Runge Kutta 4 - A Numerical Example

Oscillations and vibrations of structural elements are commonly represented as a 2^nd order ODE (derived from Newton’s law). In his paper, Piche (An L-stable Rosenbrock Algorithm for Step-By-Step Time Integration in Structural Mechanics, Computational Methods in Applied Engrg., Vol 126, 1995, pp 343-354) presents one such example of a two-degree of freedom mass-spring-damper system with the following ODEs:

 
with the initial values and constants defined as:

 

Second order ODE’s can be converted to first order ones by substituting equivalent expressions for the first order ones. For example, the above expression is based on Newton’s law, where y1 and y2 represent the motion of the system. The first derivative of displacement is velocity, and the first derivative of velocity is acceleration. Thus, each equation above gets split up into two first order eqns. as shown below:

 



A similar substitution is performed for the other equation also, thus resulting in four 1^st order ODE’s. A numerical integration scheme can then used to solve for y1, v1, y2 and v2.

The RK4 method was used to solve these equations with a step-size of 0.001, and the results for y1 and y2 are shown below.