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.

Algorithmic Representation

Numerical solution of DE’s is accomplished by feeding in first order differential form of the series of equations to be solved to an algorithm like Runge Kutta or Rosenbrock. First order ODE’s can be easily represented in this manner. With higher order ODE’s, the higher orders are first converted into 1^st order ones before an algorithm can be implemented.

Take for example the 2^nd order mass-spring-damper system model:

This equation can be numerically solved using the conversion

      

 

To be more specific, let us assume that the variables to be solved are var1 and var2. The algorithmic form would then become:

 

For a series of higher order ODE’s, one has to convert the successive higher orders into first order equivalents and then use the ODE solvers.