DAMPED VIBRATIONS 



(b) Let the initial conditions be x <*> 8 and v - when t - 0. 



x - e-*(C cos & 4- D sin 3t) 



dr 

 v-~x <H(C cos 8t + D sin 8t) + <H( - 8 C sin 8t + 8D cos 8t) 



But when / - 0, x - 3. Then C = 8 

 Also, when t =- 0, v = 0. Then 8D - C - 0, or D = 1 

 x = e~*(3 cos 3t + sin 3t) 



- VlOe- 1 sin (3< + a), where a = tan -1 8 



- 8-162 e- 1 sin (+ 1-248) 



(c) Let the initial conditions be x = 8 and t> = 9 when t = 0. 



8 = <H(cos 3* + D sin 3t) 



fir 

 v = ^ = - e-*(cos 8 + D sin ft) + <H( - 3C sin 3t + 3D cos 3t) 



But when < = 0, a? = 3. Then C = 3 

 Also, when < = 0, v = 9. Then 3D - C - 9, or D = 4 

 x = e~* (3 cos 3t + 4 sin 3<) 



o 



= 5e~* sin (3f + p), where ^ = tan- 1 - 

 sin (3t + 0-6435) 



FIG. 122. 



Fig. 122 shows the three relations plotted for values of t be- 

 tween and 8. They each represent periodic functions of con- 



