## Abstract

Theoretically, spherical involutes are used as one of the base topographies for straight bevel gears. Actual bevel gears, however, have deviations from their intended topographies due to manufacturing errors, heat treatment deviations, and finishing processes. Measuring the physical parts with coordinate measuring machines (CMMs), this study proposes a new approach to capture such deviations. The measured deviations from spherical involute are expressed in form of a third-order two-dimensional (2D) 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). At the end a sample straight bevel gear set is measured and utilizing the proposed approach its predicted TCA is compared to the experimental TCA obtained from roll tester. The results show very 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 also be analyzed. In the proposed approach, the difference between other base geometries and spherical involutes can be treated as deviations from spherical involutes and can be taken into account to perform TCA.

## Introduction

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 constructions [2–5] as well. They have wide range of shaft angles especially in marine applications; however, right angle is the most common one.

Gleason utilized a planer with indexing head [6] to introduce an efficient straight bevel gear manufacturing method of its time. Development of more efficient manufacturing methods for straight bevel gears was followed by introduction of Revacycle [5] and Coniflex [7] cutting methods. In the last two decades, forging has emerged as one more efficient and economical way of straight bevel gear manufacturing [8]. Due to forging limitations in tonnage and accuracy, this trend aims more at 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 some material and machining cost and also improve strength due to improved grain flow [5] in the material. Using forging as manufacturing method will also give the possibility of having web feature between the teeth [9].

Due to ease of manufacture and insensitivity to change of center distance, involute curves are dominant in cylindrical gears. By the same analogy, spherical involute would bring 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 mainly a legacy of the time, when forging manufacturers were trying to duplicate cut gears to prove these gears can also be near-net forged. Other than some fundamental definitions [10], most of the basic formulae of today's cut straight bevel gear geometry were established by Wildhaber [11–14]. Al-Daccak et al. [15] proposed a method for spherical involute geometry calculation; Figliolini and Angeles [16] also introduced a calculation approach for both spherical involute and octoidal bevel gears. The purpose of both the studies was to calculate surface coordinates and tooth modeling while Kolivand [17] added required formulae to compute normal to the surface. The normal vectors to the surfaces are needed to construct ease-off, conduct TCA, and measure the tooth topography using CMMs; all aim at proving quality and performance of designs, prototypes, and production parts.

Quality is usually defined/measured through measurements of pitch line run-out and tooth thickness variation. Performance characteristics are usually evaluated through durability, strength, and noise level analyses/tests; among all, 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 [18–23]. The instantaneous contact on each surface is determined by satisfying two contact conditions of (a) coincidence of position vector and (b) colinearity of the normals. The second method is based on the ease-off topography [24,25]. In one such study, Kolivand and Kahraman [25] 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 Ref. [25]. 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 system of nonlinear equations [21,22,26,27]. 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 TCA, which use the “real” surfaces. Here, the term *real* refers to the actual surfaces of the parts that can be measured on CMM machines. In one such study, Gosselin et al. [28] proposed an approach to compute tooth contact of spiral bevel gear surfaces having deviations. They interpolated the measured surfaces with rational functions to predict their unloaded contact pattern and TE. Zhang et al. [29] proposed an approach to analyze unloaded tooth contact of real hypoid gears based on generalization of the work by Kin [30,31] 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 the above mentioned studies have been utilized in spherical involute bevel gears and none also were validated through roll testing of actual parts. This study aims at addressing both the shortcomings. Here, after CMM measurements, the surface deviations are expressed in the form of continuous third-order two-dimensional (2D) polynomials, 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 in Sec. 4. The fact that the interpolation is only done on surface deviations (and not on the surface itself) brings high level of accuracy and efficiency to this approach. Spherical involute and deviation coefficients computations are required prior to TCA.

Surface coordinates and normals of spherical involute updated with deviation surfaces are then used to establish ease-off topography and surface of roll angle [25]. An example of 8 × 13, 24 deg pressure angle automotive straight bevel gear set is analyzed to show details of the developed approach as well as its efficiency and accuracy. The developed approach is unique in developing ease-off and surface of roll angle based on actual measured surfaces and conducting TCA using the developed ease-off. To reach this, the overall goal-specific objectives of this study are as follow:

- (i)
to compute bevel gear geometry based on spherical involute including surface coordinates and normal to the surface,

- (ii)
to develop a simple and effective third-order 2D least square-based approach to capture and represent surface deviations of actual measured bevel gears,

- (iii)
to perform an ease-off based TCA to compute contact pattern and TE, and

- (iv)
to 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 (3D) equivalent of the established 2D involute curve commonly used in cylindrical gears [10,15–17,26]. Some studies defined spherical involute by planar involute through gradually changing base circle radius of a planar involute from inner to outer cone section of bevel teeth [32]. In this study, a similar approach to Al-Daccak et al. [15] is used to establish spherical involute. However, normal to the surface is computed based on approach proposed by Kolivand [17].

In Figs. 1(a) and 1(b), spherical involute is defined as a 3D curve $\Gamma $ traced by a point $P$ on a taut chord $BP\u2322$ unwrapping from base circle $Cb$ that lies on sphere $\Theta $ with origin at $C$ and radius $\rho $. Base circle $Cb$ is an intersection of sphere $\Theta $ and base cone $\Lambda $. At the early stage of bevel gear design, base cone angle (cone angle of cone $\Lambda $) 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 unwraps and since it unwraps from base circle $Cb$, arc length of the great circle $BP\u2322$ is equal to the arc length of base circle $Cb$, which is $MB\u2322$, therefore

Hence, for any given angle $\varphi $, angle $\theta $ is directly found from Eq. (13), since $\beta $ is known. Angles $\theta $ and $\varphi $ are replaced in Eq. (15) to find $\eta $. Another variation of this procedure with forward approach is to find a unique $\varphi $ value for any given $\theta $ through solution of nonlinear Eq. (13) and replacing $\varphi $ and $\theta $ again in Eq. (15). The latter approach is the forward approach; however, it is mathematically easier to use $\varphi $ as given in Eq. (13) and solve for roll angle $\theta $.

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

From Eqs. (16) and (19), coordinates and unit normal vectors of any point $P$ on spherical involute surface are calculated based on two independent surface parameters of $r$ and $\varphi $.

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

## Inspection

Any point $P$ located at $(Rp,Lp)$ on the bevel gear flank has a unique set of associated $(r,\varphi )$ values that relate them by Eqs. (13)–(16); for each point $P$ also there 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 bevel pinion tooth border with dashed-line area specified for measurement. This area can be meshed with certain number of rows and columns of points, 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 machine to measure the actual parts against this theoretical file. The CMM machine 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 CMM probe radius. Usually the deviation at the middle point of the grid ($O'$ in Fig. 3) is set to zero and deviations of the other points are calculated and reported individually with respect to middle point. The tooth thickness is specified with a difference angle $\alpha $ 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” stands for *left flank* and “RFl” stands for *right flank*. Here, the theoretical surfaces are exact spherical involutes. It is very common to deviate from exact spherical involute in the design stage to optimize contact pattern and robustness of designs with respect to their application [9,26]. 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 $\alpha $ between two sides of 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

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 TCA between pinion and gear. The contact points between pinion and gear surfaces usually do not fall exactly at the measured points, and interpolations are needed to estimate the deviations at contact points.

Equation (27) is a set of ten linear equation with ten unknowns of $ak(k=1,...,10)$, which will minimize the sum of the squared of errors of fitting polynomial function in the form of Eq. (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 described above. 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 represent 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, then microgeometry modifications from spherical involute can be introduced through $bkp$ and $bkg(k=1,...,10)$, respectively, for pinion and gear.

## Unloaded Contact Pattern and TE

In UTCA of straight bevel gears, the goal is to calculate (i) the contact point path and the zone on each of tooth surfaces that are separated by the specified separation distance $\delta $ and (ii) the function of TE between two gear axes. In this study, UTCA is calculated through ease-off approach, which has several advantages over the conventional approach used in Refs. [29] and [34]. The list of the advantages is provided in the works by Kolivand and Kahraman [25,35] and Artoni et al. [36]. In this study, ease-off is defined as deviations of real gear surface from the conjugate of its real mating pinion surface [25]. This section briefly explains how ease-off is constructed. The details of the approach are specified in Refs. [24], [25], and [36].

Here, $raij$ is action point of $rpij$, $R=N2/N1$ is gear ratio, $N1$ and $N2$ are pinion and gear number of teeth, respectively, and $i\u2208[1,I]$ and $j\u2208[1,J]$ are indices in lengthwise and profile directions of tooth surface with maximum of $I$ and $J$, respectively. In Eq. (28), for every points on pinion surface $rpij$, respective point on action surface $rpij$ is calculated through solution of one nonlinear trigonometric equation (Eq. (28)) and one unknown (roll angle $\psi pij$), which is the required angle traveled around axis $ap$ by point $rpij$ to match condition of Eq. (28) for point $raij$. Each point on pinion surface $Sp$ needs certain roll angle travel $\psi pij$ to match with its respective action point $raij$. By analogy to roll angle term of cylindrical gears, this travel angle distribution on $Sp$ can be shown as a surface of roll angle $\psi $ that can be used to locate instantaneous contact lines on the tooth surface [35].

where $reij$, $i\u2208[1,I]$, $j\u2208[1,J]$ form the ease-off surface.

Having ease-off surface and surface of roll angle $\psi $ in hand, UTCA can be conducted as follows. For a specific pinion roll angle $\eta i$, intersection of the plane $z=-\eta i$ and the $\psi $ surface defines *x* and *y* coordinates of all points that have same roll angle, stating theoretically that they lie on the same contact line $C(\eta i)$. If the minimum ease-off on $C(\eta i)$ happens at point $H$; this minimum ease-off divided by distance of the point $H(\eta i)$ to gear axis is instantaneous unloaded TE $TE(\eta i)$. Across each instantaneous contact line $C(\eta i)$, the ease-off value is determined from ease-off surface $E$. Moving in both directions from point $H(\eta i)$ along $C(\eta i)$ within a preset separation distance $\delta $ gives the unloaded contact line length $S(\eta i)$. Repeating this procedure for every pinion, roll angle increment, unloaded TE curve $TE(\eta )$, and the unloaded tooth contact pattern are computed. This approach is explained in great details in works by Kolivand and Kahraman [25].

## Numerical Example and Experimental Setup

A sample automotive differential gear set, whose basic dimensions are as in Table 1, is used for numerical example and experimental setup. For base design of Table 1, microgeometry modifications are defined in form of Table 2 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) TE computed based on these defined modification coefficients. With this, the designed microgeometry contact pattern is placed at about center of the tooth with a contact width of %25 -%30 of effective face width.

The sample parts are then measured against these modified theoretical surfaces that were defined based on Table 2. The measurement results are as shown in Fig. 7 that shows the actual part differs from their intended designs. However, they can be represented by third-order polynomial coefficients if the approach proposed in Sec. 4 is utilized. There also exists tooth thickness difference between intended design and actual measurements that are shown by parameter $\alpha $ in Table 3.

Applying the proposed approach of Sec. 4, deviation coefficients of $ak2p$ and $ak2g(k=1,...,10)$ are calculated as presented in Table 3 to capture the difference between intended and actual microgeometry. These new sets of coefficients can be superimposed (simply added here) to design coefficients of $bk1p$ and $bk1g(k=1,...,10)$ introduced in Table 2 to represent the actual parts microgeometries. 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 ten steps are usually enough to reach the minimum $SSQ$.

Using this approach, the final design coefficients for current example are as shown in Table 5. The difference between theoretical surfaces of gear set is defined in Tables 1 and 5, and the actual measured part is as shown in Fig. 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 *μ*m with $SSQp=51$ and $SSQg=30$, which practically is very small, and 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 the rest of the presented CMM charts are based on that single measurement.

In preparation of CMM data file and measurement procedure, one should try to measure 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 microgeometry of Table 5 using separation value of $\delta =6.3\mu m$. This TCA is comparable with actual TCA one expects to see on roll tester.

Roll tester stand of Fig. 10(a) with perpendicular and intersecting axes is used to roll the sample gear set. Gauges of Fig. 10(b) are made to gauge pinion and gear relative positions to assure pinion and gear are located at correct relative positions. Each gauge has two ground surfaces (references/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 Fig. 9 and actual roll tester pattern of Fig. 11 shows good correlations in terms of contact location, size, and shape, which proves the effectiveness of the proposed approach.

When each individual tooth-to-tooth contact is compared (depending on gear quality) sometimes obvious variations can be observed. Therefore, if 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, 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 since 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 that 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 individual tooth. Here, in this example, both the gear and pinion samples have AGMA quality ten or higher [37].

## Conclusion

Spherical involute geometry including surface coordinates and normal to the surface formulae have been developed and used as 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 2D 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 TCA and calculate motion TE [25]. A sample straight bevel gear set was measured utilizing the proposed approach, and its predicted TCA was compared to the experimental TCA obtained from the bench roll tester. The results show very 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 also analyzed. At minimum, this analysis can be used to:

- (i)
Capture each individual bevel gear tooth topography,

- (a)
digitize master gears,

- (b)
use the captured geometry for loaded analysis, and

- (a)
- (ii)
Predict tooth contact pattern of bevel gears using their CMM results, which is useful especially when the required tooling to run on roll tester is not available.

## References

## Nomenclature

- $ag$ =
gear axis vector

- $ap$ =
pinion axis vector

- $ak,bk$ =
modification/deviation coefficients $(k=1,10)$

- $eij$ =
deviation at point

*ij*- $E$ =
ease-off surface

- $Fw$ =
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

- $R$ =
gear ratio

- $\u211c$ =
theoretical surface topography

- $\u2003\xaf\u211c$ =
actual surface topography

- $ri$ =
inner cone distance

- $ro$ =
outer cone distance

- $rij$ =
coordinate of point

*ij*of spherical involute surface- $r\xafij$ =
coordinate of point

*ij*of conjugate surface- $SSQ$ =
sum of the squared of deviations, in tenth of thousands squared

- $\alpha $ =
difference angle between two flanks at midpoint of measurement grid

- $\beta $ =
half of base cone angle

- $\delta $ =
separation distance

- $\delta b$ =
base cone angle

- $\delta f$ =
face cone angle

- $\delta p$ =
pitch angle

- $\varphi $ =
pressure angle

- $\psi $ =
surface of roll angle

- $\psi ij$ =
roll angle value of point

*ij*on surface of roll angle- $\theta $ =
roll angle