Varying stiffness and load distributions in defective ball bearings: Analyticalformulation and application to defect size estimation

Dick Petersena, Carl Howarda, Zebb Primea

aSchool of Mechanical Engineering, The University of Adelaide, Adelaide SA 5005, Australia

Abstract

This paper presents an analytical formulation of the load distribution and varying effective stiffness of a ball bearingassembly with a raceway defect of varying size, subjected to static loading in the radial, axial and rotational degreesof freedom. The analytical formulation is used to study the effect of the size of the defect on the load distribution andvarying stiffness of the bearing assembly. The study considers a square-shaped outer raceway defect centered in theload zone and the bearing is loaded in the radial and axial directions while the moment loads are zero. Analysis ofthe load distributions shows that as the defect size increases, defect-free raceway sections are subjected to increasedstatic loading when one or more balls completely or partly destress when positioned in the defect zone. The stiffnessvariations that occur when balls pass through the defect zone are significantly larger and change more rapidly at thedefect entrance and exit than the stiffness variations that occur for the defect-free bearing case. These larger, morerapid stiffness variations generate parametric excitations which produce the low frequency defect entrance and exitevents typically observed in the vibration response of a bearing with a square-shaped raceway defect. Analysis of thestiffness variations further shows that as the defect size increases, the mean radial stiffness decreases in the loadedradial and axial directions and increases in the unloaded radial direction. The effects of such stiffness changes on thelow frequency entrance and exit events in the vibration response are simulated with a multi-body nonlinear dynamicmodel. Previous work used the time difference between the low frequency entrance event and the high frequency exitevent to estimate the size of the defect. However, these previous defect size estimation techniques cannot distinguishbetween defects that differ in size by an integer number of the ball angular spacing, and a third feature of the vibrationresponse is therefore required to distinguish between such defects. It is hypothesised and validated through simulationsthat the third distinguishing feature is the characteristic frequencies of the low frequency event.

Keywords: Ball bearing, varying stiffness, defect, vibration, condition monitoring

1. Introduction

Ball bearings are used in a wide variety of rotating machinery and their failure is one of the most common reasonsfor machine breakdowns. Excessive bearing vibrations can be caused by distributed defects, such as surface rough-ness [1, 2], waviness [3–10], misaligned races, and off-size balls [3], or localized defects caused by pitting or spallingof the raceways or balls [11–22]. Raceway spalls are a common failure mode in ball bearings [23] and are initiatedby sub-surface fatigue cracks which appear after some time even when the bearing is properly lubricated, aligned,and loaded. Eventually, the sub-surface cracks grow and break through to the surface causing a spall or crack. Overtime, the spall develops into more significant damage extending across a larger section of the raceway [24]. Thispaper considers the effect of a raceway defect of varying circumferential extent on the varying effective stiffness andvibration response of a defective bearing assembly.

There are two excitation mechanisms that generate the vibration response of a bearing with a raceway defect.The first excitation mechanism is the force generated by the ball mass striking the raceway in the defect zone, e.g.upon exiting the defect, which typically produces a high frequency event in the vibration response [25]. The secondexcitation mechanism is the parametric excitation caused by the varying stiffness of the defective bearing assembly,which leads to low frequency events in the vibration response when balls enter or exit the defect [25, 26]. A methodfor calculating the varying effective stiffness of a bearing assembly with a raceway defect was presented in Ref. [26]

Preprint submitted to Journal of Sound and Vibration May 22, 2014

but this work only considered radially loaded bearings, resulting in the formulation of a 2 × 2 stiffness matrix whichonly included radial stiffness terms. This paper extends the method to formulate a fully populated 5×5 stiffness matrixfor a defective ball bearing which includes the stiffness terms for the radial as well as the axial and rotational degreesof freedom. Additionally, this paper presents a comprehensive study of the effect of the circumferential extent of thedefect on the varying stiffness whereas previous work [26] only considered two defect circumferential extents. Theextension to a 5×5 stiffness matrix is based on the bearing stiffness matrix formulations for defect-free single row ballbearings developed in [27–31]. The analytical method can be adapted to other types of bearings, such as double rowor self-aligning ball bearings, in a similar manner by utilising stiffness matrix formulations for the defect-free bearingcase [32–36].

To estimate the size of a raceway defect, previous studies [25, 37] have used the time difference between the lowand high frequency events typically observed in the vibration response of defective bearings. However, for defects thatdiffer in circumferential extent by the ball angular spacing, the time difference between these events will be the samewhich means the developed defect size estimation techniques [25, 37] cannot distinguish between them. It is shownhere, based on an analysis of the varying stiffness as well as the rigid body modes of the defective bearing assemblywhich are excited when balls enter and exit the defect [26], that such defects can be distinguished based on thedifference in the characteristic frequencies of the low frequency vibration events they produce. This is demonstratedby simulating and analyzing the vibration response of a defective bearing assembly for the case of two defects thatdiffer in size by the ball angular spacing. The simulated vibration response is generated using a previously developedmulti-body nonlinear dynamic model of a defective bearing on a fan test rig [19, 26], which does not consider themass of the balls since the aim here is to analyze the vibration response caused by the varying stiffness excitations.The presented simulations and analysis also demonstrate how the analytical methods developed here can be used toobtain greater insight into the vibration response predicted by other multi-body nonlinear dynamic models of defectivebearings [17, 18, 21, 22]. Additionally, the presented analytical methods will be of benefit to future research into theas yet unknown relative importance of the two excitation mechanisms in defective bearings, i.e. varying stiffnessversus ball mass striking the raceway.

This paper is outlined as follows. Section 2 presents the analytical formulation of the varying stiffness and loaddistribution of a ball bearing assembly with a raceway defect of varying circumferential extent, depth, and surfaceroughness. In Section 3, the presented analytical formulation is applied to study the effect of the circumferential extentof the defect on the varying bearing stiffness and load distribution for the case of a square-shaped outer raceway defect.Section 4 considers a radially loaded bearing with square-shaped outer raceway defects that differ in circumferentialextent by the ball angular spacing and analyzes the variations in the bearing stiffness and natural frequencies of therigid body modes of a defective bearing assembly. The results are correlated to the simulated vibration response forthe two defective bearings which demonstrates that the characteristic frequency of the low frequency event can beused to distinguish the two defects when estimating their size. Concluding remarks are presented in Section 5.

2. Formulation of varying stiffness of a defective ball bearing

This section presents the analytical formulation of the load distribution and varying effective stiffness of a ballbearing assembly with a raceway defect of varying circumferential extent, depth, and surface roughness. The varyingstiffness is calculated as a function of cage angular position while assuming the cage is stationary. An outer racewaydefect is considered in this paper and the method can easily be adapted to the case of an inner raceway defect.

2.1. Bearing macro-geometry and kinematicsFigure 1 presents a diagram of a ball bearing with an outer raceway defect as well as the coordinate system and

nomenclature used in the formulation of the bearing stiffness matrix. The relative displacements and rotations betweenthe inner and outer raceways are contained in a vector q defined as

q = [δx δy δz θx θy]T, (1)

where δx, δy and δz are the relative displacements in the x, y and z directions, and θx and θy are the relative rotationsaround the x and y axes. The bearing is subjected to static loads and moments which are contained in a vector Fdefined as

F = [Fx Fy Fz Mx My]T, (2)

2

fj

ws

wc

y

x

dy

Fy

Dff

dxFx

r

Mx bx

My

by

dzFz

z

ai

aodzj

drj

A0

A

ra0

a

ff

d(f)

Figure 1: Diagram illustrating the ball bearing and the coordinate system used to describe relative displacements (δx,δy,δx) androtations (θx,θy) between the inner and outer raceways; (left) radial view; (right) axial view.

where Fx, Fy and Fz are the static force loads in the radial (x, y) and axial (z) directions, and Mx and My are the staticmoment loads around the x and y axes. The shaft in Figure 1 rotates at a run speed ωs = 2π fs. For a bearing withpitch diameter Dp, a loaded contact angle α, a ball diameter Db, the resulting nominal cage speed ωc = 2π fc is givenby

ωc =ωs

2

(1 − Db cosα

Dp

). (3)

The angular position φ j of ball j shown in Figure 1 is defined as

φ j = φc +2π( j − 1)

Nb, j = 1 to Nb, (4)

with φc the cage angular position and Nb the number of balls. Equation (4) defines the cage angular position tocoincide with ball j = 1. For the case of an outer raceway defect considered here, the defect frequency is given by theouter raceway ball pass frequency

fbpo = Nb fc. (5)

2.2. Defect depth profile

In the diagram of the bearing shown in Figure 1, the outer raceway defect has a circumferential extent defined by∆φ f and is centered at an angle φ f . The defect location φ f is constant for an outer raceway defect but rotates at theshaft speed ωs for an inner raceway defect, such that φ f (t) = φ f (0) +ωst. The parametric study presented in Section 3considers square-shaped outer raceway defects of depth h for which a defect depth profile d(φ) is generated as

d(φ) =

min

(h, r2

b −√

r2b − 0.25R2

o(φ − φ f + 0.5∆φ f )2)

if φ f − 0.5∆φ f ≤ φ < φ f

min(h, r2

b −√

r2b − 0.25R2

o(φ f + 0.5∆φ f − φ)2)

if φ f < φ ≤ φ f + 0.5∆φ f

0 if φ f + 0.5∆φ f ≤ φ ≤ φ f − 0.5∆φ f

(6)

where rb is the ball radius, and Ro the outer raceway radius. Equation (6) effectively describes the path of the centerof the ball as it traverses the square-shaped defect [19]. Defects with more complicated defect depth profiles, such asextended defects with significant surface waviness features, can be generated as described in Refs. [20, 26]. Figure 2

3

presents example defect depth profiles for square-shaped defects of depth h = 50 µm and circumferential extents∆φ f = 5 and 20◦, with the solid and dashed lines indicating the actual and modelled defect depth profiles, respectively.Note that in the example, the ball does not reach the bottom of the defect for ∆φ f = 5◦.

Actual defect depth profile

Modelled defect depth profile

Def

ect

dep

th(µ

m)

Angular position φ (◦)255 260 265 270 275 280 285

−10

0

10

20

30

40

50

60

70

Figure 2: Defect depth profiles for square-shaped defects of depth h = 50 µm and circumferential extents ∆φ f of 5◦ (black) and20◦ (gray); (solid line) actual defect depth profile; (dashed line) modelled defect depth profile.

2.3. Hertzian contact deformations and forces

In this section, the Hertzian contact deformations for the defect-free bearing case [31] are modified to account forthe presence of a raceway defect with a depth profile d(φ). The contact deformation δ j for ball j is given by

δ j =

A − A0, δ j > 00, δ j 6 0,

(7)

where A and A0 are the loaded and unloaded relative distance between the inner and outer raceway groove curvaturecenters ai and ao, respectively, as illustrated in Figure 1. The loaded relative distance A is defined as

A =

√δ∗2r j + δ∗2z j , (8)

where

δ∗r j = A0 cosα0 + δr j (9)δ∗z j = A0 sinα0 + δz j. (10)

In Equation (9), the effective displacement δr j and δz j of ball j in the radial and axial directions are calculated fromthe relative bearing displacements q as

δr j = δx cos φ j + δy sin φ j − rL − d(φ j) cosα0 (11)δz j = δz + rd(θx sin φ j − θy cos φ j) − d(φ j) sinα0, (12)

4

where d(φ j) is the defect depth profile evaluated at the ball angular position φ j, rL the radial clearance, and rd theradial distance of the inner raceway groove curvature center. The Hertzian contact force Q j associated with thecontact deformation δ j is defined by the load-deflection relation

Q j = Kδ1.5j , (13)

where the load-deflection factor K (units of N/m1.5) depends on the curvatures and material properties of the surfacesin contact [38]. The contact force Q j acts along the loaded contact angle α j which is defined as

tanα j =δ∗z j

δ∗r j. (14)

2.4. Load distribution

For a defective ball bearing subjected to static loads F defined in Equation (2), static equilibrium is achieved whenthe sum of the loads and moments generated by the contact forces Q j over all balls equals the applied static loads.Using Equations (7) to (14), the loads carried by the balls are thus found by solving the following set of nonlinearalgebraic equations as a function of the cage angular position φc

Fx

Fy

Fz

Mx

My

=

Nb∑j=1

Q j

cosα j cos φ j

cosα j sin φ j

sinα j

rd sinα j sin φ j

−rd sinα j cos φ j

=

Nb∑j=1

Fx j

Fy j

Fz j

Mx j

My j

. (15)

A Newton-Raphson method can be used to solve Equation (15) at each considered cage angular position, with theball positions φ j depending on the cage angular position as defined by Equation (4). The bearing displacements androtations that solve Equation (15) are defined as

q = [δx δy δz θx θy]T. (16)

Once the relative displacements and rotations q are solved as a function of the cage angular position, the correspondingcontact deformations are calculated using Equations (7) to (14), with solutions denoted by (·) similar to Equation (16).For a defect-free bearing, the solutions are found by solving Equation (15) while setting d(φ) = 0.

2.5. Bearing stiffness variations

The bearing stiffness variations are calculated by linearizing the force-displacement relationship defined by Equa-tions (7) to (14) at the relative displacements and rotations q that solve Equation (15). The resulting symmetric bearingstiffness matrix Kb is defined as

Kb =

kxx kxy kxz kxθx kxθy

· kyy kyz kyθx kyθy

· · kzz kzθx kzθy

· · · kθxθx kθxθy

· · · · kθyθy

=

∂Fx∂δx

∂Fx∂δy

∂Fx∂δz

∂Fx∂θx

∂Fx∂θy

· ∂Fy

∂δy

∂Fy

∂δz

∂Fy

∂θx

∂Fy

∂θy

· · ∂Fz∂δz

∂Fz∂θx

∂Fz∂θy

· · · ∂Mx∂θx

∂Mx∂θy

· · · · ∂My

∂θy

q=q

(17)

where analytical expressions for each element of the stiffness matrix are provided in Appendix A [27]. Each elementof the stiffness matrix Kb varies with cage angular position even for the case of a defect-free bearing for whichd(φ) = 0, leading to the well-known varying stiffness vibrations [31, 39]. However, a raceway defect typically causesmuch larger and faster stiffness variations compared to a defect-free bearing, and may also reduce the average stiffnessover a single cage rotation, especially when multiple balls are in the defect at once. This will be demonstrated by theparametric study presented in the next section.

5

3. Effect of defect size on the varying stiffness and load distribution of a bearing assembly

The study presented in this section considers the same angular contact ball bearing used by Liew & Lim [31] intheir study of the varying stiffness of a defect-free bearing. This bearing has Nb = 12 rolling elements, a ball radiusrb = 3 mm, a radius of the inner raceway groove curvature center of rd = 19.65 mm, an unloaded distance betweenthe inner and outer raceway groove curvature centers of A0 = 50 µm, an unloaded contact angle α0 = 30◦, and a radialclearance of 0.05 µm. The load-deflection factor for the Hertzian contacts is K = 1.45 · 1010 N/m1.5. The bearingis subjected to a static load Fy = −2 kN in the vertical radial direction and Fz =0.5 kN in the axial direction. Thehorizontal radial and moment static loads are zero. The bearing has a square-shaped outer raceway defect of depthh = 50 µm centered in the load zone at φ f = 270◦. The study analyzes the effect of the circumferential extent ∆φ f ofthe defect on the load distribution and varying stiffness of the defective bearing assembly, with the defect depth profiled(φ) defined by Equation (6). Note that this is different from the study presented in Ref. [31] where the bearing wasdefect-free and was preloaded in the radial direction rather than being subjected to a static load.

3.1. Load distribution in a defective ball bearing assembly

Figure 3 presents the load distribution for defect circumferential extents ∆φ f = 5◦, 40◦ and 70◦. The number ofballs positioned in the defect zone at any one time varies between none or one for ∆φ f = 5◦, one or two for ∆φ f = 40◦,and two or three for ∆φ f = 70◦. The results in Figure 3 illustrate the typical effects of a raceway defect on the staticload distribution. The load distribution for a defect-free bearing is shown as the gray line in Figure 3 for comparison,and the gray-shaded rectangles indicate the circumferential extent of the defects.

(c) Load distribution for ∆φ f=70◦C

on

tact

forc

eQ

j(k

N)

Ball angular position φ j (◦)

(b) Load distribution for ∆φ f=40◦

Co

nta

ctfo

rce

Qj

(kN

)

Ball angular position φ j (◦)

(a) Load distribution for ∆φ f=5◦

Co

nta

ctfo

rce

Qj

(kN

)

Ball angular position φ j (◦)0 45 90 135 180 225 270 315 360 0 45 90 135 180 225 270 315 360 0 45 90 135 180 225 270 315 360

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Figure 3: Load distribution for a defect-free bearing (gray) and a defective bearing (black) with a square-shaped outer raceway de-fect of depth h = 50 µm and circumferential extents ∆φ f = 5◦, 40◦ and 70◦. The gray-shaded rectangles indicate the circumferentialextent of the defect.

Figure 3 shows that when a ball is positioned in the defect, it destresses and loses all or part of its load carryingcapacity, and the load it carried is redistributed to the balls outside the defect zone. This indicates that defect-freeraceway sections are subjected to increased static loading when balls lose all or part of their load carrying capacitywhen positioned in the defect zone. In general, the load carrying capacity that is lost and the associated load redis-tribution will largely depend on the defect geometry, but also on the applied load and the radial clearance. When thedefect circumferential extent ∆φ f is smaller than the ball angular spacing 360◦/Nb = 30◦, the load distribution onlydeviates from the defect-free case when a single ball is in the defect zone, as shown in Figure 3(a). When the defectgrows larger such that ∆φ f > 30◦, multiple balls are in the defect at once and the load distribution deviates entirelyfrom the defect-free case, as shown in Figure 3(b) and (c). For this case, there is a large increase in the load carriedby the balls located outside the defect zone. The redistribution of the load results in large variations of the effectivestiffness of the bearing assembly due to the nonlinear nature of the Hertzian contact stiffness, which becomes stifferas the contact force Q j becomes larger.

6

3.2. Varying stiffness of a defective ball bearing assemblyFigure 4 compares the stiffness variations as a function of cage angular position φc for defect-free and defective

bearings with defects of circumferential extents ∆φ f = 5◦, 40◦ and 70◦, for which the corresponding load distributionswere shown in Figure 3. To further illustrate the typical effects of the circumferential extent of the defect on thevarying bearing stiffness, ∆φ f was varied from 0◦ to 90◦ and the results are presented in Figure 5. Figure 6 illustratesthe number of loaded balls as a function of cage angular position and the circumferential extent ∆φ f of the defect, andis included to explain the diamond-shaped patterns observed in the varying stiffness results included in Figure 5.

Figures 4 and 5 show that the stiffness varies periodically for both the defective and defect-free bearing cases, withthe fundamental period defined by the ball angular spacing of 30◦. This periodicity results in a parametric excitation atthe outer raceway defect frequency fbpo defined in Equation (5). Compared to the defect-free case, the bearing stiffnessvariations are larger and change more rapidly when a defect is present. These larger, more rapid stiffness variationsgenerate larger parametric excitations which, in combination with the ball mass striking the defective raceway, producethe low and high frequency events typically observed in the vibration response of defective bearings [25, 37]. Whenthe circumferential extent of the defect is smaller than the ball angular spacing, the varying stiffness differs fromthe defect-free case only when a ball is positioned in the defect, as shown in Figure 4 for ∆φ f = 5◦. When thecircumferential extent of the defect is greater than the ball angular spacing, one or more balls are positioned in thedefect at any one time such that the varying stiffness deviates entirely from the defect-free case, as shown in Figure 4for ∆φ f = 40◦ and 70◦.

The general appearance of the diamond-shaped patterns observed in Figure 5 is determined by the ball angularspacing of 30◦, with the downwards and upwards sloped lines corresponding to a ball entering and exiting the defect,respectively. The finer details of the stiffness variations within each diamond shape correspond to the changes in thenumber of loaded balls illustrated in Figure 6. As the circumferential extent of the defect increases, the number ofballs carrying the load reduces up to ∆φ f ≈ 64◦, which is the circumferential extent at which three balls exactly fitwithin the defect when taking into account the ball diameter. For larger defects, Figure 6 shows that the number ofloaded balls begins to increase again. This occurs because balls positioned in the defect zone become partly loadedas indicated by the load distribution for ∆φ f = 70◦ in Figure 3(c). Because the presence of the defect reduces thenumber of loaded balls compared to the defect-free bearing case, the balls positioned outside the defect zone need tocarry more of the load compared to the defect-free case, which was observed in Figure 3.

Figures 4 and 5(a–c) show that as the defect circumferential length ∆φ f increases, the mean translational stiffnessin the unloaded radial direction (kxx) increases up to ∆φ f < 64◦, while it decreases in the loaded radial and axialdirections (kyy and kzz). Similarly, Figures 4 and 5(m) and (n) show that the mean rotational stiffness around thex-axis (kθxθx ) reduces while it increases for rotation around the y-axis (kθyθy ). For larger defects with ∆φ f > 64◦,the mean translational stiffness kyy begins to increase again because, as was observed in Figure 6, the number ofloaded balls increases again due to balls positioned in the defect only losing part, rather than all, of their load carryingcapacity, as was observed in Figure 3(c). The mean translational stiffnesses change because, although the changes inthe carried load compared to the defect-free case observed in Figure 3 sum to zero over all balls to maintain staticforce equilibrium, this zero net effect does not occur for the contact stiffness changes due to the nonlinear nature ofthe Hertzian load-deflection relationship defined by Equation (13). The observed changes in the radial stiffness withdefect size will be utilized in Section 4 to distinguish between two defects that differ in size by the ball angular spacingwhen estimating their size.

The cross-coupling stiffness terms define how well excitation in one degree of freedom couples to produce rigidbody vibration of the bearing assembly in another degree of freedom. For example, the radial vibration response inthe x direction due to excitation in the radial y direction is defined by the cross-coupling stiffness kxy, which will bedemonstrated in Section 4. Figures 4 and 5 show that as the circumferential extent of the defect increases, the cross-coupling stiffness terms become increasingly important relative to the radial, axial, and rotational stiffness terms. Thismeans that as the defect grows in size, excitation in one degree of freedom will result in increased rigid body vibrationof the bearing assembly in the other degrees of freedom.

4. Effect of defect size on the low frequency vibration event used in defect size estimation

A number of bearing defect size estimation techniques have been developed [25, 37] which rely on estimatingthe time difference between the low and high frequency vibration events that occur as balls pass through a defect.

7

Defect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-free

Defect-freeDefect-freeDefect-freeDefect-free

Defect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-free

Defect-freeDefect-freeDefect-freeDefect-free

Defect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-free

Defect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-freeDefect-free

Defect-freeDefect-freeDefect-freeDefect-free

Defect-freeDefect-freeDefect-freeDefect-free

Defect-freeDefect-freeDefect-freeDefect-free

(o) kθxθy

Stiff

nes

s(k

Nm/r

ad)

Cage angular position φc(◦)

(n) kθyθy

Stiff

nes

s(k

Nm/r

ad)

Cage angular position φc(◦)

(m) kθxθx

Stiff

nes

s(k

Nm/r

ad)

Cage angular position φc(◦)

(l) kzθx

Stiff

nes

s(M

N/r

ad)

Cage angular position φc(◦)

(k) kyθx

Stiff

nes

s(M

N/r

ad)

Cage angular position φc(◦)

(j) kxθy

Stiff

nes

s(M

N/r

ad)

Cage angular position φc(◦)

(i) kzθy

Stiff

nes

s(M

N/r

ad)

Cage angular position φc(◦)

(h) kyθy

Stiff

nes

s(M

N/r

ad)

Cage angular position φc(◦)

(g) kxθx

Stiff

nes

s(M

N/r

ad)

Cage angular position φc(◦)

(f) kyz

Stiff

nes

s(M

N/m

)

Cage angular position φc(◦)

(e) kxz

Stiff

nes

s(M

N/m

)

Cage angular position φc(◦)

(d) kxy

Stiff

nes

s(M

N/m

)

Cage angular position φc(◦)

(c) kzz

Stiff

nes

s(M

N/m

)

Cage angular position φc(◦)

(b) kyy

Stiff

nes

s(M

N/m

)Cage angular position φc(◦)

(a) kxx

Stiff

nes

s(M

N/m

)

Cage angular position φc(◦)

∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦

∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦

∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦

∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦

∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦

∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦

∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦

∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦

∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦∆φ f = 5◦

40◦40◦40◦40◦40◦40◦40◦40◦

40◦40◦40◦40◦

40◦40◦40◦40◦40◦40◦40◦40◦

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70◦70◦70◦70◦70◦70◦70◦70◦

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70◦70◦70◦70◦

15 22.5 30 37.5 45 15 22.5 30 37.5 45 15 22.5 30 37.5 45

15 22.5 30 37.5 45 15 22.5 30 37.5 45 15 22.5 30 37.5 45

15 22.5 30 37.5 45 15 22.5 30 37.5 45 15 22.5 30 37.5 45

15 22.5 30 37.5 45 15 22.5 30 37.5 45 15 22.5 30 37.5 45

15 22.5 30 37.5 45 15 22.5 30 37.5 45 15 22.5 30 37.5 45

100

140

180

220

260

100

140

180

220

260

100

140

180

220

260

−30

−15

0

15

30

−30

−15

0

15

30

−30

−15

0

15

30

−0.3

−0.15

0

0.15

0.3

−0.3

−0.15

0

0.15

0.3

−0.3

−0.15

0

0.15

0.3

−1.35

−0.9

−0.45

0

0.45

0.9

−0.9

−0.45

0

0.45

0.9

1.35

−0.9

−0.45

0

0.45

0.9

1.35

10

20

30

40

50

10

20

30

40

50

−5

−2.5

0

2.5

5

Figure 4: Varying stiffness for a defect-free bearing and a defective bearing with a square-shaped outer raceway defect of depthh = 50 µm and circumferential extent ∆φ f = 5◦, 40◦ or 70◦.

8

Figure 5: Bearing stiffness variations for a square-shaped outer raceway defect of depth h = 50 µm and circumferential extent ∆φ f .

9

Figure 6: Variation in number of loaded balls with cage angular position φc for a square-shaped outer raceway defect of depthh = 50 µm and circumferential extent ∆φ f .

The high frequency event occurs when a ball exits the defect and corresponds to an excitation of the high frequencybearing resonances. The low frequency event occurs when a ball gradually destresses upon entering the defect [40] andcorresponds to a parametric excitation of the rigid body modes of the bearing assembly [26]. The previously developeddefect size estimation techniques [25, 37] cannot distinguish between defects that differ in circumferential extent by(an integer times) the ball angular spacing since the time difference between the low and high frequency vibrationevents will be identical for such defects. To distinguish between such defects when estimating their size, a thirdfeature of the vibration response is therefore required. This section hypothesises that under the same static loadingconditions, the third feature is the characteristic frequency of the low frequency event. As observed in Figure 5(a) and(b), once the circumferential extent of the defect becomes larger than the ball angular spacing, substantial changesoccur in the varying radial stiffness and therefore in the natural frequencies of the rigid body modes of the bearingassembly. Thus, defects that differ in circumferential extent by (an integer times) the ball angular spacing are expectedto produce low frequency events with substantially different characteristic frequencies. This hypothesis is validatedhere by simulating the vibration response for such defects using a previously developed multi-body nonlinear dynamicmodel of a defective bearing on a fan test rig [25, 26]. Future experimental work will be undertaken to further validatethe hypothesis.

4.1. Multi-body nonlinear dynamic model of the defective bearing assembly

The fan test rig used in a previous study [25] into defect size estimation techniques includes a fan disk with 19blades fitted on a shaft supported by ball bearings. These bearings have Nb = 14 balls, a pitch radius rd = 22.5 mm, aball radius rb = 4 mm, and an unloaded contact angle α0 = 0◦. The bearings are mounted on sleeves and are containedwithin plummer blocks. Square-shaped defects were inserted on the outer raceway of one bearing and the verticalvibration response was measured using an accelerometer mounted on the plummer block.

Figure 7 presents a diagram of the multi-body nonlinear dynamic model of the defective bearing on the fan testrig [19, 26]. Only radial displacements and bearing stiffness variations are considered because the ball bearing isradially loaded and the unloaded contact angle α0 = 0◦. The inner raceway and shaft are modelled as a single rigidbody of mass mi and its displacement is defined by xi and yi. The outer raceway and bearing support structure aremodelled as a single rigid body of mass mo and its displacements is defined by xo and yo. The high-frequency bearingresonance that was included in the model presented in Refs. [19, 26] is excluded here because the emphasis is varyingstiffness excitation of the rigid body modes of the bearing assembly. Further details of the model can be found in

10

fj

ws

wc

y

x

Fy

Dff

Fx

r

ff

d(f)

kox

cox

koy coy

xoxi

yi

yo

mo

mi

Figure 7: Multi-body nonlinear dynamic model of the defective bearing on the fan test rig used in a previous study into defect sizeestimation techniques [25]. The bearing has a square-shaped outer raceway defect of depth h = 50 µm and circumferential extent∆φ f = 12.9◦ or 38.6◦.

Refs. [19, 26]. The nonlinear equations of motion are now given by [26]

Mx(t) + Cx(t) + Kx(t) +

Nb∑j=1

[Kδ j(t) + cδ j(t)]R j(t) = F. (18)

For the considered case of a radially loaded bearing with an unloaded contact angle α0 = 0◦, the equations forthe contact deformations δ j(t) given in Equations (8) to (11) simplify to the equation given in Refs. [19, 26] sinceδz j = δ∗z j = 0, such that

δ j(t) = δx(t) cos φ j(t) + δy(t) sin φ j(t) − rL − d(φ j(t)), (19)

with the relative displacements between the inner and outer raceways given by δx(t) = xi(t) − xo(t) and δy(t) =

yi(t) − yo(t). In Equation (18), the static load vector F only contains the radial static loads Fx and Fy, and the statevector x(t) in Equation (18) is defined as

x(t) =[

xi(t) yi(t) xo(t) yo(t)]T. (20)

The matrix R j(t) in Equation (18) defines a transformation from orthogonal to radial coordinates and is formulated as

R j(t) =[

cos φ j(t) sin φ j(t) − cos φ j(t) − sin φ j(t)]T. (21)

The mass, stiffness and damping matrices in Equation (18) are defined as

M =

[Mi 00 Mo

]K =

[0 00 Ko

]C =

[0 00 Co

], (22)

with

Mi =

[mi 00 mi

]Mo =

[mo 00 mo

]Ko =

[kox 00 koy

]Co =

[cox 00 coy

]. (23)

11

The multi-body nonlinear dynamic model was implemented in Simulinkr and the equations of motion were solvedusing the ordinary differential equation solver (ode45) which is based on a Runga-Kutta method. The radial clearancewas set to rL = 3 µm and the bearing was subjected to a static radial load Fy = −100 N in the vertical directionwhile the static radial load in the horizontal direction Fx = 0 N. The other parameter values used in the modelare included in Table 1. The modeled outer raceway defects were centered in the load zone at φ f = 270◦ and had

Table 1: Parameter values used in the multi-body nonlinear dynamic model of the defective bearing on the fan test rig.

Hertzian contacts Mass Stiffness DampingK = 5453 MN/m1.5 mi = 2 kg kox = 212.2 MN/m cox = 1648.2 Ns/m

mo = 2 kg koy = 212.2 MN/m coy = 1648.2 Ns/m

circumferential extents of ∆φ f = 12.9◦ and 38.6◦ such that they exactly differ in size by the ball angular spacing360◦/Nb = 25.7◦. The running speed was set to fs = 7 Hz which results in a nominal outer raceway defect frequencyfbpo = Nb fc = 40.3 Hz. Initial conditions were set to the static displacements corresponding to the cage positionat time t = 0. The continuous time-domain results were discretized using a sample frequency Fs = 65, 536 Hz andacceleration spectra were calculated from 1.4 seconds of simulated data using a hanning window and a frequencyresolution of 32 Hz.

4.2. Rigid body modes of the linearized dynamic model of the defective bearing assembly

The nonlinear dynamic model defined by Equation (18) is now linearized at the quasi-static displacements x whichare calculated as described in Section 2.4, and the natural frequencies and damping ratios of the modes of the resultinglinearized system are defined. The linearized equations of motion are given by

M ¨x(t) + C ˙x(t) + Kx(t) = 0, (24)

where x(t) = x(t)− x are small displacements around the quasi-static solution x calculated as described in Section 2.4.The linearized stiffness and damping matrices in Equation (24) are formulated as

K =

[Kb −Kb

−Kb Kb + Ko

]C =

[Cb −Cb

−Cb Cb + Co

]. (25)

For the considered case of a radially loaded bearing with an unloaded contact angle α0, the expressions defined inAppendix A for the radial stiffness terms of the stiffness matrix Kb simplify to [26]

Kb =

[kxx kxy

kxy kyy

]= nK

Nb∑j=1

δn−1j

[cos2 φ j cos φ j sin φ j

cos φ j sin φ j sin2 φ j

](26)

and the damping matrix Cb can be formulated as [26]

Cb =

[cxx cxy

cxy cyy

]= c

Nb∑j=1

[cos2 φ j cos φ j sin φ j

cos φ j sin φ j sin2 φ j

]. (27)

The bearing stiffness matrix Kb in Equation (26) is calculated as described in Section 2 and varies with cage angularposition φc as well as the circumferential extent ∆φ f of the defect. The viscous contact damping constant c in Equa-tion (18) is adjusted such that the bearing damping terms cxx and cyy in Equation (27) for the defect-free case are inthe order of 0.25-2.5 × 10−5 times the linear bearing stiffness terms kxx and kyy, respectively [17].

The undamped natural frequencies ωn and damping ratios ζn of the modes of the linearized dynamic systemdefined by Equation (24) are calculated by solving the following generalized eigenvalue problem [32][

0 −K−M 0

]v = −λ

[M C0 M

]v. (28)

12

This results in eight eigenvalues λn (four complex conjugate pairs) and eigenvectors vn from which the undampednatural frequencies and damping ratios are calculated as

ωn = |λn|, ζn =|Re(λn)||λn| , n = 1, 2, . . . , 8 (29)

These natural frequencies vary periodically at the ball pass frequency due to the varying bearing stiffness Kb, and arealso dependent on the circumferential extent ∆φ f of the defect as demonstrated in the next section.

4.3. Variation in natural frequencies of rigid body modes of linearized dynamic system

Figure 8 compares the radial stiffness variations Kb and the corresponding variations in the natural frequenciesωn of the rigid body modes of the defective bearing assembly, for the two considered defects with circumferentialextents of ∆φ f = 12.9◦ and 38.6◦ as well as the defect-free case. The defect depth profiles d(φ) are presented as well.The bearing stiffness and natural frequencies of the rigid body modes vary periodically with the fundamental perioddefined by the ball angular spacing of 25.7◦. Figure 8(e) shows that the radial cross-coupling stiffness kxy is muchsmaller than the radial stiffnesses kxx and kyy presented in Figure 8(a) and (c). As a result, analysis of the eigenvectorsvn for each of the four rigid body modes shows that two modes have dominant motion in the x direction and two inthe y direction. The corresponding natural frequencies of these modes have been labelled as ωx1, ωx2, ωy1, and ωy2and their variation with cage position φc is presented in Figure 8(b) and (d). The results in these figures illustrate thatdefects with circumferential extents that differ by the ball angular spacing produce different variations in the naturalfrequencies of the rigid body modes. The mean natural frequencies ωy1 and ωy2 are lower for the larger defect with∆φ f = 38.6◦ because the mean radial stiffness kyy is lower compared to the smaller defect with ∆φ f = 12.9◦, as shownin Figure 8(c). The mean natural frequencies ωx1 and ωx2 are higher for the larger defect because the mean radialstiffness kxx is larger in Figure 8(a). As demonstrated in the next section, this results in low frequency events in thesimulated vibration response that have different characteristic frequencies.

4.4. Simulated vibration response

Figure 9(a–d) present the simulated vibration responses in the horizontal and vertical directions for the considereddefects with circumferential extents ∆φ f = 12.9◦ and 38.6◦. A short time segment is shown to illustrate a single defectentrance and exit event in the vibration response. Since a high frequency bearing resonance was not included in themodel, the simulated results do not contain the high frequency event that normally occurs in conjunction with a lowfrequency event when a balls exits the defect. Figure 9(e) and (f) present the contact forces for the balls that enterand exit the defect within the time segment shown, with the thin lines indicating the static contact forces Fx j and Fy j,which were calculated as described in Section 2.4, and the thick lines indicating the dynamic contact forces Fx j(t) andFy j(t), which fluctuate around and decay to the static contact forces as balls enter and exit the defect.

For the smaller defect with ∆φ f = 12.9◦, Figure 9(a) and (c) show that the horizontal vibration response has amuch lower amplitude than the vertical vibration response. This occurs because the bearing assembly is predominantlyexcited in the y direction due to the locations of the defect entrance and exit, and this excitation does not producesignificant rigid body motion in the x direction since the radial cross-coupling stiffness kxy is small as was observed inFigure 8. For the larger defect with ∆φ f = 38.6◦, Figure 9(b) and (d) show that the amplitudes of the horizontal andvertical vibration responses are more similar because the bearing assembly is excited in both the x and y directionsdue to the larger extent of the defect.

For the shorter defect with ∆φ f = 12.9◦, the contact forces presented in Figure 9(e) show that the entrance and exitevents are caused by a single ball ( j = 9) de- and restressing between the raceways as it passes through the defect. Forthe larger defect with ∆φ f = 38.6◦, the contact force results illustrated in Figure 9(f) show that the entrance and exitevents are caused by two different balls; one ball ( j = 9) destresses as it enters the defect and another ball ( j = 10)restresses between the raceways as it exits. The time difference ∆t = mod(∆φ f , 25.7◦)/ωc = 0.013s between theentrance and exit events is the same for both defects because the circumferential extents of the two defects differ bythe ball angular spacing of 25.7◦. The simulated results thus illustrate that a third feature is required to distinguishbetween the two defects when estimating their size.

Figure 10 presents the spectral densities of the simulated vibration responses and illustrates that the third featurethat can be used to distinguish between the two defects is the difference in the characteristic frequencies of the low

13

∆φ f =38.6◦

∆φ f =12.9◦Defect-free

∆φ f =38.6◦

∆φ f =12.9◦Defect-free

∆φ f =38.6◦

∆φ f =12.9◦Defect-free

∆φ f =38.6◦

∆φ f =12.9◦Defect-free

∆φ f =38.6◦

∆φ f =12.9◦Defect-free

∆φ f =38.6◦

∆φ f =12.9◦Defect-free

(f) Defect depth profile d(φ)

Angular position φ(◦)

Def

ect

dep

th(µ

m)

(d) Natural frequencies ωy1,2

Cage angular position φc(◦)

Fre

qu

ency

(kH

z)

(b) Natural frequencies ωx1,2

Cage angular position φc(◦)

Fre

qu

ency

(kH

z)

(e) Radial cross-coupling bearing stiffness kxy

Stiff

nes

sk

xy

(MN/m

)

Cage angular position φc(◦)

(c) Radial bearing stiffness kyy

Stiff

nes

sk y

y(M

N/m

)

Cage angular position φc(◦)

(a) Radial bearing stiffness kxx

Stiff

nes

sk

xx

(MN/m

)

Cage angular position φc(◦)0 5 10 15 20 25

0 5 10 15 20 25

0 5 10 15 20 25

0 5 10 15 20 25

0 5 10 15 20 25

180 210 240 270 300 330 360

0

10

20

30

40

50

0

10

20

30

40

50

−3

−2

−1

0

1

2

3

0.3

0.6

0.9

1.2

1.5

1.8

2.1

0.3

0.6

0.9

1.2

1.5

1.8

2.1

0

25

50

75

Figure 8: Variations in bearing stiffness and natural frequencies of the rigid body modes of the linearized dynamic system for thecase of a square-shaped outer raceway defect of depth h = 50 µm and circumferential extent ∆φ f = 12.9◦ and 38.6◦ centered atφ f = 270◦. (a) Radial bearing stiffness kxx. (b) Natural frequencies of modes with dominant motion in x-direction. (c) Radialbearing stiffness kyy. (d) Natural frequencies of modes with dominant motion in y-direction. (e) Radial cross-coupling bearingstiffness. (f) Modeled defect depth profiles.

14

j=10

j=9

j=9

← exit event

← entrance event← exit event← entrance event

(f) Contact force ∆φ f =38.6◦

Co

nta

ctfo

rce

Qj

(N)

Time (s)

(e) Contact force ∆φ f =12.9◦

Co

nta

ctfo

rce

Qj

(N)

Time (s)

(d) Vertical vibration response ∆φ f =38.6◦A

ccel

erat

ion

(m/s

2)

Time (s)

(c) Vertical vibration response ∆φ f =12.9◦

Acc

eler

atio

n(m/s

2)

Time (s)

(b) Horizontal vibration response ∆φ f =38.6◦

Acc

eler

atio

n(m/s

2)

Time (s)

(a) Horizontal vibration response ∆φ f =12.9◦

Acc

eler

atio

n(m/s

2)

Time (s)

0.745 0.75 0.755 0.76 0.765 0.77 0.775 0.745 0.75 0.755 0.76 0.765 0.77 0.775

0.745 0.75 0.755 0.76 0.765 0.77 0.775 0.745 0.75 0.755 0.76 0.765 0.77 0.775

0.745 0.75 0.755 0.76 0.765 0.77 0.775 0.745 0.75 0.755 0.76 0.765 0.77 0.775

−2

−1

0

1

2

−2

−1

0

1

2

−4

−2

0

2

4

−4

−2

0

2

4

0

10

20

30

40

50

0

10

20

30

40

50

60

Figure 9: Simulated vibration response and contact forces for the case of a square-shaped outer raceway defect of depth h = 50 µmand circumferential extent ∆φ f = 12.9◦ and 38.6◦ centered at φ f = 270◦. (a) Horizontal acceleration for ∆φ f = 12.9◦. (b)Horizontal acceleration for ∆φ f = 38.6◦. (c) Vertical acceleration for ∆φ f = 12.9◦. (d) Vertical acceleration for ∆φ f = 38.6◦. (e)Contact force Q j(t) for ball j = 9 (thick line) passing through defect with ∆φ f = 12.9◦ with thin line indicating the static loadsolution Q j. (f) Contact forces Q j(t) (thick lines) for balls entering ( j = 9) and exiting ( j = 10) defect with ∆φ f = 38.6◦ with thinlines indicating the static load solution Q j. 15

frequency events. The dark (∆φ f = 12.9◦) and light (∆φ f = 38.6◦) gray rectangles in Figure 10 indicate the frequencyranges within which the natural frequencies ωx1, ωx2, ωy1 and ωy2 of the rigid body modes of the bearing assemblyvary, as illustrated in Figure 8(b) and (d).

∆φ f =38.6◦

∆φ f =12.9◦

∆φ f =38.6◦

∆φ f =12.9◦

(b) y-direction acceleration spectrum

Acc

eler

atio

n(d

Bre

1m

2/s

4/H

z)

Frequency (kHz)

(a) x-direction acceleration spectrum

Acc

eler

atio

n(d

Bre

1m

2/s

4/H

z)

Frequency (kHz)

0.3 0.6 0.9 1.2 1.5 1.8 2.1 0.3 0.6 0.9 1.2 1.5 1.8 2.1−80

−70

−60

−50

−40

−30

−55

−50

−45

−40

−35

−30

−25

−20

Figure 10: Simulated vibration spectra for a square-shaped outer raceway defect of depth h = 50 µm and circumferential extents of∆φ f = 12.9◦ and ∆φ f = 38.6◦. (a) Horizontal vibration response with the dark and light gray rectangles indicating the variationsin ωx1 and ωx2 for ∆φ f = 12.9◦ ∆φ f = 38.6◦, respectively. (b) Vertical vibration response with the dark and light gray rectanglesindicating the variations in ωy1 and ωy2 for ∆φ f = 12.9◦ ∆φ f = 38.6◦, respectively.

Figure 10(a) shows that as the defect grows in size from ∆φ f = 12.9◦ to 38.6◦, the characteristic frequencies ofthe low frequency events in the horizontal vibration response increase which correlates to the variations in the naturalfrequencies ωx1 and ωx2 observed in Figure 8(b). For the vertical vibration response, Figure 10(b) shows that thecharacteristic frequencies of the low frequency event decrease as the defect grows in size from ∆φ f = 12.9◦ to 38.6◦

which correlates to the variations in the natural frequencies ωy1 and ωy2 observed in Figure 8(b). Thus, the shifting ofthe characteristic frequencies of the low frequency events observed in the vibration spectra can be used to distinguishbetween the two defects when estimating their size. Future work will investigate if more sophisticated time-frequencyanalysis techniques can be used to detect the changes in the characteristic frequencies of the low frequency vibrationevent.

5. Conclusion

An analytical formulation was presented of the load distribution and varying effective stiffness of a ball bearingassembly with a raceway defect of varying circumferential extent, depth, and surface roughness, subjected to staticloading in the radial, axial and rotational degrees of freedom. The analytical formulation was used to study the effectof the circumferential extent of a defect on the load distribution and varying stiffness of the bearing assembly. Asquared-shaped outer raceway defect centered in the load zone was considered and the defective bearing was loadedin both the radial and axial directions while the moment loads were zero. The study highlighted that in comparison tothe defect-free bearing, the stiffness variations for a defective bearing are significantly larger and change more rapidlyas ball enter and exit the defect. These larger, more rapid stiffness changes generate larger parametric excitationswhich, together with the ball mass striking the defective raceway, produce the low and high frequency events typicallyobserved in the vibration response [25, 26]. The calculated load distributions for the defective bearing showed that asthe circumferential extent of the defect increases, defect-free raceway sections are subjected to increased static loading

16

when one or more balls completely or partly destress when positioned in the defect zone. When the circumferentialextent of the defect increases such that one or multiple balls are in the defect at any one time, the mean radial stiffnessdecreases in the loaded radial and axial directions and increases in the unloaded radial direction. The effects of thesestiffness changes on the rigid body vibration of a defective radially loaded bearing assembly were investigated. Thesimulated vibration results demonstrated that the characteristic frequency of the low frequency events observed in thevibration response, which are caused by varying stiffness excitation of the rigid body modes that occurs when a ballenters or exits the defect [26], changes with the circumferential extent of the defect. This means that the characteristicfrequencies of the low frequency event can be used as a feature in defect size estimation techniques in order toovercome the inability of current techniques [25, 37] to distinguish between defects that differ in circumferentialextent by the ball angular spacing. To further validate this hypothesis, future experimental work will be undertaken inaddition to the simulated results presented here.

6. Acknowledgement

The work presented in the paper was funded by Australian Research Council (ARC) Linkage Project LP110100529.

17

Appendix A. Bearing stiffness expressions

kxx = KNb∑j=1

(A − A0)n cos2 φ j

(nAδ∗2r j

A−A0+ A2 − δ∗2r j

)A3 (A.1)

kxy = KNb∑j=1

(A − A0)n sin φ j cos φ j

(nAδ∗2r j

A−A0+ A2 − δ∗2r j

)A3 (A.2)

kxz = KNb∑j=1

(A − A0)nδ∗r jδ∗z j cos φ j

(nA

A−A0− 1

)A3 (A.3)

kxθx = rdKNb∑j=1

(A − A0)nδ∗r jδ∗z j sin φ j cos φ j

(nA

A−A0− 1

)A3 (A.4)

kxθy = rdKNb∑j=1

(A − A0)nδ∗r jδ∗z j cos2 φ j

(1 − nA

A−A0

)A3 (A.5)

kyy = KNb∑j=1

(A − A0)n sin2 φ j

(nAδ∗2r j

A−A0+ A2 − δ∗2r j

)A3 (A.6)

kyz = KNb∑j=1

(A − A0)nδ∗r jδ∗z j sin φ j

(nA

A−A0− 1

)A3 (A.7)

kyθx = rdKNb∑j=1

(A − A0)nδ∗r jδ∗z j sin2 φ j

(nA

A−A0− 1

)A3 (A.8)

kyθy = rdKNb∑j=1

(A − A0)nδ∗r jδ∗z j sin φ j cos φ j

(1 − nA

A−A0

)A3 (A.9)

kzz = KNb∑j=1

(A − A0)n(

nAδ∗2z j

A−A0+ A2 − δ∗2z j

)A3 (A.10)

kzθx = rdKNb∑j=1

(A − A0)n sin φ j

(nAδ∗2z j

A−A0+ A2 − δ∗2z j

)A3 (A.11)

kzθy = rdKNb∑j=1

(A − A0)n cos φ j

(δ∗2z j−nAδ∗2z j

A−A0− A2

)A3 (A.12)

kθxθx = r2dK

Nb∑j=1

(A − A0)n sin2 φ j

(nAδ∗2z j

A−A0+ A2 − δ∗2z j

)A3 (A.13)

kθxθy = r2dK

Nb∑j=1

(A − A0)n sin φ j cos φ j

(δ∗2z j−nAδ∗2z j

A−A0− A2

)A3 (A.14)

kθyθy = r2dK

Nb∑j=1

(A − A0)n cos2 φ j

(nAδ∗2z j

A−A0+ A2 − δ∗2z j

)A3 (A.15)

18

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