A simplified multi-degree of freedom system model of an automobile suspension system is shown in Figure 1. The automobile traveling along a level road at a constant horizontal speed v0 encounters bumps in the road shown of the shapes shown in Figure 2.
The vehicle's suspension system (front and rear springs and shock absorbers) is modeled by linear springs and dampers, and the compliance of the tires is modeled by front and rear springs. The vehicle motion is limited to heave in the vertical direction and a small amount of pitch Θ of the vehicle's longitudinal axis. The tires are assumed to remain in contact with the road surface at all times.
The road profile is responsible for the system's input u Θ [uf ur] T where u f and ur are the height of the road (with respect to some reference) underneath the front and rear tires, respectively. The system has three translational degrees of freedom, y , y f yr which are the vertical displacements of the vehicle and both front and rear axles from their equilibrium positions. The lone rotational degree of freedom is the pitch angle Θ
1. Assuming small motions, derive the differential equations for the pitch angle Θ and the vertical (heave) displacement y , yf , yr of the centroid of the vehicle using uf (t) and ur(t) as the inputs.
2. Write the equations of motion in matrix form and identify the mass matrix [M] the stiffness matrix [K] and the damping matrix [C].
3. Define a suitable set of state space variables and write the equations of motion in state space form.
{q}=[A]{q}+[B]{u}
{q}=[C]{q}+[D]{u}
4. Find the natural frequencies and mode shapes of the system.
5. For each of the road profile shown in Figure 2, plot the responses y , yf , yr and Θ using MATLAB.