4.3 Periodic Motion, Vibration, Harmonics & Resonance

Periodic Motion, Vibration, Harmonics & Resonance

Aircraft and engines are full of parts that move back and forth or rotate and shake: control surfaces flutter in extreme cases, panels buzz, rotors whirl, and instrument needles can quiver. Module 2 groups these under periodic motion, vibration, harmonics, and resonance. You need clear definitions, the idea of natural frequency, and why matching a drive frequency to that natural frequency is a maintenance and design hazard.

Periodic Motion

Periodic motion repeats itself in equal time intervals. The period T is the time for one complete cycle (seconds). The frequency f is the number of cycles per second:

f = 1 / T  (unit: hertz, Hz)

One hertz means one cycle per second. A panel that completes 20 full oscillations each second has f = 20 Hz and T = 0.05 s. Angular frequency ω = 2πf (rad/s) appears when relating SHM to circular motion projections, but period and frequency in hertz are the usual Module 2 language.

Vibration is oscillatory motion about an equilibrium (mean) position — a special case of periodic or nearly periodic motion important in airworthiness. Amplitude is the maximum displacement from equilibrium; larger amplitude means more severe vibration for a given frequency.

Pendular Motion

A simple pendulum is a point mass on a light inextensible string swinging under gravity. For small angles, the motion is approximately simple harmonic, and the period is:

T ≈ 2π √(L / g)

where L is length (m) and g is gravitational acceleration (m/s²). Notice mass does not appear: for an ideal simple pendulum, period is independent of the bob’s mass. Longer pendulums swing more slowly; stronger g shortens the period.

Worked example. A pendulum of length 1.00 m on Earth (g = 9.81 m/s²):

T ≈ 2π √(1/9.81) ≈ 2π × 0.319 ≈ 2.01 s  (f ≈ 0.50 Hz)

Pendulum thinking also appears in approximate models of hanging cables, suspended tools, and some instrument damping concepts — the key exam points are the √(L/g) dependence and mass independence for the ideal simple case.

A physical (compound) pendulum has distributed mass; its period depends on moment of inertia and distance from pivot to centre of mass. Module 2 mainly expects the simple-pendulum idea and the broader SHM concept rather than heavy compound-pendulum algebra.

Simple Harmonic Motion (SHM)

Simple harmonic motion is the ideal oscillatory motion in which:

• Displacement from equilibrium varies sinusoidally with time
• Restoring force is proportional to displacement and always directed toward equilibrium: F = −kx (Hooke’s law spring is the prototype)
• Consequently acceleration a = −(k/m)x is also proportional to −x

For a mass–spring system (horizontal, frictionless idealisation):

T = 2π √(m / k)  f = (1/2π) √(k / m)

Stiffer spring (larger k) ⇒ higher natural frequency; larger mass ⇒ lower natural frequency. Small-angle pendulum motion matches this SHM pattern with an effective restoring torque from gravity.

Worked example — mass–spring frequency. A 0.50 kg mass on a spring of stiffness k = 200 N/m:

f = (1/2π) √(200/0.50) = (1/2π) √400 = (1/2π) × 20 ≈ 3.18 Hz  (T ≈ 0.31 s)

SHM assumptions fail at large amplitudes, with friction, or when materials go plastic — but the proportional restoring-force model is the conceptual core for natural frequency discussions.

Natural Frequency

Every elastic structure or mass–spring-like system has one or more natural frequencies — frequencies at which it “likes” to vibrate if displaced and released (free vibration), or at which forced vibration grows large. Natural frequency depends on:

• Stiffness (geometry, material modulus, supports)
• Mass (and mass distribution)
• For pendulums: length and g
• Boundary conditions (clamped, free, simply supported panels behave differently)

A wing, a fuselage skin bay, a turbine blade, and an engine mounting each have their own set of natural frequencies (modes). Designers and maintainers care because operating speeds (rpm, blade-pass frequency, gust encounter rates) must not sit on those frequencies without adequate damping or redesign.

Harmonics

A pure sine wave has a single frequency — the fundamental. Real vibrations and sound waves are often superpositions of several frequencies. Harmonics are components at integer multiples of a fundamental frequency f₀:

• 1st harmonic / fundamental: f₀
• 2nd harmonic: 2f₀
• 3rd harmonic: 3f₀
• …

A shaft rotating at frequency f can generate vibration content at f, 2f, 3f, etc., especially with imbalance, misalignment, or gear meshing. Blade-pass frequency (number of blades × shaft frequency) is a classic forcing frequency in fans and compressors. Recognising harmonics matters when a spectrum analysis shows peaks not only at running speed but at multiples — and when a structure’s natural frequency coincides with any of those peaks.

Resonance

Resonance occurs when a system is driven at (or very near) a natural frequency. Energy is fed in phase with the motion cycle after cycle, so amplitude builds — limited only by damping and non-linear effects. Everyday analogies: pushing a swing at the right rhythm; a note shattering a glass at its natural frequency.

On aircraft, resonance can:

• Amplify propeller or rotor vibration into the airframe
• Excite panel buzz or control-surface oscillations
• Stress engine mounts, gearboxes, and bearings through synchronous vibration
• Contribute to fatigue even when single-cycle loads look modest

Avoidance and control include: changing stiffness or mass (shift natural frequency), adding damping, balancing rotating parts, isolating mounts, imposing rpm avoid bands (“critical speeds”), and design rules that keep operating ranges clear of major modes. Critical speed of a shaft is essentially a resonance of the rotor–support system with rotation frequency.

Worked conceptual example. An unbalanced cooling fan runs at 50 Hz (3000 rpm). A nearby access panel has a natural frequency of about 50 Hz. At that rpm the panel amplitude grows sharply (resonance). Solutions might include balancing the fan (reduce drive force), stiffening the panel (raise natural frequency), adding a damping treatment, or changing the operating speed — any change that detunes drive frequency from natural frequency or reduces energy input.

Linking Back to Circular Motion

A point in SHM can be viewed as the shadow (projection) of a point in uniform circular motion. That geometric link explains why ω = 2πf appears in both topics and why the same vocabulary of period and frequency applies. Imbalance on a rotating propeller is a rotating force at frequency f = rpm/60; if that frequency hits a structural natural frequency, resonance follows.

Maintenance Mindset

Vibration is a symptom as well as a physics topic: new buzz after a prop change, a rattle at a specific rpm, or fretting wear at fasteners can indicate imbalance, loose hardware, or a path into resonance. Module 2 does not replace vibration analysis courses, but it supplies the language: period, frequency, amplitude, natural frequency, harmonic, and resonance danger.

Summary Table

If you can explain why a structure “sings” at certain rpm and how stiffening or balancing detunes that response, you own the Module 2 intent of this subsection.