The basic equations for elastic point and line contact stresses are as follows:

	Sphere-Plane	Cylinder-Plane
Contact Diameter/Width	b = 1.44·[DCEF]1/3	b = 1.60·[DCEF/L]1/2
Contact Area	A = πb2/4	A = b·L
Deformation	x = 0.5·(b2/D)	x = 0.124·(b2/D)·ln(2.791·D/b)
Maximum Stress	σ = 1.5·F/A	σ = 1.277·F/A
Load Limit	Fmax = (σy/0.918)3(D2CE2)	Fmax = (σy/0.798)2(LDCE)

source:
Roark's Formulas for Stress and Strain, Warren C. Young and Richard Budynas, 2001
(equations have been rearranged)

---

Both cone and V-groove are charcetrized solely by their angle:

The top part of the interface consists of a contact line between a sphere and a cone, which is equivalent to a cylinder-plane contact line with length L_cone, and contact diameter D.

The bottom part of the interface consists of two cylinder-plane contact lines, with a total length of 2L or 2k_LD, and contact diameter D. To equate the cylinder contact line with the cone, we define:

---

The forces are also affected by the angles of the cones and v-grooves:

The maximum allowed vertical force F, as determined by each of the contact line interfaces is:

At the cone-sphere interface, the cone angle determines both the force-amplification and the length of the line of contact, resulting in a double-angle term, and so the maximum load-carrying capacity, the cone angle is at θ_c = 90°.

In the case of the sphere, the two sphere-to-groove contacts are always the weak point, and so a wider-angle V-groove can carry more load, though this comes at the expense of weaker centering forces.

In the case of the Spherolinder, the weak spot is eliminated, and the load capacity of both interfaces can be equated by controlling the combination of L and θ_v. For θ_v = θ_c = 90°,

L_opt ≈ 1.11·D

This is short enough that in our product line, we opted to keep the V-groove at 90° rather than make L = D by making the V-groove shallower. All of our mounting blocks, therefore, have right-angle features.

The respective load capacities for the sphere and the length-optimized Spherolinders are therefore:

---

Under the allowed load, within the elastic regime, the rigidity (or more correctly, the effective spring constant) is defines as the ratio of load to deformation.

Since the contact area is a variable, the rigidity is not constant (the system is not linear), and so we want to calculate two rigidity figures:

• K_DC - static rigidity - The ratio of the total load to the entire deformation relative to a no-load condition. This figure is important, for example, when designing a tilting application.

• K_AC - dynamic rigidity - The ratio, under a given load, of a small increase in load to the corresponding increase in deformation. This figure is important when analyzing vibrations.

Using the same geometry, the cone or v-groove angle amplifies the normal deformation by the same factor it has magnified the force by. (If the system was linear, it would have resulted in an overall sin²(θ) amplification.)

---

The above calculations are for a single mounting site - a complete mount contains 3 mounting sites, so the load has to be adjusted accordingly. For a horizontal symmetrical application, the load is evenly distributed, and only a third of it acts on each mounting site. To carry out these calculations, please use our web calculator.