Oscillatory motion
The oscillating spring and the RLC circuit obey the same differential equation. Whoever understands one understands the other, and mechanical resonance is literally the same as that of a radio tuner.
01Simple harmonic motion
Whenever a restoring force is proportional to the displacement and points in the opposite direction, the resulting motion is simple harmonic. Hooke’s law, F = −k·x, combined with the second law gives:
| System | Natural angular frequency | Condition |
|---|---|---|
| Mass and spring | \( \omega _{0} = \sqrt{k/m} \) | Ideal spring |
| Simple pendulum | \( \omega _{0} = \sqrt{g/L} \) | Small angles |
| Physical pendulum | \( \omega _{0} = \sqrt{m\cdot g\cdot d/I} \) | Small angles |
| LC circuit | \( \omega _{0} = 1/\sqrt{L\cdot C} \) | No resistance |
The energy is exchanged between kinetic and elastic potential, and their sum stays constant: E = ½·k·A². In the LC circuit exactly the same thing happens between the capacitor and the inductor.
02Damped oscillations
With a viscous friction force proportional to the velocity, F = −b·v, the equation is completed:
| Regime | Condition | Behavior |
|---|---|---|
| Underdamped | \( b < 2\sqrt{m\cdot k} \) | Oscillates with an amplitude that decays exponentially |
| Critically damped | \( b = 2\sqrt{m\cdot k} \) | Returns to rest in the shortest possible time, without oscillating |
| Overdamped | \( b > 2\sqrt{m\cdot k} \) | Returns slowly, without oscillating |
Q = √(m·k)/b measures how many oscillations survive before the amplitude drops appreciably. It is the same Q as in a resonant circuit: a high Q means little loss, a sharp resonance and selectivity. A quartz crystal reaches a Q of 105, and that is why it sets the frequency of a clock or of a microcontroller.
Critical damping is what is sought in a car’s shock absorber, in the motion of the needle of an analog instrument and in the response of a switching power supply: reaching the final value quickly without overshooting.
03Forced oscillations and resonance
If a periodic force F₀·cos(ωt) is also applied, once the transient has died out the system oscillates at the frequency of the excitation, not its own, and with an amplitude that depends on how close one is to the other:
When ω approaches ω₀ the denominator becomes small and the amplitude grows: that is resonance. With little damping the amplitude can become destructive —it is what shatters glasses with sound and what forces engineers to study the wind on a bridge—, but put to good use it is what lets you tune in a station among all those that reach the antenna.
Top, the response in time; bottom, the resonance curve. It is the same lab as an RLC circuit, with different names.
04The analogy worth memorizing
| Mechanical | Electrical (series RLC) | Role |
|---|---|---|
| Mass m | Inductance L | Inertia: opposes change |
| Damping b | Resistance R | Dissipates energy |
| Constant k | \( \dfrac{1}{C} \) | Restoring: stores energy |
| Position x | Charge q | State variable |
| Velocity v | Current i | Its derivative |
| Force F | Voltage V | Excitation |
| \( \omega _{0} = \sqrt{k/m} \) | \( \omega _{0} = 1/\sqrt{L\cdot C} \) | Natural frequency |
| \( Q = \sqrt{m\cdot k}/b \) | \( Q = \dfrac{1}{R}\cdot \sqrt{L/C} \) | Selectivity |
It is not a metaphor: the two differential equations are identical, so everything that is proved in one domain holds in the other. Whoever understood the response of a spring has already understood that of a filter, and vice versa.
05Where it shows up in electronics
- Quartz crystals and MEMS resonators. A mechanical oscillator of very high Q that sets the timebase of almost everything digital.
- Filters and tuning. The resonance curve in this lab is the frequency response of a bandpass filter.
- Piezoelectric transducers. Ultrasonic emitters work at their mechanical resonance frequency, where the efficiency is highest.
- Vibration-isolation mounting. Onboard equipment is mounted on elastic elements chosen so that its natural frequency stays far from that of the excitation.
- Response of a switching power supply. The feedback loop is compensated aiming for a behavior close to critical: fast but without overshoot.
06In the lab
Time twenty oscillations of a pendulum of known length and solve for g from T = 2π√(L/g). Repeat with amplitudes of 5° and 30° to see where the approximation stops holding.
Measure k statically with weights, then let the mass oscillate and film it. From the decay of the amplitude you get b, and with it Q. Verify that the measured ω₀ matches √(k/m).
Drive a cantilevered strip with a speaker sweeping through frequencies and record the vibration with an accelerometer. Plot the amplitude against frequency and measure the −3 dB bandwidth: Q = f₀/Δf, the same definition as in a resonant circuit.
07Common mistakes
- Believing the period of a pendulum depends on the mass. It does not; it depends only on the length.
- Using the pendulum formula with large amplitudes. It holds as long as sinθ ≈ θ.
- Confusing frequency f with angular frequency ω. ω = 2π·f.
- Assuming that in the forced regime the system oscillates at its natural frequency. It oscillates at that of the excitation; the natural one appears only in the transient.
- Believing that more damping is always better. Beyond critical, the response becomes slower.
- Thinking resonance is always a problem. Without resonance there would be no radio.
08Self-assessment
A 0.5 kg mass on a 200 N/m spring. What is its natural frequency?
ω₀ = √(200/0.5) = 20 rad/s, that is f₀ = 20/2π ≈ 3.18 Hz.
What must b be for critical damping in that system?
b = 2√(m·k) = 2√(100) = 20 N·s/m.
How long is a pendulum that takes 2 s for each complete oscillation?
L = g·T²/(4π²) = 9.8·4/39.48 ≈ 0.99 m. It is the classic seconds pendulum.
What is the electrical equivalent of mass?
Inductance: both oppose abrupt changes in their variable —velocity and current— and store energy in the same form, ½·m·v² and ½·L·i².
A system has Q = 50 and f₀ = 1 kHz. What is its bandwidth?
Δf = f₀/Q = 20 Hz. The same calculation as for a resonant circuit.
Why does a quartz crystal define a frequency so well?
Because its extremely high Q makes the resonance curve very sharp: only an extremely narrow band of frequencies produces appreciable oscillation.
09Further reading
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics, Volume 1. 9th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage Learning, 2014. Chapter 15 covers simple, damped and forced harmonic motion.
- A. P. French. Vibrations and Waves. W. W. Norton, 1971 (Spanish edition, “Vibraciones y ondas”, Reverté, 1974). An entire book devoted to the subject, with the mechanical–electrical parallel developed in earnest.
- Richard P. Feynman. The Feynman Lectures on Physics, Volume I. Addison-Wesley, 1963 (Spanish edition, “Física, volumen I”, Addison-Wesley / Pearson). Chapters 21 to 24 treat the oscillator and resonance as a single phenomenon with many faces.
- Paul A. Tipler and Gene Mosca. Physics for Scientists and Engineers, Volume 1. 6th ed., W. H. Freeman, 2008 (Spanish edition, “Física para la ciencia y la tecnología”, Reverté, 2010). With good numerical examples of quality factor and bandwidth.