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Assumed Frequency ft) rad/sec

Assumed Frequency ft) rad/sec been correctly assumed, then the total torque in column 6 at the last mass will be zero. If not, another value for co is taken and the table repeated. For the currently assumed value for «column 3 then becomes the values for J„ times to2. Column 4 It is assumed that the amplitude of movement in the plane of the first mass is ±1 radian. So when the table is finally balanced column 4 gives the relative amplitudes at all the masses. In following rows the value for 6 is the value for the preceding row less the change A0for the preceding row obtained in column 8.

Column 5 is the product of J„a?6„ for the mass Jn being considered which gives the torque required to vibrate that mass. Column 6 This column really starts with torque 0 prior to mass A so the total torque in the plane of the first mass A is 0 + 7, a?6{ = J ¡of as Of has been made equal to unity in column 4. In successive lines the total torque is qual to that in the previous line plus the vibration torque in the line being considered. The final value when ft) has been correctly assumed to be that of a natural frequency for the system should be zero. It should be noted that the total torque may become negative depending on the position of the node or nodes in the system corresponding to the mode whose frequency is being sought. Column 7 Lists the stiffnesses of the shaft between the masses. Column 8 This is the shaft twist between the two adjacent masses being considered and is obtained by dividing the total torque by the shaft stiffness between the two masses. So AO = Z Ja?d/C. The value so obtained is then subtracted from the deflection 0„ in that row to give the value for 0n+i in column 4 for the deflection in the plane of the next mass 7n+1.

If the value for co is incorrect a positive or negative residual torque will be left at the bottom of column 6. Assuming a number of frequencies will give a number of residual torques which, if plotted, result in a curve. Such a plot will help the location of the correct values of the required frequencies.

This process can be speeded up by using a method suggested by Mahalingam (Proc. R.Ae. S. Oct. 1966 p. 953). If, since the system is free at both ends, the system is considered as a rigid body, a more accurate approximation for a9 can be calculated by adding a compensating term for the resulting change in the maximum kinetic energy. This results in a second order term R6 being added to the 'Rayleigh quotient'. It can be shown that the maximum strain energy for the vibrating system is

Ey¡cok2n)e2 -R8n

This is Rayleigh's quotient.

Here R is the amount the total torque at the end of the Holzer Table differs from zero; 6„ = the deflection of the last mass n; coj;(l) = initial assumption for frequency (rad/sec) squared; ®k(2) = better assumption for ft)2;

Mahalingam's method first derives an ampitude

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