Why separate sprung and unsprung masses?

The body rides on the suspension while the wheel–tire assembly has its own mass and resonates against the tire spring. Treating them as one lump misses the wheel-hop mode near the tire natural frequency. A common misconception is that stiffer suspension always improves ride; it often worsens body acceleration while mainly affecting handling.

What does the damper Cₛ mainly control?

Cₛ dissipates energy between sprung and unsprung masses and sets how quickly body motion settles after a bump. Too little damping leaves ringing; too much couples the body tightly to the wheel so road roughness transmits upward. In this linear model the damper does not generate force from absolute velocity alone—only from relative velocity across the suspension.

What if the road frequency matches a natural frequency?

Near resonance the corresponding mode amplitude grows until limited by damping. Matching the body mode worsens ride; matching wheel hop stresses the tire contact. The simulator’s sinusoidal road and RK4 traces make that frequency dependence visible without nonlinear tire liftoff.

Quarter-Car Suspension

¼-vehicle vertical model: sprung vs unsprung masses, Kₛ, Cₛ, tire Kₜ, sinusoidal road — RK4 time histories.

Live graphs

About this model

This simulator implements the classic quarter-car vertical dynamics model: a sprung mass (body) and unsprung mass (wheel/hub) linked by suspension spring Kₛ and damper Cₛ, with the tire represented as a spring Kₜ contacting a sinusoidal road profile. The two-DOF equations of motion are integrated with RK4 to produce time histories of body and wheel displacement, velocity, and acceleration. Assumptions include linear springs and damper, purely vertical motion, no pitch/roll, and a prescribed road input (no tire damping or nonlinear bushings). You can vary sprung and unsprung masses, Kₛ, Cₛ, Kₜ, and road amplitude/frequency to see how ride comfort and tire contact respond.

Who it's for: Advanced undergraduate vehicle dynamics, mechanical vibration, and automotive engineering courses.

Key terms

quarter-car model
sprung mass
unsprung mass
suspension damping
tire stiffness
ride comfort

How it works

Classic quarter-car vertical model: sprung mass (body), unsprung mass (wheel assembly), suspension spring-damper, and tire spring under a sinusoidal road profile. RK4 integration at 120 Hz.

Key equations

Mₛ zₛ″ = Kₛ(zᵤ−zₛ) + Cₛ(żᵤ−żₛ)

Mᵤ zᵤ″ = Kₜ(zᵣ−zᵤ) − Kₛ(zᵤ−zₛ) − Cₛ(żᵤ−żₛ)

Frequently asked questions

Why separate sprung and unsprung masses?: The body rides on the suspension while the wheel–tire assembly has its own mass and resonates against the tire spring. Treating them as one lump misses the wheel-hop mode near the tire natural frequency. A common misconception is that stiffer suspension always improves ride; it often worsens body acceleration while mainly affecting handling.
What does the damper Cₛ mainly control?: Cₛ dissipates energy between sprung and unsprung masses and sets how quickly body motion settles after a bump. Too little damping leaves ringing; too much couples the body tightly to the wheel so road roughness transmits upward. In this linear model the damper does not generate force from absolute velocity alone—only from relative velocity across the suspension.
What if the road frequency matches a natural frequency?: Near resonance the corresponding mode amplitude grows until limited by damping. Matching the body mode worsens ride; matching wheel hop stresses the tire contact. The simulator’s sinusoidal road and RK4 traces make that frequency dependence visible without nonlinear tire liftoff.

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Live graphs

About this model

How it works

Key equations

Frequently asked questions

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Live graphs

About this model

How it works

Key equations

Frequently asked questions