Worked exercise — Mass, spring and damping

Problem

A mass of $200$ g is attached to a spring ($k = 80$ N/m) in a viscous fluid. The amplitude halves every $4{,}0$ s. Calculate: (a) the undamped natural frequency, (b) the damping constant $\gamma = b/m$, (c) the effective frequency.

Solution

(a) Undamped natural frequency. $\omega_0 = \sqrt{\frac{k}{m}} = \sqrt{\frac{80}{0{,}2}} = \sqrt{400} = 20 \ \text{rad/s}$

(b) Damping constant. The amplitude envelope decays as $A(t) = A_0\,e^{-(\gamma/2)t}$. Halving in $t_{1/2} = 4{,}0$ s requires $e^{-\gamma t_{1/2}/2} = 1/2$, from which $\gamma = \frac{2\ln 2}{t_{1/2}} = \frac{2\cdot 0{,}693}{4} \approx \ev{0{,}347 \ \text{s}^{-1}}$

(c) Effective frequency. $\omega = \sqrt{\omega_0^2 - (\gamma/2)^2} = \sqrt{400 - 0{,}030} \approx \ev{20 \ \text{rad/s}}$

The damping is very light: the frequency remains practically unchanged, and the envelope decreases with time constant $\tau = 2/\gamma \approx 5{,}8$ s.

Links

Topics: Oscillazioni e moto armonico Concepts: Oscillazioni smorzate · Moto armonico semplice Skills: Lettura dei grafici Objects: Molla