The State Space Method (and Matrix Representations)

The 2^nd order differential equations describing the 3-DOF KCM system can be condensed into a matrix format as shown below: 

 

Once the component matrices for Mass, Stiffness, Damping and the Forcing Functions are defined, the overall system can be further condensed into a matrix format as: 

 

Letters in bold-italic are now vectors in matrix form. The response vectors and forcing functions are single-column matrices, while the KCM ones are square. Working with these matrices along the same lines as mentioned earlier, one can chop it up into two first order ODE’s, which can be “matricized” as follows:

 

The logic behind the above representation becomes fairly apparent when one expands the matrices to two separate equations. To state the obvious, what was originally a single set of matrix-equation in 2^nd order has now doubled.

Performing one last condensing, the above matrix then takes the form : 

 

  

where

           

The result is a matrix of 1^st order ODE’s that can be numerically integrated to obtain the solution of vector U. Such a representation of the system of 2^nd order equations is commonly referred to as the State-Space Method.