motorbike suspension

Engineering Failure Analysis 17 (2010) 1173–1187

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Engineering Failure Analysis

journal homepage: www.elsevier .com/locate /engfai lanal

Recent improvements and design formulae applied to frontmotorbike suspensions

D. Croccolo *, M. De Agostinis, N. VincenziDiem, University of Bologna, Viale Risorgimento, 2 , 40136 Bologna, Italy

a r t i c l e i n f o

Article history:Received 13 January 2010Received in revised form 4 February 2010Accepted 5 February 2010Available online 11 February 2010

Keywords:MotorbikeMotorcycleSuspensionFormulaeDesign

1350-6307/$ - see front matter � 2010 Elsevier Ltddoi:10.1016/j.engfailanal.2010.02.002

Abbreviations: DOE, Design of Experiment; FEM,* Corresponding author. Tel./fax: +39 051209341

E-mail address: [email protected] (D. Crocc

a b s t r a c t

The aim of this paper is to provide a methodology useful for the structural design and opti-mization of front motorbike suspensions. Two different types of shaft–hub couplings areused to assembly the whole suspension: interference fit couplings and bolted joints. Somemathematical models and engineering design formulae are proposed in order to calculatethe tensile state and the fundamental design parameters of the main couplings, such as, forexample, the fork–steering shaft and the leg–wheel pin couplings. Both experimental tests(based on the Design of Experiment approach) and numerical analyses (based on the FiniteElements Method) have been carried out to obtain the proposed results. All the researchfindings culminate in an innovative software (Front Suspension Design�) which is usefulto design and to verify the whole front motorbike suspension, by applying correct andeffective results, obtained for different geometries and materials combinations.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The front motorbike suspension, reported for example in Fig. 1, is a mechanical component realised by the assembly ofabout 200 parts, whose design should take into account the accurate knowledge of the cinematic and dynamic behaviour ofthe vehicle [1]. The main structure of front suspensions (standard and upside down) is made up of components joined eachother by shaft–hub couplings. Starting from the upper part of the suspension (Fig. 1) it is possible to highlight the followingdifferent joints:

� bolted joint between the upper part of the steering shaft and the upper fork;� bolted joint between the upper fork and the outer tube;� interference fit coupling between the steering shaft and the lower fork;� bolted joint between the wheel clamp and the wheel pin;� bolted joint for the brake caliper mount.

For the aforementioned couplings a quick fulfilment of the structural optimization is requested by the shortening time ofthe vehicle lifecycle (about 2–3 years) and also by the constant increase in performances and reliability these componentsmust provide. This paper aims at providing some engineering formulae and some design improvements useful for the struc-tural optimization of the whole front motorbike suspension. The present couplings are often impossible to be studied by the

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Finite Elements Method; ANOVA, Analysis of Variance; F-Tests, Fisher’s Tests.3.olo).

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WWheel

Inn

clam

O

ner

mp

St

Oute

tube

teeri

r tub

e

ing

be

shafft

Braake cmou

calipunt

per

Foorkss

Fig. 1. CAD model of an upside down front motorbike suspension.

1174 D. Croccolo et al. / Engineering Failure Analysis 17 (2010) 1173–1187

De Saint Venant or Lamè or similar theories [2–4], because geometries are not axially symmetric or do not have an overridingdimension with respect to the others. For these reasons, the well known theoretical formulas have been corrected by intro-ducing some ad hoc coefficients, defined by performing a lot of numerical investigations, via FEM. Furthermore, shaft–hubcouplings are strongly influenced by friction, wear and lubrication conditions. Therefore, experimental tests, based on theDOE approach, have been carried out in order to calculate the friction coefficients as functions of the type of coupling (inter-ference fit or bolted joints), of the materials in contact (aluminium alloy or steel alloy) and of the typical or conventionalproduction and assembly parameters (as an instance lubricating oil, resting time, number of assembly on the samecomponent).

2. The joint design: engineering formulae

Two kinds of shaft–hub couplings must be studied when designing front motorbike suspensions: interference fit cou-plings, between the steering shaft and the lower fork, and bolted joints, both between the forks and the outer tube and be-tween the wheel clamp and the wheel pin. The components under investigation are typically made of steel (S355 J2 G3) andof aluminium (G-AlSi5 or AlSi7) alloys, whose mechanical properties are reported in Table 1.

2.1. Interference fit couplings: tensile state definition

The design of the fork-steering shaft coupling is uncertain mainly because of the poor knowledge about the static frictioncoefficient lll as well as of the mean coupling pressure pF. The axial releasing force Fll = lll�pF�A, which is the fundamentaldesign parameter, depends on the two aforementioned factors, usually unknown, and on the coupling surface A, usuallyknown. Because of the increase in power and in weight of the motorbikes, the axial releasing force Fll has also been muchincreased in the recent years in order to guarantee the driver’s safety. For this reason, the amount of interference U whichis proportional to the coupling pressure pF, is a strategic design parameter because it must be high enough to exceed thereleasing tests but not too much to overcome the Yielding of the components. Thus, it was decided to develop a generalizedmethodology useful to calculate lll and pF parameters accurately. The static friction coefficient lll was determined by apply-ing the DOE method in order to maximize the information about the experimental data [5,6]: three different combinations ofcoupled materials (steel–steel, steel–aluminium, and aluminium–aluminium) have been investigated, as shown in the nextparagraph. The main issue is that the geometry of the fork is not axially symmetric, as shown in Fig. 2: the stiffness changeswith the h coordinate around the central bush. For this reason the correct definition of the tensile state on the coupling arearequires the help of FEM analyses.

Our investigations were dedicated to find out an overall mathematical law, function of some geometric parameters,shared by every type of fork, and able to correct the theoretical formulas. The FEM investigation is necessary because thesolution provided by the congruence and equilibrium equations [2,4] is not effective when applied to asymmetric elements,such as the fork. The tensile state on the coupling area proves to be neither constant nor appreciable as average value if theThick Walled Cylinders theory (Lamè equations) is applied. As a matter of fact the results of FEM analyses, performed withthe Ansys Code, and reported in Fig. 3, show that radial (rr) and hoop (rt) stresses on the coupling surfaces are not constantin the h coordinate. Therefore, it was decided to perform a set of FEM analyses on 15 different fork–pin couplings in order to

Table 1Materials mechanical properties.

Material Ultimate stress Su (MPa) Yielding point Sy (MPa) Elongation at break A (%)

G-Alsi5 T6 304–343 196–235 5–9S355 J2 G3 510–680 >355 >18

Fig. 2. An example of fork geometry.

Fig. 3. Radial and hoop stress distribution of coupling surface via FEM.

D. Croccolo et al. / Engineering Failure Analysis 17 (2010) 1173–1187 1175

evaluate the pF and rt trend and to define two ad hoc parameters to be inserted in the theoretical formulas (1) proposed in[7], that can still be used.

pF ¼ n

1EA�

1þQ2A

1�Q2A

þmA

� �þ 1

EI�

1þQ2I

1�Q2I

�mI

� �rr ¼ �pF

rt ¼ pF �1þQ2

A

1�Q2A

8>>>>><>>>>>:rr < 0; rt > 0; jrr j < rt

ð1Þ

n is the ratio between the actual interference Z and the nominal coupling diameter DF, E and m are the Young’s Modulusand the Poisson’s ratio of the hub (EA, mA) and the shaft (EI, mI), while QA is the ratio between the internal and the externaldiameter of the hub and QI is the ratio between the internal and the external diameter of the shaft. The nominal interferenceU (2) is evaluated as the difference between the external shaft diameter DIa and the internal hub diameter DAi. In case of pressfit couplings, the U value must be decreased by the G contribution [8,9], normally suggested by some standards as a functionof the roughness of the hub (RzA, RpA, RaA) and of the shaft (RzI, RpI, RaI) [10,11]: it is therefore possible to calculate the actualinterference Z (3).

U ¼ DIa � DAi ð2Þ

Z ¼ U � G

G ¼0:8 � ðRzA þ RzIÞ2 � ðRpA þ RpIÞ3 � ðRaA þ RaIÞ

8><>:

ð3Þ

As reported in [12] a parametric analysis has been performed to study the stress field: the internal diameter of the forkhas been set equal to six different values (Di_f : 25, 27, 29, 31, 33, 35 mm) while the central bush thickness s equal to five dif-ferent values (6, 7.3, 8, 8.5, 9 mm) within the most frequent production range: an example of the differences in the forkgeometries is reported in Fig. 4. Therefore, it is possible to develop and analyze 15 groups of 30 (6 � 5) forks each, as a com-plete combination of the internal diameter Di_f and of the central bush thickness s. In each group the stiffening ribs (Fig. 2)around the central bush were left unchanged. After 450 (15 � 30) FEM analyses an overall function b, able to correct the the-oretical formulas, has been found out. Hence, by applying the b coefficients, it is possible to design the coupling and to

Fig. 4. Pictorial representation of two forks having different values of Di-f (blue) and s (red): (a) Di-f = 25 mm, s = 6 mm; (b) Di�f = 35 mm, s = 9 mm. (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)


compare some different solutions without performing again the FEM analyses. The theoretical unmodified formulas (1) donot provide accurate results (in some cases errors are higher than 60%) because of the different hoop stiffness of each fork,which is influenced only by the geometrical parameters located near the coupling zone as shown in Fig. 5, both in thelongitudinal and in the transversal section. The trend of the mean coupling pressures and hoop stresses on the fork, boththeoretical (pF, rt) and numerical (pF_FEM, rt_FEM), have been analyzed by changing the combinations of the internal couplingdiameter Di_f and of the central bush thickness s [12]. The theoretical values have been calculated by applying Eq. (4),according to [2]. The fork has been considered as a thick walled cylinder, with the internal diameter Di_f and the externaldiameter De_f = Di_f + 2s, subjected to a radial displacement Z.

pF ¼Z

Di f� EA

1þQ2A

1�Q2Aþ mA

� � ; rt ¼ pF �1þ Q 2

A

1� Q 2A

; Q A ¼Di f

Di f þ 2 � s ð4Þ

The pF_FEM and rt_FEM were computed as the averages of the pressures and hoop stresses evaluated on the nodes belongingto the coupling surface (Fig. 3). It was noticed that, for each one of the 15 fork groups, the trend of the theoretical and FEMstresses are the same if they are plotted as a function of Di_f and s: they increase in the same way while Z increases. Thisoccurrence can be explained considering that the same fork group has the same stiffness in the h direction (Fig. 2). For thisreason, each value of the mean coupling pressure obtained via FEM was compared with those obtained by applying the the-oretical formulas [12]. The br and bt (5) coefficients were therefore calculated as the ratio between the FEM stress (radial andhoop, in average) and the theoretical one: they depend on the internal diameter Di_f and on the central bush thickness s.

br ¼rr FEM

rr¼ pF FEM

pF; bt ¼

rt FEM

rtð5Þ

Furthermore, as b ratio trends are similar to planes [12], they have been interpolated in order to obtain some linear func-tions that are able to provide the corrective coefficients with errors always lower than 4% with respect to the actual value.Fig. 6 shows br and bt trends as functions of Di_f and s, while Table 2 reports their mathematical expressions.

The final step is to define two overall mathematical functions, which are able to interpolate all the 15 different br and bt

coefficients with acceptable errors. By analyzing Table 2 it is possible to highlight that all the planes of bt are very close each

Fig. 5. Fundamental geometric parameters for the computation of the b coefficients.

Table 2br and bt mathematical expressions.

Fork ID br (Di_f , s) bt (Di_f , s)

1 1.067 + 0.006 Di_f � 0.008s 1.044 + 0.001 � Di_f � 0.055 � s2 1.953 � 0.005 Di_f � 0.068s 0.868 + 0.002 � Di_f + 0.004 � s3 1.503 � 0.004 Di_f � 0.046s 0.989 + 0.001 � Di_f + 0.003 � s4 1.492 + 0.0018 Di_f � 0.048s 0.977 + 0.002 � Di_f + 0.001 � s5 1.174 + 0.074 Di_f � 0.018s 1.028 + 0.001 � Di_f � 0.005 � s6 2.090 � 0.007 Di_f � 0.803s 0.889 + 0.003 � Di_f + 0.005 � s7 1.800 � 0.005 Di_f � 0.078s 0.938 + 0.001 � Di_f + 0.004 � s8 2.640 � 0.009 Di_f � 0.122s 0.789 + 0.004 � Di_f + 0.008 � s9 2.732 � 0.001 Di_f � 0.125s 0.777 + 0.005 � Di_f + 0.009 � s

10 2.468 � 0.005 Di_f � 0.108s 0.734 + 0.006 � Di_f + 0.011 � s11 2.551 � 0.005 Di_f � 0.124s 0.808 + 0.004 � Di_f + 0.008 � s12 2.493 � 0.001 Di_f � 0.129s 0.855 + 0.002 � Di_f + 0.007 � s13 2.647 � 0.002 Di_f � 0.134s 0.802 + 0.003 � Di_f + 0.008 � s14 2.016 + 0.009 Di_f � 0.093s 0.861 + 0.003 � Di_f + 0.003 � s15 3.102 � 0.010 Di_f � 0.146s 0.798 + 0.006 � Di_f + 0.010 � s

Fig. 6. br and bt planes as a function of Di_f and s.


other and therefore they have been easily interpolated with a mean (Eq. (6)), which provides errors that are always lowerthan 5%.

bt ¼ 0:831þ 0:003 � Di f þ 0:007 � s ð6Þ

On the contrary, a unique interpolation of br coefficient by a mean plane, which depends only on Di_f and s, would causediscrepancies up to 30%, because the difference in the fork stiffness is not appreciable only with these two parameters. Bydeeply studying the shape of the forks around the central bush we pointed out two more geometrical parameters, which areable to highlight the different stiffness of each fork group and that can help in defining the overall mathematical function. Itwas noticed that the stiffness is influenced both in the transversal direction (Y direction in Fig. 5) and in the longitudinaldirection (X direction in Fig. 5) by the amount of material located around the central bush. Therefore, according to [12]and referring to Fig. 5, two additional parameters j and k were defined (7) and added to the analytical expression of br inorder to interpolate all the 15 planes with the same equation. Hence the overall br expression (8) is capable to providethe corrective coefficient with mean errors always lower than 10%.

j ¼ Ls

Lc� ps �

Di f þ 2 � s2

� �k ¼ s

Ldð7Þ

br ¼ 2:001þ 0:002 � Di f � 0:071 � sþ 0:037 � j� 0:460 � k ð8Þ

A stiff fork, that is a fork with a large amount of material around the central bush (typically installed on a high perfor-mances vehicle) presents br coefficient within the range 1.7–2, while a light fork, that is similar to a true bush, has br coef-ficient within the range 1.1–1.4.

As mentioned before, in the case of fork–pin couplings, Eq. (1) leads to wrong results because of the asymmetry of thefork. Therefore, it is necessary to correct the congruence and equilibrium equations by applying b coefficients in order toevaluate the actual tensile state. The theoretical formulas [4] have been rewritten, according to [12], in order to obtainthe formulae reported in Eq. (9).


pF actual ¼ n

1EA �br

�1þQ2

A1�Q2

A

þmA

� �þ 1

EI�

1þQ2I

1�Q2I

�mI

� �rr actual ¼ �pF actual

rt actual ¼ rt � bt ¼ pF �1þQ2

A

1�Q2A� bt ¼ pF actual � bt

br� 1þQ2

A

1�Q2A

8>>>>><>>>>>:

ð9Þ

In detail it is possible to calculate the actual mean coupling pressure pF_actual by taking into account the asymmetric shapeof the hub (the shaft is always axially symmetric); besides, it is possible to calculate the actual hoop stress rt_actual on thecoupling surface, and therefore to evaluate the equivalent stress due to the assembly operation by applying the Tresca orthe Von Mises criterion (10).

req Tresca ¼ rt actual � rr actual ¼ rt actual þ pF actual

req VonMises ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

t actual þ r2r actual � rr actual � rt actual

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

t actual þ p2F actual þ pF actual � rt actual

q8>>><>>>:

¼ ð10Þ

2.2. Interference fit couplings: friction coefficients definition

Once the tensile state is completely and precisely defined, in order to calculate the axial releasing force Fll (the fundamen-tal design parameter) it is necessary to evaluate the static coefficients of friction, in axial direction, lll between the matingmaterials. The coupling process is a longitudinal compression-fit carried out by means of a standing press (maximum staticload equal to 250 kN), as reported in Fig. 7. No heat expansion of the hub is usually considered for the assembly operation.The knowledge of the most important and significant parameters, which can influence and maximize the static friction coef-ficients is strategic. After some screening tests [13], three appropriate input factors and their levels were identified [14].Since steering pins are not immediately assembled and they may be stored in open metal boxes for some days, their surfacesmay be covered by some rusted spots: the presence of rust is therefore the first parameter. The presence of lubricating oil,because steering pins might be protected with a thin film of lubricating oil before being assembled with forks is the secondparameter. Finally the resting time is the third parameter, because several authors [15] state that the greatest static frictioncoefficient is reached some hours after the parts assembly. The DOE method was applied in case of steering pins and forksmade both of steel, both of aluminium and in case of pins made of steel and forks made of aluminium. In order to reduce thenumber of tests, two levels for each factor were chosen: low level (0) vs. high level (1). In case of rust and lubricating oilpresence, low level means the lack of the factor, while high level means the presence of the factor; in case of resting timethe low level implies that the pins are immediately disengaged from the forks after the insertion, while the high level meansthat the resting time is, at least, equal to 72 h: a complete 23 factorial plane was obtained. To reduce the influences of thenoise and of any non investigated factors, it was decided to repeat each test three times, for a total of 24 tests for each cou-pling type, and to keep constant the coupling and decoupling speed rate (16 mm/s), the environment temperature and theshape of the coupling elements throughout the tests. The experimental tests were randomized as a necessary condition foran accurate application of the method. The DOE analysis was performed during both the coupling and the decoupling phasein order to calculate also the sliding friction coefficient lrl besides the static friction coefficient lll for each of the three types

Fig. 7. The tools useful for the assembly and decoupling operation.


of materials in contact. Thus, a total of six DOE and six mathematical models were determined for each different frictioncoefficient, which has been calculated by applying Eq. (11):

lrl;ll ¼Frl;ll

pF actual � A¼ Frl;ll

pF actual � p � DF � LFð11Þ

LF is the coupling length, while Frl and Fll are the axial forces given by the standing press during the coupling and decou-pling phase, respectively. The actual pressure value has been calculated by applying the previous expression (9) and it hasalso been checked by means of a strain gauge applied on the external surface of the fork (Fig. 7) which is able to read themono-axial hoop deformation (on the external surface the radial stress is equal to zero): this methodology is very accurate,as well demonstrated in [14,16]. An example of the coupling and decoupling run r (r within 0 � LF) is reported in Fig. 8.

The axial coupling force Frl results always lower than the axial releasing force Fll because the sliding friction coefficient lrl,which governs the coupling phase, is lower than the static friction coefficient lll, which influences the peak releasing force.

Once the 144 experimental tests (3 � 23 � 6) have been performed, all the calculated friction coefficients have been ana-lyzed by applying the ANOVA [6] in order to evaluate the significance of each factor and its interactions and to identify whatdoes impact on the friction coefficient values. By means of the Statgraphics Plus software, F-test was executed and Eqs. (12)–(14) have been defined.

lrl St�St ¼ 0:144� 0:032 � oillll St�St ¼ 0:168þ 0:026 � timeþ 0:019 � rust� 0:028 � oil

�ð12Þ

lrl St�Al ¼ 0:351� 0:302 � oilþ 0:110 � rust � oillll St�Al ¼ 0:468� 0:381 � oilþ 0:140 � time � oilþ 0:174 � rust � oil

�ð13Þ

lrl Al�Al ¼ 0:210� 0:126 � oillll Al�Al ¼ 0:286� 0:197 � oil

�ð14Þ

In case of steel–steel couplings the friction coefficients values, calculated by applying Eq. (12), are influenced by the maineffect oil, that reduces both of them, whereas rusted spots and resting time influence only the static friction coefficient byincreasing it. In case of steel–aluminium couplings, the friction coefficients values, calculated by applying Eq. (13), are influ-enced by the main effect oil, that reduces both of them, and by the interaction rust–oil, that increases both of them, whereasthe interaction time–oil raises the static friction coefficient only. Finally, in case of aluminium–aluminium couplings the fric-tion coefficients values, calculated by applying Eq. (14), are influenced only by the main effect oil that reduces both of them.This occurrence may be explained considering the corrosion resistance of the pins made of aluminium. For further details seeRef. [14]. Friction coefficients in case of dry or lubricated surfaces (resting time greater than 72 h and rust absence) are re-ported in Table 3: the values of the sliding friction coefficients are lower than the values of the static friction coefficients(their ratio is about 0.75 for dry surfaces), as widely demonstrated in literature and, particularly, in Refs. [2,15].

2.3. Interference fit couplings: recent improvements

The standard procedure to assembly the steering shaft and the fork is provided by a press fit with a standing press. Inorder to guarantee the axial releasing force, high interferences are used (up to Z = 0.1 mm with coupling diametersDF = 30 mm). This assembly condition could produce a dangerous tensile state on the components, in particular referringto the fatigue behaviour: the interference fit operation results in a stress concentration factor both on the shaft and on

Fig. 8. An example of the coupling and decoupling run.

Table 3Friction coefficient in fork–pin compression fit couplings [14].

Steel–steel Steel–aluminium Aluminium–aluminium

lrl 0.14 (0.11) 0.35 (0.05) 0.21 (0.08)lll 0.19 (0.17) 0.47 (0.23) 0.29 (0.09)lrl/lll % 74% (65%) 75% (22%) 73% (89%)

Dry (lubricated) surfaces [resting time >72 h, no rust].


the hub, according to [15,17]. For these reasons the authors have evaluated the possibility of realizing a hybrid joint (inter-ference fit and adhesively bonded), by applying anaerobic adhesive before the assembly operation [18]. This allows thereduction of the amount of interference (Z = 0.02–0.05 mm), and therefore the tensile field on the components, by takingadvantage of the adhesive strength. In fact, referring to the whole coupling surface, the dry interference involves about20–30% of contact surfaces whereas the anaerobic adhesive is able to fill the voids increasing the actual contact surfaceup to 100% [19]. The strength of hybrid joints is affected by various factors such as, for example, the coupling pressure,the type and the way of assembling, the type of materials in contact, the curing time and the curing methodology, the oper-ating temperature and the loading type [20,21]: experimental tests have been performed in order to obtain the actualmechanical performances of the joint. The total axial decoupling force Ftot can be evaluated as the addition of the interferencecontribution Fll with the adhesive contribution Fad (15).

Ftot ¼ Fll þ Fad ¼ lll � pF � Aþ sad � A ð15Þ

where sad is the adhesive static shear strength. Eq. (15) is effective if also sad is accurately evaluated. Thus, a set of couplingand decoupling tests have been carried out: the shaft and the hub fitted and adhesively bonded have been cured for 48 h at40 �C. The static strength of the adhesive Fad and its shear stress sad have been derived by Eq. (16) in which the interferencecontribution Fll has been previously defined and the total axial decoupling force Ftot is provided by the load cell of the stand-ing press.

Fad ¼ Ftot � Fll ¼ Ftot � lll � pF � Asad ¼ Ftot

A � lll � pF

(ð16Þ

The adhesive used is the Loctite� 648 type, whereas the surface cleaner is the Loctite� 7061 type [22]. An example of theforce trends plotted as functions of the run during the coupling and decoupling tests are reported in Fig. 9 (comparison be-tween dry and bonded surfaces). During the coupling phase (coupling speed always set at 16 mm/s) the adhesive does not

Fig. 9. An example of the force trend during the coupling and decoupling process.


act as a lubricant because the total axial coupling forces in the hybrid joints (interference and adhesive) are always greaterthan those given by the dry interference ones. Conversely during the decoupling phase the adhesive contribution is wellhighlighted by the different maximum force of the curves.

A strong correlation between the coupling pressure pF and the adhesive shear stress sad (16) has been observed: byincreasing the coupling pressure, the adhesive shear stress decreases with a power law (17), in the pressure range consideredas reported in Fig. 10.

sad ¼ 2680 � p�1:55F pF 2 ½25—60�MPa ð17Þ

Once the static strength has been defined, some fatigue tests have been carried out with the purpose of evaluating theresidual strength of the joint at 106 tension–tension fatigue cycle. As the interference contribution Fll is constant, each spec-imen has been loaded with a minimum force Fmin equal to the interference contribution and with a maximum force Fmax

equal to a percentage (within the range 25–50%) of the estimated static strength of the adhesive, added to Fmin: an exampleof the fatigue cycles is reported in Fig. 11. In case of surviving of the joint after 106 cycles the specimen has been disengagedand the axial decoupling force Ftot_D_fatigue has been obtained: the decrease of the adhesive strength AD% and the residualstrength of the adhesive AR% (Fig. 11) has been evaluated according to Eq. (18).

The residual strength decreases as the fatigue load amplitude increases [18,19]: the residual strength of the adhesive is100% until the stress on the adhesive is lower than 25% of its static strength (17). If the stress on the adhesive reaches the 50%of its static strength (17) the fatigue cycles disengage the specimens before 106 cycles, so that the adhesive residual strengthis equal to 0%. In conclusion, as a fatigue design rule, it is possible to consider the following adhesive fatigue limit in case ofinterference fit and adhesively bonded joints (19).

sad 106 � 0:25 � sad ¼ 0:25 � ð2680 � p�1:55F Þ ð18Þ

AD% ¼ ðFtot D fatigue

A �lll �pF Þ�ðFtot

A �lll �pF ÞðFtot

A �lll �pF Þ

��% ¼ Ftot D fatigue�Ftot

Ftot�Fll

�� %AR% ¼ 1� AD%

ð19Þ

2.4. Bolted joints: structural optimization and tensile state definition

The design and the optimization of clamps, such as those reported in Fig. 12, are difficult to be carried out with theoreticalformulas mainly because of the shape of the hub, which does forbid to define the maximum bending stress easily. Our inves-

Fig. 10. The adhesive shear stress as a function of the coupling pressure.

Fig. 11. Example of fatigue cycles and adhesive residual strength.


tigation was firstly referred to eight different wheel clamps, five with two tightening bolts, according to [23], and three withone tightening bolt. Secondly, the methodology was extended to the joint realised between the fork and the leg outer tube(Figs. 1 and 12).

The fundamental idea was to define the structural behaviour of the clamp during the tightening phase. The aim was tofind the relationship between the bolt pretension load FV and the maximum bending stress by means of the definition of atheoretical stress concentration factor Kt, which derives from the presence of the bolts holes and spot facings. The Kt defini-tion was obtained by performing non-linear FEM analyses, in order to take into account the contact between the externalsurface of the shaft and the internal surface of the clamp. Referring to Fig. 13, five types of clamped joints with two tight-ening bolts, with the same slot width t (2 mm) and the same slot location e have been investigated. Conversely the clampshave spot facings realised with different depth, in order to hide partially or fully the bolt head, and also different distances vbetween the bolts axes.

As deeply demonstrated in Ref. [23], the clamp portion between the bolt axis and the G–G rectangular (b � h) cross sec-tion, according to Fig. 13, is loaded by a bending stress. The maximum bending value is achieved in correspondence of the G–G section.

Thus, in order to reduce the tensile state and so optimize the clamp, the section modulus Wb shall be increased by reduc-ing the e distance. The optimal location for the slot axis is when it matches the pin axis, because the section modulus Wb

grows up till its maximum convenient value [23]. The contour plots of the FEM analyses are reported in Fig. 14 where dif-ferent stress fields of the original and the optimized clamp can be appreciated.

Regarding the maximum stress produced by the spot facings perturbation located near the coupling zone, the perturba-tion effects can be defined by comparing several non-linear FEM results (rmax_FEM) with the theoretical values (rth) given byEq. (20), in which a (Fig. 13) is the lever arm of FV, n the number of bolts and Wb the bending section modulus. The first anal-yses regarded five wheel clamps with two bolts (n = 2); later three more wheel clamps with only one bolt (n = 1) have beenexamined. A stress concentration factor Kt_W (21) has been calculated and related to some geometrical parameters, which

Fig. 13. Geometrical dimensions and parameters of the clamp.

Fig. 12. Example of clamps: wheel and fork clamp.

Fig. 14. Von Mises contour plots of the original and optimized wheel clamp.


could influence the perturbation. Referring to Fig. 13, Kt_W is influenced by the spot facings diameter ds�f (depending on thebolt type), by the spot facing height hs�f, by the distance a between the bolt axes and the G–G cross section, and, finally, bythe bolt axes distance v. A parametric analysis has been performed on the five different types (five groups) of clamped jointwith two bolts and with the optimal slot position. The two dimensionless parameters j1 and j2 (22) have been therefore chan-ged on three levels within their acceptable or conventional ranges.

rth ¼n � FV � a

Wb¼ n � FV � a � 6

b � h2 ð20Þ

Kt W ¼rmax FEM

rthð21Þ

j1 ¼hs�f

hj2 ¼

ads�f

ð22Þ

j1 is the ratio between the spot facing height hs�f and the total cross section height h, while j2 is the ratio between thedistance a and the spot facing diameter ds�f. Each different group was realised with two different bolt types (M6 and M8)and with three different j1 and j2 parameters values, obtained by changing hs�f and a. The present discrete parameterization(18 combination of parameters for each of the five groups) covers the most frequent production range. Hence, 90 differentgeometries were realised and 90 non-linear FEM analyses were performed. As the trend of the Kt_W values is similar and al-most linear they have been interpolated with a first order function depending on j1 and j2. Five different equations have beendetermined for each group of clamped joints with the M6 and the M8 bolt type. The total of 10 equations are able to inter-polate the theoretical stress concentration factor values with errors that are always lower than 5%. All the Kt_W planes for twobolts have a similar trend and moreover they are very close to each other. The difference between the planes position is dueto the different distance between the bolt axes v: more in detail the upper plane, which is the plane with the highest Kt_W

values, has the shorter bolt distance v. In order to define an overall mathematical function for Kt_W it was decided to intro-duce also the j3 dimensionless parameter (23) obtaining a unique expression. Therefore two different mathematical modelshave been determined (24): the first is suitable for the M6 bolt type while the second for the M8. Errors between the FEMvalues of Kt_W and the proposed ones given by Eq. (24) are always lower than 10% for all the clamped joints underinvestigation.

j3 ¼v

ds�fð23Þ

Kt W 2Bolts M6 ¼ 2:438þ 0:548 � j1 � 1:131 � j2 � 0:393 � j3

Kt W 2Bolts M8 ¼ 4:085þ 1:590 � j1 � 3:308 � j2 � 1:024 � j3

�ð24Þ

The same procedure has been applied to three groups of single bolt wheel clamps (M8). The chosen dimensionless param-eters, according to Fig. 13, are j1 (22), j4 and j5 (25). The overall mathematical function for Kt_W in case of clamps with one boltis reported in Eq. (26).

j4 ¼ab

j5 ¼ds�f

bð25Þ

Kt W 1Bolt M8 ¼ 2:982þ 1:839 � j1 � 9:749 � j4 þ 1:373 � j5 ð26Þ

j4 is the ratio between the distance a and the clamp width b, while j5 is the ratio between the spot facing diameter ds�f andthe clamp width b. The total number of FEM analyses performed in this case is equal to 81 (34) as we considered a combi-nation of the three parameters j1, j4 and j5 on three levels (33) for the three different clamp groups (33�3 = 34). Errors betweenthe FEM values of Kt_W and the proposed ones given by Eq. (26) are always lower than 5% for all the clamped joints underinvestigation.


Thus the maximum stress in wheel clamps (rmax_W) can be easily computed by multiplying the De Saint Venant theoret-ical stress rth, Eq. (20), by the ad hoc stress concentration factor Kt_W value, in order to obtain the design formulae reported inEq. (27), where n (1 or 2) stands for the number of bolts and i (6 or 8) for the bolt diameter.

rmax W ¼ Kt W nBolt Mi � rth ¼ Kt W nBolt Mi �n � FV � a � 6

b � h2 ð27Þ

Finally, according to [23,24] the mean coupling pressure pF can be computed by applying Eq. (28), based on the cantileverscheme proposed in Fig. 15. The first support (B) is located on the pin apex, while the second support (A) is located behind thecoupling region, where FEM analyses reveal the stress to be zero. The A support position can be evaluated considering thatthe distance l1 between the two supports is always within the range 0.9�Df � 1.1�Df.

FV � l2 ¼ FB � l1 ) FB ¼ FV �l2l1

pF ¼ FBDf �b¼ FV �l2

l1 �Df �b

ð28Þ

The same methodology was also applied to investigate the clamped joint between the fork and the leg outer tube (Figs. 1and 16). Six different groups of fork clamps were studied: three with one bolt and three with two bolts. The chosen dimen-sionless parameters are, in this case, j1 (22), j6, j7 and j8 (29). Six groups were therefore created and analyzed, every one withthree different levels for each parameter, to obtain an appropriate stress concentration factor expression (30) also for thesegeometries.

j6 ¼ac

j7 ¼b

ds�fj8 ¼

vds�f

ð29Þ

Fig. 15. Cantilever scheme to evaluate the mean coupling pressure pF.

Fig. 16. Geometrical dimensions and parameters of the fork clamp.

Fig. 17. Comparison between numerical and experimental results.


Kt F 1Bolt ¼ 3:330þ 2:404 � j1 � 1:444 � j6 þ 1:047 � j7

Kt F 2Bolts ¼ 3:608þ 1:481 � j1 � 1:384 � j6 � 1:280 � j8

ð30Þ

j6 is the ratio between the distance a and the distance c, j7 is the ratio between the clamp width b and the spot facingdiameter ds�f and j8 is the ratio between the distance v and the spot facing diameter ds�f. The total number of FEM analysesperformed in this case is equal to 54 as we considered a combination of three parameters (j1, j6 and j7 in case of one bolt, j1, j6

and j8 in case of two bolts) on three levels, for two different clamp groups (33�2 = 54). Discrepancies between the FEM valuesof Kt_F and the proposed ones given by (Eq. (30)) are always lower than 5% for all the clamped joints under investigation.

Fig. 18. Input windows for interference fit couplings.

Fig. 19. Output window for interference fit couplings.


Thus the maximum stress in fork clamps (rmax_F) can be easily computed by multiplying the De Saint Venant theoreticalstress rth, Eq. (20), by the ad hoc stress concentration factor Kt_F value Eq. (30), in order to obtain the design formulae re-ported in Eq. (31), where n (1 or 2) stands for the number of bolts.

rmax F ¼ Kt F nBolt � rth ¼ Kt F nBolt �n � FV � a � 6

b � h2 ð31Þ

Some experimental tightening tests performed on the components under investigation have recently confirmed the pro-posed results. In Fig. 17 a comparison between FEM prediction and the actual failure is reported.

3. Front Suspension Design� software

All the proposed design formulae presented in this paper have been implemented into an innovative software, Front Sus-pension Design�, realised in Visual Basic programming language, with the aim of designing and verifying the main compo-nents and joints of the front motorbike suspension. As a matter of fact the software is useful to design or to verify, in a guidedway, the basic elements and the fundamental couplings of the front motorbike suspension, in a very short time, and withoutapplying FEM analyses or experimental tests anymore. The input windows (Figs. 18 and 20) have to be filled with the geo-metrical parameters and with the information about the materials, the surface finishing, and the production/assembly con-

Fig. 20. Input window for wheel clamped joints.

Fig. 21. Output window for wheel clamped joints.


ditions. The output windows (Figs. 19 and 21) show all the project parameters calculated and provided by the program. Indetail, concerning the interference fit couplings, the program provides the maximum and the minimum value of the inter-ference, the maximum and the minimum value of the axial releasing force, the ISO Standard suggested coupling, the max-imum stress in the coupling, and safety coefficients referred to the yield and ultimate points. At any time it is possible toperform the calculation by applying the proposed mathematical models or the theoretical formulae. Concerning the clampedjoints definition, the program provides the maximum actual stress on the clamp, the mean coupling pressure between theshaft and the hub, and the safety coefficients referred to the yield and ultimate points.

4. Conclusions

This paper deals with the structural design and optimization of front motorbike suspensions. Two different types of shaft–hub couplings have been taken into account: the interference fit coupling between the fork and the steering shaft and theclamped joint between the leg and the wheel pin and between the fork and the leg. Different fork and leg geometries havebeen analyzed in light of some fundamental design parameters. The analytical study has been supported by several numer-ical analyses and experimental tests, since all the coupling geometries are not axial symmetric and therefore the theoreticalformulae provide unacceptable errors. The numerical results have been compared with the theoretical values in order to de-fine some coefficients that are able to correct the Thick Walled Cylinders theory and the De Saint Venant formulae. A detailedDOE has also been applied in order to define the static and sliding coefficients of friction in the interference couplings fordifferent types of materials combinations: steel–steel, steel–aluminium, and aluminium–aluminium. The new developedmodels are useful to optimize and to verify the basic components and the fundamental types of couplings of front motorbikesuspensions without performing any complex numerical analyses. The effectiveness of the models has been proved by sev-eral experimental tests carried out in cooperation with Paioli Meccanica S.p.A, which produces front motorbike suspensions.The results have been implemented in an innovative software (Front Suspension Design�) realised by the authors via VisualBasic programming language. This software can be used for designing or comparing different geometries and materialscombinations.

References

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[10] Yang GM, Coquille JC, Fontaine JF, Lambertin M. Influence of roughness on characteristics of tight interference fit of a shaft and a hub. Int J Solids Struct2001;38:7691–701.

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[13] Croccolo D, Reggiani S. Modello matematico del coefficiente di attrito in accoppiamenti stabili. Tecniche Nuove 2002;1:45–55.[14] Croccolo D, Cuppini R, Vincenzi N. Friction coefficient definition in compression fit couplings applying the DOE method. Strain 2008;44:170–9.[15] Niemann G, Winter H, Hohn BR. Maschinenelemente Bd. I. Berlin: Springer-Verlag; 2005.[16] Croccolo D, Cuppini R. A methodology to estimate the adhesive bonding defects and the final releasing moments in conical joints based on the acoustic

emissions technique. Int J Adhes Adhes 2006;26(7):490–7.[17] Croccolo D, Vincenzi N. Stress concentration factors in compression fit couplings. Proc Inst Mech Eng C: J Mech Eng Sci, in press. doi:10.1243/

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[19] Sekercioglu T, Gulsoz A, Rende H. The effects of bonding clearance and interference fit on the strength of adhesively bonded cylindrical components.Mater Des 2005;26:377–81.

[20] Sekercioglu T. Shear strength estimation of adhesively bonded cylindrical components under static loading using the genetic algorithm approach. Int JAdhes Adhes 2005;25:352–7.

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[22] Loctite worldwide design handbook; 1995.[23] Croccolo D, Cuppini R, Vincenzi N. Design improvement of clamped joints in front motorbike suspension based on FEM analysis. Finite Elem Anal Des

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