A cardan (universal) joint at an angle does not turn the output evenly: it speeds up and slows down twice every turn. See how much, and how a second joint with the right phasing cancels it.
Layout
Joint 1 angle
°
Joint 2 angle
°
°
0° means both yokes on the intermediate shaft are in line (correct). 90° is the classic assembly mistake.
RPM
N·m
Hands turn with each shaft: watch the output gain and lose
How it is calculated
Hooke's (cardan) joint with bend angle β. θ₁ is the input shaft angle, measured from the plane of the input yoke.
Output angle tan θ₂ = tan θ₁ / cos β
Speed ratio ω₂ / ω₁ = cos β / (1 − sin²β · cos²θ₁)
Range cos β ≤ ω₂/ω₁ ≤ 1 / cos β, fluctuation = 1/cos β − cos β
Peak acceleration α₂ ≈ ω₁² · sin²β · cos β · sin 2θ₁ / (1 − sin²β cos²θ₁)²
Torque T₂ = T₁ · ω₁ / ω₂ (power stays the same, torque rises when speed falls)
Two joints output is even only if β₁ = β₂ and the intermediate yokes are in phase (Z or W layout)
Rules of thumb: keep each angle under about 3° for long life at high speed, and no more than about 6° on car driveshafts; slow industrial shafts accept up to 15–25°. A tiny angle (under about 0.5°) is also bad, because the needle bearings never roll and brinell the cross. Two joints must have equal angles within about 0.5–1° to cancel well.