Joint and Cardan Shafts

168 169 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 M dII = cos ß 1 M dI max cos ß 2 ( ) 2.3 Motion and torque characteristic of a universal driveline as a function of deflection angles ß 1 and ß 2 Section 2.2 illustrates that angular velocity and torque at the output of a single joint follow a sinusoidal pattern with a 180° cycle. Maximum angular velocity  II max max coin- cides with minimum torque M d II min and vice versa. From this it can be deduced that a uniform output is possible, when a second joint, with a 90° phase shift is connected to the first joint by means of a shaft. Then, the non-uniform motion of the first joint can be balanced by the non-uniform motion of the second joint. The required 90° phase shift is always met, when the two inner forks happen to be in the deflection plane of their respecti- ve joints. Moreover, the two deflection angles ß 1 und ß 2 of both joints must be the same. (See also Section 1.1 and 1 .4). With unequal deflection angles, complete compensation is not possible. For ß 2  ß 1 the following applies: The transmitted power, however, is constant, if you disregard friction losses in the bearings. Therefore, the following applies: N I = N II = Constant M dI ·  I = M dII ·  II = Constant M dI =  II = cos ß M dII  I 1 – cos 2  1 · sin 2 ß For fork position  1 = 0° we obtain: M dI = 1 =  II max M dII min cos ß  I and for fork position  1 = 90°: M dI = cos ß =  II min M dII max  I M dI =  II  I = M dII M dII  I  II M dI  II min = cos ß 1  I max cos ß 2 ( )  II min = cos ß 2  I min cos ß 1 ( ) 3. Fluctuation rate 3.1 Single joint As explained under 2.1, on a single joint the output velocity deviates from the input velocity. This means, the speed ratio is not uniform. This non-uniformity (fluctuation) can be calculated as a dimensionless value: Fluctation rate U =  2 max –  2 min = 1 – cos ß  1 cos ß 3.2 Universal driveline (2 joints connected in series) If the preconditions listed in Chapter 1 for ob- taining a complete motion compensation cannot be met, it must be aimed for that: U   0,0027. Joint 1 + Joint 2 + Joint 3 – Since the rate of fluctuation is a function of deflection angle ß, a limiting condition can be set in regard to the resulting deflection angle ß res ß res =   ß 2 1  ß 2 2  ß 2 3  3° ß res corresponds to the deflection angle of a single joint if it were to replace the entire driveline. M dII = cos ß 2 M dI min cos ß 1 ( ) 3.3 Universal driveline with more than two joints Design requirements might dictate the use of a universal driveline that employs more than 2 joints. This universal driveline, how- ever, must then incorporate an intermediate bearing. Here, also, the condition applies: U Res   0,0027. Here, U Res expresses the total fluctuation of the driveline. Observe, when determining U Res : a) Joints with the same fork position get the same sign. b) The fluctuation rate of each joint must be calculated individually U 1 , U 2 , U 3 . c) The signs must be observed when adding: U Res =  U 1  U 2  U 3 Antrieb Antrieb Gabelstellung Input Output fork position Length extension