Damped Free Vibration Questions and Answers

Answer: D

The damping factor is closest to 2.0%.

Logarithmic decrement \(\delta\) and damping factor \(\zeta\) are related by

\[ \begin{aligned} \delta &= \frac{2\pi\zeta}{\sqrt{1 - \zeta^{2}}} \\ \zeta &\approx \frac{\delta}{2\pi} = \frac{0.125}{6.283} \\ &= 0.0199 \approx 2\% \end{aligned} \]

The exact form \(\zeta = \delta/\sqrt{4\pi^{2} + \delta^{2}}\) gives the same 1.99%. 1.0% is what dividing by \(4\pi\) instead of \(2\pi\) gives, and 0.5% and 1.5% are smaller and larger than any reading of the formula allows.

Answer: A

The energy dissipated per cycle is \(\pi c\omega_{d}X^{2}\).

The damping force is \(c\dot{x}\), so the energy lost over one cycle of \(x = X\cos\omega_{d}t\) is

\[ \begin{aligned} W_{d} &= \oint c\dot{x}\,dx = \int_{0}^{2\pi/\omega_{d}} c\dot{x}^{2}\,dt \\ &= c\omega_{d}^{2}X^{2}\int_{0}^{2\pi/\omega_{d}}\sin^{2}\omega_{d}t\,dt \\ &= \pi c\omega_{d}X^{2} \end{aligned} \]

Energy must scale with amplitude squared and with the damping constant, so \(\pi\omega_{d}X^{2}\) (no c) and the two forms linear in X cannot be right, and \(\omega_{d}^{2}\) has the wrong dimensions for energy.

Answer: C

The matching is P - 3, Q - 4, R - 2, S - 1.

• Q oscillates with a constant amplitude that never decays: undamped (4).
• R oscillates about zero with a shrinking amplitude: underdamped (2).
• P returns to zero without crossing it and faster than any non-oscillating curve: critically damped (3).
• S also creeps back without oscillating, but much more slowly than P: overdamped (1).

The key distinction is between P and S. Critical damping is the smallest damping that prevents oscillation and gives the quickest non-oscillatory return; more damping than that (overdamping) only slows the return. Options that call P overdamped and S critically damped have them the wrong way round, and those that swap Q and R confuse constant with decaying oscillation.

Answer: B

The damping ratio is \(\frac{c}{2\sqrt{km}}\).

The damping ratio is the actual damping coefficient divided by the critical damping coefficient, the value at which the system just stops oscillating:

\[ \begin{aligned} c_{c} &= 2\sqrt{km} = 2m\omega_{n} \\ \zeta &= \frac{c}{c_{c}} = \frac{c}{2\sqrt{km}} \end{aligned} \]

\(c/\sqrt{km}\) is twice the damping ratio. The square-root forms are dimensionally wrong: c/(mk) has units of s³/kg, so its square root is not a pure number.

Answer: C

The statement that is always true is that the time period of free oscillations will increase.

A second damper in parallel adds to the damping coefficient, so the damping ratio ζ rises while the mass and stiffness, and hence the natural frequency ω_n, are unchanged. The damped frequency is \(\omega_{d} = \omega_{n}\sqrt{1-\zeta^{2}}\), which falls as ζ rises, so the damped period \(2\pi/\omega_{d}\) gets longer. Because the system stays underdamped, it still oscillates, and this holds whatever the size of the extra damper.

Transmissibility does not change one way always: more damping lowers it for frequency ratios below √2 but raises it above √2. So neither transmissibility option is always true, and a shorter period is the opposite of the correct effect.

Answer: A

The damping coefficient is 4.41 N.s/m.

From two successive amplitudes, the logarithmic decrement and damping ratio are

\[ \begin{aligned} \delta &= \ln\frac{7.75}{7.20} = 0.0736 \\ \zeta &= \frac{\delta}{\sqrt{4\pi^{2} + \delta^{2}}} = 0.01171 \end{aligned} \]

The natural frequency is \(\omega_{n} = 2\pi \times 15 = 94.25\) rad/s, so

\[ \begin{aligned} c &= 2\zeta m\omega_{n} \\ &= 2 \times 0.01171 \times 2 \times 94.25 = 4.41 \end{aligned} \]

Forgetting to convert 15 Hz to rad/s would give a value about six times too small, and the other options do not follow from the measured amplitudes.

Answer: A

The logarithmic decrement is about 0.63.

The damping ratio is \(\zeta = c/c_{c} = 35/350 = 0.1\). Then

\[ \begin{aligned} \delta &= \frac{2\pi\zeta}{\sqrt{1 - \zeta^{2}}} \\ &= \frac{0.6283}{0.995} = 0.631 \end{aligned} \]

For light damping \(\delta \approx 2\pi\zeta = 0.628\) anyway. 1.26 and 1.89 are two and three times the correct value, and 0.32 is half of it, as if \(\pi\zeta\) were used.