A SDOF undamped system is set into free vibration with an initial displacement and an initial velocity. The mass of the system is given as 10 lb-s 2 /in and the stiffness of the system is known to be 160 lb/in. The initial displacement (at time t= 0 is given as 4 in. and the displacement at t = 2 seconds is also 4 inches. Find:
a. The initial velocity in in/sec
b. The displacement in inches at T = 4 seconds
c. The amplitude of the vibration

Explanation:

stiffness k = 160

m = 10

angular frequency ω =

=

= 4

ω  = 4

Let x = 4 - A sinωt

when t = 0

x = 4 in

when t = 2 s , x = - 4

- 4 = 4 - A sinωt

8  = A sin 4 x 2

8 = A sin8

A = 8 / sin 8

= 8 / .989

= 8.09 in .

x = 4 - A sinωt

dx / dt = - Aω cosωt

v =  - Aω cosωt

for t = 0

v = - Aω

= - 8.09 x 4

= - 32.36 in / s

initial velocity v = - 32.36 in /s

displacement x for t = 4s

x = 4 - 8.09 sin 4 x 4

= 4 - 8.09 sin 16

= 4 - 8.09 x - .2879

= 4 + 2.33

= 6.33  in.

c ) Amplitude of vibration A = 8.09 in .as calculated above .

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