46 MECHANICAL VIBRATING SYSTEMS 



3.7. Torsional Vibration of Bars 21. 22. _ a solid bar or tube may be 

 twisted about the axis of the rod in such a manner that each transverse 

 section remains in its own plane. If the section is not circular there will 

 be motion parallel to the axis of the bar. Consider an entirely free rod 

 of homogeneous material and circular cross section. The simplest or fun- 

 damental mode of torsional vibration occurs when there is a node in the 



())) M t M t t w u M >)) 



FUNDAMENTAL flRST HARMONIC 



( )) ) M t . . . V U ) ) ) M > > - M ) } )) ^ 



FIRST OVERTONE SECOND HARMONIC 



(Y) t ^ » M n M - ' ^ M ) } M » M u) 



SECOND OVERTONE THIRD HARMONIC 



Fig. 3.6. Modes of torsional vibration of a free rod. The nodes and loops are indicated by 



N and L. 



middle and a loop at each end, that is, when the length of the rod is one 

 half wavelength. The fundamental resonant frequency. Fig. 3.6, is given by 



•^^ = 2l\Ma + 1) ^'^ 



where / = length of the rod, in centimeters, 



p = density, in grams per cubic centimeter, 

 ^ = Young's modulus, in dynes per square centimeters, and 

 a = Poisson's ratio. See Table 3.1. 

 The overtones, as in the case of longitudinal vibrations, are harmonics of 

 the fundamental. That is, /2 = Vufz = 3/i, /4, = 4/i, etc. The nodes 

 and antinodes for the various harmonics are formed as in the case of longi- 

 tudinal vibrations. 



Torsional vibrations may be set up in bars by any means which applies 

 tangential forces to the free end. From a comparison of the longitudinal 

 and torsional vibrations in the same bar, Poisson's ratio may be determined. 



21 Wood, " A Text Book of Sound," Bell and Sons, London. 



22 Rayleigh, " Theory of Sound," Macmillan and Co., London. 



