Question

A free vibration test is conducted on an empty elevated wat er tank. A cable attached to the tank applies a lateral (horiz ontal) force of 16.4 tons and pulls the tank horizontally by 2 cm. The cable is suddenly cut and the resulting free vibra tion is recorded. At the end of four complete cycles, the tim e is 2.0 sec and the amplitude is 1 cm. From these data co mpute the following: (a) damping ratio; (b) natural period of u ndamped vibration; (

Answer 1

The damping ratio is approximately 0.041 and the natural period of undamped vibration is 1 second.

Step-by-step explanation:

To find the damping ratio, we can use the formula: \(\zeta = \frac{ln(A_1/A_2)}{4\pi}\), where \(A_1\) and \(A_2\) are the amplitudes at the end of four complete cycles. Plugging in the values, we get: \(\zeta = \frac{ln(1/0.01)}{4\pi} \approx 0.041\).

The natural period of undamped vibration is given by the formula: \(T_0 = \frac{2\pi}{\omega}\), where \(\omega\) is the angular frequency. Since the time for four complete cycles is 2 seconds, the angular frequency is \(\omega = \frac{4\pi}{2} = 2\pi\). Plugging in the values, we get: \(T_0 = \frac{2\pi}{2\pi} = 1\) second.