Student_Guide_Harmonic_Cantilever.docx

Harmonic Oscillation of a Cantilever
Precision Period Analysis and Interference
Scientific Inquiry with AI
Student Guide
Detailed Instructions & Course Material
Why This Lesson
A strip of tape measure, clamped at one end with a small magnet on the tip, is a cantilever —
and a cantilever swinging near a smartphone's magnetometer is a precision timing instrument, if
you analyze the signal carefully. In Session A, you will record your own cantilever's decaying
oscillation and pull a period and frequency out of it four independent ways — starting with your
own hand-measurement, no AI at all — so you can judge which method to trust and why. In
Session B, two cantilevers oscillate at once near the same sensor, and the signal you record is
their sum. You will learn to read that sum in the frequency domain — where each cantilever's
frequency is instantly visible — split the combined signal back into its two parts, and finally
check your work against a completely independent measurement: each cantilever recorded
oscillating alone.
Learning Goals
Upon completion of this investigation, students should be able to:
Phenomena
• Explain the physics of simple and damped harmonic motion in a cantilever and connect it to
the equations that describe it.
• Explain linear superposition and how a complex combined waveform can be decomposed
into its constituent frequencies.
Experimental Practices
• Use AI to analyze time-series data and determine period/frequency by multiple methods,
and evaluate which method to trust.
• Use AI to convert a signal to the frequency domain and to separate a combined signal into
its components with a bandpass filter.
Habits of Mind
• Predict before you analyze, and verify one method against another before trusting a number.
• Critically assess what AI made possible here, and where your own judgment did the work AI
could not.

What You'll Need
• One paid AI chatbot — ChatGPT, Gemini, or Claude — with the ability to run code or
analyze uploaded data (a spreadsheet/table it can compute on, not just describe). Anyone is
fine.
• A smartphone with the free Phyphox app installed, for its magnetometer sensor.
• A cantilever beam: a thin, flexible metal strip — a small retractable tape measure works well
— plus a rigid way to clamp one end (a table clamp, a heavy book, or a taped-down block all
work; the clamp just needs to not move).
• A small cube magnet to attach near the free tip of the cantilever (for Session B, a second
cantilever of a visibly different length, with its own magnet).
• A ruler or tape measure to record cantilever length.
• A web browser, for the postlab simulation (a single .html file you open like any web page —
no install, no account).
The Core Idea
• Damping shrinks amplitude over time but, to good approximation, leaves frequency alone —
the two are independent properties of the same oscillation.
• A fitted estimate of when a peak occurred is more precise than just reading off the single
largest data point, especially as noise rises or your effective sampling rate (samples per
oscillation) drops.
• A method earns your trust by agreeing with an independently obtained number — another
method, or a signal that recombines correctly after being split apart.
• Superposition means independent signals simply add. The frequency domain (via an FFT) is
often the easiest place to see that a mixed-looking signal is actually built from a small
number of pure frequencies.
Worked Example: What a Good Analysis Looks Like
The LLNL team's own reference recordings (yours will differ — these are here to show what a
clean, trustworthy result looks like, not a target to match):
Single
cantilever
Segmented-fit average frequency ≈ 2.551 Hz, cross-checked against a
hand-measured baseline, an AI peak-picking estimate, and a full-record
damped-oscillator fit — close agreement across all four is what made this
number trustworthy.
Dual cantilever FFT of the combined signal showed two clean peaks: a long cantilever ≈
1.123 Hz and a short cantilever ≈ 2.722 Hz. Bandpass filters of about ±0.15
Hz around each center separated the two, the separated pair summed
back to the original signal, and both frequencies matched closely against
each cantilever recorded oscillating alone.

Session A — Single Cantilever: Pre-Class Preparation (≈
1–1.5 h)
• A. Socratic AI Tutoring Session (~25 min). Paste the Session A Prelab Tutor into your AI
chatbot and work through it. It will ask you questions rather than lecture — answer honestly,
in your own words. Share the chat log with your instructor.
• B. Orientation to the Instrument (~20 min). Install Phyphox. Open the magnetometer
sensor and confirm you can start, stop, and export a recording (as CSV, or by copying the
data table).
• C. Prediction Exercise (~20 min). Without AI: sketch what you expect a lightly damped
cantilever's magnetic-field-vs-time trace to look like. Write one sentence — do you expect
the period to change as the amplitude visibly shrinks? Why or why not?
• D. Build-Guide Review (~10 min). Confirm you can get a CSV or data table into your AI
chat (upload or paste).
Session A — In-Class Investigation, Stage by Stage (≈ 2 h)
Stage 1 — Build and Record
Clamp your cantilever rigidly at one end — a loose clamp is the most common source of a
messy signal. Fix the magnet near the free tip. Position your phone with the magnetometer
running as close to the tip's swing path as you can. Displace the tip a small, consistent amount,
release, and record until the oscillation has clearly decayed.
Stage 2 — Look Before You Analyze
Why it matters: catching a bad recording (too little damping range, a signal that is mostly noise,
a clamp that visibly moved) now is much cheaper than discovering it after three analysis
methods. Look at the raw trace. Does peak spacing look constant even as height shrinks,
matching your prediction?
Stage 3 — Method 1: Hand Measurement with Phyphox, No AI
Why it matters: this sets your own baseline, honestly obtained, before any AI touches the data
— every later method gets checked against it. Use Phyphox's own data-cursor feature to tap
along your recorded graph and read off the times of several successive peaks by hand.
Compute a period and frequency from these hand-read times. Don't worry if this feels imprecise
— that's expected, and worth noticing.
Stage 4 — Method 2: Peak-Picking with Local Fits
Why it matters: precision in how you ask for the peak time is the difference between a noisy
estimate and a trustworthy one. Ask your AI to find the local maxima in your recording, then to
refine each maximum's time with a short local fit rather than the single largest sample point.
Compute period and frequency between consecutive peaks across the whole record and plot
both vs. time. If the plot is scattered, ask the AI to try an upsampled spline fit or a moving
average and compare. This addresses under-sampling, not noise itself.

Stage 5 — Method 3: Segmented Harmonic-Oscillator Fit
Why it matters: fitting many short windows is exactly the kind of repetitive, precise task AI does
quickly, and you would not want to do by hand twenty-five times. Ask your AI to split your
recording into roughly twenty-five segments and fit each to x(t) = A cos(2πft + φ) + C, reporting
the residual sum of squares (RSS) — a smaller RSS means a better fit. Tabulate and average
the per-segment frequencies.
Stage 6 — Method 4: Full-Record Damped-Oscillator Fit
Why it matters: This is the only method that also gives you a damping rate, but it is the hardest
to get to converge cleanly. Ask your AI to fit your entire recording at once to x(t) = A e^(−γt)
cos(2πft + φ) + C. Compare this frequency to your Method 3 average — they should be close.
Only attempt this after Method 3 has succeeded, and only if your recording has enough clean
oscillations and low enough noise for a full-record fit to converge — it will not work for every
dataset, and that's fine.
Stage 7 — Compare and Choose
Why it matters: a number nobody checked against anything else is not yet a result. Put all four
frequency estimates side by side — hand-measured, peak-picking, segmented fit, full damped
fit. Which do you trust most, and what would make you distrust it?
Reflect
• Which of your four estimates do you trust most, and why?
• Where did comparing methods catch something a single method would have missed?
Session A — Everything You Turn In
• Your raw recording (or a link/export of it), your hand-measured baseline (Method 1), and the
three AI-assisted frequency/period estimates with the prompts you used to generate each.
• The RSS or other fit-quality values reported for Method 3, and a note on what they tell you.
• Your frequency-vs-time and period-vs-time plots from Method 2, with an under-sampling
remediation (spline or moving average) applied and compared.
• A short written comparison of all four methods: where they agree, where they diverge, and
which you trust most.
Session B — Dual Cantilever: Pre-Class Preparation (≈ 45
min–1 h)
• A. Socratic AI Tutoring Session (~20 min). Paste the Session B Prelab Tutor into your AI
chatbot and work through it.
• B. Prediction Exercise (~15 min). A longer cantilever oscillates more slowly than a shorter
one. Sketch what you expect the combined signal from two simultaneously oscillating
cantilevers of different lengths to look like in time, and separately, what you expect an FFT of
that same signal to show.

• C. Review (~10 min). Refresh Session A's vocabulary: period, frequency, fit, RSS.
Session B — In-Class Investigation, Stage by Stage (≈ 2 h)
Stage 1 — Build and Record: Combined and Separate
Set up two cantilevers of clearly different lengths near the same phone's magnetometer.
Displace both, release together, and record the combined signal. Then, using the same setup,
record each cantilever oscillating alone, one at a time — keep these solo recordings; you will
need them at the end.
Stage 2 — Time Domain First
Why it matters: You need to see what the problem looks like before the frequency domain
solves it for you. Plot the raw combined signal, and zoomed views at the start, middle, and end.
Does it look simply periodic, or does it look more complex than Session A's single-cantilever
trace, matching your prediction?
Stage 3 — Convert to the Frequency Domain
Why it matters: this is the payoff — information invisible in the time domain becomes immediate
here. Ask your AI to compute and plot the FFT (magnitude spectrum) of your combined signal.
Identify the two dominant peaks and read off each cantilever's frequency directly.
Stage 4 — Separate the Components
Why it matters: an unverified separation is a guess, not a result. Ask your AI to design and apply
two narrow-bandpass filters, one centered on each identified frequency, and plot the original
(zero-mean) signal alongside the two filtered components on shared axes. Then add the two
filtered components back together and confirm the sum tracks your original signal — this check
is required, not optional.
Stage 5 — Cross-Check Against Your Solo Recordings
Why it matters: the combined signal alone can only check itself against itself — your solo
recordings are a completely independent measurement. Run a quick AI peak-picking pass
(Session A's Method 2) on each solo recording to get an independent frequency for each
cantilever. Compare all three: the solo-measured frequency, the FFT peak from the combined
signal, and the bandpass-separated component. Do they agree?
Stage 6 — Push the Method
Why it matters: Understanding where a method breaks is as important as using it when it works.
What do you think would happen to the separation if the two cantilevers' frequencies were much
closer together? Note your prediction. You can test it directly in the postlab simulation.
Reflect
• What did the FFT make obvious that the raw time-domain trace did not?
• Where does the analogy between this cantilever pair and other superposed-wave situations
(sound, light, radio tuning) start to break down?

Session B — Everything You Turn In
• Your combined-signal plot, its FFT with both peaks labeled in Hz, and the two
bandpass-filtered component signals.
• The recombination check: the two separated components are summed and plotted against
the original.
• Your two solo recordings, their AI peak-picked frequencies, and a three-way comparison
against the FFT peaks and bandpass-separated frequencies.
• A short written answer comparing what the time-domain and frequency-domain views each
made easy or hard to see.
Post-Class Work (Both Sessions) (≈ 1 h each)
• A. Report Review. Prompt your AI to draft a short report of your results, then check every
number against your own analysis, correcting anything overstated or invented. The review,
not the generation, is the graded work.
• B. Postlab Simulation (Session B only). Open
Dual_Cantilever_Superposition_Simulation.html in a browser. Adjust each cantilever's
frequency, amplitude, and damping; watch the combined signal and its live FFT update; use
the bandpass control to pull each component back out. Try bringing the two frequencies
close together and watch separation degrade — this is the edge case your physical setup
may not have let you reach.
• C. Retrieval Practice. About 30 minutes with the course's spaced-repetition review tool,
seeded toward this lesson's content.
Appendix A — Prompt Templates
Adapt the bracketed parts to your own recording. Use them roughly in order. Method 1 (hand
measurement) uses no AI at all — just Phyphox's own data cursor — so it has no prompt below;
start with Method 2.
• Peak-picking: "Here is my time-series data of magnetic field vs. time from a decaying
cantilever oscillation [attach/paste data]. Find the local maxima, refining each peak's time
with a local fit rather than just the largest sample point. Compute the period and frequency
between consecutive peaks and plot both vs. time."
• Under-sampling remediation: "The frequency-vs-time plot above looks scattered. Try an
upsampled spline fit through the data, and separately a [N]-point moving average, and
compare both to the original."
• Segmented fit: "Split this same dataset into about 25 equal-length segments. Fit each
segment to x(t) = A cos(2*pi*f*t + phi) + C and report f and the residual sum of squares
(RSS) for each segment. Tabulate and average f."
• Full-record damped fit: "Now fit the entire dataset at once to x(t) =
A*exp(-gamma*t)*cos(2*pi*f*t + phi) + C. Report f, gamma, A, phi, and C, and plot the fit
over the data."

• FFT: "Here is my combined dual-cantilever time-series [attach/paste data]. Compute and
plot the FFT magnitude spectrum. Identify the frequencies of the dominant peaks in Hz."
• Bandpass separation: "Design two bandpass filters, each centered on one of the two
frequencies you identified, with a half-width of about 0.15 Hz. Apply both to the original
(zero-mean) signal and plot the original alongside the two filtered components on shared
y-axes. Then add the two filtered components together and plot that sum against the original
to check they match."
• Cross-check against solo recordings: "Here is my time-series data from cantilever [1 or 2]
recorded oscillating alone [attach/paste data]. Find the local maxima, refining each peak's
time with a local fit, and compute the frequency between consecutive peaks. Report the
average frequency."
Appendix B — Building the Cantilever
• A thin, flexible metal strip works best — a small retractable tape measure (the metal blade,
extended and clamped) is light and lightly damped, so you get many oscillations before the
signal fades into sensor noise.
• Clamp one end rigidly. Anything that lets the base flex or slip will show up as a messy,
inconsistent signal — check this first if your recording looks wrong.
• Attach a small cube magnet near the free tip with tape; the exact position affects signal
strength but not the physics — experiment to get a clean-looking trace.
• Place your phone so the magnetometer (usually near the phone's camera module — check
Phyphox's live reading while moving the magnet near different parts of the phone to find the
sensitive spot) is close to the tip's swing path, but not in the way of the swing itself.
• For Session B, build a second cantilever with a visibly different length from the first — length
sets natural frequency, so a clearly different length gives you two well-separated frequencies
to resolve.