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Journal of the Chinese Institute of Engineers

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On-machine quasi-3D scanning measurement ofbevel gears on a five-axis CNC machine

Yi-Pei Shih & C. H. You

To cite this article: Yi-Pei Shih & C. H. You (2017) On-machine quasi-3D scanning measurementof bevel gears on a five-axis CNC machine, Journal of the Chinese Institute of Engineers, 40:3,207-218, DOI: 10.1080/02533839.2017.1303404

To link to this article: http://dx.doi.org/10.1080/02533839.2017.1303404

Published online: 27 Mar 2017.

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Journal of the Chinese institute of engineers, 2017Vol. 40, no. 3, 207–218http://dx.doi.org/10.1080/02533839.2017.1303404

On-machine quasi-3D scanning measurement of bevel gears on a five-axis CNC machine

Yi-Pei Shih and C. H. You

Department of Mechanical engineering, national taiwan university of science and technology, taipei, taiwan, roC

ABSTRACTMost commercialized measuring machines for bevel gears have a four-axis structure plus a 3D scanning probe. Because these probes are highly efficient, a large number of high-precision gear grinding and complex machines now apply them to detect tooth surface contour error. Nevertheless, such probes are still too expensive to implement on a milling machine. This paper, therefore, proposes a quasi-3D probe as a better solution for cost reduction. More specifically, the study develops an on-machine measurement system on five-axis machines which are needed for producing bevel gears. It employs a quasi-3D scanning probe whose sensor can detect in xyz directions while outputting only a one-dimensional (1D) deviation. This novel method enables measurement using solely a four-axis movement, meaning that only four coordinates are needed to measure pitch and topography. These coordinates are derived by first establishing mathematical models of bevel gear measurement. Also developed is an evaluation program that communicates between the PC and CNC controller. The proposed mathematical models are then applied to a five-axis CNC (computer numerical control) machine equipped with a Siemens 840Dsl controller and verified by comparing the results of this measurement and evaluation with the accuracy report for the same gear measured on a dedicated gear measurement machine.

1. Introduction

Bevel gears, found extensively in gearboxes and rear-axle drives, are very important elements for power transmission. However, because strength, noise, vibration, efficiency, and lifetime of the gear pair are also affected by the accuracy of the gear pro-file, during gear manufacturing, each individual gear must be transferred to a gear measuring machine for accuracy inspec-tion. Because the loading and unloading of each workpiece is extremely time consuming, many manufacturers have started equipping gear cutting machines with probes which can mon-itor the accuracy of gear manufacturing. Two types of prob-ing systems are common in industry: the touch trigger probe, in which discrete points are gathered on the surface, and the analog scanning probe, which acquires large amounts of surface data. The latter is thus a good choice for measuring complicated forms.

Because the tooth geometry of bevel gears depends on the cutting system, the positions and normals of the tooth surface must be provided before gear measurement. There are two main types of cutting systems, face milling and face hobbing, both of which are widely investigated in the literature. Litvin and Gutman (1981a, 1981b, 1981c) and Litvin et al. (1988), for example, pro-posed mathematical models of a face-milled bevel gear based on format, helixform, and cutter-tilt cutting systems. Fong (2000) later established a universal mathematical model of a bevel gear based on a universal hypoid generator with supplemental kine-matic flank modification motions. Shih, Fong, and Lin (2006) then

proposed a universal mathematical model of a face-hobbed bevel gear and applied ease-off and tooth contact analyses to evaluate contact condition. According to the above mathematical models, the positions and normal vectors of the tooth surface can be fur-ther derived for gear measurement. The mathematical model of topographic measurement for bevel gears using a 3D scanning probe, however, has not been revealed in the literature because of commercial considerations. Nor has a mathematical model been disclosed for a quasi-3D scanning probe, which has yet to be implemented in industry. In terms of measurement software, Wang (2011) used Visual Basic to develop a measuring program via Fanuc’s FOCAS and OpenCNC. More recently, Lin (2013) integrated a one-dimensional scanning probe with a Siemens controller, and used Visual C# to develop a measuring program via Siemens RPC software. In this paper, the above research is used to develop a four-axis quasi-3D scanning measurement for bevel gears.

Specifically, this study aims to develop a scanning measuring system based on a five-axis machine tool equipped with both a Siemens 840Dsl controller and a Blum TC76 sensor. This sen-sor is a quasi-3D scanning system that outputs a one-dimension deviation but can synchronously detect xyz directions. The only gear measurement needed is thus a four-axis motion. Once the four-axis coordinates of the measuring path are derived based on the detected positions of the bevel gear’s tooth surface, the cor-rectness of the proposed mathematical model is tested by apply-ing the method to a five-axis machine (Quaser Machine Tools Inc., UX300 series). The experiment comprises a measurement

ARTICLE HISTORYreceived 6 June 2016 accepted 3 March 2017

KEYWORDSspiral bevel gears; on-machine measuring system; quasi-3D scanning probe

© 2017 the Chinese institute of engineers

CONTACT Yi-Pei shih shihyipei@mail.ntust.edu.tw

208 Y.-P. SHIH AND C. H. YOU

2.2. Form deviations and tooth thickness

Form deviations are measurable using a coordinate measur-ing machine (CMM), which calculates normal deviation after measuring each topographic point. Figure 2, which depicts the measurement of flank topographic errors, shows the theoreti-cal position RT and normal vector nT of a measured tooth sur-face. Rotating the actual tooth surface RM at angle Δα makes it coincide with the theoretical tooth surface at the mean point to remove backlash. Normal deviation, expressed in Equation (1), is the dot product of the normal vector and position deviation between the actual and theoretical points.

where (i,  j) indicates the grid number. The usual indicator for evaluating a produced gear is the sum of squared errors, which is defined in Equation (2) and should be below 3000 μm2 in the case of gear cutting.

The tooth thickness error is the sum of the tooth thickness devi-ations on the convex (l) and concave (r) flanks, which are deter-mined by the product of the deviation angle Δα and the mean radius rm of the gear:

2.3. Pressure angle and spiral angle errors

The pressure angle error Δα and the spiral angle error Δβ, shown in Figure 3, are angular deviations of the profile and the longitudinal directions, respectively, measured at the mean tooth point. These errors can be obtained by calculating the angle between two tangent lines of the actual and theoretical flanks along the profile direction and longitudinal direction, respectively. Tangent lines to the surface constructed from organized points can be determined using the least squares method.

(1)e(i,j)n = (R(i,j)

M� −R(i,j)

T) ⋅ n

(i,j)

T,

(2)S =

nc∑i=1

nr∑j=1

(e(i,j))2.

(3)ΔT (f )a = Δ�

(f )rm, f = l, r.

evaluation of the spiral bevel gear; in particular, a tooth inspec-tion that includes pitch errors and flank topographic deviations. The experimental results are then compared with the accuracy report for the same gear measured by a Klingelnberg P40 gear measurement machine.

2. Measurement method

According to ANSI/AGMA 2009-B01 (ANSI/AGMA 2009), the measurement of bevel gear geometry should address (1) ele-mental accuracy, (2) tooth thickness, and (3) composite accu-racy. The first two are measured for each individual gear, but the latter is inspected using an additional mated gear. Because this paper focuses on the measurement of individual gears, elemen-tal accuracy and tooth thickness includes primarily pitch errors and form deviations, which are discussed in more detail below.

2.1. Pitch errors

Pitch errors, depicted in Figure 1, occur in two forms: single pitch variation and cumulative pitch variation. The former, designated by fp, is the deviation between the actual and theoretical circular pitches at the same tooth, while the latter, represented by Fp, is the difference between summations of the actual and theoreti-cal circular pitches measured over a k teeth interval.

Figure 1. Pitch errors.

Figure 2. flank topographic errors.

JOURNAL OF THE CHINESE INSTITUTE OF ENGINEERS 209

3. Mathematical model of the spiral bevel gear

Although the reference list cites several mathematical models of face-milled and face-hobbed bevel gears (Litvin and Gutman 1981; Litvin et al. 1988; Fong 2000; Shih, Fong, and Lin 2006), this analysis uses as its example the SGDH spiral bevel gear (Gleason Works 1971), which employs a face-milling method. This method needs a shorter machining time than other cutting methods because two flanks are cut in one operation. It also achieves better contact bearing of a gear pair through a helical motion modification to the tooth flanks.

3.1. Face-milling cutter

As shown in Figure 4, the cutting edge in a face-milling cutter is generally straight lined with a round fillet, and the coordinate systems Sl and St are rigidly connected to the cutting edge and cutter head, respectively. The cutter parameters are the profile angle αb, cutter radius r0, and fillet radius ρb.

The position equation of the cutter is as follows:

The coordinates (xn, zn) of the straight lined edge are

where u is the line parameter, ± indicates the inner and outside blades, and β is the rotation angle of the cutter.

3.2. Mathematical model of the tooth surface

The gear blank parameters can be calculated according to the given basic gear parameters and the formulas listed in ANSI/

(4)rt(u) =[xn cos � xn sin � zn 1

]T.

(5)

{xn(�b;u

)= r

0± u sin �b

zn(�b;u

)= u cos �b

,

Figure 3. Pressure angle and spiral angle errors.

Figure 4. Coordinate systems of a face-milling cutter.

Figure 5. Coordinate systems for the universal bevel gear generator (shih, fong, and lin 2006).

210 Y.-P. SHIH AND C. H. YOU

Figure 6. scanning measuring system.

Figure 7. Coordinate systems of the measuring motion.

JOURNAL OF THE CHINESE INSTITUTE OF ENGINEERS 211

4. Scanning measurement for a four-axis movement

4.1. Scanning measuring system

As shown in Figure 6, the scanning measuring system includes (1) a Blum TC76 scanning probe sensor, (2) a Siemens 840Dsl controller, and (3) a personal computer (PC). The bevel gear measurement program is developed using Visual C#. The sensor output signal voltage level is 2–8 V, dependent on probe detect-ing displacement. The PC acquires and analyzes the sensor data through a National Instruments USB-6212 data acquisition (DAQ) device while also gathering controller information via Ethernet cards. The NC is transferred to and from the controller

AGMA ISO 23509-A08 (ANSI/AGMA 2008). Bevel gear manu-facturing is often explained using the concept of an imaginary virtual generating gear whose teeth are formed by the locus of the cutter edges but whose tooth number is not necessar-ily an integer. Letting the same generating gear roll with both the pinion and the gear allows the tooth surfaces of both to conjugate completely with each other. In real gear manufactur-ing, however, a few adjustments are made on the pinion tooth surface to reduce the influence of manufacturing and assembly errors.

Figure 5 shows the coordinate systems for the universal cradle-type bevel gear cutting machine often used to derive mathematical models of a bevel gear. Coordinate systems St and S1 are rigidly connected to the cutter and work gear, while Sa to Sf are auxiliary coordinate systems that describe the rela-tive motion between them. ϕc and ϕ1 are the rotation angles of the cradle and work gear, while the generating gear is fixed on the cradle coordinate system Sc. Nine machine settings control the positions of the cutter and work gear: (1) tilted angle i, (2) swivel angle j, (3) radial distance SR, (4) initial cradle angle setting θc, (5) vertical offset Em, (6) sliding base ΔB, (7) machine root angle γm, (8) increment of machine center to back ΔA, and (9) roll ratio Ra.

Equation (6) expresses the generation of the cutting tool locus from the transformation matrices from St to S1 (Shih, Fong, and Lin 2006).

Because the generating gear rolls with the work gear at a fixed speed ratio during the generating process, the rotation angles of the cradle and work gear should be satisfied by the following equation:

Substituting Equation (7) into Equation (6) yields the position equation as a function of variables (u, � ,�

c). According to differ-

ential geometry (Litvin and Fuentes 2004), two tangent vectors on the tooth surface can be obtained by partially differentiat-ing Equation (6) with u and β, respectively. The cross product of these two vectors is the normal vector of the tooth surface:

The tooth surface is the envelope for the family of surfaces gen-erated by the cutter, for whose existence the equation of mesh-ing is a necessary condition determinable as follows:

Equation (6) has three variables (u, � ,�c), which can be solved

using two boundary equations of the gear blank and the equation of meshing. The theoretical positions and unit nor-mal vectors of the topographic points for measurement can be obtained either from the above equations or from the nominal data generated by commercial bevel gear design software.

(6)r1(u, � ,�c ,�1

) = M1g(�1

)Mgf (�m,ΔA)Mfe(em,ΔB)Med

× (�c)Mdc(�c)Mcb(j, SR)Mba(i)rt(u, �)

= [x1y1z11]

T.

(7)�1= Ra�c .

(8)n1(u, � ,�

c) =

�r1(u,� ,�c )

�u×

�r1(u,� ,�c )

��

‖‖‖�r

1(u,�)

�u×

�r1(u,�)

��

‖‖‖= [n

1x n1y n

1z]T.

(9)f1(u, � ,�c) = n

1⋅

(�r

1

��c

)⋅

�c = n1⋅ v

(1t)

1= 0.

Figure 8. Positions of the probe and work gear at the measuring point.

Figure 9. Pitch measurement.

Figure 10. topographic measurement.

212 Y.-P. SHIH AND C. H. YOU

vectors of both surfaces coincide at each measured point. Using coordinate transformation, the position and normal vector of the work gear can be represented in coordinate system Sp as follows:

(10)

{rp(Cx , Cy , Cz ,�c) = Mp1(Cx , Cy , Cz ,�c)r1 = [xp yp zp 1]

T

np(�c) = Lp1(�c)n1= [npx npy npz]

T.

using Siemens’ RPC SINUMERIK computer link software package for PC. Also installed on the PC is an RPC software package, which includes (1) MCIS RPC sl, (2) HMI sl, and (3) an MCIS_RPC.OCX interface. The first two connect to the Siemens 840Dsl, while the latter enables Windows development software (e.g. Visual C#) to access the RPC SINUMERIK interface using COM objects.

4.2. Measuring coordinate systems

Existing gear measuring machines – for example, the Gleason GMS series and Klingelnberg P series – have a four-axis structure and are equipped with a scanning probe that outputs three-axis analog signals. The Blum TC76 sensor, in contrast, outputs only one signal but can detect xyz directions. The 3D probe enables bevel gear measurement using a four-axis movement, meaning that only four coordinates are needed to program the path for pitch and topographic measurement.

Figure 7 shows the coordinate systems of a measuring machine that includes three translational axes (Cx,  Cy,  Cz) and one work gear rotation angle (ϕc). Coordinate systems Sp and S1 are rigidly connected to the probe and work gear. Here, we use a five-axis machine (of the double rotary table type) as the experi-mental machine. Ky and Kz are the machine constants −0.022 and 0.085, respectively, which should be calibrated before machin-ing; Md is the gear mounting distance; and Hf is the fixture height.

The position r1 and normal vector n1 of the tooth surface can be derived either from bevel gear design software or from the mathematical model of the tooth surface expressed in Equations (4)–(9). During measurement, the probe and work gear are moved by four axes so that the positions and normal

Figure 11. repeatability of the scanning probe.

Figure 12. Calibration of probe displacement.

Table 1. relationship between the output voltage and probe displacement (mm).

θx 2 V 3 V 4 V 5 V 6 V 7 V 8 V0 −0.2445 −0.1631 −0.0834 0.0000 0.0879 0.1879 0.292914.4 −0.2466 −0.1674 −0.0881 0.0000 0.0929 0.1933 0.300328.8 −0.2365 −0.1591 −0.0802 0.0000 0.0863 0.1886 0.294643.2 −0.2325 −0.1548 −0.0764 0.0000 0.0873 0.1863 0.291657.6 −0.2382 −0.1602 −0.0819 0.0000 0.0881 0.1863 0.290872.0 −0.2401 −0.1604 −0.0823 0.0000 0.0881 0.1863 0.290686.4 −0.2402 −0.1607 −0.0821 0.0000 0.0885 0.1873 0.2916100.8 −0.2405 −0.1608 −0.0822 0.0000 0.0887 0.1874 0.2915115.2 −0.2447 −0.1652 −0.0867 0.0000 0.0874 0.1867 0.2914129.6 −0.2419 −0.1617 −0.0825 0.0000 0.0890 0.1873 0.2922144.0 −0.2380 −0.1586 −0.0802 0.0000 0.0862 0.1852 0.2898158.4 −0.2470 −0.1664 −0.0863 0.0000 0.0892 0.1898 0.2930172.8 −0.2416 −0.1637 −0.0833 0.0000 0.0884 0.1878 0.2910187.2 −0.2376 −0.1621 −0.0827 0.0000 0.0886 0.1893 0.2945201.6 −0.2446 −0.1606 −0.0820 0.0000 0.0885 0.1873 0.2928216.0 −0.2426 −0.1604 −0.0819 0.0000 0.0883 0.1920 0.3026230.4 −0.2371 −0.1592 −0.0812 0.0000 0.0871 0.1848 0.2886244.8 −0.2387 −0.1604 −0.0820 0.0000 0.0867 0.1842 0.2885259.2 −0.2381 −0.1598 −0.0815 0.0000 0.0869 0.1846 0.2883273.6 −0.2418 −0.1629 −0.0847 0.0000 0.0870 0.1839 0.2880288.0 −0.2430 −0.1654 −0.0870 0.0000 0.0916 0.1903 0.2938302.4 −0.2397 −0.1611 −0.0831 0.0000 0.0869 0.1847 0.2885316.8 −0.2420 −0.1618 −0.0821 0.0000 0.0844 0.1832 0.2954331.2 −0.2442 −0.1638 −0.0834 0.0000 0.0888 0.1880 0.2932345.6 −0.2462 −0.1642 −0.0835 0.0000 0.0884 0.1876 0.2928

JOURNAL OF THE CHINESE INSTITUTE OF ENGINEERS 213

The transformation matrix Mp1 is

where the Lp1 is the upper 3 × 3 rotation matrix of Mp1. When all coordinates are zero, the origin op and program zero Q coin-cide. The measuring path is obtained by solving the coordinates whose measuring points are detected by the probe. As shown in Figure 8, when the probe contacts measured point rp, the normal vector of the work gear must pass through the probe’s center, whose distance to the measured point is ρp.

Because the probe center is positioned at origin op, the three coordinates in Equation (12), each of which produces its own Equation (13), should equal zero. An additional equation is needed to solve four coordinates.

4.3. Coordinates for pitch measurement

To measure the pitch (see Figure 9), the probe moves to a spe-cific position according to the coordinates(Cx,  Cy,  Cz) and then measures the mean points of the convex and concave flanks by rotating the work gear axis (φc). All teeth must be sequentially inspected to evaluate pitch error. Assuming that Cy = −3.498 mm for appropriate y position, the coordinates (Cx,  Cy,  φc) can be solved from Equation (13).

(11)

Mp1(Cx , Cy , Cz ,�c) =

⎡⎢⎢⎢⎢⎣

1 0 0 0

0 1 0 −(Cy + Ky)

0 0 1 −(Cz + Kz + Hf )

0 0 0 1

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

1 0 0 −Cx

0 1 0 0

0 0 1 0

0 0 0 1

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

1 0 0 0

0 1 0 Ky0 0 1 (Kz + Hf +Md)

0 0 0 1

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

cos�c sin�c 0 0

− sin�c cos�c 0 0

0 0 1 0

0 0 0 1

⎤⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎣

1 0 0 Cx

0 1 0 −Cy

0 0 1 −Cz

0 0 0 1

⎤⎥⎥⎥⎥⎦

,

(12)op = rp + �pnp = [0 0 0]

T.

(13)⎧⎪⎨⎪⎩

−Cx + (x1+ n

1x�p) cos�c + (n1y�p + y

1) sin�c = 0

−Cy − (x1+ n

1y�p) sin�c + (n1y�p + y

1) cos�c = 0

−Cz +Md + z1+ n

1z�p = 0

.

Figure 13. Part relationship between the output voltage and probe displacement calibrated on yz plane.

Figure 14. topographic points on the tooth surfaces.

Table 2. Basic parameters of the measured bevel gear.

Items Unit Pinion GearModule at heel met mm 3.500number of teeth zn – 10 36spiral angle βm deg 35.000 l.h 35.000 r.h.Pressure angle αn deg 20.000face angle δa deg 19.583 77.500Pitch angle δp deg 15.524 74.475outer diameter da mm 43.541 126.855face width b mm 17.500 17.500Mounting distance Md mm 66.000 27.500

Table 3. Cutter parameters and machine settings.

Pinion Gear

Items Convex Concave Convex Concave

(A) Cutter parametersCutter radius r0 mm 56.134 58.223 56.527 57.830Profile angle αb deg 23.000 17.000 23.000 17.000

(B) Machine settingstilt angle i deg 1.369 1.251swivel angle j deg 294.666 209.751initial cradle angle

settingθc deg 65.750 −64.750

radial setting SR mm 51.794 51.749Vertical offset Em mm 0.000 0.000increment of

machine center to back

ΔA mm 0.000 0.000

sliding base feed setting

ΔB mm 0.840 1.727

Machine root angle γm deg 11.917 68.883roll ratio Ra – 3.735500 1.037639

214 Y.-P. SHIH AND C. H. YOU

tooth surface at every measuring point, it can obtain normal devi-ations directly, thereby enabling bevel gear measurement using a quasi-3D probe. Letting npx(φc) = 0(on the plane yz), the rotation angle of work gear φc can be derived from Equation (10), after which coordinates (Cx, Cy, Cz) can be solved using Equation (13). The result is shown by the following equations:

4.5. Repeatability of the scanning probe

Repeatability of the scanning probe is measured using a datum plane. Here, the probe touches the same position 120 times to assess the accuracy of the repeatability on the five-axis machine. Figure 11 shows the results, in which the deviations are less than 1 μm.

4.6. Calibration of probe displacement

The Blum TC76 probe is a quasi-3D touch probe that can detect in xyz directions while only outputting a one analog voltage deviation. This sensor, which has a measuring range of about 0.5  mm, has two output signals: a 24  V trigger signal and an analog stroke (with 2–8  V analog voltage). The stroke can be converted from the output analog voltage, and the relation between them is close to linear (see the left side of Figure 12). To improve measurement precision, the probe displacement must be calibrated in 3D space, a task that is both difficult and time consuming. This paper, therefore, develops a new method in which the probe measuring direction is restricted to the yz or

(14)

⎧⎪⎪⎪⎨⎪⎪⎪⎩

�c = tan−1[x, y] = tan

−1[∓n

1y ,±n1x]

Cx =∓n

1y (x1+n1x�p)±n1x (y1+n1y�p)√n2

1x+n2

1y

Cy =∓n

1x (x1+n1x�p)∓n1y (y1+n1y�p)√n2

1x+n2

1y

Cz = z1+Md + n

1z�p

.

4.4. Coordinates for topographic measurement

The topographic measurement inspects form error on the tooth surface, as well as tooth thickness, pressure angle, and spiral angle errors. The probe’s detection directions, selected for their higher accuracy and stability, should be on (the detecting) plane yz, per-pendicular to the probe’s cantilever bar (see Figure 10). The probe sensor must thus be calibrated on the detecting plane before measurement. Because the probe detects the normal vector of the

Table 4. Position and normal of topographic points for the pinion.

Col. Row

Position Normal

XP YP ZP XN YN ZN

Convex

1 1 15.2202 7.8895 −62.0627 0.7777 −0.4040 −0.48161 3 16.2511 9.8447 −61.5474 0.8669 −0.2321 −0.44111 5 16.8938 12.2323 −61.0322 0.9112 −0.0698 −0.40605 1 15.3837 2.0325 −54.7283 0.6173 −0.6665 −0.41805 3 16.6264 3.1741 −54.3366 0.7423 −0.5517 −0.38035 5 17.7329 4.6637 −53.9450 0.8322 −0.4324 −0.34729 1 13.5999 −2.8303 −47.3939 0.3912 −0.8502 −0.35229 3 14.6656 −2.3519 −47.1259 0.5125 −0.7963 −0.32139 5 15.7256 −1.6759 −46.8578 0.6143 −0.7325 −0.2935

Concave

1 1 16.6611 4.0381 −62.0627 −0.0322 0.7409 0.67081 3 18.7006 3.3617 −61.5474 0.1404 0.6897 0.71031 5 20.7446 2.1660 −61.0322 0.2545 0.6237 0.73905 1 15.3738 −2.1068 −54.7283 0.2957 0.7137 0.63495 3 16.6264 −3.1741 −54.3366 0.4145 0.6158 0.67015 5 17.7613 −4.5548 −53.9450 0.4903 0.5229 0.69739 1 12.0927 −6.8363 −47.3939 0.5927 0.5306 0.60609 3 12.6056 −7.8556 −47.1259 0.6461 0.4254 0.63379 5 12.9996 −9.0064 −46.8578 0.6771 0.3320 0.6567

Table 5. four-axis coordinates for topographic measurement.

Col. Row Cx Cy Cz φc

Convex

1 1 14.0181 −10.7450 3.4557 62.54551 3 13.7121 −14.0502 4.0115 75.01401 5 13.4862 −16.8247 4.5618 85.62235 1 12.6671 −9.8712 10.8537 42.80825 3 12.4652 −12.3762 11.2831 53.38065 5 12.3151 −14.5225 11.7078 62.54159 1 11.1720 −9.1915 18.2539 24.70639 3 11.0593 −10.8618 18.5528 32.76579 5 10.9728 −12.3446 18.8487 39.9821

Concave

1 1 16.8209 4.0516 4.6081 2.49171 3 17.6540 7.7289 5.1629 −11.50801 5 18.3893 10.5156 5.7068 −22.19505 1 15.0095 4.7103 11.9066 −22.50405 3 15.5650 7.3938 12.3335 −33.94705 5 16.0727 9.5416 12.7523 −43.15209 1 13.1591 5.2459 19.2121 −48.16609 3 13.4929 6.9827 19.5078 −56.64109 5 13.8100 8.4604 19.7989 −63.8780

Table 6. four-axis coordinates for pitch measurement.

Items Cy Cx Cz φc

Convex −3.4980 17.2138 11.2831 20.0726Concave −3.4980 16.8732 12.3335 3.1742

JOURNAL OF THE CHINESE INSTITUTE OF ENGINEERS 215

xy (detecting) plane. Calibration is then only necessary on the detecting plane (see Figures 10 and 12, right side). In the sev-eral tests conducted, the probe moves in the radial direction of a datum sphere 25 mm in diameter while displacement and out-put voltage are recorded simultaneously within a voltage range from 2 to 8 V. The calibrations are made on the yz and xy planes to measure the pinion and gear, respectively. The results for θx, listed in Table 1, range from 0° to 360° on the yz plane. Partial results are also graphed in Figure 13.

5. Experimental results and discussion

5.1. Measurement data

For concision, the experimental example is restricted to only the pinion results for the bevel gear pair produced using an

Figure 15. simulation of pitch and topographic measurement using software VeriCut.

Figure 16. Picture for pinion measurement.

Figure 17. Pitch errors for the pinion’s convex flank (a) our result (b) Klingelnberg P40 result.

216 Y.-P. SHIH AND C. H. YOU

5.2. Experimental results

The scanning measurement program developed using Visual C# has the following three functions: (1) programing NC codes for pitch and topographic measurements, (2) acquiring sensor data (output voltage) and NC controller information (four coor-dinates), and (3) evaluating gear accuracy. Figure 16 illustrates the measuring experiment for the pinion, while Figures 17 and 18 compare the results for single and cumulative pitch errors in both measurement machines with those obtained using a Klingelnberg P40 measurement machine. Table 7, which lists the pitch errors and their DIN accuracy class, reveals a large dif-ference between the two measurements in terms of cumulative pitch error. This difference may be the result of clamping runout, for which the proposed method dos not compensate.

Figure 19 shows the average topographic errors for P40 ver-sus the experimental measurements, displaying a maximum dif-ference of about 20 μm. Table 8 lists the remaining form errors including (1) sum of squares, (2) tooth thickness deviation, and (3) pressure and spiral angle errors, all with differences of between 15 and 60%. Although the topographic measurements differ from the P40 results because of a lack of fixture calibration in the experiment, both sets of outcomes show a similar tendency.

SGDH cutting system, whose basic parameters are listed in Table 2. The manufacturing parameters, listed in Table 3, are determined from the calculation tables provided by Gleason Works (1971) and comprise the gear blank parameters, cutter parameters, and machine settings. These parameters are sub-stituted into Equations (4)–(9) to enable calculation of the posi-tions and normals of 9  ×  5 topographic points on the tooth surface (see Figure 14). The positions and normal vectors of the topographic points, which can also be obtained from bevel gear design software, are referred to as nominal data, a por-tion of the data is listed in Table 4. The tooth thickness angle is 14.384° measured at the mean points. In the topographic measurement, when all detecting directions of the probe are on plane yz (detecting plane), the x component of the normal vector np is equal to zero. Hence, Equation (13) and npx(φc) = 0 enable solution of the four coordinates of each measured point listed in Equation (14) (see Table 5 for the most relevant data). For pitch measurement, when Cy = −3.498 mm, the other three coordinates can be solved using Equation (13). The results are listed in Table 6 for convex and concave flanks. The four coordi-nates calculated are then used to program the NC codes, and their correctness is further verified using VERICUT simulation (see Figure 15).

Figure 18. Pitch errors for the pinion’s concave flank (a) our result (b) Klingelnberg P40 result.

Table 7. Pitch errors and Din accuracy class.

Items

P40 results Our results

Percentage error (%)Error Quality Error Quality

Convex (um)

Max. t. s. index errorfp 35.3 9 37.79 10 7.1Max. tooth spacing error fu 30.8 9 27.49 8 10.7t. s. total index error Fp 112.1 9 256.73 11 129.0

Concave (um)

Max. t. s. index errorfp 25.3 6 36.3 10 43.5Max. tooth spacing error fu 31.5 6 25.9 8 17.8t. s. total index error Fp 145.8 6 288.2 11 97.7

JOURNAL OF THE CHINESE INSTITUTE OF ENGINEERS 217

machine. The correctness of the proposed mathematical model is validated experimentally and also using VERICUT simulation. Nevertheless, because the proposed method neither accounts for clamping runout nor completely calibrates the machine tools, some gaps still exist between the experimental measure-ments and those obtained with a Klingelnberg P40.

Nomenclature

Cx, Cy, Cz rectilinear axes of the five-axis machineLij upper left 3  ×  3 submatrix of the transformation

matrix MijMij homogeneous transformation matrix from coordi-

nate system Sj to coordinate system Sinp surface unit normal of the work gear in the work-

piece coordinate system S1np surface unit normal of the work gear in the probe

coordinate system Spr1 position vector of the work gear tooth surface in the

workpiece coordinate system S1rp position vector of the work gear tooth surface in the

probe coordinate system Spρp radius of the probeφc workpiece rotational axis of the five-axis machine

6. Conclusions

This paper successfully constructs a quasi-3D probe meas-uring system that meets the low cost requirement of milling machines. The mathematical models of pitch and topographic measurements for bevel gears are developed based on a four-axis movement. To simplify calibration of the quasi-3D probe, every measuring point of the work gear must be rotated to ensure its normal vector to the tooth surface on the detect-ing plane. The Visual C#-based measuring program integrates both a Siemens 840Dsl controller and the scanning probe sen-sor used for measuring purposes by the five-axis experimental

Figure 19. average topographic errors (a) our result (b) Klingelnberg P40 result.

Table 8. other form errors.

Items Unit P40 results Our resultsPercentage

error (%)sum of squares μm2 37,648.4 22,143.2 41.2thickness deviation mm 0.316 0.214 32.3Pressure angle

error, convexdeg −0.659 −0.509 22.8

Pressure angle error, concave

deg −0.707 −0.601 15.0

spiral angle error, convex

deg −0.06 −0.044 26.7

spiral angle error, concave

deg −0.213 −0.086 59.6

218 Y.-P. SHIH AND C. H. YOU

Litvin, F. L., and Y. Gutman. 1981a. “Methods of Synthesis and Analysis for Hypoid Gear-Drives of ‘Formate’ and ‘Helixform’ – Part 1. Calculations For Machine Settings For Member Gear Manufacture of the Formate and Helixform Hypoid Gears.” Journal of Mechanical Design 103 (1): 83–88. doi:10.1115/1.3254890.

Litvin, F. L., and Y. Gutman. 1981b. “Methods of Synthesis and Analysis for Hypoid Gear-Drives of ‘Formate’ and ‘Helixform’ – Part 2. Machine Setting Calculations for the Pinions of Formate and Helixform Gears.” Journal of Mechanical Design 103 (1): 89–101. doi:10.1115/1.3254891.

Litvin, F. L., and Y. Gutman. 1981c. “Methods of Synthesis and Analysis for Hypoid Gear-Drives of ‘Formate’ and ‘Helixform’ – Part 3. Analysis and Optimal Synthesis Methods For Mismatch Gearing and its Application For Hypoid Gears of ‘Formate’ and ‘Helixform’.” Journal of Mechanical Design 103 (1): 102–110. doi:10.1115/1.3254837.

Litvin, F. L., Y. Zhang, M. Lundy, and C. Heine. 1988. “Determination of Settings of a Tilted Head Cutter for Generation of Hypoid and Spiral Bevel Gears.” Journal of Mechanisms, Transmissions, and Automation in Design 110 (4): 495–500. doi:10.1115/1.3258950.

Shih, Y.-P., Z.-H. Fong, and G. C.-Y. Lin. 2006. “Mathematical Model for a Universal Face Hobbing Hypoid Gear Generator.” Journal of Mechanical Design 129 (1): 38–47. doi:10.1115/1.2359471.

Wang, P.-Y. 2011. “A Study on the Human Machine Interface of the Fix-axis CNC Gear Profile Grinding Machine.” Master dissertation, National Taiwan University of Science and Technology.

Disclosure statementNo potential conflict of interest was reported by the authors.

ReferencesANSI/AGMA. 2009. Bevel Gear Classification, Tolerances, and Measuring

Methods. American Gear Manufacturers Association Standard. ANSI/AGMA 2009-B01. Alexandria, Virginia, USA: American Gear Manufacturers Association.

ANSI/AGMA ISO. 2008. Bevel and Hypoid Gear Geometry. American Gear Manufacturers Association Standard. ANSI/AGMA ISO 23509-A08. ANSI/AGMA 2009-B01. Alexandria, Virginia, USA: American Gear Manufacturers Association.

Fong, Z.-H. 2000. “Mathematical Model of Universal Hypoid Generator With Supplemental Kinematic Flank Correction Motions.” Journal of Mechanical Design 122 (1): 136–142. doi:10.1115/1.533552.

Gleason Works. 1971. Calculation Instructions: Generated Spiral Bevel Gears, Duplex-Helical Method, Including Grinding. Rochester, NY: Gleason Works.

Lin, S.-S. 2013. “On-machine Scanning Measurement of Bevel Gears Based on the Five-axis CNC Machine.” Master dissertation, National Taiwan University of Science and Technology.

Litvin, F. L., and A. Fuentes. 2004. Gear Geometry and Applied Theory. 2nd ed. Cambridge: Cambridge University Press.

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