// AFAPL-TR-78-6 Part III ROTOR-BEARING DYNAMICS TECHNOLOGY DESIGN GUIDE Part III Tapered Roller Bearings r -T 106ý A. B. Jones J. M. McGrew, Jr. Shaker Research Corporation Northway 10 Executive Park Ballston Lake, New York 12019 February 1979 CD C..) Interim Report for Period April 1977 - December 1978 L Approved for public release; distribution unlimited. AIR FORCE AERO PROPULSION LABORATORY AIR FORCE WRIGHT AERONAUTICAL LABORATORIES AIR FORCE SYSTEMS COMMAND WRIGHT-PATTERSON AIR FORCE BASE, OHIO 45433 REPRODUCED FROM 'B BEST AVAILABLE COPY7N0 J 2
Shaker Research CorporationNorthway 10 Executive ParkBallston Lake, New York 12019
February 1979
CD
C..) Interim Report for Period April 1977 - December 1978
L Approved for public release; distribution unlimited.
AIR FORCE AERO PROPULSION LABORATORYAIR FORCE WRIGHT AERONAUTICAL LABORATORIESAIR FORCE SYSTEMS COMMANDWRIGHT-PATTERSON AIR FORCE BASE, OHIO 45433
REPRODUCED FROM 'BBEST AVAILABLE COPY7N0 J 2
NOTICE
When Government drawings, specifications, or other data are used for anypurpose other than in connection with a definitely related Government pro-curement operation, the United States Government thereby incurs no responsi-bility nor any obligation whatsoever; and the fact that the Government mayhave formulated, furnished, or in any way supplied the said drawings, speci-fications, or other data, is not to be regarded by implication or otherwiseas in any manner licensing the holder or any other person or corporation, orconveying any rights or permission to manufacture, use, or sell any patentedinvention that may in any way be related thereto.
This report has been reviewed by the Information Office (01) and is releas-able to the National Technical Information Service (NTIS). At NTIS, it willbe available to the general public, including foreign nations.
This technical report has been reviewed and is approved for publication.
JHB.SCHRAND HOWARD W-~JONESProject Engineer Chief, Lubrication Branch
FOR THE COMMANDER
BLACKWELL C. DUNNAMChief, Fuels and Lubrication Division
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UNrt.A~qTWTmSECURTV~ CL.MSIFICATION Of T4IS PAGE Veto.. DX. n~qw.
READ) INSTtUCTIONS(//)2REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM
REQTU;R-~GOVT ACCESSION No. 3.ECIPIENOT'S rrOGNM
4. T ITLE (E dSu.btitle) S. TYPE OF REPO03IEEWi*OOCOV9*E* -
x.20. ABSTRACT (Co..tirt. - revW~o @(o l. f necooo awl idotflt br block nowbow)
'This report is an update of the original Part IV of the Rotor-BearingDynamics Design Technology Series, AFAPL-TR-65-45 (Parts I through X).A computer program is given for preparation of tapered roller bearingstiffness data input for rotordynamic response programs. The completestiffness matrix is calculated including centrifugal effects. Consider-ations such as elastohydrodynamic and cage effects are not included sincethey have little influence on the calculation of tapered roller bearingst~ffni q.gg. TP rPaiilti-onrpjra m i raqo~hi ~11 and Paqv to une.
,,A 73~V 1473 EDITION OF I NOV 66BIS COBSLETES/N 0102-014- 6601 1UNCLASSIFT~fl
SECURITY CLASSIFICATION OF THIS PAGE (WRO,.Dt o .i0
FOREWORD
This report was prepared by Shaker Research Corporation under USAF
Contract No. AF33615-76-C-2038. The contract was initiated under Project
3048, "Fuels, Lubrication, and Fire Protection," Task 304806, "Aerospace
Lubrication," Work Unit 30480685, "Rotor-Bearing Dynamics Design."
The work reported herein was performed during the period 15 April
1977 to 15 November 1978, under the direction of John B. Schrand (AFAPL/
SFL) and Dr. James F. Dill (AFAPL/SFL), Project Engineers. The report
was released by the authors in December 1978.
ilor
iViS
TABLE OF CONTENTS
Section I INTRODUCTION 1
Section II ANALYSIS 3
2.1 General Bearing Model and Coordinate System 3
2.2 General Bearing Support Characteristics -------- 7
Figure 13 Sample Problem Data Input ------------- 44
Figure 14 Output Data for Load Condition #1 - - - - 45
Figure 15 Output Data for Load Condition #2 49
vi
NOMENCLATURE
Symbol Description Units
b Semi-width of contact ellipse at x in.
B Corner break at roller small end in.
B2 Corner break at roller big end in.
B ij Damping component, change of force in i direction lb-secdue to velocity in j direction; i - x, y, z; inj = x, y, Z.
B Damping matrix lb-sec(N lineal 0 0 in
0 0
0 0
(BN)angular
0 0
(O)lineal Damping matrix due to lateral velocities ib-sec
B in
[:yx :YY1ineal
(#)angular Damping matrix due to angular velocities in-lb-secangular radian
S Ylngular
11 for i = 1C. A constant, Ci 1-1 for i = 2
d Roller diameter at midpoint of effective length in.of roller
d Roller diameter at x in.
X
E Pitch diameter at midpoint of effective length in.
EE Modulus of elasticity for roller lbs/in2
vii
ER Modulus of elasticity for race body lbs/in2
E Pitch diameter at x in.x
F Roller centrifugal force lbs.c
Fi External applied force, i " x, y, z lbs.
F' Reaction force, positive in direction opposite lbs.to displacements, i = x, y, z
F Force Matrix = F lbs.
z
C Distance along roller cone element from extreme in.end of effective length to point where crown dropis measured
H Roller crown radius minus the rise of the arc at in.midpoint of effective length
I Moment of inertia about roller center of gravity Ibs-in2
cg
K Roller-race stiffness ibs/in.
K ij Stiffness component, change of force in i direction Ibs/in.due to displacement in j direction.
i = x, y, z; j x, y, z
=KN Stiffness matrix
"K lineal 0 00 0
0 0 (0 angular
L0 0
(KWlineal Stiffness matrix due to lateral displacements lbsfin.
yx YYllineal
(K)angular Stiffness matrix due to angular rotations in-lbK K tad
Sn aYYangular
viii
Perpendicular distance from the line of action in.of flange reaction, P3, to roller centerline atmidpoint of effective length
Ie Effective length of roller load carrying surface in.
IF Length of flat portion of roller measured along in.roller cone element
LT Total length of roller measured parallel to roller in.axis between sharp intersections of end faces with
roller cone elements
m Mass of roller lbs.
MG aller gyroscopic moment lbs-in.
Mi External applied moment, i x, y, z lbs-in.
Mi Reaction moment, i = x, y, z lbs-in.M1 outer race/roller contact moment ibs-in.
M2 Inner race/roller contact moment lbs-in.
n Number of rollers
N1 Outer ring rotational speed rad/sec.
N2 Inner ring rotational speed rad/sec.
Px Contact unit loading lbs/in.
Px Current estimate of contact unit loading lbs/in.
PD Diametral clearance in.
P1 Outer contact load on qth roller lbs.q
P2 Inner contact load on qth roller lbs.q
P3 Flange reaction on qth roller lbs.q
q Roller position index
R Roller crown radius in.c
RE Roller big-end spherical radius in.
V Radial distance from roller center line to in.flange reaction
ix
6K
6SColumn Y
Matrix 0X
0y
xyz Bearing coordinate system in.
x Static component of displacement in.
xv Dynamic component of displacement in.
XA Big-end extremity of contact pattern measuredparallel to roll axis from the midpoint of the
effective length
XB Small and extremity of contact pattern measured in.parallel to roller axis from the midpoint of theeffective length
XA Maximum permissible distance of big-end pattern in.extremity from midpoint of effective lengthmeasured along race
XB Maximum permissible distance of small-end pattern in.
extremity from midpoint of effective lengthmeasured along race
-N Impedance matrix %N + i N
Other notations as defined in text.
X
GREEK SYMBOLS
Angle between roller axis and line of action of radiansflange reaction
8 Outer ring contact angle radians
Y I d x cosO/Ex
S2 d x cos($-T)/Ex
6 Displacement in.
6 Linear displacement in x direction in.x
6 Linear displacement in y direction in.Y
6 Linear displacement in z direction in.z
A Approach of inner race to outer race at midpoint in.of effective length
A Approach of roller to race at x in.x
A Approach of roller to outer race (cup) at qth in.Alq roller
A2q Approach of roller to inner race (cone) at qth in.roller
C i Residues of simultaneous equations
4(1 - v) in 2
Roller elastic constant = EE
24(l - vR)
nR Race elastic constant = ER
0 Angular rotation about x axis radians, 0
x
O Angular rotation about y axis radians, 0YO Angular rotation about z axis radians, 0
Frequency of rotation rad/sec.
VE Poisson's ratio for roller
xi
VR Poisson's ratio for race
Material density lbs/in3
Included roller cone angle radians, 0
SCircumferential roller position radians, 0
WR Angular velocity of roller about its own center rad/sec.
Orbital velocity of roller rad/sec.
V Crown drop in.
xii
SUBSCRIPTS
Symbol Description
b Refers to bearing
cg Refers to center of gravity
E Refers to roller
F Refers to flat
g Refers to gyroscopic
i Index, i = 1, 2, 3 or i = x, y, z
i,j Refers to index of stiffness matrix; i.e.,force in i direction due to displacementin j direction
p Refers to pedestal
q Refers to roller circumferential position
R Refers to roller
T Refers to total
x Refers to x direction
y Refers to y direction
z Refers to z direction
1 Refers to outer race
2 Refers to inner race
xiii
\
SECTION I
INTRODUCTION
The original Rotor-Bearing Dynamics Design Technology Series AFAPL-TR-
65-45 (Parts I through X) included a volume, Part IV(l), which presented
design data for typical deep-groove and angular contact ball bearings. The
data was presented in graphical form and consisted of direct radial stiff-
ness, load carrying capacity, and load levels. In addition design guide-
lines and limitat 3ns were discussed. The major deficiencies of this
original volume were that centrifugal effects due to high speed were
ignored, and axial and angular stiffness information were omitted.
Subsequent to the publication of Part IV, several extensive treatments of
rolling element bearings including elastohydrodynamic, thermal, and cage
effects have been published. The computer program of Mauriello, LaGasse,
and Jones (2) considers both eiastohydrodynamic and cage effects for ball
bearings. The more recent computer based design guide prepared by
Crecelius and Pirvics (3) treats elastohydrodynamic, thermal, and cage
effects for a system of ball and roller bearings.
Thus, very sophisticated analytical tools are available for the design and
application of rolling element bearings. Neither of these tools, however,
provide the user with the stiffness matrix required for solution of rotor
dynamics problems. In addition both computer programs are very large and
require an extensive computer facility for use.
Part 11(4) of the revised series provided an update of the original
Part IV(l). Those aspects of the original Part IV(l) which treated general
design aspects of ball bearings, load capacity, speed limitations, etc.
were deleted since their coverage is superficial compared to the more
sophisticated computer tools now available (2,3). Only those parts
directly connected with preparation of input for the rotordynamic response
I
programs (Part 1(5) of the revised series) were retained. The stiffness
data included in the original Part IV were also updated.
The present volume (Part III of the revised series) extends the treatment
of rolling eJement bearings to the tapered roller bearing. The complete
stiffness matrix is calculated including centrifugal effects. Considera-
tions such as elastohydrodynamic and cage effects are not included since
they have little influence on the calculation of tapered roller bearing
stiffness. The resulting program (Appendix) is reasonably small z-nd easy
to use.
SECTION I1
ANALYSIS
2.1 General Bearing Model and Coordinate System
Accurate calculation of the lateral dynamic response of a high-speed rotor
depends on realistic characterization of the support bearings. In the most
general case, both linear and angular motions are restraintd by the support
bearings at the attachment location. In the analytical model, the reaction
force and the reaction moment of each bearing are felt by the rotor through
a single station of the rotor axis As schematically illustrated in Figure
la, a coil spring restraining the lateral displacement and a torsion spring
which tends to oppose an inclination are attached to the same point of the
rotor axis. A complete description of the characteristics of the support
bearings, however, involves much more than the specification of the two
spring constants. This is because:
The lateral motion of the rotor axis is concerned with two
displacement components and two inclination components.
The restraining characteristics may include cross coupling
among various displacement/inclination coordinates.
The restraining force/moment may not be temporally in phase
with the displacement/inclination.
The restraining characteristics of the bearing may be
dependent on either the rotor speed or the frequency of
vibration, or both.
Bearing pedestal compliance may not be negligible.
To accommodate the above considerations, the support bearing characteristics
are described in Reference 5 by a four-degrees-of-freedom impedance matrix as
defined in Equation (1):
3
I
(FIG la) Bearing Stiffness Model
y
"-Z ROTOR SPIN
STATIC LOAD
(FIG Ib) Bearing Location Coordinate
System
4
iN
where W is a column vector containing elements which are the two lateral
displacements (6X9 6y) and the two lateral inclinations (G., 0 y) of the
rotor axis at the bearing station N.
Employing a right-handed Cartesian representation in a lateral plane as
depicted in Figure lb, the z-axis is coincident with the spin vector of the
rotor. The x-axis is oriented in the direction of the external static load,
and the y-axis is perpendicular to both z and x axes forming the right-handed
triad (x, y, z). (6 x, 6 y) are respectively lateral lineal displacement
components of the rotor axis along the (x, y) directions. (0x, 0 y) are
lateral inclination components respectively in the (z-x, z-y) planes. Note
that 0 is a rotation about the y-axis, while 0, is a rotation about they
negative x-axis.
ZN is a complex (4 x 4 matrix), and in accordance with the common notation
for stiffness and damping coefficients, may be expressed as
N KN+ ivB (2)
where %K is the stiffness matrix and B. is the damping matrix. v is the
frequency of vibration. Most commonly, lateral lineal and angular displace-
ments do not interact with each other so that the non-vanishing portions of
and B are separate 2 x 2 matrices. That is
lineal
0 0 0 (3)
[ 0 0 angular
5
E (B)0 0N#
(N lineal 00 0
EN 10 0 (4)
0 0 (-N)L angular
Accordingly, a total characterization of a support bearing would include
sixteen coefficients which make up the 4 (2 x 2) matrices:
K Kxx xy
K K= lieal L j linal(5)
Lyx YY- lineal
B B[xx xy(B) lineal (6)
=y linealL yx By lineal
FK Kxx xy(_)angular (7
LYX yy jangular
B Bxx xy( angular L- (8)
B BByx Byy angular
In the event that the pedestal compliance is significant, then the effective
support impedance can be calculated from
Z (Z + Z -i (9)=N ---- P
where subscripts "p" and "b" refer to the pedestal and bearing respectively.
Note that both pedestal inertia and damping may be included in Z=P
6
2.2 General Bearing Support Characteristics
The function of a bearing is to restrict the rotor axis to a nominal axis
under realistic static and dynamic load environments. Deviation of any
particular point of the rotor axis from the nominal line can be character-
ized by three lineal and two angular displacements. These may be designa-
ted as (6 x, 6 y, 6z, Ox, 0 y) in accordance with a right-handed Cartesian
reference system. The z-coordinate is coincident with the reference axis
and is directed toward the spin vector. (Ox, 0 y) are rotor axis inclina-
tions respectively in the z-x and z-y planes. The x-coordinate is directed
toward the predominant static load; e.g., earth gravity. Ideally, the
bearing would resist the occurrence of any displacement so that the
reaction force system imparted by the bearing to the rotor is generally
expressed in matrix notation as
F - Z . x (10)
F is a column vector comprising the five reaction components (Fx, Fy, Fz,
MX, My), while x is the displacement vector (6x, 6y, 6z, 0,0 ). Z is
a (5 x 5) matrix containing the elements Zij with both indices (i, j)
ranging from 1 to 5. The values of Zij characterize how rotor displacements
are being resisted by the bearing.
From the standpoint of dynamic perturbation, distinction is made between
a static equilibrium component and a dynamic perturbation component for
both the displacements and the reactions. Thus,
x = x + x'; F = F + F' (11)
(x', F') are respectively presumed to be infinitesimal in comparison with
(x , F ). Accordingly, Zij are regarded as dependent on x-x but not on x'.
To illustrate the idea of perturbation linearization, one may examine the
one-dimensional load-displacement curve shown in Figure 2.
7
F ' - ,° .t a ,,
I
x-SxTA
8'o
Figure 2. Linearization of Tapered Roller Bearing Stiffness
8
As illustrated, the load-displacement relationship is a 10/9 power law in
accordance with the Hertzian point contact formula. It is not possible
to describe the entire range by a linear approximation. However, if a
small dynamic perturbation is taken around a static equilibrium point,
6' < 6 , the small segment of the load-displacement curve can bex x
approximated by a local tangent line. The corresponding force increment
isaF
F' 0 _x 6' (12)x 36 x
where 6' is the incremental displacement. 3F /36 will depend on thex x x
amplitude of 6 .0
The question of history dependence is resolved by regarding x' as periodic-
motions at any frequency v of interest, and Z accordingly would have both
real and imaginary parts and may also be dependent on both the rotor speed w
and the vibration frequency v.
To avoid notational clumsiness, the primes will be dropped from (F', x')
which are understood to be dynamic perturbation quantities unless the sub-
script "o" is used to designate the static equilibrium condition.
2.3 Tapered Roller Bearing Characterization
In many ways the tapered roller bearing is much simpler to model from. a rotor
dynamic point of view than a fluid film bearing. In general, the following
two simplifications can be made:
The restraining characteristics do not include cross coupling
among the various displacement/inclination coordinates.
The restraining force/moment is normally temporally in phase
with the displacement/inclination.
9
Figurc. 3 shows a tapered roller bearing referred to in an orthogonal xyz
coordinate system. The outer ring is fixed but the inner ring may move
with respect to the coordinate system. Both rings are free to rotate about
their axes.
Three lineal displacements, 6X, 6 y, 6z, and two angular displacements, Ox,
0y, are required to define the spatial position and attitude of the inner
ring when it is displaced from its initial position. For purposes of deriva-
tion the initial situation is that existing when the bearing's end play is
just taken up in the thrust direction. Figure 4 shows these displacements
in the positive sense. Figure 5 establishes the convention of the roller-
position index q.
2.3.1 Stiffness
The total characterization of a tapered roller bearing's stiffness
can be expressed by the matrix.
FaF 3F aF~ aF aFjDx ay az M0x a0
aF aF aF DF aFI
ax ay ax a® a®x y
aF 3F aF aF aF[K] = (1y 3) 30 (13)
x y
am aM am aM aMX X X X X
ax ay az a® acx y
am aM am aM am
ax ay az a0 ® 0
10
Y
CONE
z
I • CUP
Figure 3. Tapered Roller Bearing
1i
x
Crz"iN cn
to
N~S-I
120
q--q ( q -4
Figure 5. Tapered Roller Bearing Index, q
13
The lineal and angular stiffness matrices (Equations 5 and 7) can
be derived from Equation (13). For example:
aF DFx x
ax ay
(_lineal (14)
3F aF-y y
Lax ay_
am am 1
x xao ao
x y
-~angular (15)
am amao 30x Y
Note that although the axial components of stiffness are not utilized
by the lateral rotor dynamics program (5), they have been retained in
the general tapered roller hearing stiffness matri:,, Equation (13). The
axial stiffness would be required, for example, if the reader was calculat-
ing the axial natural frequency of a tapered roller bearing mounted shaft.
23.2 aming
An extensive search of the literature revealed no experimental damping
data for tapered roller bearings. As the current state-of-the-art
does not permit accurate calculation of tapered roller bearing damping,
no damping data is included in this report.
14
2.4 Tapered Roller Bearing Under Combined Loading
Solution for the stiffness matrix of a tapered roller bearing under com-
bined loading is a tedious problem and requires the use of a digital
computer. In this section, the derivation of the solution is described.
A computer program for obtaining the solution is included in the Appendix.
2.4.1 Bearing Applied Forces and Moments
As the result of the five displacements described previously in
Figures 3 and 4, there are the reactions FI, F', F;, and M' andx y z xM'. F'x, F', and F' are forces. M'I and M' are moments. All arey x y zx yshown in their positive sense in Figure 4. External forces F and
F may be applied at the inner ring center. The senses of thezI
signs are the same as for the reactions F' and F'.x z
2.4.2 Roller Geometry
Figure 6 shows the boundary dimensions of a typical tapered roller.
Roller mass, moment of inertia, and location of the center of
gravity are calculated assuming the roller is a flat-ended, trun-
cated cone bounded by R1 , R2 , and t
In general, the big-end face of the roller is not flat but is a
sphere having the radius Re which is generally a proportion of the
slant height, Zs, of the untruncated roller cone. Roller crown and
corner breaks are also omitted from mass and moment of inertia
calculations as their contributions are second order.
Figure 7 is a more complete sketch of the roller showing the details
of the roller crown. The big-end spherical surface is neglected
here also.
T is the included angle of the roller cone and is obtained by itera-
tion of
r -l d- tan {4 sin( - (16)
2E2
-4
SToE-4
0
00
16
i64
/ U
S -- 4
0'
L • , -• -
2 L ' -'-4
, a,
17
From Figure 7
cH V2_)2 (17)
where R is the crown radius and I the length of the flat portionc Fof the roller profile. In a fully crowned roller, the flat length
is zero.
Se is the effective length of the roller load-carrying surface.e
The actual working length for any loading must lie within e" B 1
and B2 are the corner breaks. Their shapes are unimportant as long
as they blend smoothly into the crowned surface.
V is the drop of the crown and is measured at the extremes of the
effective length of the roller.
V H - JR ~ (18)
dH (19)
0 2tan(T)
1 H cos () + Vsin(j) B1 (20)I H0 -- -- cos1
H2 = H0 + cos (1) + Vsin(½) + B2 (21)
R1 = Hltan(T) (22)
R = H2 tan( ) (23)
Let the cone corresponding to H1 have the mass mI and a moment of
inertia about its center of gravity I1cg
Let the cone corresponding to H2 have the mass m2 and a moment of
inertia about its center of gravity 12cg
18
irR2 H P2 3x386.4 (24)
3mI R2 H2I I I ) (26)
cg 3M R2 2
3m2 R2 H2""'21. (26)
2 212 "-• (27)+
cg
where p is the material density in lb/in3
Then the distance X from the big end of the roller at H2 to the
center of gravity of the roller is X'
m2 H2 3H1
4 - ml (H2 - (28)
2 -1
The moment of inertia I of the tapered roller about its center ofcggravity at X is
REPRODUCED..,,€ n FR.....O..M :•:-.....,-,BEST AVAILABLE COPY4.. 4...
APPENDIX
COMPUTER PrOGRAMFOR
CALCULATING STIFFNESS MATRIXOF
TAPERED ROLLER BEARING
52
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REFERENCES
1. Lewis, P. and Malanoski, S.B., "Rotor Bearing Dynamics Design Technology.Part IV: Ball Bearing Design Data Technical Report," AFAPL-TR-65-45,Part IV. Air Force Aero Propulsion Laboratory, Wright Patterson AFB,Ohio.
2. Mauriello, J.A., LaGasse, and Jones, A.B., "Rolling Element BearingRetainer Analysis," DAAJ02-69-C-0080, TR105.7.10, USAAMRDL-TR-72-45.
3. Crecelius, W.T. and Pirvics, J., "Computer Program Operation Manual on"SHABERTH" a Computer Program for the Analysis of the Steady State andTransient Thermal Performance of Shaft Bearing Systems," AFAPL-TR-76-90,Air Force Aero Propulsion Laboratory, Wright Patterson AFB, Ohio,October 1976.
4. Jones, A.B. and McGrew, J.M., "Rotor Bearing Dynamics Technology DesignGuide--Part II: Ball Bearings," AFAPL-TR-78-6, Part II, February 1978,Air Force Aero Propulsion Laboratory, Wright Patterson Air Force Base,Ohio.
5. Pan, C.H.T., Wu, E.R., and Krauter, A.I., "Rotor Bearing DynamicsTechnology Design Guide: Part I, Flexible Rotor Dynamics," AFAPL-TR-78-6,Part I, June 1978, Air Force Aero Propulsion Laboratory, Wright-PattersonAir Force Base, Ohio.
