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.

Transfer Function

Laplace transform (eq. 1) and (eq. 2):

Solve for the 2 unknowns:

Solve for output/input:


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 αΊ‹(t) and y(t) in terms of x(t) and u(t):

Organize into matrix format:


Note: this system is based off of an example in a textbook[1]. It is modified and extended with additional calculations.

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