Embed Size (px)
DESCRIPTION
The Finite Element Method and an Introduction to Kart Chassis Dynamics. Contents. Finite Element Method What is it? Formulation Chassis Dynamics Importance Modeling Data and Results Beyond…. 42 ℏ. FEM and Chassis Dynamics. Who has ever heard of FEM?. Who has ever used FEM?. - PowerPoint PPT Presentation
Citation preview
The Finite Element Method and an Introduction to Kart Chassis
Dynamics
The Finite Element Method and an Introduction to Kart Chassis
Dynamics
Contents
• Finite Element Method– What is it?– Formulation
• Chassis Dynamics– Importance– Modeling– Data and Results– Beyond…
42 ℏ FEM and Chassis Dynamics
Who has ever heard of FEM?
Who has ever used FEM?
Finite Element Method – What is it?
• The Finite Element Method (FEM) is a numerical method of solving systems of partial differential equations (PDEs)
• It reduces a PDE system to a system of algebraic equations that can be solved using traditional linear algebra techniques.
42 ℏ FEM and Chassis Dynamics
Heat Conduction Structural Mechanics E&M
Finite Element Method - Formulation
• There are several different formulations of FEM involving variational calculus and weighted residuals.
• The variational formulation is based on our beloved Green’s Identity.
42 ℏ FEM and Chassis Dynamics
dSvdVvudVvu nu
2 ..2 CBdVvudVvu
42 ℏ FEM and Chassis Dynamics
Finite Element Method - Formulation
0)1()0(
)(
yyEI
Mxy
1
0
)( )()()( vudxxvdxxvxy EIxM )()( 1
0 Hxv
Partial Differential Equation
Plus a set of boundary conditions
dxxvdxxvxy EIxM )()()( )( Multiply by a test function v(x)
and integrate over the domain
Apply the Green’s Formula to derive the so-called weak formulation.
Require test functions to vanish on the boundary so that the boundary conditions disappear.
42 ℏ FEM and Chassis Dynamics
nn v
v
v
v
v
EI
M
x
x
x
x
x
h
......
2100...0
1210......
01......00
......1210
0...0121
0...0012
4
3
2
1
4
3
2
1
dxxvdxxvxy EIxM )()()( )(
Finite Element Method - Formulation
Using the Lax-Milgram Theorem, principles of variational calculus, and principles of elliptical equations the problem can then be reduced to an algebraic system where xn are the displacements at the nodes.
)(),( hhh vlvua
n
iiihjh xuv
0
1ix ix 1ix
)(),(0
j
n
iiji lxa
Chassis Dynamics
42 ℏ FEM and Chassis Dynamics
Chassis Dynamics-Motivation
Chassis Dynamics is a huge part of how a kart works.
A greater understanding leads to better performance.
42 ℏ FEM and Chassis Dynamics
Chassis Dynamics - ModelingUsed a program called VisualFEA
42 ℏ FEM and Chassis Dynamics
• Simplifications
– Doesn’t account for small miscellaneous pieces
– Large masses centered at COM
– Wheels are pinned at center
– No steering geometry
– No bent sections
42 ℏ FEM and Chassis Dynamics
Chassis Dynamics - Modeling
Testing 1 2 3…
• Reproduce Commonly known facts– Weight is transferred to the outside wheels in
a turn– The rear track width effects handling– The drivers VCOM effects handling.
• Static constant centripetal acceleration turns. (0-1.5g)
42 ℏ FEM and Chassis Dynamics
42 ℏ FEM and Chassis Dynamics
Weight Shift
0
50
100
150
200
250
300
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
g-turn
lbs
total weight
right weight
left weight
42 ℏ FEM and Chassis Dynamics
Individual Weights (left/right)
-20
0
20
40
60
80
100
120
140
160
180
0 0.5 1 1.5 2
G-turn
wei
gh
ts (
lbs)
RF-z-left
LF-z-left
LR-z-left
RR-z-left
RF-z-right
LF-z-right
LR-z-right
RR-z-right
42 ℏ FEM and Chassis Dynamics
Required Coefficient of Friction (x+y)
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
turn acceleration
co
eff
icie
nt
of
Fri
cti
on
LF
LR
RR
RF
42 ℏ FEM and Chassis Dynamics
Corner Weights varying driver virticle COM (0.5g left turn)
45
55
65
75
85
95
105
115
9 10 11 12 13 14
COM-z (in)
wei
gh
t (l
bs) RF-z
LF-zLR-zRR-z
42 ℏ FEM and Chassis Dynamics
Corner Weights Varying Rear Track Width(0.5g left turn)
45
55
65
75
85
95
105
115
35 36 37 38 39 40 41
Rear Track [center to center] (in)
We
igh
ts (
lbs
)
RF-zLF-zLR-zRR-z
Beyond…
• Next Generation Models– Improved frame geometry– Steering geometry– Inclusion of smaller objects– Extension of larger objects
• Tire Modeling and Measurements• Roll Cage Modeling
Complete Functional Model
42 ℏ FEM and Chassis Dynamics
http://physics.bgsu.edu/Quantum_Racing
or on the links page of the Physics Department Website
Thank You!
• A special thanks goes out to:– My advisors on this project
Dr. Boughton
Dr. Chou (Dept. of Mathematics and Statistics)
Dr. Sun (Dept. of Mathematics and Statistics)
– The Quantum Racing advisorAlex Hann
– The Department of Physics and Astronomy
