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 [25] 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 [1114]. 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 [1823]. 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 [10,1517,26]. 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 Γ traced by a point P on a taut chord BP unwrapping from base circle Cb that lies on sphere Θ with origin at C and radius ρ. Base circle Cb is an intersection of sphere Θ and base cone Λ. At the early stage of bevel gear design, base cone angle (cone angle of cone Λ) 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 is equal to the arc length of base circle Cb, which is MB, therefore

Fig. 1
Spherical involute surface definitions
Fig. 1
Spherical involute surface definitions
Close modal
BP=MB
(1)
and hence
ρτ=Rb(θ+ϕ)=ρsinβ(θ+ϕ)
(2)
Here, θ is roll angle and assuming Σ=θ+ϕ in Eq. (2)
τ=Σsinβ
(3)
Since in spherical trigonometry, arcs are represented with their angles [33]; arcs BP, OB, and OP are represented by their angles, respectively, τ, β, and η. Considering the right angle spherical triangle OPB, spherical law of sine is
sinτ/sinϕ=sinη/sin90deg=sinβ/sinυ
(4)
Hence,
sinη=sinτ/sinϕ
(5)
sinυ=sinβ/sinη
(6)
And from spherical law of cosines
cosη=cosτcosβ+sinτsinβcos90deg=cosτcosβ
(7)
cosτ=cosηcosβ+sinηsinβcosϕ
(8)
cosβ=cosηcosτ+sinηsinτcosυ
(9)
By replacing sinη and cosη, respectively, from Eqs. (5) and (7) into Eq. (8)
cosτ=cosτcos2β+sinτsinβcosϕ/sinϕ
(10)
and rearranging Eq. (10)
tanτ=sinβtanϕ
(11)
and substituting τ from Eq. (3) into Eq. (11)
tan(Σsinβ)=sinβtanϕ
(12)
and from Σ=θ+ϕ and Eq. (12)
θ=Σ-ϕ=tan-1(sinβtanϕ)/sinβ-ϕ
(13)
2β is usually known and given as base cone angle of the straight bevel gear and θ is calculated based on given ϕ from Eq. (13). This equation is also an exact spherical involute function of ϕ and, by analogy to planar involute functions where θ=tanϕ-ϕ, it is called spherical involute function. Equation (13) is the first parametric equation of spherical involute. The second parametric equation is found through substituting Σ=θ+ϕ in Eq. (3)
τ=(θ+ϕ)sinβtanτ=tan[(θ+ϕ)sinβ]
(14)
Replacing sinτ from Eq. (5) and cosτ from Eq. (7) into Eq. (14)
tanη=(tan[(θ+ϕ)sinβ])/(sinϕcosβ)
(15)

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

Referring to Fig. 1, coordinates of any point P that is generated at roll angle θ from base cone Λ at cone distance r and base cone angle 2β are (where T represents matrix transpose)
P=[-rsinβsinθ-rsinβcosθrcosβ]T
(16)

Note that Fig. 1 is established only for one cross section of r=ρ, 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 the coordinate system xyz around its z axis as much as -θ, such that arc OP coincides with plane y'-z'. Coordinate system x"y"z" is established by rotating the coordinate system x'y'z' around x' axis as much as +η, such that line CP coincides with z" axis. With this in spherical triangle OPB if tangent to arc OP (shown as n' in Fig. 1(a)) at point P is rotated around z" axis for angle +ϑ, it results in normal to the curve Γ (shown as n in Fig. 1(a)). In coordinate system x"y"z", unit tangent vector to arc OP at point P is n'=[0-10]T (superscript T means transpose); therefore, n in x"y"z" is
nx"y"z"=[sinϑ-cosϑ0]T
(17)
Hence, unit normal vector n in xyz coordinate system is
nxyz=[cosθsinθ0-sinθcosθ0001]×[1000cosη-sinη0sinηcosη]×[sinϑ-cosϑ0]
(18)
or, after simplification
nxyz=[cosθsinϑ-sinθcosηcosϑsinθsinϑ-cosθcosηcosϑ-sinηcosϑ]
(19)

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 ϕ.

Points of spherical involute surface lie between gear base cone with cone angle δb=2β and face cone with cone angle δf 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

Fig. 2
Bevel gear flank sketch
Fig. 2
Bevel gear flank sketch
Close modal
x2+y2=Rp2;z=Lp
(20)

Inspection

Any point P located at (Rp,Lp) on the bevel gear flank has a unique set of associated (r,ϕ) 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 α 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).

Fig. 3
Typical bevel gear flanks; area within dashed-border lines will be measured by CMM
Fig. 3
Typical bevel gear flanks; area within dashed-border lines will be measured by CMM
Close modal

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 α 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.

Fig. 4
Measurements of a sample parts against exact spherical involute for both pinion and gear. LFl stands for left flank and RFl stands for right flank.
Fig. 4
Measurements of a sample parts against exact spherical involute for both pinion and gear. LFl stands for left flank and RFl stands for right flank.
Close modal

Correction of Theoretical Surfaces to Represent Actual Measured Surfaces

The proposed approach in this study is to capture the surface deviations of both flanks of both members in the form of a 2D third-order polynomial. To simplify the representations of the deviation surfaces, coordinate system XYZ (of Fig. 3) is defined. In this coordinate system, Z is in the normal to the surface direction and 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 Fig. 3, and is calculated as
O'=[(ro-FW2)sinδp(ro-FW2)cosδp0]T
(21)
where ro is gear outer cone distance, FW is gear face width, and δp is gear pitch angle. With this, the difference between actual ¯ and the theoretical surfaces can be represented up to third-order as
Z=a1+a2X+a3Y+a4X2+a5XY+a6Y2+a7X3+a8X2Y+a9XY2+a10Y3(22)
(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 such that it matches with the actual measured surface. This is done by finding a set of ak(k=1,...,10) of Eq. (22), which best describes the measured error surface. The tooth thickness can also be corrected by modifying α angle. Coefficient a1 can also be used to correct tooth thickness but it is in direction normal to the tooth surface, as opposed to α that changes tooth thickness in tangential direction. Having the ak(k=1,...,10), α, and theoretical spherical involute surface (or modified spherical involute as is the case here in this study), the actual measured surface can be obtained by superimposing Z values on as
¯=+Zn
(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 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.

Next focus is on finding ak(k=1,...,10) coefficients that best describe the measured surface. Assuming that the ak(k=1,...,10) are known; for every measured points Xij,Yij,Zij(i=1,I;j=1,J) of actual surface ¯, the estimated deviation Z'ij by the polynomial fit is
Z'ij=a1+a2Xij+a3Yij+a4Xij2+a5XijYij+a6Yij2+a7Xij3+a8Xij2Yij+a9XijYij2+a10Yij3i=1,I;j=1,J
(24)
Since the actual measured value of deviation is Zij, the amount of error of estimation is
eij=Zij-Z'ij(i=1,I;j=1,J)
(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:
SSQ=i=1mj=1neij2
(26)
to minimize the value of SSQ with respect to ak(k=1,...,10)
ddakSSQ=0(k=1,...,10)ddaki=1mj=1neij2=0(k=1,...,10)
(27)

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 δ 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].

Having point rpij(r,ϕ) and its normal npij(r,ϕ) of pinion spherical involute surface Sp (with pinion axis ap) calculated, its associated action point on gear spherical involute Sg, raij (with gear axis ag) is calculated by applying fundamental equation of meshing as [24]
R(ap×rpij)·npij=(ag×raij)·npij,i[1,I],j[1,J]
(28)

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[1,I] and j[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 ψpij), 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 ψpij 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 ψ that can be used to locate instantaneous contact lines on the tooth surface [35].

The angle ψgij=ψpij/R corresponds to the amount of rotation from the surface of action to reach the conjugate of theoretical pinion surface. Therefore
r¯gij=τz(ψgij)raij(i[1,I],j[1,J])
(29)
where τz(ζ) is rotation matrix around z axis by angle ζ, defined as
τz(ζ)=[cosζsinζ00-sinζcosζ0000100001]
(30)
If this conjugate surface of the pinion was to match perfectly with the real gear surface at any point, then a perfect meshing condition with zero unloaded TE 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.,
reij=rgij-r¯gij(i[1,I],j[1,J])
(31)

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

Having ease-off surface and surface of roll angle ψ in hand, UTCA can be conducted as follows. For a specific pinion roll angle ηi, intersection of the plane z=-ηi and the ψ 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(ηi). If the minimum ease-off on C(ηi) happens at point H; this minimum ease-off divided by distance of the point H(ηi) to gear axis is instantaneous unloaded TE TE(ηi). Across each instantaneous contact line C(ηi), the ease-off value is determined from ease-off surface E. Moving in both directions from point H(ηi) along C(ηi) within a preset separation distance δ gives the unloaded contact line length S(ηi). Repeating this procedure for every pinion, roll angle increment, unloaded TE curve TE(η), 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.

Fig. 5
Designed modifications of pinion and gear based on microgeometry coefficients defined in Table 2
Fig. 5
Designed modifications of pinion and gear based on microgeometry coefficients defined in Table 2
Close modal
Fig. 6
TCA of gear set of Table 1 if pinion and gear microgeometry are as defined in Table 2: (a) ease-off, (b) single tooth contact as seen on the gear, (c) multiple tooth contact as seen on the gear, and (d) TE as seen on the gear axis
Fig. 6
TCA of gear set of Table 1 if pinion and gear microgeometry are as defined in Table 2: (a) ease-off, (b) single tooth contact as seen on the gear, (c) multiple tooth contact as seen on the gear, and (d) TE as seen on the gear axis
Close modal
Table 1

Example of 8 × 13, 24 deg pressure angle automotive straight bevel gear

PinionGear
Number of teeth813
Shaft angle (deg)90.0
Inner cone distance (mm)27.0
Outer cone distance (mm44.0
Pitch cone angle (deg)31.6158.39
Base cone angle (deg)28.6151.08
Face cone angle (deg)42.9064.80
Root cone angle (deg)25.2047.10
Outer pitch diameter (mm)46.1274.94
Pressure angle (deg)24.0
PinionGear
Number of teeth813
Shaft angle (deg)90.0
Inner cone distance (mm)27.0
Outer cone distance (mm44.0
Pitch cone angle (deg)31.6158.39
Base cone angle (deg)28.6151.08
Face cone angle (deg)42.9064.80
Root cone angle (deg)25.2047.10
Outer pitch diameter (mm)46.1274.94
Pressure angle (deg)24.0
Table 2

Designed microgeometry modifications for gear set of Table 1 

PinionGear
a100
a2250
a3035
a4700
a500
a6550
a700
a8200
a9−200
a1000
PinionGear
a100
a2250
a3035
a4700
a500
a6550
a700
a8200
a9−200
a1000

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 α in Table 3.

Fig. 7
CMM measurements of sample pinion and gear against design represented in Tables 1 and 2
Fig. 7
CMM measurements of sample pinion and gear against design represented in Tables 1 and 2
Close modal
Table 3

Polynomial coefficients of initial microgeometry measurement for gear set defined in Tables 1 and 2 

PinionGear
a1−430
a2−7221
a312658
a418−1
a52829
a6−86−83
a71817
a8−19−25
a911102
a1077−56
α−0.9895+0.3221
PinionGear
a1−430
a2−7221
a312658
a418−1
a52829
a6−86−83
a71817
a8−19−25
a911102
a1077−56
α−0.9895+0.3221

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.

Table 4

Sum of squared of errors (SSQ) for pinion and gear at different iteration steps

StepPinionGear
160662019
2108052
315930
410430
55130
66330
75830
StepPinionGear
160662019
2108052
315930
410430
55130
66330
75830
At each iteration step, the new design microgeometry coefficients are then calculated by adding computed deviation coefficients to previous design coefficients as
bkip=bk(i-1)p+akip,bkig=bk(i-1)g+akig,(k=1,...,10)
(32)

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 μm 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.

Fig. 8
Residual error between theory and actual parts for pinion and gear after correction of theoretical surfaces to represent actual parts
Fig. 8
Residual error between theory and actual parts for pinion and gear after correction of theoretical surfaces to represent actual parts
Close modal
Table 5

Final microgeometry modifications of measured/actual pinion and gear that associate with minimum SSQ reported in Table 4 

PinionGear
a1−54−12
a2−5915
a310387
a485−2
a54525
a6−31−81
a71616
a8−2−25
a9−1896
a1061−45
PinionGear
a1−54−12
a2−5915
a310387
a485−2
a54525
a6−31−81
a71616
a8−2−25
a9−1896
a1061−45

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 δ=6.3μm. This TCA is comparable with actual TCA one expects to see on roll tester.

Fig. 9
Predicted TCA of actual parts: (a) ease-off, (b) single tooth contact as seen on the gear, (c) multiple tooth contact as seen on the gear, and (d) TE as seen on the gear axis
Fig. 9
Predicted TCA of actual parts: (a) ease-off, (b) single tooth contact as seen on the gear, (c) multiple tooth contact as seen on the gear, and (d) TE as seen on the gear axis
Close modal

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.

Fig. 10
Experimental setup: (a) roll tester arbors (b) master gauges that replicate pinion and gear mounting distances
Fig. 10
Experimental setup: (a) roll tester arbors (b) master gauges that replicate pinion and gear mounting distances
Close modal
Fig. 11
Actual contact pattern on (a) pinion and (b) gear
Fig. 11
Actual contact pattern on (a) pinion and (b) gear
Close modal

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].

Fig. 12
Contact patterns on gear consecutive gear teeth
Fig. 12
Contact patterns on gear consecutive gear teeth
Close modal

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,

    1. (a)

      digitize master gears,

    2. (b)

      use the captured geometry for loaded analysis, and

  • (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

1.
ANSI/AGMA Standard 2005-D03
,
2005
,
Design Manual for Bevel Gears
,
American Gear Manufacturers Association
,
Alexandria, VA
.
2.
AGMA 932-A05
,
2005
,
Rating the Pitting Resistance and Bending Strength of Hypoid Gears
,
American Gear Manufacturers Association
,
Alexandria, VA
.
3.
ANSI/AGMA 2003-C10
,
2010
,
Rating the Pitting Resistance and Bending Strength of Generated Straight Bevel, Zerol Bevel, and Spiral Bevel Gear Teeth
,
American Gear Manufacturers Association
,
Alexandria, VA
.
4.
Buckingham
,
E.
,
2011
,
Analytical Mechanics of Gears, Dover Publications
,
Springfield
,
New York
.
5.
Townsend
,
D. P.
,
1991
,
Dudley's Gear Handbook
,
McGraw-Hill
,
New York.
6.
Gleason
,
W.
,
1876
, “
Improvement in Machines for Cutting the Teeth of Metal Gears
,” U.S. Patent No. 175,859.
7.
Stadtfeld
,
H. J.
,
2007
, “
Straight Bevel Gear Cutting and Grinding on CNC Free Form Machines
,”
AGMA Fall Technical Meeting
,
Detroit, MI
, Oct. 7–9, Paper No. 07FTM16.
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
,”
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 Gear Works
,
Lexington, MA
.
11.
Wildhaber
,
E.
,
1945
, “
Relationships of Bevel Gears
,”
Am. Mach.
,
89
(
20
), pp.
99
102
.
12.
Wildhaber
,
E.
,
1945
, “
Relationships of Bevel Gears
,”
Am. Mach.
,
89
(
21
), pp.
118
121
.
13.
Wildhaber
,
E.
,
1945
, “
Relationships of Bevel Gears
,”
Am. Mach.
,
89
(
22
), pp.
122
125
.
14.
Wildhaber
,
E.
,
1956
, “
Surface Curvature
,”
Prod. Eng.
,
27
(
5
), pp.
184
191
.
15.
Al-Daccak
,
M. J.
,
Angeles
,
J.
, and
González-Palacios
,
M. A.
,
1994
, “
The Modeling of Bevel Gears Using the Exact Spherical Involute
,”
ASME J. Mech. Des.
,
116
(
2
), pp.
364
368
.
16.
Figliolini
,
G.
, and
Angeles
,
J.
,
2004
, “
Algorithms for Involute and Octoidal Bevel-Gear Generation
,”
ASME J. Mech. Des.
,
127
(
4
), pp.
664
672
.
17.
Kolivand
,
A.
,
2014
, “
Involute Straight Bevel Gear Surface and Contact Lines Calculation Utilizing Ease-Off Topography Approach
,” SAE Technical Paper No. 2014-01-1765.
18.
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
,”
ASME J. Mech. Des.
,
103
(
1
), pp.
83
88
.
19.
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
,”
ASME J. Mech. Des.
,
103
(
1
), pp.
89
101
.
20.
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’
,”
ASME J. Mech. Des.
,
103
(
1
), pp.
102
110
.
21.
Fan
,
Q.
,
2006
, “
Enhanced Algorithms of Contact Simulation for Hypoid Gear Drives Produced by Face-Milling and Face-Hobbing Processes
,”
ASME J. Mech. Des.
,
129
(
1
), pp.
31
37
.
22.
Simon
,
V.
,
1996
, “
Tooth Contact Analysis of Mismatched Hypoid Gears
,”
7th International Power Transmission and Gearing Conference
,
San Diego, CA
, Oct. 6–9, pp.
789
798
.
23.
Wang
,
X. C.
, and
Ghosh
,
S. K.
,
1994
,
Advanced Theories of Hypoid Gears/Book and Disk
,
Elsevier Science
,
New York
.
24.
Stadtfeld
,
H. J.
,
1993
,
Handbook of Bevel and Hypoid Gears: Calculation, Manufacturing, Optimization
,
Rochester Institute of Technology
,
Rochester, NY
.
25.
Kolivand
,
M.
, and
Kahraman
,
A.
,
2009
, “
A Load Distribution Model for Hypoid Gears Using Ease-Off Topography and Shell Theory
,”
Mech. Mach. Theory
,
44
(
10
), pp.
1848
1865
.
26.
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
.
27.
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
,”
ASME J. Mech. Des.
,
135
(
3
), p.
034504
.
28.
Gosselin
,
C.
,
Cloutier
,
L.
, and
Brousseau
,
J.
,
1991
, “
Tooth Contact Analysis of High Conformity Spiral Bevel Gears
,”
JSME International Conference on Motion and Power Transmission
,
Hiroshima, Japan
, Nov. 23–26, pp.
725
730
.
29.
Zhang
,
Y.
,
Litvin
,
F. L.
,
Maruyama
,
N.
,
Takeda
,
R.
, and
Sugimoto
,
M.
,
1994
, “
Computerized Analysis of Meshing and Contact of Gear Real Tooth Surfaces
,”
ASME J. Mech. Des.
,
116
(
3
), pp.
677
682
.
30.
Kin
,
V.
,
1992
, “
Tooth Contact Analysis Based on Inspection
,”
3rd World Congress on Gearing and Power Transmissions
,
Paris, France
, Feb. 12–14, pp.
1108
1113
.
31.
Kin
,
V.
,
1994
, “
Computerized Analysis of Gear Meshing Based on Coordinate Measurement Data
,”
ASME J. Mech. Des.
,
116
(
3
), pp.
738
744
.
32.
Marambedu
,
K. R.
,
2009
, “
Development of a Procedure for the Analysis of Load Distribution, Stresses and Transmission Error of Straight Bevel Gears
,” M.Sc. thesis, The Ohio State University, Columbus, OH.
33.
Donnay
,
J. D. H.
,
1945
,
Spherical Trigonometry
,
Interscience Publishers
,
New York
.
34.
Litvin
,
F. L.
, and
Fuentes
,
A.
,
2004
,
Gear Geometry and Applied Theory
,
Cambridge University Press
,
Cambridge, UK
.
35.
Kolivand
,
M.
, and
Kahraman
,
A.
,
2011
, “
A General Approach to Locate Instantaneous Contact Lines of Gears Using Surface of Roll Angle
,”
ASME J. Mech. Des.
,
133
(
1
), p.
014503
.
36.
Artoni
,
A.
,
Kolivand
,
M.
, and
Kahraman
,
A.
,
2009
, “
An Ease-Off Based Optimization of the Loaded Transmission Error of Hypoid Gears
,”
ASME J. Mech. Des.
,
132
(
1
), p.
011010
.
37.
AGMA 390.03A:1980 (R1988)
,
1988
,
Handbook—Gear Classification, Materials and Measuring Methods for Bevel, Hypoid, Fine Pitch Wormgearing and Racks
,
American Gear Manufacturers Association
,
Alexandria, VA
.

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

  •  
  • =

    theoretical surface topography

  •  
  • ¯ =

    actual surface topography

  •  
  • ri =

    inner cone distance

  •  
  • ro =

    outer cone distance

  •  
  • rij =

    coordinate of point ij of spherical involute surface

  •  
  • r¯ij =

    coordinate of point ij of conjugate surface

  •  
  • SSQ =

    sum of the squared of deviations, in tenth of thousands squared

  •  
  • α =

    difference angle between two flanks at midpoint of measurement grid

  •  
  • β =

    half of base cone angle

  •  
  • δ =

    separation distance

  •  
  • δb =

    base cone angle

  •  
  • δf =

    face cone angle

  •  
  • δp =

    pitch angle

  •  
  • ϕ =

    pressure angle

  •  
  • ψ =

    surface of roll angle

  •  
  • ψij =

    roll angle value of point ij on surface of roll angle

  •  
  • θ =

    roll angle