# Internal Loading Distribution in Statically Loaded Ball Bearings Subjected to an Eccentric Thrust Load

Authors: Mário César Ricci
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

Rolling-element bearings are simple machine elements of great
utility used both in simple commercial devices as in complex
engineering mechanisms. Because of being a very popular machine
element, there is a lot of literature on the subject. With regard
to the behavior of internal loading distribution, elastic
deformations at point or line contacts, and geometric parameters
under loading, although there are many works describing the
parameters variation models, few works show such variations in
practice, even under simple static loadings. In an attempt to
cover this gap some studies are being developed in parallel.
Particularly in this work, anew, iterative computational
procedure is introduced which calculates internal normal ball
loads in statically loaded single-row, angular-contact ball
bearings, subjected to a known thrust load which is applied to a
variable distance (lever arm or eccentricity) from the geometric
bearing center line. Numerical examples results for a 218
angular-contact ball bearing have been compared with those from
the literature. Fifty figures are presented showing geometrical
features and the following parameters variations as functions of
the thrust load and eccentricity: contact angle, contact ellipse
parameters, normal ball loads, distances between groove curvature
centers, normal and axial deflections, and loading zones.


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## Body

## 1. Introduction

Ball and roller bearings, generically calledrolling bearings, are commonly used machine elements. They are employed to permit rotary motions of, or about, shafts in simple commercial devices such as bicycles, roller skates, and electric motors. They are also used in complex engineering mechanisms such as aircraft gas turbines, rolling mils, dental drills, gyroscopes, and power transmissions.The standardized forms of ball or roller bearings permit rotary motion between two machine elements and always include a complement of ball or rollers that maintain the shaft and a usually stationary supporting structure, frequently called ahousing, in a radially or axially spaced-apart relationship. Usually, a bearing may be obtained as a unit, which includes two steel rings; each of which has a hardened raceway on which hardened balls or rollers roll. The balls or rollers, also called rolling elements, are usually held in an angularly spaced relationship by a cage, also called a separator or retainer.There are many different kinds of rolling bearings. This work is concerned withsingle-row angular-contact ball bearings (Figure 1) that are designed to support combined radial and thrust loads or heavy thrust loads depending on the contact angle magnitude. The bearings having large contact angle can support heavier thrust loads. Figure 2 shows bearings having small and large contact angles. The bearings generally have groove curvature radii in the range of 52%–53% of the ball diameter. The contact angle does not usually exceed 40o.Figure 1
An angular-contact ball bearing (courtesy of SKF Industries).Angular-contact ball bearings.
(a)
Small angle(b)
Large angleThis work is devoted to the study of the internal loading distribution in statically loaded ball bearings. Several researchers have studied the subject as, for example, Stribeck [1], Sjoväll [2], Jones [3], and Rumbarger [4], to cite a few. The methods developed by them to calculate distribution of load among the balls and rollers of rolling bearings can be used in most bearing applications because rotational speeds are usually slow to moderate. Under these speed conditions, the effects of rolling-element centrifugal forces and gyroscopic moments are negligible. At high speeds of rotation these body forces become significant, tending to alter contact angles and clearance. Thus, they can affect the static load distribution to a great extension.Harris [5] described methods for internal loading distribution in statically loaded bearings addressing pure radial, pure thrust (centric and eccentric loads), combined radial and thrust load, which uses radial and thrust integrals introduced by Sjoväll [2], and for ball bearings under combined radial, thrust, and moment load, initially due to Jones [3].The method described by Harris for eccentric thrust load, initially due to Rumbarger [4], is an approximate, direct method, based in a single-row, 90o thrust bearing and in thrust and moment integrals whose values are obtained from tables and graphics, as functions of eccentricity and pitch diameter. The maximum ball load is given directly and no computer is necessary. Although it is not entirely appropriate, the method was used by Harris to find approximations for the maximum ball load magnitude and for the extension of the loading zone in the 218 angular-contact ball bearing.We can see that there are many works describing the parameters variation models under static loads but few show such variations in practice, even under simple static loadings. The author believes that the lack of practical examples in the literature is mainly due to the inherent difficulties of the numerical procedures that, in general, deal with the resolution of several nonlinear algebraic equations that must be solved simultaneously.In an attempt to cover this gap studies are being developed in parallel [6–14]. Particularly in this work is described a new, precise method for internal load distribution computation in statically loaded, single-row, angular-contact ball bearings subjected to a known external thrust load which is applied to a variable distance (lever arm or eccentricity) from the geometric bearing center line. It must be solved iteratively using a digital computer and can be thought as a particular case of the Jones method, with null external radial load and external moment load given by the product of the thrust load by the eccentricity. Unlike Rumbarger’s method, it is adequate to angular-contact bearings, and theoretically and numerically more precise. The novelty of the method is in the choice of the set of the nonlinear equations, which must be solved simultaneously. The author did not find in the literature the resolution of this problem using the same set of equations.The difference between the method described here and the method described by Harris for eccentric thrust load mainly comes from the fact that Rumbarger’s method, for sake of simplicity, makes use of the pitch radius,de/2, as lever arm, instead of the inner contact radius, dcj/2, in the r.h.s. of the moment equation—see (4.19) for comparison— and secondarily by the fact that it uses the pitch radius instead of the locus of the centers of the inner ring raceway groove curvature radii, Ri, in the computations of the load distribution factor, ε, in (4.10) and of the extension of load zone, ψl, in (4.11). These approximations are guarantee of the straightforwardness but obviously they introduce errors in the normal ball loads determination. However, at first glance appears that the method for thrust bearing is more attractive than the method of this paper because it supplies results more directly whereas no computer is necessary. But, despite the simplicity of the former, comparative analyses between the results show significant differences in the magnitudes of the maximum ball load and extension of the loading zone.


## 2. Geometry of Ball Bearings

In this section, the principal geometrical relationships for an unloaded ball bearing are summarized. The radial cross section of a single-row ball bearing shown in Figure3 depicts the diametral clearance and various diameters. The pitch diameter, de, is the mean of the inner- and outer-race diameters di and do, respectively, and is given by
(2.1)de=12(di+do).
The diametral clearance, Pd, can be written as
(2.2)Pd=do-di-2D.Race conformity is a measure of the geometrical conformity of the race and the ball in a plane passing through the bearing axis (also named center line or rotation axis), which is a line passing through the center of the bearing perpendicular to its plane and transverse to the race. Figure 4 depicts a cross section of a ball bearing showing race conformity expressed as
(2.3)f=rD.
Radial bearings have some axial play since they are generally designed to have a diametral clearance, as shown in Figures 5(a) and 5(b), that shows a radial bearing with contact due to the axial shift of the inner and outer rings when no measurable force is applied. The radial distances between the curvature centers of the two races are the same in Figures 5(a) and 5(b). Denoting quantities which referred to the inner and outer races by subscripts iand o, respectively, this radial distance value can be expressed as A-Pd/2, where A=ro+ri-D is the curvature centers distance in the shifted position given by Figure 5(b). Using (2.3) we can write A as
(2.4)A=BD,
where B=fo+fi-1 is known as the total conformity ratio and is a measure of the combined conformity of both the outer and inner races to the ball.Figure 3
Radial cross-section of a single-row ball bearing.Figure 4
Cross-section of a ball and an outer race showing race conformity.Cross-section of a radial ball bearing showing ball-race contact due to axial shift of inner and outer rings.
(a)
Initial position(b)
Shifted positionThecontact angle, β, is defined as the angle made by a line, which passes through the curvature centers of both the outer and inner raceways and that lies in a plane passing through the bearing rotation axis, with a plane perpendicular to the bearing axis of rotation. The free-contact angle, βf, (Figure 5(b)) is the contact angle when the line also passes through the points of contact of the ball and both raceways and no measurable force is applied. From Figure 5(b), the expression for the free-contact angle can be written as
(2.5)cosβf=A-Pd/2A.
From (2.5), the diametral clearance, Pd, can be written as
(2.6)Pd=2A(1-cosβf).Free endplay, Pe, is the maximum axial movement of the inner race with respect to the outer when both races are coaxially centered and no measurable force is applied. Free endplay depends on total curvature and contact angle, as shown in Figure 5(b), and can be written as
(2.7)Pe=2Asinβf.
Considering the geometry of two contacting solids (ellipsoids) in a ball bearing, we can arrive at the two quantities of some importance in the analysis of contact stresses and deformations. The curvature sum, 1/R, and curvature difference, Γ, are defined as
(2.8)1R=1Rx+1Ry,Γ=R(1Rx-1Ry),
where
(2.9)1Rx=1rax+1rbx,1Ry=1ray+1rby,
with rax, rbx, ray, and rby being the radii of curvature for the ball-race contact.A cross section of a ball bearing operating at a contact angleβ is shown in Figure 6. Equivalent radii of curvature for both inner- and outer-race contacts in, and normal to, the direction of rolling can be calculated from this figure. Considering x the direction of the motion and y the transverse direction, the radii of curvature for the ball-inner-race contact are
(2.10)rax=ray=D2,rbx=de-Dcosβ2cosβ,rby=-fiD=-ri.
The radii of curvature for the ball-outer-race contact are
(2.11)rax=ray=D2,rbx=-de+Dcosβ2cosβ,rby=-foD=-ro.
Let
(2.12)γ=Dcosβde.
Then
(2.13)rbx=D21-γγ,1R|i=1rax+1rbx+1ray+1rby=1D(4-1fi+2γ(1-γ)),Γi=R(1rax+1rbx-1ray-1rby)=1/fi+2γ/(1-γ)4-1/fi+2γ/(1-γ),
for the ball-inner-race contact, and
(2.14)rbx=-D21+γγ,1R|o=1rax+1rbx+1ray+1rby=1D(4-1fo-2γ1+γ),Γo=R(1rax+1rbx-1ray-1rby)=1/fo-2γ/(1+γ)4-1/fo-2γ/(1+γ),
for the ball-outer-race contact.Figure 6
Cross-section of a ball bearing.


## 3. Contact Stress and Deformations

When two elastic solids are brought together under a load, a contact area develops; the shape and size of which depend on the applied load, the elastic properties of the materials, and the curvatures of the surfaces. For two ellipsoids in contact the shape of the contact area is elliptical, witha being the semimajor axis in the y direction (transverse direction) and b being the semiminor axis in the x direction (direction of motion).Theelliptical eccentricity parameter, k, is defined as
(3.1)k=ab.
From Harris [5], k can be written in terms of the curvature difference, Γ, and the elliptical integrals of the first and second kinds K and E, as
(3.2)J(k)=2K-E(1+Γ)E(1-Γ),
where
(3.3)K=∫0π/2[1-(1-1k2)sin2φ]-1/2dφ,E=∫0π/2[1-(1-1k2)sin2φ]1/2dφ.
A one-point iteration method which has been used successfully in the past [15] is used, where
(3.4)kn+1=J(kn).
When the ellipticity parameter, k, the elliptic integrals of the first and second kinds, K and E, respectively, the normal applied load, Q, Poisson’s ratio, ν, and the modulus of elasticity, E, of the contacting solids, are known, we can write the semimajor and semiminor axes of the contact ellipse and the maximum deformation at the center of the contact, from the analysis of Hertz [16], as
(3.5)a=(6k2EQRπE′)1/3,b=(6EQRπkE′)1/3,(3.6)δ=K[92ER(QπkE′)2]1/3,
where
(3.7)E′=2(1-υa2)/Ea+(1-υb2)/Eb.


## 4. Static Load Distribution under Eccentric Thrust Load

Methods to calculate distribution of load among the balls and rollers of rolling bearings statically loaded can be found in various papers [5, 17]. The methods have been limited to, at most, three degrees of freedom in loading and demand the solution of a simultaneous nonlinear system of algebraic equations for higher degrees of freedom. Solution of such equations generally necessitates the use of a digital computer. In certain cases, however—for example, applications with pure radial, pure thrust, or radial and thrust loading with nominal clearance—the simplified methods will probably provide sufficiently accurate calculational results.Having defined a simple analytical expression for the deformation in terms of load in the previous section, it is possible to consider how the bearing load is distributed among the rolling elements. Most rolling-element bearing applications involve steady-state rotation of either the inner or outer race or both; however, the speeds of rotation are usually not so great as to cause ball or roller centrifugal forces or gyroscopic moments of significant magnitudes. In analyzing the loading distribution on the rolling elements, it is usually satisfactory to ignore these effects in most applications. In this section the load-deflection relationships for ball bearings are given, along with a specific load distribution consisting of an eccentric thrust load of statically loaded rolling elements.


### 4.1. Load-Deflection Relationships for Ball Bearings

From (3.6) it can be seen that for a given ball-raceway contact (point loading)
(4.1)Q=Kδ3/2,
where
(4.2)K=πkE'2ER9K3.
The total normal approach between two raceways under load separated by a rolling element is the sum of the approaches between the rolling element and each raceway. Hence
(4.3)δn=δi+δo.
Therefore,(4.4)Kn=[11/Ki2/3+1/Ko2/3]3/2,(4.5)Q=Knδn3/2.


### 4.2. Ball Bearings under Eccentric Thrust Load

Let a ball bearing with a number of balls,Z, symmetrically distributed about a pitch circle according to Figure 7, be subjected to an eccentric thrust load. Then, a relative axial displacement, δa, and a relative angular displacement, θ, between the inner and outer ring raceways may be expected. Let ψ=0 be the angular position of the maximum loaded ball.Figure 7
Ball angular positions in the radial plane that is perpendicular to the bearing’s axis of rotation;Δψ=2π/Z,ψj=2π/Z(j-1).Figure8 shows the initial and final curvature centers positions at angular position ψ, before and after loading, whereas the centers of curvature of the raceway grooves are fixed with respect to the corresponding raceway. If δa and θ are known, then the total axial displacement, δt, at angular position ψ, is given by
(4.6)δt(ψ)=δa+Riθcosψ,
where
(4.7)Ri=de2+(fi-0.5)Dcosβf
expresses the locus of the centers of the inner ring raceway groove curvature radii.Figure 8
Initial and final curvature centers positions at angular positionψ, with and without applied load.Also,(4.8)δmax≡δt(0)=δa+Riθ.From (4.6) and (4.8), one may develop the following relationship:
(4.9)δt=δmax[1-12ε(1-cosψ)]
in which
(4.10)ε=12(1+δaRiθ).The extension of theloading zone is defined by
(4.11)ψl=cos-1(-δaRiθ).From Figure8,
(4.12)β(ψ)=cos-1(A-Pd/2A+δn),(4.13)δt(ψ)=(A+δn)sinβ-Asinβf.From (2.5) and (4.12), the total normal approach between two raceways at angular position ψ, after the thrust load has been applied, can be written as
(4.14)δn(ψ)=A(cosβfcosβ-1).From Figure8 and (4.14) it can be determined that s, the distance between the centers of the curvature of the inner and outer ring raceway grooves at any rolling-element position ψ, is given by
(4.15)s(ψ)=A+δn=Acosβfcosβ.From (4.6), (4.13), and (4.14), yields, for ψ=ψj,(4.16)δa+Riθcosψj-Asin(βj-βf)cosβj=0,j=1,…,Z.From (4.5), and (4.14) one yields, for ψ=ψj,(4.17)Qj=KnjA3/2(cosβfcosβj-1)3/2,j=1,…,Z.If the external thrust load,Fa, is applied at a point distant e from the bearing’s axis of rotation, then for static equilibrium to exist
(4.18)Fa=∑j=1ZQjsinβj,(4.19)M=eFa=12∑j=1ZdcjQjsinβjcosψj,
where dcj≡de-Dcosβj.Substitution of (4.17) into (4.18) yields
(4.20)Fa-A3/2∑j=1ZKnjsinβj(cosβfcosβj-1)3/2=0.Similarly,(4.21)eFa-A3/22∑j=1ZKnjdcjcosψjsinβj(cosβfcosβj-1)3/2=0.Equations (4.16), (4.20), and (4.21) are Z+2 simultaneous nonlinear equations with unknowns δa, θ, and βj, j=1,…,Z. Since Knj and dcj are functions of final contact angle, βj, the equations must be solved iteratively to yield an exact solution for δa, θ, and βj.


## 4.1. Load-Deflection Relationships for Ball Bearings

From (3.6) it can be seen that for a given ball-raceway contact (point loading)
(4.1)Q=Kδ3/2,
where
(4.2)K=πkE'2ER9K3.
The total normal approach between two raceways under load separated by a rolling element is the sum of the approaches between the rolling element and each raceway. Hence
(4.3)δn=δi+δo.
Therefore,(4.4)Kn=[11/Ki2/3+1/Ko2/3]3/2,(4.5)Q=Knδn3/2.


## 4.2. Ball Bearings under Eccentric Thrust Load

Let a ball bearing with a number of balls,Z, symmetrically distributed about a pitch circle according to Figure 7, be subjected to an eccentric thrust load. Then, a relative axial displacement, δa, and a relative angular displacement, θ, between the inner and outer ring raceways may be expected. Let ψ=0 be the angular position of the maximum loaded ball.Figure 7
Ball angular positions in the radial plane that is perpendicular to the bearing’s axis of rotation;Δψ=2π/Z,ψj=2π/Z(j-1).Figure8 shows the initial and final curvature centers positions at angular position ψ, before and after loading, whereas the centers of curvature of the raceway grooves are fixed with respect to the corresponding raceway. If δa and θ are known, then the total axial displacement, δt, at angular position ψ, is given by
(4.6)δt(ψ)=δa+Riθcosψ,
where
(4.7)Ri=de2+(fi-0.5)Dcosβf
expresses the locus of the centers of the inner ring raceway groove curvature radii.Figure 8
Initial and final curvature centers positions at angular positionψ, with and without applied load.Also,(4.8)δmax≡δt(0)=δa+Riθ.From (4.6) and (4.8), one may develop the following relationship:
(4.9)δt=δmax[1-12ε(1-cosψ)]
in which
(4.10)ε=12(1+δaRiθ).The extension of theloading zone is defined by
(4.11)ψl=cos-1(-δaRiθ).From Figure8,
(4.12)β(ψ)=cos-1(A-Pd/2A+δn),(4.13)δt(ψ)=(A+δn)sinβ-Asinβf.From (2.5) and (4.12), the total normal approach between two raceways at angular position ψ, after the thrust load has been applied, can be written as
(4.14)δn(ψ)=A(cosβfcosβ-1).From Figure8 and (4.14) it can be determined that s, the distance between the centers of the curvature of the inner and outer ring raceway grooves at any rolling-element position ψ, is given by
(4.15)s(ψ)=A+δn=Acosβfcosβ.From (4.6), (4.13), and (4.14), yields, for ψ=ψj,(4.16)δa+Riθcosψj-Asin(βj-βf)cosβj=0,j=1,…,Z.From (4.5), and (4.14) one yields, for ψ=ψj,(4.17)Qj=KnjA3/2(cosβfcosβj-1)3/2,j=1,…,Z.If the external thrust load,Fa, is applied at a point distant e from the bearing’s axis of rotation, then for static equilibrium to exist
(4.18)Fa=∑j=1ZQjsinβj,(4.19)M=eFa=12∑j=1ZdcjQjsinβjcosψj,
where dcj≡de-Dcosβj.Substitution of (4.17) into (4.18) yields
(4.20)Fa-A3/2∑j=1ZKnjsinβj(cosβfcosβj-1)3/2=0.Similarly,(4.21)eFa-A3/22∑j=1ZKnjdcjcosψjsinβj(cosβfcosβj-1)3/2=0.Equations (4.16), (4.20), and (4.21) are Z+2 simultaneous nonlinear equations with unknowns δa, θ, and βj, j=1,…,Z. Since Knj and dcj are functions of final contact angle, βj, the equations must be solved iteratively to yield an exact solution for δa, θ, and βj.


## 5. Numerical Results

A numerical method (the Newton-Rhapson method) was chosen to solve the simultaneous nonlinear equations (4.16), (4.20), and (4.21). Choosing the rolling bearing, input must be given the geometric parameters di, do, D, Z, ri, and ro, in accordance with Figures 3 and 5, and the elastic properties Ea, Eb, νa, and νb. Next, the following parameters must be obtained: fi, fo, B, A, ψj(j=1,…,Z), E′, de, Pd, βf,and Ri.The interest here is to observe the behavior of an angular-contact ball bearing under a known thrust load which is to be applied statically to avariable distance (lever arm or eccentricity), e, from the geometric bearing center line. Then, given a thrust load and the initial estimates for δa, θ, and βj, j=1,…,Z, for each distance e, varying from zero up to a given maximum eccentricity, the values 1/R|i, 1/R|o, Γi, Γo, ki, ko, Ki, Ko, Ei, Eo, Ki, Ko, and Kn are calculated for each ball, according to previous sections, and new values for δa, θ, and βj are obtained. The new βj values are compared with old ones, and if the difference is greater than a minimal error, then new values for 1/R|i, 1/R|o, Γi, Γo, ki, ko, Ki, Ko, Ei, Eo, Ki, Ko, and Kn are calculated for each ball, and again new values for δa, θ, and βj are obtained. If the difference is lesser than the error then a new value for e is taken. If e is the last valid value, then a new thrust load value is acquired and the procedure is repeated up to the last valid thrust load value, when the program ends.To show an application of the theory developed in this work, a numerical example is presented here. It was chosen the 218 angular-contact ball bearing that was also used by Harris[5]. Thus, the results generated here can be compared to a certain degree with Harris results. The input data for this rolling bearing were the following:inner raceway diameter:di=0.10279 m,outer raceway diameter:do=0.14773 m,ball diameter:D=0.02223 m,ball number:Z=16,inner groove radius:ri=0.01163 m,outer groove radius:ro=0.01163 m,modulus of elasticity for both balls and races:E=2.075×1011 N/m2 ,poisson’s ratio for both balls and races:υ=0.3.The remaining parameters have been calculated yielding:inner race conformity:fi=0.523166891587944,outer race conformity:fo=0.523166891587944,total conformity ratio:B=0.046333783175888,initial curvature centers distance:A=0.00103 m,effective elastic modulus:E′=228021978021.978 N/m2 ,angular spacing between rolling elements:Δψ=22.5°,angular position of rolling elements:ψj=22.5°(j-1),  j=1,…,16,bearing pitch diameter:de=0.12526 m,diametral clearance:Pd=0.00048 m,free-contact angle:βf=39.915616407992260°,radius of locus of inner raceway groove curvature centers:Ri=0.063025 m.For each thrust load value, the initial estimates forδa, θ, and βj were the following:axial deflection:δa=10-5 m,misalignment angle:θ=10-2 rd,contact angle:βj=1.1βf,  j=1,…,16.


### 5.1. Numerical Results for a 17,800 N Thrust Load

Since it is the qualitative behavior of solutions that is the interest, the results are presented here in graphical form.Initially, for comparative purposes with the Harris work, a specific thrust loadFa=17,800 N was chosen to be applied, and the following graphical results are presented as functions of the moment, M=Fae:(i)
relative angular displacement,θ (Figure 9),(ii)
partial axial displacement,Riθcosψ (Figure 10),(iii)
relative axial displacement,δa (Figure 11),(iv)
total relative axial deflection,δt (Figure 12),(v)
loading zone,ψl (Figure 13),(vi)
distance between loci of inner and outer raceway groove curvature centers,s (Figure 14),(vii)
maximum elastic compression at the ball/inner-race contact,δi (Figure 15),(viii)
maximum elastic compression at the ball/outer-race contact,δo (Figure 16),(ix)
total normal ball deflection,δn (Figure 17),(x)
ball-raceway normal load,Q (Figure 18),(xi)
contact angle,β (Figure 19),(xii)
semimajor axis of the ball/inner-race contact area,ai (Figure 20),(xiii)
semiminor axis of the ball/inner-race contact area,bi (Figure 21),(xiv)
semimajor axis of the ball/outer-race contact area,ao (Figure 22),(xv)
semiminor axis of the ball/outer-race contact area,bo (Figure 23),(xvi)
elliptical eccentricity parameter for ball/inner-race contact,ki (Figure 24),(xvii)
elliptical eccentricity parameter for ball/outer-race contact,ko (Figure 25).Figure 9
Relative angular misalignment,θ, for 17,800 N thrust load, as a function of the Moment, M.Figure 10
Partial axial deflection,Riθcosψ, for each ball and 17,800 N thrust load, as a function of the Moment, M.Figure 11
Axial deflection,δa, for 17,800 N thrust load, as a function of the Moment, M.Figure 12
Total axial deflection,δt, for 17,800 N thrust load, as a function of the Moment, M.Figure 13
Loading zone,ψl, for 17,800 N thrust load, as a function of the Moment, M.Figure 14
Distance between curvature centers,s, for 17,800 N thrust load, as a function of the Moment, M.Figure 15
Maximum normal elastic compression at the ball/inner race contact,δi, for 17,800 N thrust load, as a function of the Moment, M.Figure 16
Maximum normal elastic compression at the ball/outer-race contact,δo, for 17,800 N thrust load, as a function of the Moment, M.Figure 17
Total ball deflection,δn, for 17,800 N thrust load, as a function of the Moment, M.Figure 18
Normal ball load,Q, for 17,800 N thrust load, as a function of the Moment, M.Figure 19
Contact Angle,β, for 17,800 N thrust load, as a function of the Moment, M.Figure 20
Semimajor axis of the ball/inner-race contact area,ai, for 17,800 N thrust load, as a function of the Moment, M.Figure 21
Semiminor axis of the ball/inner-race contact area,bi, for 17,800 N thrust load, as a function of the Moment, M.Figure 22
Semimajor axis of the ball/outer-race contact area,ao, for 17,800 N thrust load, as a function of the Moment, M.Figure 23
Semiminor axis of the ball/outer-race contact area,bo, for 17,800 N thrust load, as a function of the Moment, M.Figure 24
Elliptical eccentricity parameter for ball/inner-race contact,ki, for 17,800 N thrust load, as a function of the Moment, M.Figure 25
Elliptical eccentricity parameter for ball/outer-race contact,ko, for 17,800 N thrust load, as a function of the Moment, M.The graphics above, with exception of Figures 9, 11, and 13, show one curve for each ball angular position.Figures9 and 10 show the relative angular misalignment, θ,  and the partial axial deflection for each ball, Riθcosψ, respectively. It is observed that there is an approximately linear relationship between the misalignment angle, θ, and applied moment, M, for moment values ranging from zero up to about 600 Nm, which corresponds to a distance e of approximately 33.7 mm. Keeping the load constant and increasing the lever arm, e, above this value, it can be observed a deeper increase in the misalignment angle and, therefore, in the resultant axial deflection, Riθ.From Figure9 it can be observed that for an applied moment of 900 Nm (e≅50.6 mm) the angular misalignment can be as high as a quarter of degree.As already been waited for, from Figure10 it can be observed that the partial axial deflection is symmetrical with respect to the horizontal axis (null displacement) and that the displacement is null for the balls located at ψ=±90°.Figure11 shows the axial deflection, δa. It is observed that the axial deflection, δa, is approximately constant for moment values where the relationship between θ and M is approximately linear, that is, from zero up to about 600 Nm (e≅33.7 mm). For higher moment values the axial deflection falls abruptly and becomes negative in the vicinity of 800 Nm (e≅44.9 mm). The deeper increase in θ due to the increase in the lever arm forces the decrease of δa to preserve the force and moment static balances.Figure12 shows the total axial deflection, δt. It can be observed that the total axial deflection, δt, is the axial deflection, δa, in two situations: under centric thrust load (e=0), where all balls have the same axial deflection (3.6011095400455×10-5 m), and under eccentric thrust load for balls located at ψ=±90°. Increasing from zero the lever arm, an almost linear increase (decrease) in the total axial deflection is observed for the balls whose angular positions satisfy |ψ|<90° (|ψ|>90°). This relation is approximately linear up to vicinity of M=600 Nm when the ball located at ψ=180° occurs to be unloaded, that is, δt(ψ=180°)=0 for M=588.9687 Nm (e=3.3088×10-2 m).From Figure12 it is observed that for eccentricity of about 50 mm the total axial deflection of the most heavily loaded ball can reach one tenth of millimeter.The Figure13 shows the loading zone, ψl. The increase of the moment above 587.4 Nm (or lever arm above 3.3×10-2 m) causes the decrease of the loading zone from initial value ψl=±180°, with the successive unloading of the balls pairs located at ψ=±157.5° (M=609.448 Nm), ψ=±135° (M=661.1407 Nm), ψ=±112.5° (M=729.9584 Nm), ψ=±90° (M=803.9741 Nm), and ψ=±67.5° (M=873.7125 Nm), respectively. Going ahead cause the unloading of the balls pair located at ψ=±45º. However, it is not advisable to go beyond M=900 Nm, once the radial displacements between curvature centers start to acquire micrometer order values and they cannot more be disregarded.Figure13 shows a substantial difference between results found in this work and  those found by Harris. While Harris found a loading zone of 92.86º (p. 252) for an eccentricity of 50.8 mm, this work found a loading zone of 53.66º. Considering the last result as reference, this represents an error of +73% in the loading angle, meaning that Harris calculation has underestimated the effect of the moment M.Figure14 shows the distance between loci of inner and outer raceway groove curvature centers, s. It can be observed that the distance, s, under centric thrust load (e=0), is the same for all balls (1.053468971830×10-3 m). Increasing from zero the lever arm, an almost linear increase (decrease) in the distance, s, is observed for the balls whose angular positions satisfy |ψ|<90° (|ψ|>90°). This relation is approximately linear up to vicinity of M=600 Nm when the ball located at ψ=180° occurs to be unloaded, that is, s(ψ=180°)=A for M=588.9687 Nm (e=3.3088×10-2 m).The increase of the moment above 588.9687 Nm (or lever arm above3.3088×10-2 m) causes the decrease of the loading zone, as already explained, with the successive unloading of the ball pairs. At the points where the unloading occurs it is observed that the distance s falls below of the distance between centers of curvature, A, for the unloaded bearing.Figures15 and 16 show the maximum normal elastic compressions at the ball/inner-race and ball/outer-race contacts, δi and δo, respectively. It can be observed that δi and δo, under centric thrust load (e=0), are the same for all balls (1.18852986717367×10-5 m for δi and 1.15836731583185×10-5 m for δo) and that the deformation for the maximum loaded ball, in both cases, can reach values as high as 36 μm for moment about 900 Nm.Figure17 shows the total normal ball deflection, δn, that can be obtained by summing the maximum normal elastic compressions on the inner and outer races, δi and δo, or by subtracting A from s, once δn=s-A>0 also. It can be observed that δn, under centric thrust load (e=0), is the same for all balls (2.3468971830055×10-5m) and that the total normal elastic deformation for the maximum loaded ball can reach values as high as 70 μm for moment about 900 Nm.Figure18 shows the normal ball load, Q. It can be observed that the normal ball load, Q, under centric thrust load (e=0), is the same for all balls (1, 681.663561507042 N). Increasing from zero the lever arm, an almost linear increase (decrease) in the normal ball load is observed for the balls whose angular positions satisfy |ψ|<90° (|ψ|>90°). This relation is approximately linear up to vicinity of M=600 Nm when the ball located at ψ=180° occurs to be unloaded, that is, Q(ψ=180°)=0 for M=589.18 Nm (e=3.31×10-2 m).Figure18, as well as Figure 13, shows a substantial difference between results found in this work and those found by Harris. While Harris found a 5, 878 N magnitude for the maximum normal ball load (p. 252), for an applied load eccentricity of 50.8 mm, this work found a 9, 445 N maximum normal ball load. This represents an error of −62.2% in the normal load, meaning that the Harris calculation has underestimated the normal load for the maximum loaded ball.Figure19 shows the contact angle, β. It can be observed that the contact angle, β, under centric thrust load (e=0), is the same for all balls (41.417986227161386o). Increasing from zero the lever arm, an almost linear increase (decrease) in the contact angle is observed for the balls whose angular positions satisfy |ψ|<90° (|ψ|>90°). This relation is approximately linear up to vicinity of M=600 Nm when the ball located at ψ=180° occurs to be unloaded, that is, β(ψ=180°)=βf for M=589.18 Nm (e=33.1 mm).Figure19, as well as Figure 18 and Figure 13, shows a substantial difference between results found in this work and to those found by Harris. While Harris has assumed a contact angle magnitude of 41.6o for all balls (p. 252), under a 50.8 mm applied load eccentricity, contact angles ranging from 44.31727851159821o to 16.16919216282055o were found in this work while ψ were varied from ψ=0° to ±180o, respectively. This represents errors between −6.1% and +157.3% in the contact angles determination, meaning that the Harris calculation has underestimated (strongly overestimated) the contact angles for balls located at angular positions satisfying |ψ|<45°(|ψ|>45°).Figures20 and 22 show the semimajor axes of the ball/inner-race and ball/outer-race contact areas, ai and ao, respectively. It can be observed that ai and ao, under centric thrust load (e=0), are the same for all balls (2.069901480072 mm for ai and 2.025827993682 mm for ao) and that the major axes for the maximum loaded ball, in both cases, can reach values as high as 7.4 mm for moment about 900 Nm.Figures21 and 23 show the semiminor axes of the ball/inner-race and ball/outer-race contact areas, bi and bo, respectively. It can be observed that bi and bo, under centric thrust load (e=0), are the same for all balls (0.254108993896064 mm for bi and 0.293013306181356 mm for bo) and that the major axes for the maximum loaded ball, in both cases, can reach values as high as 0.9 mm for moment about 900 Nm.Figures24 and 25 show the elliptical eccentricity parameters for ball/inner-race and ball/outer-race contact, ki and ko, respectively. It can be observed that ki and ko, under centric thrust load (e=0), are the same for all balls (8.1457 for ki and 6.9138 for ko). Increasing from zero the lever arm, an almost linear increase (decrease) in the parameter ki is observed for the balls whose angular positions satisfy |ψ|>90°(|ψ|<90°), and an almost linear increase (decrease) in the parameter ko is observed for the balls whose angular positions satisfy |ψ|<90°(|ψ|>90°). These relations are approximately linear up to vicinity of M=600 Nm when the ball located at ψ=180° occurs to be unloaded. It can be observed that ki(ψ=180°)≅8.1631 and ko(ψ=180°)≅6.9024 when M=588.9687 Nm (e=3.3088×10-2 m).The increase of the moment above 588.9687 Nm (or lever arm above3.3088×10-2 m), causes the successive unloading of the ball pairs. At the points where the unloading occurs the values of the parameters ki and ko remain roughly equal to those indicated in the preceding paragraph for ψ=180°. So, it can be observed that the contact ellipse of the inner race is slightly more eccentric than that of the contact ellipse of the outer race. For M=900 Nm, for example, while ki varies numerically from 8.11, for the most heavily loaded ball, to 8.37, for the minimum loaded ball, ko varies from 6.83 to 6.37, respectively.


### 5.2. Numerical Results for Thrust Load Ranging from 0 up to 20, 000 N

Graphics for various thrust loads also are shown. The following graphics present curves for thrust loads ranging from 0 up to 20,000 N as functions of lever arm,e:(i)
partial axial displacement for the maximum loaded ball,Riθ (Figure 26),(ii)
relative axial displacement,δa (Figure 27),(iii)
total relative axial deflection for the maximum loaded ball,δt(ψ=0) (Figure 28),(iv)
total relative axial deflection for the minimum loaded ball,δt(ψ=180°) (Figure 29),(v)
loading zone,ψl (Figure 30),(vi)
distance between loci of inner and outer raceway groove curvature centers for the maximum loaded ball,s(ψ=0) (Figure 31),(vii)
distance between loci of inner and outer raceway groove curvature centers for the minimum loaded ball,s(ψ=180°) (Figure 32),(viii)
total normal ball deflection for the maximum loaded ball,δn(ψ=0) (Figure 33),(ix)
total normal ball deflection for the minimum loaded ball,δn(ψ=180°) (Figure 34),(x)
ball-raceway normal load for the maximum loaded ball,Q(ψ=0) (Figure 35),(xi)
ball-raceway normal load for the minimum loaded ball,Q(ψ=180°) (Figure 36),(xii)
contact angle for the maximum loaded ball,β(ψ=0) (Figure 37),(xiii)
contact angle for the minimum loaded ball,β(ψ=180°) (Figure 38),(xiv)
semimajor axis of the ball/inner-race contact area for the maximum loaded ball,ai(ψ=0)  (Figure 39),(xv)
semiminor axis of the ball/inner-race contact area for the maximum loaded ball,bi(ψ=0)  (Figure 40),(xvi)
semimajor axis of the ball/outer-race contact area for the maximum loaded ball,ao(ψ=0)  (Figure 41),(xvii)
semiminor axis of the ball/outer-race contact area for the maximum loaded ball,bo(ψ=0)  (Figure 42),(xviii)
semimajor axis of the ball/inner-race contact area for the minimum loaded ball,ai(ψ=180°)  (Figure 43),(xix)
semiminor axis of the ball/inner-race contact area for the minimum loaded ball,bi(ψ=180°)  (Figure 44),(xx)
semimajor axis of the ball/inner-race contact area for the minimum loaded ball,ao(ψ=180°)  (Figure 45),(xxi)
semiminor axis of the ball/inner-race contact area for the minimum loaded ball,bo(ψ=180°)  (Figure 46),(xxii)
elliptical eccentricity parameter of the ball/inner-race contact area for the maximum loaded ball,ki(ψ=0) (Figure 47),(xxiii)
elliptical eccentricity parameter of the ball/outer-race contact area for the maximum loaded ball,ko(ψ=0) (Figure 48),(xxiv)
elliptical eccentricity parameter of the ball/inner-race contact area for the minimum loaded ball,ki(ψ=180°)  (Figure 49),(xxv)
elliptical eccentricity parameter of the ball/outer-race contact area for the minimum loaded ball,ko(ψ=180°)   (Figure 50).Figure 26
Partial axial displacement for the maximum loaded ball,Riθ, as a function of lever arm, e.Figure 27
Relative axial displacement,δa, as a function of lever arm, e.Figure 28
Total relative axial deflection for the maximum loaded ball,δt(ψ=0), as a function of lever arm, e.Figure 29
Total relative axial deflection for the minimum loaded ball,δt(ψ=180°), as a function of lever arm, e.Figure 30
Loading zone,ψl, as a function of lever arm, e.Figure 31
Distance between loci of inner and outer raceway groove curvature centers for the maximum loaded ball,s(ψ=0), as a function of lever arm, e.Figure 32
Distance between loci of inner and outer raceway groove curvature centers for the minimum loaded ball,s(ψ=180°), as a function of lever arm, e.Figure 33
Total normal ball deflection for the maximum loaded ball,δn(ψ=0), as a function of lever arm, e.Figure 34
Total normal ball deflection for the minimum loaded ball,δn(ψ=180°), as a function of lever arm, e.Figure 35
Ball-raceway normal load for the maximum loaded ball,Q(ψ=0), as a function of lever arm, e.Figure 36
Ball-raceway normal load for the minimum loaded ball,Q(ψ=180°), as a function of lever arm, e.Figure 37
Contact angle for the maximum loaded ball,β(ψ=0), as a function of lever arm, e.Figure 38
Contact angle for the minimum loaded ball,β(ψ=180°), as a function of lever arm, e.Figure 39
Semimajor axis of the ball/inner-race contact area for the maximum loaded ball,ai(ψ=0), as a function of lever arm, e.Figure 40
Semiminor axis of the ball/inner-race contact area for the maximum loaded ball,bi(ψ=0), as a function of lever arm, e.Figure 41
Semimajor axis of the ball/outer-race contact area for the maximum loaded ball,ao(ψ=0), as a function of lever arm, e.Figure 42
Semiminor axis of the ball/outer-race contact area for the maximum loaded ball,bo(ψ=0), as a function of lever arm, e.Figure 43
Semimajor axis of the ball/inner-race contact area for the minimum loaded ball,ai(ψ=180°), as a function of lever arm, e.Figure 44
Semiminor axis of the ball/inner-race contact area for the minimum loaded ball,bi(ψ=180°), as a function of lever arm, e.Figure 45
Semimajor axis of the ball/outer-race contact area for the minimum loaded ball,ao(ψ=180°), as a function of lever arm, e.Figure 46
Semiminor axis of the ball/outer-race contact area for the minimum loaded ball,bo(ψ=180°), as a function of lever arm, e.Figure 47
Elliptical eccentricity parameter of the ball/inner-race contact area for the maximum loaded ball,ki(ψ=0), as a function of lever arm, e.Figure 48
Elliptical eccentricity parameter of the ball/outer-race contact area for the maximum loaded ball,ko(ψ=0), as a function of lever arm, e.Figure 49
Elliptical eccentricity parameter of the ball/inner-race contact area for the minimum loaded ball,ki(ψ=180°), as a function of lever arm, e.Figure 50
Elliptical eccentricity parameter of the ball/outer-race contact area for the minimum loaded ball,ko(ψ=180°), as a function of lever arm, e.Due to the size quite extensive of the paper comments about the figures will be omitted from now on.


## 5.1. Numerical Results for a 17,800 N Thrust Load

Since it is the qualitative behavior of solutions that is the interest, the results are presented here in graphical form.Initially, for comparative purposes with the Harris work, a specific thrust loadFa=17,800 N was chosen to be applied, and the following graphical results are presented as functions of the moment, M=Fae:(i)
relative angular displacement,θ (Figure 9),(ii)
partial axial displacement,Riθcosψ (Figure 10),(iii)
relative axial displacement,δa (Figure 11),(iv)
total relative axial deflection,δt (Figure 12),(v)
loading zone,ψl (Figure 13),(vi)
distance between loci of inner and outer raceway groove curvature centers,s (Figure 14),(vii)
maximum elastic compression at the ball/inner-race contact,δi (Figure 15),(viii)
maximum elastic compression at the ball/outer-race contact,δo (Figure 16),(ix)
total normal ball deflection,δn (Figure 17),(x)
ball-raceway normal load,Q (Figure 18),(xi)
contact angle,β (Figure 19),(xii)
semimajor axis of the ball/inner-race contact area,ai (Figure 20),(xiii)
semiminor axis of the ball/inner-race contact area,bi (Figure 21),(xiv)
semimajor axis of the ball/outer-race contact area,ao (Figure 22),(xv)
semiminor axis of the ball/outer-race contact area,bo (Figure 23),(xvi)
elliptical eccentricity parameter for ball/inner-race contact,ki (Figure 24),(xvii)
elliptical eccentricity parameter for ball/outer-race contact,ko (Figure 25).Figure 9
Relative angular misalignment,θ, for 17,800 N thrust load, as a function of the Moment, M.Figure 10
Partial axial deflection,Riθcosψ, for each ball and 17,800 N thrust load, as a function of the Moment, M.Figure 11
Axial deflection,δa, for 17,800 N thrust load, as a function of the Moment, M.Figure 12
Total axial deflection,δt, for 17,800 N thrust load, as a function of the Moment, M.Figure 13
Loading zone,ψl, for 17,800 N thrust load, as a function of the Moment, M.Figure 14
Distance between curvature centers,s, for 17,800 N thrust load, as a function of the Moment, M.Figure 15
Maximum normal elastic compression at the ball/inner race contact,δi, for 17,800 N thrust load, as a function of the Moment, M.Figure 16
Maximum normal elastic compression at the ball/outer-race contact,δo, for 17,800 N thrust load, as a function of the Moment, M.Figure 17
Total ball deflection,δn, for 17,800 N thrust load, as a function of the Moment, M.Figure 18
Normal ball load,Q, for 17,800 N thrust load, as a function of the Moment, M.Figure 19
Contact Angle,β, for 17,800 N thrust load, as a function of the Moment, M.Figure 20
Semimajor axis of the ball/inner-race contact area,ai, for 17,800 N thrust load, as a function of the Moment, M.Figure 21
Semiminor axis of the ball/inner-race contact area,bi, for 17,800 N thrust load, as a function of the Moment, M.Figure 22
Semimajor axis of the ball/outer-race contact area,ao, for 17,800 N thrust load, as a function of the Moment, M.Figure 23
Semiminor axis of the ball/outer-race contact area,bo, for 17,800 N thrust load, as a function of the Moment, M.Figure 24
Elliptical eccentricity parameter for ball/inner-race contact,ki, for 17,800 N thrust load, as a function of the Moment, M.Figure 25
Elliptical eccentricity parameter for ball/outer-race contact,ko, for 17,800 N thrust load, as a function of the Moment, M.The graphics above, with exception of Figures 9, 11, and 13, show one curve for each ball angular position.Figures9 and 10 show the relative angular misalignment, θ,  and the partial axial deflection for each ball, Riθcosψ, respectively. It is observed that there is an approximately linear relationship between the misalignment angle, θ, and applied moment, M, for moment values ranging from zero up to about 600 Nm, which corresponds to a distance e of approximately 33.7 mm. Keeping the load constant and increasing the lever arm, e, above this value, it can be observed a deeper increase in the misalignment angle and, therefore, in the resultant axial deflection, Riθ.From Figure9 it can be observed that for an applied moment of 900 Nm (e≅50.6 mm) the angular misalignment can be as high as a quarter of degree.As already been waited for, from Figure10 it can be observed that the partial axial deflection is symmetrical with respect to the horizontal axis (null displacement) and that the displacement is null for the balls located at ψ=±90°.Figure11 shows the axial deflection, δa. It is observed that the axial deflection, δa, is approximately constant for moment values where the relationship between θ and M is approximately linear, that is, from zero up to about 600 Nm (e≅33.7 mm). For higher moment values the axial deflection falls abruptly and becomes negative in the vicinity of 800 Nm (e≅44.9 mm). The deeper increase in θ due to the increase in the lever arm forces the decrease of δa to preserve the force and moment static balances.Figure12 shows the total axial deflection, δt. It can be observed that the total axial deflection, δt, is the axial deflection, δa, in two situations: under centric thrust load (e=0), where all balls have the same axial deflection (3.6011095400455×10-5 m), and under eccentric thrust load for balls located at ψ=±90°. Increasing from zero the lever arm, an almost linear increase (decrease) in the total axial deflection is observed for the balls whose angular positions satisfy |ψ|<90° (|ψ|>90°). This relation is approximately linear up to vicinity of M=600 Nm when the ball located at ψ=180° occurs to be unloaded, that is, δt(ψ=180°)=0 for M=588.9687 Nm (e=3.3088×10-2 m).From Figure12 it is observed that for eccentricity of about 50 mm the total axial deflection of the most heavily loaded ball can reach one tenth of millimeter.The Figure13 shows the loading zone, ψl. The increase of the moment above 587.4 Nm (or lever arm above 3.3×10-2 m) causes the decrease of the loading zone from initial value ψl=±180°, with the successive unloading of the balls pairs located at ψ=±157.5° (M=609.448 Nm), ψ=±135° (M=661.1407 Nm), ψ=±112.5° (M=729.9584 Nm), ψ=±90° (M=803.9741 Nm), and ψ=±67.5° (M=873.7125 Nm), respectively. Going ahead cause the unloading of the balls pair located at ψ=±45º. However, it is not advisable to go beyond M=900 Nm, once the radial displacements between curvature centers start to acquire micrometer order values and they cannot more be disregarded.Figure13 shows a substantial difference between results found in this work and  those found by Harris. While Harris found a loading zone of 92.86º (p. 252) for an eccentricity of 50.8 mm, this work found a loading zone of 53.66º. Considering the last result as reference, this represents an error of +73% in the loading angle, meaning that Harris calculation has underestimated the effect of the moment M.Figure14 shows the distance between loci of inner and outer raceway groove curvature centers, s. It can be observed that the distance, s, under centric thrust load (e=0), is the same for all balls (1.053468971830×10-3 m). Increasing from zero the lever arm, an almost linear increase (decrease) in the distance, s, is observed for the balls whose angular positions satisfy |ψ|<90° (|ψ|>90°). This relation is approximately linear up to vicinity of M=600 Nm when the ball located at ψ=180° occurs to be unloaded, that is, s(ψ=180°)=A for M=588.9687 Nm (e=3.3088×10-2 m).The increase of the moment above 588.9687 Nm (or lever arm above3.3088×10-2 m) causes the decrease of the loading zone, as already explained, with the successive unloading of the ball pairs. At the points where the unloading occurs it is observed that the distance s falls below of the distance between centers of curvature, A, for the unloaded bearing.Figures15 and 16 show the maximum normal elastic compressions at the ball/inner-race and ball/outer-race contacts, δi and δo, respectively. It can be observed that δi and δo, under centric thrust load (e=0), are the same for all balls (1.18852986717367×10-5 m for δi and 1.15836731583185×10-5 m for δo) and that the deformation for the maximum loaded ball, in both cases, can reach values as high as 36 μm for moment about 900 Nm.Figure17 shows the total normal ball deflection, δn, that can be obtained by summing the maximum normal elastic compressions on the inner and outer races, δi and δo, or by subtracting A from s, once δn=s-A>0 also. It can be observed that δn, under centric thrust load (e=0), is the same for all balls (2.3468971830055×10-5m) and that the total normal elastic deformation for the maximum loaded ball can reach values as high as 70 μm for moment about 900 Nm.Figure18 shows the normal ball load, Q. It can be observed that the normal ball load, Q, under centric thrust load (e=0), is the same for all balls (1, 681.663561507042 N). Increasing from zero the lever arm, an almost linear increase (decrease) in the normal ball load is observed for the balls whose angular positions satisfy |ψ|<90° (|ψ|>90°). This relation is approximately linear up to vicinity of M=600 Nm when the ball located at ψ=180° occurs to be unloaded, that is, Q(ψ=180°)=0 for M=589.18 Nm (e=3.31×10-2 m).Figure18, as well as Figure 13, shows a substantial difference between results found in this work and those found by Harris. While Harris found a 5, 878 N magnitude for the maximum normal ball load (p. 252), for an applied load eccentricity of 50.8 mm, this work found a 9, 445 N maximum normal ball load. This represents an error of −62.2% in the normal load, meaning that the Harris calculation has underestimated the normal load for the maximum loaded ball.Figure19 shows the contact angle, β. It can be observed that the contact angle, β, under centric thrust load (e=0), is the same for all balls (41.417986227161386o). Increasing from zero the lever arm, an almost linear increase (decrease) in the contact angle is observed for the balls whose angular positions satisfy |ψ|<90° (|ψ|>90°). This relation is approximately linear up to vicinity of M=600 Nm when the ball located at ψ=180° occurs to be unloaded, that is, β(ψ=180°)=βf for M=589.18 Nm (e=33.1 mm).Figure19, as well as Figure 18 and Figure 13, shows a substantial difference between results found in this work and to those found by Harris. While Harris has assumed a contact angle magnitude of 41.6o for all balls (p. 252), under a 50.8 mm applied load eccentricity, contact angles ranging from 44.31727851159821o to 16.16919216282055o were found in this work while ψ were varied from ψ=0° to ±180o, respectively. This represents errors between −6.1% and +157.3% in the contact angles determination, meaning that the Harris calculation has underestimated (strongly overestimated) the contact angles for balls located at angular positions satisfying |ψ|<45°(|ψ|>45°).Figures20 and 22 show the semimajor axes of the ball/inner-race and ball/outer-race contact areas, ai and ao, respectively. It can be observed that ai and ao, under centric thrust load (e=0), are the same for all balls (2.069901480072 mm for ai and 2.025827993682 mm for ao) and that the major axes for the maximum loaded ball, in both cases, can reach values as high as 7.4 mm for moment about 900 Nm.Figures21 and 23 show the semiminor axes of the ball/inner-race and ball/outer-race contact areas, bi and bo, respectively. It can be observed that bi and bo, under centric thrust load (e=0), are the same for all balls (0.254108993896064 mm for bi and 0.293013306181356 mm for bo) and that the major axes for the maximum loaded ball, in both cases, can reach values as high as 0.9 mm for moment about 900 Nm.Figures24 and 25 show the elliptical eccentricity parameters for ball/inner-race and ball/outer-race contact, ki and ko, respectively. It can be observed that ki and ko, under centric thrust load (e=0), are the same for all balls (8.1457 for ki and 6.9138 for ko). Increasing from zero the lever arm, an almost linear increase (decrease) in the parameter ki is observed for the balls whose angular positions satisfy |ψ|>90°(|ψ|<90°), and an almost linear increase (decrease) in the parameter ko is observed for the balls whose angular positions satisfy |ψ|<90°(|ψ|>90°). These relations are approximately linear up to vicinity of M=600 Nm when the ball located at ψ=180° occurs to be unloaded. It can be observed that ki(ψ=180°)≅8.1631 and ko(ψ=180°)≅6.9024 when M=588.9687 Nm (e=3.3088×10-2 m).The increase of the moment above 588.9687 Nm (or lever arm above3.3088×10-2 m), causes the successive unloading of the ball pairs. At the points where the unloading occurs the values of the parameters ki and ko remain roughly equal to those indicated in the preceding paragraph for ψ=180°. So, it can be observed that the contact ellipse of the inner race is slightly more eccentric than that of the contact ellipse of the outer race. For M=900 Nm, for example, while ki varies numerically from 8.11, for the most heavily loaded ball, to 8.37, for the minimum loaded ball, ko varies from 6.83 to 6.37, respectively.


## 5.2. Numerical Results for Thrust Load Ranging from 0 up to 20, 000 N

Graphics for various thrust loads also are shown. The following graphics present curves for thrust loads ranging from 0 up to 20,000 N as functions of lever arm,e:(i)
partial axial displacement for the maximum loaded ball,Riθ (Figure 26),(ii)
relative axial displacement,δa (Figure 27),(iii)
total relative axial deflection for the maximum loaded ball,δt(ψ=0) (Figure 28),(iv)
total relative axial deflection for the minimum loaded ball,δt(ψ=180°) (Figure 29),(v)
loading zone,ψl (Figure 30),(vi)
distance between loci of inner and outer raceway groove curvature centers for the maximum loaded ball,s(ψ=0) (Figure 31),(vii)
distance between loci of inner and outer raceway groove curvature centers for the minimum loaded ball,s(ψ=180°) (Figure 32),(viii)
total normal ball deflection for the maximum loaded ball,δn(ψ=0) (Figure 33),(ix)
total normal ball deflection for the minimum loaded ball,δn(ψ=180°) (Figure 34),(x)
ball-raceway normal load for the maximum loaded ball,Q(ψ=0) (Figure 35),(xi)
ball-raceway normal load for the minimum loaded ball,Q(ψ=180°) (Figure 36),(xii)
contact angle for the maximum loaded ball,β(ψ=0) (Figure 37),(xiii)
contact angle for the minimum loaded ball,β(ψ=180°) (Figure 38),(xiv)
semimajor axis of the ball/inner-race contact area for the maximum loaded ball,ai(ψ=0)  (Figure 39),(xv)
semiminor axis of the ball/inner-race contact area for the maximum loaded ball,bi(ψ=0)  (Figure 40),(xvi)
semimajor axis of the ball/outer-race contact area for the maximum loaded ball,ao(ψ=0)  (Figure 41),(xvii)
semiminor axis of the ball/outer-race contact area for the maximum loaded ball,bo(ψ=0)  (Figure 42),(xviii)
semimajor axis of the ball/inner-race contact area for the minimum loaded ball,ai(ψ=180°)  (Figure 43),(xix)
semiminor axis of the ball/inner-race contact area for the minimum loaded ball,bi(ψ=180°)  (Figure 44),(xx)
semimajor axis of the ball/inner-race contact area for the minimum loaded ball,ao(ψ=180°)  (Figure 45),(xxi)
semiminor axis of the ball/inner-race contact area for the minimum loaded ball,bo(ψ=180°)  (Figure 46),(xxii)
elliptical eccentricity parameter of the ball/inner-race contact area for the maximum loaded ball,ki(ψ=0) (Figure 47),(xxiii)
elliptical eccentricity parameter of the ball/outer-race contact area for the maximum loaded ball,ko(ψ=0) (Figure 48),(xxiv)
elliptical eccentricity parameter of the ball/inner-race contact area for the minimum loaded ball,ki(ψ=180°)  (Figure 49),(xxv)
elliptical eccentricity parameter of the ball/outer-race contact area for the minimum loaded ball,ko(ψ=180°)   (Figure 50).Figure 26
Partial axial displacement for the maximum loaded ball,Riθ, as a function of lever arm, e.Figure 27
Relative axial displacement,δa, as a function of lever arm, e.Figure 28
Total relative axial deflection for the maximum loaded ball,δt(ψ=0), as a function of lever arm, e.Figure 29
Total relative axial deflection for the minimum loaded ball,δt(ψ=180°), as a function of lever arm, e.Figure 30
Loading zone,ψl, as a function of lever arm, e.Figure 31
Distance between loci of inner and outer raceway groove curvature centers for the maximum loaded ball,s(ψ=0), as a function of lever arm, e.Figure 32
Distance between loci of inner and outer raceway groove curvature centers for the minimum loaded ball,s(ψ=180°), as a function of lever arm, e.Figure 33
Total normal ball deflection for the maximum loaded ball,δn(ψ=0), as a function of lever arm, e.Figure 34
Total normal ball deflection for the minimum loaded ball,δn(ψ=180°), as a function of lever arm, e.Figure 35
Ball-raceway normal load for the maximum loaded ball,Q(ψ=0), as a function of lever arm, e.Figure 36
Ball-raceway normal load for the minimum loaded ball,Q(ψ=180°), as a function of lever arm, e.Figure 37
Contact angle for the maximum loaded ball,β(ψ=0), as a function of lever arm, e.Figure 38
Contact angle for the minimum loaded ball,β(ψ=180°), as a function of lever arm, e.Figure 39
Semimajor axis of the ball/inner-race contact area for the maximum loaded ball,ai(ψ=0), as a function of lever arm, e.Figure 40
Semiminor axis of the ball/inner-race contact area for the maximum loaded ball,bi(ψ=0), as a function of lever arm, e.Figure 41
Semimajor axis of the ball/outer-race contact area for the maximum loaded ball,ao(ψ=0), as a function of lever arm, e.Figure 42
Semiminor axis of the ball/outer-race contact area for the maximum loaded ball,bo(ψ=0), as a function of lever arm, e.Figure 43
Semimajor axis of the ball/inner-race contact area for the minimum loaded ball,ai(ψ=180°), as a function of lever arm, e.Figure 44
Semiminor axis of the ball/inner-race contact area for the minimum loaded ball,bi(ψ=180°), as a function of lever arm, e.Figure 45
Semimajor axis of the ball/outer-race contact area for the minimum loaded ball,ao(ψ=180°), as a function of lever arm, e.Figure 46
Semiminor axis of the ball/outer-race contact area for the minimum loaded ball,bo(ψ=180°), as a function of lever arm, e.Figure 47
Elliptical eccentricity parameter of the ball/inner-race contact area for the maximum loaded ball,ki(ψ=0), as a function of lever arm, e.Figure 48
Elliptical eccentricity parameter of the ball/outer-race contact area for the maximum loaded ball,ko(ψ=0), as a function of lever arm, e.Figure 49
Elliptical eccentricity parameter of the ball/inner-race contact area for the minimum loaded ball,ki(ψ=180°), as a function of lever arm, e.Figure 50
Elliptical eccentricity parameter of the ball/outer-race contact area for the minimum loaded ball,ko(ψ=180°), as a function of lever arm, e.Due to the size quite extensive of the paper comments about the figures will be omitted from now on.


## 6. Conclusions

The importance of this work lies in the fact that it uses a new procedure for gettinnng numerically, accurately, and quickly the static load distribution of a single-row, angular-contact ball bearings, subjected to a known thrust load which is applied to a variable distance from the geometric bearing center line. Precise applications, as for example, space applications, require a precise determination of the static loading. Models available in literature are approximate and often are not compatible with the desired degree of accuracy. This work can be extended to determine the loading on high-speed bearings where centrifugal and gyroscopic forces are not discarded. The results of this work can be used in the accurate determination of the friction torque of the ball bearings, under any operating condition of temperature and speed.



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*Source: 471804-2010-03-03.xml*