Quarter car model

The quarter car model can be used to observe the vertical behaviour of a car on uneven road surfaces.
This model gives a good understanding of system dynamics.
The quarter car model has two degrees of freedom, namely the displacement of the wheel and the displacement of the car body.
In the figure below the free body diagram of the quarter car model is shown. Mass m_{s} is the mass of the total car body divided by four. Mass m_{u} is called the unsprung mass.
Unsprung mass consists of the wheel, the brake disc and a part of the axle.
The forces between the body and the wheel are transmitted via a spring and a damper.
The tyre characteristics can be modelled as a spring either.
The level of comfort experienced by people in a car is mainly determined by the magnitudes of the vertical accelerations.
Suppose a mapping of the height variations of a certain route was available.
Given the suspension properties and the weight of the sprung and unsprung mass, the average comfort level could be quantified.
Optimization algorithms can be used in order to obtain the optimal stiffness values for the springs, which gives us the optimal comfort for this route.

As is known from general dynamics, the acceleration of a body is equal to the sum of all the forces acting on it, divided by its mass. In other words, the net force in a certain direction acting on the body is equal to the mass multiplied by the acceleration in that direction.