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LXXXIII. Jfote on the Analysis of J Jumped Vibrations. 

 By H..S. Rowkll*. 



THE two primary difficulties in the analysis of damped 

 vibrations are the nature of the friction and the position 

 of the zero. In most cases it is sufficient to assume that the 

 friction is a combination of so-called solid friction — a 

 constant, and of fluid friction, proportional to the velocity. 

 In the ordinary view it is inconceivable that these two kinds 

 of friction can coexist at the same time and interface, for 

 the conditions supposed to produce these two kinds of friction 

 are essentially different ; dry and wet, or molecular film and 

 measurable film. In practical cases, however, the two kinds 

 of friction can coexist in a system as, for example, where a 

 body slides or turns on dry surfaces and is damped by fluid 

 friction. Thus the equation of motion may be taken as 



mx + ka;'+c 2 a:±F = 0, ...... (I.) 



where the signs of F and of iv are the same ; put 



and the solution of equation (I.) is 



,c =+ F /c 2 -A,e- /d2 " 1 cos n't; . . . (II.) 

 where n = \/ ri 2 — k' 2 / 4:in 2 



and n 2 = c 2 /m, 



write F/c- 2 = S 



and kirj2mn —\ ; so that e~ K =$, 



where 8 is the logarithmic decrement for half periods. 



Assume that a datum line is drawn at a distance E from 

 the true time axis, and let R* be the reading from this datum 

 corresponding to the ?th half swing ; then 



So = E-S-A , (1) 



1{ 1 = E-S+A o\ (2) 



R 2 = E + S-A S 2 , (3) 



R ;; = E-S + A o :5 , (4) 



R 4 = E + S-A S 4 (;>) 



* Communicated by the Author. 



