Math for ME · Chapter 15 of 19 · Advanced
Fourier Series, Frequency, and Signals
Any repeating signal is a sum of sines. Frequency thinking explains why machines resonate, how sensors filter, and what a spectrum analyzer shows.
The thread: A transform handles a single transient. A repeating signal needs a different lens: every vibration is a sum of pure tones, and this chapter learns to read that spectrum.
Readiness check
From Trigonometry, Integrals, and ODEs. Tick only what you can do closed-notes.
- Sketch A sin(ωt + φ) and name amplitude, frequency, phase.
- Integrate products of sines over a period (or accept the orthogonality result).
- Convert between f (Hz), ω (rad/s), and period T.
- Explain resonance from ODEs qualitatively.
- Read a bar chart as values at discrete points.
The core idea
Every periodic signal is a recipe of sines: a fundamental plus harmonics.
f(t) = a₀ + Σ aₙcos(nωt) + bₙsin(nωt)The Fourier coefficients are the recipe quantities, extracted by integration. The frequency domain lists them as a spectrum: which frequencies, how strong. A machine responds to each ingredient separately, which is why one buried harmonic can find a resonance.
The skills, taught in order
15.1 Any periodic signal is a sum of sines
A repeating signal of period T equals a constant plus sinusoids at the fundamental frequency ω = 2π/T and its integer multiples:
f(t) = a₀ + Σ [aₙ cos(nωt) + bₙ sin(nωt)]The coefficients aₙ, bₙ are extracted by integrating the signal against each sine and cosine over one period; they are the recipe amounts of each ingredient.
15.2 Harmonics and the spectrum
The component at n times the fundamental is the nth harmonic. Plotting the coefficient sizes against frequency gives the spectrum: a list of which frequencies are present and how strong. The same signal seen as a spectrum often reveals what a time trace hides.
15.3 Reading a spectrum diagnoses machines
Rotating machinery announces its health in harmonics. A peak at the rotation frequency means imbalance; peaks at the blade or tooth count times the rotation speed are normal; a peak that is not a harmonic of the rotation points to a resonance, a bearing defect, or mains hum. Frequency content is the fingerprint.
15.4 Sampling, the DFT, and aliasing
The discrete Fourier transform (computed fast by the FFT) gives the spectrum of a sampled signal, but only up to half the sampling rate, the Nyquist frequency. Anything faster folds back as a false low-frequency peak:
fNyquist = fsample/2The cure is to sample at least twice the highest frequency present, with margin, or to filter out the high content before sampling.
15.5 Filtering and convolution
A linear system reshapes a signal's spectrum by multiplying it, frequency by frequency, by the system's frequency response. That multiplication in frequency is a convolution in time. It is why a low-pass filter quietly removes high harmonics and why every sensor has a bandwidth.
Engineering connection: Vibrations, Heat Transfer, Measurements, Signal Processing.
Worked example: what a square wave is made of
A test rig drives a structure with a ±1 square wave at frequency ω. Its Fourier series is f(t) = (4/π)[sin ωt + sin 3ωt/3 + sin 5ωt/5 + …]. Find the first three harmonic amplitudes and explain the engineering consequence.
- ProblemQuantify the harmonic content of the square wave in Figure 1.
- Given / findf(t) = (4/π)Σ sin(nωt)/n over odd n. Find the amplitudes of n = 1, 3, 5 and interpret.
- AssumptionsIdeal ±1 square wave; linear structure under test.
- ModelThe series coefficients are the spectrum: amplitude of harmonic n is 4/(nπ) for odd n, zero for even n.
- Equationsbₙ = 4/(nπ), n odd
- Solveb₁ = 4/π = 1.273; b₃ = 4/(3π) = 0.424; b₅ = 4/(5π) = 0.255. Even harmonics vanish by symmetry.
- CheckAmplitudes fall like 1/n, matching the spectrum bars of the concept figure. Summing the three terms already reproduces the square shape closely (Figure 1), so the truncation is behaving.
- ConclusionThe "single-frequency" square drive actually injects energy at ω, 3ω, 5ω, and beyond. If the structure has a resonance near 3ω, a square-wave test will excite it at 42% of the fundamental's strength: a classic source of mystery vibration in test rigs.
Worked example 2: a blade-pass peak that lies to you
A cooling fan rotates at 1200 rpm and has 7 blades. A data logger samples its vibration at 200 Hz. Find the rotation and blade-pass frequencies, and determine what the blade-pass component appears as in the spectrum.
- Given / find1200 rpm, 7 blades, sampling rate 200 Hz. Find the rotation frequency, the blade-pass frequency, and its displayed frequency.
- Rotation frequencyf = 1200/60 = 20 Hz.
- Blade-pass frequencyeach revolution passes 7 blades, so f_blade = 7 × 20 = 140 Hz.
- Nyquist limitf_Nyquist = 200/2 = 100 Hz. Since 140 Hz exceeds 100 Hz, it cannot be represented honestly and will alias.
- Aliased frequencythe 140 Hz content folds about the sampling rate: 200 − 140 = 60 Hz. The spectrum shows a confident peak at 60 Hz that the machine never produced.
- Check60 Hz is not an integer multiple of the 20 Hz rotation, yet it could easily be mistaken for a real feature; that is exactly why aliasing is dangerous.
- FixSample at least twice the highest real frequency, above 280 Hz (use a comfortable margin such as 500 Hz), or add an anti-alias filter that removes content above 100 Hz before sampling.
- ConclusionReading a spectrum is worthless without knowing the sampling rate and its Nyquist limit first. The most dangerous measurement is a clean-looking false peak.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| "One signal, one frequency" | Surprise at energy appearing at 3ω | "Is my signal a pure sine, or merely periodic?" | Only pure sines are single-frequency. Everything else carries harmonics. |
| Spectrum read as a time plot | "The signal rises then falls" said of a spectrum | "Is the horizontal axis time or frequency?" | A spectrum is a recipe list, not a history. Label the axis before reading. |
| More terms expected to fix corners instantly | Frustration at the overshoot near jumps | "What happens at a discontinuity?" | The ringing at jumps (Gibbs effect) narrows but never vanishes. It is a feature of truncation, not an error. |
| FFT treated as magic | Peaks trusted at impossible frequencies | "What is my sampling rate, and half of it?" | The DFT only sees below half the sampling frequency; faster content folds back (aliasing). Sample fast enough. |
Practice ladder
A machine runs at 1800 rpm. Find the fundamental in Hz and rad/s, and the frequencies of the first three harmonics.
Show answer
1800 rpm = 30 Hz = 188.5 rad/s. Harmonics: 30, 60, 90 Hz. Bearing and blade-pass defects announce themselves at such multiples.
A sawtooth wave has harmonic amplitudes that fall as 1/n. By what factor is the 4th harmonic weaker than the fundamental?
Show answer
By a factor of 4: the 4th harmonic has 1/4 the fundamental's amplitude. Slow 1/n decay means many harmonics carry real energy.
Using the square-wave series, estimate f(t) at ωt = π/2 with one, two, and three terms, and compare with the true value 1.
Show answer
One term: 1.273. Two: 1.273 − 0.424 = 0.849. Three: 0.849 + 0.255 = 1.104. The sums oscillate around 1, tightening as terms accumulate.
You must capture vibration up to 2 kHz cleanly. What is the minimum sampling rate, and a sensible practical choice?
Show answer
The minimum is 2 × 2000 = 4000 Hz (Nyquist). In practice engineers sample several times higher, say 8 to 10 kHz, with an anti-alias filter, to leave margin.
A vibration sensor samples at 1000 Hz. A shaft actually vibrates at 800 Hz. What frequency will the DFT display, and why is the measurement worthless?
Show answer
The limit is 500 Hz; 800 Hz aliases to 1000 − 800 = 200 Hz. The spectrum shows a confident, completely false 200 Hz peak. Anti-alias filtering or faster sampling is mandatory.
Use a phone spectrum-analyzer app on a real machine (fan, drill, washing machine). Identify the fundamental from the rpm, find two harmonics, and explain one peak that is not a harmonic of the rotation.
What good work looks like
A screenshot with the fundamental matched to rpm within a few percent, harmonics marked, and a sensible attribution for the extra peak (blade count times rpm, mains hum, resonance).
Working with AI, and proving it yourself
Use AI as an examiner, not a solver
Portfolio task
Produce a one-page spectrum report on a real machine (your Level 4 measurement): the time trace, the spectrum, a table linking each labeled peak to its physical source, and one recommendation (balance, isolate, or change speed).
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. What claim does the Fourier series make?
Any reasonable periodic signal equals a constant plus a sum of sines and cosines at integer multiples of the fundamental frequency.
2. What is a harmonic, and why do engineers track them?
A component at n times the fundamental. Faults and nonlinearities put energy into specific harmonics, so spectra diagnose machines.
3. What does the DFT do, in one sentence?
It computes the harmonic recipe of a sampled signal: the FFT is just its fast algorithm.
4. State the aliasing rule.
Content above half the sampling frequency folds back as false low frequencies; sample at least twice the highest frequency present.
5. What is convolution used for, conceptually?
Filtering: a system's output spectrum is the input spectrum times the system's frequency response: multiplication in frequency, convolution in time.
Textbook mapping
| Item | Mapping |
|---|---|
| Main sources | Kreyszig, Advanced Engineering Mathematics, Ch 11 (Fourier analysis). Partial differential equations follow in Ch 12 |
| Core topics | 15.1 Periodic functions · 15.2 Sine-cosine representation · 15.3 Series idea · 15.4 Harmonics · 15.5 Frequency domain · 15.6 DFT (conceptual) · 15.7 Convolution (conceptual) · 15.8 Vibration, heat, signal applications |
| Engineering connection | Vibrations (spectra, resonance), Heat Transfer (periodic boundary conditions), Measurements and Signal Processing. |
| Skip on first pass | Complex-exponential form proofs, convergence theory, fast-algorithm internals. |
| Read next | Partial Differential Equations. |