DANA
Industrial Drive Shafts Catalogue
| Drive Shafts | |
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| General fundamental theory | ||
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| Kinematics of Hooke's joints
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. |
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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). |
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A measure for the non-uniformity is the
difference of the rotation angles
and
or the transmission ratio of the angular
speeds  and
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Expressed by an equation, that means:
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 a) rotation angle difference (also called gimbal error)  b) Gear ratio  |
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The following diagram shows the gear ratio i = /
for a full revolution of the universal joint
for β = 60°. |
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The degree of non-uniformity U is
defined by: U = i max. - i min. = tan β * sin β where:  |
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The diagram shows the course of the
degree of nonuniformity U and of the angular difference
 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 β = 0°. A synchronous or homokinematic running can be achieved by a suitable combination or connection of two or more joints. |
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| 2. The universal shaft 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: |
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2) The two joints must have a kinematic
angular relationship of 90° (
/ 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. |
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