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Leonardo Electronic Journal of Practices and Technologies
ISSN 1583-1078
Issue 33, July-December 2018
p. 139-158
139
http://lejpt.academicdirect.org
Engineering, Environment
Static stress analysis and displacement of a tricycle rear shock absorber
using finite element analysis
Temitope Olumide OLUGBADE 1*, Tiamiyu Ishola MOHAMMED 2, Oluwole Timothy
OJO1
1 Department of Industrial and Production Engineering, Federal University of Technology,
PMB 704, Akure, Ondo State, Nigeria
2Department of Mechanical Engineering, Federal University of Technology, PMB 704, Akure,
Ondo State, Nigeria
E-mails: tkolugbade@futa.edu.ng1*, timohammed@futa.edu.ng2, otojo@futa.edu.ng1
*Corresponding Author, Phone: +2348068859326
Received: August 14, 2018 / Accepted: November 19, 2018 / Published: December 30, 2018
Abstract
The static stress analysis and displacement of a tricycle rear shock absorber using finite
element analysis (FEA) were carried out. The design of spring in the suspension system
was also studied. A physical model (pressure model and flow model) was adopted and
implemented for the modelling of the tricycle shock absorber. The pressure model
determines the internal pressures while flow model calculates the oil flow between the
tricycle chambers. Static stress analysis and displacements were performed by varying
materials for suspension spring and considering the tricycle weight at a loading capacity
of 1500 kg as well as when the loading capacity was exceeded by 200 kg. The analysis
was done to validate the strength and determine the best spring material for this
application. A comparison was made between the existing spring material Model and
other available spring materials. The results showed that the spring made of Inconel X-
750 has the least stress value when compared to other spring materials. This makes it a
better material in terms of load bearing capability. However, the material retained its
position as the most considerable material for making springs when the loading capacity
of the tricycle was exceeded and when there is unexpected overloading.
Static stress analysis and displacement of a tricycle rear shock absorber using finite element analysis
Temitope Olumide OLUGBADE, Tiamiyu Ishola MOHAMMED, Oluwole Timothy OJO
140
Keywords
Static stress analysis; Finite element analysis; Shock absorber; Tricycle
Introduction
In vehicle dynamics, shock absorbers are part of the suspension system and the
importance of suspension cannot be overemphasized. It is needed to guarantee good handling,
braking, and comfort. Due to its complexity, it is sometimes challenging to design and model a
shock absorber of a suspension. It is therefore important to consider some factors while
designing for shock absorber, which includes geometry, strength, durability, and functionality.
A shock absorber is a mechanical assembly which is majorly used to support the weight, absorb
road shock and maintain tire contact as well as the proper wheel to chassis relationship in
automotive industries and structure analysis [1-2]. It does not only smooth out or damp shock
impulse but also absorb and dissipate the impact kinetic energy in such a way that accelerations
imposed upon the airframe are reduced to a tolerable rate [3]. Major problems associated with
tricycle operation are low ride quality, discomfort when over rough ground and unnecessary
disturbances. Also, the spring is compressed too quickly most times when a tricycle is traveling
on a level road and the wheels strike a bump. These problems can be resolved by designing a
good shock absorber for the tricycle using finite element analysis (FEA). This was
accomplished by designing a shock absorber using FEA which will, in turn, control the spring
and suspension movement.
Nowadays, several methods are available to model suspension system in a tricycle. The
applicability of these methods, discrete or diffuse depends on how they account for the supposed
discontinuity. To model a shock absorber, node splitting, cohesive surfaces, hybrid discrete and
finite element analysis are the common techniques using discrete methods [4-8]. Finite element
analysis (FEA) has been considered as an effective tool in shock absorber design to reduce the
error caused by the simplification of equations [9]. It has been presented to solve many
problems and equations numerically, for examples, modelling for chloride binding in mesoscale
concrete [10], phase-field solution for modelling brittle fracture [11], dynamics of flexible
beams [12], modelling hybrid all terrain trike [13], solid-shell formulation [14], determining
the reference geometry [15] and modelling helical compression spring for vehicle automotive
front suspension [16].
Leonardo Electronic Journal of Practices and Technologies
ISSN 1583-1078
Issue 33, July-December 2018
p. 139-158
141
Recently, studies on shock absorber and the suspension system have been a subject of
investigation. It has been established that without suspension, the tires could lose contact with
the road, traction would be lost, the ride would be uncomfortable, and chassis would as well be
subjected to damaging shock loads (9, 17-19). Finite element analysis (FEA) tools and
analytical solutions were used to compare the different stress and deflection values in shock
absorber components. Świtoński et al. [20] studied and analysed stiffness and damping of shock
absorber system. The results emphasized that the stiffness and damping value for shock
absorber is strongly related to the capacity of the shock absorber that was considered. Some
other factors such as damping force, control energy, and time constant are also important when
designing for shock absorber [21-23]. These factors and others like material selection and
strength, impact energy, toughness and durability are essential when designing for engineering
components [36-42].
Despite the various work done, few or little work has been done on the static stress
analysis and displacement of suspension spring of a tricycle. The comparison of different spring
materials that can be used for the design and analysis of the suspension spring of a tricycle has
also not been properly investigated.
In the present work, static stress analysis and displacement of a tricycle rear shock
absorber using finite element analysis (FEA) was carried out by varying coil springs made of
different materials. Firstly, a shock absorber used in the rear of a tricycle was designed and a
3D model was created. In addition, a comparison was made between the current spring material
(Monel K-500) and four other common spring materials to determine the best for making
suspension springs in tricycle shock absorber. The physical damper model is adopted for the
pressure model and flow model of the tricycle.
Materials and methods
Design calculations for the suspension spring shock absorber
The suspension spring shock absorber comprises of coil spring (helical), damper, piston
and fluid (oil). The analytical calculation for the considered suspension assembly was
performed using the basic design calculations and the same was compared with the ANSYS
results. Figure 1 illustrates the working algorithms of the designed tricycle rear shock absorber.
Static stress analysis and displacement of a tricycle rear shock absorber using finite element analysis
Temitope Olumide OLUGBADE, Tiamiyu Ishola MOHAMMED, Oluwole Timothy OJO
142
Figure 1. Working algorithm of the designed tricycle rear shock absorber.
In Figure 1, the working algorithm of the designed tricycle rear shock absorber include
the materials selections, design of shock absorber parts, design models, analysis and
calculations, and validation of models. The design of the coil-over helical spring was done using
the following vehicle data and information. Wet weight of tricycle (W) is 1200 kg (weight
include oil, fuel etc.). Loading capacity of tricycle is 1500 kg (weight includes added masses
such as persons). The rear axle bears the 63% of total weight (i.e.945 kg with load and 756 kg
without load) while front axle bears the remaining 37%.
Using the above vehicle data, the force on rear axle with and without load was obtained
to be 9450N and 7560 N respectively. The tricycle has two rear shock absorbers. With load, the
force exerted on each rear shock absorber was 4725 N while it was 3870 N without load. The
vertical force (with load) was 4725 N and the vertical displacement of the tire ground contact
with respect to chassis (with load) was 220 mm. By calculation, the ride rate of the tricycle with
load (vertical force per unit vertical displacement of the tire ground contact with respect to
chassis) was obtained to be 21.47 Hz. The ride rate of the tricycle with and without load are
21.47 Hz and 17.18 Hz respectively and the difference in the ride rate with and without load is
Comparison of Analytical and FEA results
Materials Selection and Considerations
Design of Shock Absorber Parts
Design Models and Analysis
Design Calculations
Static Stress Analysis and Displacement
Validation of Models
Leonardo Electronic Journal of Practices and Technologies
ISSN 1583-1078
Issue 33, July-December 2018
p. 139-158
143
4.29 Hz. This implies that the rate at which tricycle operates and moves (when fully loaded) is
more convenient and comfortable to users compared to the tricycle without load, which agrees
with the previous findings [19, 24-25].
Design considerations for the suspension spring (Helical)
In the present study, proper design of compression springs was taken into consideration
which requires the knowledge of both the potential and the limitations of available materials
together with simple formulae. The design of a coil spring involves the following basic
considerations: space into which the spring must fit and operate, working forces and deflections,
accuracy, reliability, tolerances, ride height, environmental conditions such as temperature, cost
and quality. These factors were used to select a material and suitable values for the wire size,
the number of turns, the coil diameter and the free length, type of ends and the spring rate which
is needed to meet the working force deflection requirements. The primary design constraints
are that the wire size should be commercially available and that the stress at the solid length be
no longer greater than the torsional yield strength.
The spring rate was calculated using the primary features which include the inside
diameter, wire diameter, coil mean diameter and number of active coils. The implication of
spring rate is that the greater the rate, the stiffer the spring and the more pounds of load it takes
to compress the spring one inch, Eq. [1], [17].
K = G × (d)4
8 × Na × (D)3 (1)
Where: K - spring rate, G - modulus of rigidity, d - wire diameter, D - coil mean diameter, Na
- number of active coils.
Evident from Eq. (1), suspension spring dimension including the wire diameter, mean
coil diameter, and the number of active coils is the major factors affecting the spring rate of a
steel coil. In the design analysis, value for modulus of rigidity was 2.124E+5 MPa, mean
diameter of a coil (D) was 80 mm, total no of coils (C) was 10, height (h) was 220 mm, diameter
of the wire (d) was 10 mm and the outer diameter of spring coil (D) was 90 mm, which was
obtained by the addition of inner and outer diameter of the spring coil. The axial load exerted
on the spring using the data in the design analysis was obtained to be 4725 N. The model of the
shock absorber assembly comprises the bottom part, top part and coil spring.
Static stress analysis and displacement of a tricycle rear shock absorber using finite element analysis
Temitope Olumide OLUGBADE, Tiamiyu Ishola MOHAMMED, Oluwole Timothy OJO
144
The physical damper model for the tricycle
The physical damper model adopted [26] was used for the modelling and analysis of the
tricycle rear shock absorber. The damper model was split into pressure model and flow model.
The pressure model predicts the internal chamber pressures while the flow model determines
the oil flow (Q) between the chambers in function of the pressure drop ∆p through a set of static
equations. From the rebound and compression chamber pressures, friction and bumper force,
the damper force is obtained, Eq. [2].
F = (Apt − Arod)Preb − AptPcom + Friction + Fbumper (2)
Where: F- damper force, Apt – area of pressure tube, Arod – area of damper rod, Preb – rebound
pressure, Pcom – compression chamber pressure, Fbumper - bumber force
Pressure model
Through a set of differential equations, the pressure model determines the dynamic
features of the damper. At low frequencies, the set of differential equations are related to the
adiabatic compression of the gas present in the reserve tube. It is important to note that some
undesirable effects such as the mixture of gas and oil in the pressure tube [27] can directly affect
the value of the bulk compressibility. The presence of such effects can greatly affect the damper
efficiency due to low bulk modulus. Double-tube and mono-tube design are considered under
the pressure model. This is because the difference in modelling of both types is most relevant
to pressure model. For double-tube design, the set of differential equations which can predict
the built-up of pressure in the pressure tube are presented in Eq. [3-4].
Preb =(x(Apt−Arod)−Qpv)(1−∝Preb)
(Lpt−x−x0)(Apt−Arod)α (3)
Pcom =(xApt+Qbv−Qpv)(1−∝Pcom)
(x+x0)Aptα (4)
Where: Ṗreb – rebound pressure (differentiation), Ṗcom – compression pressure (differentiation),
Qpv – flow rate of piston valve, Lpt – length of pressure tube, α – compressibility (Pa-1), x -
displacement (m), ẋ - differentiation of x, x0 – initial displacement, bv – base valve, pv – piston
valve
Leonardo Electronic Journal of Practices and Technologies
ISSN 1583-1078
Issue 33, July-December 2018
p. 139-158
145
For the reserve tube, the compressibility of the oil is infinitesimal compared to the
present gas. Hence, the reserve tube can be predicted from a polytrophic compression or
expansion, during the compression and rebound stroke, Eq. [5].
Prt = Prt,0 (Vrt,gas,0
Vrt,gas,0+ArodX)
γ
(5)
Where: Prt - pressure of reserve tube, Prt.0 - pressure of reserve tube, at original point, Vrt.gas.0 –
volume of reserve tube, gas at original point, x-displacement (m), γ- polytropic exponent.
For mono-tube design, only one tube is present and the change of volume inside the
damper body is compensated by a volume of gas at high pressure. In the rebound chamber, the
pressure remains the same while the pressure in the compression chamber is determined by the
compressed gas, Eq. [6-7].
Preb =(x(Apt−Arod)−Qpv)(1−∝Preb)
(Lpt−x−x0)(Apt−Arod)α (6)
Pcom = Pcom,0 (Vcom,gas,0
Vcom,gas,0+ArodX)
γ
(7)
Where: Pcom.0 – compression pressure, at original point, Vcom.gas.0 – compression volume, at
original point.
Flow model
The flow model shows the relationship between the flow rates through the valve
assemblies and the pressure drops. Pressure flow and intake flow are the two major models
under flow model. The pressure flow is responsible for the build-up of pressure drop across the
piston while the intake flow does not build up any important pressure drop across the valve
assembly, Eq. [8].
∆p = ∑ Bimi=1 Qri with ri ∈ R (8)
Where: Δp -pressure drop, Bi – global restriction coefficient, Qri – flow rate restriction
coefficient.
The series is truncated at two terms (m=2) with r1 = 1 and r2 mostly lies between 1 and
2. From literatures [28-29] r2 is usually fixed as 2 and calculate the coefficient B2, Eq. [9].
B2 =ρ
2(CDAV)2 (9)
Where: B2 - coefficient, ρ – oil density (kg/m3), CD – discharge coefficient, AV – valve area.
Static stress analysis and displacement of a tricycle rear shock absorber using finite element analysis
Temitope Olumide OLUGBADE, Tiamiyu Ishola MOHAMMED, Oluwole Timothy OJO
146
The discharge of coefficient CD is usually a function of the Reynolds number, the
Cauchy number, the acceleration number and the thickness to length ratio of the restriction [17].
Using B2 = 1.75, Reybrouck [30] argues that, Eq. [10].
∆p = KleakV1
4⁄ Qleak
74⁄
(10)
Where: Kleak – orifice stiffness, V1/4 – volume (m3), Qleak7/4 – orifice flow rate.
Considering the blow-off condition, Eq. [11].
∆Pblow−off ≥ ∆P0 (11)
Where: ΔPblow-off – blow-off pressure drop, ΔP0 – specific pressure developed
Blow-off valve can be modelled using Eq. [12]:
KspringQblow−off = (∆Pblow−off − ∆P0)√∆Pblow−off (12)
Where: Kspring – stiffness of blow-off spring, Qblow-off – blow-off flow rate
Based on literatures [17, 29, 31-33], Kspring and ∆Po can be obtained with Eq. [13, 14].
Kspring =Kspring
CDПdvAv√
ρ
2 (13)
∆Po =Fpreload
Av (14)
Where: CD – discharge coefficient, П – constant (3.142), dv – valve diameter, Av – valve area
The total valve feature is determined by taking into consideration, Eq. (15-17).
∆Ptot = ∆Pport + ∆Pparallel (15)
∆Pparallel = ∆Pblow−off = ∆Pleak (16)
Qtot = Qleak + Qblow−off (17)
Where: ΔPtotal – total pressure drop, ΔPport – pressure drop in port channel, ΔPparallel – pressure
drop in parallel, ΔPleak – pressure drop in orifice, Qtot - total flow rate, Qleak - orifice flow rate,
Qblow-off - blow off flow rate.
In series, the pressure drop across two elements is the addition of the pressure drop
across each element (Eq. [15]). The pressure drop in parallel, across two elements is the same
to the drops across each element (Eq. (16)) while the total flow of two valves in parallel is equal
to the sum of the flows across individual elements (Eq. (17)).
Leonardo Electronic Journal of Practices and Technologies
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Issue 33, July-December 2018
p. 139-158
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Finite element approach (FEA)
The model was validated by comparing with the results obtained from Pinjarla and
Lakshmana [3] and Sudarshan et. al [19]. The parameters used for the validation (compares
between the developed model and the published results) are the loading capacity, force exerted,
analytical stress, boundary conditions and material properties. The percentage error between
the analytical stress obtained from this work compared with the previous result [3, 19] is 2.11
% while the percentage error in terms of loading capacity is 1.98 %, which is similar with the
previous work [19, 24]. FEA was used to predict the behaviour of a model in response to loads
that it will be subjected to. FEA allows for model development to be carried out at low cost and
can result in the removal unnecessary mass should it be determined that the design would still
retain enough strength properties without it.
Results and Discussion
Static stress analysis and displacement of selected wire materials
The analytical and FEA results obtained from the static stress analysis and
displacements of selected shock absorber wire material are presented in Tables 1-4.
Table 1. Static stress analysis and displacements of selected shock absorber wire material
No Material Young’s modulus
(E) MPa
Density
(Kg/m3)
Force
(N)
1 Nickel Base Alloy (Monel K500) 1.79×109 8440 4725
2 High-carbon Steel (ASTM A 228) 2.1×109 7850 4725
3 Alloy steel (ASTM A401) 2×109 7850 4725
4 Stainless Steel (AISI 302/304) 1.93×109 7860 4725
5 High temperature alloy (Inconel X-750) 2.124×109 8280 4725
Table 2. Summary of FEA result at tricycle loading capacity (4725N) for different shock
absorber wire materials
No Wire
material
Minimum
deflection (mm)
Maximum
deflection (mm)
Maximum
stress (MPa)
Density
(Kg/m3)
1 Monel K500 0.0064707 0.010246 37.213 8440
2 ASTM A 228 0.0066108 0.010461 37.761 7850
3 ASTM A401 0.0067469 0.010678 38.534 7850
4 AISI 302/304 0.0068492 0.010841 39.130 7860
5 Inconel X-750 0.0064742 0.010244 36.978 8280
Static stress analysis and displacement of a tricycle rear shock absorber using finite element analysis
Temitope Olumide OLUGBADE, Tiamiyu Ishola MOHAMMED, Oluwole Timothy OJO
148
Table 3. Summary of FEA result at tricycle overloaded capacity (5000N)
No Wire
material
Minimum
deflection (mm)
Maximum
deflection (mm)
Maximum
stress (MPa)
Density
(Kg/m3)
1 Monel K500 0.008932 0.013518 43.434 8440
2 ASTM A 228 0.009174 0.013761 44.217 7850
3 ASTM A401 0.009337 0.014006 45.000 7850
4 AISI 302/304 0.009499 0.014250 45.782 7860
5 Inconel X-750 0.009012 0.013498 43.043 8280
Table 4. Comparison of numerical (FEA) and theoretical values of maximum deflection
(loading capacity of 4725N)
No Method
loading (4725 N)
Monel
K500
ASTM
A228
ASTM
A401
AISI
302/304
Inconel
X-750
1 Numerical (FEA) 0.010246 0.010461 0.010678 0.010841 0.010244
2 Theoretical 0.015394 0.013122 0.013778 0.014277 0.012973
Loading (5000 N)
1 Numerical (FEA) 0.013518 0.013761 0.014006 0.014250 0.013498
2 Theoretical 0.0162905 0.013885 0.014579 0.015108 0.013728
As shown in Tables 1-4, the selected shock absorber wire materials are nickel base alloy
(Monel K500), high-carbon steel (ASTM A228), alloy steel (ASTM A401), stainless steel
(AISI 302/304) and high temperature alloy (Inconel X-750) (Table 1). The different stress and
deflection values in shock absorber wire material was obtained using ANSYS FEA tool. The
results obtained at loading capacity (4725 N) for different shock absorber wire materials are
summarized in the Table 2. The results obtained when the loading capacity was exceeded for
different shock absorber wire materials were also summarized in Table 3. The absorber
geometries such as the thickness of spring have influence on the deflection in stiffness and
damping of shock absorber system. In addition, the stiffness and damping value for shock
absorber are strongly related to the capacity of the shock absorber. The geometric study was
considered putting into consideration the damping force, control energy and time constant.
Moreover, Table 4 shows the results obtained when the loading capacity of the tricycle
was exceeded. This is required for load bearing structural members to determine their behaviour
when there is unexpected overloading. From the results obtained, Inconel X-750 retained its
position as the most considerable material for making springs.
Leonardo Electronic Journal of Practices and Technologies
ISSN 1583-1078
Issue 33, July-December 2018
p. 139-158
149
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Static stress analysis and displacement of a tricycle rear shock absorber using finite element analysis
Temitope Olumide OLUGBADE, Tiamiyu Ishola MOHAMMED, Oluwole Timothy OJO
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(j)
Figure 2. The magnitude of maximum deflection in the spring and the equivalent stress in the
shock absorber (a) and (b) nickel base alloy (Monel K500), (c) and (d) high carbon steel
(ASTM A228), (e) and (f) alloy steel ASTM A401, (g) and (h) stainless steel (AISI 302/304),
(i) and (j) high temperature alloy (Inconel X-750).
Figure 2 a-j shows the magnitude of maximum deflection in the spring and the
equivalent stress in the shock absorber of the five (5) selected shock absorber wire materials.
For nickel base alloy (Monel K500), the magnitude of maximum deflection in the spring is
0.010246 mm, as shown in Figure 2a and the maximum shear stress in the shock absorber is
37.213MPa, as shown in Figure 2b. Also, from Figure 2b, the values of deflection in Monel K-
500 wire under static load of 4725 N force are obtained within the range of 0.0064707 mm and
0.010246 mm.
The magnitude of maximum deformation in the spring for high carbon steel (ASTM
A228) is as shown in Figure 2c while Figure 2d shows the maximum shear stress in the shock
absorber. The magnitude of maximum deformation in the spring is 0.010461 mm and the
maximum shear stress in the shock absorber is 37.761 MPa. Within the range of 0.0066108 mm
and 0.010461 mm, from Figure 2d, the values of deflection in ASTM A228 wire under static
load of 4725 N force are obtained within the range of 0.0066108 mm and 0.010461 mm. Figure
2e shows the magnitude of maximum deformation in the spring for alloy steel ASTM A401.
Figure 2f shows the maximum shear stress in the shock absorber. The magnitude of maximum
deformation in the spring is 0.010678 mm and the maximum shear stress in the shock absorber
is 38.534 MPa. In Figure 2f, the values of deformation in ASTM A401 wire under static load
of 4725N force are obtained within the range of 0.0067469 mm and 0.010678 mm.
In Figure 2g, the magnitude of maximum deformation in the spring for stainless steel
(AISI 302/304) is illustrated. Figure 2h shows the maximum shear stress in the shock absorber.
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The magnitude of maximum deformation in the spring is obtained to be 0.010841 mm and the
maximum shear stress in the shock absorber is given as 39.13 Mpa. Figure 2h also showed that
the values of deformation in AISI 302/304 wire under static load of 4725 N force are obtained
within the range of 0.0068492 and 0.010841 mm. The magnitude of maximum deformation in
the spring for high temperature alloy (Inconel X-750) is illustrated in Figure 2i while Figure 2j
shows the maximum shear stress in the shock absorber. The magnitude of maximum
deformation in the spring is 0.0064742 mm and the maximum shear stress in the shock absorber
is 36.978 MPa. The values of deformation in Inconel X-750 wire under static load of 4725N
force are obtained within the range of 0.0064742 mm and 0.010244 mm, as evident in Figure
2j.
From Table 3, it can be deduced that the shock absorber made of spring material Inconel
X-750 has the least stress value of 36.978 MPa which makes it a better material in terms of load
bearing capability, which agrees with the previous results [25, 34]. In terms of deflection or
deformation, Monel K-500 has the least deformation with value of 0.0064707mm. This
compared with deformation in Inconel X-750 with value of 0.0064742mm is very insignificant
and not noticeable. Though, materials with larger deflection value has tendency to buckle than
materials with less deflection value, but Inconel X-750 has less deflection value compared to
the rest of the materials, as established from previous studies [35].
Weight is also an important aspect of the product development, and every attempt must
be made to adhere to the specified weight of the shock absorber. The springs of the various
materials are of equal volume with different densities. Inconel X-750 has a density which is
lesser than that of Monel K-500 translating to the fact that it is lighter. Compared with the
previous reports [17, 28], the shock absorber design is safe because the equivalent stress
obtained with the entire materials is lesser than their respective yield strength. Inconel Alloy
has higher resistance to deformation and failure because it has the least stress value than Monel
K-500; hence, it can be used as an alternative material for making helical springs in tricycle
shock absorber, similar conclusion was made [24-25].
Analytical approach
To validate the above result, theoretical calculation is being carried out below for Monel
K500, ASTM A 228, ASTM A401, AISI 302/304, Inconel X-750. Consequently, for all the
Static stress analysis and displacement of a tricycle rear shock absorber using finite element analysis
Temitope Olumide OLUGBADE, Tiamiyu Ishola MOHAMMED, Oluwole Timothy OJO
152
spring materials, the spring constant, maximum deflection (for loading capacity of 4725 N and
overloaded capacity 5000 N), spring index and maximum shear stress are calculated using Eq.
(18-22):
k = G × (d)4
8 × n× (D)3 (18)
δ = F
k (19)
C = D
d (20)
W =4C−1
4C−4+
0.615
C (21)
τ = 8 ×W× D×F
πd3 (22)
Where: G - modulus of rigidity, d - wire diameter, D - mean coil diameter, n - no of active coils,
δ - maximum deflection, F - maximum force, k – spring constant, C - spring index, τ - maximum
shear stress, W - wahl correction factor.
The maximum deflection and stress values obtained from the analytical approach are
summarized and compared with the numerical values as shown in Table 4.
Comparison of numerical (FEA) and theoretical values of maximum deflection
Figure 3 represents the numerical results of maximum deflection at loading capacity of
4725 N and overloaded capacity of 5000N for different suspension spring materials while the
maximum shear stress numerical results for various spring materials at different loading
capacity are shown in Figure 4.
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
Monel K500 ASTM A228 ASTM A401 AISI 302 INCONEL X-750
Ma
xim
um
de
flectio
n (
mm
)
Loading capacity - 4725N
Loading capacity - 5000N
Figure 3. ANSYS results of maximum
deflection at loading capacity of 4725 N and
overloaded capacity of 5000 N for different
suspension spring materials
37.213 37.761 38.534 39.1336.978
43.434 44.217 45 45.78243.043
37.213 37.761 38.534 39.1336.978
43.434 44.217 45 45.78243.043
0
10
20
30
40
50
60
Monel K500 ASTM A228 ASTM A401 AISI 302 INCONEL X-750
Maxim
um
str
ess (
MP
a)
Loading capacity - 4725N
Loading capacity - 5000N
Figure 4. ANSYS results of maximum stress
for different suspension spring materials at
loading capacity of 4725 N and overloaded
capacity of 5000 N
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153
In both the numerical and theoretical analysis, AISI302/304 (among the suspension
spring materials) has the highest maximum deflection and stress at loading capacity of 4725 N
and 5000 N. At 4725 N, the maximum deflection for numerical and theoretical analysis are
0.010841 mm and 0.014277 mm respectively (Figure 3) while it is 0.014250 mm and 0.015108
mm for numerical and theoretical analysis respectively, at 5000 N (Figure 4).
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Monel K500 ASTM A228 ASTM A401 AISI 302 INCONEL X-750
Maxim
um
deflection (
mm
)
Numerical (FEA) deflection
Theoretical deflection
Figure 5. Comparison of numerical (FEA)
and theoretical values of maximum
deflection for different suspension spring
materials at loading capacity of 4725 N
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
0.022
Monel K500 ASTM A228 ASTM A401 AISI 302 INCONEL X-750
Maxim
um
deflection (
mm
)
Numerical (FEA) deflection
Theoretical deflection
Figure 6. Comparison of numerical (FEA)
and theoretical values of maximum
deflection for different suspension spring
materials at loading capacity of 5000N
The results obtained from numerical (FEA) and theoretical analysis or both maximum
deflection and maximum shear stress are compared, as shown in Figure 5 and Figure 6. The
comparison was done when the loading capacity is 4725 N and when there is overloaded
capacity (5000N) for the different suspension spring materials.
As evident in Figure 5 and Figure 6, for all the suspension spring materials except Monel
K500, it is observed that the numerical and theoretical values for both maximum deflection and
shear stress are close when the loading capacity is 4725 N and 5000 N. The significant
difference between the numerical and theoretical values of Monel K500 is a little bit more
compared to the differential values of the remaining four (4) suspension spring materials.
Conclusions
Statistical stress analysis of suspension spring of a tricycle was carried out to solve the
major problems associated with tricycle operation which include low ride quality, discomfort
Static stress analysis and displacement of a tricycle rear shock absorber using finite element analysis
Temitope Olumide OLUGBADE, Tiamiyu Ishola MOHAMMED, Oluwole Timothy OJO
154
when over rough ground, unnecessary disturbances and quick compression of springs. This was
done by designing a good shock absorber for the tricycle using finite element analysis (FEA),
which in turn control the spring and suspension movement. The essential points of this work
are as follows:
(1) The physical damper model which consist of pressure model and flow model, was adopted
for modelling the tricycle, to determine the internal pressure and calculate the oil flow
between the tricycle chambers. Static stress analysis of five (5) different suspension spring
materials was done. Two different analysis methods are implemented: the numerical
analysis using finite element method and theoretical analysis which was employed for
validation purpose.
(2) The analysed stress values are less than their respective yield stress values. This makes the
design distinct. By comparing the results for the five (5) materials, the spring made of
Inconel X-750 has the least stress value, deformation and a density of 8280 kg/m3 which is
lesser than that of Monel K-500. It can be concluded that Inconel X-750 is the best material
for making helical springs in a tricycle rear shock absorber because of its low deformation
value, less weight and cost. This conclusion was confirmed in line with the computations
for numerical examples that were studied using the developed model.
(3) The method of finite element analysis was employed to evaluate the performance of
different coil spring materials, to determine the material with the least deformation under a
given condition (loading).
In future work, the combination of experimental, numerical and theoretical approach
will be adopted for the structural and modal analysis of shock absorber. In addition, finite-
element formulation based on interpolation of strain measures will be developed and dynamics
of flexible spring following the same approach.
Acknowledgements
Special thanks to Fashade Olalekan for his assistance during this research. This research did
not receive any specific grant from funding agencies in the public, commercial or profit sectors.
Leonardo Electronic Journal of Practices and Technologies
ISSN 1583-1078
Issue 33, July-December 2018
p. 139-158
155
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