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Aluminium alloy wheels are increasingly popular for their light weight and good thermal conductivity. Cooling Holes (CH) are introduced to reduce their weight without compromising structural integrity. Literature is sparse on the effect of aspect ratio (AR) of CHs on wheels. This, work, therefore, attempts to undertake a parametric study of the effect of aspect ratio (AR) on the mechanical response of an aluminium alloy wheel with triangular, quadrilateral and oval-shaped CHs. Three-dimensional wheel models (6JX14H2ET42) with triangular, quadrilateral and oval shaped CH (each with CH area of 2229 mm
^{2}) were generated, discretized, and analyzed by FEM using Creo Elements/Pro 5.0 to determine the mechanical response at the inboard bead seat at different ARs of 1, 0.5, 0.33 and 0.25, each for quadrilateral-CH and oval-CH, at a static Radial Load of 4750 N and Inflation Pressures of 0.3 and 0.15 MPa, respectively. The study shows that the magnitude of stress and displacement is affected by shape and AR of CH. From the results, it could be established that oval-shaped-CH wheel at AR of 0.5 offers greater prospect in wheel design as it was least stressed and deformed and, that the CH combination with highest integrity was the oval-CH and quadrilateral-CH at AR of 0.5.

Automobile wheels are vital structural members of the vehicular suspension system that sustain both static and dynamic loads encountered in operation [

Aluminium alloy wheels are increasingly popular for their light weight and good thermal conductivity. However, there are efforts to introduce CHs to reduce their weight without compromising structural integrity. Cooling holes are very vital in automobile wheels as they help in weight reduction, aesthetics and aid in dissipation of heat. Varieties of cooling holes are in use [

A lot of experimental, numerical and parametric studies have been undertaken to study the mechanical response in terms of stress and displacement of wheels using different FEM analytical tools.

Analogy from thick ring theory in the development of loadings on links and eye-bar has been analyzed [

Without compromising fatigue resistance and other mechanical properties, using Finite Element analysis, radial fatigue and damage analysis for weight optimization of aluminum alloy wheel were examined [

A parametric study of radial and spiral models of Al 356.2 and ZK60A alloy, respectively was undertaken. The design and analysis were executed using CREO and ANSYS, respectively. Results showed that models with spiral flexure offer greater resistance to stress than radial flexure [

The effect of various materials and detailed fatigue life of the automotive wheel rim by using finite element analysis and radial load testing was reviewed. Modeling of the wheel was with CATIA and imported into ANSYS for analysis. In the modern technological advancement, considerable attempts are being made to develop Al and Mg alloy wheels bearing in mind factors such as strength to weight ratio, low cost and better fuel consumption [

The through process modelling methodology to predict the fatigue life of A356 automobile wheel subjected to bending fatigue was investigated [

Assessment of the wheel performance as a function of the rim and disc thickness was carried out using fatigue analysis and employing NASTRAN [

Experimental and numerical simulation of dynamic impact loading of cast aluminum alloy wheel owing to collision with the curb of the road or large obstacle was studied [

The studies above show that for dynamic analysis of wheels, the most stressed location of the wheel is the bolt holes and the arms, while for static analysis, the most severe location is the inboard bead seat. However, literature is sparse on the effect of aspect ratio (AR) of CHs on wheels, hence, an attempt is made on this sturdy.

In the case of linearly elastic isotropic three-dimensional solid [

σ = { σ x x σ y y σ z z σ x y σ y z σ z x } = D ε (1)

D = { ε x x ε y y ε z z ε x y ε y z ε z x } (2)

where the matrix D is given by:

[ D ] = E ( 1 + ν ) ( 1 − 2 ν ) [ 1 − ν ν ν 0 0 0 ν 1 − ν 0 0 0 0 ν ν 1 − ν 0 0 0 0 0 0 ( 1 − 2 ν ) / 2 0 0 0 0 0 0 ( 1 − 2 ν ) / 2 0 0 0 0 0 0 ( 1 − 2 ν ) / 2 ]

(3)

For solids of revolution (axisymmetric solids), [D] is given as:

[ D ] = E ( 1 + ν ) ( 1 − 2 ν ) [ 1 − ν ν ν 0 ν 1 − ν ν 0 ν ν 0 0 0 0 0 ( 1 − 2 ν ) / 2 ] (4)

For the tetrahedron element is shown in

u ( x , y , z ) = α 1 + α 2 x + α 3 y + α 4 z (5a)

v ( x , y , z ) = α 5 + α 6 x + α 7 y + α 8 z (5a)

w ( x , y , z ) = α 9 + α 10 x + α 11 y + α 12 z (c)

where, α 1 , α 2 , ⋯ , α 12 are constants. By using the nodal coordinates,

u = Q 3 i − 2 ; v = Q 3 i − 1 ; w = Q 3 i at ( x i , y i , z i ) (6a)

u = Q 3 j − 2 ; v = Q 3 j − 1 ; w = Q 3 j at ( x i , y i , z i ) (6b)

u = Q 3 k − 2 ; v = Q 3 k − 1 ; w = Q 3 k at ( x i , y i , z i ) (6c)

u = Q 3 l − 2 ; v = Q 3 l − 1 ; w = Q 3 l at ( x i , y i , z i ) (6d)

From it is obtained

u ( x , y , z ) = N i ( x , y , z ) Q 3 i − 1 + N j ( x , y , z ) Q 3 j − 1 + N k ( x , y , z ) Q 3 k − 1 + N l ( x , y , z ) Q 3 l − 1 (7)

where, N_{i}, N_{j}, N_{k} and N_{l} are the shape functions.

The field variable is expressed, as follows, in matrix form

u = { u ( x , y , z ) v ( x , y , z ) w ( x , y , z ) } = [ N ] Q ( e ) (8)

where,

N = [ N i 0 0 N j 0 0 N k 0 0 N l 0 0 0 N i 0 0 N j 0 0 N k 0 0 N l 0 0 0 N i 0 0 N j 0 0 N k 0 0 N l ] (9)

noting that all the six strain components are relevant in three-dimensional analysis, the strain-displacement relation can be expressed using equation as,

ε = { ε x x ε y y ε z z ε x y ε y z ε z x } = { ∂ u ∂ x ∂ v ∂ y ∂ w ∂ z ∂ u ∂ y + ∂ v ∂ x ∂ v ∂ z + ∂ w ∂ y ∂ w ∂ x + ∂ u ∂ z } = [ B ] Q ( e ) (10)

[ B ] = 1 6 V [ b i 0 0 b j 0 0 b k 0 0 b l 0 0 0 c i 0 0 c j 0 0 c k 0 0 c l 0 0 0 d i 0 0 d j 0 0 d k 0 0 d l c i b i 0 c j b j 0 c k b k 0 c l b i 0 0 d i c i 0 d j c j 0 d k c k 0 d l c l d i 0 b i d j 0 b j d k 0 b k d l 0 b i ] (11)

The stiffness matrix of the element in the global system is written as,

[ K ] e = ∭ [ B ] T [ D ] [ B ] d v (12)

where, [D] is as expressed by Equation (3).

The research work presents (FE) analysis of a selected automobile aluminum alloy wheel (6JX14H2; ET 42) for a passenger car which was loaded with a combination of inflation pressure of 0.3 MPa and radial load of 4750 N. The radial load was spread within a contact patch angle of 30˚ symmetric about the point of contact of the wheel with the ground. This angle was chosen based on literature values [^{2}) were generated, discretized into elements and analyzed by the FEM using Creo Elements/Pro 5.0. This area was chosen because this was the largest sized equilateral triangular-CH area that can be accommodated between the hub and the inner face of the wheel. The model consists of 38,493 tetrahedral elements. The wheel was constrained at the bolt holes. Static Radial load of 4750 N was then applied at an inflation pressure of 0.3 MPa to determine the stress and displacement distribution at the inboard bead seat of the wheel at circumferential angles (between 0˚ and 180˚) symmetric about the wheels’ point of contact with the ground. Young’s Modulus, yield stress and Poison’s ratio of the wheels are 22.29 GPa, 222.50 MPa and 0.42, respectively.

A parametric study was carried out at the inboard bead seat at different ARs for the CHs. The ARs considered for the triangular cooling hole were 1 and 0.5 respectively, while for the quadrilateral and oval CHs, the AR each were 1, 0.5, 0.33, and 0.2, respectively. AR of 0.33 and 0.25 were not considered for the triangular cooling hole because beyond aspect ratio of 0.5, the arm of the wheel snapped leaving the wheel as a hollow cylinder. The characteristic displacement and stress curves were compared with those of [

The Aspect ratios (AR_{s}) of 1, 0.5, 0.33 and 0.25 were considered for the cooling holes. For triangular cooling hole, the AR was terminated at 0.5 because it was observed that beyond the equilateral triangular-CH, an AR of 0.5 and beyond lead to greater stress and displacement values and overlapping of CHs leading to snapping-off of the web or arm of the wheel, thus, resulting to an open ended cylinder. Figures 1-4 show quadrilateral shape cooling hole and oval shape cooling hole each at aspect ratio of 1, 0.5, 0.33 and 0.25, respectively.

Figures 6-8 show the displacement curves for the CHs. It could be seen that at

ARs of 1, 0.5, 0.33 and 0.5, the maximum displacement values occur at the point of contact of the wheel with the ground, that is, at 0˚ circumferential angle. The

character of the curves is in good agreement with that established by literature. The, seemingly, slight variation in the shape of the curves of Figures 6-8, when compared with

Figures 10-12 represent the magnitude of the stress as depicted by the shapes of the curves. It could be seen that at AR of 1, the curves exhibited the same character as that of

point of contact with the ground, which is also in good agreement with

Having analyzed the various CH geometries of triangular, quadrilateral and oval, the effect of CH combination was investigated at the inboard bead seat at 4750N radial load and 0.3 MPa inflation pressure. Figures 13(a)-(c) show the CHs’ combination, viz: triangular-CH at AR 1 and oval-shape-CH at AR of 0.5; triangular-CH at AR1 and quadrilateral-CH at 0.5 AR and, quadrilateral-CH and oval-shape-CH at 0.5 AR.

or stress curves.

A parametric study on the effect of aspect ratio on the mechanical response of an automobile aluminium alloy wheel was undertaken. Results show that at the inboard bead seat:

1) The magnitude of stress and displacement and, shape of their curves are affected by shape and AR of CH.

2) Considering the extreme values, the triangular CH wheel is most deformed, while the oval CH is least deformed at AR of 0.5. The maximum displacement of the wheels occurs at point of contact of the wheel with the ground, which agrees favorably with literature findings. From the results, it could be established that oval-shaped-CH wheel at AR of 0.5 offers greater prospect in wheel design as it was least stressed and deformed.

3) For the wheels at AR of 1, the maximum value of the Von-Mises stress is at point of contact with the ground. The same attribute was established for the

triangular CH wheel at AR of 0.5. Maximum Von-Mises values ranging between 40˚ and 80˚ contact angle both for the oval CH and quadrilateral CH wheels at AR OF 0.5, 0.33 and 0.25. The drift in maximum displacement values may probably be due to the ARs and shape of the CH. The CH combination with highest integrity was the oval and quadrilateral at AR of 0.5.

4) This study, in itself, is not exhaustive in view of the CH shapes and ARs considered. Further studies in this area are encouraged.

The authors declare no conflicts of interest regarding the publication of this paper.

Igbudu, S.O. and Fadare, D.A. (2021) Parametric Study on the Effect of Aspect Ratio of Selected Cooling Hole Geometries on the Mechanical Response of an Automobile Aluminium Alloy Wheel. Open Journal of Applied Sciences, 11, 41-57. https://doi.org/10.4236/ojapps.2021.111004