Mass Spring Damper System

Transfer function and state space model are developed for system shown below.

(input) f(t) = external force applied on mass
(output) z(t) = position unknown
m = mass
k = spring constant
b = damping coefficient

Differential Equation

1 unknown so 1 equation is needed.

Newton's 2nd law:

(eq. 1)

Transfer Function

Laplace transform (eq. 1):

Solve for Y(s)/F(s):


Block Diagram

State Space Model

When there is a mass in a system, its position and velocity are commonly chosen as state variables. Also, position, velocity, and force (input) are sufficient to determine this system's future position (output). For these reasons, position and velocity are chosen as state variables.

State vector:

Input vector:

Output vector:

Solve for αΊ‹(t) and y(t) using (eq. 1):

Organize into matrix format: