Calculation models for the maximum pressure in the ball joint (spherical pair)

Variant I

Calculation assumption: the pressure on the contact surface is uniform, p₀, which is possible for spherical pairs with small clearances.

With ds = r dφ, l(φ) = 2πr sin φ and dA = ds l(φ):

F₀ = cos φ = = = 2π r² p₀ 2 = 2π r² p₀ = π r² p₀.

From the relation obtained,

F₀ = π r² p₀,

the pressure is

p₀ = .

Variant II

Calculation assumption: the pressure on the contact surface has a cosine distribution, p(φ) = p_max cos φ, which is possible for spherical pairs with larger clearances.

With ds = r dφ, l(φ) = 2πr sin φ and dA = ds l(φ):

F_r = cos φ = = = 2π r² p_max 2 = π r² p_max.

From the relation obtained,

F_r = π r² p_max,

the maximum pressure is

p_max = .

Note. The Variant II model is suitable for towing-coupling joints with large clearances.