FRICTION COEFFICIENTS DEFINITION IN COMPRESSION-FIT
COUPLINGS APPLYING THE DOE METHOD
D. Croccolo, R. Cuppini and N. Vincenzi
DIEM
University of Bologna
Viale Risorgimento, 2 – 40136 – Bologna – Italy
ABSTRACT
The fundamental goal of this work is to define a generalized methodology which is useful to calculate the starting friction
coefficient µll in the fork-pin compression fit couplings of front motorbike suspensions. The starting friction coefficient µll for
these applications was obtained applying the Design of Experiment methodology on several specimens and performing some
Finite Element Method analyses in combination with a mathematical model. Couplings are different both for the material of the
elements in contact and for the coupling surfaces condition. The possible combinations investigated are: the fork in aluminium
and the pin in steel and both the fork and the pin in aluminium. The second goal of this work is to update, applying the µll
mathematical models, an innovative software (Fork Design©), realized by the authors in Visual Basic® programming
language, which is useful to perform the design and the verification of the fork-pin couplings of front motorbike suspensions,
differentiated by the geometry (that means by the stiffness of the fork) and by the materials of the coupled parts.
Introduction
The design of the fork-pin compression-fit couplings in front motorbike suspensions, like the one reported in Figure 1, is
actually uncertain mainly because of the poor knowledge of the starting friction coefficient µll, and of the mean coupling
pressure p. In these mechanical applications, friction coefficient is one of the variable parameters because it depends on
coupling condition, while the mean coupling pressure p is uncertain because the geometry of the fork isn’t a symmetric one,
thus it’s impossible to define p by means the high thickness pipe theory that needs a polar axial symmetry. The axial releasing
force Fll=µll·p·A, which is the fundamental design parameter for these couplings, indeed depends on the mentioned two factors,
usually unknown, and on the coupling area A, usually known.
Figure 1. A front motorbike suspension.
Figure 2. A strain gauge applied on the external surface of the fork.
In a previous work [1] a mathematical model useful to calculate the mean contact pressure p in the fork-pin coupling has been
defined by introducing an overall coefficient β, depending on some geometric parameters, which is able to correct the
theoretical formulas valid only for axial-symmetric elements. In another work [2] an additional mathematical model useful to
calculate the starting friction coefficient µll has been defined as a function of the production and assembly specifications in
couplings with both the fork and the pin in steel.
The aim of the present work is to complete the definition of the starting friction coefficient µll applying the Design of Experiment
(DOE) methodology [3-4] with the approaches presented in [2], for the pins in steel and the forks in aluminium, and both for the
pins and for the forks in aluminium. It was also updated an innovative software (Fork Design©), realized by the authors in
Visual Basic® programming language, which is useful to perform the design and the verification of the whole fork-pin
couplings.
Fork-pin compression fit couplings
The coupling under investigation is realized between the pin (symmetrical circular and hollow section) and the fork
(asymmetrical section) of a motorbike front suspension. The coupling process is a longitudinal compression-fit realized by a
standing press. The fundamental design parameter is the axial releasing force Fll=µll·p·A. Because of the power and the weight
increasing of motorbikes, the axial releasing force Fll is also increased a lot in the recent years in order to guarantee the driver
safety. For this reason, the amount of interference is an important design parameter, proportional to the pressure p, because it
must be high enough to exceed the releasing tests but not too high in order to keep low the tensile state of the components,
which may overcome the yield stress. The contact surface A is also an important parameter, easy to define, but often limited
by the design constraints. Concerning µll parameter, the next two steps are actually strategic for the coupling design:
1. the right knowledge of the friction coefficients;
2. the definition of the most important parameters which influence and maximize the starting friction coefficient.
Factors and levels chosen for the Design Of Experiment (DOE)
According to [2], after a screening test, were identified the appropriate input factors and their levels that are:
•
the rust presence, because pins are not immediately assembled but they are stored in open metal boxes for some
days and, therefore, their surfaces might be covered by some rusted spots;
•
the lubricating oil presence, because the pins might be protected with a little film of lubricating oil before they are
assembled with the forks;
•
the resting time, because several authors [5-6-7] state that the greatest starting friction coefficient is reached some
hours later the connection of the parts.
We must consider that the DOE methodology is applied both in the case of the pin in steel and the fork in aluminium, and in
the case of the pin and the fork in aluminium, obtaining different analyses on each type of coupling.
In order to reduce the number of the tests, two levels for each factor are chosen: low and high. In the case of the rust
presence and the lubricating oil presence the low level means the lack of the factor, while the high level means the presence
of the factor on the pin; in the case of the resting time the low level means that and the pin is immediately disengaged from
the fork after the insertion, while the high level means that the resting time is, at least, equal to 72 hours.
3
Therefore, a complete 2 factorial plane is obtained. In order to reduce the influences of the noise and of the not investigated
factors, it was decided to repeat each test three times, for a total of 24 tests for each coupling type (steel-aluminium and
-1
aluminium-aluminium) and to keep constant the coupling and decoupling speed (16mmws ), the environment temperature and
the shape of the coupling elements during all the tests. We performed the DOE during both the coupling and the decoupling
phase in order to calculate the sliding friction coefficient µrl, and the starting friction coefficient µll respectively for each of the
two type of coupling. Thus, a total of four DOE (coupling and decoupling phases for steel-aluminium and aluminiumaluminium) with 96 tests were performed, and four mathematical models were determined for each different friction coefficient.
Methodology
First of all, the fork and the pin geometrical dimensions are evaluated (the internal diameter of the fork DAi, the external
diameter of the pin DIa, the collar thickness of the fork s, the external diameter of the fork DAa= DAi+2s, the roughness of the
fork RpA and the roughness of the pin RpI). According to [5] the actual interference of the coupling Z is a function of the nominal
value of the interference U and of the total roughness RpA,I, Z=U-2(RpA+ RpI). It’s known that the actual interference Z is
proportional to the contact pressure p, both in symmetrical and in asymmetrical geometries. Even if it has been demonstrated
[2] that friction coefficients are independent from the interference Z we decided to couple the forks and the pins which provide
a similar interference level; this decision was assumed both for the steel-aluminium and for the aluminium-aluminium
couplings. In order to calculate the friction coefficient, it’s possible to distinguish the phase of “coupling” from the phase of
“decoupling”: during the first one the sliding friction coefficient µrl is determined related to the maximum axial coupling force,
while during the second one the starting friction coefficient µll is determined related to the first peak of the axial releasing force.
Both the coupling and the decoupling forces may be evaluated by the following formula:
Frl,ll = µ rl,ll ⋅ p⋅ A
where
F is the axial force
p is the mean coupling pressure
(1)
A is the coupling area equal to π ⋅
(DAi + DIa )
⋅ LF = π ⋅ DF ⋅ LF , in which LF is the
2
coupling length and DF is the coupling diameter.
The coupling length LF is the total interference run of the pin inside the fork while the snapshot force is F=µ·p·(π·DF·r), where r
is the snapshot run which is included within the 0 and LF range. Obviously, if the surfaces are cone-shaped the actual area
must be accurately evaluated.
In the equation (1) the mean coupling pressure p is another unknown parameter. As a matter of fact, the coupling pressure is
not directly definable through the high thickness pipe theory because of the asymmetry of the fork. An effective approach
useful to correct the theory, is proposed in [1] where the authors introduced a coefficient βr (2) into the well known theoretical
formula (3).
βr = 2,0008 + 0,0022 ⋅ DAi − 0,0714 ⋅ s + 0,0372 ⋅ j − 0,4597 ⋅ k
(2)
where
j and k are two geometric parameters able to estimate the variable stiffness of the
fork around the central bush in the transversal and in the longitudinal direction
respectively.
The introduction of the βr coefficient provides errors always lower than 10% for the evaluation of the mean contact pressure p.
p=
Z
2
⎛
⎞ D
1 + QA
DF
⋅ ⎜⎜
+ ν A ⎟⎟ + F
2
E A ⋅ βr ⎝ 1 − QA
⎠ EI
⎛ 1 + QI2
⎞
⋅ ⎜⎜
− νI ⎟⎟
2
⎝ 1 − QI
⎠
(3)
where
QA,I is the ratio between the internal and the external diameter of the fork (A) and the
pin (I)
DF is the coupling diameter, as the average between DAi and DIa
Z is the diametrical interference between the fork and the pin
EA,I and νA,i are the Young’s modulus and the Poisson’s ratio of the fork (A) and the
pin (I)
Once the mean coupling pressure p is computed, it’s possible to calculate the interference level of the fork ZAi (4).
Z Ai =
⎛ 1 + Q2A
⎞
DAi
⋅ p⋅ ⎜
+ νA ⎟
2
⎜
⎟
EA
⎝1 − QA
⎠
(4)
Applying the same interference level ZAi on the fork in a Finite Element Method (FEM) software it’s possible to calculate the
mean coupling pressure p and to verify the congruence of the value obtained by (3). The software used to mesh and solve the
model is Ansys 10.0. The model has 82,000 tetrahedral elements (Solid 92) with 10 nodes per element. The FEM analysis
provides both the mean coupling pressure and the complete stress and strain state. On the external surface of the fork there is
only the circumferential stress σt because the stress σr is equal to zero. In order to compare the FEM results with the 48
different coupling pressures we calculated a correlation coefficient γ [MPa] (5). The γ coefficient is useful to correlate the
circumferential strain εt to the mean coupling pressure p using both the FEM analysis and the experimental results. The tested
coupling strain εt_COUPLING was evaluated applying a strain gauge on the external surface of the fork (as the one reported in
Figure 2) which is capable to measure only the circumferential strain. The circumferential strain εt_FEM and the mean coupling
pressure pFEM obtained by the FEM analysis were calculated analyzing the strain state on the nodes corresponding to the
position of the strain gauge (Figure 3) and the mean pressure value on the coupling surface. Thanks to the γ coefficient, when
the εt_COUPLING is evaluated, it’s possible to calculate the actual mean coupling pressure pCOUPLING (6) on each specimen, and to
define the actual friction coefficients using the equation (1).
γ=
pFEM
ε t_FEM
pCOUPLING i = ε t_COUPLING i ⋅ γ
(5)
i=1,…,48
(6)
Pin in Aluminium – Fork in Aluminium
For the fork and the pin in aluminium the value of the correlation factor γAl-Al is 63,934MPa.
Figure 3. Meshing and stress results on the forks.
•
Sliding friction coefficient µrl_Al-Al
The coupling process was carried out with a standing press; for each of the 24 tests an acquirement equipment (data
acquisition system) provided the force Frl trend and the strain εt trend in function of the run r. As shown in Figure 4 two different
types of diagram were obtained: the first one has an expected trend (Figure 4.a), according to [5], the second one has,
instead, an unexpected trend (Figure 4.b).
Frl_peak
Frl_complete_run
Lreal
LF
(a)
Lconicity
(b)
Figure 4. Trends of the axial coupling force Frl for aluminium-aluminium couplings: expected trend (a) and unexpected one (b).
For couplings that follows the trend (a) of Figure 4 the coupling area is calculated considering the complete axial run (LF),
therefore, the maximum force is at the end of the run (Frl_complete_run): the sliding friction coefficient is calculated using the
equation (7):
µ rl =
Frl_complete_run
p⋅ π ⋅ DF ⋅ LF
(7)
Instead, for couplings that follow the trend (b) of Figure 4 the maximum force is not at the end of the coupling length LF. The
decreasing of the force in the final part of the run is imputable to a run (or area) loss due to the cone-shape of the specimens:
as a matter of fact the run r must increase while the sliding friction coefficient µrl and the mean coupling pressure p must be
approximately constant (some fluctuations are possible because of the material plasticity due to the high interference). In fact,
some measurements on the internal fork diameter DAi pointed out that the end part of the coupling surfaces is cone-shaped,
probably due to some metalworking imperfections. This occurrence obviously reduces the releasing force because the actual
contact area is smaller than the ideal one. In this case, therefore, the actual run is taken equal to Lreal which coincides with the
peak of the force (Frl_peak) during the coupling phase. The sliding friction coefficient is calculated by equation (8):
µ rl =
Frl_peak
p⋅ π ⋅ DF ⋅ Lreal
(8)
•
Starting friction coefficient µll_Al-Al
The decoupling process was carried out using the same standing press and the same equipment used for the coupling
process. Two different types of diagrams were obtained also during the decoupling phase, as shown in Figure 5. They reflect
the two different ones obtained during the coupling phase with an expected trend (Figure 5.a), according to [5], and the
unexpected one (Figure 5.b): thus there is a strong correlation between the coupling and the decoupling process.
Fll_max
Fll_decoupling
(a)
(b)
Figure 5. Trends of the axial releasing force Fll for aluminium-aluminium couplings: expected trend (a) and unexpected one (b).
For specimens that follow the first trend (a) of Figure 5 there is the peak (Fll_max) at the beginning of the decoupling run r which
may be considered the axial releasing force. In this case the coupling area A must be evaluated considering the entire axial
run (LF) and the starting friction coefficient may be calculated by the equation (9).
µ ll =
Fll_max
(9)
p⋅ π ⋅ DF ⋅ LF
Instead for specimens that follow the second trend (b) of Figure 5 the highest decoupling force doesn’t occur at the starting
point of the run. This occurrence is due to the cone-shape of the specimens which provides an increase of the coupling run, or
of the coupling area, during the decoupling phase. In fact comparing the decoupling diagram (Figure 5.b) with the coupling one
(Figure 4.b), the zone of the force increasing for the decoupling phase is the same of the force decreasing for the coupling
phase. Thus the value of the axial releasing force, which is useful to calculate the starting friction coefficient, is the relative
peak of force Fll_decoupling and the actual run, or the actual area, to be used in (10) is smaller than the total one Lreal previously
calculated for the coupling phase. The actual run is, in fact, equal to Lreal minus Lconicity because Lconicity is the decoupling run
that occurs at the end of the coupling phase. The starting friction coefficient may be, therefore, calculated using the equation
(10).
µ ll =
Fll_decoupling
(
p⋅ π ⋅ DF ⋅ Lreal − Lconicity
)
(10)
Pin in Steel – Fork in Aluminium
For the specimens with the pin in steel and the fork in aluminium the value of the correlation factor γSt_Al is 49,831MPa.
•
Sliding friction coefficient µrl_St-Al
The coupling and decoupling processes were carried out in the same way used for aluminium-aluminium specimens. During
the coupling process were obtained, again, two different types of diagram, as shown in Figure 6 (a) and (b). For couplings that
follow the trend (a) of Figure 6 (the expected trend), the µrl was calculated using equation (7). Instead, for the couplings that
follow the trend (b) of Figure 6 (the unexpected trend) the µrl was calculated using equation (8). The trend of Figure 6.b is due
to the achievement of the yield field in the fork which is in aluminium. In fact the high interference level provides the
plasticization of the material and, therefore, a small decreasing of coupling pressure occurs. For this type of couplings the
maximum force (Frl_peak) and the correspondent run (Lreal) are indicated in the diagram. All the decoupling processes, indeed,
have an expected decoupling diagram, like that of Figure 5 (a). For this reason, the small decreasing of the force at the end
part of the run r is not imputable to the cone-shape of the specimens.
•
Starting friction coefficient µll_St-Al
As mentioned before, in the steel-aluminium case all the releases diagrams have an expected trend, as that shown in Figure 5
(a). The maximum force (Fll_complete_run) is obtained at the beginning of the decoupling run r which may be considered the axial
releasing force. Also in this case the coupling area A must be evaluated considering the entire axial run (LF) and the starting
friction coefficient can be calculated using the equation (9).
Frl_peak
Frl_complete_run
Lreal
LF
(a)
(b)
Figure 6. Trends of the axial coupling force Frl for the case St-Al.
Analysis of the Results
Once all the experimental tests have been performed, were obtained 24 values for each friction coefficient and for both the
material combinations. The data set values of the results are summarized in Tables 1 (for Aluminium-Aluminium) and Table 2
(for Steel-Aluminium) in which are reported the coupling identification, the factors and the levels of the DOE, the coupling
diameter, the loads, the actual run, the actual area, the mean coupling pressure and the friction coefficients. The friction
coefficient values (µll and µrl) have been, then, analyzed applying the Analysis Of Variance (ANOVA) [3, 4]. The ANOVA is a
method based on a statistical approach, useful to evaluate the significance both of each single factors and of their interactions
in order to point out which ones really influence the friction coefficient values. By means of the software Statgraphics PLUS
5.1, the ANOVA tables (Tables 3 (a) and (b)) were generated and the Fisher Test (F-Test) was executed. The F0 values were
determined subdividing the Mean Square of the single effect with the Mean Square of the Error. The error probability α was set
equal to 0.05 and the Fisher’s values (Fν1; ν2; α) were obtained from the Fisher’s tables [3, 4] referred to the degrees of freedom
of the Source of Variation (ν1) and of the Error (ν2). The effects with the F0 higher than the corresponding Fν1; ν2; α are
significant in the analysis whereas the others have to be ignored in the response model. Thus the ANOVA and the F-Test
provide the equations (11), (12), (13), and (14) which are the mathematical models of the sliding friction coefficient and of the
starting friction coefficient for the Aluminium-Aluminium coupling and for the Steel-Aluminium coupling respectively. Equations
(12) and (14) are plotted in the diagrams of Figure 7 (a) and (b).
µrl_Al − Al = 0,210333 − 0,126 ⋅ oil
(11)
µll_Al − Al = 0,2585 − 0,019733 ⋅ oil
(12)
µrl_St − Al = 0,351167 − 0,3015 ⋅ oil + 0,11033 ⋅ rust ⋅ oil
(13)
µll_St − Al = 0,46817 − 0,38066 ⋅ oil + 0,14 ⋅ time⋅ oil + 0,1743 ⋅ rust ⋅ oil
(14)
As shown in (11) and (12), for the Aluminium-Aluminium coupling, the friction coefficients are influenced only by the main
effect oil, that reduces both the friction coefficients; neither the presence of rusted spots on the pin, nor the resting time, nor
the interactions between factors influence the friction coefficient values. This occurrence may be explained considering the
corrosion resistance of the aluminium that is covered by an oxide protection film.
Instead, for the Steel-Aluminium coupling, as shown in (13) and (14), the interactions time-oil and rust-oil are significant in
addition to the main effect oil: the presence of the oil reduces both the friction coefficients, while the interaction between the
presence of the oil and the rusted spots increases both friction coefficients and the interaction between the resting time and
the oil increases the starting friction coefficient.
As shown by equations (11), (12), (13) and (14) the sliding friction coefficients are, in average, lower than the starting friction
coefficients as demonstrated in [5] and [7].
Table 1. Test results for the Al-Al specimens
ID
a1-12
a16-17
a7-9
a5-15
a10-23
a14-11
a2-1
a17-4
a13-16
a11-6
a15-21
a6-20
a3-3
a22-18
a19-24
a8-13
a12-5
a18-8
a24-19
a4-7
a23-22
a20-10
a21-2
a9-14
Time
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
Oil
0
0
0
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
1
1
1
1
1
1
Rust
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
1
1
1
DF [mm]
30,54
30,53
30,54
30,53
30,53
30,53
30,54
30,53
30,53
30,54
30,54
30,54
30,53
30,54
30,53
30,54
30,53
30,54
30,53
30,54
30,54
30,53
30,54
30,54
Frl [kN]
32,30
27,67
22,11
52,52
52,72
30,22
14,22
11,67
12,72
12,43
12,78
10,89
44,87
21,78
23,34
60,95
45,47
26,09
14,23
14,66
15,38
11,31
12,71
12,74
Run [mm]
26,00
18,85
26,22
17,20
18,38
26,55
26,10
26,57
26,88
26,68
26,83
26,42
17,17
21,62
22,52
18,32
17,65
26,37
26,72
26,17
26,79
26,40
26,70
26,76
2
A [mm ]
2495
1808
2515
1650
1763
2547
2504
2549
2578
2560
2574
2535
1647
2074
2160
1758
1693
2530
2563
2511
2570
2532
2561
2567
p [MPa]
62
68
64
80
85
53
64
61
67
55
60
76
72
69
67
139
86
57
61
66
67
68
67
65
µrl
0,20984
0,22416
0,13707
0,3961
0,3517
0,2247
0,08899
0,0746
0,07335
0,08771
0,08306
0,05647
0,37744
0,15194
0,16204
0,24915
0,3137
0,17963
0,09123
0,08868
0,08931
0,06529
0,0742
0,07609
Fll [kN]
21,99
32,28
24,92
34,94
31,57
42,70
13,43
8,18
14,13
9,66
14,08
14,45
26,50
24,80
23,07
23,62
28,09
27,76
15,15
21,04
16,94
12,96
13,99
21,42
2
Run [mm]
25,46
11,16
26,22
21,00
10,22
26,55
26,10
26,57
26,88
26,68
26,83
26,42
7,80
16,70
18,50
10,10
8,76
26,37
26,72
26,17
26,79
26,40
26,70
26,76
A [mm ]
Run [mm]
16,26
11,91
15,59
15,60
18,89
12,90
21,23
21,48
21,49
21,14
21,29
21,08
16,15
18,50
15,88
11,92
13,33
13,45
21,18
22,00
17,22
18,87
17,82
21,30
A [mm2]
2443
1070
2515
2014
980
2547
2504
2549
2578
2560
2574
2535
748
1602
1774
969
840
2530
2563
2511
2570
2532
2561
2567
µll
0,14591
0,44187
0,15449
0,21583
0,37891
0,3175
0,08405
0,05229
0,08148
0,06816
0,09151
0,07493
0,49095
0,22403
0,19501
0,17521
0,39065
0,19113
0,09713
0,12727
0,09837
0,07481
0,08167
0,12794
Table 2. Test results for the St-Al specimens
ID
Time
a28-28
0
a38-37
0
a41-47
0
a30-33
0
a44-44
0
aT2-P3
0
a32-46
0
a37-39
0
a42-45
0
a33-43
0
a43-48
0
aT1-P1
0
a27-40
1
a39-31
1
a45-34
1
a29-36
1
a35-30
1
a48-35
1
a31-41
1
a40-42
1
a46-25
1
a34-26
1
a36-27
1
aT3-32
1
Oil
0
0
0
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
1
1
1
1
1
1
Rust
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
1
1
1
DF [mm]
30,13
30,13
30,13
30,13
30,13
30,12
30,12
30,12
30,13
30,12
30,13
30,12
30,12
30,13
30,13
30,12
30,12
30,13
30,13
30,13
30,13
30,12
30,12
30,13
Frl [kN]
58,65
58,30
55,71
58,74
60,80
44,36
9,28
16,77
11,28
10,82
9,81
9,46
47,29
67,48
55,11
32,51
29,57
41,46
8,90
23,29
21,67
6,84
7,83
42,41
Run [mm]
16,26
11,91
15,59
15,60
18,89
12,90
21,23
21,48
21,49
21,14
21,29
21,08
16,15
18,50
15,88
11,92
13,33
13,45
21,18
22,00
17,22
18,87
17,82
21,30
A [mm2]
1539
1128
1475
1477
1787
1221
2009
2033
2034
2001
2015
1994
1528
1751
1503
1128
1262
1273
2004
2082
1630
1786
1686
2016
p [MPa]
113
124
94
118
113
117
103
106
121
111
125
112
113
134
91
108
127
104
115
102
89
110
107
100
µrl
0,3385
0,41658
0,40119
0,33778
0,29991
0,31184
0,04486
0,07749
0,04581
0,04862
0,03885
0,04229
0,27481
0,28807
0,40363
0,26627
0,1848
0,31376
0,03859
0,10924
0,14885
0,03486
0,04324
0,21032
Fll [kN]
75,60
71,54
66,43
65,18
67,17
54,43
15,70
20,22
17,95
33,70
33,50
27,68
63,37
72,30
66,81
41,34
33,55
54,27
16,34
50,87
40,52
48,55
37,66
53,45
1539
1128
1475
1477
1787
1221
2009
2033
2034
2001
2015
1994
1528
1751
1503
1128
1262
1273
2004
2082
1630
1786
1686
2016
µll
0,43634
0,51122
0,47836
0,37485
0,33133
0,38265
0,07588
0,09343
0,07286
0,15136
0,13267
0,12378
0,36823
0,30864
0,48931
0,33863
0,20965
0,41072
0,07083
0,23859
0,27841
0,24732
0,20798
0,26507
Table 3. Results of ANOVA for µll: Al-Al specimens (a) and St-Al specimens (b)
Source of Variation
A:Time
B: Oil
C: H2O
AB
AC
BC
Error
Total
Sum of Squares
DoF
0.001176
1
0.213193
1
1.67E-07
1
0.00084017
1
0.0059535
1
0.000054
1
0.158793
17
23
0.380009837
(a)
Mean Square
0.001176
0.213193
1.67E-07
0.000840167
0.0059535
0.000054
0.009340765
F0
0.1259
22.8239
1.80E-05
0.08995
0.63737
0.00578
1
Source of Variation
A: Time
B: Oil
C: H2O
AB
AC
BC
Error
Total
Sum of Squares
DoF
0.00248067
1
0.299713
1
0.0030375
1
0.0294
1
0.000384
1
0.0455882
1
0.069604
17
0.45020737
23
(b)
Mean Square
0.0025
0.2997
0.003
0.0294
0.0004
0.0456
0.0041
F0
0.60588
73.2016
0.74188
7.18062
0.09379
11.1344
1
Time=1
µ
µ
Oil
Oil
(a)
Rust
(b)
Figure 7. The response surfaces for µll: Al-Al specimens (a) and St-Al specimens (b)
Fork Design
An original software Fork Design©, realized by the authors in Visual Basic® programming language has been updated through
the new mathematical model. It can be successfully used in order to calculate the interferences of fork-pin coupling realized in
steel or in aluminium, to define the critical value of the axial releasing force, and to calculate the values of the interference on
the coupled parts. Furthermore, it is possible to find out, in a very short time, the minimum value necessary to overcome the
releasing tests imposed by the Standards, and the range of the allowed releasing values during the assembly phase of the
components. Thanks to these results, it is possible to complete the design of the coupling also evaluating the local tensile
state.
Conclusions
In this work four mathematical models were defined with the aim of calculating accurately the sliding friction (µrl) and the
starting friction (µll) coefficients in fork-pin compression-fit couplings. By means of the proposed mathematical models it’s
possible to define the optimal parameter combinations useful to maximize the starting friction coefficient and the axial releasing
force Fll, on the base of the design specifications. The results discussed above pointed out that the optimal combination for the
Aluminium-Aluminium and the Steel-Aluminium specimens is the cleaned and dry surfaces.
Finally, an original software, Fork Design©, realized by the authors in Visual Basic® programming language, have been
updated. This software can be successfully used in order to complete the design of the fork-pin compression fit couplings of
front motorbike suspensions.
Acknowledgments
The authors would like to acknowledge the Paioli Meccanica S.p.A. for the contribution to this paper.
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