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Physics for ME · Chapter 10 of 16 · Intermediate

Oscillations, Mechanical Waves, and Resonance

Anything with mass and stiffness has a favorite frequency. Drive it there and amplitudes grow: the most important warning in mechanical engineering.

Readiness check

From Chapters 4 and 6, Math Chapters 2 and 10. Tick only what you can do closed-notes.

Use the spring model F = ks with units.
Sketch A sin(ωt + φ) and name its anatomy (Math Chapter 2).
Trade kinetic and potential energy in an audit.
Recall ω = √(k/m) from the math course's oscillator.
Convert between f, ω, and T.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsReview Math Chapter 10; the oscillator ODE lives there.

3 or more weak itemsStep back to Chapter 6 and Math Chapter 2.

The core idea

Restoring force plus inertia equals oscillation, at one natural frequency.

ω_n = √(k/m)T = 2π√(m/k)

Simple harmonic motion is the spring law inside Newton's second: x(t) = A sin(ω_nt + φ). Damping drains the swing; periodic driving feeds it. When the drive frequency meets ω_n, energy arrives in step every cycle: resonance.

The skill works when: amplitudes are small and the restoring force is roughly linear: springs, pendulums, shafts, machine mounts.

The skill breaks down when: amplitudes grow nonlinear or several modes interact; Mechanical Vibrations (with Rao) takes over there.

The concept. The resonance curve: harmless response away from ω_n, dangerous growth at it. Damping is the only thing capping the peak.

What this chapter covers

10.1 Simple harmonic motion: the spring-mass archetype.
10.2 Energy in oscillation: kinetic and potential trading at 2f.
10.3 Pendulums: T = 2π√(L/g) for small swings.
10.4 Damping: under, critical, over (Math Chapter 10's trio).
10.5 Forced vibration and resonance: the peak and its dangers.
10.6 Mechanical waves: disturbance traveling at v = fλ.
10.7 Standing waves and natural modes: why parts have several ω_n.

Engineering connection: direct preparation for Mechanical Vibrations (bridge reference: Rao); machine mounts, rotor whirl, acoustics.

Worked example: the suspension's favorite speed

A quarter-car model carries m = 300 kg on a spring of k = 30 000 N/m. Find the natural frequency. The road has expansion joints every 12 m: at what speed does the car resonate?

Figure 1. The governing model: a sprung mass excited once per road joint. Resonance when the joint rate equals 1.59 Hz.

ProblemFind f_n and the resonant road speed in Figure 1.
Given / findm = 300 kg, k = 30 000 N/m, joint spacing λ = 12 m.
AssumptionsQuarter-car single degree of freedom; light damping; one bump impulse per joint.
Modelω_n = √(k/m); the road forces the system at frequency v/λ; resonance when they match.
Equationsω_n = √(k/m) f_n = ω_n/2π v = f_n·λ
Solveω_n = √100 = 10 rad/s, so f_n = 10/2π = 1.59 Hz (T = 0.63 s). Resonant speed v = 1.59 × 12 = 19.1 m/s ≈ 69 km/h.
CheckReal cars ride at 1 to 2 Hz: the 1.59 Hz result is in the right band. Faster or slower than 69 km/h moves the excitation off the peak: matching everyday experience of a "bad speed" on jointed roads.
ConclusionThe shock absorber exists to flatten exactly this peak (the damping of Math Chapter 10). The same match-the-frequency audit governs machine mounts, pipeline supports, and rotor critical speeds.

Result. f_n = 1.59 Hz; resonance near 69 km/h on 12 m joints; dampers cap the peak.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Amplitude in the frequency formula	"Bigger swing, slower swing" claimed for SHM	"Does A appear in ω = √(k/m)?"	It does not: small-amplitude SHM keeps time regardless of amplitude. That is why pendulum clocks work.
Resonance read as a force amplifier	"The road hits harder at 69 km/h"	"What grows: the input, or the response?"	The input is unchanged; arriving in phase each cycle lets small inputs accumulate a large response.
Damping expected to shift ω_n strongly	Retuning claimed from a fitted damper	"How far does ω_d = ω_n√(1−ζ²) move for small ζ?"	Barely, for realistic ζ. Damping mainly caps the peak; mass and stiffness set the tune.
Wave speed confused with particle speed	v = fλ applied to the bobbing material	"Is this the pattern's speed or the medium's?"	v = fλ is the pattern. The medium oscillates locally and goes nowhere.

Practice ladder

Level 1 · Direct skill

A 4 kg mass hangs on a spring that stretches 50 mm under its weight. Find k, ω_n, and f_n.

Show answer

k = mg/s = 39.24/0.05 = 785 N/m. ω_n = √(785/4) = 14.0 rad/s; f_n = 2.23 Hz. The static deflection alone fixed the tune: ω_n = √(g/s).

Level 2 · Mixed concept

A pendulum clock runs slow by 2 minutes per day. Should its pendulum be lengthened or shortened, and by what fraction?

Show answer

Slow means T too long, so shorten. T ∝ √L: the period error is 2/1440 = 0.14%, so L must shrink by about 0.28% (twice the period fraction). Tiny adjustments, big timekeeping: the √L sensitivity from Math Chapter 4.

Level 3 · Independent problem

A 50 kg machine on its mounts has ω_n = 25 rad/s. Its rotor runs at 1450 rpm. Is the operating point safely above resonance, and what happens during every start-up?

Show answer

Rotor frequency = 1450 × 2π/60 = 151.8 rad/s: about 6 times ω_n, comfortably supercritical. But every start and stop sweeps the speed through 25 rad/s (239 rpm), so the machine crosses resonance twice per cycle: dampers and quick run-up exist for that moment.

Level 4 · Transfer to real engineering

Measure a real natural frequency: pluck a ruler clamped to a desk, time ten oscillations at two clamp lengths, and test the stiffness scaling (shorter overhang, higher frequency). Optionally verify with the phone spectrum app from Math Chapter 13.

What good work looks like

Two measured frequencies with timing method shown, the qualitative L-scaling confirmed, and one sentence connecting the experiment to turbine-blade or PCB vibration testing.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Here is my resonance audit (ω_n, excitation source, frequencies compared). Challenge the excitation frequency I identified."

"Give me three systems; I will estimate each ω_n by √(k/m) or √(g/s) reasoning before any data."

"What is the natural frequency?" Extracting k and m from the hardware is the modeling skill.

"Is resonance a problem here?" Comparing excitation against ω_n is the design judgment to own.

Portfolio task

Write a one-page resonance audit of a real system (washing machine, fan on a shelf, footbridge video): the estimated ω_n, the excitation source and frequency, the margin between them, and one fix (stiffen, soften, damp, or change speed).

Must include: both frequencies with their estimates justified, the start-up sweep issue addressed, and the chosen fix defended in one sentence.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Write ωn and T for a spring-mass system and a pendulum.

Spring-mass: ω_n = √(k/m), T = 2π√(m/k). Pendulum: T = 2π√(L/g), small angles.

2. What does damping change, and what does it barely change?

It caps the resonance peak and decays free vibration; it barely shifts the frequency for small ζ.

3. State the resonance condition and why it is dangerous.

Drive frequency ≈ ω_n: each cycle's energy arrives in phase, so amplitude builds until damping or failure stops it.

4. Relate wave speed, frequency, and wavelength.

v = fλ: the pattern's speed, set by the medium; f set by the source.

5. Why do continuous parts have many natural frequencies?

Each standing-wave pattern (mode) that fits the geometry has its own ω_n: the mode shapes of Math Chapter 9, made physical.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-solve the suspension example from memory.

+3 daysOne supercritical-machine check (Level 3 style) with new numbers.

+7 daysMixed set: an ω_n estimate, a Chapter 6 energy trade, a spectrum read.

+30 daysOpen Rao's first chapter (or the Vibrations course when it arrives): this chapter is its runway.

Textbook mapping

Item	Mapping
Main source	OpenStax University Physics Vol. 1, Oscillations and Waves chapters
Benchmark / reference / bridge	MIT 8.01 · Halliday, Resnick and Walker · Rao, Mechanical Vibrations (the destination course's text)
Core topics	10.1 SHM · 10.2 Oscillation energy · 10.3 Pendulums · 10.4 Damping · 10.5 Forced vibration and resonance · 10.6 Waves · 10.7 Standing waves and modes
Engineering connection	Mechanical Vibrations, machine mounts, rotor dynamics, acoustics; pairs with Math Chapter 10 and Chapter 13.
Read next	Chapter 11: Thermal Physics: Temperature, Heat, and Material Response.