Stiffness of Conical Springs

For biconical springs, which are applied in the Bonnell-type spring systems, the torsion moment Ms is not a value dependent only on the load, but a function of the changing length of the coil radius Ms = f(R). This radius depends on the angle of the unstretching of the spring a, R = f(a) (Fig. 8.26). The angle a is the angle between the intermediate radius R (variable) and the upper radius of the spring coil R1 depending on the number of the spring coils n. Therefore, the increase in the length of the intermediate radius can be written as:


R1 the largest radius of the coil,

R2 the smallest radius of the coil, a the angle of the unstretching of the spring and n number of the spring coils.

We calculate the strength of the wire in the conical spring by entering into the equation for maximum stresses Tmax the values of the intermediate radius R. When determining the values of the conical spring deflections, the change in the length of the intermediate radius should also be taken into account:

And because the torsion moment Ms is a function of the intermediate radius, the deflection of the spring under the load P can be therefore written as:

where as it can be seen from Fig. 8.25, ds = Rda.

By assuming the borders of variation 0 < a < 2nn as the integration boundaries, we eventually obtain the equation for the deformation of the conical spring:

For conical springs, the stiffness k can be written as:

This coefficient has significant importance in the calculation of the stiffness of whole spring systems.