 7 Ð n2 7 sin ð u Þ r ð u Þdu 2 1 1 5 2p 0
ð1Þ
0
rð0Þ rðu1 Þ where r(u1 ) = a(1 e2 )=1 e cos (n1 u1 ) is the polar radius of pitch curve of non-circular gear, a is the semimajor axis of ellipse curve, e is the eccentricity of ellipse, n1 is the order of ellipse, u1 is the polar angle of ellipse, u2 is the turned angle of curve-face gear when noncircular gear turns u1 , and n2 is alternation number of the teeth of curve-face gear in the range of 0 2p.
Tooth profile of the curve-face gear pair Tooth profile of non-circular gear can be obtained by generating motion of cylindrical gear on pitch curve of non-circular gear. Similarly, tooth profile of curve-face gear can be obtained by generating motion of noncircular gear on pitch curve of curve-face gear. Since, each tooth profile of non-circular gear is different, and its surface equation is complex. It is very complicated
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to derive surface equation of curve-face gear from surface equation of non-circular gear. Assuming that there is a cylindrical gear engaging both of the non-circular gear and the curve-face gear at the same time, then cylindrical gear tooth profile can envelope the surface of curve-face gear tooth profile. The tooth profile of cylindrical gear is very simple; thus, this issue is simplified. The tooth profile of curve-face gear was obtained by virtual machining with envelope method, while the gear shaper cutter is a standard involute cylindrical gear. The family of surfaces in the coordinate f2 can be expressed as follows ~ rk ðuk , uk Þ r2 ðuk , uk , u1 Þ = M2k~
ðsÞ
~ v2k ðsÞ
= cos ( u1 uk + uk + ak ) 3 (uk + R) i12 3 (rbk + A 3
Number of teeth of the non-circular gear, z1 Number of teeth of the curve-face gear, z2 Module, m (mm) Order of ellipse, n1 Alternation number of the curve-face gear in one cycle, n2 Eccentricity of ellipse, e Inner radius of the curve-face gear, R1 (mm) Outer radius of the curve-face gear, R2 (mm)
18 36 4 2 4 0.1 65 78
rotation angle of gear cutter which rotates around its own axis.
ð2Þ
where ~ rk (uk , uk ) is the equation of gear shaper cutter tooth profile, and M2k is the transformation matrix from coordinate fk of gear shaper cutter to coordinate f2 of curve-face gear. According to the conjugate relationship of tooth surfaces between the curve-face gear and the gear cutter, the projection at the normal line of the relative velocity vector of tooth surface in the mesh point equals to zero. Convert the mesh point from the curve-face gear’s coordinate system to the gear cutter; thus, the meshing equation of curve-face gear can be obtained.19 The equation of meshing can be expressed as follows ~ f ðuk , uk , u1 Þ = N
Table 1. Basic parameters of the curve-face gear pair.
ð3Þ
cos ( u1 uk + uk + ak + l)) ~(s) is the normal vector of gear cutter tooth where N (s) is the relative velocity of surface in coordinate S, ~ v2k gear cutter tooth surface and curve-face gear in coordinate S, au is the pressure angle of gear cutter, ak is the pressure angle on any point k, rbk is the radius of gear base circle of gear cutter, L1 = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 (u1 ) + rk2 2rk r(u1 ) sin m, rk = rbk = cos ak , l =
Measuring of curve-face gear pair The geometric error detection of gear refers to the deviations of the theory model and measured points or curves which are obtained by the detecting instruments. For the normal cylindrical gears, bevel gears, and so on, there already exist mature detection methods, testing instruments, and quality evaluation standards, around the world.20,21 For this curve-face gear pair, the contour scanning software of the German Klingelnberg P26 automatic CNC-controlled gear measuring center was used to measure the coordinates of tooth profile points of the gear pair. Three curve-face gear pairs processed by five-axis CNC machining, additive manufacture, and composite machining were measured in this article. The basic parameters of these curve-face gear pairs are shown in Table 1. The measurement coordinate system is shown in Figure 2. The method to establish measurement coordinate system is as follows: 1.
2.
arccos (L21 + r2 (u1 ) rk2 =2L1 r(u1 )), and uk = inv(ak ). Finally, according to the principle of gear engagement, the equation of the surface of curve-face gear can be expressed as follows 3. ~ r2 ðuk , u1 Þ = 2
3 L1 sin l sin u2 A cos u2 + rbk sin u2 ðsin fs + uk cos fs Þ 4 L1 sin l cos u2 A sin u2 rbk cos u2 ðsin fs + uk cos fs Þ 5 rð0Þ L1 cos l rbk ðcos fs + uk sin fs Þ
ð4Þ where A = rbk + L1 cos (fs l)=i21 cos fs , fs = j + u1 uk uok , and j = p=2 + h u1 m; j is the
Acquisition of more than three points on end face of non-circular gear and curve-face gear artifacts. Obtain a plane by these points as the datum plane XM OM YM and select the normal direction of it as direction of ZM axis. Acquisition of more than four points on cylindrical surface of shaft hole of non-circular gear and curve-face gear. Obtain a circle by these points and select the projection of the circle center on datum plane as the origin OM of the coordinate system. Acquisition of a point on semi-major of noncircular gear and crest of curve-face gear, project this point on datum plane, and name it P. Select the direction of OM P as direction of XM axis. Then, YM axis can be determined according to the right-hand rule.
In order to facilitate the measurement of the whole tooth flank, it is necessary to plan measurement path.
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Figure 2. Measurement of the gear artifacts.
pitch curve and mainly affects the smoothness of gear running. Total accumulative pitch deviation is the maximum absolute value of the difference between actual pitch and nominal pitch of any two corresponding flanks, which mainly affect the kinematic accuracy of gear pair. The pitch of curve-face gear and the noncircular gear is defined as the arc length of the pitch curve between two adjacent corresponding flanks, as shown in Figure 4. The theoretical pitch of the non-circular gear and the curve-face gear pair is the same Figure 3. Measurement path.
To improve measurement efficiency, reduce alignment and calibration time; that is, keep the probe position invariant along tooth width direction and the probe moving along tooth depth direction relative to tooth surface in each measurement process. After one circle is completed, change probe position along tooth width direction and continue the next circle. The measurement path is shown in Figure 3. As can be seen in Figure 6, during the measurement, the gear artifacts rotate around Z1 axis with workbench, and then, the CNC system makes the probe move along the specified direction according to feedback signal from sensor in probe, so that the probe can remain in contact with the tooth surface. At the same time, the CNC system records the absolute position of each axis in real time and converts it to coordinate. For example, in the measurement process of curve-face gear, X2 and Y2 axes remain unchanged and the probe moves along Z2 axis. After one circle is completed, the probe moves DX along X2 axis and continues the next circle.
Pt1 = Pt2 = pm = 12:5664 mm
ð5Þ
In accordance with the definition of pitch, the actual pitch can be confirmed as the arc length between two pitch points on corresponding flank, where pitch point is the intersection of pitch curve and the fitted curve of the measured coordinate points, as shown in Figure 5. It is easy to derive coordinates of pitch points based on measured points and equations of the pitch curves. With these coordinates, u11 and u12 can be determined according to the polar radius r(u1 ) of pitch curve of non-circular gear. Similarly, u21 and u22 can be determined according to the pitch curve equation of curveface gear, as shown in formula (1). The actual pitch can be obtained by calculation of curvilinear integral of pitch curve between the two adjacent corresponding pitch points, which can be expressed as follows P01i
uð12
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 (u1 ) + r0 2 (u1 )du1
ð6Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x0 2 (u2 ) + y0 2 (u2 ) + z0 2 (u2 )du2
ð7Þ
= u11
P02i =
uð22 u21
Pitch deviation Single pitch deviation is the maximum absolute value of difference between the actual pitch and the theoretical pitch, which reflects the uniformity of teeth in the
where i is the tooth number, P01i is the actual pith of non-circular gear, P02i is the actual pith of curve-face gear, r(u1 ) is the polar radius of pitch curve of noncircular gear, r0 (u1 ) is the derivative of r(u1 ) with respect to u1 , ½ x(u2 ) y(u2 ) z(u2 ) T is the equation of
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Figure 4. Definition of pitch deviation of curve-face gear pair: (a) non-circular gear and (b) curve-face gear.
Figure 5. Schematic representation of the calculation of pitch deviations.
Table 2. Equations of pitch deviation. Individual pitch deviation Single pitch deviation Individual accumulative pitch deviation Total pitch accumulative error
fpi = Pi0 Pt fp = max jfpi j P Fpk = ki= 1 fpi Fp = max Fpi min Fpi
the pitch curve of curve-face gear, and x0 (u2 ) y0 (u2 ) z0 (u2 ) is the derivative of x(u2 ) y(u2 ) z(u2 ) with respect to u2 . According to the definitions in related standards, the value of pitch deviations can be calculated through equations in Table 2.
Profile error The profile error ff refers to the projection of the deviate distance between the measured tooth surfaces from the theoretical point on the unit normal vector at any given point of the tooth surface. The location of any point on the surface of tooth flank is to be measured
Figure 6. Positions of grid points on one tooth.
and compared to the specified coordinate, and then, the deviation is calculated. In order to facilitate the measurement of the whole tooth profile, grid division of the tooth surface is needed. The number relates to the sample accuracy of tooth flank, while too much grid will take excessive measurement time. A grid of 5 3 7 points (5 places root-to-tip, 7 places toe-to-heel) is used in this article. The grid boundary was determined by the limitation of gear peak sharpening and undercut, that is, 10%–15% inward indentation,22 as shown in Figure 6. The positions of reference points on different flanks vary with the spatial pitch curve of curve-face gear, but the same is that the reference point is at the tolerance diameter and it is the midpoint of the grid. For curveface gear, tolerance diameter is the sum of inner radius and outer radius of the curve-face gear. In order to cover the whole tooth profile, Z coordinate of the reference point is the same as the intersection point of pitch curve and tooth flank. A point can be defined on an axial plane of the curve-face gear with the Z coordinate and radius of reference point, and then, define a grid
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Advances in Mechanical Engineering reference points) can be derived with Z coordinate, radius, and equation of tooth surface easily. The coordinates of actual measuring points were defined as the intersection of the normal vectors of the corresponding theoretical coordinates and the measured curve, as shown in Figure 8. Thus, in order to obtain the measured coordinates, the unit normal vectors should be worked out first. According to the equation of the surface of curve-face gear ~ r2 , the unit normal vector ~ en (xi , yi , zi ) of theoretical point (xi , yi , zi ) was expressed as follows ∂~ r2 ∂uk 3 ~ en ðxi , yi , zi Þ = ½ai , bi , ci = ∂~ r2 3 ∂u k
Figure 7. Derivation of grid points.
∂~ r2 ∂u1
∂~ r2 ∂u1
ð8Þ
2 3 sin u2 ½r0 k sin (u1 j 6 uk ) 6 rk u0 k cos (u1 j 6 uk ) ∂~ r2 = 4 cos u2 ½r0 k sin (u1 j 6 uk ) 6 rk u0 k cos (u1 j 6 uk ) 5 ∂uk r00 k cos (u1 j 6 uk ) 6 rk u0 k sin (u1 j 6 uk )
ð9Þ ∂~ r2 = ∂u 2 1 6 6 6 6 4
3 0 u0 2 cos u2 ½rk sin (u1 j 6 uk ) + R sin l + sin u2 ½rk (1 j ) cos (u1 0 0 0 7 j 6 uk ) + R sin l + Rl cos l + u 2 sin u2 (uk + R2 ) 7 7 cos u2 ½rk (1 j0 ) cos (u1 j 6 uk ) + R0 sin l + Rl0 cos l 7 5 u0 2 sin u2 ½rk sin (u1 j 6 uk ) + R sin l 0 0 0 rk (1 j ) sin (u1 j 6 uk ) R cos l + Rl sin l
ð10Þ
Figure 8. Schematic representation of the calculation of profile errors.
Finally, the coordinate (x0i , y0i , z0i ) of corresponding measuring point can be derived from equation (9) xx
i = yy bi = f (x, y, z) = 0 i
ai
zzi ci
ð11Þ
where f (x, y, z) = 0 represents the surface fitted from the measured points. Thus, the profile error ff can be expressed as follows ff = Dd =
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx0 i xi Þ2 + ðy0 i yi Þ2 + (z0 i zi )2 ð12Þ
Calculation of deviations
Figure 9. Schematic representation of the coordinate systems.
with this point as the center. Each grid point is projected by rotation around the curve-face gear axis onto the theoretical tooth flank, see Figure 7. Thus, theoretical coordinates of grid points on tooth flank (including
The measurement coordinate system is established by the method introduced in section ‘‘Measuring of curveface gear pair.’’ It is clear that the measurement coordinate system S(XM , YM , ZM ) does not coincide with the design coordinate system S(XD , YD , ZD ). Thus, the rotating and shifting relation between the two systems cannot be avoided. The deviations are calculated by measured points and theoretical points, which should be expressed in the same coordinate system. As shown in Figure 9, z represents the shift amount along ZD direction, and d represents the rotation amount around ZD direction. The coordinate
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cos d 6 sin d M =6 4 0 0
sin d cos d 0 0
0 0 1 0
3 0 07 7 z5 1
ð13Þ
The coordinate of measured points in design coordinate system can be expressed as follows Figure 10. Pitch deviation of non-circular gear artifacts.
½ XD
YD
ZD
1 T = M ½ XM
YM
ZM
1 T
ð14Þ
Pitch deviation
Figure 11. Pitch deviation of five-axis CNC machined curve-face gear artifacts.
Based on the calculation method of pitch deviation presented in Table 2, the values of pitch deviations of the curve-face gear pair artifacts can be worked out with the revised coordinate data. The single pitch deviation of non-circular gear artifacts is shown in Figure 10. The single pitch deviation of curve-face gear artifacts is shown in Figures 11–13. Since there are not any measurement and accuracy standards for non-circular gear and curve-face gear, ISO 1328-1:2013 (cylindrical gears-ISO system of flank tolerance classification), ISO 17485:2006 (bevel gearsISO system of accuracy), and GB/T 11365:1989 (accuracy of bevel and hypoid gears (in Chinese)) are referred to evaluate the accuracy grades of the gear pairs. With reference to ISO 1328-1:2013, single pitch tolerance of non-circular gear shall be calculated using equation (15). Total cumulative pitch tolerance of noncircular gear shall be calculated using equation (16) as follows fpT = ð0:001d + 0:4mn + 5Þ
pffiffiffi(A5) 2
ð15Þ
pffiffiffi(A5) pffiffiffi 2 FpT = 0:002d + 0:55 d + 0:7mn + 12 Figure 12. Pitch deviation of additive manufactured curve-face gear artifacts.
Figure 13. Pitch deviation of composite machined curve-face gear artifacts.
ð16Þ where d is the reference diameter; for non-circular gear, choose the sum of major axis radius and minor axis radius of pitch curve as d; and A is the number of the required tolerance grade. The pitch deviation tolerance of non-circular gear in this size is presented in Table 3. So, values of pitch deviation and the corresponding accuracy grade of non-circular gear artifacts are presented in Table 4. With reference to ISO 17485:2006, single pitch tolerance of curve-face gear shall be calculated according to equation (17). Total cumulative pitch tolerance shall be calculated according to equation (18) pffiffiffi fptT = ð0:003dT + 0:3mn + 5Þ( 2)(B4)
ð17Þ
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Table 3. Pitch deviation tolerance of non-circular gear. Accuracy grade
4
5
6
7
8
9
10
11
fpT (mm) FpT (mm)
4.7 14
6.5 20
9 28
13 39
19 55
27 78
38 111
53 157
Table 4. Accuracy grade of non-circular gear artifacts. Machining method
Single pitch deviation (mm)
Accuracy grade
Total cumulative pitch deviation (mm)
Accuracy grade
Five-axis CNC machined Additive manufactured Composite machined
22.3 43.5 23.1
9 11 9
44.1 80.7 42.8
8 10 8
CNC: computer numerical control.
Table 5. Pitch deviation tolerance of curve-face gear. Accuracy grade
4
5
6
7
8
9
10
11
fpT (mm) FpT (mm)
6.5 24
9.5 34
13 48
19 67
27 95
37 134
53 190
74 269
CNC: computer numerical control.
Table 6. Accuracy grade of curve-face gear artifacts. Machining method
Single pitch deviation (mm)
Accuracy grade
Total cumulative pitch deviation (mm)
Accuracy grade
Five-axis CNC machined Additive manufactured Composite machined
39.8 119.7 48.8
10 –a 10
79.9 234.5 84.2
8 11 8
CNC: computer numerical control. a Pitch deviation beyond the maximum tolerance of grand 11.
FpT = ð0:025dT + 0:3mn + 19Þ
pffiffiffi(B4) 2
ð18Þ
where dT is the tolerance diameter; for curve-face gear, the tolerance diameter is the sum of inner radius and outer radius of the curve-face gear; and B is the number of the required tolerance grade. The pitch deviation tolerance of curve-face gear in this size is presented in Table 5. Same as non-circular gear, values of pitch deviation and corresponding accuracy grade are presented in Table 6.
Profile error Similar to the pitch deviation, based on the calculation method of profile error introduced in equations (6)– (10), it is easy to work out the profile errors of curveface gear artifacts with the revised coordinate data. One
gear tooth is chosen as an example in this article. The profile errors on three curve-face gear artifacts are presented in Tables 7–9. In comparison with the numerical results, a simplified graphical output can provide a better overview of the course of deviations across the tooth flank. So, a kind of three-dimensional (3D) graph output of profile errors is plotted, as shown in Figures 14–16. In these figures, the theoretical tooth flank is represented by the grid in plane, and the nodes of grid represent theoretical points on tooth flank. The measured points deviate from the grid nodes, and deviations are illustrated by line segments vertical to grid plane, where the length of line segment represents the value of deviation. Thus, the grid connected by measured points can reflect actual tooth flank. With reference to GB/T 11365:1989, profile error tolerance of curve-face gear is presented in Table 10. So, values of profile error and the corresponding
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Table 7. Profile error of five-axis CNC machined curve-face gear artifact.
Right flank
1 2 3 4 5 5 4 3 2 1
Left flank
A
B
C
D
E
F
G
14.2 6.0 27.1 24.7 7.5 3.5 5.6 7.3 23.5 5.9
13.5 8.0 11.2 6.5 23.5 24.3 5.8 7.2 25.8 8.8
16.4 8.1 25.8 8.1 24.8 5.3 12.6 11.0 26.6 14.1
8.2 14.3 0.0 26.1 6.3 26.8 10.3 25.9 9.3 5.2
12.6 24.0 7.1 6.5 25.8 25.7 2.8 4.3 25.1 5.7
8.0 5.9 7.3 7.0 26.7 4.6 22.9 23.5 21.9 9.0
14.8 8.5 27.7 2.8 1.9 26.9 24.4 22.9 5.4 8.4
Table 8. Profile error of additive manufactured curve-face gear artifacts.
Right flank
1 2 3 4 5 5 4 3 2 1
Left flank
A
B
C
D
E
F
G
74.3 65.1 74.2 84.8 75.4 77.9 83.3 82.3 70.7 72.4
73.6 88.0 81.2 75.4 78.4 77.3 67.5 65.1 70.2 76.8
69.4 55.4 57.6 54.3 54.1 77.1 78.9 81.8 59.3 236.9
34.9 52.1 0.0 48.1 52.3 58.2 38.7 74.4 246.3 63.1
221.5 41.0 23.1 97.6 87.0 52.6 237.4 241.6 234.4 240.6
41.0 61.9 41.9 248.2 222.6 221.9 247.7 245.0 42.2 73.5
254.8 246.6 232.2 12.8 244.9 62.4 233.4 238.4 80.7 50.9
Table 9. Profile error of composite machined curve-face gear artifacts.
Right flank
1 2 3 4 5 5 4 3 2 1
Left flank
5LJKWIODQN & % $
'
2XWHU HQG
$
(
A
B
C
D
E
F
G
19.7 11.5 10.3 8.1 10.8 8.5 5.3 20.2 2.0 6.2
19.1 2.1 1.1 20.1 22.2 10.0 6.8 26.4 4.4 8.7
18.5 12.1 8.5 2.5 8.7 19.0 12.6 24.8 26.6 14.1
18.4 14.1 0.0 6.1 212.6 18.6 210.6 22.8 213.4 216.4
17.6 15.4 12.3 11.1 25.8 212.5 6.1 29.4 29.7 8.4
18.4 16.0 9.2 7.0 29.5 211.7 5.8 23.8 24.9 11.5
17.4 10.4 23.6 212.2 216.8 216.1 4.8 5.4 4.5 9.6
)
*
(
)
,QQHU HQG
7RS ODQG
%
& ' /HIWIODQN
*
XP
Figure 14. 3D graph output of five-axis CNC machined curve-face gear artifacts.
Figure 15. 3D graph output of additive manufactured curve-face gear artifacts.
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Figure 16. 3D graph output of composite machined curve-face gear artifacts.
Figure 17. Average profile error of lines on right flank. Table 10. Profile error tolerance of curve-face gear. Accuracy grade
7
8
9
10
11
12
A C fc = 0:84(Amn + Bd + C) B = 0:0125A
0.84 6.7 10
1.34 8.4 14
2.1 13.4 21
3.35 21 34
5.3 34 54
8.4 53 85
Table 11. Accuracy grade of non-circular gear artifacts. Machining method
Profile error (mm)
Accuracy grade
Five-axis CNC machined Additive manufactured Composite machined
16.4 97.6 19.7
9 –a 9
CNC: computer numerical control. a Profile error beyond the maximum tolerance of grand 12.
accuracy grades of curve-face gear artifacts are presented in Table 11. For further analysis of profile error difference of curve-face gear processed by three methods, different parts of the tooth flank are selected for analysis. As shown in Figure 6, six lines are selected. Lines 1–3 are placed toe-to-heel, representing districts around tip, intermediate, and root. Lines 4–6 are placed root-totip, representing districts around outer flank, medial flank, and inner flank. Average profile errors of these lines are presented in Figures 17 and 18. As can be seen from Figures 14–18, profile error of the curve-face gear processed by additive manufactured method is much larger than the other two methods; profile error of the curve-face gear artifact processed by composite machining method is slightly larger than the one processed by five-axis CNC machining. Since the tooth flank of curve-face gear is complex spatial surface, step effect will be produced in the process according to the basic principle of additive manufacturing. This leads to relatively large profile error in curve-face gear artifact processed by additive manufacturing. For
Figure 18. Average profile error of lines on left flank.
composite machined curve-face gear artifact, accuracy is mainly guaranteed by five-axis CNC machining in the second step. The secondary clamping, which produces new error, leads to a slightly larger profile error than five-axis CNC machining.
Conclusion In this article, a method based on gear measuring center has been developed to detect and evaluate geometric errors of curve-face gear artifacts processed by three different ways: 1.
2.
A method to find out the actual measured points that corresponds to a theoretical point has been put forward. By fitting the measured points into a curved surface, the normal vector of theoretical point intersects with the fitted curved surface, and the intersection can be regarded as the actual measured points. Pitch deviations of curve-face gear pair artifacts processed by three methods are derived and accuracy grade of pith deviation is evaluated with reference to the accuracy standards for cylindrical gear and bevel gear.
Lin et al. 3.
4.
Profile errors of curve-face gear artifacts are derived, 3D graph is plotted to provide a better overview of the course of deviations across the tooth flank, and accuracy grade is evaluated with reference to the accuracy standard for bevel gear. Overall, geometric error of curve-face gear artifact processed by additive manufactured method is much larger than the other two methods because of step effect; and the geometric error of the curve-face gear artifact processed by composite machining method is slightly larger than the one processed by five-axis CNC machining owing to secondary clamping.
Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is supported by the National Natural Science Foundation of China (51275537) and Chongqing University Postgraduates’ Innovation Project (CYS15008).
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