This document describes a custom MATLAB simulation of 3D rigid body physics for modeling a novel roller coaster suspension system. Key points: - The simulation models point masses connected by rigid links and springs/dampers to represent a roller coaster car and its suspension moving along 3D spline tracks. - A double four-bar linkage and pivoting joint allow vertical and yaw movement to give sensations of "floating" and "drifting". - The simulation is validated against acceleration data from real roller coasters and used to tune spring/damper values for desirable suspension behavior. - MATLAB was chosen for its solver options, accuracy/speed control, and integration with coaster design software.

1. Custom MATLAB 3D Rigid
Body Physics Engine
Alex Hsia

2. Project Overview
• Dynamic simulation of a Star Wars-themed roller
coaster with a novel suspension, built at low-
level from ground up (Simulink not used)
• Simulation is used to tune spring and damper
values to produce desirable suspension behavior
• Double four bar linkage allows vertical
translation, giving feeling of “floating”
• Pivoting joint allows yaw, giving feeling of
“drifting”
• MATLAB was chosen for following reasons:
• Existing team expertise
• Many options for type of solver used
• Good control over simulation accuracy and
speed
• Very customizable integration with other
roller-coaster design software
Double four-bar
linkage
Pivoting joint (springs
and dampers not
shown for yaw DOF)

3. Physical Model
• Motivation: Dynamic simulation
of a novel suspension on a roller
coaster car
• Simulates any number of point
masses connected by rigid
massless links, or springs, or
dampers
• Constrains any number of points
to 3D cubic splines representing
roller coaster track rails
Point masses
Spring and damper (red)
Triangle masses are
constrained to roller coaster
rails, which are represented
as 3D cubic splines
Spring and
damper
(red)

4. Yaw DOF: Pivoting Joint
Velocity in car X frame indicates “drift”
0 0.5 1 1.5 2 2.5
-0.5
0
0.5
1
1.5
2
Velocity, Car Frame
Time (s)
Velocity(m/s)
X
Y
Z
0 0.5 1 1.5 2 2.5
-0.3
-0.2
-0.1
0
0.1
Acceleration, Car Frame
Time (s)
Acceleration(Gs)
X
Y
Z
https://www.youtube.com/watch?v=eouZZfH-8a8

5. Heave DOF: 4 Bar Linkage
0 0.5 1 1.5 2 2.5
-1
0
1
2
3
Velocity, Car Frame
Time (s)
Velocity(m/s)
X
Y
Z
0 0.5 1 1.5 2 2.5
-1
0
1
2
Acceleration, Car Frame
Time (s)
Acceleration(Gs)
X
Y
Z
https://www.youtube.com/watch?v=LMSTMYtSI58

6. Full-Scale Simulation Validation
- Acceleration data taken for actual roller coasters using smartphones
- Acceleration time history compared to MATLAB simulation
- Kingda Ka simulation shown below

7. Small Scale (1:20) Simulation Validation

8. Final Coaster Track Design
0 20 40 60 80 100 120 140 160 180 200
-20
0
20
Velocity, Car Frame
Time (s)
Velocity(m/s)
X
Y
Z
0 20 40 60 80 100 120 140 160 180 200
-5
0
5
Acceleration, Car Frame
Time (s)
Acceleration(Gs)
X
Y
Z
0 20 40 60 80 100 120 140 160 180 200
-50
0
50
Yaw Angle (deg)
Time (s)
YawAngle(deg)(m/s)
A