© The Authors. CC-BY 4.0 Int. Licencee: SV-JME  Strojniški vestnik - Journal of Mechanical Engineering   ▪   VOL 71   ▪   NO 1-2   ▪  Y 2025
44 DOI: 10.5545/sv-jme.2024.1020
A Mathematical Model of the Dimensional Chain  
for a Generation 2 Wheel Hub Unit 
Stanisław Adamczak
1
 – Marek Gajur
2
 – Krzysztof Kuźmicki
2 
1 Kielce University of Technology, Department of Mechatronics and Mechanical Engineering, Poland
2 Fabryka Łożysk Tocznych Kraśnik S.A., Poland
 kkuzmicki@flt.krasnik.pl
Abstract This article overviews wheel hub design solutions and proposes a mathematical model of the dimensional chain and a tolerance formula for 
calculating axial clearance for a generation 2 wheel hub assembly with ball bearings. The dimensional chain analysis and its synthesis are carried out using 
three partial interchangeability methods. The possibility of manufacturing the hub bearing using selection compensation was proposed. The considerations 
made provide an alternative to the current method of process design based on numerous trials and considerable cost. 
Keywords rolling-element bearings, dimensional chain, tolerance formula, axial clearance, wheel hub unit
Highlights
 ▪ A mathematical model of the dimensional chain and the tolerance equations for axial clearance was developed.
 ▪ Axial clearance and its limit deviations were compared to the constructor’s values, revealing significant discrepancies. 
 ▪ Calculations for component dimension tolerances show that achieving partial interchangeability in production is not feasible.
 ▪ Calculated tolerances of independent dimensions using selection compensation into 9 selection groups.
1  INTRODUCTION
The main objective of this paper is to present an approach to the 
design of manufacturing processes using a mathematical model of 
the dimensional chain to achieve the expected axial clearance. In 
previous practice when designing bearing hubs, designers did not 
build mathematical models of dimensional chains. Tolerances were 
selected intuitively based on the knowledge and experience of the 
designer. They were often suboptimal and economically unjustified. 
Therefore, work was undertaken to build a mathematical model of the 
dimensional chain of the axial clearance of a second-generation ball 
bearing hub assembly, in order to use it to verify the hub design and 
to plan the optimal manufacturing process. The hypothesis was made 
that in cases where the tolerance values obtained using the model 
would be less than the tolerances adopted by the designer, it would 
be necessary to change the tolerances of the independent dimensions 
and manufacture the product under interchangeable conditions using 
a selective compensation approach.
Tolerance design is very important in product development and 
manufacturing processes. It is particularly important when applying 
the concept of concurrent engineering (CE). This ensures that 
manufacturing costs are minimised with maximum product quality 
[1]. Many articles address the issues of tolerancing the dimensions of 
independent parts as well as assembling them into assemblies with 
the indication of problems in building tolerance chain models [2]. 
Singh et al [3] and [4] characterise various studies and theories on 
tolerance analysis and synthesis, which can be divided into traditional 
and advanced ones. Traditional tolerance design approaches do not 
consider the impact of tolerance on manufacturing cost as opposed 
to advanced ones. The authors of the theoretical considerations have 
not supported their application in industry with examples. Tolerance 
optimisation for products with multidimensional chains was described 
by Tsung [5] using the example of a ball bearing incorporated into a 
bicycle bottom bracket using specialised software. Nonetheless, the 
measurement of components at the manufacturing stage and their 
association at the assembly stage do not reflect the problems of 
determining the independent dimensional tolerances themselves. 
There are many methods of tolerance analysis in use today, 
both manual and software-based. The use of computer software 
is constantly being developed and the results obtained can be used 
for analysis and optimal design [6] and [7]. In this paper, however, 
manual methods have been used to determine the dimensional chain 
model, perform its analysis and synthesise it for the axial clearance of 
a generation II hub. 
Fig. 1.  Wheel hub unit with two tapered roller bearings
Bearing hubs have undergone extensive modifications over the 
years in order to reduce their manufacturing costs and meet market 
expectations. Decisions made at the design stage help minimize the 

SV-JME   ▪   VOL 71   ▪   NO 1-2       Y 2025   ▪   45
Structural Design
risks of product failure, which may result from errors associated 
with drawing interpretation, product manufacturing, or its primary or 
secondary assembly. The major factors responsible for these errors 
are the designer’s insufficient knowledge of the processes involved in 
the manufacturing of the product and a low level of mechanization or 
automation of the product assembly [8]. Designers making decisions 
on tolerance sizes generally rely on intuition and experience. Such an 
approach, however, may not be correct because if tolerances are too 
small, the manufacturing costs increase substantially.
Wheel hub units for motor vehicles are precision products 
where appropriate dimensioning and tolerancing of components are 
critical aspects of engineering design. The simplest wheel hub unit 
configuration, depicted in Fig. 1, features a pair of tapered roller 
bearings in an adjusted face-to-face preload arrangement.
Another design solution is the first-generation wheel hub unit, 
fitted with a double-row angular contact ball bearing (Fig. 2) or a 
double-row tapered roller bearing (Fig. 3), lubricated for life with 
defined and preset clearance.
The second generation unitized or integrated wheel hub unit 
illustrated in Fig. 4 is a solution in which the bearing housing is 
replaced by an outer ring with a flange to which the rotating parts 
of the brake system and the road wheels are attached. It comes in 
two variants based either on the ball bearing or tapered roller bearing 
designs [9].
A more technologically advanced solution is the third-generation 
wheel hub unit with ball or tapered roller bearings (Fig. 5), where 
the outer flanged ring (1) takes over the function of the housing, and 
the inner flanged ring (2) allows the attachment of the brake disc and 
the road wheel. The wheel hub unit design has undergone various 
modifications to suit specific purposes. For example, the unit shown 
in Fig. 6 features a torque transmission gear system with an inner 
flanged ring (2) and a halfshaft gear (3).
When designing wheel hub units, engineers need to determine 
the tolerance limit for each toleranced dimension. This requires 
robust design knowledge and previous experience. They also analyze 
all the relevant measurement and simulation data. In some cases, 
however, tolerances assumed by the designer may not ensure that 
parts will function correctly [10]. Measurement methods used at 
the manufacturing stage play an essential role [11] and [12]. These 
days, geometrical tolerances can also be used to precisely define 
the product, such as for the design of gears [13]. This methodology, 
called geometrical product specification (GPS), described by EN 
ISO 8015:2012 [14] and ISO 492:2014 [15] has supplemented the 
traditional techniques for determining the geometrical requirements 
of engineering products. GPS helps specify linear and angular 
dimensional tolerances, geometrical tolerances, and surface texture 
tolerances [16] and [17]. 
Fig. 2.  Generation I wheel hub unit with a double-row angular contact ball bearing;  
1 outer ring, 2 inner ring, and 3 seal
Fig. 3.  Generation I wheel hub unit with a double-row tapered roller bearing; 1 outer ring, 2 
inner ring and 3 clamping ring
Fig. 4.  Generation II wheel hub unit based on the ball bearing design; 1 outer flanged ring, 2 
inner rings, and 3 seal 
Fig. 5.  Generation III wheel hub unit based on the tapered roller bearing design; 1 outer flanged 
ring, 2 inner flanged ring and 3 standard inner ring
Fig. 6.  Generation III wheel hub unit with a torque transmission gear system; 1 outer flanged 
ring, 2 inner flanged ring with a face gear and 3 halfshaft gear with meshing teeth

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46   ▪   SV-JME   ▪   VOL 71   ▪   NO 1-2   ▪   Y 2025
Fig. 7.  Generation II wheel hub unit with a double-row angular contact ball bearing
In the study, the GPS dimensioning principles were employed 
to develop the design of a second generation wheel hub unit 
incorporating a double-row angular contact bearing (Fig. 7) [18].
2  A MATHEMATICAL MODEL
A mathematical model of the tolerance formula for the axial clearance 
of the generation 2 wheel hub unite with a double-row angular 
contact ball bearing is presented. The components of a double-row 
angular contact bearing, such as the inner rings, the outer ring, and 
the balls need to be carefully selected to ensure correct axial and 
radial clearance. In practice, however, for this type of bearing, only 
the axial clearance is measured under axial load, which is exerted in 
the axial direction on the outer ring alternately on either face, with 
the inner rings held stationary. 
Assuming zero axial load, we obtained a dimensional chain with 
14 components:
•	 m is axial distance between the raceways of the outer ring,
•	 n
l
 axial distance between the raceway of the left inner ring and its 
narrow face,
•	 n
p
 axial distance between the raceway of the right inner ring and 
its narrow face,
•	 r
wl
 radius of the left inner ring raceway,
•	 r
wp
 radius of the right inner ring raceway,
•	 r
zl
 radius of the left outer ring raceway,
•	 r
zp
 radius of the right outer ring raceway,
•	 d
wl
 diameter of the left inner ring raceway,
•	 d
wp
 diameter of the right inner ring raceway,
•	 d
zl
 diameter of the left outer ring raceway,
•	 d
zp
 diameter of the right outer ring raceway,
•	 α contact angle,
•	 d
l
 diameter of the left raceway ball,
•	 d
p
 diameter of the right raceway ball.
Thus, the dimensional chain equation for axial clearance is a 
function [19]:
L= f (m, p
l
, p
p
, r
wl
, r
wp
, r
zl
, r
zp
, d
wl
, d
wp
, d
zl
, d
zp
, α, d
l
, d
p
).
Because of the symmetry, the number of dimensions was reduced 
to half. Thus, the dimensional chain was simplified to an eight-
component chain. The function had the form:
L = f (m, n, r
w
, r
z
, d
w
, d
z
, α, d),
where m is axial distance between the raceways of the outer ring, n 
axial distance between the raceways of the inner rings, r
w
 radius of 
the inner ring raceway, r
z
 radius of the outer ring raceway, d
w
 diameter 
of the inner ring raceway, d
z
 diameter of the outer ring raceway, α 
contact angle, and d ball diameter. In the study, a simplified design of 
the wheel hub unit (Fig. 8) was considered.
Fig. 8.  Simplified design of the wheel hub unit with a double-row angular contact ball bearing
The dimensional chain equation for the axial clearance was 
written as:
m 	= 	n 	+ 	2a	 –	 L. (1)
Hence,
L 	= 	n 	– 	m 	+ 	2a, (2)
ab c 
22
, (3)
b 	= 	r
w
	+ 	r
z
	– 	d,	 (4)
c 	= 	r
w
	+ 	r
z
	– 	0.5( dz	 –	 dw),	 (5)
a
2
	= (r
w
	+ 	r
z
	– 	d)
2
	– 	(r
w
	+ 	r
z
	– 	0.5(d
z
	– 	d
w
))
2
.	 (6)
Let 
R = r
w
	+ 	r
z
      and      S = 0.5(d
z
	– 	d
w
). (7)
After substituting Eq. (7) into Eq. (6), we obtaine
a
2
 = (R 	– 	d)
2
	– 	(R 	– 	S)
2
, (8)
and after the conversion, we had
a
2
 = (d 	– 	S)( d	 +	 S	 –	2R). (9)
The equation of the dimensional chain for the axial clearance 
was thus given as
Lnmd Sd SR  22 () () . (10)
The tolerance formula was derived using the partial 
interchangeability method. It was assumed that the coefficient of 
variation for all the components of the dimensional chain was equal 
to 1. The tolerance formula for this complex chain was determined 
from Eq. (11):
T
L
A
TA
L
i i
k
i
2
1
1
2
2












. (11)
The partial derivatives were calculated as:






L
n
L
m
11 ,, (12)



 
   
L
d
dR
dSdS R
2
2
, (13)



 
   
L
S
RS
dSdS R
2
2
, (14)

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L
R
Sd
dSdS R
2
2
. (15)
Since
     dSdS R 2, (16)
the tolerance equation took the following form:
TTTd RT RSTS dT
Lnmd SR
222
2
2
2
2
2
2
2
4
     





,
 (17)
TT Td RT RSTS dT
Ln md SR
      




22
2
2
2
2
2
2
2
4

.(18)
3  VALIDATION OF THE MATHEMATICAL MODEL
In this paper, the validation of the mathematical model of the axial 
clearance dimension chain is reduced to analysis and synthesis to 
confirm that the equation is mathematically and physically correct 
and meets the specified functional requirements. 
The nominal value of the axial clearance in the analysed wheel 
hub unit was calculated using Eq. (18), derived under the assumption 
of partial interchangeability. The assumption was made because there 
were eight components in the dimensional chain and the designer’s 
past experience in the design and production of similar products 
indicated such an approach was correct. Table 1 summarizes the 
nominal values of the dimensions shown in Fig. 8 together with their 
tolerances.
Table 1.  Nominal values and tolerances determined for the dimensions provided in Fig. 8
Dimension Unit Nominal value Tolerance
n	 = n
l
	 +n
p
[mm] 22.772 ± 0.020
m [mm] 23.488 ± 0.010
r
z
[mm] 6.730 ± 0.025
r
w
[mm] 6.540 ± 0.025
d [mm] 12.700 ± 0.00975
α [º] 40 -
d
z
[mm] 65.878 ± 0.02
d
w
[mm] 40.211 ± 0.02
L [mm] 0.015 ± 0.010
Due to the complex nature of the eight-component dimensional 
chain, small tolerances, and measurement difficulties, the tolerance 
of the contact angle α was assumed to be zero.
3.1  Dimensional Chain Analyis
Eq. (7) was used to calculate the values of R and S with tolerances.
R	= 	6.54
±0.025
	+ 	6.73
±0.025
	= 	13.27
±0.05
 mm,
S	= 	0.5(65.878
±0.02
	–	40.211
±0.02
) 	= 	12.834
±0.02
 mm.
The numerical values from Table 1 and the calculated values of R 
and S from Eq. (7) were substituted into Eq. (10) to get the nominal 
value of the axial clearance, L = 0.0183 mm
L
  
22 772 23 488
21 27 12 834 12 71 2 834 26 54 0 0133
..
(. .) (. .. ). mm m.
The method of differential calculus, Eqs. (19) and (20), was 
employed to calculate the limit deviations.
x
L
A
a
L
B
b
i i
k
i
i i
n
i 2
1
2
1
1








, (19)
x
L
A
a
L
B
b
i i
k
i
i i
n
i 1
1
1
1
2








. (20)
Then, the values of the partial derivatives were determined using 
Eq. (12 to 15) and Eq. (16) in order to calculate β.
      12 71 2 834 12 71 2 834 26 64 0 3672 .. .. .. .
Thus



 




  L
d
L
S
21 27 13 27
0 3672
3 105
21 32 71 2 837
0 3672
..
.
.,
..
.
 



 

2 375
21 2 834 12 7
0 3672
07 3
.,
..
.
..
L
R
The upper limit deviation x
2 
was:
x
2
 = 0.02 + 2.375×0.02 + 0.73×0.05 + 0.01 + ( –3.105)×( –0.00975) 
    = 0.144 mm, 
while the lower limit deviation x
1 
was:
x
1
 = –0.02 + 2.375×( –0.02) + 0.73×( –0.05) – 0.01
    +( –3.105)×0.00975 = –0.144 mm.
The axial clearance determined from the tolerances of the 
components of the dimensional chain assumed by the designer was:
L = 0.0183 
±0.144 
mm 
(with L = 0.015
±0.010 
mm assumed by the designer).
As can be seen, the calculated axial play was 3.3 µm greater than 
that assumed by the designer. The calculated tolerance value did not 
correspond to the nominal value of the axial clearance.
3.2  Synthesis of the Dimension Chain
The tolerances of the component dimensions were calculated for 
partial interchangeability using the method of equal tolerance, equal 
tolerance class and equal influence [20]. The experience of bearing 
manufacturers shows that performing these calculations for total 
interchangeability gives tolerance results of independent dimensions 
so small that it is not possible to meet them during technological 
process execution.
3.2.1  Method of Equal Tolerance
By substituting the numerical data into the tolerance equation; Eq. 
(18) and assuming T
n
	= 	T
m
	= 	T
d
	= 	T
S
	= 	T
R
	= 	T the result was obtained:
TT Td RT RSTS dT
TT d
Ln md SR
      




 
22
2
2
2
2
2
2
2
22
2
4
4


      




      

RT RSTS dT
Td RR SS d
2
2
2
2
2
2
2
222
2
4

 


  

T
T
2
4
0 135
0 325 01 90 018
00 2
4 218
0 005
.
.. .,
.
.
. mm.
TT Td RT RSTS dT
TT d
Ln md SR
      




 
22
2
2
2
2
2
2
2
22
2
4
4


      




      

RT RSTS dT
Td RR SS d
2
2
2
2
2
2
2
222
2
4

 


  

T
T
2
4
0 135
0 325 01 90 018
00 2
4 218
0 005
.
.. .,
.
.
. mm.
3.2.2  Method of Equal Tolerance Class for a Complex Chain
Eqs. (21) and (22) were used to calculate tolerances using the equal 
tolerance class method.
kT ak
L
A
A
LL A
i i
n
i
i
22 22
1
1
2
3
2








 



, (21)
Ta A
Ai
i
=
3
. (22)
By substituting the values from Table 1 into Eq. (22), the results 
were:
T
n
	= 	2.834a,	T
m
	= 	2.862a,	T
d
	= 	2.333a,	T
S
	= 	2.341a,	T
R
	= 	2.368a.

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48   ▪   SV-JME   ▪   VOL 71   ▪   NO 1-2   ▪   Y 2025
After substitution into Eq. (21), the following was obtained
T
L
2
	= 	a
2
(1×2.834
2
 + 1×2.862
2
 + 9.641×2.333
2
 + 5.641×2.341
2
 
             + 0.533×2.368
2
) = a
2
×102.6, hence,
a
T
L
= =
1026
000197
.
. mm.
So the tolerances of the component dimensions are respectively: 
T
n
 = 0.006 mm, T
m
 = 0.006 mm, T
d
 = 0.005 mm, T
S
 = 0.005 mm, and 
T
R
 = 0.005 mm.
3.2.3  Method of Equal Impact
Based on Eq. (23) for the composite chain and assuming that the 
components of the chain have a normal distribution i.e. k	 =	 1, the 
result was obtained:
k
L
A
Tk
L
A
Tk
L
A
AA AA A
n
n 11 22
2
1
2
22
2
2
22























...
 



2
22
Tm
A
n
. (23)
hence T
m
L
A
m
L
A
A
n
n
n












2
2
.
For five dimensions of the chain T
L
2
 = 5×m
2
, 
hence m
T
L
= = =
5
002
2236
0009
.
.
. mm. 
The tolerances of the individual dimensions are therefore 
T
n
 = 0.009 mm, T
m
 = 0.009 mm, T
d
 = 0.003 mm, T
S
 = 0.004 mm, and 
T
R
 = 0.012 mm.
Table 2.  Summary of results for three methods of calculating tolerances for partial 
interchangeability
Tol. specified by 
constructor 
Tol. calculated 
using the equal 
tolerance method 
Tol. calculated using 
the equal tolerance 
class method
Tol. calculated 
using the equal 
impact method 
T
n
 [mm] 0.04
0.005
0.006 0.009
T
m
 [mm] 0.02 0.006 0.009
T
d
 [mm] 0.0195 0.005 0.003
T
S
 [mm] 0.04 0.005 0.004
T
R
 [mm] 0.10 0.005 0.012
Table 2 summarises the results for the three methods mentioned 
above and the values adopted by the designer. No calculations have 
been made for the minimum cost method because the cost functions 
of making the rings as a function of changes in the dimension chain 
values are not known.
3.3  Partial Interchangeability for Axial Clearance with Selective 
Compensation
For small tolerances of the dependent dimension, it often turns out 
that the tolerances of the independent dimensions determined under 
the assumption of partial interchangeability are very small. Meeting 
them results in a significant increase in manufacturing costs or is 
even impossible. In turn, their enlargement generates an unacceptable 
number of defects. In such cases, the most economical solution is 
to make the individual parts within the tolerances achievable under 
the given production conditions. The tolerance of the dependent 
dimension will then be bigger than the tolerance assumed by the 
designer, and additional work must be done during assembly to 
achieve the resulting dimension within the assumed limits. This kind 
of interchangeability, in which additional operations are needed to 
obtain the correct product, is called conditional interchangeability. 
One of its types is the selection of parts with the right dimensions 
- selection compensation. It involves measuring all the parts to 
be made and dividing them into several dimensional groups and 
associating them appropriately together with the rolling elements 
during assembly [21] In the case of selection interchangeability, 
dividing all the parts from a given lot into m selection groups allows 
their production tolerance to be increased m times. We can therefore 
write that the actual closure dimension tolerance T
L
'
 is equal to 
the product of the number of selection groups m and the closure 
dimension tolerance in a given selection group T
L
 [22].
T
L
'
	= 	m×T
L
. (24)
For dimension chains with multiple dimensions, they should be 
divided into two chains I and II, trying to obtain an equal sum of the 
tolerances of their component dimensions using Eq. (25).
chain I           chain II





 

L
A
T
L
A
T
i
A
i
p
i
A
ip
m
ii
11
, (25)
where p is the number of component dimensions of one of the 
two chains obtained from splitting the chain under consideration, 
and m the number of independent dimensions of the chain under 
consideration.
In the case of the wheel hub unit under consideration for the axial 
clearance tolerance equation this looks like the following: 














L
n
T
L
S
T
L
m
T
L
d
T
L
R
T
nS mdR
.
The equality of parts I and II of the chains could not be achieved, 
as the following calculation illustrates.
1×0.04+2.375×0.04≈1×0.02+3.105×0.0195+0.73×0.1,
0.135 mm ≠ 0.154 mm.
Actual tolerance of the closing dimension T
L
' = 0.135 + 0.154 = 
0.289 mm, and the desired tolerance is T
L
 = 0.020 mm.
In order to meet the condition that the tolerance in a selection 
group should not be greater than the tolerance of the closing 
dimension using Eq. (24), the number of selection groups was 
calculated, which should be no less than 15 [22]:
• executive deviations: x
2
' 	= 	0.144 mm and x
1
' 	= 	–0.144 mm,
• desired deviations: x
2
	= 	0.01 mm and x
1
	= 	–0.01 mm.
For the coordinates of the centre of the area of variation the 
following condition must be met:
M
L
'	=	M
L
, (26)
where M
L
'	is coordinate of the centre of the performance tolerance 
variation area, and M
L
 coordinate of the centre variation area of the 
tolerance desired. Using the formulas:
M
L
 = 0.5(x
1
 + x
2
)  and M
L
'
M
L
A
M
L
i i
m
A
i
' 










1
,
 (27)
where x
1
, x
2
 
are desired deviations, and M
A
i
 are coordinates of the 
centres of the area of variation of the component dimensions before 
selection M
L
 = 0.015 mm, and M
L
'	= 0 mm.
A zero value was obtained due to symmetrically distributed 
deviations for each component dimension. In the present case, the 
condition of equality of the coordinates of the centres of the areas of 
variation M
L
' = M
L
 is not satisfied. The fulfilment of the conditions 
of equality of the coordinates of the area of variation Eq. (26) and 
equality of tolerance Eq. (25) gives confidence in the constancy of 
the resultant dimensions in each selection group [22]. In order to meet 
this requirement, the dimensional deviations of some component 
cells were changed and recalculated. New deviation values for the 
ball diameter d were adopted on the basis of experience and deviation 
calculations were carried out for R and S. Table 3 summarises the 
previous and new tolerance values. 

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Structural Design
M
L
’









 
    


05
05 03 105 0 005 07 3
1
12
1
.
.. ..
L
A
xx
x
i i
m
ii
R RR
x    
2
0 015 . mm,
and T
R
 =  0.08 = x
2R
 – x
1R
, we get x
1R
 = –0.03 mm and x
2R
 = 0.05 mm.
Table 3.  Previous and new tolerance values
Dimension Nominal value Design tolerance New tolerance
r
z
 [mm] 6.730 ± 0.025 +0.025, –0.015
r
w
 [mm] 6.540 ± 0.025 +0.025, –0.015
R = r
w
+r
z
 [mm] 13.270 ± 0.050 +0.050, –0.030
d [mm] 12.700 ± 0.0097 +0.001, –0.006
d
z
 [mm] 65.878 ± 0.020 ± 0.013
d
w
40.211 ± 0.020 ± 0.013
S = 0.5(d
z
–d
w
) [mm] 12.834 ± 0.020 ± 0.0125
After this change, equal values of the coordinates of the centres of 
the tolerance variation areas were obtained. Rechecking the condition 
of equality of tolerance of the two parts of the split dimensional chain.
1×0.04 + 2.375 × 0.04 ≈ 1 × 0.02 + 3.105 × 0.007 + 0.73 × 0.08, 
0.135 ≠ 0.1.
The condition of equality of tolerance is not fulfilled, which means 
that the axial clearance will not reach the value between 0.05 and 
0.025 mm. To avoid this, the tolerance of dimension S was changed 
without changing the equality of values of the centres of the tolerance 
variation areas of part I of the chain. For this purpose, the necessary 
tolerance for S was calculated.
The tolerance of Part I is equal to 1×0.04 + 2.375×T
S
' = 0.1 so  
T
S
' = 0.025. Then S will be 12.834
±0.0125 
mm.
Calculation of actual tolerance for new tolerances on independent 
dimensions T
L
' = 1×0.04+1×0.02+2.375×0.0125+3.105×0.007+0.73
×0.08 = 0.17 mm. For an actual axial clearance tolerance of 0.17 mm, 
the number of selection groups will be 9.
4  DISCUSSION
Wheel hub units have numerous benefits and as such they are 
replacing traditional wheel bearings in motor vehicles. To design 
them, engineers now commonly apply GPS principles since these 
help them better reflect their intentions and prevent drawing 
interpretation errors. The mathematical model proposed in this article 
was used to determine the amount axial clearance and the optimal 
tolerances of toleranced dimensions for a generation 2 wheel hub unit 
with a dual-row angular contact ball bearing. In this case the axial 
clearance determined using the dimensional chain equation was 3.3 
µm greater than the value assumed by the designer. The tolerance 
values obtained with the model were validated by comparing them 
with the values assumed by the designer for this product. These are 
significantly smaller than the tolerances adopted by the designer. This 
indicates that it is necessary to change the tolerances of the component 
dimensions and manufacture the product under interchangeability 
conditions using the selective compensation approach. In the case 
under consideration, the division of the independent dimensions into 
nine selection groups allows components to be manufactured with 
tolerances that are achievable, economically justified and assembly 
guaranteed to meet the designer’s requirements. 
5  CONCLUSIONS
The proposed solution of using a mathematical dimensional chain 
model in the design of a second-generation bearing hub is an 
alternative to the frequently used product and process design method 
based on designer experience, numerous tests and at considerable 
cost. Its use will reduce the time and cost associated with designing 
a bearing hub and putting it into production. The proposed solution 
will still be documented and validated in series production of bearing 
hubs which is planned in the near future. 
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Structural Design
50   ▪   SV-JME   ▪   VOL 71   ▪   NO 1-2   ▪   Y 2025
Received 2024-04-13, revised 2024-09-12, accepted 2024-10-03 
Original Scientific Paper.
Data availability Data supporting the proposed solution will be available 
from authors K Kuźmicki and M Gajur.
Author contribution S Adamczak contributed to conceptualization, 
supervision, writing – review & editing; M Gajur performed formal 
analysis, validation, writing – review & editing; K Kuźmicki provided formal 
analysis, funding acquisition, project administration, resources, validation, 
visualization, writing – original draft, writing – review & editing
Matematični model dimenzijske verige  
za enoto kolesnega pesta 2. generacije
POVZETEK Članek predstavlja pregled konstrukcijskih rešitev pesta kolesa 
ter predlaga matematični model dimenzijske verige in enačbo za izračun 
tolerance za izračun osne zračnosti za sklop pesta kolesa 2. generacije s 
krogličnimi ležaji. Analiza dimenzijske verige in njena sinteza sta izvedeni 
z uporabo treh metod delne zamenljivosti. Predlagana je bila možnost 
izdelave ležaja pesta z uporabo izbirne kompenzacije. Opravljeni razmisleki 
predstavljajo alternativo sedanji metodi načrtovanja procesa, ki temelji na 
številnih poskusih in precejšnjih stroških.
Ključne besede kotalni ležaji, dimenzijska veriga, tolerančna formula, osna  
zračnost, enota kolesnega pesta