UNIVERSITY OF CALIFORNIA, SAN DIEGO
Weight Reduction Techniques Applied to Formula SAE Vehicle Design:
An Investigation in Topology Optimization
A thesis submitted in partial satisfaction of the
requirements for the degree Master of Science
in
Engineering Sciences (Mechanical Engineering)
by
Lucas V. Fornace
Committee in Charge:
Professor Frank E. Talke, Chair
Professor David J. Benson
Professor Hidenori Murakami
2006
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Copyright
Lucas V. Fornace, 2006
All rights reserved
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The thesis of Lucas V. Fornace is approved:
_______________________________________
_______________________________________
_______________________________________
Chair
University of California, San Diego
2006
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This work is dedicated to my younger sister and recent UCSD admit,
Gabrielle Angeline Fornace
May she continue along and improve upon the path set forth by her big brother, with
confidence and determination aplenty.
In addition, I would like to dedicate the completion of this thesis, and thus the
conclusion of my academic career to
Carrie Rose Martin
A dear friend whose companionship surely propelled my own drive and confidence,
and to whom I credit most of my successes. For this, I am forever grateful.
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“Any car which holds together for a whole race is too heavy.”
Colin Chapman, Founder, Lotus Engineering Co.
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TABLE OF CONTENTS
Signature Page……………………………………………………………………..….iii
Dedication…………………………………………………………………………......iv
Epigraph………………………………………………………………………………..v
Table of Contents…………………………………………………………………..….vi
List of Figures and Tables………..………………………………………………….viii
Acknowledgements………………………………………………………………….....x
Abstract………………………………………………………………………………..xi
I. Introduction…………………………………………………………………………..1
A. Formula SAE Competition……………………………...………………………..1
B. Recognition of Need for Reduced Vehicle Mass…...…...……………………….3
C. Concepts of Topology Optimization……………………...……………………...5
D. Problem Definition………………………………………...……………………..7
II. Pre-processing……………………………………………………………………..12
A. Load Prediction……………………………………………...………………….12
1. Multi-body Dynamics……………………………………..………………….12
2. Load Cases…………………………………………………..………………..14
B. Design Space…………………………………………………...……………….16
III. Topology Optimization…………………………………………………………...17
A. Design Space Mesh……………………………………………………………..17
1. Geometry Clean-up…………………………………………………………...17
2. Tetra-Meshing………………………………………………………………...17
B. Boundary Conditions……………………………………………………………19
1. Treatment of Bolt Holes………………………………………………………19
2. Constraints………………..…………………………………………………...19
3. Loads…………………………..……………………………………………...21
C. Optimization Statement…………………………………………………………23
1. Optimization Responses………………………………………………………23
2. Material Definition……………………………………………………………25
3. Manufacturing Constraints……………………………………………………25
IV. Post Processing…………………………………………………………………...27
A. Topology Optimization Results………………………………………………...27
B. Geometry Interpretation………………………………………………………...30
V. Design Validation………………………………………………………………….32
A. Finite Element Analysis………………………………………………………...32
B. Bell Crank Prototype Fabrication……..…………………………...…….……...33
C. Physical Testing………………………………………………………………...35
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1. Hydraulic Test Fixture Design………………………………………………..35
2. Experimental Results…………………………………………………………40
VI. Conclusion………………………………………………………………………..42
A. Discussion of Experimental Results………………………………….………...42
B. Overall Vehicle Performance and Awards…………..…………………………44
Bibliography…………………………………………………………………………..47
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LIST OF FIGURES AND TABLES
Figure 1: Diagram of a typical pushrod and bell crank suspension……………………9
Figure 2: Bell crank as installed on 2005 vehicle…………..………………………...11
Figure 3: Close-up view of bell crank as installed on 2005 vehicle………………….11
Figure 4: MBD model of rear suspension………………………………………….…13
Figure 5: Bell crank load case in left vehicle roll condition………………………….15
Figure 6: Bell crank load case in right vehicle roll condition………………………...15
Figure 7: Design space and spring/damper system.…………………………………..16
Figure 8: Geometry cleaning in preparation for meshing...…………………………..18
Figure 9: Tetra-meshed design space…………..…..…………………………………18
Table 1: Summary of bell crank constraints……………………………..…………..21
Table 2: Summary of bell crank load cases…………………………..….…………..22
Figure 10: Applied boundary conditions………………..…………………………...22
Figure 11: 2005 bell crank finite element analysis displacement results….…………24
Figure 12: Topology optimization results, side view………………………………...27
Figure 13: Topology optimization results, isometric view…………………………..28
Figure 14: Topology optimization results with draw constraint, isometric view……29
Figure 15: Optimized geometry interpretation, isometric view……………………...31
Figure 16: von-Mises stress distribution and exaggerated displacement.……………33
Figure 17: MasterCAM X tool paths………………………………………………...34
Figure 18: Bell crank prototype fabrication………………………………………….34
Figure 19: Bell crank prototype finished product……………………………………35
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Figure 20: Hydraulic test fixture CAD assembly…………………………………….37
Figure 21: Hydraulic test fixture prepared for experiment…………………………..39
Table 3: Summary of experimental test results………………………………………41
Figure 22: Physical testing of 2006 optimized bell crank…………………………...41
Table 4: Comparison of 2005 and 2006 UCSD FSAE vehicle performance………...45
Figure 23: 2006 UCSD FSAE vehicle at competition………….……….…………...45
Figure 24: Close-up of bell crank on 2006 vehicle…………………………………..46
Figure 25: 2006 vehicle in motion…………………………………………………...46
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ACKNOWLEDGEMENTS
I would like to thank Professor Frank E. Talke for advising this work and for
allowing me the freedom to pursue this optimization project.
His advice and
leadership was plentiful and much appreciated.
I would also like to acknowledge Terence Smith of Altair Engineering for his
relentless support of the relationship with UCSD and UCSD Formula SAE.
Furthermore, a thank you is in order to my colleagues in the CMRR for their
support; namely, Maik, Bart, Aravind, D.E., Ralf, John, Paul, Max, Mathias and
Thorsten. Thank you all for your suggestions and friendship.
Robert Shanahan deserves a huge thank you for his unparalleled dedication to
the Formula SAE team at UC San Diego as community advisor, as does Professor
Keiko K. Nomura for her contributions as Faculty Advisor. The team would not be
able to operate at the current level without their tremendous efforts.
Furthermore, I would like to acknowledge Billy Wight for his skilled work in
fabricating the prototype bell cranks for this project, not to mention just about every
other component on the 2004, 2005 and 2006 vehicles. In addition, I want to thank
Tom Chalfant and Dave Lischer in the MAE machine shop for their continued support
of the Formula SAE team at UCSD.
Lastly I want to thank my family and friends for putting up with my hectic
schedule, trusting my decisions, and supporting me along the way—a sincere thank
you to everybody who contributed.
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ABSTRACT OF THE THESIS
Weight Reduction Techniques Applied to Formula SAE Vehicle Design:
An Investigation in Topology Optimization
by
Lucas V. Fornace
Master of Science in Engineering Sciences (Mechanical Engineering)
University of California, San Diego, 2006
Professor Frank E. Talke, Chair
In the quest for reduced vehicle mass without sacrificed integrity, Computer
Aided Engineering (CAE) topology optimization software was investigated and
utilized in the design of the 2006 UC San Diego Formula SAE vehicle as a means to
determine the optimum material distribution within a component for a given set of
loading and boundary conditions. This paper looks at the design of a rear suspension
bell crank component using modern topology optimization techniques and compares
the end product to that of the 2005 model bell crank component, which was designed
using more traditional techniques.
A hydraulic load cell system was created to
simulate the vehicle suspension forces and was used to physically test the original and
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optimized parts to failure.
Through the use of Altair OptiStruct® topology
optimization software, a weight savings of 24.3% coupled with an increase in yield
strength of 29.7% was realized in the optimized design of the 2006 bell crank.
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I. Introduction
A. Formula SAE Competition
Formula SAE is an international design competition held annually by the
Society of Automotive Engineers, and is described by the organization as follows:
The Formula SAE® competition is for SAE student members to conceive,
design, fabricate, and compete with small formula-style racing cars. The
restrictions on the car frame and engine are limited so that the knowledge,
creativity, and imagination of the students are challenged. The cars are built
with a team effort over a period of about one year and are taken to the annual
competition for judging and comparison with approximately 120 other
vehicles from colleges and universities throughout the world. The end result is
a great experience for young engineers in a meaningful engineering project as
well as the opportunity of working in a dedicated team effort [1].
This competition first began in Texas in the year 1981 with only four vehicle
entries. The event has since grown and moved to Detroit, Michigan, with a
consistently filled capacity of 140 vehicles that arrive annually to compete in the four
day affair.
Over the years, the competition has seen a tremendous growth in
popularity. To satisfy the demand of the students, similar competitions have since
been created in Great Britain, Australia, Brazil, Italy and California. Even with the
addition of the new events, registration of the 140 available slots for the 2006 Detroit
competition filled in just 34 minutes. Needless to say, the event has grown quite
competitive.
The scoring of the competition is broken down into eight distinct events: the
first three of which are static, leaving the final five categories for dynamic
performance evaluation.
Before any scoring begins, however, the student built
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vehicles must first pass a thorough technical inspection which includes the vehicle
being laterally tilted to 61 degrees.
This angle simulates a lateral acceleration
equivalent to 1.5 g’s to ensure that the car does not show any tendency to roll over or
leak fuel during hard cornering. Next the cars must demonstrate adequate braking
performance by being able to lock all four tires while driving at a moderate velocity.
Afterwards, the vehicle must pass a noise inspection to ensure compliance with the
sound level regulations. Fulfillment of this exercise marks the end of the technical
inspection.
Static scoring events include a marketing presentation, a cost feasibility
analysis, and most importantly, a design presentation in which the overall vehicle is
scrutinized by experienced automotive engineers. It is here that the students have a
chance to explain and defend their design rationale and field questions from the
various judges.
The dynamic scoring events include an acceleration event, a skid-pad event, an
autocross event, and an endurance event that is coupled with a fuel economy
measurement.
The acceleration event is a standing 75 meter sprint that is meant to
characterize the vehicle’s straight-line acceleration capabilities, which is a measure of
the vehicle’s inherent power-to-weight ratio and traction capacity.
To measure the car’s lateral acceleration abilities, a “skid-pad” circular path is
used in which the cars are in a constant radius turn at essentially steady state
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conditions. This is a measure of the team’s ability to design and tune the vehicle
suspension for maximum lateral balance and traction.
The Autocross event is a timed driving event that takes place on a temporary
track typically laid out with pylons on a large paved surface in the formation of very
tight and technically challenging course. These autocross-style courses effectively
measure the overall performance package of the vehicle, including braking,
acceleration and cornering.
The largest and most heavily weighted of the contests is the endurance and fuel
economy event.
Here the vehicles compete on a larger autocross-style course
approximately one kilometer in length. Twenty laps mark the duration of the event,
with a timed driver change on the 11th lap. In addition, the fuel level is measured, both
before and after the event, to obtain the effective fuel economy of the vehicle—further
adding to the challenge.
B. Recognition of Need for Reduced Vehicle Mass
Typical in high performance automotive and aerospace applications is the
demand for reduced vehicle mass while maintaining adequate performance and safety.
Formula SAE competition is no exception. The nature of the autocross style course
favors vehicles with good acceleration capabilities, and both the fuel economy and
acceleration events add to the desire for a lightweight vehicle design.
To increase the acceleration potential of a vehicle, or racecar in this case,
physics tells us that the traction force needs to be increased and/or the mass decreased
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[2]. In Formula SAE competition, safety mandates require all of the engine’s air
supply to pass through a 20 mm diameter restrictor in an effort to limit the intake
volumetric flow rate. This puts a ceiling on the engine’s power capability, thus
reduction in vehicle mass becomes increasingly important, if not the only option in the
drive for improved acceleration performance. Furthermore, FSAE competition has no
minimum weight requirement, so the weight reduction benefits essentially have no
bound. Even in classes of racing where minimum weight mandates do exist, having
an underweight vehicle entry allows the engineer to put ballast in strategic locations,
such as down low on the floor pan or to a targeted location in an effort to aide or
correct vehicle handling and balance.
The aerospace world has an even greater desire for weight reduction, as any
given reduction in mass can be directly correlated to increased fuel savings and
increased vehicle range, as well as improved performance and agility [3]. Overall, as
long as the reduced mass does not pose any threats to the integrity of the design—be it
aircraft or racecar—there are rarely any drawbacks associated with decreased vehicle
mass.
For the 2006 UC San Diego Formula SAE team, the drive for weight reduction
stemmed from the 2005 competition, where the majority of the top-placing vehicles
were below 227 kg (500 lbs) in weight. The 2005 UC San Diego entry was slightly
heavy at 246 kg (543 lbs), thus it was made the primary goal to achieve a 2006 design
that was less than 227 kg (500 lbs). This would represent a weight loss of roughly
10% of the entire vehicle.
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C. Concepts of Topology Optimization
Structural design problems require the fulfillment of specific objectives while
satisfying a set of performance constraints [4]. In the automotive and aerospace
worlds, this traditionally requires numerous iterations using finite element analysis
results that drive design changes, and the final design is often arrived at via a trial-anderror approach [5]. This iteration process can prove to be a very time-consuming and
thus costly endeavor. While this technique has obviously been effective in meeting
design requirements, the final solution does not necessarily represent the best
solution—only one that has successfully met the objective. Furthermore, the quality
of the final design relies heavily on the quality and potential of the initial design
attempts. This is due to the fact that, as the design progresses, the freedom to make
significant changes diminishes over time.
Therefore, in the interest of time and
quality, it is very important to have a good initial design solution early in the process.
Another drawback of the traditional design method is the fact that engineers
tend to think intuitively. Sometimes the optimum solution can be quite counterintuitive, and thus a great solution can go justifiably overlooked because it does not
seem plausible or reasonable.
On the contrary, a design can include an overly
complicated network of reinforcing ribs, for example, that were deemed necessary by
the design engineer, when the ideal optimized solution is actually much simpler [6].
Topology optimization is a relatively new numerical method used to determine
the optimum shape and distribution of material within a given design space for a given
set of design constraints based on responses obtained from a finite element analysis
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[7]. For structures under static loading, the basic finite element equation that is solved
can be expressed as
Kd=f
(1.1)
where K is the effective stiffness matrix of the structure, d is the displacement vector,
and f is the load vector as applied to the structure. This equation is often referred to as
the matrix equivalent form of the Bubnov-Galerkin equation, and it represents
equilibrium of external and internal forces. Once the unknown nodal displacements
are solved for, Hooke’s law can be used to calculate the material stresses for
deformations in the elastic range. Hooke’s Law can be stated as
σ=Cε
(1.2)
where σ is the stress vector, C is the material elasticity matrix, and ε is the strain
vector. Interested readers should consult [8] for further discussion of the linear static
finite element method.
The OptiStruct® topology optimization algorithm solves a structural
optimization problem in which the goal is to minimize an objective function subject to
constraint functions comprised of finite element responses [7]. Formally, the problem
can be written as follows:
Objective:
Minimize W(x)
(1.3)
Constraints:
g(x)-gupper ≤ 0
(1.4)
Design Variables:
xlower ≤ x ≤ xupper
(1.5)
where the objective function W and the constraint functions g are structural responses.
Typical structural responses used to define the objective and constraint functions
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include mass, volume fraction, compliance, frequency and displacement. Commonly,
the objective for a topology optimization study is to minimize the mass subject to
either nodal displacement or eigenfrequency constraints, with element density as the
design variable. By allowing the normalized element density to vary between 0 and 1,
the end result of the topology optimization should be a mesh in which the elements
take on a density value of either 0 or 1. A low density value represents a void, and a
high density value indicates solid material. By masking elements of low density, the
shape of the optimized structure is revealed. Overall, the optimization algorithm
involves discretizing the design space into a finite element mesh, calculating the
elemental material properties, and iteratively altering the material distribution
(element density) and re-calculating until convergence is reached at a solution that
best meets the objective function.
This technique can be used very early in the design stage to ensure that the
final design of the structure not only meets the requirements, but represents the
mathematically best solution based on the design constraints. Today the method is
widely accepted for bracket-type structures and has already proven to be a huge
benefit to many of the large aerospace and automotive corporations [5-6].
D. Problem Definition
With the need for weight reduction in mind as well as the desire to explore
topology optimization techniques, the general problem statement for the 2006 UCSD
Formula SAE team was to redesign and thus remove mass from an existing Formula
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SAE vehicle component through the use of modern optimization software. The 2005
rear-suspension bell crank component was chosen as the test specimen for this project.
A bell crank is defined as a type of lever whose two arms form a right angle, or
nearly a right angle, having its fulcrum at the apex of the angle [9]. It received the
name from its first use which was to change the vertical pull of a rope into a horizontal
pull on the striker of a bell [10]. Bell cranks can also be found in the suspension
system of formula-style racecars.
Because formula-style vehicles have their
suspension control arms exposed by protruding through the body panels, there are
aerodynamic reasons for relocating the spring and damper assembly to a location on
the vehicle that is contained within the bodywork. This set-up is distinctly different
from that of a standard production car (where the springs and dampers are traditionally
attached directly to the suspension control arms), and requires the use of a push or pull
rod and bell crank to transmit the suspension forces from the control arm to the spring
and damper. In addition to the aerodynamic benefits associated with relocating the
spring/damper assembly, the use of a bell crank gives the suspension design engineer
added control over the vehicle’s effective spring rate via manipulation of the bell
crank’s geometric parameters. A diagram showing a typical pushrod and bell crank
suspension is given in Figure 1.
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Figure 1: Diagram of a typical pushrod and bell crank suspension [11].
The 2005 UCSD Formula SAE rear-suspension bell crank component was
chosen as the test specimen for the initial investigation of the topology optimization
for many reasons:
1) Because it transfers all of the suspension spring and damper forces for one
corner of the car as part of a four-bar linkage, the bell crank component is exposed to
high stresses, making the optimization results both critical and exciting.
2) Due to the 2005 rear suspension packaging (which remained the same for
2006), the bell crank geometry required a cut-out to clear the spring and damper
assembly. This feature makes the design problem more challenging, as the cut-out lies
in a potential load path between the spring connection and the connection for the rear
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anti-roll bar pushrod. It was reasoned that this could possibly highlight a counterintuitive optimization result.
3) As part of a four-bar linkage, the bell crank is connected to truss-like
members of the suspension system, thus all of the loads can be thought of as simple
force vectors which can be predicted and represented with relative ease. This puts the
focus on the power of the software as opposed to the team’s ability to gather boundary
conditions (which can be quite complicated, as in the case of a wheel, for example).
4) Being that the rear suspension design is of the pushrod type, the rear bell
cranks are located in a position that is highly visible on the racecar, making it easy for
a person to identify the research component of interest (as opposed to the front
suspension bell cranks, which lie hidden under the drivers legs due to the pull-rod
style front suspension). Figures 2 and 3 show the rear bell cranks as designed and
installed on the 2005 competition-year vehicle.
5) Finally, the 2005 bell cranks were designed using a traditional trial-anderror finite element approach, therefore making for a good comparison to the methods
utilizing topology optimization early in the design process.
After deciding upon a test article, the problem statement was to reduce the
mass of the rear bell crank component by at least 10% while maintaining the yield
strength of the previous design. Furthermore, due to the criticality of the component,
physical testing was deemed necessary to validate and objectively asses the integrity
of the design under static loading conditions similar to and beyond those seen in actual
use.
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Figure 2: Bell crank as installed on 2005 vehicle (circled).
Figure 3: Close-up view of bell crank as installed on 2005 vehicle.
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II. Pre-processing
A. Load Prediction
1. Multi-body Dynamics
To get a better understanding of the loading conditions at the component of
interest, a Multi-body Dynamic model (MBD) was created to accurately capture the
geometry of the vehicle’s rear suspension. Altair MotionView® software was used as
a mechanical system simulation tool with which the entire rear suspension of the
racecar was created in a 3-D environment. This enabled the extraction of the spring
and anti-roll bar force vectors on the bell crank component due to different vehicle
suspension conditions, such as full spring compression or full body roll.
The model was created using 3-D suspension and chassis coordinates from the
Formula SAE suspension design team inserted manually into MotionView® as points
in space [12]. The MBD software then easily allows for the creation of cylindrical
bodies between the points, and so the suspension linkages and control arms were made
in this way, and the 2005 model bell crank solid model file was imported as a graphic
file.
The next step was to add joints between the bodies of interest, thus revolute
joints were used at the location of the suspension bearings to give the system the
necessary travel. Furthermore, both linear and torsional springs were added to the
model to act as the suspension and anti-roll bar springs, respectively. The UCSD
Formula SAE suspension design team specified the compression spring to have a
linear stiffness value of 701 N/cm (400 lb/in) and the anti-roll torsion bar to have a
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stiffness of 4.91 N*m/deg (3.62 ft*lbs/deg) with an arm length of 6.35 cm (2.50 in), so
this data was programmed into the model accordingly. Figure 4 shows the MBD
model complete with the springs and linkages.
Figure 4: MBD model of rear suspension.
While the original intent of the MBD model was to solve for the dynamic loads
at the bell crank due to a time-dependent disturbance at the vehicle’s wheel, solver
capabilities and Formula SAE timeline constraints limited the analysis to only include
static forces. Therefore, damping and inertial forces were not accounted for in this
model, and only spring forces were captured. Obviously the components on the
vehicle are subjected to dynamic loads in actual use, so it was reasoned that a larger
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factory of safety be designed into the parts to compensate for the unknown dynamic
forces.
2. Load Cases
Figure 5 shows the spring loads on the left bell crank due to a worst-case
scenario of the vehicle body rolling completely to the left such that the suspension
travel is at a maximum compression on the left side and maximum droop on the right.
In this case the linear spring is completely compressed and the anti-roll bar is at
maximum twist. Figure 6 depicts the load case in the event that the suspension is
loaded in the configuration that the vehicle is rolled over to the right. This would put
the linear spring in a relaxed position, with the anti-roll bar again at maximum twist,
but in the opposite direction. With the spring relaxed, there is no spring force at the
bell crank node C; however, the mass of the wheel system would be pulling on the
pushrod, and hence pulling on node C with roughly the weight of the vehicle’s
unsprung mass at one corner.
As an engineering approximation, this force was
estimated to be 445 N (100 lbf) at a maximum. Thus the load case was updated to
include this force.
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Figure 5: Bell crank load case in left vehicle roll condition.
Figure 6: Bell crank load case in right vehicle roll condition.
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B. Design Space
Before the topology optimization study can begin, it is first necessary to
determine the maximum amount of volume that the geometry can safely occupy. This
is known as the design space, and it represents the volume that will be meshed into
finite elements and iterated upon while the optimization algorithm is working. If the
finalized component needs clearance or a pass-through for wires, for example, the
design space must reflect this so that the software does not try to use that space for
load-bearing elements. In the case of the UCSD Formula SAE bell crank, the design
space necessitated a notch for clearance to the spring, as well as cut-outs for the
attachment of the spring assembly and pushrods. This notch can be clearly seen in the
bell crank design space and spring/damper system of Figure 7. In addition, holes for
the bearing and bolt clearance are present.
Figure 7: Design space and spring/damper system.
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III. Topology Optimization
A. Design Space Mesh
1. Geometry Clean-up
Part of the Altair Hyperworks® software package, Altair HyperMesh® was
used as the finite element meshing utility in preparation for the optimization study.
The design space previously created in a CAD program was imported as an IGES
(Initial Graphics Exchange Specification) file, and the geometry was “cleaned” to
prepare for meshing. This means that some of the lines in the imported model were
toggled from edge lines to suppressed (or manifold) lines so that they would not
represent an artificial edge that would force the finite elements to unnecessarily align
themselves to [13]. The misreading of lines happens at the locations of fillets and
radii features created in CAD models, as the features get falsely interpreted as distinct
surfaces in the IGES transformation. An example of the line types can be seen in
Figure 8, which shows the geometry cleaning phase for the design space meshing
process. Note: if viewed in color, the suppressed lines are depicted in blue, and the
edge lines are green.
2. Tetra-Meshing
Once the geometry was cleaned, the design space volume was filled with
tetrahedral elements using the auto-mesh features of HyperMesh®. This was done
with a volume-tetra element with a nominal minimum size of 2.54 mm (0.100 in), and
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curvature and proximity adaptation enabled to refine the mesh in the regions of more
complex geometry. The resulting mesh that was used as the design space for the
topology optimization study can be seen in Figure 9.
Figure 8: Geometry cleaning in preparation for meshing.
Figure 9: Tetra-meshed design space.
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B. Boundary Conditions
1. Treatment of Bolt Holes
As in any finite element analysis, proper boundary conditions (BC’s) are
crucial if the results are to be of any significance, and this is especially true for
topology optimization studies since these BC’s will be the basis for the resulting
distribution of material.
An example of this lies in the treatment of the boundary
condition around the region of a bolted or pinned joint. Bolts and pins can only
transfer compressive load (unless, of course, they are bonded in place), thus they can
only push on another surface that is in contact. In a finite element model, however,
this phenomenon is somewhat difficult to capture as it requires the use of non-linear
gap elements which have a very low stiffness in tension to simulate the lack of
connectivity [7]. For the sake of time and simplicity, the bolt holes in this model were
filled with rigid “spiders” (RBE2 elements) at the acknowledged cost of reduced
accuracy of topological results in the vicinity of the bolt holes.
These rigid spiders
connect all the perimeter nodes of the bolt hole to a single node in the center at which
the loads and constraints are applied.
2. Constraints
As installed on the vehicle, the rear bell crank pivots about a needle roller
bearing (node A of Figure 5) that is captured by a bolt through welded tabs to the
vehicle chassis. Furthermore, needle thrust bearings on both faces keep the bell crank
located with respect to the axis of the bearing bolt, and also serve to support any
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moments out of the plane of the intended motion. Thus the only remaining degree of
freedom at the bearing is a rotation about the bolt, which in the local coordinate
system represents a rotation about the z-axis (refer to Figure 5 for coordinate system).
Because we have an idea of what the forces are at the spring nodes (nodes C
and D), it makes sense to apply known loads at these nodes, and constrain the pushrod
node (node B) in accordance with the actual set-up on the vehicle. The pushrod
transmits the suspension force from the lower control arm of the vehicle suspension to
the bell crank via spherical bearings (sometimes called ball-joints or rod-ends), thus
the node at which it connects to the bell crank is almost completely free to translate
and rotate. The pushrod is essentially a purely tension/compression member and so it
can only support translation along its axis. Because of this, the pushrod node at the
bell crank (node B) was allowed all of its degrees of freedom except translation in the
pushrod direction. Although this direction changes slightly as the suspension moves
through the range of motion, it is nominally at a position that corresponds to the ydirection in the bell crank local coordinate system, and thus this degree of freedom
was constrained for the topology study. Defining DOFs 1, 2, and 3 as translational
degrees of freedom in x, y, and z; and DOFs 4, 5, and 6 as rotational degrees of
freedom about the x, y, and z axes; the constraints can be summarized as seen in Table
1. This set of constraints was denoted SBC and was used for all load cases. Refer to
Figure 5 for the bell crank node naming convention and coordinate system.
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Table 1: Summary of bell crank constraints.
Bell crank
Node
DOFs
constrained
A
B
C
D
12345
2
N/A
N/A
3. Loads
Using the two previously determined load cases from the MBD model, we
have updated the finite element model to include these load vectors applied to the rigid
spiders of nodes C and D. Furthermore, two additional “out of plane” load cases were
created in which 445 N (100 lbf) transverse forces were applied to nodes C and D
perpendicular to the intended direction of travel. This was done to ensure that the
optimized geometry could accommodate realistic but unforeseen conditions and 445 N
(100 lbf) was chosen as an estimate for the worst-case condition. Table 2 shows a
summary of the four load cases used in the optimization of the 2006 UCSD Formula
SAE rear bell crank design. These values represent the forces seen at the left rear bell
crank during four distinct conditions. Note that there are no applied moments to the
system, and all forces are shown in units of Newtons with pound-force in parenthesis.
Load case 1 represents the vehicle in a worst-case right roll, load case 2 represents a
worst-case left roll, and load cases 3 and 4 are the assumed worst-case out of plane
loads. Figure 10 shows the aforementioned boundary conditions as applied to the
design space in preparation for the topology optimization study.
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Table 2: Summary of bell crank load cases.
Load
Case
{A}
{B}
1
0
0
2
0
0
3
0
4
0
Cx
Cy
400N
(90 lbf)
-3438
(-773)
-187
(-42)
-921
(-207)
0
0
0
0
0
0
Cz
0
0
445
(100)
-445
(-100)
Dx
Dy
311
(70)
-1259
(-283)
-1748
(-393)
1259
(283)
0
0
0
0
Figure 10: Applied boundary conditions.
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Dz
Const.
Set
0
SBC
0
SBC
445
(100)
-445
(-100)
SBC
SBC
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C. Optimization Statement
1. Optimization Responses
The objective function of any optimization problem is to minimize or
maximize a certain response while meeting a prescribed set of constraints. For this it
is necessary to program the software to solve for the desired responses, and then
choose limits to these responses [14].
In the case of the bell crank optimization project, the objective function was to
minimize the mass while maintaining the integrity of the original design;
consequently, the response to minimize was chosen as the volume fraction. This value
represents the normalized volume of the design space after elements have been
iteratively eliminated or turned “off” in an effort to meet the design constraints, and
with a constant material density, the volume fraction directly correlates to the mass.
Obviously this mass minimization cannot continue without bound, thus it is necessary
to define responses that have upper and/or lower limits as constraints. For the design
of a highly stressed component it would be ideal to define the elemental stress level as
a response and limit the stress to a set factor of safety below the material yield stress.
Unfortunately, this response is not yet offered in the current version of HyperWorks®
7.0 (which was used for this research); however, it will be available in 8.0. In lieu of
this feature, the constraints applied in the optimization of the bell crank were
compliance driven as opposed to stress driven. Namely, three nodal displacements
were defined as responses, and the magnitude of these displacements was limited at a
level that compared with previous displacement results from a finite element analysis
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of the 2005 model bell crank, the results of which can be seen in Figure 11. Because
the 2005 bell cranks went through an entire season of racing without any problems, it
was reasoned that the design was adequate enough to justify using the FEA
deformation results as a constraint for the new design. With this constraint set, the
optimized design would be at least as stiff as its predecessor.
Figure 11: 2005 bell crank finite element analysis displacement results.
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2. Material Definition
The last input parameter required before implementation of the topology
optimization study is the material identification. In the case of the bell crank, the
material was specified as 7075-T6 aluminum; as a result, the material was defined as
linear isotropic (MAT1) and the values for Young’s modulus and Poisson’s ratio were
entered. These values used for the study were 71.7 GPa (1.04e7 psi) and 0.33,
respectively.
3. Manufacturing Constraints
Although the optimization problem is now fully defined, the resulting topology
design proposal will likely be very difficult to manufacture because the algorithm will
tend to make hollow structures with a lot of holes [15]. To ensure that the optimized
geometry can be realistically manufactured, the software includes algorithms for
implementing manufacturing constraints such as minimum size control and prescribed
draw directions. With minimum size control enabled, the optimization software will
not create geometry that is smaller than the desired size. This feature reduces the
number of small ribs and “blobs” that can complicate the interpretation and creation of
the optimized geometry. For the bell crank project, the minimum member size was set
to 2.54 mm (0.100 in), as this was the thinnest rib that the machinist felt comfortable
fabricating.
Because the bell crank was to be machined from solid aluminum, it was also
deemed beneficial to enable a draw direction, as if the part was going to be cast in a
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mold. This eliminates the formation of a hollow structure, giving the part more of a
two-dimensional quality, thereby lending to its manufacturability. The bell crank
design space was given a single draw direction outward from the x-y plane in the zdirection. For comparison purposes, however, optimization results were obtained with
and without the draw direction enabled, and the differences will be discussed in the
following chapter.
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IV. Post Processing
A. Topology Optimization Results
The optimization analysis was conducted using Altair OptiStruct® 7.0 and the
jobs were run on a 3.40 GHz Pentium 4 processor with 512 Mb RAM. The average
run-time was about one hour. The resulting geometry proposal can be seen in Figure
12 with the previously stated boundary conditions, minimum member size control, and
symmetry constraint. Notice that there are clearly defined groupings of elements that
form ribs in an “x” pattern within the outer structure of the proposed bell crank design,
resulting in a rib geometry that is very different from that of the 2005 design. It is
especially interesting to see that the right hand portion of said rib group seems to curve
and meet the upper truss that forms the support for the anti-roll bar node. This was a
surprise, as intuitive thinking would say that typical truss members should be straight.
Figure 12: Topology optimization results, side view.
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The isometric view of the proposal (Figure 13) reveals a hollow geometry that
would be very hard to machine from billet material. In particular, the design looks
like it would lend itself to fabrication from two separate plates spaced out from each
other. While this was an option for the design of the 2006 bell crank, the original goal
was to have the parts machined from billet aluminum; therefore, the study was run
again with the draw direction constraint enabled, and the results were much different.
Figure 13: Topology optimization results, isometric view.
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Figure 14 shows the optimized material distribution when the same analysis
was run but also included a prescribed draw direction constraint. The algorithm
involved with this manufacturing constraint is complex and beyond the scope of this
Master’s Thesis, but suffice it to say that the constraint suppresses the formation of
cavities and undercuts [15]. The first obvious difference in this result is the lack of
two separate “plates” forming on each side, and instead the proposal is much more
solid looking—more closely resembling that of a cast and/or machined component.
The second important feature to notice is the reoccurring formation of an “x” shaped
brace in the center; however, this time the ribs are not separated into two distinct
planes. Instead of being spaced apart, the ribs have converged into one thin center
brace section.
Figure 14: Topology optimization results with draw constraint, isometric view.
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An interesting result of both studies is the removal of material near the curved
notch region of the bell crank design space (see note of Figure 13). Because this cutout is in the path between the spring and anti-roll bar nodes, it was anticipated that all
of this material would remain in an effort to maximize the radius of the notch to
reduce the stress concentration factor in that region. Instead, almost all of the material
is removed, save for a small triangular brace for the anti-roll bar connection node.
It is important to note the lack of material around the bolt and bearing holes
due to the previously mentioned difficulties of accurately modeling the bolt
connection. Obviously, without any material surrounding the bolt-hole, the part could
not support a load placed at the bolt, thus one must pay attention to this fact in the
geometry interpretation phase of the topology study and design in the appropriate
amount of material around the connection holes.
B. Geometry Interpretation
Although the topology results appear reasonable, the design is definitely not
ready to hand over to the machine shop for fabrication. The results of the topology
studies are merely rough geometric proposals, and some interpretation is required to
create the final design. OptiStruct has the ability to export the topology results as an
IGES file using an export feature so that the geometry can be opened and perhaps
traced over in a CAD environment. For the interpretation of the 2006 UCSD bell
crank design, the optimized shape was created using SolidWorks® 2006 CAD
software. To capture the organic shape of the proposed design, splines were utilized
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in the creation of the CAD model, and the interpretation can be seen in Figure 15. The
design retains the features proposed by OptiStruct: such as the curved “x” brace and
the truss-like anti-roll bar node support, and the final design is somewhat of a hybrid
inspired by both of the analyses. Appropriately sized fillets were utilized to reduce
stress concentrations and to aid manufacturability using standard tooling.
Figure 15: Optimized geometry interpretation, isometric view.
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V. Design Validation
A. Finite Element Analysis
To validate the structural integrity of the newly designed bell crank, a finite
element analysis was performed using HyperMesh® 7.0 as a qualitative test to ensure
that the design did not have any inherent stress concentrations or fatal flaws. The
boundary conditions and mesh parameters were the same as used in the optimization
analysis, and the load case used was load case two of Table 2. This load case was
chosen as it represents the worst-case scenario of the linear suspension spring being at
maximum compression, and the anti-roll bar at maximum twist. Furthermore, this is
in the direction of loading that the bell crank will see due to impact loading, such as
the racecar driving over a pothole. Figure 16 depicts the von-Mises stress distribution
and 10-fold exaggerated displacement plot for the optimized 2006 design. Note that
the stresses are fairly homogenous within the structure. This is precisely the goal of
the topology optimization, and is the result of the algorithm which mimics the growth
behavior of biological load carriers like trees and bones, where the structures always
tend to grow into shapes that have homogeneous surface stress [15].
From the finite element results of Figure 16, there does not appear to be any
severe stress concentrations that would indicate a faulty design, and the highest stress
levels appear to be on the order of 140-210 MPa (20 to 30 ksi). With a material yield
stress of 500 MPa (73 ksi), these FEA results indicate a factor of safety of roughly 2.4
to 3.6.
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Figure 16: von-Mises stress distribution and exaggerated displacement
B. Bell Crank Prototype Fabrication
With the initial finite element analysis validation complete and successful, it
was then necessary to build prototype bell crank components in preparation for the
next stage of the validation: physical testing. For this, MasterCAM X computer aided
manufacturing software was used to program the tool paths of a Hass VF-0 three axis
mill, and the machining required five mill operations and eight different tools. Using
7075-T6 alloy aluminum, four prototype components were fabricated. The tool paths
of the MasterCAM environment can be seen in Figure 17, as well as the fabrication
and finished product in Figures 18 and 19.
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34
Figure 17: MasterCAM X tool paths
Figure 18: Bell crank prototype fabrication.
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35
Figure 19: Bell crank prototype finished product.
C. Physical Testing
1. Hydraulic Test Fixture Design
As a means to further validate the design in a more practical sense, a hydraulic
load cell test frame capable of simulating the static suspension forces was designed
and assembled with the goal of testing the original and optimized components first to
the nominal load condition specified in load case two, and then to failure. Failure in
this case was defined as the yield point at which the bell crank would exhibit a
diminishing load carrying capacity. Before any testing could begin, however, the
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36
experimental procedure first had to be established, and for this it was necessary to look
at the physical arrangement of the racecar suspension.
First of all, the anti-roll bar is a torsional type spring, and the force it exerts on
the anti-roll bar pushrod and thus bell crank is a function of its angle of twist, which is
a function of the vehicle suspension travel. Because the suspension has a finite
amount of travel (limited by the damper travel), the anti-roll bar force that can be
exerted to the bell crank is also finite, and this maximum level of anti-roll bar force is
precisely what is captured in load case two. Furthermore, aside from some small
structural value, the torsion bar spring does not have any associated damping
characteristics, thus dynamic impact will not drastically increase this force. The linear
suspension spring/damper assembly, however, is more complex. First, it does have a
viscous damper, thus dynamic travel will most definitely create a significant damping
force in addition to the static spring force. The spring/damper assembly also has a
finite amount of travel. In fact, it is what limits the travel of the vehicle suspension.
Under vehicle bottom-out conditions, for example, the damper shaft runs out of
compressive travel, and the bottom-out bump-rubber comes into contact with the
damper body, thus increasing the effective spring rate. If the bottom-out bump-rubber
should become completely compressed, the damper assembly would simply run out of
travel, thereby making the entire unit infinitely stiff (compared to the spring rate).
Because of this, the load that can be seen at the bell crank due to the linear spring
essentially has no bound, and will be responsible for the failure of the component in
actual use. Therefore it was decided for the experiment to first load the anti-roll bar
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node to the worst-case load as depicted in load case two (1780 N), and then load the
linear spring node to the point of yield failure.
Hydraulic load cells with an acting bore diameter of 34.9 mm (1.375 in) were
used to supply the load to the connection pins associated with the linear spring as well
as the anti-roll bar attachment, as can be seen in the CAD model of Figure 20 below
[16]. In this manner hydraulic pressure could be directly correlated to force, so
pressure was recorded as a function of time for the physical tests.
This was
accomplished by means of both analog pressure gauges as well as digital pressure
transducers, and the information was captured via videotape and LabVIEW data
acquisition, respectively. Hydraulic pressure was supplied with a hand pressure pump
capable of 10300 kPa (1500 psi).
Figure 20: Hydraulic test fixture CAD assembly.
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In order to ensure that the experiments matched the conditions of both the
finite element model and the actual vehicle installation, care was taken in the
treatment of the test frame’s boundary conditions.
For example, to mimic the
constraint at the pushrod attachment node (node B), a steel beam was used to represent
the pushrod, and it was loosely attached to the test frame base plate so that it could
rotate in a similar fashion as the actual vehicle component. In addition, the bell crank
bearing node (node A) was allowed to rotate only in the intended direction of motion,
and this was accomplished by capturing the bell crank with a bolt that was welded to
the test frame base plate. Furthermore, the hydraulic load cells were attached to the
bell crank using the same type of hardware that would be used to attach the
spring/damper assembly and anti-roll bar pushrod.
For the experiment to be a success, the bell crank components needed to be
tested to failure, thus the hydraulic cylinders had to be capable of supplying enough
force. With a nominal bore diameter of 34.9 mm (1.375 in), the cylinders have a
piston area of 958 mm2 (1.485 in2). With a maximum pressure rating of 10300 kPa
(1500 psi), the cylinders could each exert a maximum force of 9.92 kN (2230 lbf).
While this was obviously sufficient to supply the 1780 N (400 lbf) needed at the antiroll bar node, it was unknown how much force was needed at the spring node to fail
the part; therefore, two hydraulic cylinders were installed in parallel for this purpose,
doubling the maximum force available. A ball-valve was installed on the lines so that
the cylinder associated with the anti-roll bar node could first be pumped up to the test
pressure, and then closed while the other two cylinders were increasingly loaded.
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Granted, it was realized that the pressure in the first cylinder would fall as the bell
crank deformed, but his was deemed negligible based on the very small deformation
seen in the finite element model. Furthermore, it was hypothesized that the small
force from this cylinder would be relatively insignificant as compared to the final
force at failure required from the tandem cylinders.
In preparation for the experiment, the test fixture was covered with a
polycarbonate safety shield, and the reservoir was filled with hydraulic fluid. Figure
21 shows the device loaded with a 2006 model bell crank ready to test.
Figure 21: Hydraulic test fixture prepared for experiment.
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2. Experimental Results
To accurately compare the 2006 optimized bell crank design to the 2005
model, one of each component type was tested to static failure using the hydraulic test
fixture described in the previous section. Furthermore, both the 2005 and 2006 test
specimens were machined by the same person using the same equipment and alloy of
aluminum.
Weighing in at 185 grams, the 2005 model bell crank required a hydraulic
pressure of 6380 kPa (925 psi) before deforming plastically. Given the effective bore
area of the tandem cylinders, this pressure equates to a load of 12.2 kN (2750 lbf).
The final mass of the 2006 optimized rear bell crank design as tested was measured at
140 grams—a weight savings of 24.3%; moreover, the optimized design supported a
hydraulic pressure of 8270 kPa (1200 psi). This critical pressure represents a load of
15.9 kN (3560 lbf) and thus a 29.7 % increase in strength as compared to the 2005
model. Table 3 summarizes these results, and Figure 22 shows the 2006 design bell
crank after testing. Note the similarities in the deformation compared to the finite
element prediction of Figure 16.
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Table 3: Summary of experimental test results.
Mass
Yield Load
2005 model
185 grams
12.2 kN (2750 lbf)
2006 model
140 grams
15.9 kN (3560 lbf)
Percent change
-24.3%
+29.7%
Figure 21: Physical testing of 2006 optimized bell crank.
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VI. Conclusion
A. Discussion of Experimental Results
Due to the very lightweight look and feel of the topology optimized bell crank,
many observers were initially dubious as to the structural integrity of the prototype
component, and justifiably so, as the parts did look fairly slim—especially when
compared to the already slender 2005 component. This meant that a lot was riding on
the physical testing of the bell crank—both literally and figuratively. Especially
considering the fact that during the time of physical testing, the 2006 Formula SAE
West competition was about a week away, and thus the results of the physical testing
would determine whether the final components found their way onto the 2006 UCSD
competition vehicle entry. Furthermore, the team’s overall confidence in topology
optimization also hinged upon the result of the physical tests. Needless to say, seeing
a nearly 30% increase in strength coupled with an equally impressive decrease in mass
surely had many people excited about the computational powers of the topology
optimization algorithm. And the parts made it on the car in time for competition—
anodized to match the wheels.
It is understood that these numbers might be on the high side of the weight loss
spectrum for similar studies, and this can most likely be attributed to the fact that the
2005 model to which the study is compared was not a very developed part. This is to
be expected for a student-run vehicle design team, where development time is at a
premium.
Even so, some large companies are claiming similar weight savings
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numbers. Volkswagon Design Development claimed a 23.0% weight reduction in the
optimized design of an engine bracket in a presentation given at the Optimization
Technology Conference (OTC) in Troy, Michigan in 2005 [6], and it seems that values
in the range of 10% -30% are fairly common amongst other users that have shared
their results.
Aside from the newfound confidence in topology optimization, another result
of the testing was the discovery of a relatively high measured factor of safety in the
tested bell crank component.
While the lack of a complete and thorough
understanding of the dynamic loading conditions at the part necessitated an overly
designed part, the loads required at failure were higher than anticipated. Knowing this
information, some strength could have been sacrificed for further decreased mass;
namely, the bell cranks could have been crafted from magnesium as opposed to high
strength aluminum, bringing an additional 30% weight savings.
While the results of the study as presented in this thesis are indeed optimistic,
they merely represent the first attempt in applying topology optimization schemes in
vehicle design at UCSD. Hopefully future students will further exploit the powers of
the topology optimization algorithm through the use of increasingly developed and
demanding applications.
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B. Overall Vehicle Performance and Awards
The 2006 UCSD Formula SAE vehicle represented the third successive vehicle
designed and developed by engineering students at the University of California, San
Diego, and it proved to be an entry of high caliber. Unfortunately, a fuel pump failure
during the final laps of the endurance/fuel economy event prevented the team from
placing well overall; however, many of the goals set forth by the team were met. For
example, the final wet weight of the 2006 entry was 223 kg (492 lbs), meeting the goal
of being under 227 kg (500 lbs), which was no small feat considering that the vehicle
was equipped with a supercharger (a feature which added roughly 7 kg (15 lbs) to the
vehicle package). Furthermore, a goal for the 75 meter acceleration event was set at
4.350 seconds, which would represent a drastic improvement from the team’s best of
4.634 seconds with the 2005 car [17].
Beyond anyone’s expectations, the 2006
vehicle shattered the goal by putting down an astonishing time of 4.137 seconds—a
time that not only exceeded our goals, but was good enough for 3rd place in the entire
field [18]. In fact, the time is one of the top ten fastest in Formula SAE competition
over the past five years—representing a field of over 700 vehicles! Table 4 shows a
performance comparison between the 2005 and 2006 UC San Diego Formula SAE
entries, and Figure 23 shows the 2006 vehicle in final form at the Formula SAE West
competition.
In addition to the satisfaction of meeting our own goals, the 2006 Formula
SAE team at UCSD was twice invited up to the stage to accept awards involving the
vehicle performance and design. For the acceleration event, the team was awarded the
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3rd place prize for the Honda Acceleration Award. The team also won 1st prize in the
Altair Engineering’s William R. Adam Engineering Award, for the “top team that
exhibits innovative engineering concepts in vehicle design” [19], and this award was
given for the team’s effective use of topology optimization software in the design of
the rear suspension bell cranks. Figure 24 shows the anodized bell cranks as installed
on the 2006 vehicle at competition, and the racecar in motion can be seen in Figure 25.
Table 4: Comparison of 2005 and 2006 UCSD FSAE vehicle performance
Mass
75m Acceleration
Peak Torque
2005 Vehicle
246 kg (543 lbs)
4.634 s
58 N*m (43 ft*lbs)
2006 Vehicle
223 kg (492 lbs)
4.137 s
71 N*m (52 ft*lbs)
Percent Change
-9.4%
-10.7%
+21%
Figure 22: 2006 UCSD FSAE vehicle at competition.
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Figure 23: Close-up of bell crank on 2006 vehicle.
Figure 24: 2006 vehicle in motion.
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