The Finite Element Method and an Introduction to Kart Chassis Dynamics

1. The Finite Element Method and an Introduction to Kart Chassis Dynamics

2. Contents • Finite Element Method • What is it? • Formulation • Chassis Dynamics • Importance • Modeling • Data and Results • Beyond… 42 ℏ FEM and Chassis Dynamics

3. Who has ever heard of FEM? Who has ever used FEM?

4. 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. Heat Conduction Structural Mechanics E&M 42 ℏ FEM and Chassis Dynamics

5. 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

6. Finite Element Method - Formulation Partial Differential Equation Plus a set of boundary conditions 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

7. 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. 42 ℏ FEM and Chassis Dynamics

8. Chassis Dynamics 42 ℏ FEM and Chassis Dynamics

9. 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

10. Chassis Dynamics - Modeling Used a program called VisualFEA 42 ℏ FEM and Chassis Dynamics

11. Chassis Dynamics - Modeling • 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

12. 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

13. 42 ℏ FEM and Chassis Dynamics

14. 42 ℏ FEM and Chassis Dynamics

15. 42 ℏ FEM and Chassis Dynamics

16. 42 ℏ FEM and Chassis Dynamics

17. 42 ℏ FEM and Chassis Dynamics

18. 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

19. http://physics.bgsu.edu/Quantum_Racing or on the links page of the Physics Department Website

20. 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 advisor Alex Hann • The Department of Physics and Astronomy