The spring forces described thus far assumed the forces to be proportional to displacement. If the spring has non-linearities in it, then the restoring force is no longer just proportional to displacement but is a squared or a cubic function. As noted by Fay & Graham, the restoring spring force can be modeled as:
The ODE’s then become:
The following values were used for the simulation:
The initial values used were:
The patterns for this set of ODE’s are plotted below. What started as a relatively “simple” 2-DOF system has resulted in a fairly intricate displacement-web with a slight non-linearization of the restoring forces.
[…] is one last simulation for the mass-spring-damper system, with a non-linear spring. The following values were used for the […]