Characteristics of tapered roller bearing subjected to combined radial and moment loads
INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING-GREEN TECHNOLOGY Vol. 1, No. 4, pp. 323-328
OCTOBER 2014 / 323
DOI: 10.1007/s40684-014-0040-1
Characteristics of Tapered Roller Bearing Subjected to
Combined Radial and Moment Loads
Van-Canh Tong1 and Seong-Wook Hong1,#
1 Department of Mechatronics, Kumoh National Institute of Technology, Daehak-ro 61, Gumi, Gyeongbuk, South Korea, 730-701
# Corresponding Author / E-mail: , TEL: +82-54-478-7644, FAX: +82-54-478-7319
KEYWORDS: Tapered roller bearing(TRB), Radial load, Moment load, Contact force, Roller profile, Stiffness matrix
This paper investigates characteristics of the TRB such as displacements, contact forces between roller and inner ring, outer ring and
flange, contact angle between roller and flange, load distribution along roller, and stiffness matrix, when the TRB is subjected to
combined radial and moment loads. Understanding of these characteristics deserves attention for developing more sustainable TRBs.
To this end, a five-degree-of-freedom model of TRB is employed. Unlike other studies, this paper takes TRB displacements as unknown
variables and determines them by iteratively solving the roller and bearing equilibrium equations. A new formula for load variation
in rollers is also presented by using an integration technique. The developed method is validated by comparing preliminary results
with those from a reference program. The characteristics of TRBs subjected to combined radial and moment loads are simulated as
a function of roller profile and rotational speed.
Manuscript received: July 3, 2014 / Revised: September 9, 2014 / Accepted: September 12, 2014
1. Introduction
Tapered roller bearings (TRBs) have a very noticeable feature of
high load capacity against axial and radial loads. By nature, they can
also support moment load. Hence, TRBs have been particularly
considered in high load supporting applications such as automobile
wheel hub assembly, gas turbine engine, milling machine spindle, etc.
Generally, rolling element bearing stiffness is an essential factor that
affects dynamic behavior of rotating spindle. Because bearing stiffness
is needed for analyzing the dynamic characteristics, for example,
natural frequencies, mode shapes, and vibration amplitudes. The
contact force and load distribution in roller of TRBs have been found
to significantly influence their lubrication regimes, life time, or
reliability. Therefore, accurate estimation of these characteristics is very
valuable for better design so as to improve the performance and fatigue
lives of TRBs, as well as predicting the performance of rotating
systems supported by TRBs.
The fundamental theories of rolling element bearing were early
outlined by Palmgren,1 Jones2 and Harris.3 Studies on bearing were
further extended by many researchers because of increasing demand
toward higher efficiency. Regarding TRB, various investigations have
been carried out that focused on determination of bearing
characteristics. Andreason4 analyzed the load distribution in a TRB
© KSPE and Springer 2014
under radial and axial loads. Although he effectively used a vector
method to determine the elastic deformation of the roller and raceways,
the centrifugal force and gyroscopic moment in TRB were not taken
into account. Andreason’s model4 was further improved by Liu5 so as
to investigate the effect of given misalignment angles on the
performance of TRB under combined loads. He considered the TRB
characteristics under more realistic conditions, e.g., high speed, and
actual direction of flange-roller contact force, which were neglected in
the Andreason’s research.4 However the TRB displacements in the
researches of both Liu5 and Andreason4 were assumed to be known a
priori, rather than taken as unknown variables. This obviously limits
the applicability of the model.
De Mul et al.6,7 presented a general theory for determination of
stiffness matrix and displacements of ball and roller bearings. Cretu et
al.8 used the de Mul’s model to investigate the dynamic characteristics
of TRBs under lubricating conditions. His study was confined to
straight roller and raceway profiles. This implies impracticability
because the roller or raceway profiles have been modified considerably
to reduce the pressure concentration at the ends.9
Recently, Houpert improved the previous study10,11 to perform an
analytical approach for determining loads and moments of TRB as a
function of given displacement with neglecting the influence of
centrifugal and gyroscopic moment. A transition from point contact to
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INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING-GREEN TECHNOLOGY Vol. 1, No. 4
line contact between roller and raceways with the increasing load has
also been considered.
In this paper, the characteristics of TRBs with modified roller
profile such as displacements, contact forces between roller and inner
ring, outer ring and flange, load distribution along roller, and stiffness
matrix were investigated. First, based on the theory of de Mul et al.6,7
an alternative integration method was proposed to replace the slicing
technique. Subsequently, the developed model was verified by
comparing the computational displacement and stiffness of the TRB at
difference rotational speeds with those from a reference code.12 Finally,
the effects of combined radial and moment loads on the characteristics
of TRB were reported. The presented results on TRBs subjected to
practical loading conditions are believed to contribute to developing
more sustainable TRBs.
2. Derivation of Dynamic Equations for TRB
{u} = [Rφ].{δ}
where [Rφ] is the global transformation matrix,
[Rφ ] =
2.2 TRB Coordinate Systems, Loads and Displacements
Fig. 1 shows the TRB coordinate systems and loading. The bearing
global coordinate system (x, y, z), external force vector {F}T = {Fx, Fy,
Fz, Mx, My} and the inner ring displacement vector {δ}T = {δx, δy, δz, γx,
γy} are shown in the Fig. 1(a). In the local cylindrical coordinate system
(r, φ, z) which is fixed with a particular roller at azimuth angle φ, the
inner ring contact force and displacement vectors are {Q}T = {Qr , Qz,
M} and {u}T = {ur , uz, θ}, respectively. The roller displacement is
expressed by vector {v}T = {vr , vz, ψ} (Fig. 1(b)). The relationship
between vectors {u} and {δ} is stated as
cosφ sinφ 0 –zP sinφ zP cosφ
0
0 1 rP sinφ zP cosφ
0
0 0 –sinφ
(2)
cosφ
The contact load vector {Q} depends on both the roller and inner
ring displacements, i.e.,
{Q} = {Q({u},{v})}
(3)
In the inclined coordinate system (ξ, ζ, η) (Fig. 1(c)), the
corresponding contact force and displacement vectors indexed by
subscript κ are {Qκ}T= {Qξ, Qζ, M}, {uκ}T= {uξ, uζ, θ}, and {vκ}T= {vξ,
vζ, ψ}. By using local transformation matrix [K], one can get
In order to obtain bearing equilibrium equations, it is necessary to
solve the equilibrium equations of all rollers which are related to roller
contact forces. The deriva (...truncated)
