38 



MECHANICAL VIBRATING SYSTEMS 



supports, the effect of damping and the effect of different types of excita- 

 tion. These factors of course alter the form of vibration and the overtones. 

 3.3. Transverse Vibration of Bars^'^'^'^. — In the preceding section 

 the perfectly flexible string was considered where the restoring force due 

 to stiffness is negligible compared to that due to tension. The bar under 

 no tension is the other limiting case, the restoring force being entirely due 

 to stiffness. For the cases to be considered it is assumed that the bars 

 are straight, the cross section is uniform and symmetrical about a central 

 plane, and as in the case of the string, only the transverse vibrations will 

 be considered. 



FUNDAMENTAL 



FUNDAMENTAL 



FUNDAMENTAL 



FIRST OVERTONE 



FIRST OVERTONE 



FIRST OVERTONE 



SECOND OVERTONE 



SECOND OVERTONE 



SECOND OVERTONE 



THIRD OVERTONE 



THIRD OVERTONE 



THIRD OVERTONE 



Fig. 3.2. Modes of transverse vibrations of a bar clamped at one end and free at the other, 

 of a free bar and a bar supported at both ends. 



A. Bar Clamped at One End. — Consider a bar clamped in a rigid sup- 

 port at one end with the other end free (Fig. 3.2). The fundamental 

 frequency is given by 



/i 



/2 



.5596 Wk 



3.2 



where 



/ = length of the bar, in centimeters, 



p = density, in grams per cubic centimeter, 



^ = Young's modulus, in dynes per square centimeter, see Table 3.1, and 



K = radius of gyration. 

 For a rectangular cross section the radius of gyration is, 



K = 



