# Car Suspension System

Transfer function and state space model are developed for car suspension system shown below. This system is an applied version of the mass spring damper system. (input) f(t) = external force applied on tire (output) z1(t) = displacement of body unknown z2(t) = displacement of wheel M1 = car body mass M2 = wheel mass K1 = suspension spring constant K2 = tire elastance B = shock absorber damping coefficient

## Differential Equation

2 unknowns so 2 equations are needed.

• Newton's 2nd law on body From above free body diagram,
(eq. 1) • Newton's 2nd law on wheel From above free body diagram,
(eq. 2) ## Transfer Function

Laplace transform (eq. 1) and (eq. 2): Solve for the 2 unknowns: Solve for output/input:

(solution) ## Block Diagram ## State Space Model

When there is a mass in a system, its position and velocity are commonly chosen as state variables. For this system, position and velocity of both masses and force (input) are sufficient to determine any future ouput value (body's position). For these reasons, position and velocity of both masses are chosen as state variables.

State vector: Input vector: Output vector: Rewrite (eq. 1) and (eq. 2) in these new notations: Rearrange equations to express ẋ(t) and y(t) in terms of x(t) and u(t): Organize into matrix format:

(solution) Note: this system is based off of an example in a textbook. It is modified and extended with additional calculations.

Phillips, Parr (2011) Feedback Control Systems 5th EditionFeedback Control Systems