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1. The joints
In the theory of mechanics the cardan joint or Hooke's joint is defined as a spatial or spherical drive unit with a non-uni form gear ratio or transmission. The transmission behaviour of this joint is described by the equation.
In this equation
the momentary rotation angle of the driven shaft 2.
The motion behaviour of the driving and the driven ends is shown in the following diagram.
The asynchronous and / or non-homokinematic running of the shaft 2 is shown in the periodical
oscilation of the asynchronous line round the synchronous line
(dotted line).
and
or the transmission ratio of the angular speeds
and 
.
a) rotation angle difference
(also called gimbal error)
b) Gear ratio
The following diagram shows the gear
ratio i =
/
for a full revolution of the universal joint for ß = 60°.
as a function of the deflection angle of the joint from 0 to 45°.From the motion equation it is evident that a homokinematic motion behaviour corresponding to the dotted line under 45° - as shown in the diagram - can only be obtained for the deflection angle ß = 0°. A synchronous or homokinematic running can be achieved by a suitable combination or connection of two or more joints.
The rotation angle difference
or the gimbal error of
a deflected universal joint can be offset under certain
installation conditions with a second universal joint.
The constructive solutions are the following:
1) The deflection angles of both joints must be equal, i.e.
Two arrangements are possible:
/ 2), i.e. the yokes of the connecting shaft are in one plane.
For a more intensive study of universal shaft kinematics we refer you to the VDI-recommendation 2722 to the relevant technical literature and especially to the book ,,Kardangelenkgetriebe und ihre Anwendung" (Cardan joint drives and their application) by Florian Duditza, published by VDI.