Why is the operating angle important in universal joints?

A single universal joint (Hooke's joint) causes cyclic speed variation in the output shaft - at operating angle theta, output speed varies as omega_out = omega_in x cos(theta)/(1-sin^2(theta)sin^2(phi)), creating 2x frequency fluctuation. This causes vibration and torsional stress. To eliminate this, use two universal joints with equal angles in the proper phasing arrangement (yokes in same plane). Maximum recommended angle is 25-30 degrees for continuous operation. Critical in automotive driveshafts.