306 Prof. H. H. Jeffcott on Lateral Vibration of Loaded 



line in the disk such as EM makes with a line fixed in space 

 such as OX. It would not be correct in the statement of 

 the problem to assume that the inclination of the radii 

 vectores OE or OM chances as cot. It will be shown later 

 that the path of E, when steady motion is established, is a 

 circle described with angular velocity co. Hence is the 

 instantaneous centre, and the inclinations of the lines OE 

 and OM change as cot. 



The equation of motion of m parallel to OX is 



d 2 / x 7 dx 



m -j-j {x + a cos cot) + b -j- + ex =■ 0, 



where b is the coefficient of damping due to viscous resistances, 

 and c is the elastic force of restitution at unit displacement. 

 Hence 



ma 4- bx + cx = maw 2 cos cot. 



The solution of this equation is well known to be 



- — . N maco 2 



x = Ae ,„.sm (gt + a) + v . ( 7- W) , + w cos («t-0), 



where Q bay \Zkmc—b 2 



tan/3 = 2 ; q = 5 ; 



c — mar 2m 



and A and a are arbitrary constants. 



In like manner the equation of motion parallel to OY is 



d 2 , • v . -.dy , . 



m jr^y +a sin at ) + h dt "*" cy ' 



and the solution is 



_ii . maco 2 . . ON 



y = A'e 2 ™sin( g t + «) + sin (o*-ft), 



where A' and a are new arbitrary constants. 



The first term in the solution represents an oscillatory 

 motion of amplitude 



_bt _ bt 



Ae 2m or A'e 2m . 



This amplitude diminishes with increase of t, so that the 

 term becomes negligible. 



The second term persists, and is a vibration of amplitude 



</(c—ma> 2 ) 2 + b 2 co* 



This forced vibration is caused by the disturbing action of 



the eccentric mass during rotation. 



