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.

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 (M_s): The car body (quarter of total body mass)
• Unsprung Mass (M_us): Wheel, tire, brake, and suspension components
• Suspension Spring (k_s): Connects body to wheel
• Shock Absorber (c_s): Damper parallel to spring
• Tire (k_t): 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:

State-Space Representation

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

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:

    ┌─────────┐
    │   M_s   │  Car Body (Sprung Mass)
    └────┬────┘
         │ k_s, c_s  (Suspension Spring + Damper)
    ┌────┴────┐
    │  M_us   │  Wheel (Unsprung Mass)
    └────┬────┘
         │ k_t       (Tire Spring) ← Road enters HERE!
    ═════╧═════════  y_r = Road Height (prescribed input)
                        

The tire force is calculated as:

• When y_r increases (bump rises): tire compresses → pushes wheel UP
• When y_r decreases (pothole): tire extends → pulls wheel DOWN

Why Base Excitation Instead of Direct Force?

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 — softer tires absorb more bumps!

Note: y_r(t) is a prescribed input, not a degree of freedom. The system has only 2 DOF (y_s and y_us); 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:

Damping and Ride Quality

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

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

Natural Frequency

The undamped natural frequency determines how fast the system oscillates:

• Passenger cars: f_n ≈ 1-1.5 Hz (comfortable)
• Sports cars: f_n ≈ 1.5-2.5 Hz (responsive)
• Race cars: f_n ≈ 3-5 Hz (maximum grip)

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:
   • ▶ Run / ⏸ Pause: Toggle continuous simulation
   • ◀ Step / Step ▶: Single-step backward/forward for detailed analysis
   • 💥 Bump: Apply instant impulse to wheel
   • ↺ 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 — 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

Road Surface Types

Key Experiments

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:

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 — the physics detected the bump at that exact moment!

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

RK4 Integration Algorithm

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

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:

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

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 (▶): Advances simulation by one timestep (with substeps for stability)
• Step Bwd (◀): 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

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