1
Derive the oscillator equation and its frequency
Write m·x″ + c·x′ + k·x = 0 from Newton's law and Hooke's law, compute ω₀² = k/m exactly, and give the period as a symbol and as a rounded number.
Course · Differential Equations
Oscillators, Damping and Resonance
The previous course solved a·y″ + b·y′ + c·y = 0 as a piece of algebra: find the roots of one quadratic, and the roots decide the shape. This course gives the letters their meanings. m is a mass, k a spring, c a drag, and the roots of the quadratic become frequencies and decay rates you can name and compare. The questions change with the meaning: how fast does it swing, how big, how long until it is still, and what happens when something keeps pushing. The lab for the whole course is one oscillator whose exact quantities appear as tiles — ω₀², c² − 4mk, the critical damping, the amplitude squared, the forced response — with the motion itself drawn by stepping in floating point and labelled as a drawing. A figure that involves π, a square root or an exponential is rounded to six figures and marked ≈; everything else is a fraction.
m·x″ + c·x′ + k·x = F₀·cos(ωt)
ω₀² = k/m period 2π/ω₀
c² − 4mk: > 0 over, 0 critical, < 0 under
ω_d² = k/m − c²/(4m²) envelope e^(−ct/(2m))
A = F₀/(m·(ω₀² − ω²)) fails at ω = ω₀
ω_r² = k/m − c²/(2m²), not k/m
What this course is about
- Subject
- Differential Equations
- Lessons
- 9
- Format
- Interactive
Overview
What you will be able to do
By the end you can take a mass, a damper and a spring from their three numbers to the shape of the motion without solving anything, and you can predict how a steady push is answered. Each outcome is something you can compute and check against the lab.
2
Convert and read free motion
Turn C₁cos(ω₀t) + C₂sin(ω₀t) into A·cos(ω₀t − φ), compute A² exactly, and state the conserved energy and the phase-plane ellipse it draws.
3
Classify the damping
Decide from the sign of c² − 4mk whether the motion is overdamped, critical or underdamped, compute the critical damping 2√(mk), and write each solution form.
4
Compute what damping does to an oscillation
Find the pseudo-frequency, the envelope rate and the ratio of successive peaks, and say whether a critically damped motion crosses zero.
5
Fit the response to a forcing
Compute the undamped forced amplitude exactly, read its sign, write the resonant solution that grows linearly, and find a beat period.
6
Locate the peak of a damped response
Compute the steady amplitude squared and the resonant frequency of a damped oscillator, and say why the peak is not at the natural frequency.
Lessons
9 lessons
Three modules of three lessons each. Free oscillation first, because it fixes the frequency, the amplitude and the energy that everything later is compared with. Damping second: one sign decides the kind of motion, and the two lessons after it say what each kind does and which one a designer wants. Forcing last, where the natural frequency meets a frequency from outside.
The Mass–Spring ModelTurn Newton's law and Hooke's law into m·x″ + c·x′ + k·x = 0, and read the frequency ω₀² = k/m and the period off it.Not marked
Amplitude, Phase and PeriodRewrite C₁cos(ω₀t) + C₂sin(ω₀t) as a single cosine A·cos(ω₀t − φ), and compute A² = x₀² + v₀²/ω₀² exactly.Not marked
Energy and the Phase EllipseCompute E₀ = ½m·v₀² + ½k·x₀², show it does not change along the motion, and read the ellipse it draws in the plane of position against velocity.Not marked
Overdamped, Critical and UnderdampedClassify the motion from the sign of c² − 4mk, write the solution form for each case, and compute the critical damping 2√(mk).Not marked
Underdamped Motion and the EnvelopeCompute the pseudo-frequency ω_d² = k/m − c²/(4m²) exactly, the envelope rate c/(2m), and the ratio of successive peaks.Not marked
Critical Damping and DesignWrite the critically damped solution (C₁ + C₂·t)·e^(rt), decide exactly whether and when it crosses zero, and say why a door closer is set at critical damping.Not marked
Forced OscillationFit x_p = A·cos(ωt) to m·x″ + k·x = F₀·cos(ωt), compute A = F₀/(m·(ω₀² − ω²)) exactly, and read its sign.Not marked
Resonance and BeatsShow that the fitted amplitude fails when ω = ω₀, write the resonant solution (F₀/(2m·ω₀))·t·sin(ω₀t), and compute the beat period 2π/|ω₀ − ω| near resonance.Not marked
Damped Forcing and the Amplitude CurveCompute the steady amplitude squared A² = F₀²/((k − m·ω²)² + (c·ω)²) exactly, the resonant frequency ω_r² = k/m − c²/(2m²), and the peak amplitude.Not marked
Recommended background
Second-Order Linear Equations, which supplies the characteristic equation, the three kinds of roots and the solutions they give, and the fitting of two initial conditions, and through it First-Order Linear Equations for undetermined coefficients. From Algebra, Quadratics and Complex Numbers for the quadratic formula and i.
Practice suggestions
Say what the letters mean before you compute with them. m is a mass, k a spring constant, c a damping constant and F₀ the size of a push. Units are left out, because any consistent set will do, but every answer in this course is a statement about a physical motion, and a negative k or m is a refusal rather than a curiosity.
Predict the class of the motion from c² − 4mk before you press anything. The lab prints that number as a tile, and a surprise in it is almost always a slip in 4mk.
Read the tier of every number. Fractions in a tile are exact. A figure behind ≈ is rounded to six figures, and a surd such as √5 is exact and printed with its rounded value beside it. The curve itself is drawn by stepping in floating point, and nothing in a tile comes from that drawing.
Change one number at a time. The labs take any rational m, c and k, and the lessons are written so that moving a single one — the mass, the damping, the forcing frequency — shows a single effect.
What it does not cover
Real springs. The model assumes a force exactly proportional to the stretch and a drag exactly proportional to the speed. Real springs stiffen or soften at large stretch and real drag often grows with the square of the speed, and either change makes the equation nonlinear and outside this course.
Forcing other than a cosine. A square wave, an impulse or a force that switches on and off needs either a Fourier series, which is not here, or the Laplace transform, which is the last course. The cosine is the case undetermined coefficients can verify exactly.
The transient in a forced damped system. The lab reports the steady amplitude, which the motion settles to; the way it settles, and how long it takes, is the damped free solution of the earlier lessons added to it.
Coupled oscillators and normal modes. Two masses on three springs is a system of two equations, and the next course is where systems begin.
Modelling as a skill. The spring, the door closer and the bridge are stated with their equations given. Choosing m, c and k for a real structure is the work of an engineer with measurements, not something a lab can tell you.