A simple model of the vertical vibrations of a car is the "Quarter Car", which consists of one mass for the car and a smaller mass for the tyre that is connected elastically to the car and the ground. Driving on uneven ground leads to excitations of the system that are here modelled by a harmonic function for the base point position u(t)

Besides the excitations x_{i}(t) one is mainly interested in the
vertical acceleration a_{2} of the car (and the driver) and in the
total force R that is transmitted to the ground.

Use the applet to find for fixed car parameters the excitation
frequencies where a_{2} or R are maximal. Then change the
suspension parameters c_{i}, b_{i} to minimize either
a_{2} or R. How could one define an "optimal" suspension
system?

Use the standard procedures of vibration theory to verify your findings.