Modern shaft calculation, taking into consideration non-linear effects from bearing inner geometry
Modern Shaft Calculation taking into Consideration non-linear Effects from
the Bearing inner Geometry
M. Raabe, KISSsoft AG., Hombrechtikon, Switzerland
Summary
Most of the time, shaft calculations are carried out considering bearings either as rigid
boundary conditions or as simple springs. For the calculation of bearings with a big contact
angle, there are different procedures available. For a more exact calculation of deflection,
bearing forces or critical frequencies and also the consideration of the correct bearing
stiffness based upon the bearing’s inner geometry load is often demanded. Some application
problems will be presented, showing the big benefits of non-linear bearing stiffness in a
modern shaft calculation.
1 Introduction and Problem Definition
Shafts are mostly rotating, cylindrical machine elements for torque transmission. Strength is a
criterion in shaft sizing. Further criteria are, for instance, deflection and critical speed. The shaft
deflection will be required, for example, for the sizing of gearing corrections. The shaft calculation
also provides the loading upon the involved bearings.
With most calculations, out of simplicity, only stiff bearings are considered. This is mostly
sufficient for the calculation of the strength of statically defined bearings. On the other hand, for
statically over defined bearings (multiple supports), or for the calculation of critical speeds,
bearing stiffness is often taken into account. These stiffnesses are non-linear for both journal and
roller bearings. They are mostly determined for a working point and then used for a linear
calculation. Stiffness data are seldom available and estimated values are used instead.
Bearing manufacturers internally have for a long time used calculation programs that also consider
the non-linear bearing stiffness, which are partly available to their customers. One can also ask the
manufacturers for bearing stiffness figures; however, this will be delaying the sizing procedure. A
few publicly available calculation programs include the calculation of bearing stiffness from the
bearing inner geometry and the company KISSsoft Inc. has just incorporated this procedure in its
calculation program for machine design. DIN ISO281, Supplement 4 [1], describes the possibility
of bearing stiffness calculation and the corresponding international standard [2] is being prepared.
The certification of wind turbine gearboxes already prescribes the calculation of bearing lifetime
taking in consideration the bearing inner geometry [1].
2 Shaft Calculation Methods
A beam model is usually used when calculating shafts. Since a beam model is one-dimensional
only, 2D or 3D FE shaft calculations has to be used, particularly in case of high loaded shafts or in
case of not-common notch effects.
Public
KISSsoft AG
Uetzikon 4
8634 Hombrechtikon
Switzerland
www.KISSsoft.ch
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Title: Shaft & bearing analysis
No.:
Date: 1.10.07
Manager: Markus Raabe
@: markus.raabe@KISSsoft.ch
Version: 0
Autor: MR
Date: 1.10.07
Approved: MR
Date: 1.10.08
W:\Artikel-Papers-Konferenzen\050-DMK2007-Wellenberechnung\DMK2007_Raabe_E[2].doc
2.1 Beam model
A one-dimensional element, which can be subject to forces and torques in three axes, is called a
beam. Loads are considered as punctual loads or, for forces, also as linear loads. Constraints are
applied at individual points.
The most currently used beam element is the Euler-Bernoulli-Beam. This model is based on some
simplifications: A cross-section remains always straight and a proportionality of the curvature and
the bending moment is supposed. With small displacements, the second derivative of the
deflection will be used instead of the curvature.
d 2w
EI 2 = − M ( x)
dx
A more extensive beam model like Timoshenko’s, additionally considers shear strain. Because of
shear strain, the cross-section is no more perpendicularly to the beam axis and the shear strain
alone will cause deflection. This is to be taken into consideration for short, relatively thick beams
if strain is relevant. Because of the shear strain, it must be distinguished between shaft inclination
and the bending angle ψ that differ by the shear strain [9]:
With dynamic calculations, the Euler-Bernoulli-Beam uses only one linear mass while
Timoshenko’s also considers inertia due to rotation. This can also lead to a difference for short,
thick beams.
[9] gives a rule-of-thumb for the validity range, so that the Euler-Bernoulli-Beam is usable for a
shaft length bigger than the quintuple of the beam height, whereas Timoshenko’s is adequate up to
a shaft length equal to the beam height.
For shafts, a strength calculation is often produced according to the nominal stress concept. The
beam model supplies the nominal stress concept that, combined with notch factors, allows a
verification according to DIN 743 [5], the FKM-guidelines [6] or other similar methods.
2.2 Finite Element Calculations
While the beam theory is sufficient for the calculation of strain for most shafts, an FE calculation
is often necessary for the strength calculation with complex notch effects or a high utilization of
the strength. However, for the proof of fatigue resistance, the problem exists to get the admissible
stresses. The FKM guideline [6] offers an approach for the proof with local stresses. When FE
calculations is used for fatigue proof of a shaft, it is always very difficult to get a reliable result
without comparing with measurements on a test rig.
2.3 Influence of the Bearing on the Shaft
Bearings are taken into account in the shaft calculation as border or transition conditions. Bearings
are often considered stiff in the framework of the calculation with a beam model. There, the
deflection is set to zero. For linearly elastic bearings, an additional equation per bearing and
coordinate will be used.
With bearings, the transition condition is a non-linear function of all displacements and rotational
components. That is why, for a working point, it is possible to determine a stiffness leading to the
same displacement. However, for critical speed calculation, the important factor is the tangent at
the force-to-deflection characteristic in the working point. Depending upon the purpose of the
calculation, another stiffness is to be chosen.
3 Bearing Calculation according to DIN ISO 281, Supplement 4
DIN ISO 281, Supplement 4 [1] describes a method for the calculation of the bearing lifetime
using the bearing inner geometry and a general load. The load distribution on the single rotating
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element (ball or pin) is determined from the non-linear stiffness between rotating element and
bearing (inner/outer) ring.
For ball bearings, the normal force Q upon the ball is deduced from the displacement of the ball
by:
−3

 2
ρ
ρ
×
π
2
E
∑
∑
i
e

×
×  K (κ i ) × 3 2
+ K (κ e ) × 3 2
Q = c p × δ 2 with c p =
3
1 + υ E2 
κ i × E (κ i )
κ e × E (κ e ) 


The values κ and ρ emerge from the bearing inner geometry for which, besides the rotating
element diameter and their number also the radius of bearing rings is required.
For roller bearings, the normal force Q on a pin element of length Lwe is given by:
3
8
10
Q = c L × δ 9 with c L = 35948 × Lwe9
In order to be able to consider the inclinations in the bearing and the rotating element crowning,
the element is separated in several slices (Fig. 1, at right). DIN ISO 281, Supplement 4, does not
consider the influence of axial loads leading to an unequal load distribution in roller bearings.
You can find here information, for instance, under Harris [8].
Figure 1: Models of Ball- and Roller Bearing according to DIN ISO 281, Supplement 4 [1]
With given displacements, the resultant bearing forces and moments can be determined by adding
up the forces in the individual contacts. If these displacements are unknown, but the external load
given, an iterative solution must be found.
For the calculation of the load distribution, it is necessary to consider in addition to the radiuses for
the calculation of the Hertzian pressure also the bearing clearance. Once again, the bearing
clearance will be influenced by the initial bearing clearance, the pressure between the bearing and
the shaft, as well as thermal expansions.
For the shaft system of equations, a non-linear equation emerges for the bearing force:
Fx , Fy , Fz , M x , M z = f (u x , u y , u z , ϕ x , ϕ z )
Solving this system of equations, for example, with a Newton method, the following stiffness
matrix will be obtained:
T
T
Fx , Fy , Fz , M x , M z = f (u x 0 , u y 0 , u z 0 , ϕ x 0 , ϕ z 0 ) + C × ∆u x , ∆u y , ∆u z , ∆ϕ x , ∆ϕ z
[
[
]
]
[
]
The stiffness matrix is normally fully populated. A tilting of the radial loaded ball bearing around
the center point, for example, causes an axial force that, because of the axial clearance through the
radial displacement on the load side, will be smaller than on the opposite side. If not calculated
with displacement increments, only a linear calculation can be executed, and the equation can be
rewritten:
T
T
T
Fx , Fy , Fz , M x , M z = f (u x 0 , u y 0 , u z 0 , ϕ x 0 , ϕ z 0 ) − C × u x 0 , u y 0 , u z 0 , ϕ x 0 , ϕ z 0 + C × u x , u y , u z , ϕ x , ϕ z
[
]
[
]
[
]
This way, every algorithm for a shaft calculation that admits a definition of stiffness can be used.
In addition to the stiffness, an external load will be introduced and the system iteratively
calculated. With the solution, it must be noticed that the bearing stiffness in the initial
configuration without displacements is zero and that, with the stiffness calculation for excessively
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big displacements problems could emerge during the iterative solution. If, during the iterations, the
displacement becomes bigger than the rotating element diameter, the equations will deliver
unexpected results.
According to DIN ISO 281, Supplement 4 [1], with the loads on the individual rotating elements
the nominal reference lifetime L10r as well as the modified reference lifetime of Lnmr, can be
calculated. Lnmr also contains the factors a1 and aDIN for the probability of failure and the influence
of the lubrication.
The question is; where does one get the bearing data for the calculation with the inner bearing
geometry now? For individual bearings, one can get this information from the bearing
manufacturers. Since the results with approximate data are still much better than those for stiff
bearings, the data can also be estimated from the catalog data. According to ISO 281 [3] and ISO
76 [4] one has two equations for the load ratings C and C0, from which the two parameters,
number of rotating elements Z and rotating element’s diameter Dw, can be determined, if one
assumes the standard values for the radius of the inner/outer rings (see also [11]). For roller
bearings, the rotating element length Lwe is added. Here, only an estimation based on the bearing
width is possible.
4 Shaft Calculation with radial Bearings and big Contact Angle
Radial bearings with big contact angles like angular contact ball bearings or bevel roller bearings
cannot be only radially loaded. Through the normal force direction on the rotating element, there
always appears an axial force. This axial force is given in bearing catalogs or in [7] as generated
by the radial force Fa=0.5xFr/Y. Further on, for the calculation of the radial load, the bearing is
displaced from the geometrical position in the contact center point for the calculation of the radial
loads.
For the shaft calculation with contact angle ball bearings, there are four possibilities for the
bearing modeling:
1. The bearings are considered as a constraint at the geometrical center point
2. The bearings are considered as a constraint in the contact center point
3. The bearings are considered as a constraint at the geometrical center point with the
introduction of an additional bending moment
4. The bearings are considered as a constraint at the geometrical center point with non-linear
stiffness.
The first case is the simplest to take into account in the calculation. However, wrong bearing
forces will result.
The second case: The displacement of the bearings into the contact center point, requires only
additional data about the bearing itself, that is to be found in bearing catalogs. The bearing forces
are calculated right, however, the load introduction takes place at the wrong place on the shaft. The
contact center point can be at the other side of the shaft shoulder, by which the nominal stresses at
the shoulder could appear as zero. For bearings in O-disposition, it can happen that the contact
center points are outside the shaft. With a X-disposition, with contact center points joining in the
same point, the calculation is not possible because of a missing constraint. The bending, for stiff
bearings at the contact center point, is assumed to be zero (this does not correspond to the reality).
The third case: It is possible to get the correct bearing forces at the right place if the loads on the
bearing are at the geometrical right place and an additional bending moment is considered(see also
[10]). If it is assumed that the bearing force passes through the contact center point, then it can also
be displaced to the geometrical bearing position along its line of action. The now eccentrically
acting axial load causes a bending moment on the shaft. This action delivers the same bearing
forces, and the force flows are equal except within the area between the bearing center and the
contact center points. As long as the contact center point is within the bearing width, the
differences between case 2 and 3 are hardly noticeable. Fig. 2 shows the moment flow in case 3. In
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case 2, with the bearings in the contact center point, the negative bending moments would be zero,
the shaft shoulders, therefore, free of stress.
A new problem appeared with this third calculation variation. If an angular contact ball bearing
with 40° pressure angle in direction X lays on a shaft shoulder, there will be a relatively big
bending moment on it. As a result, there are partly small calculation safeties for the shaft shoulder.
In reality, the axial force is not set in the center of the bearing but through the inner bearing ring on
the shaft shoulder. The bending moment is therefore introduced exactly at the shoulder without
creating a bending stress condition in the shoulder groove. The calculation with nominal stresses is
actually not possible at this point. It is necessary to carry out an FE calculation for an exact
calculation of this case.
Figure 2: Bending Moment Distribution for Case 3 (lower line) and 4 (upper line)
The fourth case is the calculation with non-linear stiffnesses at the place of the geometrical support
place. This delivers a bending moment like in case 3, furthermore the pressure angle is loaddependent correctly considered and the stiffness appear automatically. In Fig. 2, the small
deviation of the bending moment distribution between case 3 and 4, is load-dependent and will
increase with the load because the pressure angle changes under load. The same problem as in case
3 arises with the shaft strength calculation; should the pressure center point lie outside the bearing.
5 Influence of the non-linear Stiffness: Examples
The influence of the bearing stiffness on shaft’s behaviour will be shown in the several following
examples. For the time being, the selection is limited to contact angle ball bearings and cylindrical
roller bearings.
5.1 Contact Angle Ball Bearings with different Preload Forces
Contact angle ball bearings are employed when stiff support is required. The system stiffness is
increased by an axial preload. However, a very high preload force is not desirable because it would
increase the friction and reduce the bearing lifetime.
Radial Force Fr / [N]
Bearing Center Offset depending upon the Preload Force
Preload
Radial Bearing Offset ur / [mm]
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Figure 3: Radial Bearing Offset for a Contact Angle Ball Bearing 7208E in Dependence of the Preload Force ua
in µm.
Fig. 3, shows the dependence of the stiffness on the preload force. The preload will be placed
between the inner and outer rings through an axial offset (ua in µm). Now, for the same radial
load, different radial bearing displacements emerge. With a load of 2000N, the bearing
displacement is reduced by 50% when the preload increases from 1 to 20 µm. With bigger radial
loads, the Force-/Deviation paths run parallel.
Natural Mode for 1st Frequency
(1717 Hz, with 20 µm Preload)
Deviation
Deviation
Natural Mode for 1st Frequency
(1191 Hz, without Preload)
Figure 4: Natural- Modes and Frequencies of different Bearing Preloads for Contact Angle Ball Bearing 7206B.
(Loads act in the Z-direction; Axis is in the Y-direction).
The stiffness increased through the preload can also be recognized in the natural frequencies. Fig.
4 shows the natural- mode and frequency for the shaft in Fig. 2 for two different preloads. Because
of the strong preload, the frequency has considerably increased and the amount of the axial
movement in the natural mode is reduced a little.
As a whole, the natural mode in Fig. 4 is somewhat uncommon for a classic shaft calculation. The
axial- and the bending oscillations are here coupled. Should the shaft move itself axially on a
bearing, then, it will be forced to move opposite of the load direction due to the inclined bearing
raceway. On the second bearing, the radial backlash increases due to the axial relief and the shaft
can further move in the load direction. So, axial vibrations cause also a tilting movement in the
radial direction.
5.2 Spindle Bearing Package
Several spindle bearings are often used side by side in manufacturing machines. The bearing
forces, already in a simple tandem arrangement, cannot be calculated assuming stiff bearings
because high reaction forces originate by the smallest inclination of the shaft in the stiff bearings.
Here, either the stiffness for each bearing should be considered, or a single bearing is used for the
shaft calculation and the load is later separately distributed by the other bearings during the
bearing calculation.
Figure 5: Bending Moment and Shear Force Distribution for a Spindle Bearing Package calculated considering
the inner Bearing Geometry
The same solution can also be obtained with simple spring elements corresponding to a suitable
selection of bearing stiffnesses, as with the consideration of the inner bearing geometry. Such an
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example can be seen in Fig. 5 with the bending moment and the shear force distribution. However,
the additional effort is that of determining the load-dependent stiffness.
5.3 Bending Moment induced in Cylindrical Roller Bearings
Cylindrical roller bearings are sensitive against shaft inclination. However, not only the bearing
lifetime decreases with an inclination, but also the bearing reacts against the inclination. The
cylindrical roller bearing generates a reaction moment, that alters the distribution of the bearing
loads.
Moment Reaction and Lifetime for tilted
Cylindrical Roller Bearings
Lifetime
Bending Moment
Angle [ ]
Figure 6: Effect of Tilting on Lifetime and Reaction Moment of a cylindrical Roller Bearing NU226 (Fr=200kN)
The bearing catalog indicates for a cylindrical roller bearing an inclination limit of 2 – 4 minutes
of angle. Fig. 6 shows almost no reduction of the bearing lifetime for an inclination of 2 minutes of
angle.
5.4 Planet Support with Needle Bearings
Planets or idler gears in transmissions are often supported with needle bearings. Here also, only
one single bearing can be employed. The bearing load is directly given for spur gears, but what
happens with helical gears? A bending moment can not be absorbed with only one punctual
assumed bearing. As a remedy, instead of one, two half width bearings can be used for the
calculation and this would constitute a rough approximation. It can also be done using the DIN
ISO 281, Supplement 4, bearing slice model that uses many thin slices (see Fig. 1). With it, the
single bearing also absorbs a bending moment and the lifetime is calculated taking into account the
border stresses.
Figure 7: Planet Support with Needle Bearing, Bending Moment and Deflection Line
Fig. 7 shows the bending line and bending moment of a planet gear with a needle rim. The Figure
shows again a comparison between the calculation with inner bearing geometry and with the
assumed stiffnesses of a punctual bearing. Since bearings, for the calculation, are still assumed to
be punctual, there is a bending moment jump in the center of the bearing. In reality there is a
trapezoidal-shaped line load. Also the deflection has a minimum at the theoretical bearing position
and presents higher values at the borders. Better results could be achieved considering the bearing
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width in the shaft calculation. However, this also presupposes elastic bearing rings, which exceeds
the calculation according to DIN ISO 281, Supplement 4.
With this calculation at least two important values will result: The bending angle (here barely two
minutes of arc) and the bearing lifetime.
5.5 Multiple Support with different radial Clearance
In very long shafts, or also in vehicle gearboxes, more than two bearings are frequently used. This
is required to prevent high bending of the shaft. In order to influence the load distribution upon the
bearings, bearings with different radial backlash could be employed.
The calculation must consider the inner bearing backlash, in order to be able to evaluate the effect
of bearing backlash, which also depends upon interference fits and operational temperature.
6 Summary and Outlook
Different application examples were shown, in which the consideration of the load-dependent,
non-linear bearing stiffness brings advantages for the shaft calculation. Only with such a
calculation method, effects such as preload force, shaft inclination or different bearing backlash by
multiple supports can be examined. Since the calculation of non-linear stiffness is very costly, it
must be done in a numerical way. KISSsoft delivers a user-friendly tool that accomplishes this
task. After the first version of the KISSsoft shaft calculation considering the inner bearing
geometry, further features, such as influence of the centrifugal force, will be added, and it is
planned to extend the calculation of coaxial shafts to any system of coupled shafts. Furthermore,
the number of the supported bearing types will be increased.
7 Bibliography
[1]
DIN ISO 281, Supplement 4 (2003): Dynamic Load Ratings and nominal Lifetime
Procedures for the Calculation of the modified Reference Lifetime for universally loaded
Bearings
[2] ISO/CD TS 16281, under development,: Rolling Bearing Methods for Calculating the
modified Reference Rating Life for universally loaded Bearings
[3] ISO 281 (2007): Rolling Bearings-Dynamic Load Ratings and Rating Life
[4] ISO 76 (2006): Rolling bearings-Static load ratings
[5] DIN 743 (2000): Load Proof of Capacity for Shafts and Axles
[6] Research Board of Trustees: Mechanical Engineering: Arithmetical Proof of Strength for
Mechanical Engineering Parts; VDMA Publishing House, Frankfurt/Main; 5. Edition 2003
[7] Brändlein, Eschmann, Hasbargen, Weigand: The Bearing Praxis; United Publishing Houses
Ltd., Mainz; 3. Edition 1998
[8] T. A. Harris: Rolling Bearing Analysis; John Wiley & Sons Inc., New York; Fourth Edition
2001
[9] Gross, Hauger, Schnell, Wriggers: Technical Mechanics 4; Jumper Verlag; 4. Edition 2002
[10] KISSsoft INC.: Handbook for the Mechanical Engineering Calculation Software;
www.KISSsoft.ch; 2006
[11] M. Breuer: Theoretical and experimental Regulation of the Bearing Stiffness; Advance
Reports ASSOCIATION OF GERMAN ENGINEERS Series 1, No., 241; Association of
German Engineers Publishing House Ltd. 1994
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