Free Mechanical Vibrations
Differential Equations · Section 3.7 · generated practice set
Mass, spring and damper: natural frequency from a hanging stretch, amplitude and phase, and the three damping cases.
Practice this set → Fresh numbers on every attempt. No account needed.
Method
A mass on a spring with damping obeys $$mu'' + gamma u' + ku = 0,$$ with $m$ the mass, $gamma ge 0$ the damping coefficient and $k$ the spring constant. It is the constant-coefficient second-order equation from 3.1–3.4 wearing physical clothes.
Getting $k$ from a hanging mass
If the mass stretches the spring by $L$ at rest, the forces balance: $$kL = mg quadRightarrowquad k = rac{mg}{L}.$$ Then $omega_0 = sqrt{k/m} = sqrt{g/L}$ — the mass cancels. Watch units: $L$ in metres with $g = 9.8$, or inches with $g = 32$ ft/s².
Undamped motion
$u'' + omega_0^2u = 0$ gives $u = c_1cosomega_0t + c_2sinomega_0t$, which is the single oscillation $$u = Rcos(omega_0t - delta), qquad R = sqrt{c_1^2 + c_2^2}, qquad andelta = rac{c_2}{c_1}.$$ Period $T = 2pi/omega_0$. The amplitude never decays.
Damped motion — three cases
Everything is decided by $gamma^2$ against $4mk$:
- $gamma^2 > 4mk$ — overdamped. Two negative real roots, no oscillation.
- $gamma^2 = 4mk$ — critically damped. Repeated root, no oscillation, fastest return.
- $gamma^2 < 4mk$ — underdamped. Complex roots; oscillation inside the envelope $e^{-gamma t/2m}$ at quasi-frequency $mu = dfrac{sqrt{4mk - gamma^2}}{2m}$.
In all three cases $u o 0$. Damping decides only whether it oscillates on the way.
Definitions and theorems
Worked example
A 2 kg mass stretches a spring 0.098 m. It is pulled down 0.05 m and released from rest. Find the motion, period and amplitude.
- $k = dfrac{mg}{L} = dfrac{2(9.8)}{0.098} = 200$ N/m.
- $2u'' + 200u = 0 Rightarrow u'' + 100u = 0$, so $omega_0 = 10$ rad/s.
- $u = c_1cos 10t + c_2sin 10t$. Measuring $u$ downward positive: $u(0) = 0.05$ gives $c_1 = 0.05$; $u'(0) = 0$ gives $c_2 = 0$.
- $u = 0.05cos(10t)$ m.
- Amplitude $R = 0.05$ m, period $T = 2pi/10 approx 0.628$ s. Released from rest at maximum displacement, so the phase is zero.
Common mistakes
- Mixing unit systems. $g = 9.8$ m/s² with metres and kilograms, or $g = 32$ ft/s² with feet and slugs. A stretch given in centimetres or inches must be converted first.
- Using the weight as the mass. In pounds, weight $= mg$, so $m = W/32$ slugs. In SI a mass given in kilograms is already the mass.
- Computing $R$ from the initial conditions directly. $R = sqrt{c_1^2 + c_2^2}$ uses the coefficients; $c_2 = u'(0)/omega_0$, not $u'(0)$.
- Getting the phase in the wrong quadrant. $arctan$ returns values in $(-pi/2, pi/2)$; check the signs of $c_1$ and $c_2$ and add $pi$ when needed.
- Calling a damped system's oscillation frequency $omega_0$. Damping lowers it to the quasi-frequency $mu < omega_0$.
Practice it
Reading the method is not the same as being able to run it under time pressure. This set generates a new problem with new coefficients every attempt, marks each part separately, and shows the full worked solution.