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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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
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-and-
error 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 counter-
intuitive, 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 counter-
intuitive 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-and-
error 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 y-
direction 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 A B C D
DOFs constrained 1 2 3 4 5 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} Cx Cy Cz Dx Dy Dz Const.
Set
1 0 0 400N (90 lbf)
-187 (-42) 0 311
(70) -1748 (-393) 0 SBC
2 0 0 -3438 (-773)
-921 (-207) 0 -1259
(-283) 1259 (283) 0 SBC
3 0 0 0 0 445 (100) 0 0 445
(100) SBC
4 0 0 0 0 -445 (-100) 0 0 -445
(-100) SBC
Figure 10: Applied boundary conditions.
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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 z-
direction. 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.
Lockheed Martin Aeronautics Company.” Optimization Technology Conference. 27-28 September 2005. Troy, Michigan, USA.
[4] Talke, F.E., “Optimization: Computer Aided Analysis and Design.” Class Notes. MAE 292. UC San Diego, Spring 2006.
[5] Schneider, Detlef, and Erney, Thomas. “Combination of Topology and
Topography Optimization for Sheet Metal Structures.” OptiCON 2000 Conference Proceedings.
[6] Meyer-Pruessner, Rainer. “Significant Weight Reduction by Using Topology Optimization in Volkswagen Design Development.” Optimization Technology Conference. 27-28 September 2005. Troy, Michigan, USA.
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[10] Wikipedia. “The Free Encyclopedia: Bell Crank Definition.” 2006. Online posting. <http://en.wikipedia.org/wiki/Bell_crank>
[11] Wright, P., “Formula 1 Technology.” Warrendale: SAE International, 2001. [12] Altair Engineering. “Altair MotionView: Pre-and Post Processing for Multi-
Body Dynamics, Volume I.” HyperWorks Training Manual, 2004.
[13] Altair Engineering. “Altair HyperMesh: Introduction to FEA: Pre-Processing Volume I.” HyperWorks Training Manual, 2004.
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[14] Altair Engineering. “Altair OptiStruct: Concept Design Using Topology and Topography Optimization.” HyperWorks Training Manual, 2004.
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[16] Ariely, A., Doring, P., Erkebaev, T., and Tawatao, M.L., “Suspension
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[17] SAE International. “Collegiate Design Series: Formula SAE Results.” 2005.
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