Universal Joint - Equation of Motion

Equation of Motion

Output shaft rotation angle, versus input shaft rotation angle, for different bend angles, of the joint

The Cardan joint suffers from one major problem: even when the input drive shaft axle rotates at a constant speed, the output drive shaft axle rotates at a variable speed, thus causing vibration and wear. The variation in the speed of the driven shaft depends on the configuration of the joint, which is specified by three variables:

The angle of rotation for axle 1
The angle of rotation for axle 2
The bend angle of the joint, or angle of the axles with respect to each other, with zero being parallel or straight through.

These variables are illustrated in the diagram on the right. Also shown are a set of fixed coordinate axes with unit vectors and and the planes of rotation of each axle. These planes of rotation are perpendicular to the axes of rotation and do not move as the axles rotate. The two axles are joined by a gimbal which is not shown. However, axle 1 attaches to the gimbal at the red points on the red plane of rotation in the diagram, and axle 2 attaches at the blue points on the blue plane. Coordinate systems fixed with respect to the rotating axles are defined as having their x-axis unit vectors ( and ) pointing from the origin towards one of the connection points. As shown in the diagram, is at angle with respect to its beginning position along the x axis and is at angle with respect to its beginning position along the y axis.

is confined to the "red plane" in the diagram and is related to by:

$\hat{\mathbf{x}}_1=$

is confined to the "blue plane" in the diagram and is the result of the unit vector on the x axis being rotated through Euler angles

$\hat{\mathbf{x}}_2 =$

A constraint on the and vectors is that since they are fixed in the gimbal, they must remain at right angles to each other:

$\hat{\mathbf{x}}_1 \cdot \hat{\mathbf{x}}_2 = 0$

Thus the equation of motion relating the two angular positions is given by:

$\tan\gamma_1=\cos\beta\tan\gamma_2\,$

with a formal solution for :

The solution for is not unique since the arctangent function is multivalued, however it is required that the solution for be continuous over the angles of interest. For example, the following explicit solution using the atan2(y,x) function will be valid for :

The angles and in a rotating joint will be functions of time. Differentiating the equation of motion with respect to time and using the equation of motion itself to eliminate a variable yields the relationship between the angular velocities and :

$\omega_2=\frac{\omega_1\cos\beta}{1-\sin^2\beta\cos^2\gamma_1}$

As shown in the plots, the angular velocities are not linearly related, but rather are periodic with a period twice that of the rotating shafts. The angular velocity equation can again be differentiated to get the relation between the angular accelerations and :

$a_2 = \frac{a_1 \cos\beta }{1-\sin^2\beta\,\cos^2\gamma_1}-\frac{\omega_1^2\cos\beta\sin^2\beta\sin 2\gamma_1}{(1-\sin^2\beta\cos^2\gamma_1)^2}$

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