Quarter-Car Suspension Dynamics & Road-Disturbance Response

This MATLAB project studies the vertical response of a two-degree-of-freedom quarter-car suspension under a deterministic road disturbance. The public version preserves the archived four-state passive model and its source-derived parameters, then extends the analysis with modal checks, suspension-travel and tire-deflection metrics, direct road-to-body frequency response, and a clearly labelled damping-sensitivity study.

The complete MATLAB implementation, source audit, and generated numerical results are available here: GitHub repository

The archived simulation sets the inter-mass force input to zero. This page therefore presents a passive suspension study; it does not claim an active or optimal suspension controller.

Source-derived quarter-car model

The state vector is x = [zs, żs, zu, żu]ᵀ, where zs is sprung-mass displacement and zu is unsprung-mass displacement. The two model inputs are an inter-mass force and road displacement; the archived simulation uses zero force input.

Parameter	Value
Sprung mass	100 kg
Unsprung mass	10 kg
Suspension stiffness	200 N/m
Tire stiffness	200 N/m
Suspension damping	500 N·s/m
Control-force input	0 N

Passive stability

The source-derived nominal state matrix is asymptotically stable. The passive poles are approximately −54.2597, −0.4345, and −0.1529 ± 1.2935j; all real parts are negative.

Road disturbance

The archived indexed input is represented by its continuous-time equivalent: a 0.10 m half-sine bump over 0–20 s followed by a 0.06 m half-sine bump over 40–60 s.

Two half-sine road bumps used as the suspension disturbance — **Figure 1. Source-derived road disturbance.**The cleaned runner reproduces the intended two-bump excitation over the 100 s experiment.

Body and wheel displacement response

Road, sprung-mass, and unsprung-mass displacements — **Figure 2. Passive quarter-car displacement response.**The sprung and unsprung masses follow the slow road input with lightly damped transients after each bump.

Nominal response metric	Value
Maximum road displacement	0.100000 m
Maximum sprung displacement	0.104200 m
Maximum unsprung displacement	0.102653 m
Maximum suspension travel	0.003710 m
Maximum tire deflection	0.010590 m
Maximum \|body acceleration\|	0.019350 m/s²
RMS body acceleration	0.004920 m/s²
Peak body/road displacement ratio	1.0420

Suspension travel and tire deflection

Vertical acceleration

Phase-plane response

Sprung-mass displacement versus velocity phase plane — **Figure 5. Sprung-mass phase plane.**The trajectories contract after each road event, consistent with the stable passive poles.

Road-to-body frequency response

The modernization adds a direct numerical evaluation of H(jω) = C(jωI − A)⁻¹Br from road displacement to sprung-mass displacement. This analysis is new; it was not part of the archived script.

Road-displacement to sprung-displacement frequency response — **Figure 6. Road-to-body frequency response.**The source-derived academic parameter set produces a peak magnitude of about 4.619 near 0.2036 Hz.

Frequency-response metric	Value
Peak road-to-body magnitude	4.619
Peak magnitude	13.29 dB
Frequency at peak	0.2036 Hz

These resonance values are properties of the archived academic parameter set and are not presented as validated production-vehicle suspension characteristics.

New damping-sensitivity analysis

The archived nominal damping is c = 500 N·s/m. The cleaned project adds a sweep from 100 to 1200 N·s/m to expose the tradeoff between ride acceleration and suspension travel.

RMS body acceleration and suspension travel versus damping — **Figure 7. New damping-sensitivity analysis.**The damping value that minimizes RMS body acceleration is not the same value that minimizes suspension travel.

Damping-sweep result	Value
Sweep range	100–1200 N·s/m
Archived nominal damping	500 N·s/m
Minimum-RMS-acceleration damping	200 N·s/m
Minimum RMS body acceleration	0.004288 m/s²

For this specific road test, damping near 200 N·s/m minimizes RMS body acceleration, while increasing damping continues to reduce suspension travel. The archived 500 N·s/m value is therefore best interpreted as a compromise point, not as a uniquely optimal damping choice.

Scope and limitations

This is a linear two-degree-of-freedom quarter-car model using archived academic parameters. Suspension geometry, tire nonlinearity, bump stops, actuator dynamics, load transfer, and experimental parameter identification are outside its scope. The stiffness values are also much lower than typical automotive values, so the time scales and resonance should be interpreted only within this model. The archived control-force channel is present in the state-space equations but is fixed to zero in the validated experiment.

Reproducibility

The public repository contains the original archived suspension.m file as legacy material and a self-contained MATLAB runner that reproduces the passive response, metrics, frequency response, damping sweep, and figures without requiring additional toolboxes. The reported results were validated in MATLAB R2022b.