Algorithmic representation – An Example

Consider, for example, a 3-degree of freedom mass-spring-damper system as shown below. 

Each of the masses (m1 thru’ m3) is connected to springs (K1-K3) and dampers (C1-C3), with K3 and C3 connected to ground. Each of these masses can be subject to a time-dependent forcing function (F1-F3). The resultant displacement of each of these masses due to the forcing functions can be described by the 2^nd order ODE’s by applying Newton’s laws of motion. 

 

  

Using the conversion mentioned earlier, the above three 2^nd order ODE’s result in the following six 1^st order ODE’s:

 

 

 

 

 

By rearranging the above 1^st order ODE’s to solve for and , the numerical solutions can be readily obtained.