Presets
Playback
Joint geometry
Joint angle b is the bend between the shafts. Choose the layout, then tune the second joint and the yoke phase.
Drive and load
Real speed and torque set the torque ripple; slow-motion only slows the picture.
Layers
Model checks
The science
The joint
A Cardan (Hooke) joint joins two shafts that meet at an angle b with a cross-shaped spider. The input fork carries one axis of the cross and the output fork the other. The axes stay perpendicular, which forces the kinematic relation tan(th2) = cos(b) tan(th1).
Speed ripple
Differentiating with the input at constant speed gives w2 / w1 = cos(b) / (1 - sin^2(b) sin^2(th1)). The output therefore swings between w1 cos b and w1 / cos b twice per turn; the peak-to-peak ripple is 1/cos b - cos b = sin b tan b (about 3 % at 10 deg, 12 % at 20 deg, 54 % at 40 deg). The average over a turn is exactly 1: the output never gains or loses a revolution.
Torque
The joint is lossless in this model, so power is conserved: T1 w1 = T2 w2, i.e. T1 = T2 * (w2 / w1). When the output is slowest the joint multiplies the torque and vice versa. With an output inertia J the output shaft also needs J * alpha2, with alpha2 = w1^2 d(w2/w1)/d(th1) (about b^2 w1^2 for small b), so the ripple grows with the square of the speed. This is the 2x-per-turn torsional excitation that engineers must keep away from driveline resonances.
Double joint
Two joints in series give w3/w1 = r(th1, b1) * r(th_mid + 90 deg + phi, b2). For equal angles, both bends in one plane (Z: shafts end parallel; W: bends the same way) and the two intermediate-shaft forks in one plane (phi = 0) the second ripple is exactly the inverse of the first and the output turns uniformly: this is the basis of the double-Cardan constant-velocity driveshaft. A phase error or unequal angles leave a residual ripple.
What is simplified
Rigid bodies, no friction, no backlash, the input is driven at an exactly constant speed (a stiff motor) and the intermediate shaft is massless in the kinematics. Real propeller shafts also have secondary bending couples of order T tan b on the bearings, which this model does not resolve. Constant-velocity joints used on front-wheel-drive cars (Rzeppa) are a different mechanism and are not modelled.
What to try
- Start with the 5 deg car driveshaft: the ripple is only about 0.8 %. Then drag the joint angle to 40 deg: the speed swings by more than 50 %.
- Choose the Z double joint and keep the phase error at 0: the output follows the ghost marker exactly. Drag the phase error to 90 deg and the ripple roughly doubles.
- Make the two angles unequal (20 / 12 deg): the residual ripple equals the difference of the two single-joint ripples.
- Raise the speed to 3000 rpm with a large inertia: the torque swing grows with the square of the speed.