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 ConductionStructural MechanicsE&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

42 ℏ FEM and Chassis Dynamics 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 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 x n are the displacements at the nodes.

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 - Modeling Used 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

42 ℏ FEM and Chassis Dynamics

42 ℏ FEM and Chassis Dynamics

42 ℏ FEM and Chassis Dynamics

42 ℏ FEM and Chassis Dynamics

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

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 advisor Alex Hann –The Department of Physics and Astronomy