10 1 0 0 0 2 0 0 0 3 double dot vector xt 5 3 1 0 1 4 3 0 3

10 [1 0 0 0 2 0 0 0 3] double dot vector x(t) + 5 [3 -1 0 -1 4 -3 0 -3 3] dot vector x (t) + 20 [7 -3 0 -3 5 -2 0 -2 2] vector x(t) = {5 cos 2t 0 0} Suppose, the forcing function \"5cos2t\" is now replaced with an impulse of rho_0 delta(tau). Find the forced vibration of above system.

Solution

If you were to take a guitar string and stretch it to a given length and a given tightness and have a friend pluck it, you would hear a noise; but the noise would not even be close in comparison to the loudness produced by an acoustic guitar. On the other hand, if the string is attached to the sound box of the guitar, the vibrating string is capable of forcing the sound box into vibrating at that same natural frequency. The sound box in turn forces air particles inside the box into vibrational motion at the same natural frequency as the string. The entire system (string, guitar, and enclosed air) begins vibrating and forces surrounding air particles into vibrational motion. The tendency of one object to force another adjoining or interconnectedobject into vibrational motion is referred to as a forced vibration. In the case of the guitar string mounted to the sound box, the fact that the surface area of the sound box is greater than the surface area of the string means that more surrounding air particles will be forced into vibration. This causes an increase in the amplitude and thus loudness of the sound.