Kinematic displacement analysis of a double-cardan-joint driveline

The input-output displacement relations of two Cardan joints arranged in series on a driveline has been investigated in detail, including the effects of unequal joint angles, the phase angle between the two Cardan joints and also such manufacturing tolerance errors as nonright angle moving link lengths and offset joint axes, A combined Newton-Raphson method and Davidon-Fletcher-Powell optimization algorithm using dual-number coordinate-transformation matrices was employed to perform the analysis. An experiment was conducted to validate the results of the analysis. The apparatus consisted of a double-Cardan-joint driveline whose rotations were measured by optical shaft encoders that were sampled by a computer data-acquisition system. The equipment was arranged so that the phase angle between the joints and the offset angles between the shafts at each of the two joints could be readily varied. The "relative phase angle," the difference between the phase angle of the two joints and the angle between the planes defined by the input and intermediate and the intermediate and output shafts, was found to be the significant factor. If the offset angles at both Cardan joints are equal, the double- Cardan-joint driveline functions as a constant-velocity coupling when the magnitude of the relative phase angle is zero. If the offset angles at the two Cardan joints are unequal, a condition prevailing in the important front-wheel-drive automobile steering-column application, then fluctuation in output velocity for a constant input velocity is minimized although not eliminated for zero relative phase angle.