Project Overview
Objective: Model and analyze a quarter-car suspension system to study how stiffness and damping parameters affect vehicle comfort and stability. Using Newton’s Second Law, governing differential equations were derived for both the sprung and unsprung masses. These were implemented in MATLAB to simulate vehicle responses to road disturbances.
The project aimed to replicate real-world suspension dynamics by comparing three distinct vehicle types: a Formula 1 car, a trophy truck, and a Mazda Miata. Each system’s stiffness, damping, and tire elasticity were adjusted to reflect realistic suspension setups. This analysis demonstrated how suspension tuning can balance ride comfort and handling performance under varying speeds and terrain conditions.
Model Development
The model used a two-degree-of-freedom system—the sprung mass (vehicle body) and the unsprung mass (wheel assembly). The following forces were considered:
- Suspension spring and damper forces between the body and wheel
- Tire stiffness force between the wheel and the road
- External input from a road bump at 1 second into the simulation
From Newton’s Second Law, the derived governing equations were:
mₛx¨ₛ + cₛ(ẋₛ − ẋᵤ) + kₛ(xₛ − xᵤ) = 0
mᵤx¨ᵤ − cₛ(ẋₛ − ẋᵤ) − kₛ(xₛ − xᵤ) + kₜ(xᵤ − xᵣ) = 0
where mₛ and mᵤ are the sprung and unsprung masses, kₛ the suspension stiffness, cₛ the damping coefficient, and kₜ the tire stiffness. These relationships form the foundation of the simulation.
Simulation and Results
The MATLAB simulation visualized the system’s response to a sudden road disturbance for each vehicle type at three velocities: 10 m/s, 25 m/s, and 45 m/s. Displacement and velocity of both masses were analyzed over time.
Key findings include:
- The Formula 1 car exhibited minimal oscillation and near-identical movement of sprung and unsprung masses due to its high damping and stiffness, prioritizing handling stability.
- The Trophy Truck and Mazda Miata showed greater oscillation and delayed settling, consistent with comfort-oriented suspension systems.
- Increasing vehicle velocity amplified oscillation magnitude and reduced damping effectiveness, emphasizing the need for precise parameter tuning across speed ranges.
Parameter Sensitivity Analysis
Each vehicle’s suspension parameters were defined as follows:
- Normal Car (Miata): mₛ = 250 kg, mᵤ = 40 kg, kₛ = 15 000 N/m, cₛ = 1000 N·s/m, kₜ = 200 000 N/m
- Formula 1 Car: mₛ = 200 kg, mᵤ = 14 kg, kₛ = 30 000 N/m, cₛ = 3000 N·s/m, kₜ = 270 000 N/m
- Trophy Truck: mₛ = 800 kg, mᵤ = 68 kg, kₛ = 8000 N/m, cₛ = 2000 N·s/m, kₜ = 180 000 N/m
The damping ratio for each configuration was also calculated to quantify system behavior. These values validated that performance-oriented suspensions prioritize rapid damping and control, while off-road setups permit greater travel and oscillation for impact absorption.
Conclusion
The quarter-car simulation provided critical insights into how vehicle suspension tuning affects ride quality and handling. The F1 car achieved the lowest oscillation amplitude and fastest recovery, ideal for high-speed cornering. The Miata offered the most balanced ride comfort, while the Trophy Truck prioritized compliance and shock absorption. These findings highlight how stiffness and damping optimization directly influence performance tradeoffs between comfort and control.
Project Report
For full equations, MATLAB code, and simulation plots, you can view the complete project report below:
📄 View Full Report (PDF)