Joint and Cardan Shafts

170 171 Application Guidelines Calculation data Technical Annex Technical Annex  Service Phone Europe  + 49 (0) 71 42 / 353-0  Service Phone North America  + 1 – 2 6 9 / 6 3 7 7 9 9 9  Catalog  Spare parts  Drive-Shaft Calculation  Installation and Maintenance  www.elbe-group.com 5.3 Caused by axial displacement forces If a driveline with an adjustable spline is being changed in length while under torque, in both cases, Z- or W configuration, addition bearing loads are introduced, resulting from the friction caused in the spline. The axial dis- placement force Pa responsible for these bearing loads is calculated as follows: d m is the spline pitch diameter, Ü the spline overlap. Depending on configuration and lubri- cation, the coefficient of friction for steel on steel must be assumed to range from 0.11 to 0.15. Plastic coated splines have considerably better sliding characteristics. Here, the fric- tion value is approximately 0.08. Rilsan coated splines are available from size 0.109 up. On drivelines with three-dimensional deflec- tion angles, input and output shaft are not lo- cated in one plane. This results, if no special measures are taken, in a non-uniform output motion. The constantly repeating accelerati- on and deceleration unleashes inertia forces which can greatly reduce the life of the joints. However, not only the driveline, the driven equipment also is subjected to these forces and vibration caused by them. To avoid this, the inner forks must be offset relative to each other such that each fork ends up in the plane of deflection of its joint. The angle between both deflection planes is called offset angle  and it can be obtained as follows. 4. Offset angle Example 1 tan  1 = tan ß h1 ; tan  2 = tan ß h2 tan ß v1 tan ß v2 Offset angle  =  1 –  2 Example 2 tan  1 = tan ß h1 ; tan  2 = tan ß h2 tan ß v1 tan ß v2 Offset angle  =  1 +  2 As shown by the graphic illustrations, in both examples two directions of rotation are possible: Example 1: a) Rotate joint 1 counter clockwise by the offset angle b) Rotate joint 2 clockwise by the offset angle. The direction for viewing is, in both cases, from joint 1 to joint 2. Example 2: a) Rotate joint 1 counter clockwise by the offset angle b) Rotate joint 2 clockwise by the offset angle. The direction for viewing is, in both case, from joint 1 to joint 2. To determine the turning direction of the off- set angle, you always have to take the graphic illustration. Only in this way is it possible to find the right direction of rotation and to determine whether the offset angle  1 and  2 have to be summed or have to be subtracted In Section 2.2 it was shown that the torque is transmitted only in the spider plane and that depending on the fork position, the spider can be perpendicular either to the input axis or the output axis. What additional forces and moments this causes on the driveline as well as on the bea- rings of the input and output shaft, is explained briefly in the following chapter. 5. Additional moments on the drive line; Bearing loads on the input and output shaft 5.1 With Z-Arrangement The adjacent illustration shows the location and direction of the additional forces and moments on drivelines having a Z-arrange- ment, in particular for yoke angles  1 = 0° and  1 = 90°. This shows clearly, that the driveline center part is stressed by the tor- que which fluctuates between M dI · cos ß and M dI / cos ß in torsion and by the additio- nal periodically alternating, moment M ZII in bending. (See also Section 6.8). Likewise, input and output shaft are stressed by M ZI and M ZIII periodically alternating in bending. The resulting bearing loads A and B vary twice per revolution between O and maximum value. 5.2 With W-Arrangement According to the adjacent illustration, with the W-arrangement, an additional force, “S“ is introduced, caused by the additional mo- ments M ZII acting in the same direction. The maximum force value occurs at fork position  1 = 0° , and it is transmitted to the input and output shaft by the faces of the spider pins. Side view Top view Side view Top view Side view Top view Side view Top view Driveline-center part stressed in bending A = 2 . M dl . sin ß . b L . a B = 2 . M dl . sin ß . (a+b) L . a Driveline-center part stressed in bending A = B = M dl . tan ß a [N] A max = B max = Mdl . tan ß a [N] Input and output shaft stressed in bending Input and output shaft stressed in bending A = B = 0 Bearing loads on input and output shaft with Z-arrangement Bearing loads on input and output shaft with W-arrangement P a = 2 · M dI ·  1 + sin ß dm Ü [N] ( ) Vertical plane Horizontal plane Vertical plane Horizontal plane starting position