Vibration of Bars treated simply. 587 



Table II. — Tones for Free-Free or for Fixed-Fixed Bar. 



Values of m. 



Relative 

 Frequencies, 



Intervals from 

 Prime to each 

 Higher Tone. 



Approximate 

 Notes. 



m =0. 

 m x —4i'73 



1 



2-756 



5-404 



8-933 



13-345 



Octaves. Equal 

 Semitones. 



i 



C 

 F'* 



F" 



D"' 



A'" 



m 2 — 7-85 



1 and 5*52 



2 and 5*23 



3 and 191 



3 and 8-86 



m s = 10-996 



m 4 = 14-137 



m.= 17-279 



From equation (14) and Table IL, first column, we can 

 obtain the actual frequency of any tone for a given bar. 

 Thus,, let the prime tone be required for a free-free bar of 

 steel of rectangular section and of thickness a cm. and 

 length I cm., then 



K=ajjs/l2j v / q/p=b=523,7Q0 cm./sec, 

 whence 



X 1 = 538,400 (a/l 2 ) per second. 



So if a = l cm. and Z = 100 cm., we have 

 Xi = 53*84 per second. 



Multiplying this value by the relative frequencies in 

 column 2 of Table II. , we have the absolute values of the 

 frequencies of the other tones for the same bar. 



Fixed-Fixed Bar. — On applying the proper conditions, 

 equation (6) in Table I. for a bar fixed at each end, it will 

 be found that the same series of tones is obtained as for one 

 free at each end. 



Nodes for Free-Free Bar by Tlieory and, Experiment. — 

 By inserting any one value of m in equation (19), the dis- 

 placement curve at any instant may be obtained for that type 

 of vibration. Hence, also, the nodes or points of no displace- 

 ment can be found. The theoretical positions for a thin bar 

 as thus determined by Lord Eayleigh are shown in Table III. 

 in comparison with the actual positions found by the writers 

 experiments on a steel bar 29 inches long, I5 inch wide, and 

 \ an inch thick. The actual and theoretical frequencies are 

 also given as a further illustration of how slightly a bar of 

 sensible thickness departs from the behaviour theoretically 



