﻿510 



Dr. J. H. Vincent and Mr. C. W. Jude on 



Four Frequencies; Pairs of Opposite Pendulums 

 beating. 



We will now give a few examples from what we have 

 called the fourth class of those figures in which oppositely- 

 placed pendulums are beating. In this class four frequencies 

 are involved, simple relations subsisting between the means 

 of the frequencies of opposite pendulums. 



Index of Diagrams 47 to 54. 



No. 



Freq. of 1 

 A&C. ; 



[ 47 1 



2p 

 and 



2p + 2h , 



(3. 49 



p+h 



and 



p — h 



foO) 





y{ 



51 



5 2 ► 

 53 



2p+k 



and 

 2p+3h 



2p+h 



2p + 2h 



2Ji 



2h 



Freq. of 



Mean freq. 

 of B & D. 



Diff. of 

 B&D. 



Initial Oon( 



litions. 



P 



and 

 p+h 



2p+h 



2 



"{ 



All negative. ^ 



A & B negative 

 C & D positive 





p+2k 



and 



p-2h 



P 



\h j 



A & B negative 

 & D positive 



All 







{ 



. o -r, ,• , starting 

 A & B negative ° 



C & D positive [ together. 



P+2k 

 and 



P + h 



f 



1 



2h 



Ail negative 



but C. 

 All negative. 



P 







All negative. J 







i 



All negative but D. 



B releases C, D releases A. 



«. Mean frequencies and differences of frequencies of 

 opposite pendulums as two to one. 



To satisfy these conditions the ratio of the frequencies will 

 be 2p, p, 2p + 2h.p + h. These frequencies are employed in 

 figs. 47 and 48. The initial amplitudes in both cases were 

 equal. In fig. 47 the pendulums were started with their 

 bobs in the extreme outward position. The tracing-point 

 was put down on the plate when the bobs of the pendulums 

 B and D had first reached their inward positions, so that 

 counting time from this instant the equations to the trace 

 are 



x — a cos [2pt + 7r]+a cos { 2/> + 2h } t + 7r] , 



y = acospt + a cos {p + h}t, 



the trace lasting until t — ir/'h. In this figure h is again so 

 small with respect to p that the effects of friction are notice- 

 able. Disregarding these effects, however, the excursions 

 in the approximate parabolic path are limited bv the initial 



