Universal Joint (Cardan) Speed Variation

The Cardan error of a single U-joint: the twice-per-revolution output speed fluctuation it introduces at an angle.

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Checked against published worked examples from: Mustansiriyah University, College of Engineering.

Details, formula, and sources

ScopeThe output ranges between omega cos(beta) and omega/cos(beta), a peak-to-peak swing of 1/cos(beta) - cos(beta) of the input. A joint at 10 degrees on a 1,000 rpm shaft swings the output 985-1,015 rpm (3.1% peak-to-peak); open it to 30 degrees and it explodes to 866-1,155 rpm (29%), which no driveline tolerates. That is why U-joint working angles are held to a few degrees and the two ends of a shaft are angled equally - a second joint phased 90 degrees at an equal angle (double-Cardan) cancels the error. Enter an input torque for the matching torque pulsation (200 lb-ft at 10 degrees swings 197 to 203) and the peak output angular acceleration; the secondary shaking couple and CV joints are separate. A design aid; Machinery Handbook / Shigley and the driveline maker govern.

Formula or tablemax output = omega/cos(beta); min output = omega cos(beta); fluctuation = 1/cos(beta) - cos(beta) = sin(beta) tan(beta), twice per revolution.

Source and editionThe Cardan (Hooke) universal-joint velocity relation, output speed ranging between omega cos(beta) and omega/cos(beta) (Machinery's Handbook; Shigley, Mechanical Engineering Design), by name.

Free accessThe Cardan velocity relation is a standard published kinematics result; the joint angle and input speed are the user's inputs.

What governsEstimate. AHJ and licensed professional govern.

API field namesjoint_angle_deg, input_speed_rpm, max_output_rpm, min_output_rpm, fluctuation_pct

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