Editing Universal joint (section)

== Equation of motion ==
[[Image:UJoint.png|thumb|right|Diagram of variables for the universal joint. Axle 1 is perpendicular to the red plane and axle 2 is perpendicular to the blue plane at all times. These planes are at an angle &beta; with respect to each other. The angular displacement (rotational position) of each axle is given by <math>\gamma_1</math> and <math>\gamma_2</math> respectively, which are the angles of the unit vectors <math>\hat{x}_1</math> and <math>\hat{x}_2</math> with respect to their initial positions along the x and y axis. The <math>\hat{x}_1</math> and <math>\hat{x}_2</math> vectors are fixed by the gimbal connecting the two axles and so are constrained to remain perpendicular to each other at all times.]]

[[File:UJoint_3D.png|alt=A sample universal joint colour-coded to the diagrams about the equation of motion. The red and blue planes are visible.|right|thumb|A sample universal joint colour-coded to the diagrams about the equation of motion. The red and blue planes are visible.]]

{{multiple image
 | width = 300
 | direction = vertical
 | image1 = UJoint1.png
 | caption1 = Angular (rotational) output shaft speed <math>\omega_2\,</math> versus rotation angle <math>\gamma_1\,</math> for different bend angles <math>\beta\,</math> of the joint
 | image2 = UJoint2.png
 | caption2 = Output shaft rotation angle, <math>\gamma_2\,</math> versus input shaft rotation angle <math>\gamma_1\,</math> for different bend angles <math>\beta\,</math> 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:
# <math>\gamma_1</math> the angle of rotation for axle 1
# <math>\gamma_2</math> the angle of rotation for axle 2
# <math>\beta</math> 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 <math>\hat{\mathbf{x}}</math> and <math>\hat{\mathbf{y}}</math> and the [[Plane of rotation|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 (<math>\hat{\mathbf{x}}_1</math> and <math>\hat{\mathbf{x}}_2</math>) pointing from the origin towards one of the connection points. As shown in the diagram, <math>\hat{\mathbf{x}}_1</math> is at angle <math>\gamma_1</math> with respect to its beginning position along the ''x'' axis and <math>\hat{\mathbf{x}}_2</math> is at angle <math>\gamma_2</math> with respect to its beginning position along the ''y'' axis.

<math>\hat{\mathbf{x}}_1</math> is confined to the "red plane" in the diagram and is related to <math>\gamma_1</math> by:

<math display=block>\hat{\mathbf{x}}_1 = \left[\cos\gamma_1\,,\, \sin\gamma_1\,,\,0\right]</math>

<math>\hat{\mathbf{x}}_2</math> is confined to the "blue plane" in the diagram and is the result of the unit vector on the ''x'' axis <math>\hat{x} = [1, 0, 0]</math> being rotated through [[Euler angles]] <math>\left[\tfrac{\pi}{2}\,,\, \beta\,,\, \gamma_2\right]</math>:

<math display=block>
\hat{\mathbf{x}}_2 = \left[-\cos\beta\sin\gamma_2\,,\, \cos\gamma_2\,,\, \sin\beta\sin\gamma_2\right]
</math>

A constraint on the <math>\hat{\mathbf{x}}_1</math> and <math>\hat{\mathbf{x}}_2</math> vectors is that since they are fixed in the [[gimbal]], they must remain at [[Orthogonality|right angles]] to each other. This is so when their [[dot product]] equals zero:

<math display=block>
\hat{\mathbf{x}}_1 \cdot \hat{\mathbf{x}}_2 = 0
</math>

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

<math display=block>
\tan\gamma_1 = \cos\beta\tan\gamma_2\,
</math>

with a formal solution for {{nowrap|1=<math>\gamma_2</math>:}}

<math display=block>\gamma_2 = \arctan\left[\tan\gamma_1 \sec\beta\right]\,</math>

The solution for <math>\gamma_2</math> is not unique since the arctangent function is multivalued, however it is required that the solution for <math>\gamma_2</math> be continuous over the angles of interest. For example, the following explicit solution using the [[atan2]](''y'',''x'') function will be valid for <math>-\pi < \gamma_1 < \pi</math>:

<math display=block>\gamma_2 = \operatorname{atan2}\left(\sin\gamma_1, \cos\beta\, \cos\gamma_1\right)</math>

The angles <math>\gamma_1</math> and <math>\gamma_2</math> 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 <math>\omega_1 = \frac{d\gamma_1}{dt}</math> and {{nowrap|1=<math>\omega_2 = \frac{d\gamma_2}{dt}</math>:}}

<math display=block>
\omega_2 = \omega_1\left(\frac{\cos\beta}{1 - \sin^2\beta\,\cos^2\gamma_1}\right)
</math>

As shown in the plots, the angular velocities are not linearly related, but rather are periodic with a period half that of the rotating shafts. The angular velocity equation can again be differentiated to get the relation between the angular accelerations <math>a_1</math> and {{nowrap|1=<math>a_2</math>:}}

<math display=block>
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}{\left(1 - \sin^2\beta\,\cos^2\gamma_1\right)^2}
</math>