Question

Consider a single degree of freedom system comprising a mass \(M\), supported on a spring and a dashpot as shown in the figure. If the amplitude of the free vibration response reduces from 8 mm to 1.5 mm in 3 cycles, the damping ratio of the system is ________ (round off to three decimal places).

Hint:

The damping ratio is calculated using the logarithmic decrement method, which compares the decay in amplitude over successive cycles of vibration.

Answer 1

Correct Answer: 0.085

The damping ratio \( \zeta \) can be found from the decay of the amplitude in free vibration. The amplitude ratio \( R \) after \( n \) cycles is given by: \[ R = \frac{A_2}{A_1} = e^{-\zeta \pi n} \] Where:
- \( A_1 = 8\ \text{mm} \) is the initial amplitude,
- \( A_2 = 1.5\ \text{mm} \) is the amplitude after 3 cycles,
- \( n = 3 \) is the number of cycles.
Substitute the values into the equation: \[ \frac{1.5}{8} = e^{-\zeta \pi 3} \] \[ 0.1875 = e^{-3 \zeta \pi} \] Take the natural logarithm of both sides: \[ \ln(0.1875) = -3 \zeta \pi \] \[ -1.673 = -3 \zeta \pi \] \[ \zeta = \frac{1.673}{3 \pi} = \frac{1.673}{9.4248} = 0.177 \] Thus, the damping ratio is: \[ \boxed{0.085} \]