Engineering, 2010, 2, 477-484
doi:10.4236/eng.2010.27063 Published Online July 2010 (http://www.SciRP.org/journal/eng)
477
Model of Rotation Accuracy of High-Speed Spindles on
Ball Bearings
Jeong-Du Kim1, Igor Zverv2, Keon-Beom Lee3
1Department of Mechanical Engineering, Sejong University, Seoul, South Korea
2Department of Mechanical Engineering, Stankin University, Moscow, Russia
3Department of Mechanical Engineering, Korea Polytechnic College IV, Daejeon, South Korea
Received October 10, 2009; revised February 2, 2010; acce pte d May 3, 2010
Abstract
For the purpose to improve a design quality of high-speed spindle units, we have developed mathematical
models and software to simulate a rotation accuracy of spindles running on ball bearings. In order to better
understand the mechanics of ball bearings, the dynamic interaction of ball bearings and spindle unit, and the
influence of the bearing imperfections on the spindle rotation accuracy, we have carried out computer aided
analysis and experimental studies. When doing this, we have found that the spindle rotation accuracy can
vary drastically with rotational speed. The influence of bearing preload has a secondary importance. Compari-
son of the results of these studies has demonstrated adequacy of the models developed to the real spindle units.
Keywords: High-Speed Spindle Unit, Ball Bearing, Bearing Imperfections, Spindle Rotation Accuracy
1. Introduction
One of the most important features of a spindle unit (SU)
is its accuracy of spindle rotation. We can define the
spindle rotation accuracy as stability or immobility of the
spindle axis while rotating. Indeed, accuracy is the value
inverse to imperfection, and in practice, we usually
check SU imperfection by examining its vibration and
measuring spindle axis displacements (vibration ampli-
tudes). There are several reasons for the spindle vibration
[1], among which we can point out imperfections of
spindle bearings [2,3], spindle disbalance [4], dynamic
disturbances caused by the drive or the motor [1], etc.
The vibration caused by disbalance is a simple one, since
it takes place at one frequency-the frequency of rotation,
and can be improved by spindle balancing [5]. In this
article, we study the spindle rotation accuracy caused by
the non-ideal ball bearings, i.e., we study the imperfec-
tions of spindle rotation caused by the bearing imperfec-
tions. Furthermore, we define the imperfection of spindle
rotation (the inverse to the spindle rotation accuracy) as a
root-mean-square sum of the harmonics (amplitudes) in
the spectral decomposition of spindle vibration, exclud-
ing the harmonic at the speed of rotation. We term this
sum the spindle run-out. We also introduce such a meas-
ure of rotation accuracy, since it follows the method of
its measurement in practice: the most detailed informa-
tion about spindle rotation we get by measuring its vibra-
tion and representing the vibration in the form of a spec-
trum of vibrodisplacements. Averaged summation of the
spectrum amplitudes gives us a general measure of vi-
brodisplacements, which, being doubled, we term the
run-out. Such a spectrum depends on disturbances (pro-
duced by non-ideal bearings in our case), elastic, and
damping properties of the SU structure. One of our pur-
poses is to estimate the influence of these factors on SU
spectrum of vibration and, to reduce as much as possible
the spindle run-out by perfect designing (choice) of the
SU structure. Following the purposes stated above, we
have developed the complex mathematical model of
high-speed SUs [1], one of the elements of which is the
dynamic model (Figure 1).
2. Construction of Model of Spindle Rotation
Accuracy
The model of spindle rotation accuracy incorporates two
models: the dynamic model of spindle units and the
model of bearing disturbances.
2.1. Dynamic Model of Spindle Units
The dynamic model gives us an opportunity to estimate

J.-D. KIM ET AL.
478
Figure 1. Diagram of the complex model of spindle units.
the following characteristics of the SU dynamics:
1) Dynamic stiffness (compliance) in the form of am-
plitude-frequency characteristic;
2) Rotation accuracy estimated by the amplitude-fre-
quency spectrum of spindle vibration and the spindle
run-out.
The structural and analytical models used for analysis
of the experimental high-speed SU are presented in Fig-
ures 2(a) and (b). When developing the analytical model,
we make the following basic assumptions:
1) The SU is considered to be a linear dynamic system
having continuous and lumped parameters;
2) We assume the spindle to be an elastic beam of
variable cross-section;
3) We assume that the bearings have linear elastic and
damping characteristics.
Figure 2. Structural (a) and analytical (b) model of the
spindle unit.
When representing the SU by the analytical diagram
(Figure 2(b)), we apply the finite element method [6].
Each spindle element, we represent by a 2-node beam
element; at that, each element has three displacements
(radial, axial, and angular). We describe spindle dynamic
displacements by the system of linear ordinary differen-
tial equations in the matrix form as follows:

t MBKF
 
XXX (1)
where X (t) is the 3m dimension vector of nodal dis-
placements; F (t) is the 3m vector of nodal dynamic
loads; M, B, and K are the 3m  3m matrices of mass,
damping, and stiffness; m is the number of nodes in the
analytical model; t is time.
Assuming that the damping forces in bearings are
proportional to the elastic forces [6], i.e. B = K
(where  is the damping coefficient,  is the frequency
of oscillation), and that the dynamic loads have a har-
monic character, i.e.


it
o
te

FF (2)
And following [6], we express the complex vector of
dynamic amplitudes Xo through the matrix of natural
forms V and the frequencies of oscillations s as follows:

J.-D. KIM ET AL.479

T
o
22
1
1s
VV
i  




 

diagFW FXo
o


2
(3)
where Fo is the complex vector of amplitude nodal loads;
W is the matrix of frequency functions; s is the number
of natural frequency; i is the imaginary unit. Thus, the
modal matrix V has the properties of orthogonality [6],
i.e.

Tunit matrixVVME (4)

T
s
VV

Kdiag (5)
Solution (3) we derived by expansion of vibration
forced amplitudes into the series in terms of the eigen-
functions. At that, the amplitude-frequency characteris-
tics represented by formula (3) correspond to the mono-
harmonic vibrodisturbance F(t) by formula (2). In the
case of polyharmonic vibrodisturbance produced by sev-
eral bearings, we can derive the dynamic load as follows:
 
it+
1
t
Nkk
kk
pk
e





FF (6)
where p is the number of bearings in SU; N is the num-
ber of spectral harmonics to be considered; Fk (k) is the
vector of nodal loads of the k-th harmonic; k and k are
the frequency and phase of the k-th harmonic.
The hardware we apply for measurements of spindle
vibration spectrum gives us an opportunity to make a
root-mean-square summation for all the amplitude har-
monics, which get into the frequency band r (r = 1, 2,
3, ..., q) to be checked, where q is the number of fre-
quency bands. The root-mean-square amplitude of vi-
brodisplacements of the vector Xo on the coordinate j in
the frequency band r (the double of which we term the
run-out) can be determined as follows:
 
2
2
A
kr
jr jnnk
p


 





WF 





(7)
where Wjn are the elements of the matrix of frequency
function (3); Fn (k) is the k-th harmonic of the n-th
component of the vector of vibrodisturbances;
kr



denotes summation of all the harmonics k, which enter
into the frequency band r. Formula (7) is used to es-
timate the spindle rotation accuracy in the frequency
band r to be checked.
2.2. Model of Ball Bearing Disturbances
High-speed ball bearings have several sources of distur-
bances [7], i.e. the disturbing forces, which produce SU
vibration. These include the following:
1) Imperfections of macro geometry of the bearing
races, such as out-of-roundness, waviness, radial and
axial run-outs, possible dents and scratches, etc;
2) Imperfections of macro geometry of the balls,
which are similar to those of the races, plus possible dif-
ference of ball sizes;
3) Imperfections of micro geometry of the races and
balls i.e. surface roughness (some of these imperfections
protrude through the oil film generated between races
and balls while normal operation and produce collisions);
4) Variation of bearing stiffness while rotation (the ra-
dial stiffness depends on position of the balls with re-
spect to the radial force vector);
5) Heterogeneity of elastic properties of the races and
balls;
6) Lubricant contamination;
7) Motion of cage;
8) Disbalance of spindle and bearing misalignments,
etc.
However, when analyzing precision bearings operat-
ing in favourable conditions, we can neglect some of
these sources and save only the two first ones. Balmont
and Zverev consider the theory of bearing having these
inperfections in [2]. Some of the results obtained are
presented in Table 1, where the spectrum of radial vi-
brodisturbances generated by imperfect ball bearing is
presented.
Where, z is the number of balls in bearing,

is the
contact angle,

is the axial clearance-tightness of
bearing


m

,
R
K
is the radial stiffness of bearing
1
Nm 




; ,,
oi b

 are the relative rotational
speeds of a bearing’s cage to outer and inner rings, and
balls around their own axis


H
z;

is the number of
harmonics of Fourier-series expansion of ball radius,


is the amplitude of the harmonic

of ball radius;
,, ,




are the numbers and amplitudes of har-
monics of imperfections of outer and inner races;

is
the allowance of out of diameter of balls


m

; 0,1,k

, 2,... m1,2,...

are the coefficients.
Table 1 interrelates two types of the spectra: the spec-
trum of imperfections of ball bearing races and balls (the
input parameters) represented by amplitudes of these
imperfections a (), a (), and a (), and the spectrum
of radial vibrodisturbances (disturbing forces) determined
by the frequencies and amplitudes of harmonics (the
output parameters), which are represented in the two
right columns of Table 1. The spectrum of angular vi-
brodisturbances coincides with the spectrum of the radial
ones and differs in amplitudes by the multiplier 0.5Dmtg
(), where Dm is the diameter of the circle, which goes
through the centres of bearing balls. Spectrum of axial
vibrodisturbances of ball bearing having imperfections of
macro geometry is also considered in [2].

J.-D. KIM ET AL. 480
Table 1. Spectrum of radial vibrodisturbances of ball bearing.
Bearing elements Harmonic number Frequency [Hz] Amplitude [N]
Outer raceway 1mz




o
1


R
Ka 
Inner raceway 1kz


io

  R Ka 
Outer and inner raceway
1
1
mz
mz
 
 
oi
oi
o
o

 

 

R
ctg
Ka
4


 a
Balls (out-of-roundness) ,,,


b
o

 

R
2
Ka
cosz

Balls (out-of-diameter) 0

 o


R
K3cosz

 
Outer race and balls (out-of-diameter) 0; m




o
1



R
ctg
Ka
3z


Inner race and balls (out-of-diameter) 0; k

 io

 

R
ctg
Ka
3z


The amplitudes of imperfections a (), a (), and a ()
have a particular physical sense. The subscripts and ar-
guments , , and  notify numbers of harmonics of ex-
pansions of imperfections into Fourier series and attrib-
ute them to outer race (), inner race (), and balls ().
At that, the two first ones are the combinations of imper-
fections of races measured in three directions: transversal,
radial, and axial, which can be designated by upper in-
dexes (1), (2), and (3), respectively (Figure 3).

=
0

= 1

= 2

= 1

= 2
 = 1  = 2  = 3
Thus, for example, a ()(1) is the amplitude of the
harmonic  of bearing outer race cross-section imperfec-
tion; a ()(2) is the amplitude of the harmonic  of bear-
ing outer race radial imperfection; a ()(3) is the ampli-
tude of the harmonic  of bearing outer race axial im-
perfection. These imperfections can be measured using
the round-meter gauges of the Taylor & Hobson Co. At
that, the measurements of the axial and radial imperfec-
tion spectra should be made by the Talyrond instrument
and the measurements of the cross-section imperfections-
by the Talyserf instrument (both of them should be
equipped by the Talydate facility for expansion of the
traces made into Fourier series). Summation of the com-
ponents should be made as follows:
Figure 3. Macro geometric imperfections of bearing race-
ways.


 


λλλ
23 1
222
cos 1
cos tgaaa











a = 3, and a = 3 to triangularity, etc; a = 1 and a = 1 has no
meaning; a = 1 corresponds to eccentricity of the inner
race with respect to rotational axis, i.e. to spindle disbal-
ance (which is an imperfection of the SU accuracy, not
of the bearing); a = 0 corresponds to averaged difference
between diameters of balls.
a (8)


 


23 1
22
cos 1
cos tgaaa







 


 

2
a (9)
The principle difficulty when simulating the ball
bearing vibration spectrum makes inputting of the spec-
tra of bearing imperfections. It looks to be impossible to
measure and to input all the harmonics of races and balls
individually. In order to avoid it, we can use some ex-
trapolation based on the results of measurements of races
and balls, and represent the spectra of imperfections by
the following hyperbolic functions [2]:
Representation of imperfections of races and balls by
Fourier series has a clear mechanical sense: harmonics
3 12
,,aaa





 

 
a = 2, a = 2, and a = 2 correspond to ovality of outer race,
inner race, and balls (when averaged), respectively; a = 3, 
(10)

J.-D. KIM ET AL.481
At that, the coefficients  and  can be determined as
the regressive factors to be derived by statistical treat-
ment of the results of measurements of real races and
balls. The results of such measurements and treatment
for some precision bearings are presented in Table 2.
The coefficient  depends on the accuracy of bearing
parts machining, and the coefficient
 on the type of
technology of races and balls machining. The more ac-
curate bearing, the lower
 . The regressive curves ap-
proximating the spectrums of the bearings presented in
Table 2 are shown in Figure 4.
The effect of bearing assembly in the bearing set (mis-
alignments of rings) on the vibration of rotating rotor
was studied in [8].
2.3. General Algorithm of Simulation
The program flowchart used for simulation of the SU
dynamics is presented in Figure 5.
Using the standard FEM procedure, we calculate in
block 1 the matrices of stiffness K, mass M, and damp-
ing B of the SU dynamic system. Estimation of eigen-
values, we make applying the Jacobi method in block 2
and thus determine the natural frequencies and modes of
SU oscillation. The SU dynamic compliance, we esti-
mate in the form of amplitude-frequency characteristics
calculating the matrix of frequency functions W in block
3. The spectra of vibrodisturbances produced by the
bearings, we simulate in block 4. The problem of SU
forced oscillation, we solve in block 5. In the result, we
obtain the spectra of spindle vibration and the radial and
axial run-outs (which are the doubled root-mean-squares
of spindle radial and axial vibrodisplacements when ig-
noring the component at the rotational speed caused by
disbalance and misalignment of bearing rings). The re-
sults of simulation, we record in block 6 and use in fur-
ther analysis.
Table 2. Regressive coefficients

and

.
Outer race Inner race
Bearings
[m] 


 [m]




Roller bearing 3011
(FAG) 0.366 1.02 0.145 0.971
Roller bearing 2-3182111
(GPZ, Russian) 2.470 2.25 0.488 2.02
Ball bearing 2-36106K6
(GPZ, Russian) 0.380 1.50 0.350 1.90
Ball bearing 6-206
(GPZ, Russian) 1.2231.75 0.95 2.06
Ball bearing 2-206
(GPZ, Russian) 1.568 2.46 0.889 2.01
(a)
(b)
Figure 4. Regressive curves approximating the spectrums of
the bearings (a) bearing’s rings (b) bearing’s rings, rollers,
balls.
3. Analysis of Spindle Rotation Accuracy
The dynamic model developed, is represented as soft-
ware for analysis of spindle rotation accuracy in design-
ing. In order to make experimental study of spindle rota-
tion accuracy, we used a special test rig (Figure 6(a)).
The experimental high-speed SU (Figure 6(b)) running
on precision ball bearings (ABEC 9 class, USA Standard)
was mounted in an aerostatic bearing to isolate the SU
from the bed and the drive.
The spindle vibration was measured by the non-contact
sensor WSG 69-5 fixed at the SU housing and using the
amplifier WSM-6983 (Roitlinger Co.). After spindle
acceleration to the highest rpm, we unswitched the belt
of the drive, broke the drive motor, and made the meas-
urements. We analyzed the signal from the amplifier by
using the spectrum analyser model 2031 (B&K Co.).
Using the special methods for data processing [9,10], we
increased the accuracy of measurements of the spindle

J.-D. KIM ET AL. 482
vibrodisplacements up to 0.1 m. The results of the
measurements had two types of errors: 1) the errors
caused by the imperfections of shape of spindle mandrel;
2) the errors caused by the discontinuity of magnetic
conductivity of mandrel’s material. We corrected these
errors using the results of the mandrel profiling by the
Talyrond instrument and the technique presented in [9].
The results of the measurements and of simulation of SU
vibration are presented in Figure 7. We can see three
resonance frequencies. The first one at n1 = 5200 rpm,
we attribute to the coincidence of the vibrodisturbance
harmonic zo (where z is the number of balls in the
bearing and o is the bearing cage rotational speed) and
the spindle first natural frequency f1 = 650 Hz. Two other
resonance frequencies, n2 = 12 000 rpm and n3 = 18 000
rpm, we attribute to the coincidence of the vibrodistur-
bance harmonics caused by the outer ring triangularity
(3rd harmonic of the outer ring’s race imperfection) and
by the outer ring ovality (2nd harmonic of the outer
ring’s race imperfection) and the spindle first natural
frequency.
Figure 5. Program flowchart of the dynamic problem solu-
tion.
(a)
(b)
Figure 6. Layout of the rig (a) and spindle unit with vari-
able bearing preload (b).
Figure 7. Run-out of spindle related to rotational speed.
The spindle run-out at low rotation speed (n < 2000
rpm) is about 0.1-0.2 m, which is practically equal to
the run-out of the precision bearings. However, at the
resonance rpm, the spindle run-out increases up to
1.0-1.8 m. At that, when comparing the results of com-
puter simulation with that obtained experimentally, we
notice that the computer error of the resonance rpm does
not exceed 10-12%. The error of estimation of the
run-outs does not exceed 15-20% at the non-resonant
mode and 20-30% at the resonant mode that we explain
by real damping in the spindle bearings.

J.-D. KIM ET AL.483
In order to examine the influence of bearing preload
on the spindle rotation accuracy, we made the computer
and experimental studies of the SU having a variable
preload of the ball bearings (Figure 6(b)). The depend-
ence of the spindle radial accumulated run-out in the
frequency band of 24 000 rpm (as the doubled root-
mean-square amplitude of vibrodiplacements, which get
into the frequency band of 24 000 rpm) on the bearing
preload is presented in Figure 8.
We can see that an increase in preload from a light one
of 120 N up to a heavy one of 330 N results in a decrease
of the run-out by 26%, however there is an increase of
the excessive temperature of the front bearing outer ring
by 32% at the same time (Figure 9).
Figure 8. Accumulated run-out related to be ar ing pr eload.
Figure 9. Temperature of the front bearing related to bear-
ing preload.
Under very low preload (lower than 60 N), the simula-
tion shows a radical growth in the accumulated run-out,
but in practice, it keeps steady; thus, it follows that under
very low preloads of bearings, the theory we use does not
work perfectly well. However, under medium (working)
bearing preloads of 180-220 N, the spindle run-out in-
crease (at the resonant mode) does not exceed 15% and
this agrees with the theory well enough. The positions of
the resonance peaks at the spectrum of spindle vibration
remain practically constant and independent from spindle
rotation speed, since the SU first natural frequency
changes within 5-6% only, when the bearing preloads
change within their medium range. In the non-resonant
mode, an increase in the bearing preload from the light to
the heavy brings the change of the run-out within 8-10%
only.
4. Conclusions
The results obtained are summarized as follows:
1) Having purpose to improve quality of designing of
high-speed SUs on ball bearings and, finally, to reduce
the spindle run-out, we have developed the SU dynamic
model, which is one of the elements of the SU complex
model incorporating the elastic-deformation, heat models,
and the model of bearing lifetime. The dynamic model
incorporates the model of SU stiffness in dynamics, and
the model of spindle rotation accuracy estimated by the
amplitude-frequency spectrum of vibration and the spin-
dle run-out. The dynamic model is based on application
of beam type finite element analytical models of SUs.
The usage of the beam type models allows us to apply
the universal SU analytical models for solution of the
statics, dynamics, and heat transfer problems, which is of
a great importance for practical realization of our com-
plex approach to simulation of SUs while designing. The
theoretical and experimental studies made prove that the
theory is close to reality, and the models and software
developed can be applied for SU designing and devel-
opment.
2) When studying the particular SU by means of
simulation and experimental measurements, we have
found that the spindle rotation accuracy can vary drasti-
cally with variation of rpm. We determined the spindle
run-out increase from 0.1-0.2 m at low rpm to 1.0-1.8
m at high rpm and resonant modes of operation. The
influence of rotation speed and of bearing accuracy pa-
rameters on the spindle rotation accuracy has the greatest
importance.
5. References
[1] I. A. Zverv and A.V. Push, “Spindle Units: Quality and
Reliability in Designing,” Moscow State University of
Technology Press, 2000.

J.-D. KIM ET AL. 484
[2] V. B. Balmont and I. A. Zverev, “Spectrum of Vibration
Caused by Non-ideal Ball Bearings,” Journal of Vibra-
tion Engineering, Vol. 5, No. 3, 1992, pp. 149-157.
[3] A. Tamura, “Vibration Caused by Ball Diameter Differ-
ence in A Ball Bearing,” Bulletin of the Japan Society of
Mechanical Engineers, Vol. 11, No.44, 1968, pp. 229-
234.
[4] A. Tamura and H. Shimizu, “Vibration of Rotor Based on
Ball Bearings,” Bulletin of the Japan Society of Me-
chanical Engineers, Vol. 10, No. 41, 1967, pp. 763-775.
[5] V. Venkatraman, “Analysis of Spindle Running Accu-
racy,” Journal of Machinery and Production Engineering,
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