When measuring with coordinate measuring machines, a new proposed approach is utilized to capture deviations from spherical involute bevel gears; these deviations are then used in tooth contact analysis.

Straight bevel gears are the simplest type of bevel gears that transfer power between intersecting axes [1]. They are widely used in low-speed applications or static-loading conditions. Differential gears are one such application where the speed is very low and the load type is mainly static. There are still several traditional applications of straight bevel gears in aerospace, marine, agriculture, and construction [2-5]. These have a wide range of shaft angles, especially in marine applications; however, right angle is the most common.

In the late 1800s, Gleason utilized a planer with an indexing head [6] to introduce an efficient straight bevel gear manufacturing method. Development of more efficient manufacturing methods for straight bevel gears was followed by the introduction of Revacycle [5] and Coniflex [7] cutting methods. In the last two decades, forging has emerged as one more efficient and economical method used in straight bevel gear manufacturing [8]. Due to forging limitations in tonnage and accuracy, this trend is aimed more toward smaller size bevel gears with low-speed applications such as automobile differentials, which are usually finish forged or near-net forged. Utilizing forging for such applications will save on material and machining costs and improve strength due to improved grain flow [5] in the material. Using forging as a manufacturing method will also give the possibility of having a web feature between the teeth [9].

Due to ease of manufacture and insensitivity to changes in center distance, involute curves are dominant in cylindrical gears. By the same analogy, spherical involute would bring the same benefits for straight bevel gears. However, in many of today’s applications, forging manufacturers of bevel gears imitate cut surfaces of Coniflex or Revacycle, which is a legacy of the time when forging manufacturers were trying to duplicate cut gears to prove that these gears can also be near-net forged. Other than some fundamental definitions [10], most of the basic formulas of today’s cut straight bevel gear geometry were established by Wildhaber [11, 12]. Al-Daccak [13] et al proposed a method for spherical involute geometry calculation; Figliolini [14] and Angeles also introduced a calculation approach for both spherical involute and octoidal bevel gears. The purpose of both studies were to calculate surface coordinates and tooth modeling while Kolivand [15] added required formulas to compute normal to the surface. The normal vectors to the surfaces are needed to construct ease-off, conduct tooth contact analysis (TCA), and measure the tooth topography using coordinate measuring machines (CMM), which all aim to improve quality and performance of designs, prototypes, and production parts.

Quality is usually defined and measured through measurements of pitch line run-out and tooth thickness variation. Performance characteristics are usually evaluated through durability, strength, and noise level analyses and tests, in which TCA plays a significant role. Therefore, having an accurate method to evaluate TCA becomes crucial. Unloaded tooth contact analysis (UTCA) with mismatched surfaces has been performed using two fundamentally different methods. The first method that was widely used defines the tooth surfaces as two arbitrary surfaces [16-21]. The instantaneous contact on each surface is determined by satisfying two contact conditions of (a) coincidence of position vector and (b) co-linearity of the normals. The second method is based on the ease-off topography [22, 23]. In one such study, Kolivand and Kahraman [23] proposed a formulation to construct an ease-off surface and determine the instantaneous contact curves from surface of roll angle; details of this approach are in reference [23]. TCA of bevel gears has usually been performed by considering the theoretical pinion and gear surfaces.

In general, bevel gear surface geometry is not explicitly available and has to be calculated through either cutting simulation (Coniflex, Revacycle, etc.) or implicit solutions of systems of nonlinear equations [19, 20, 24, 25]. In both cases, the surfaces are theoretical and do not include any manufacturing errors in surface topography and tooth thickness. There are only a few published studies on bevel and hypoid gear tooth contact analysis that use the real surfaces (here, the term “real” refers to the actual surfaces of the parts that can be measured on CMMs). In one such study, Gosselin [26] proposed an approach to compute tooth contact of spiral bevel gear surfaces having deviations. He interpolated measured surfaces with rational functions to predict their unloaded contact pattern and transmission error. Zhang et al [27] proposed an approach to analyze unloaded tooth contact of real hypoid gears based on generalization of Kin’s [28, 29] work for spur gears. In Kin’s approach, the measured pinion and gear tooth surfaces were divided into two vectorial surfaces, defined as the theoretical and the deviation surfaces. By separating the theoretical and the deviation surfaces, finding the theoretical surfaces through cutting simulation, and applying interpolation only to the deviation surfaces, he made his approach simpler and more accurate. None of these studies have been utilized in spherical involute bevel gears, and none were validated through roll testing of actual parts. This paper aims to address both shortcomings. Here, after CMM measurements, the surface deviations are expressed in the form of a continuous third-order two-dimensional polynomial, and the polynomial coefficients are computed, stored, and used to modify theoretical spherical involutes to represent actual measured parts. The details of this interpolation approach are given later on in this paper. The fact that the interpolation is only done on surface deviations (and not on the surface itself) brings a high level of accuracy and efficiency to this approach. Spherical involute and deviation coefficients computations are required prior to tooth contact analysis.

Surface coordinates and normals of spherical involute updated with deviation surfaces are then used to establish ease-off topography and surface of roll angle [23]. For example, an 8 x 13, 24-degree pressure angle automotive straight bevel gear set is analyzed to show details of the developed approach as well as its efficiency and accuracy. The approach is unique in developing ease-off and surface of roll angle based on actual measured surfaces and conducting tooth contact analysis using the developed ease-off. To reach this overall goal, specific objectives of this study include:

  • Compute bevel gear geometry based on spherical involute including surface coordinates and normal to the surface.
  • Develop a simple and effective third-order two-dimensional least-square approach to capture and represent surface deviations of actual measured bevel gears.
  • Perform an ease-off based tooth contact analysis to compute contact pattern and transmission error.
  • Bench roll test the measured parts and evaluate contact patterns of the actual parts at different mounting positions, and compare them to predicted ones to evaluate the effectiveness and accuracy of the approach.

Spherical Involute Surface and Normal

A spherical involute surface is a three-dimensional equivalent of the established two-dimensional involute curve commonly used in cylindrical gears [10, 13-15, 24]. Some studies defined spherical involute by planar involute through gradually changing the base circle radius of a planar involute from the inner to outer cone section of the bevel teeth [30]. In this study, a similar approach to Al-Daccak et al [13] is used to establish spherical involute. However, normal to the surface is computed based on the approach proposed by Kolivand [15].

In Figure 1 (a and b), spherical involute is defined as a three-dimensional curve G traced by a point P on a taut chord42829.jpgunwrapping from base circle Cb that lies on sphere Q with origin at C and radius r. Base circle Cb is the intersection of sphere Q and base cone L. At the early stage of bevel gear design, base cone angle (cone angle of cone L) is specified based on pitch and pressure angles [1]; pitch angle itself is directly calculated based on gear ratio [1]. Point P lies on sphere surface while it unwraps, and because it unwraps from base circle Cb, arc length of the great circle
42833.jpg is equal to the arc length of base circle Cb, which is 42849.jpg, therefore:

42208.jpg Equation 1

and hence,

42219.jpg Equation 2

Here, q is roll angle and assuming S=q+j in Equation 2,

42231.jpg Equation 3

In spherical trigonometry, arcs are represented with their angles [31], therefore, arcs 42853.jpg, 42869.jpg, and 42878.jpg are represented by their angles t, b, and h, respectively. Considering the right angle spherical triangle DOPB, spherical law of sine is:

42245.jpg Equation 4


42258.jpg Equation 5

42271.jpg Equation 6

And from spherical law of cosines:

42283.jpg Equation 7

42296.jpg Equation 8

42312.jpg Equation 9

By replacing sin h and cos h respectively from Equations 5 and 7 into Equation 8:

42322.jpg Equation 10

and rearranging Equation 10,

42330.jpg Equation 11

and substituting t from Equation 3 into Equation 11,

42342.jpg Equation 12

and from S=q+j and Equation 12,

42356.jpg Equation 13

2b is usually known and given as the base cone angle of the straight bevel gear, and q is calculated based on given j from Equation 13. This equation is also the exact spherical involute function of j and, by analogy to planar involute functions where q=tan j-j, is called the spherical involute function. Equation 13 is the first parametric equation of spherical involute. The second parametric equation is found through substituting S=q+j in Equation 3,

42368.jpg Equation 14

Replacing sin t from Equation 5 and cos t from Equation 7 into Equation 14,

42388.jpg Equation 15

Hence, for any given angle j, angle q is directly found from Equation 13, since b is known. Angles q and j are replaced in Equation 15 to find h. Another variation of this procedure with a forward approach is to find a unique j value for any given q through a solution of nonlinear Equation 13 and replacing j and q again in Equation 15. Latter approach is forward approach; however, it is mathematically easier to use j as given in Equation 13 and solve for roll angle q.

Referring to Figure 1 coordinates of any point P that is generated at roll angle q from base cone L at cone distance r and base cone angle 2b are (where T represents matrix transpose),

42403.jpg Equation 16

Note that Figure 1 is established only for one cross section of r=r, where r should vary across face width of the tooth to generate entire bevel gear surface.

To calculate normal n to the spherical involute surface at point P, coordinate systems x’y’z’ and x”y”z” with origins at the same point as coordinate system xyz are established. x’y’z’ coordinate system is established by rotation of coordinate system xyz around its z axis as much as -q such that arc 42996.jpg coincides with plane y’z’. Coordinate system x”y”z” is established by rotating coordinate system x’y’z’ around x’ axis as much as +h such that line CP coincides with z” axis. With this in spherical triangle DOPB if tangent to arc 42999.jpg (shown as n in Figure 1a) at point P is rotated around z” axis for angle +J, it results in normal to the curve G (shown as n in Figure 1a). In coordinate system x”y”z”, unit tangent vector to arc 43730.jpg at point P is n=[0 -1 0] T (superscript T means transpose), therefore, n in x”y”z” is:

42414.jpg Equation 17

hence, unit normal vector n in xyz coordinate system is: 42431.jpg

Equation 18

or, after simplification,

42446.jpg Equation 19

From Equations 16 and 19, coordinates and unit normal vectors of any point P on the spherical involute surface is calculated based on two independent surface parameters of r and j.

Points of spherical involute surface lie between the gear base cone with cone angle db=2b and the face cone with cone angle df between the inner cone ri and outer cone ro all shown in Figure 2. For any given point P shown in Figure 2 located at (Rp, Lp), coordinates are calculated through a solution to the following set of equations. From this, normal to the surface at this point can also be calculated.

42454.jpg Equation 20


Any point P located at (Rp, Lp) on the bevel gear flank has a unique set of associated (r, j) values that relate them by Equations 13-16; for each point P, there also exists a unique unit normal n. Actual surfaces should be measured against its intended theoretical surface. The theoretical surface can be spherical involute, any modifications to it, or other bevel gear surface profiles used by different manufacturers depending on their manufacturing process. An area within the tooth borders needs to be specified for measurement against theoretical surface. Figure 3 shows an example of a bevel pinion tooth border with dashed line area specified for measurement. This area can be meshed with a certain number of rows and columns of points, and at each point, coordinates and unit normal vectors are calculated. A file containing coordinates and normals for both sides of the tooth is generated and supplied to a CMM to measure the actual parts against this theoretical file. The CMM approaches each point along the unit normal direction and reports the error at that point against the theoretical surface. It should be noted that CMMs measure an offset of the given theoretical surface along local normals for as much as the CMM probe radius. Usually, the deviation at the middle point of the grid (O’ in Figure 3) is set to zero and deviations of the other points are calculated and reported individually with respect to the middle point. The tooth thickness is specified with a difference angle a that represents the angle between two flanks at a specific R and L at the flank (usually at R and L associated with the middle point). Figure 4 shows a sample CMM measurement chart for a gear set against its intended theoretical surfaces. All the numbers in this chart are in microns; LFl refers to left flank and RFl refers to right flank. Here, the theoretical surfaces are exact spherical involutes. It is common to deviate from the exact spherical involute in the design stage to optimize contact pattern and robustness of designs with respect to their application [9, 24]. The charts show the difference between theoretical and measured surfaces at each grid point along the normal direction (if the error at the middle point is set to zero). The difference angle a between the two sides of the tooth is also measured and reported. Depending on the number of teeth on the gear, usually three to four teeth are measured and the results are averaged.

Correction of Theoretical Surfaces to Represent Actual Measured Surfaces

The proposed approach in this paper is to capture the surface deviations of both flanks of both members in the form of a two-dimensional third-order polynomial. To simplify the representations of the deviation surfaces, coordinate system XYZ (of Figure 3) is defined. In this coordinate system, Z is in the normal to the surface direction, X is along lengthwise (pointing toward heel), and Y is along profile direction (pointing toward tip). The origin O of the XYZ coordinate is located at the middle of the face width and along the pitch line (as shown in Figure 3) and is calculated as:

42470.jpg Equation 21

where ro is gear outer cone distance, Fw is gear face width, and dp is gear pitch angle. With this, the difference between actual 43586.jpg and the theoretical 43598.jpg surfaces can be represented up to third order as:

42487.jpg Equation 22

independently for each side of the tooth for each member (pinion or gear). After measuring the tooth, the goal is to modify the theoretical surface so that it matches with the actual measured surface. This is done by finding a set of ak(k=1,…,10) of Equation 22 that best describes the measured error surface. The tooth thickness can also be corrected by modifying a angle. Coefficient a1 can also be used to correct tooth thickness, but it is in direction normal to the tooth surface, as opposed to a that changes tooth thickness in tangential direction. Having the ak(k=1,…,10), a, and the theoretical spherical involute surface (or modified spherical involute as is the case in this study), the actual measured surface can be obtained by superimposing Z values on 43626.jpg as:

42498.jpg Equation 23

By describing measured surface deviations through polynomial coefficients ak(k=1,…,10) as opposed to discrete grids, the actual tooth surface will be described as a continuous function, and hence, it will be easier to conduct tooth contact analysis between the pinion and gear. The contact points between the pinion and gear surfaces usually do not fall exactly at the measured points, and interpolations are needed to estimate deviations at contact points.

The next focus is on finding ak(k=1,…,10) coefficients that best describe the measured surface. Assuming the ak(k=1,…,10) are known, for every measured point XijYijZij(i=1, I; j=1, J) of actual surface 43589.jpg, the estimated deviation Z’ij by the polynomial fit is:

42522.jpg Equation 24

Because the actual measured value of deviation is Zij, the amount of error of estimation is:

42536.jpg Equation 25

In order to minimize the sum of the squared errors (SSQ) with respect to polynomial coefficients ak(k=1,…,10), the below function needs to be minimized:

42547.jpg Equation 26

to minimize the value of SSQ with respect to ak(k=1,…,10) :

42566.jpg Equation 27

Equation 27 is a set of 10 linear equations with 10 unknowns of ak(k=1,…,10), which will minimize the sum of the squared of errors of fitting polynomial function in the form of Equation 22 to the measured surface. Zero-, first-, and second-order coefficients of ak(k=1,…,10) have specific names as a1 (thickness error), a2 (spiral angle error), a3 (pressure angle error), a4(lengthwise crowning), a5 (bias error), and a6 (profile crowning). Higher order coefficients of a7, a8, a9, and a10 do not have specific names, however, there are many designs that carry modifications that can be only captured using these coefficients. The measurement is done only once and ak(k=1,…,10) for the measurement is calculated using the approach previously described. The ak(k=1,…,10) can be stored and superimposed on the basic spherical involute surface (or modified) to represent the actual surface; the resultant surface, however, is not always exactly matching the actual surface and few more iterations (using obtained values from the previous iterations) are needed to minimize the difference. At each iteration step, the sum of squared of errors SSQ represents the closeness of theoretical surface to the measured surface. The goal here is to come up with a set of ak(k=1,…,10) that represents the actual surface through changing the theoretical surface and the main criterion is SSQ, where SSQ = 0 shows the actual and theoretical surfaces are identical. Usually in real applications, SSQ cannot be minimized to zero utilizing third-order polynomial functions; a residual value always remains. Figure 4 shows a sample measurement of actual parts against exact spherical involute. In practical design, however, designers usually apply certain first-, second-, and sometimes third-order modifications to accommodate different application requirements such as loading, misalignments, housing deflections, etc. The modifications can be described in the same manner as deviations since both have the same nature; modifications are intended while deviations are not. For a typical design, micro-geometry modifications from spherical involute can be introduced through 43303.jpg and 43313.jpg respectively for pinion and gear.

Unloaded Contact Pattern and Transmission Error

In unloaded tooth contact analysis (UTCA) of straight bevel gears, the goal is to calculate (i) the contact point path (CPP) and the zone on each of the tooth surfaces that are separated by the specified separation distance d and (ii) the function of transmission error between two gear axes. In this study, UTCA is calculated through ease-off approach, which has several advantages over the conventional approach used in references [27, 34]. The list of the advantages is provided in Kolivand’s et al and Artoni’s et al works [23, 32, 33]. In this study, ease-off is defined as deviations of real gear surface from the conjugate of its real mating pinion surface [23]. The following section briefly explains how ease-off is constructed. The details of the approach are specified in references [22, 23, 32].

Having point 43323.jpg and its normal 43333.jpg of pinion spherical involute surface Sp (with pinion axis ap) calculated, its associated action point on gear spherical involute Sg, 43349.jpg (with gear axis ag) is calculated by applying fundamental equation of meshing as [22]:

42580.jpg Equation 28

Here, 43360.jpg is action point of 43371.jpg, R=N2/N1 is gear ratio, N1 and N2 are pinion and gear number of teeth respectively, and i and j [1,J] are indices in lengthwise and profile directions of tooth surface with maximum of I and J respectively. In Equation 28, for every point on pinion surface 43376.jpg, respective point on action surface 43383.jpg is calculated through a solution of one nonlinear trigonometric (Equation 28) and one unknown (roll angle 43507.jpg), which is the required angle traveled around axis ap by point 43424.jpg to match condition of Equation 28 for point 43429.jpg. Each point on pinion surface Sp needs certain roll angle travel 43509.jpg to match with its respective action point 43472.jpg. By analogy to roll angle term of cylindrical gears, this travel angle distribution on Sp can be shown as a surface of roll angle y that can be used to locate instantaneous contact lines on the tooth surface [33].

The angle 43523.jpg corresponds to the amount of rotation from the surface of action to reach the conjugate of theoretical pinion surface. Therefore:

42600.jpg Equation 29

Where tz(z) is rotation matrix around z axis by angle z, it’s defined as:

42615.jpg Equation 30

If this conjugate surface of the pinion were to match perfectly with the real gear surface at any point, then a perfect meshing condition with zero unloaded transmission error would be reached. The difference between these two surfaces (conjugate of theoretical pinion and theoretical gear surfaces) is defined as theoretical ease-off (E) topography, i.e.:

42625.jpg Equation 31

where 43544.jpg form the ease-off surface.

Having ease-off surface and surface of roll angle y in hand, UTCA can be conducted as follows. For a specific pinion roll angle hi, intersection of the plane z =hi and the y surface defines x and y coordinates of all points that have the same roll angle, stating theoretically that they lie on the same contact line C(hi). If the minimum ease-off on C(hi) happens at point H, this minimum ease-off divided by distance of the point H(hi) to gear axis is instantaneous unloaded transmission error TE(hi). Across each instantaneous contact line C(hi), the ease-off value is determined from ease-off surface E. Moving in both directions from point H(hi) along C(hi) within a preset separation distance d gives the unloaded contact line length S(hi). Repeating this procedure for every pinion, roll angle increment, unloaded transmission error curve TE(h), and the unloaded tooth contact pattern are computed. This approach is explained in great detail in works by Kolivand and Kahraman [23].

Pinion Gear
Number of teeth 8 13
Shaft angle (deg) 90.0˚
Inner cone distance (mm) 27.0
Outer cone distance (mm) 44.0
Pitch cone angle (deg) 31.61˚ 58.39˚
Base cone angle (deg) 28.61˚ 51.08˚
Face cone angle (deg) 42.90˚ 64.80˚
Root cone angle (deg) 25.20˚ 47.10˚
Outer pitch diameter (mm) 46.12 74.94
Pressure angle (deg) 24.0˚

Table 1: Example 8×13, 24˚ pressure angle automotive straight bevel gear

Numerical Example and Experimental Setup

A sample automotive differential gear set with basic dimensions as shown in Table 1 is used for numerical example and experimental setup. For base design of Table 1, micro-geometry modifications are defined in the form of Table 1 — to localize contact pattern and allow a level of robustness to inevitable misalignments. These designed modifications are superposition of several orders of corrections for pinion and only lead correction (a3) for gear. Figure 5 shows the shape of modifications for pinion and gear associated with coefficients introduced in Table 2. Figure 6 shows TCA results including a) ease-off, b) single tooth contact, c) multiple tooth contact and d) transmission error computed based on these defined modification coefficients. With this designed micro-geometry, contact pattern is placed at about center of the tooth with a contact width of 25-30 percent of effective face width.

a1 0 0
a2 25 0
a3 0 35
a4 70 0
a5 0 0
a6 55 0
a7 0 0
a8 20 0
a9 -20 0
a10 0 0

Table 2: Designed micro-geometry modifications for gear set of Table 1

The sample parts are then measured against these modified theoretical surfaces that were defined based on Table 2. The measurement results in Figure 7 show that the actual part differs from their intended designs. However, they can be represented by third-order polynomial coefficients if the proposed approach in the previous section is utilized. Also, there is tooth thickness difference between intended design and actual measurements shown by parameter a in Table 3.

Pinion Gear
a1 -43 0
a2 -72 21
a3 126 58
a4 18 -1
a5 28 29
a6 -86 -83
a7 18 17
a8 -19 -25
a9 11 102
a10 77 -56
a -0.9895 +0.3221

Table 3 (left): Polynomial coefficients of initial micro-geometry measurement for gear set defined
by Tables 1 and 2

Step Pinion Gear
1 6066 2019
2 1080 52
3 159 30
4 104 30
5 51 30
6 63 30
7 58 30

Table 4 (above): Sum of squared of errors (SSQ) for pinion and gear at different iteration steps

Applying the proposed approach, deviation coefficients of 43673.jpg and 43684.jpg are calculated as presented in Table 3 to capture the difference between intended and actual micro-geometry. These new sets of coefficients can be superimposed (simply added here) to design coefficients of 43651.jpg and 43660.jpg introduced in Table 2 to represent the actual part’s micro-geometries. The first iteration here results in SSQ1p = 6066 and SSQ1g = 2019 (units here are in tenth of thousands of inch squared for a 9×5 grid resolution) respectively for pinion and gear. Table 4 shows different iteration steps and associated SSQ values. Referring to Table 4, it can be concluded that Step 5 for pinion and Step 3 for gear and associated computed coefficients are as close as possible to the real surfaces of the parts, since beyond these steps, no further reduction of SSQ is achieved. The criterion to stop the iteration process is to reach a step where SSQ does not get reduced any further. Less than 10 steps are usually enough to reach the minimum SSQ.

At each iteration step, the new design micro-geometry coefficients are then calculated by adding computed deviation coefficients to previous design coefficients as:

42641.jpg Equation 32

a1 -54 -12
a2 -59 15
a3 103 87
a4 85 -2
a5 45 25
a6 -31 -81
a7 16 16
a8 -2 -25
a9 -18 96
a10 61 -45

Table 5: Final micro-geometry modifications of measured/actual pinion and gear that associate with minimum SSQ reported in Table 4

Using this approach, the final design coefficients for the current example is shown in Table 5. The difference between theoretical surface of gear set defined by Table 1 and Table 5, and actual measured part is shown in Figure 8, which is residual error in capturing the actual geometry and is practically negligible. Maximum error occurs at toe-tip corner of the gear and is as low as 4 microns with SSQp = 51 and SSQg = 30, which is small. It can be concluded that the geometry of real parts is captured accurately for the purpose of TCA analysis. It should be noted that each part is measured only once and that the rest of the presented CMM charts are based on that single measurement.

In preparation of CMM data file and measurement procedure, one should measure the maximum available area on the tooth, however, caution should be taken to avoid running over tip fillet area, root fillet radius, and any other edge features. Otherwise, sharp changes introduced by these local features will influence the entire surface recognition coefficients. Usually, three or more teeth are measured, averaged, and reported as measurement results. Variation of contact pattern on different tooth pairs depends on actual topography of each individual tooth and their variation from tooth to tooth. The idea of using average teeth topography versus each individual tooth owes its accuracy to the amount of this variation, which is usually evaluated by metric of gear quality. So it should be stipulated that these coefficients in Table 5 are for average tooth and its accuracy depends on consistency of the tooth topography. Figure 9 shows TCA for gear set with captured actual micro-geometry of Table 5 using separation value of d=6.3 micron. This TCA is comparable with actual TCA one expects to see on roll tester.

Roll tester stand of Figure 10(a) with perpendicular and intersecting axes is used to roll the sample gear set. Gauges of Figure 10(b) are made to gauge pinion and gear relative positions to assure the pinion and gear are located at correct relative positions. Each gauge has two ground surfaces (references or touching surfaces), which, if set in tangency, assures correct pinion and gear relative positions. In actual differential applications, however, there are always mounting errors due to housing manufacturing errors and deflection of the housing under load and thermal expansion/contractions; those errors are eliminated here by using gauges made specific for the sample gear set used in this paper. Figure 11 shows contact pattern on a) pinion and b) gear flanks. Comparison between predicted contact pattern of Figure 9 and actual roll tester pattern of Figure 11 shows good correlations in terms of contact location, size, and shape that proves the effectiveness of the proposed approach.

When each individual tooth to tooth contacts are compared (depending on gear quality), sometimes obvious variations can be observed. Therefore, if the gear and pinion are rolled together for many revolutions, the resulted contact pattern is envelope of all individual patterns, which is usually larger than each individual tooth to tooth pattern. In this study, the pinion and gear are rolled manually by hand for partial of full revolution to avoid such overlap and to have rather individual tooth to tooth patterns, and because tested gear set had rather small variations, minimal pattern variations were observed. Figure 12 shows contact pattern on gear tooth for several consecutive teeth with rather minor variation, which supports accuracy of using average tooth for the analysis; however, this will only work when gear quality is to the level that average tooth topography is representative of the individual tooth. In this example, both gear and pinion samples have AGMA quality 10 or higher [35].


Spherical involute geometry including surface coordinates and normal to the surface formulas have been developed and used as a baseline for straight bevel gears. CMM measurement results of physical parts are used to capture actual part deviations from their intended design topographies. The measured deviations from spherical involute are then expressed in the form of a third-order two-dimensional polynomial function and added to the base topography to duplicate the geometry of the actual part. Tooth thickness deviation is also accounted for and corrected through changing the theoretical tooth thickness.

The resultant surfaces are then used to construct ease-off and surface of roll angle topographies and to perform tooth contact analysis (TCA) and calculate motion transmission error (TE) [23]. A sample straight bevel gear set was measured, and utilizing the proposed approach, its predicted TCA was compared to the experimental TCA obtained from bench roll tester. The results show good correlation between the predicted and actual TCA of the parts. Utilizing the proposed methodology, the other bevel gear base profile geometries such as octoids can be analyzed as well. At a minimum, this analysis can be used:

  1. To capture each individual bevel gear tooth topography
    • To digitize master gears
    • To use the captured geometry for loaded analysis
  2. To predict tooth contact pattern of bevel gears using their CMM results, which is especially useful when the required tooling to run on roll tester is not available.


  1. ANSI/AGMA Standard, 2005, “2005-D03 Design Manual for Bevel Gears.”
  2. 2003-C10, A. / A., “932-A05 Rating the Pitting Resistance and Bending Strength of Hypoid Gears.”
  3. AGMA, 2010, “2003-C10 A. / A. Rating the Pitting Resistance and Bending Strength of Generated Straight Bevel, Zerol and Spiral Bevel Teeth.”
  4. Buckingham, E., 2011, Analytical Mechanics of Gears, Dover Publications, Springfield, Vt.; New York.
  5. Townsend, D. P., Dudley’s Gear Handbook.
  6. Gleason, W., 1876, “Improvement in machines for cutting the teeth of metal gears.”
  7. Stadtfeld, H. J., 2007, “Straight Bevel Gear Cutting and Grinding on CNC Free Form Machines,” 07FTM16, AGMA Fall Technical Meeting, Detroit, Michigan.
  8. Meidert, M., Knoerr, M., Westphal, K., and Altan, T., 1992, “Numerical and physical modelling of cold forging of bevel gears,” J. Mater. Process. Technol., 33 (1-2), pp. 75-93.
  9. Ligata, H., and Zhang, H. H., 2011, “Overview and Design of Near-Net Formed Spherical Involute Straight Bevel Gears,” SOURCE Int. J. Mod. Eng., 11 (2), pp. 47-53.
  10. Grant, G. B., 1891, Odontics, or, The theory and practice of the teeth of gears, Lexington, Mass., Lexington Gear Works.
  11. Wildhaber, E., 1945, “Relationships of Bevel Gears,” Am. Mach., 89 (20, 21, 22), pp. 99-102, 118-121, 122-125.
  12. Wildhaber, E., 1956, “Surface Curvature,” Prod. Eng., 27 (5), pp. 184-191.
  13. Al-Daccak, M. J., Angeles, J., and González-Palacios, M. A., 1994, “The Modeling of Bevel Gears Using the Exact Spherical Involute,” J. Mech. Des., 116 (2), pp. 364-368.
  14. Figliolini, G., and Angeles, J., 2004, “Algorithms for Involute and Octoidal Bevel-Gear Generation,” J. Mech. Des., 127 (4), pp. 664-672.
  15. Kolivand, A., 2014, “Involute Straight Bevel Gear Surface and Contact Lines Calculation Utilizing Ease-Off Topography Approach,” SAE International, Warrendale, Pennsylvania.
  16. Litvin, F. L., and Gutman, Y., 1981, “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,” J. Mech. Des., 103 (1), pp. 83-88.
  17. Litvin, F. L., and Gutman, Y., 1981, “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,” J. Mech. Des., 103 (1), pp. 89-101.
  18. Litvin, F. L., and Gutman, Y., 1981, “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,'” J. Mech. Des., 103 (1), pp. 102-110.
  19. Fan, Q., 2006, “Enhanced Algorithms of Contact Simulation for Hypoid Gear Drives Produced by Face-Milling and Face-Hobbing Processes,” J. Mech. Des., 129 (1), pp. 31-37.
  20. Simon, V., “Tooth Contact Analysis of Mismatched Hypoid Gears,” Int. Power Transmission and Gearing Conference, San Diego.
  21. Wang, X. C., and Ghosh, S. K., 1994, Advanced Theories of Hypoid Gears/Book and Disk, Elsevier Science Ltd, Amsterdam; New York.
  22. Stadtfeld, H. J., Handbook of Bevel and Hypoid Gears: Calculation, Manufacturing, Optimization.
  23. Kolivand, M., and Kahraman, A., 2009, “A load distribution model for hypoid gears using ease-off topography and shell theory,” Mech. Mach. Theory, (10), pp. 1848-1865.
  24. Ligata, H., and Zhang, H. H., 2012, “Geometry Definition and Contact Analysis of Spherical Involute Straight Bevel Gears,” Int. J. Ind. Eng. Prod. Res., 23 (2), pp. 101-111.
  25. Tsuji, I., Kawasaki, K., Gunbara, H., Houjoh, H., and Matsumura, S., 2013, “Tooth Contact Analysis and Manufacture on Multitasking Machine of Large-Sized Straight Bevel Gears With Equi-Depth Teeth,” J. Mech. Des., 135 (3), pp. 034504-034504.
  26. Gosselin, C., Cloutier, L., and Brousseau, J., 1991, “Tooth Contact Analysis of High Conformity Spiral Bevel Gears,” Proceedings of JSME Int. Conf. on Motion and Power Transmission, Hiroshima, Japan, pp. 725-730.
  27. Zhang, Y., Litvin, F. L., Maruyama, N., Takeda, R., and Sugimoto, M., 1994, “Computerized Analysis of Meshing and Contact of Gear Real Tooth Surfaces,” J. Mech. Des., 116 (3), pp. 677-682.
  28. Kin, V., 1992, “Tooth Contact Analysis Based on Inspection,” Proceedings of 3rd World Congress on Gearing.
  29. Kin, V., 1994, “Computerized Analysis of Gear Meshing Based on Coordinate Measurement Data,” J. Mech. Des., 116 (3), pp. 738-744.
  30. Marambedu, K. R., 2009, “Development of a Procedure for the Analysis of Load Distribution, Stresses and Transmission Error of Straight Bevel Gears,” The Ohio State University.
  31. Donnay, J. D. H., 1945, Spherical Trigonometry, Interscience Publishers Inc.
  32. Artoni, A., Kolivand, M., and Kahraman, A., 2009, “An Ease-Off Based Optimization of the Loaded Transmission Error of Hypoid Gears,” J. Mech. Des., 132 (1), pp. 011010-011010.
  33. Kolivand, M., and Kahraman, A., 2011, “A General Approach to Locate Instantaneous Contact Lines of Gears Using Surface of Roll Angle,” J. Mech. Des., 133 (1), pp. 014503-014503.
  34. Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory, Cambridge University Press.
  35. AGMA, 1988, “AGMA 390.03A:1980 (R1988) Handbook – Gear Classification, Materials And Measuring Methods For Bevel, Hypoid, Fine Pitch Wormgearing and Racks.”


  • ak, bk = modification/deviation coefficients (k = 1, 10)
  • ag = gear axis vector
  • ap = pinion axis vector
  • E = ease-off surface
  • eij = deviation at point ij
  • Fw = face width
  • nij = unit normal vector at point ij
  • N1 = number of pinion teeth
  • N2 = number of gear teeth
  • p, g = superscript for pinion (p) and gear (g)
  • r = cone distance
  • ri = inner cone distance
  • ro = outer cone distance
  • rij = coordinate of point ij of spherical involute surface
  • 42703.jpg= coordinate of point ij of conjugate surface R = gear ratio
  • 42713.jpg= theoretical surface topography
  • 42722.jpg= actual surface topography
  • SSQ = sum of the squared of deviations, in tenth of thousands squared
  • a = difference angle between two flanks at mid-point of measurement grid
  • b = half of base cone angle
  • d = separation distance
  • db = base cone angle
  • df = face cone angle
  • dp = pitch angle
  • j = pressure angle
  • q = roll angle
  • y = surface of roll angle
  • yij = roll angle value of point ij on surface of roll angle
Previous articleTooling Profile: Riten Industries
Next articleCompany Profile: Blaser Swisslube Inc.
is a bevel gear differential technical manager with AAM in Detroit. He received his Ph.D. from University of Illinois at Chicago in 1993 with a focus on developing a model for face-mill hypoid gear contact and strength analysis. He has more than 20 years’ experience working on bevel and hypoid gear design and analysis.
is a senior product engineer with AAM in Detroit. He holds a bachelor’s degree in mechanical engineering from Cornell University. He has over 35 years’ experience in net-shaped forged bevel gear design and manufacturing process development.
is the executive director of product validation and NVH for AAM in Detroit. He received his Ph.D. from The Ohio State University in 1983 in mechanical engineering. He has 38 years’ experience in gear system design for low noise.
is a mechanical engineer at GE Global Research Center (Schenectady, New York), working on the design, analysis, and testing of the rotating components for the power generation, transportation, oil and gas, and aerospace industries. Prior to General Electric, he spent five years working for AAM in Detroit on the design and development of helical and straight bevel (differential) gears. He received his Ph.D. from The Ohio State University (Gear Lab) in 2007.
is the gear design and analysis manager with American Axle & Manufacturing (AAM) in Detroit, Michigan. He received his Ph.D. from Ohio State University in 2009 with a focus on developing a model for hypoid gear contact and efficiency analysis. He has more than 10 years’ experience working on bevel and hypoid gear design, analysis, inspection, and manufacturing.