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            From Table 1. for b 120, we have a_max/w² = 0.0442. The weight, W, of the disc is p r² t y, where y = 0.283 lbs/in³
and denotes the density of steel.
            Thus W = p (6)² (0.5) (0.283) = 16 lbs. The polar mass moment of inertia, l, of the disc is given by

l =   Wr²    =   (16) (16)²    = 0.746 in-lb-sec².
         2g          (2) (386)  

     The inertia torque = l a_max = 50% of 250 in-lbs = 125 in.lbs.
    
      Hence, 0.0442 w²l = 125, i.e. (0.746) (0.0442) w² = 125,

giving w = 61.6 rad/sec. = (61.6) (60) = 588 RPM.
                                            (2)(p)
         Hence, If the inertia torque is not to exceed its limit, the max. speed of the input shaft is 588 RPM.
         For joints made with thermoplastic material, consult the catalogue, which Contain design charts for the torque rating of such joints.

6.0  SECONDARY COUPLES
In designing support bearings for the shafts of a Cardan joint and in determining vibrational characteristics of the driven system, it is useful to keep in mind the so-called secondary couples or rocking torques, which occur in universal joints.
        These are rocking couples in the planes of the yokes, which tend to bend the two shafts and rock them about their bearings. The bearings are thus cyclIcally loaded at the rate of two cycles per shaft revolution. The maximum values of the rocking torques are as follows:

        Max. rocking torque on Input shaft  = T_in tan b;
        Max. rocking torque on output shaft = T_in sin  b,

       where T_in   denotes the torque transmitted by the input shaft and b the operating angle. These couples are always
180° out of phase. The bearing force induced by these couples is equal to the magnitude of the rocking couple
divided by the distance between shaft bearings.
       For example if the Input torque, T_in, is 1000 in.lbs. and the operating angle is 20’, while the distance between support bearIngs on each shaft is 6’ the max. secondary couple acting on the input shaft is (1000) (tan 20°) = 364 in-lbs. and on the output shaft it is (1000) (sin 20°) = 342 in-lbs. The radial bearing load on each bearing of the input shaft is 364/6 = 60.7 lbs. and it is 342/6 = 57 lbs. for each bearing of the output shaft. The bearings should be selected accordingly.
      It has been observed also that due to the double frequency of these torques, the critical speeds associated with universal drives may be reduced by up to 50% of the value calculated by the standard formulas for the critical speeds of rotating shafts. The exact percentage Is a complex function of system design and operating conditions.

7.0 JOINTS IN SERIES

As mentioned in paragraph 3, universal joints can be used in series in order to eliminate velocity fluctuations, to connect offset (non-intersecting) shafts, or both. Figure 2 shows a schematic of such an arrangement.

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