applied
sciences
Article
Reverse Engineering and Topology Optimization for
Weight-Reduction of a Bell-Crank
Toh Yen Pang *
and Mohammad Fard
School of Engineering, RMIT University, Bundoora Campus East, Bundoora, VIC 3083, Australia;
mohammad.fard@rmit.edu.au
* Correspondence: tohyen.pang@rmit.edu.au; Tel.: +61-3-9925-6128
Received: 11 November 2020; Accepted: 29 November 2020; Published: 30 November 2020
Abstract: This paper describes a new design method that was developed to achieve an optimal
design method for weight reduction of a bell crank, sourced from a Louis Christen Road Racing F1
Sidecar. The method involved reverse engineering to produce a 3D model of the mechanical part.
The 3D bell crank model was converted to a finite element (FE) model to characterize the eigenvalues
of vibration and responses to excitation using the Lanczos iteration method in Abaqus software.
The bell crank part was also tested using a laser vibrometer to capture its natural frequencies and
corresponding vibration mode shapes. The test results were used to validate the FE model, which was
then analysed through a topology optimization process. The objective function was the weight and
the optimization constraints were the stiffness and the strain energy of the structure. The optimized
design was converted back to a 3D model and then fabricated to produce a physical prototype for
design verification and validation by means of FE analysis and laboratory experiments and then
compared with the original part. Results indicated that weight reduction was achieved while also
increasing the natural frequency by 2%, reducing the maximum principal strain and maximum
von Mises stress by 4% and 16.5%, respectively, for the optimized design when compared with the
original design. The results showed that the proposed method is applicable and effective in topology
optimization to obtain a lightweight (~3% weight saving) and structurally strong design.
Keywords: bell crank; topology optimization; natural frequency; reverse engineering; vibrometer;
design; Abaqus
1. Introduction
A bell crank is a mechanical component that is primarily responsible for translating the motion of
links through an angle. It acts as a link between a spring and a damper component at one end, and a
pushrod/pull rod at the opposing end. The most common limitation associated with vehicle suspension
in racing vehicles is the inability to mount the spring/damper mechanism vertically, compared with
most ordinary cars. As a result, bell cranks are incorporated into the design of racing vehicles to
translate the vertical motion of the wheel into horizontal motion to allow the suspension to be mounted
transversely or longitudinally [1].
However, in racing vehicles, there are limitations on bell cranks that relate to their weight.
When there is a limit in engine power capacity, a reduction in vehicle mass will improve the
performance aspect of a racing vehicle known as a power-to-weight ratio. Hence, engineers are
constantly trying to improve the performance of bell crank by reducing its weight, while maintaining
its structural integrity. To achieve that, engineers have been using the structural topology optimization
method to find an optimal material distribution in a structural design domain considering an objective
function in presence of constraints [2–5].
Appl. Sci. 2020, 10, 8568; doi:10.3390/app10238568
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However, only a limited number of studies in the literature examined topology optimization of
bell cranks [4,6,7]. Fornace [6] used a topology optimization method to assess weight reduction of bell
cranks for a Formula Society of Automotive Engineers (SAE) vehicle. The author considered reducing
the mass of the rear bell crank component by at least 10% while maintaining the yield strength of
the previous design. The author utilized destructive testing on three specimens under static loads of
445 N to validate their new design. Choudhury et al. [4] used a ‘Fully Stressed Design’ optimization
procedure to achieve the mass and stress optimization of bell crank of Formula SAE vehicle from SRM
University Chennai in India. The procedure allowed them to investigate and analyze the structural
stress distribution of the bell crank in real-time conditions during the damping process and the spring
actuation. The authors managed to produce an optimized computer aided design (CAD) bell crank
model with a 22% and 20% of overall weight and stress reduction, respectively.
When reducing the mass (or volume) of a mechanical component such as bell cranks, engineers
face challenges to maintain structural integrity. The structural integrity relevant to a mechanical
component is its overall stiffness. An element that directly correlates with component stiffness is its
natural frequency. The natural frequency of a certain vibration mode can be controlled in order to
reduce the displacement or deformation of a particular point in the structure [8,9]. Thompson et al. [10]
found that chassis stiffness is directly related to its flexibility when the stiffness of the chassis increases,
the twist vibration decreases and the overall vehicle handling is improved by allowing the suspension
components to control a larger percentage of the vehicle’s kinematics.
No studies in the literature have utilized a non-destructive mode shape analysis to investigate
the dynamic response of the bell crank. The benefit of using mode shape analysis is that it allows
engineers to determine the dynamic stiffness of a structure. By changing the stiffness of the structure,
engineers could minimize the amount of vibration, which is produced by the relative motion or
deformation of the structure under applied loads [11]. In the case of the bell crank, a small relative
motion in the suspension system will provide undesirable feedback to the operator in either lateral,
longitudinal or cyclic loadings [12].
The purpose of this study is to propose a methodology that incorporates a reverse engineering
technique and physical testing for accurately and efficiently assessing topology optimization problems.
2. Theory of Weight Reduction and Stiffness Matrices
2.1. Modelling Approach
In a static structural analysis, for an arbitrary structure that is subjected to external forces and
under the prescribed boundary conditions, the FE matrix equation can be expressed as [13–15]:
[F] = [K]{u}
(1)
where [F] the force vector, [K] the stiffness matrix, and {u} the displacement vector [16].
According to Guyan [13], the vector {u} can be partition into boundary degrees of freedom (DOF)
and internal DOF, and they are denoted with subscript of b and i, respectively. Equation (1) is fully
general and is applicable for two-dimensional and three-dimensional problems, with appropriate
boundary conditions. The entire system can be partitioned to [14]:
(
Fi
Fb
)
"
=
kii
kbi
kib
kbb
#(
ui
ub
)
(2)
Once the global stiffness matrix [K] is assembled, the unknown nodal displacement, {ui } and
unknown reaction forces, [Fb ] can be obtained by solving Equation (2). From the first line of equation,
we first solve for {ui }, i.e., ui = k−1
ii (Fi − kib ub ); subsequently, substitute ui into the second line of
equations, and thus Fb can be solved as Fb = kbi ui + kbb ub .
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2.2. Topology Optimization: Structural Compliance
Once the nodal displacements and reaction forces are calculated from Equation (2), the topology
optimization is to determine the optimal subset Ω* of material distribution of a given design domain,
Ω. The generalized formulation [16,17] for the compliance of the design domain is given as:
CΩ (xe ) = FTi (xe ) ui (xe ) − FTb (xe ) ub
(3)
where xe is the vector of design variables.
Assuming a structure that is subjected to the prescribed external loads and under appropriate
supports, the design domain is meshed with N finite elements. To achieve a reduction in mass,
the topology optimization statement is expressed as [5,16,18–20]:
Find : ρ = ρ1 , ρ2 , · · · , ρN
n o
Minimize : [F]Tρ uρ ,
R
)
Subject to : Ω ρ(xe )dΩ = Vol(Ω∗ ) ≤ V
: 0 < ρmin ≤ ρ(xe ) ≤ 1
(4)
(5)
where V is the total initial volume before optimization, and ρmin is the lower bound of artificial material
density, ρ(xe ) of each element in the design domain, which can be determined:
(
(xe ) =
1 if solid, xe ∈ Ω∗
0 if void, xe < Ω∗
(6)
The displacement vector uρ , which is the function ρ in Equation (4) can be obtained by solving:
[K(ρ(xe ))]{u} = [f]
(7)
where [K] is a structural global stiffness matrix and is obtained by the assemblage of element stiffness
matrix over the design domain. For a given isotropic material, the stiffness matrix of each element,
which depends on its artificial material density ρ(xe ) is given by the following Young’s modulus:
E(xe ) = ρ(xe )E
(8)
where E is Young’s modulus for the given isotropic material, which assumed to be the same for all
elements. When in equilibrium state, the energy bilinear form (i.e., the internal virtual work done by
the elastic body, within a given design domain Ω) can be expressed as [18,19]:
Z
δW(v, w) =
Ω
E(xe )εij (v)εkl (w)dV
(9)
where E(xe ) the optimal stiffness tensor that attain the isotropic material properties, V is the given
total material volume, v the internal work
at equilibrium
state, w an arbitrary virtual displacement,
R
R
∂uj
1 ∂ui
and with linearized strains εij (v) = 2 ∂x + ∂x and linearized load as l(v) = Ω fδudΩ + Γ tδudS.
j
i
T
The first integral extends over the volume of the body and the second integral extends the surface
of the body.
We know strain energy is stored within an elastic body when the body deformed under the
external forces. The objective function in Equation (4) is to achieve the optimal design via minimize
the structural strain energy, which can be expressed as:
Minimize: C = {u}T [K]u
Subject to: [K(ρ(xe ))]{u} = [f]
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2.3. Vibration and Eigenvalue Problems
The eigenvalues of vibration and responses to an excitation mechanism play an important
role in the design process. Likewise, the efficiency and accuracy of the FE analysis that solves the
eigenvalue
a crucial role in the success of present topology optimization of continuum
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structures [21]. Consider the linearized equation of motion for a vibration excitation in a discrete
undamped
structure
[15,22,23],
in our case
the bell of
crank,
can for
be expressed
in excitation
the form: in a discrete
structures [21].
Consider
the linearized
equation
motion
a vibration
undamped structure [15,22,23], in our case the.. bell crank, can be expressed in the form:
Mq + Kq = F
(10)
(10)
𝑴𝒒 + 𝑲𝒒 = 𝑭
with M and K the global mass and stiffness matrices of the structure, F is the force vector, and q is the
with Mofand
K the global
mass andDOF.
stiffness matrices of the structure, F is the force vector, and q is the
vector
unknown
displacement
vector
of solution
unknown
The
ofdisplacement
Equation (10)DOF.
results in an eigenvalue problem, with $ are the eigenvalues and
N
The
solution
of
Equation
(10)
results
eigenvalue
with 𝝕 are the eigenvalues and
φ the associated eigenvectors, which caninbeanexpressed
as problem,
[23]:
𝝓𝑵 the associated eigenvectors, which
can
be
expressed
as
[23]:
−ω2𝟐MMN
+ KMN φN = 0
(11)
(11)
−𝝎 𝑴𝑴𝑵 + 𝑲𝑴𝑵 )𝝓𝑵 = 𝟎
𝑴𝑵
𝑴𝑵
MN
whereM𝑴
mass
matrix,
the stiffness
andNM
N are The
the eigenvalues
DOFs. The
where
is is
thethe
mass
matrix,
KMN 𝑲
is theisstiffness
matrix,matrix,
and M and
areand
the DOFs.
eigenvalues
of Equation
(11)solved
are usually
using or
thesubspace
Lanczos iteration
or subspace
iteration method.
of
Equation (11)
are usually
usingsolved
the Lanczos
method.
3. Materials and Methods
3.1.
Overview of
of the
the Process
Process
3.1. Overview
The
overall design
designoptimization
optimizationprocess
processofof
proposed
study
is summarized
in Figure
1.
The overall
thethe
proposed
study
is summarized
in Figure
1. The
The
process
started
with
a physical
crank,
whichwent
wentthrough
throughnon-destructive
non-destructive experimental
experimental tests
process
started
with
a physical
bellbell
crank,
which
tests
and
a
reverse
engineering
process
to
create
an
equivalent
finite
element
(FE)
model.
The experimental
experimental
and a reverse engineering process to create an equivalent finite element (FE) model. The
data
used to
to verify
verify the
the FE
FE model.
model.The
Theoptimization
optimizationwas
was
based
generalized
process
data were
were used
based
onon
thethe
generalized
process
of
of
finding
a
solution
to
the
objective
functions,
which
were
either
to
be
maximized
or
minimized
finding a solution to the objective functions, which were either to be maximized or minimized
material
material distributions
distributions within
within aa design
design domain
domain [24].
[24]. The
The design
design domain
domain was
was subjected
subjected to
to constraints
constraints
(e.g.,
(e.g., material
material layout
layout within
within aa given
given design
design space
space (volume
(volume constraint)
constraint) or
or maximum
maximum stress
stress values)
values) to
to
be
satisfied.
be satisfied.
Figure 1.
1. Flow
stages of
of design
design optimization
optimization approach.
approach.
Figure
Flow chart
chart depicting
depicting major
major stages
3.2. Physical Part of Reverse Engineering
The detail drawings and three-dimensional (3D) model for the bell crank were not available.
Therefore, a reverse engineering technique known as 3D scanning was used to create a 3D model of
the bell crank.
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3.2. Physical Part of Reverse Engineering
The detail drawings and three-dimensional (3D) model for the bell crank were not available.
Therefore, a reverse engineering technique known as 3D scanning was used to create a 3D model of
the bell crank.
A bell cranks sourced from a Road Racing Sidecar and produced by Louis Christen Racing (LCR)
Appl. Sci. 2020, 10, x FOR PEER REVIEW
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in 1994, was used for the present study. The bell crank is made of anodized aluminum (Figure 2a)
andits
its surface
surface luster
luster isis relatively
relatively high,
high, which
which resulted
resulted in
in aa ‘noisy’
‘noisy’ scan.
scan. Hence,
Hence, the
the shiny
shiny surface
surface
and
required
a
coating
that
minimized
the
unwanted
scanning
noise.
A
product
named
‘PlastiDip’
in
required a coating that minimized the unwanted scanning noise. A product named ‘PlastiDip’ in solid
solid
Matte
White
was
used
to
coat
the
bell
crank
to
achieve
a
matte
finish
surface.
The
coated
bell
Matte White was used to coat the bell crank to achieve a matte finish surface. The coated bell crank
crank
wasplaced
then placed
the rotating
tablescan
andimages
scan images
were captured
with FlexScan
(Figure
2b).
was
then
on the on
rotating
table and
were captured
with FlexScan
(Figure
2b). All
All
scanned
images
were
merged
to
create
the
shape
of
a
3D
part
model.
In
various
locations,
FlexScan
scanned images were merged to create the shape of a 3D part model. In various locations, FlexScan
wasunable
unableto
toduplicate
duplicatethe
theexact
exactspecifications
specificationsand
anddimensions
dimensionsof
ofthe
thebell
bellcrank.
crank.Hence,
Hence,aaclean-up
clean-up
was
of
the
3D
model
was
necessary
(highlighted
in
yellow
(Figure
2c)).
The
surface
of
the
material
was
of the 3D model was necessary (highlighted in yellow (Figure 2c)). The surface of the material was
cleaned
using
a
‘defeature
tool’
within
‘Geomagic’.
Holes
were
filled
using
the
‘fill-up
tool’.
Before
the
cleaned using a ‘defeature tool’ within ‘Geomagic’. Holes were filled using the ‘fill-up tool’. Before the
modelwas
wasconverted
convertedinto
intoan
an‘igs’
‘igs’format,
format,aa‘mesh
‘meshdoctor’
doctor’tool
toolwas
wasused
usedtotoremove
removenon-manifold
non-manifoldedges,
edges,
model
self-intersections,
highly
creased
edges,
spikes
and
small
components.
Various
modifications
of
the
3D
self-intersections, highly creased edges, spikes and small components. Various modifications of the
model
were
performed
in
the
CATIA
(V5R21)
computer-aided
design
(CAD)
program
to
assure
the
3D
3D model were performed in the CATIA (V5R21) computer-aided design (CAD) program to assure
model
the accurate
dimensions
and features
of the actual
bellactual
crank.bell
Figure
2d Figure
shows 2d
the
the
3D replicated
model replicated
the accurate
dimensions
and features
of the
crank.
revised
and
final
3D
model
that
was
then
used
for
the
validation
and
optimization
tasks.
shows the revised and final 3D model that was then used for the validation and optimization tasks.
Figure 2. Bell crank scanning process: (a) Actual bell crank; (b) 3D scanning of the bell crank with
Figure 2. Bell crank scanning process: (a) Actual bell crank; (b) 3D scanning of the bell crank with
FlexScan; (c) initial scanned 3D model; (d) final 3D model.
FlexScan; (c) initial scanned 3D model; (d) final 3D model.
3.3. Experiment Method
3.3. Experiment Method
Since only one bell crank part was available, a non-destructive method was adopted. The bell
Since
bella crank
part was available,
a non-destructive
method
adopted.
The with
bell
crank was only
testedone
using
laser vibrometer
(PSV-400 Scanner,
Polytec Inc.,
Irvine, was
CA, USA,
coupled
crank
was
tested
using
a
laser
vibrometer
(PSV-400
Scanner,
Polytec
Inc.,
Irvine,
CA,
USA,
coupled
the Polytec scanning program), which uses non-contact laser measurement [25]. The schematic diagram
with
the
Polytec scanning
program),
uses non-contact
laser measurement
of the
experimental
equipment
usedwhich
for vibration
testing is shown
in Figure 3a.[25].
The The
bell schematic
crank was
diagram
of
the
experimental
equipment
used
for
vibration
testing
is
shown
in
Figure
3a.
Theholes.
bell
suspended in the air with strings, which has minimal damping characteristics, through the
crank
was
suspended
in
the
air
with
strings,
which
has
minimal
damping
characteristics,
through
the
The experiment’s approach was aimed at extracting the modes of the frequency of the bell crank
holes.
Theapplied
experiment’s
approach
wasataimed
at extracting
modes
of thewas
frequency
of the
bell
through
sinusoidal
vibration
five different
points.the
The
bell crank
excited by
a modal
crank
through
applied
sinusoidal
vibration
at
five
different
points.
The
bell
crank
was
excited
by
a
shaker (model 2007E, miniature electrodynamic shaker, Cincinnati, OH, USA) as a sinusoidal force
modal
shaker
(model
2007E,
miniature
electrodynamic
shaker,
Cincinnati,
OH,
USA)
as
a
sinusoidal
at predefined locations, indicated in Figure 3b. The tests were repeated five times at each location,
force
at predefined
locations,
in Figure
3b. The tests
were
repeated
five
times
each
so average
eigenvalues
for eachindicated
location could
be extracted.
A mesh
grid
(x = 50, y
= 20)
wasatcreated
location, so average eigenvalues for each location could be extracted. A mesh grid (x = 50, y = 20) was
created over the top face of the bell crank, as this sizing of the grid gave the greatest coverage for the
analysis (Figure 3c). Across the top face of the bell crank, the laser pointer recorded each separate
point for a response. The laser vibrometer captures the natural frequencies and the corresponding
vibration mode shapes of the bell crank in the required frequency band.
Appl. Sci. 2020, 10, 8568
6 of 15
over the top face of the bell crank, as this sizing of the grid gave the greatest coverage for the analysis
(Figure 3c). Across the top face of the bell crank, the laser pointer recorded each separate point for a
response. The laser vibrometer captures the natural frequencies and the corresponding vibration mode
shapes
the10,bell
crank
the required frequency band.
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Sci. of
2020,
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PEERinREVIEW
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Figure
3. (a)
Vibrometer
test
setup,
(b) Locations
Locations for
modal shaker
attachment,
(c)
Scanning
grid
on
Figure 3.
(a) Vibrometer
Vibrometer test
test setup,
setup, (b)
for modal
modal
shaker attachment,
attachment, (c)
(c) Scanning
Scanning grid
grid on
on
Figure
(a)
Locations for
shaker
top
face
of
bell
crank.
top face
face of
of bell
bell crank.
crank.
top
Bell Crank
Crank Model
Model Validation
Validation
3.4. Bell
Bell
3.4.
Crank Model
Validation
simulation program (Abaqus
(Abaqus v6.14
v6.14 [26]) for
The 3D
3D bell
bell crank
crank model
model was
was imported
imported into
into the
the simulation
The
program (Abaqus v6.14
[26]) for
running a natural frequency analysis. Model
Model geometry,
geometry, material properties, element type, boundary
running a natural frequency analysis. Model geometry, material properties, element type, boundary
frequency modes, and a detailed description of the modelling
modelling are
are outlined
outlined
conditions, step-to-request
step-to-request frequency
conditions,
modes, and a detailed description of the modelling are
outlined
in
the
following
sub-sections.
in the following sub-sections.
3.4.1. Finite
Finite Element
Element Mesh
Mesh and
and Material
Material Properties
Properties
3.4.1.
3.4.1.
Finite Element
Mesh and
Material Properties
The element
bell
crank
waswas
set at
10-node
quadratic
tetrahedron
element
The
elementsize
sizeofof
ofthe
the
bell
crank
set3 mm,
at 33 and
mm,the
and
the 10-node
10-node
quadratic
tetrahedron
The
element
size
the
bell
crank
was set
at
mm,
and
the
quadratic
tetrahedron
was
chosen
as
the
element
type.
element was
was chosen
chosen as
as the
the element
element type.
type.
element
Figure 44 shows
shows the
the meshed
meshed bell
bell crank,
crank, which
which contains
contains 20,827
20,827 elements
elements and
and 35,803
35,803 nodes
nodes to
to the
the
Figure
Figure 4 shows the meshed bell crank, which
contains 20,827
elements and
35,803 nodes
to the
solid
section.
The
mechanical
properties
of
AW-6082-T6
aluminum
alloy
were
assigned
to
the
bell
solid section.
section. The
The mechanical
mechanical properties
properties of
of AW-6082-T6
AW-6082-T6 aluminum
aluminum alloy
alloy were
were assigned
assigned to
to the
the bell
bell
solid
crank model
model [27,28].
[27,28].The
Thematerial
materialproperties
propertiesused
used
for
the
bell
crank
are
as
follows:
Yield
strength,
crank
for
the
bell
crank
are
as
follows:
Yield
strength,
crank model [27,28]. The material properties
used
for the bell crank are as follows: Yield strength, 𝑓𝑓
−9 kg/mm
3 , Young’s
f=y310
= 310
MPa,
density,
ρ
=
2.7
×
10
Modulus,
E =GPa
70 GPa
and Poisson’s
−9
3
MPa, density,
density, 𝜌𝜌 == 2.7
2.7 ×× 10
10−9 kg/mm
kg/mm3,, Young’s
Young’s Modulus,
Modulus,
E == 70
70
and Poisson’s
Poisson’s
ratio,ratio,
=
= 310 MPa,
E
GPa and
ratio,
𝜈𝜈 =
ν
=
0.33.
0.33.
0.33.
Figure 4. Tetra-meshed of the 3D bell crank FE model.
Figure 4. Tetra-meshed of the 3D bell crank FE model.
Figure 4. Tetra-meshed of the 3D bell crank FE model.
3.4.2. Boundary Conditions
3.4.2.
Boundary Conditions
Conditions
3.4.2.Boundary
Boundary
conditions were set up to replicate the best testing scenario previously conducted using
Boundary
conditions
were
set
upsuspended
to replicate
replicate
the
best
testing
scenario
previously
conducted
a laser
vibrometer.
The bellwere
crankset
was
bythe
the best
stringtesting
to avoid
any external
forcesconducted
impeding
Boundary
conditions
up
to
scenario
previously
using
a
laser
vibrometer.
The
bell
crank
was
suspended
by
the
string
to
avoid
any
external
forces
using a laser vibrometer. The bell crank was suspended by the string to avoid any external forces
impeding
the
modal
shaker;
hence,
the
corresponding
boundary
conditions
were
imposed
on
the
FE
impeding the modal shaker; hence, the corresponding boundary conditions were imposed on the FE
model.
The
two
holes
at
either
end
of
the
FE
model
were
restricted
in
both
linear
and
rotational
X,
Y
model. The two holes at either end of the FE model were restricted in both linear and rotational X, Y
and Z
Z directions.
directions. The
The eigenvalues
eigenvalues of
of vibration
vibration and
and responses
responses to
to excitation
excitation for
for the
the FE
FE bell
bell crank
crank was
was
and
determine
using
Equation
(11)
through
the
Lanczos
iteration
method
in
Abaqus
software.
The
results
determine using Equation (11) through the Lanczos iteration method in Abaqus software. The results
obtained from the vibrometer testing were used for validation of the eigenvalue or natural frequency
Appl. Sci. 2020, 10, 8568
7 of 15
the modal shaker; hence, the corresponding boundary conditions were imposed on the FE model.
The two holes at either end of the FE model were restricted in both linear and rotational X, Y and Z
directions. The eigenvalues of vibration and responses to excitation for the FE bell crank was determine
using Equation (11) through the Lanczos iteration method in Abaqus software. The results obtained
from the vibrometer testing were used for validation of the eigenvalue or natural frequency of the 3D
bell crank FE model.
4. Bell Crank Optimization
4.1. Optimization Problem Statement
Appl. Sci. 2020, 10, xalgorithms
FOR PEER REVIEW
7 of 16
The optimization
are based on published work [18,19,29]. Topology optimization
for a
continuum4.structure
be regarded as a material distribution problem, where the target is to find an
Bell Crank can
Optimization
optimized material distribution within the design domain, for the minimum compliance (or maximum
1
4.1. Optimization
Problem
Statement (3). Since Sti f f ness ∝
global stiffness)
according
to Equation
Compliance , minimizing the compliance
will maximizeThe
theoptimization
stiffness. algorithms are based on published work [18,19,29]. Topology optimization for
a continuum
structure
be regarded
as a material into
distribution
where
the the
targetAbaqus
is to find Topology
In this
study, the
bell can
crank
was discretized
finiteproblem,
elements
and
an optimized material distribution within the design domain, for the minimum compliance (or
Optimization Module (ATOM) was used to search for a minimum compliance design to minimize
, minimizing the
maximum global stiffness) according to Equation (3). Since Stiffness ∝
strain energy or compliance, based on the boundary conditions and forces being applied to the bell
compliance will maximize the stiffness.
crank. The bell
wasthe
discretized
intodiscretized
finite elements.
Assuming
a constant
EeTopology
for each element,
In crank
this study,
bell crank was
into finite
elements and
the Abaqus
Module
(ATOM)
was usedform
to search
minimum compliance
design to minimize
and using Optimization
Equations (4)
to (7),
the discrete
can for
be aexpressed
as:
strain energy or compliance, based on the boundary conditions and forces being applied to the bell
crank. The bell crank
was discretized
elements.toAssuming
a {u}
constant
Ee for each element,
[K·E(xe )]
Minimize
: C =into
= {F}
{F}Tfinite
{u}Subject
and using Equations (4) to (7), the discrete form can be expressed as:
where E(xe ) ∈ Ead , K =
N
P
e=1
Minimize: 𝑪 = 𝑭
𝒖
)), summing
Ke (E(xe Subject
1, . . . , N elements. The objective function
for this
(1)
to [𝑲 ∙ 𝑬e𝐱=
𝐞 )] 𝒖 = 𝑭
study waswhere
to minimize
{u},
measure
of displacements
is the
𝑬 𝐱 𝐞 ) ∈ 𝐸 the
, 𝑲overall
= ∑𝑵
e = and
1, …, a
Nglobal
elements.
The objective
function for
𝒆 𝑬 𝐱 𝐞 )), summing
𝒆 𝟏 𝑲displacements
this study
was minimizing
to minimize thethe
overall
displacements
{u}, and a global
measure(9).
of displacements is
strain energy,
thereby
strain
energy according
to Equation
the strain energy, thereby minimizing the strain energy according to Equation (9).
{F}T represents
the forces applied, [K] is the global stiffness matrix, which depends on the stiffness
{F}T represents the forces applied, [K] is the global stiffness matrix, which depends on the
of the individual
elements
Ee , whereas
is a setEadofisadmissible
stiffness
tensors
forfordesign
stiffness of the individual
elements E
Ee,ad
whereas
a set of admissible
stiffness
tensors
design problems.
The entireproblems.
design optimization
process
is summarized
in Figurein 5.
The entire design
optimization
process is summarized
Figure 5.
Figure 5. The design optimization process.
Figure 5. The design optimization process.
4.2. Topology Optimization of the Bell Crank
For the topology optimization analysis, only two design domains of the bell crank were
considered to find an optimized material distribution. The regions were highlighted in red in Figure
Appl. Sci. 2020, 10, x FOR PEER REVIEW
8 of 16
6. The regions, which cannot be affected by the topology optimization, were located at the center
bearing
and10,the
bolt holes on either end of the bell crank.
Appl.
Sci. 2020,
8568
8 of 15
Appl. Sci. 2020, 10, x FOR PEER REVIEW
8 of 16
6. The regions, which cannot be affected by the topology optimization, were located at the center
bearing and the bolt holes on either end of the bell crank.
Figure6.6.Design
Designdomains
domainstopology
topologyoptimization
optimization(highlighted
(highlightedininred).
red).
Figure
4.2.
Optimization
4.3.Topology
Boundary
Conditions of the Bell Crank
For
the topology
optimization
analysis,
two design
of the bell
crank
were considered
During
the optimization
process,
the only
bell crank
modeldomains
was subjected
to two
different
boundary
toconditions.
find an optimized
material
distribution.
The
regions
were
highlighted
in
red
in
Figure
6.
The
regions,
The first was applied to restrain all degrees of freedom at the center bearing hole.
The
which
cannot
be
affected
by
the
topology
optimization,
were
located
at
the
center
bearing
and
bolt
second consists of point loads at four different locations. The loading forces were obtained the
from
the
Design
domains topology optimization (highlighted in red).
holes
on Motorsport
either endFigure
ofCatalogue
the 6.
bell
crank.
Eibach
2014 [30], based on the 100-60-0080 spring, which is featured on the
sidecar. A force value of 5299 N was provided as the block height of the spring and, therefore, will
4.3. Boundary Conditions
be considered as the worst-case loading scenario. Due to the pushrod and suspension orientations
During
the optimization
process,
the bell crank
model wasand
subjected
two different
within
the vehicle,
these forces
were divided
into horizontal
verticaltocomponents.
Aboundary
horizontal
conditions.
The
first
was
applied
to
restrain
all
degrees
of
freedom
at
the
center
bearing
hole.
The
second
conditions.
first awas
applied
degrees
of freedom
at the center
bearing
The
force of 460The
N and
vertical
forcetoofrestrain
2609 Nall
were
applied
as a concentrated
force
at eachhole.
reference
consists
of pointindicated
loads
atloads
four
different
locations.
The loading
forces were
obtained
from thefrom
Eibach
second
consists
of
point
at four
locations.
The loading
forces
were obtained
the
point location
in Figure
7. different
Motorsport
Catalogue
2014 [30],
based
the 100-60-0080
spring, which
featured
on the sidecar.
Eibach Motorsport
Catalogue
2014
[30],on
based
on the 100-60-0080
spring,iswhich
is featured
on the
A
force value
of 5299
N of
was
provided
the blockas
height
of theheight
spring of
and,
willtherefore,
be considered
sidecar.
A force
value
5299
N wasasprovided
the block
thetherefore,
spring and,
will
be the
considered
as loading
the worst-case
Dueand
to suspension
the pushrodorientations
and suspension
as
worst-case
scenario.loading
Due to scenario.
the pushrod
withinorientations
the vehicle,
withinforces
the vehicle,
these forces
were divided
into horizontal
and vertical
components.
horizontal
these
were divided
into horizontal
and vertical
components.
A horizontal
force ofA460
N and a
force offorce
460 N
vertical
force ofas2609
N were applied
a concentrated
forcelocation
at each indicated
reference
vertical
ofand
2609a N
were applied
a concentrated
force as
at each
reference point
point
location
in
Figure
7. indicated in Figure 7.
Figure 7. Applied boundary conditions for topology optimization.
5. Results and Discussion
5.1. Experimental Testing Natural Frequency
Values of natural frequency for the bell crank at the five tested locations are presented in Table
Figure 7. Applied boundary conditions for topology optimization.
1. Note that an average natural frequency of 5046.75 Hz was obtained for all tested locations.
The value
for the stiffness of the bell crank was 417.81 MN/m, which was solved by derivation
5. Results
and Discussion
5. Results and Discussion
of the natural frequency equation. The results obtained in the vibrometer testing were then used for
5.1.
Experimental
Frequency
validation
of theTesting
3D bellNatural
crank model
before the topology optimization process.
5.1. Experimental Testing Natural Frequency
Values of natural frequency for the bell crank at the five tested locations are presented in Table 1.
1. Vibrometer
Test
Results.
Values of natural frequency forTable
the bell
crank at the
five
tested locations are presented in Table
Note that an average natural frequency of 5046.75 Hz was obtained for all tested locations.
1. Note that an average natural frequency of 5046.75 Hz was obtained for all tested locations.
The value forLocation
the stiffness of
the Peak
bell crank
was (Hz)
417.81 MN/m,Standard
which was
solved by derivation
Mean
Frequency
Deviation
The value for the stiffness of the bell crank was 417.81 MN/m, which was solved by derivation
of the natural frequency
obtained in the vibrometer testing
were then used for
1 equation. The results
5061.25
1.30
of the natural frequency equation. The results obtained in the vibrometer testing were then used for
2 crank model before5036.25
0.98
validation of the 3D bell
the topology optimization process.
validation of the 3D bell
crank model before5045.00
the topology optimization process.
Laser vibrometer3measuring has high accuracy
without causing any 1.15
damage to the test object.
However, displacements of a freely suspended test object during the acquisition process is a common
Table 1. Vibrometer Test Results.
source of minor error in the method [25]. However, in the measurement setup for this study, the bell
Location
1
2
3
Mean Peak Frequency (Hz)
5061.25
5036.25
5045.00
Standard Deviation
1.30
0.98
1.15
Appl. Sci. 2020, 10, 8568
9 of 15
crank was constrained by strings through the two side holes to ensure stability and to avoid any
Appl. displacement
Sci. 2020, 10, x FOR
REVIEW
9 of 16
large
ofPEER
the laser
focal point during the data acquisition process. Hence, the properly
constrained bell crank has enhanced the repeatability of the test results.
4
5
5053.75
5037.50
Table 1. Vibrometer
Test Results.
1.04
0.64
Laser vibrometer
measuring
has
high
accuracy(Hz)
withoutStandard
causingDeviation
any damage to the test object.
Location
Mean
Peak
Frequency
However, displacements of a freely suspended test object during the acquisition process is a common
1
5061.25
1.30
source of minor error in
However, in the measurement
2 the method [25].5036.25
0.98 setup for this study, the bell
crank was constrained3by strings through
the two side holes to ensure
stability and to avoid any large
5045.00
1.15
4
1.04
displacement of the laser
focal point 5053.75
during the data acquisition
process. Hence, the properly
5
5037.50
0.64
constrained bell crank has enhanced the repeatability of the test results.
5.2.
5.2.Finite
FiniteElement
ElementModel
ModelValidation
ValidationofofNatural
NaturalFrequency
Frequency
The
Theaccuracy
accuracyofofthe
thecreated
createdFE
FEbell
bellcrank
crankmodel
modelwas
wasinvestigated
investigatedthrough
througha aproper
properprocess
processofof
validation
minimize
any modelling
errors,errors,
including
the element
discretization
errors (meshing),
validationto to
minimize
any modelling
including
the and
element
and discretization
errors
and
the inputand
errors
boundary
and loading
conditions
[31]. conditions
Table 2 represents
the results
obtained
(meshing),
thefrom
input
errors from
boundary
and loading
[31]. Table
2 represents
the
from
the obtained
natural frequency
of the FE bell
crankof
model.
from
FE model
results
from the analysis
natural frequency
analysis
the FEThe
bellmode
crankshapes
model.
Thethe
mode
shapes
were
to findwere
the one
that best
represented
a sinusoidal
wave asaobserved
in wave
the physical
test.
fromanalyzed
the FE model
analyzed
to find
the one that
best represented
sinusoidal
as observed
Itin
was
that,
during
the
vibrometer
testing,
first natural
frequency
was
foundfrequency
in the fourth
thefound
physical
test.
It was
found
that, during
thethe
vibrometer
testing,
the first
natural
was
mode
foundshape.
in the fourth mode shape.
Table
Table2.2.Finite
Finiteelement
elementbell
bellcrank
crankmodal
modalanalysis
analysisresults.
results.
Mode
Mode
Frequency
Frequency(Hz)
(Hz)
11
22
3
3
4
1714.5
1714.5
2560.7
2560.7
2864.2
2864.2
5285.7
4
5285.7
AAsimilar
was
found
in in
thethe
FEFE
bell
crank
model
(Figure
8), 8),
which
displayed
a
similarmode
modeshape
shaperesult
result
was
found
bell
crank
model
(Figure
which
displayed
sinusoidal
motion
at
the
fourth
mode
with
a
resonant
frequency
of
5285.7
Hz.
a sinusoidal motion at the fourth mode with a resonant frequency of 5285.7 Hz.
Figure
Figure8.8.Fourth
Fourthmode
modeshapes
shapesofofthe
thephysical
physicalbell
bellcrank
crankand
andfinite
finiteelement
elementmodel.
model.
When analyzing both experimental and FE results, the difference in the natural frequency at the
When analyzing both experimental and FE results, the difference in the natural frequency at the
fourth mode of each case is below 5%; hence, the FE bell crank model achieved a satisfactory validation
fourth mode of each case is below 5%; hence, the FE bell crank model achieved a satisfactory
against the vibrometer experimental test. The validated FE bell crank model was then used for further
validation against the vibrometer experimental test. The validated FE bell crank model was then used
topology
optimization
analysis.
for further
topology optimization
analysis.
5.3. Topology Optimization
5.3. Topology Optimization
A strain energy design response and a volume constraint were assigned in the optimization
A strain energy design response and a volume constraint were assigned in the optimization
module. To improve the stiffness of the bell crank, the strain energy design response was assigned as
module. To improve the stiffness of the bell crank, the strain energy design response was assigned as
the “minimizing strain energy”. The stiffness of the structure was determined by its displacement
corresponding to the assigned boundary conditions [32]. The volume constraint was assigned as it
directly related to the overall mass of the bell crank model. In the analysis, a range of volume
Appl. Sci. 2020, 10, 8568
10 of 15
the “minimizing strain energy”. The stiffness of the structure was determined by its displacement
corresponding to the assigned boundary conditions [32]. The volume constraint was assigned as
Appl. Sci. 2020, 10, x FOR PEER REVIEW
10 of 16
it directly related to the overall mass of the bell crank model. In the analysis, a range of volume
constraints,
was
assigned
to to
thethe
FEFE
model
to
constraints, i.e.,
i.e., 95%,
95%, 90%,
90%,85%,
85%,80%
80%and
and75%
75%ofofthe
theoriginal
originalvolume,
volume,
was
assigned
model
achieve
the
optimal
design.
to achieve the optimal design.
Figure
the the
progressive
removal
and grouping
of elements
the topology
optimization
Figure 99shows
shows
progressive
removal
and grouping
of from
elements
from the
topology
analysis.
At
the
volume
constraint
of
95%,
the
top
surface
and
the
two
opposing
arm
locations
of
optimization analysis. At the volume constraint of 95%, the top surface and the two opposing
arm
the
bell crank
green contour)
demonstrated
a potentialafor
element
the
locations
of the(indicated
bell crankby
(indicated
by green
contour) demonstrated
potential
forremoval.
element Once
removal.
volume
constraint
was
reduced
to
90%,
a
significant
material
reduction
was
observed
on
the
left
arm
Once the volume constraint was reduced to 90%, a significant material reduction was observed on
region.
Fromregion.
the 85%
to 80%
constraint
results,
significant
material on
arm
the left arm
From
the volume
85% to 80%
volume
constraint
results,reductions
significantofreductions
ofboth
material
regions,
well
as on the
thethe
topedges
of the at
bell
crank,
were
The
75%achieved.
volume constraint
on both as
arm
regions,
as edges
well asaton
the
top of
theachieved.
bell crank,
were
The 75%
in
the
topology
optimization
featured
the
highest
amount
of
material
removal.
volume constraint in the topology optimization featured the highest amount of material removal.
Figure
volume constraints,
constraints, side
side view.
view.
Figure 9.
9. Topology
Topology optimization
optimization results
results with
with associated
associated volume
5.4.
Design Validation
Validation
5.4. Design
5.4.1.
Geometry Interpretation
Interpretation and
and FEA
5.4.1. Geometry
FEA
Topology
often
produces
complex
contours
in the geometry,
which may
be difficult
Topologyoptimization
optimization
often
produces
complex
contours
in the geometry,
which
may for
be
manufacturing
and/or
machining
[8,33].
The
results
obtained
from
the
iteration
process
of
the
topology
difficult for manufacturing and/or machining [8,33]. The results obtained from the iteration process
optimization
were
used as guide
geometries
to determine
where
material should
withdrawn
from
of the topology
optimization
were
used as guide
geometries
to determine
wherebe
material
should
be
the
design
domain.
The
guide
geometries
were
interpreted
toward
the
creation
of
optimal
designs
withdrawn from the design domain. The guide geometries were interpreted toward the creation of
with
suitable
shapes
features
for both
practical
purposes.
The guide
geometries
optimal
designs
withand
suitable
shapes
and machining
features forand
both
machining
and practical
purposes.
The
were
exported
from
Abaqus
into
CATIA
(V5R21),
where
their
shapes
and
features
were
traced
in the
guide geometries were exported from Abaqus into CATIA (V5R21), where their shapes and features
CAD
environment.
The optimized
CAD
design
for theCAD
volume
constraint
95%, 90%,
85%,
were traced
in the CAD
environment.
The
optimized
design
for themodels,
volume i.e.,
constraint
models,
80%,
and
75%,
were
then
created.
i.e., 95%, 90%, 85%, 80%, and 75%, were then created.
To
To overcome
overcome unrealistic
unrealistic contour
contour issues,
issues, the
the geometry
geometry was
was simplified
simplified to
to produce
produce aa practical
practical
optimized
part.
Sharp
edges
obtained
through
the
topology
optimization
iterations
optimized part. Sharp edges obtained through the topology optimization iterations were
were rounded
rounded
with
were
smoothed
out,
due
to
with fillets,
fillets, with
withaaminimum
minimumradius
radiusofof33mm
mmfor
forease
easeofofmachining.
machining.Profiles
Profiles
were
smoothed
out,
due
the
checkerboard
contours
resulting
from
thethe
removal
of elements.
An An
example
of the
CAD
design
for
to the
checkerboard
contours
resulting
from
removal
of elements.
example
of the
CAD
design
the
80%
volume
constraint
model
is
shown
in
Figure
10.
for the 80% volume constraint model is shown in Figure 10.
Re-analysis was carried out to assess the efficiency and structural integrity of the optimized bell
crank design. The same ‘static-general’ and frequency-finite element tests were conducted, as examined
on the original model, with identical set-ups of boundary and loading conditions. The maximum von
Mises, stiffness, mass and maximum principal strain for the optimal design produced in the topology
optimization with different volume constraints are shown in Figure 11.
Appl. Sci. 2020, 10, x FOR PEER REVIEW
11 of 16
Appl. Sci. 2020, 10, 8568
Appl. Sci. 2020, 10, x FOR PEER REVIEW
11 of 15
11 of 16
Figure 10. Revised CAD model for the 80% volume constraint. (Left) Optimization model, (Right) the
CAD design free of irregular and complex shapes.
Re-analysis was carried out to assess the efficiency and structural integrity of the optimized bell
crank design. The same ‘static-general’ and frequency-finite element tests were conducted, as
examined
onRevised
the original
model,
with
identical
of (Left)
boundary
and loading
conditions.
Figure
10.
model
for
the80%
80%
volume set-ups
constraint.
Optimization
model,model,
(Right) (Right)
the Thethe
Figure
10. Revised
CADCAD
model
for
the
volume
constraint.
(Left)
Optimization
maximum
von
Mises,
stiffness,
mass
and
maximum
principal
strain
for
the
optimal
design
produced
CAD design free of irregular and complex shapes.
CAD
free ofoptimization
irregular and
complex
shapes.
in design
the topology
with
different
volume constraints are shown in Figure 11.
Re-analysis was carried out to assess the efficiency and structural integrity of the optimized bell
crank design. The same ‘static-general’ and frequency-finite element tests were conducted, as
examined on the original model, with identical set-ups of boundary and loading conditions. The
maximum von Mises, stiffness, mass and maximum principal strain for the optimal design produced
in the topology optimization with different volume constraints are shown in Figure 11.
Figure Figure
11. Summary
of the
structural
of the
theoptimized
optimized
crank
design.
11. Summary
of the
structuralintegrity
integrity of
bellbell
crank
CADCAD
design.
The optimization
results
show
that
thethe90%
constraint
model
produced
of
The optimization
results
show
that
90%volume
volume constraint
model
produced
higherhigher
values values
of
a corresponding
natural
frequency of
of 5334
5334 Hz.
also
featured
a 16.5%
reduction
in
stiffness,stiffness,
with a with
corresponding
natural
frequency
Hz.It It
also
featured
a 16.5%
reduction
in
maximum
von Mises
value,
a 4.3%
reduction in
in maximum
principal
strains,
and aand
decrease
in
maximum
von Mises
stressstress
value,
a 4.3%
reduction
maximum
principal
strains,
a decrease
in
weight byFigure
10 g from
the
original
bell
crank.
The
finite
element
result
indicated
that
the
highest
stress
11.
Summary
of
the
structural
integrity
of
the
optimized
bell
crank
CAD
design.
weight by 10 g from the original bell crank. The finite element result indicated that the highest stress
concentration in the optimized model was 199.76 MPa. Based on the part that was machined out of
concentration in the optimized model was 199.76 MPa. Based on the part that was machined out of
The optimization results show that the 90% volume constraint model produced higher values of
Al6082-T6,
the corresponding
yieldnatural
stressfrequency
value was
indicated
a safety
factor
stiffness,
with a corresponding
of 310
5334 MPa,
Hz. It which
also featured
a 16.5%
reduction
in of 1.55
for the maximum
optimized
bell
crank.
the optimization
iteration
processes,
the 90%
von
Mises
stressHence,
value, athrough
4.3% reduction
in maximum principal
strains,
and a decrease
in volume
weight
by 10was
g from
the original
The finite
element result indicated that the highest stress
constraint
model
considered
asbell
thecrank.
optimal
design.
concentration
in thethat
optimized
model
was 199.76
MPa. Basedofonconverting
the part thatthe
wasoptimized
machined out
It is
worth noting
the main
technical
challenges
FEofanalysis
topological results in CAD design for further analysis and manufacturing. As highlighted previously by
other investigators [29,33] and the current study, the initial topology optimization results still required
manual process to eliminate holes, shape, irregular and complex edges, for example, see Figure 10.
The topology optimization results were manually reconstructed to smoothen the shape and complex
edges and to fill the missing pores to improve its manufacturability to the final design before the
fabrication commencing.
by other investigators [29,33] and the current study, the initial topology optimization results still
required manual process to eliminate holes, shape, irregular and complex edges, for example, see
Figure 10. The topology optimization results were manually reconstructed to smoothen the shape
and complex edges and to fill the missing pores to improve its manufacturability to the final design
Appl.
Sci.the
2020,
10, 8568
12 of 15
before
fabrication
commencing.
5.4.2. Bell Crank Prototype Fabrication
5.4.2. Bell Crank Prototype Fabrication
After validating the FEA model, the prototype of the bell crank was fabricated for the next stage
After validating
the FEA
model, the
of the5-Axis
bell crank
was fabricated
the next(Okuma,
stage of
of physical
testing and
validation.
A prototype
MU500VII-L
Vertical
MachiningforStation
physical
testing
and
validation.
A
MU500VII-L
5-Axis
Vertical
Machining
Station
(Okuma,
Charlotte,
Charlotte, NC, USA) was used to machine the prototype. Due to cost limitations and the availability
NC,
USA)
was used
to machine
thecrank
prototype.
Due to cost
and the availability
of the
of the
material,
the optimized
bell
was machined
out limitations
of Al6061 aluminum.
The mechanical
material,
the
optimized
bell
crank
was
machined
out
of
Al6061
aluminum.
The
mechanical
properties
properties of Al6082-T6 and Al6061 are very similar, and the density, modulus of elasticity and
of
Al6082-T6
andallAl6061
arethe
very
similar,
the density,
modulus
of elasticity
andstrength
Poisson’sfrom
ratio260
all
Poisson’s
ratio
remain
same.
The and
differences
were
in reductions
in yield
2
2
remain
same.
The2 differences
in reductions
yield2 to
strength
from2.260
toin
240
N/mm
N/mm2the
to 240
N/mm
and tensilewere
strength
from 310 in
N/mm
260 N/mm
TheN/mm
changes
yield
and
2
2
and
tensile
strength
310 N/mm
to of
260the
N/mm
. The changes
in yield
tensile did
strength
did not
tensile
strength
did from
not alter
the result
eigenvalue
simulation
and,and
therefore,
not alter
the
alter
the
result
of
the
eigenvalue
simulation
and,
therefore,
did
not
alter
the
validation
result
from
the
validation result from the laser vibrometer testing. The machined version of the optimized bell crank
laser
vibrometer
testing.
is illustrated
in Figure
12.The machined version of the optimized bell crank is illustrated in Figure 12.
Figure 12.
12. Prototype
Prototype of
of optimal
optimal design
design and
and CAD
CAD comparison.
comparison.
Figure
5.4.3. Experimental Validation
Validation of
of Prototype
Prototype
A
of of
thethe
optimized
bellbell
crank
was was
carried
out using
the same
A final
finalvalidation
validationafter
aftermanufacturing
manufacturing
optimized
crank
carried
out using
the
laser
vibrometer
test.
The
original
part
was
tested
again,
as
well
as
the
optimized
part,
to
confirm
same laser vibrometer test. The original part was tested again,
well as the optimized part, to
that
the optimization
of the bellofcrank
hascrank
been successful
and the same
conditions
had been
confirm
that the optimization
the bell
has been successful
and testing
the same
testing conditions
satisfied.
3 shows
naturalthe
frequency
of the original
bell
crank compared
the optimized
had been Table
satisfied.
Tablethe
3 shows
natural frequency
of the
original
bell crank with
compared
with the
prototype.
Overall,
the
optimized
prototype
showed
an
increase
in
natural
frequency
of
1.5%
compared
optimized prototype. Overall, the optimized prototype showed an increase in natural frequency of
with
original bell
Since thebell
difference
within
5% limit,
the within
optimization
task
and the
1.5%the
compared
withcrank.
the original
crank. was
Since
the the
difference
was
the 5%
limit,
machining
cantask
be deemed
successful. can be deemed successful.
optimization
and the machining
Table 3.
Table
3. Summary of laser Vibrometer
Vibrometer testing
testing of
of the
the original
original bell
bell crank
crank and
and optimized
optimized prototype.
prototype.
OriginalOriginal
Bell Crank,
Bell Crank,
Location
Standard Standard
Deviation
Location Frequency (Hz)
Frequency (Hz)
Deviation
Location
1
Location
1
Location 2
Location 2
Location 3
Location
3
Location
4
Location
5
Location 4
5055.5
5037.5
5056.9
5087.8
5029.6
Average
Location 5
Optimized
Prototype,
Optimized
Prototype, Standard
Standard
Deviation
Frequency
(Hz)
Frequency
(Hz)
Deviation
5087.8
1.45 1.45
0.70
0.70
1.48
1.15 1.48
1.41 1.15
5126.6
5126.6
5133.7
5133.7
5119.1
5119.1
5127.0
5142.4
5127.0
0.70
0.70
1.06
1.06
0.58
0.58
1.40
1.45
1.40
5053.3 5029.6
1.41
5129.8
5142.4
1.45
5055.5
5037.5
5056.9
The final mass of the optimized bell crank design achieved a weight saving of 2.6% (Table 4).
Achieving a reduction in mass of the bell crank not only benefits the overall weight of the sidecar, but also
reduces the unsprung mass of the suspension system and, therefore, improves the spring/damper
unit efficiency [12]. With a careful design interpretation, the optimized design was able to reduce the
density of elements within the design domain, while improving the overall stiffness of the bell crank.
Moreover, the optimized design achieved a higher value of natural frequency and lower von
Mises stress value under the same static load conditions compared with the original part. The resulting
topologically optimized bell crank was mounted in the LCF sidecar as shown in Figure 13.
The final mass of the optimized bell crank design achieved a weight saving of 2.6% (Table 4).
Achieving a reduction in mass of the bell crank not only benefits the overall weight of the sidecar,
but also reduces the unsprung mass of the suspension system and, therefore, improves the
spring/damper unit efficiency [12]. With a careful design interpretation, the optimized design was
able to reduce the density of elements within the design domain, while improving the overall stiffness
Appl. Sci. 2020, 10, 8568
13 of 15
of the bell crank.
Table 4. Comparison of original bell crank and optimized prototype.
Table 4. Comparison of original bell crank and optimized prototype.
Parameters
Original Part
Natural Frequency (Hz)Original
5053.3
Parameters
Part
Max von Mises (MPa)
Natural Frequency (Hz)
Stiffness (k) (MPa)
Max von Mises (MPa)
Mass (kg)
Stiffness (k) (MPa)
Max strain (-)
Mass (kg)
239.3
5053.3
452.09
239.3
0.416
452.09
3.04E-03
0.416
Prototype
% of Change
5129.8
Prototype
199.8
5129.8
455
199.8
0.405
455
2.91E-03
0.405
+1.5
% of Change
−16.5
+1.5
+0.6
−16.5
−2.6
+0.6
−4.3
−2.6
−3
−3
Moreover,Max
the optimized
a higher
value
natural
frequency
strain (-) design achieved
−4.3and lower von
3.04 × 10
2.91 of
× 10
Mises stress value under the same static load conditions compared with the original part. The
resulting topologically optimized bell crank was mounted in the LCF sidecar as shown in Figure 13.
Figure 13. Optimized bell crank in the LCF sidecar.
Figure 13. Optimized bell crank in the LCF sidecar.
The modern LCR F1 Sidecar weight is approximately 225 kg, including the engine weight
The
modern LCR F1 Sidecar weight is approximately 225 kg, including the engine weight
(approximately 50 to 55kg for current engine specifications). Considering an approximation of 30%
(approximately
55 kg for
current engine
specifications).
Considering
approximation
to 40% for50
thetosidecar’s
components
(e.g., brake
bell, steering, wheel
hub and an
spindle,
brake rotor of 30%
to 40% etc.)
for the
components
(e.g., brake
bell,methodology
steering, wheel
hub and spindle,
brake rotor
andsidecar’s
further work
using the currently
proposed
that incorporated
the topology
optimization
approach,
it was
possible
that a 5 to
10kg drop in net
weight
could be achieved.
etc.) and
further work
using
thedeemed
currently
proposed
methodology
that
incorporated
the topology
The methodology
described
in this possible
paper canthat
be used
and
aerospace
optimization
approach, it
was deemed
a 5 toin10mechanical,
kg drop inautomotive
net weight
could
be achieved.
industries for weight reduction design optimization.
The methodology described in this paper can be used in mechanical, automotive and aerospace
In producing a lightweight product, the present study also contributed to enhancing knowledge
industries
for reverse
weightengineering
reductionand
design
optimization.
of how
non-destructive
testing methods can be used effectively for structural
In topology
producing
a lightweight
product,
present
study
also contributed
to enhancing
knowledge
optimization
to solve
complexthe
designs
without
compromising
performance
and integrity.
The engineering
emerging of additive
manufacturing, especially
the 3D printing
metallic
materials,
of how reverse
and non-destructive
testing methods
can beof
used
effectively
forwill
structural
enable
the possibilities
to further
optimize
and without
manufacture
bell crank with
lattice structures
topology
optimization
to solve
complex
designs
compromising
performance
and[34],
integrity.
and hence, further work could be carried out for topological optimization of complex design, such as
The
emerging of additive manufacturing, especially the 3D printing of metallic materials,
the bell crank and other mechanical components, as miniature lattice structures, which leads to even
will enable the possibilities to further optimize and manufacture bell crank with lattice structures [34],
higher weight reduction.
and hence, further work could be carried out for topological optimization of complex design, such as
the bell crank and other mechanical components, as miniature lattice structures, which leads to even
higher weight reduction.
6. Conclusions
In this paper, a design optimization methodology is described that incorporated: (1) a non-destructive
measuring technique using a laser vibrometer; (2) a reverse engineering technique for 3D model
construction; and (3) a structural topology optimization. The design optimization methodology was
applied to a sidecar suspension bell crank to minimize its structural weight and then subjected to
constraints including volume, strain energy and von Mises stress.
A 3D bell crank model was created using a reverse engineering technique and the corresponding
FE bell crank model was validated using laser vibrometer test results. The difference between the
experimental and FE results of the natural frequency at the fourth mode was less than 5%, which was
deemed to be valid for further analysis.
Appl. Sci. 2020, 10, 8568
14 of 15
At the beginning of the optimization process, the design domain for material removal was selected.
Using the topology optimization approach, the geometry designs produced by the optimization
algorithm were transformed into CAD models for the smoothing of the geometry to eliminate holes,
irregular shapes and complex edges before fabricating the prototype for further testing. The results
obtained from the design optimization methodology demonstrated its applicability and capability
to produce an optimized bell crank, which increased its overall stiffness and natural frequency,
while reducing its weight by 3%, maximum principal strain by 4.3% and maximum von Mises stress
by 16.5%. This study demonstrated how reverse engineering and non-destructive testing methods
can be used effectively for structural topology optimization to produce a suitable lightweight product.
This is significant contribution to finding solution to complex designs without compromising structural
performance and integrity.
Further work using the current methodology could be implemented in other sidecar components
(e.g., brake bell, steering, wheel hub and spindle, brake rotor etc.), and additive manufacturing using
3D printing of lattice structures could achieve further net weight reduction.
Author Contributions: T.Y.P.: Conceptualization, methodology, software, writing—original draft preparation,
M.F.: Methodology, writing, review and editing. All authors have read and agreed to the published version of
the manuscript.
Funding: This research received no external funding.
Acknowledgments: The authors would like to acknowledge and give many thanks to Luka Grubic and Jamie
Crass for their assistance on the project. The authors also acknowledged the contribution of the Laboratory
Technician at RMIT Bundoora East Campus for their ongoing help and support in this project.
Conflicts of Interest: The authors declare no conflict of interest.
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