Quarter-car model — VehicleLab Documentation

Model IDvehiclelab.quarter-car.linear-passive

Model version0.2.0

Product release0.6.0

Physical interpretation

Sprung mass, m_s

The quarter share of body-side mass supported by the suspension.

Unsprung mass, m_u

The wheel-side mass represented below the suspension.

Suspension, k_s and c_s

A linear spring and viscous damper acting on relative displacement and velocity.

Tire, k_t

A linear vertical stiffness acting between unsprung mass and prescribed road input.

Coordinates and state

Displacements are measured from loaded static equilibrium and positive displacement is upward. The implemented first-order state order is:

\mathbf{x}=\begin{bmatrix}z_s&\dot z_s&z_u&\dot z_u\end{bmatrix}^{\mathsf T}

The generic preset starts all four states at zero. Gravity is absent because static gravitational loads are already balanced by the spring and tire deflections represented by the coordinate origin; adding it again would double-count static load.

Governing differential equations

m_s\ddot z_s=-k_s(z_s-z_u)-c_s(\dot z_s-\dot z_u)

m_u\ddot z_u=k_s(z_s-z_u)+c_s(\dot z_s-\dot z_u)-k_t(z_u-z_r)

Sprung-mass acceleration is evaluated directly from the first force balance at every output sample; it is not calculated by differentiating a plotted velocity channel.

Derived displacement definitions

Suspension travel

z_{\mathrm{travel}}=z_s-z_u

Dynamic tire deflection

z_{\mathrm{tire}}=z_u-z_r

Both are calculated in metres. Dynamic tire deflection excludes static preload.

Prescribed road inputs

FlatZero displacement throughout

Cubic smooth stepRises smoothly and remains elevated

Half-sine bumpTransient event that returns to zero

Raised-cosine bumpZero slope at onset, peak, and return

Flat road

z_r(t)=0

Cubic smooth step

z_r(t)=\begin{cases}0&t<t_0\\A(3\tau^2-2\tau^3)&t_0\le t<t_0+t_r\\A&t\ge t_0+t_r\end{cases},\quad \tau=\dfrac{t-t_0}{t_r}

The input rises smoothly to amplitude A and remains there after rise duration t_r.

Half-sine bump

z_r(t)=\begin{cases}A\sin\!\left(\dfrac{\pi(t-t_0)}{t_b}\right)&t_0\le t\le t_0+t_b\\0&\text{otherwise}\end{cases}

The input reaches amplitude A and returns to zero after bump duration t_b.

Raised-cosine bump

z_r(t)=\begin{cases}\dfrac{A}{2}\left[1-\cos\!\left(\dfrac{2\pi(t-t_0)}{t_b}\right)\right]&t_0\le t\le t_0+t_b\\0&\text{otherwise}\end{cases}

The input is exactly zero with zero slope at both event boundaries and reaches amplitude A at the midpoint. For a spatial bump of length L traversed at speed v, the event duration is t_b = L/v. Height controls amplitude; length and speed control traversal time.

Model version 0.2.0 adds this supported road-input contract. The two governing differential equations and passive force balance are unchanged.

SI units

Symbol	Quantity	Unit
z_s, z_u, z_r	Displacement	m
dot z_s, dot z_u	Velocity	m/s
ddot z_s	Sprung-mass acceleration	m/s²
m_s, m_u	Mass	kg
k_s, k_t	Stiffness	N/m
c_s	Viscous damping	N·s/m
t	Time	s

Generic passenger-car quarter preset

The preset is a transparent starting point, not data for a named production vehicle.

Input	Default
Sprung mass	300 kg
Unsprung mass	40 kg
Suspension stiffness	18,000 N/m
Suspension damping	1,500 N·s/m
Tire stiffness	180,000 N/m
Road input	Half-sine; 0.05 m; starts 0.5 s; duration 0.2 s
Solver	5 s duration; 0.001 s requested step

Motion playback boundary

A schematic view of the documented coordinates

Optional playback represents only sprung vertical displacement, unsprung vertical displacement, and prescribed road displacement. The geometry is schematic, movement is visually amplified, and the display is not drawn to geometric scale. It does not add forward motion, wheel rotation, pitch, roll, linkage kinematics, or contact loss.

Assumptions and excluded phenomena

Appropriate use boundary

Linear, time-invariant spring, damper, and tire
Rigid masses with vertical motion only and continuous tire contact
No geometry, friction, bump stops, hysteresis, tire damping, pitch, roll, or aerodynamics
No uncertainty, identification, control, or physical-vehicle correlation

The model supports transparent exploration of the stated equations. It is not evidence of production-vehicle fidelity and is not a substitute for certification, homologation, safety analysis, or correlated testing.