Universal Joints and Driveshafts H.Chr.Seherr-Thoss · F ... - Index of

6 1 Universal Jointed Driveshafts for Transmitting Rotational Movements
while the axis CL turns about the arc FB = j 2 then according to the cosine theorem
[1.10, Sect. 3.3.12, p. 92]
cos 90° = cos j 1 cos(90 + j 2) + sin j 1 sin (90 + j 2) cos b.
Since cos 90° = 0 it follows that
0 = cos j 1 (– sin j 2) + sin j 1 cos j 2 cos b.
After dividing by cos j1 sin j2 Poncelet obtained
tan j2 = cos b tan j1 (1.1a)
or
j2 = arctan (cos b tan j1). (1.1b)
For the in-phase starting position, with j1 + 90° and j2 + 90° (1.1a). The following
applies
tan (j2 + 90°) = cos b tan (j1 + 90°) fi cot j2 = cos b cot j1 or
1 1 tan j 1
0 = cos b 0 fi tan j 2 = 0 . (1.1c)
tan j 2 tan j 1 cos b
The first derivative of (1.1b) with respect to time gives the angular velocity
dj2 1 cos b dj1 7 =
dt 1
008
+ tan
92 7
2j1 cos2b cos2j1 dt
cos b dj1 =
cos
00004 7
2j1 + sin2j1(1 – sin2b) dt
dj2 cos b dj1 7 = 007 7 [1.10, Sect. 4.3.3.13, p. 107]
dt 1 – sin2j1 sin2b dt
or because dj 2/dt = w 2 and dj 1/dt = w 1
w2 cos b
5 = 006 . (1.2)
w1 1– sin2bsin2j1 Equations (1.1a–c) and (1.2) form the basis for calculating the angular difference
shown in Fig. 1.7a:
Dj = j2 – j1 and the ratio of angular velocities w 2/w 1 shown in Fig. 1.7b.
For the two boundary conditions, (1.2) gives
j 2 = 0°, j 1 = 90° fi w 2 = w min = w 1 cos b,
j 2 = 90°, j 1 = 180° fi w 2 = w max = w 1/cos b.
Figure 1.7a shows that an articulation of 45° gives rise to a lead and lag Dj about
±10°, and very unpleasant vibrations ensue. The reduction of these vibrations has
been very important generally for the mechanical engineering and the motor