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Quarter Car Suspension Model Tutorial 

This interactive simulation demonstrates the Quarter Car Model, the fundamental model for analyzing vehicle suspension systems. It shows how a car's suspension isolates the body from road disturbances using springs and dampers.

Sections

What is the Quarter Car Model?

The quarter car model represents one corner of a vehicle as a 2-DOF (degree of freedom) mass-spring-damper system:

  • Sprung Mass (Ms): The car body (quarter of total body mass)
  • Unsprung Mass (Mus): Wheel, tire, brake, and suspension components
  • Suspension Spring (ks): Connects body to wheel
  • Shock Absorber (cs): Damper parallel to spring
  • Tire (kt): Modeled as a spring connecting wheel to road

The Equations of Motion

The system is governed by Newton's Second Law applied to each mass. The sprung mass (car body) is pushed only by the suspension spring and damper:

Ms · ΓΏs = −ks(ys − yus) − cs(ẏs − ẏus)

The unsprung mass (wheel) feels those same suspension forces plus the tire spring force from the road:

Mus · ΓΏus = ks(ys − yus) + cs(ẏs − ẏus) − kt(yus − yr)

State-Space Representation

For numerical simulation, we convert to first-order ODEs:

State vector: x = [ys, ẏs, yus, ẏus]T. The four first-order derivatives are:

ẏs = vs
vΜ‡s = [−ks(ys − yus) − cs(vs − vus)] / Ms
ẏus = vus
vΜ‡us = [ks(ys − yus) + cs(vs − vus) − kt(yus − yr)] / Mus

How Road Surface Enters the System: Base Excitation

The road surface is NOT applied as a direct external force. Instead, it enters through the tire spring as a kinematic (base) excitation:

The Tire Spring Model:

    β”Œβ”€β”€β”€β”€β”€β”€β”€β”€β”€β”
    β”‚   M_s   β”‚  Car Body (Sprung Mass)
    β””β”€β”€β”€β”€β”¬β”€β”€β”€β”€β”˜
         β”‚ k_s, c_s  (Suspension Spring + Damper)
    β”Œβ”€β”€β”€β”€β”΄β”€β”€β”€β”€β”
    β”‚  M_us   β”‚  Wheel (Unsprung Mass)
    β””β”€β”€β”€β”€β”¬β”€β”€β”€β”€β”˜
         β”‚ k_t       (Tire Spring) ← Road enters HERE!
    ═════╧═════════  y_r = Road Height (prescribed input)

The tire force is calculated as:

Ftire = kt × (yus − yr)
  • When yr increases (bump rises): tire compresses β†’ pushes wheel UP
  • When yr decreases (pothole): tire extends β†’ pulls wheel DOWN
Why Base Excitation Instead of Direct Force?

Approach

Input

Physical Meaning

Base Excitation (Used)

yr(t) — road height profile

Road is a prescribed displacement; tire spring converts geometry to force

Direct Force (Not used)

Froad(t) — force input

Would require measuring force directly (impractical)

Base excitation is realistic because we know the road's geometry (height profile), not the force. The tire's deflection determines the actual force transferred to the wheel. This also allows modeling tire compliance β€” softer tires absorb more bumps!

Energy Flow Path:

Road Profile (yr) → Tire Spring (kt) converts displacement→force → Wheel (Mus) moves → Suspension (ks, cs) filters & dampens → Body (Ms) feels smoothed motion
2 DOF, not 3: yr(t) is a prescribed input, not a degree of freedom. The system has only two states that move freely (ys and yus); the road acts as a time-varying boundary condition.

Why RK4 Integration?

Spring-damper systems are stiff - simple Euler integration causes numerical instability (the simulation "explodes"). We use Runge-Kutta 4th Order (RK4) for stability:

Method

Accuracy

Stability

Use Case

Euler

O(h)

Poor

Simple problems only

RK4

O(h⁴)

Good

Most physics simulations

Implicit Methods

Varies

Excellent

Very stiff systems

Damping and Ride Quality

The damping ratio (ΞΆ) determines how the system responds to disturbances:

ccr = 2√(ks · Ms)      ζ = cs / ccr

Damping Ratio

Condition

Behavior

Feel

ζ < 1

Under-damped

Oscillates before settling

Bouncy, “floaty”

ζ = 1

Critically damped

Fastest return without overshoot

Ideal (theory)

ζ > 1

Over-damped

Slow return, no oscillation

Harsh, stiff

Engineering Goal: Most cars target ΞΆ β‰ˆ 0.2-0.4 (under-damped) for comfort, while sports cars use ΞΆ β‰ˆ 0.5-0.7 for better control.

Natural Frequency

The undamped natural frequency determines how fast the system oscillates:

fn = (1/2π) · √(ks / Ms)  Hz
  • Passenger cars: fn β‰ˆ 1-1.5 Hz (comfortable)
  • Sports cars: fn β‰ˆ 1.5-2.5 Hz (responsive)
  • Race cars: fn β‰ˆ 3-5 Hz (maximum grip)

Simulation

The interactive simulator is below. Use the controls to explore the concepts described above.

πŸš— Vehicle Preset

βš–οΈ Mass

Sprung Mass (Ms) 300kg
Unsprung Mass (Mus) 40kg

πŸ”§ Suspension

Spring Stiffness (ks) 20.0kN/m
Damping (cs) 1500NΒ·s/m
Tire Stiffness (kt) 150kN/m
Damping Status Critical
ΞΆ = 0.61 | fn = 1.3 Hz

πŸ›£οΈ Road Surface

Vehicle Speed 1m/s
Road Type

πŸ“Š State

Body Pos 0.0 cm
Wheel Pos 0.0 cm
Body Vel 0.0 cm/s
Wheel Vel 0.0 cm/s
πŸ“ˆ Position vs Time
● Body ● Wheel ● Road
Displacement vs Accel
● Body ● Wheel
Velocity vs Accel
● Body ● Wheel
Body:
MsΓΏs = -ksΞ”y - csΔẏ
Wheel:
MusΓΏus = ksΞ”y + csΔẏ - kt(yus-yr)
Ξ”y = ys - yus
Δẏ = ẏs - ẏus
Car Body (Ms)
Wheel (Mus)
Spring (ks)
Damper (cs)
Road Input (yr)

Usage Instructions

  1. Select a Vehicle Preset: Choose from sedan, SUV, sports car, etc. Each has realistic suspension parameters.
  2. Adjust Parameters:
    • Sprung Mass: Car body mass (heavier = slower response)
    • Unsprung Mass: Wheel assembly mass (lighter = better road following)
    • Spring Stiffness: Higher = stiffer ride, more responsive
    • Damping: Controls oscillation decay (watch the damping status!)
    • Tire Stiffness: Usually much higher than suspension spring
  3. Select Road Type: Different road profiles to test suspension behavior
  4. Controls:
    • β–Ά Run / ⏸ Pause: Toggle continuous simulation
    • β—€ Step / Step β–Ά: Single-step backward/forward for detailed analysis
    • πŸ’₯ Bump: Apply instant impulse to wheel
    • β†Ί Reset: Return to initial state
  5. Observe the Visualization:
    • Car Animation: Watch the car body and wheels respond independently
    • Red Dot (●): Shows exactly where physics samples the road β€” the car jumps when the red dot crosses a bump!
    • Position vs Time: Real-time plot of body, wheel, and road positions
    • Phase Plots: Displacement-Acceleration and Velocity-Acceleration reveal dynamic behavior
    • Model Diagram: Abstract mass-spring-damper view synchronized with car
    • Damping Status: Shows under-damped (bouncy), critical, or over-damped (stiff)

Vehicle Presets

Vehicle

Ms (kg)

ks (kN/m)

cs (NΒ·s/m)

Character

πŸš— Sedan

300

20

1500

Balanced comfort/handling

🏎 Sports Car

250

35

2500

Stiff, responsive, flat cornering

πŸš™ SUV

450

25

2000

Soft, comfortable, body roll

🎩 Luxury

400

18

2200

Very soft, "magic carpet" ride

🏁 Rally

280

40

3500

High damping for rough terrain

πŸ† F1

150

80

4000

Extremely stiff, low mass

Road Surface Types

Type

Description

Test Purpose

Flat

Perfectly smooth surface

Baseline - observe natural decay after bump

Waves

Sinusoidal undulations

Continuous excitation, resonance testing

Bumps

Discrete speed bumps

Impulse response, settling time

Steps

Square wave profile

Step response, overshoot analysis

Pothole

Sudden dips

Rebound behavior, bottom-out protection

Random

Realistic rough road

General ride quality assessment

Key Experiments

Experiment

Setup

What to Observe

Under-Damping

Set cs = 500 NΒ·s/m, click "Bump!"

Car body bounces multiple times before settling

Over-Damping

Set cs = 5000 NΒ·s/m, click "Bump!"

Car slowly returns without bouncing, feels "heavy"

Critical Damping

Adjust cs until ΞΆ β‰ˆ 1.0

Fastest return without overshoot

Resonance

Use "Waves" road, adjust speed to match fn

Amplitude grows dramatically at resonance!

Mass Effect

Compare Ms = 200 kg vs 600 kg

Heavier body = slower oscillation, lower fn

Tire Stiffness

Compare kt = 100 kN/m vs 300 kN/m

Stiffer tire = wheel follows road more closely

Engineering Insights

  • Trade-off: Soft suspension = comfort but body roll; Stiff = handling but harsh ride
  • Unsprung Mass Matters: Lighter wheels improve both comfort and handling
  • Damping is Critical: Too little = bouncy; too much = harsh
  • Real Cars: Use variable damping (adaptive suspension) to get best of both worlds

Visual Synchronization: The "Treadmill" Model

This simulation uses a "treadmill" coordinate system to visualize car dynamics:

Key Concept: The car doesn't move horizontally on screen. Instead, the road scrolls underneath it β€” just like running on a treadmill!

Component

Behavior

Reference Point

Car (Visual)

Fixed at screen center

Wheel aligned with ● red dot

Road (Visual)

Scrolls from right to left

Bump appears at red dot when wheel "hits" it

Physics Engine

Samples road height at current position

Exactly at the red dot location

The Red Dot (●): This marker shows exactly where the physics engine samples the road height. When you see the car "jump," it's because a bump has reached the red dot β€” the physics detected the bump at that exact moment!

Synchronization Math:

tireScreenX = canvas pixel where wheel appears
road.position = world distance traveled (meters)

For any canvas pixel 'px':
  worldMeter = road.position + (px - tireScreenX) / pixelsPerMeter

At tireScreenX: worldMeter = road.position (exact match!)

This coordinate system ensures that what you see is what the physics calculates β€” the visual bump and the physics bump are perfectly aligned.

RK4 Integration Algorithm

The simulation uses 4th-order Runge-Kutta integration:

k1 = f(t, y)
k2 = f(t + h/2, y + hΒ·k1/2)
k3 = f(t + h/2, y + hΒ·k2/2)
k4 = f(t + h, y + hΒ·k3)

yn+1 = yn + (h/6)Β·(k1 + 2k2 + 2k3 + k4)

This provides O(h⁴) accuracy and excellent stability for oscillatory systems.

Phase Space Plots

The simulation displays two phase space plots that reveal the system's dynamic behavior:

Plot

Axes

What It Shows

Displacement vs Acceleration

X: y (cm), Y: a (m/sΒ²)

Force-displacement relationship, reveals stiffness behavior

Velocity vs Acceleration

X: v (m/s), Y: a (m/sΒ²)

Damping characteristics, energy dissipation patterns

Analytical Acceleration: To ensure smooth phase plots, acceleration is calculated directly from forces (F/m) rather than differentiating velocity numerically:

Body Acceleration:
as = (-ksΒ·Ξ”y - csΒ·Ξ”v) / Ms

Wheel Acceleration:
aus = (ksΒ·Ξ”y + csΒ·Ξ”v - ktΒ·(yus - yr)) / Mus

A light moving average filter is applied to the wheel acceleration for visualization (window=5) to smooth high-frequency noise without affecting the physics.

Step-by-Step Controls

The simulation supports stepping forward and backward through time:

  • Step Fwd (β–Ά): Advances simulation by one timestep (with substeps for stability)
  • Step Bwd (β—€): Restores the previous state from history

State history is automatically saved every 3 frames to enable backward stepping without excessive memory usage.

Real-World Applications

Industry

Application

Automotive

Suspension design, ride comfort analysis, handling optimization

Rail

Train bogie design, track-vehicle interaction

Aerospace

Aircraft landing gear design, vibration isolation

Civil Engineering

Building vibration control, seismic isolation

Robotics

Legged robot compliance, shock absorption

Extensions Beyond Quarter Car

  • Half Car Model: Adds pitch dynamics (front-rear coupling)
  • Full Car Model: 7-DOF with roll, pitch, and heave
  • Active Suspension: Add actuators for adaptive control
  • Non-linear Elements: Bump stops, progressive springs

Limitations

  • One corner only: the quarter-car model captures vertical (heave) motion of a single wheel. It cannot show pitch, roll, or front–rear weight transfer — those need half- or full-car models.
  • Linear springs and damper: ks, kt, and cs are constant. Real suspensions have progressive springs, bump stops, and velocity-dependent (non-linear) dampers.
  • Tire never leaves the road: the tire is a simple spring with no separation; wheel hop and loss of contact over sharp bumps are not modelled.
  • Prescribed road input: the road profile is a fixed kinematic input; tire damping, sprung-mass aerodynamics, and chassis flex are ignored.