The equations used here can be described graphically. The ranges of spring forces and spring deformations D can be plotted on a graph. Where they overlap defines the area where permissible spring rates may pass.
To find the minimum spring rate, draw a line through the graph's origin and the {D_{min}, F_{min}} point. Likewise, to find the maximum spring rate, draw a line through the graph's origin and the {D_{max}, F_{max}} point. Both lines are shown on the following figure,
All values of k in between the k_{min} and k_{max} lines are permissible, as shown by the shaded region on the graph.
To include the effect of a manufacturing toloerance on the spring rate k, move the k_{min} and k_{max} lines together by the tolerance fraction, as shown on this figure,
If we find that k_{max} is less then k_{min}, then no solution exists.
