Delta Time (Δt)
The time step of your simulation (how much time passes per update).
If you are running at 60 frames per second:
dt = 1.0 / 60 = 0.0167 seconds
If you want substeps (I prefer this over increasing constraint iterations), and you want 4 substeps:
substeps = 4
sdt = dt / substeps = 0.00417 seconds
Angular Acceleration (α)
The rate at which angular velocity changes.
Unit: radians per second squared (rad/s²)
How to calculate
From angular velocity:
α = Δω / Δt
From torque:
α = τ / I
Intuition:
Positive α = angular speed increases
Negative α = angular speed decreases
Example (engine):
Torque: 200 N·m
Inertia: 0.2 kg·m²
α = 200 / 0.2 = 1000 rad/s²
The engine RPM rises very quickly.
Angular Velocity (ω)
How fast something rotates, measured in radians per second.
Unit: rad/s
Example:
Speed: 30 m/s
Wheel radius: 0.3 m
ω = v / r = 30 / 0.3 = 100 rad/s
To calculate angular velocity from angular acceleration (α):
1. Divide torque by moment of inertia
2. Increase angular velocity using delta time and angular acceleration
α = τ / I
ω += α * dt
Moment of Inertia (I)
Resistance to changes in rotational motion.
It is the rotational equivalent of mass.
Unit: kg·m²
Intuition:
High inertia = hard to spin up
Low inertia = spins easily
Heavy wheel = slower acceleration
Light wheel = quicker response
Torque (τ)
A rotational force that causes angular acceleration.
Unit: newton-meter (N·m)
τ = r × F
Intuition:
Bigger torque = stronger rotation
Like pushing a door:
Push near the hinge = weak torque
Push far from the hinge = strong torque
An engine produces torque to spin the wheels.
Brakes produce torque to slow the wheels.
Torque > Force
Torque at the wheel becomes pushing force at the ground.
F = τ / r
Example:
Torque = 400 N·m at the wheel
Wheel radius = 0.3 m
F = 400 / 0.3 ≈ 1333 N
The car pushes forward with 1333 newtons.
Bigger torque = more push
Bigger wheel = less force
Rotation > Speed
Wheel angular velocity determines how fast the car moves.
v = ω × r
Example:
Wheel speed: 100 rad/s
Radius: 0.3 m
v = 100 × 0.3 = 30 m/s ≈ 108 km/h
Force > Motion
Force determines acceleration based on mass.
F = m × a
or
a = F / m
Example:
Force: 1333 N
Car mass: 1000 kg
a = 1333 / 1000 = 1.33 m/s²
Same force:
Light car = faster acceleration
Heavy car = slower acceleration
Torque > Rotation
Torque changes angular velocity, and the rate of change depends on inertia.
τ = I × α
or
α = τ / I
Example:
Torque: 200 N·m
Engine inertia: 0.2 kg·m²
α = 200 / 0.2 = 1000 rad/s²
Intuition:
Low inertia = revs quickly (motorbike engine)
High inertia = slower response (truck engine)
Power relationship
Power depends on both torque and angular speed.
P = τ × ω
Example:
Torque: 200 N·m
Engine speed: 100 rad/s (~955 RPM)
P = 200 × 100 = 20,000 W = 20 kW
Now double the RPM:
Torque: 200 N·m
Speed: 200 rad/s
P = 200 × 200 = 40,000 W = 40 kW
Same torque, more speed = more power
Intuition:
Torque = “strength”
Speed = “how fast you’re using it”
Power = “how much work you’re doing”
Grip Limit
A tire can only generate a limited amount of force before it starts to slip.
F_max = μ × N
Example:
Normal load: 4000 N
μ = 1.0 (good asphalt)
F_max = 4000 N
Intuition:
More weight = more grip
Better tires = higher μ
Too much torque = slip
Convert RPM (revolutions per minute) to angular velocity (ω)
Key idea:
1 revolution = 2π radians
1 minute = 60 seconds
ω = (2π × RPM) / 60
This gives angular velocity ω in rad/s.
Convert angular velocity (ω) to RPM (revolutions per minute)
Key idea:
1 revolution = 2π radians
1 minute = 60 seconds
RPM = (ω × 60) / 2π
This converts angular velocity ω in rad/s to RPM.
Clutch Torque Capacity
Maximum torque the clutch can transmit before slipping
Unit: N·m
Clutch Slip (Δω)
difference between engine speed and gearbox input speed.
Unit: rad/s
Δω = ω_engine – ω_gearbox
Transmitted Torque
How “grippy” the clutch material is
Clutch Friction Coefficient (μ)
Actual torque passing through the clutch
Unit: N.m
Δω = ω_engine – ω_gearbox
Clutch Stiffness
How strongly clutch resists speed difference
Unit: N.m / (rad/s)
T = stiffness * Δω
(clamped by max torque)
Slip Ratio
Slip ratio measures the difference between the wheel’s surface speed and the vehicle’s forward speed.
Intuition:
Positive slip ratio = driven wheel spins faster than the car is moving
Negative slip ratio = braking slip
Too much slip = loss of traction
Slip Angle
Slip angle is the angle between the direction the wheel is pointing and the actual direction the tire is traveling.
Intuition:
A small slip angle generates lateral tire force
Too much slip angle reduces grip and causes sliding
Slip angle is very important for cornering behavior and tire handling.