DAMPED VIBRATIONS 



367 



Fig. 123 shows the three relations plotted for values of t between 

 and 2. In each case the maximum displacement of the body 

 from the equilibrium position is quickly attained, and then the 

 body goes slowly back to its equilibrium position although it 

 never reaches it. 



FIG. 123. 



Case HI. When a>b. Let a = 3, and b* = 8. 

 Then a= -31 



= - 4 or - 2 



and a? = Ae~ 2 ' + Be- 4 * 



where A and B are constants depending upon the initial conditions, 

 (a) Let the initial conditions be x = and v = 8 when t = 0. 

 x - Ae- 2 ' + Be- 4 ' 



at 



But when t - 0, x = 0, and A + B = 

 Also, when t - 0, v = 8, and - 2A - 4B = 8 

 Hence A = 4 and B = 4 



x = 4(c- 2< - e- 4 *) 



