Half Car Suspension- State Space Model

This 4-DOF half-car model simulates passive suspension dynamics for heave and pitch optimization using a state-space model.

The half-car suspension system comprises three primary inertial elements: (i) the chassis body, (ii) the front wheel, and (iii) the rear wheel. These elements are illustrated in the schematic diagram presented in Figure below:

Furthermore, the coordinate systems (CS) used to measure the displacement of each inertial element from its static equilibrium position are established. Here, z and θ denote the vertical bounce displacement and the pitch angle of the car body, respectively. Variables and represent the vertical displacements of the front and rear wheels, while and denote the road profile inputs acting on the front and rear tires.

To derive the equations of motion using Newton's second law, a free body diagram (FBD) for each inertial element is presented in Figure below:

In the FBD of the car body, the dynamic suspension forces and are generated by the front and rear spring elements () and damping elements (). Applying Newton's second law to the sprung mass (M) yields:

where the expressions for the front and rear suspension forces are defined as follows:

Substituting these force expressions into \eqref{eq:bod} yields the heave motion dynamics of the vehicle body:

Similarly, for the front wheel, the FBD indicates that the acting forces are the suspension force and the tire force , which is generated by the tire stiffness and tire damping . Applying Newton's second law to the front unsprung mass () yields:

where the expression for the tire force is given by:

Substituting the expressions for and into above equation results in the front wheel dynamics:

Applying the identical procedural steps to the rear wheel () yields the corresponding dynamics:

Finally, the angular pitching motion of the car chassis, governed by its pitch moment of inertia (), is derived by summing the moments about the center of gravity: