Vibrations and Waves
(Level 4)
So far: motion, forces, rotation. All treated as if things happen once and you solve for the result. Level 4 asks: what if the system doesn't stop? What if it bounces back, oscillates, repeats? What if disturbances travel through the system instead of just sitting there?
Vibrations mean things move back and forth. Waves mean disturbances propagate. Both show up constantly in machines, structures, vehicles, acoustics, and failure modes. If you don't understand them, you can't predict resonance, noise, fatigue, or stability.
Simple Harmonic Motion
Push a mass on a spring. Let go. It bounces back and forth. That's oscillation. The system has an equilibrium position (rest state). Displace it and a restoring force pulls it back. But it overshoots, oscillates past equilibrium, reverses direction, repeats.
Simplest case: restoring force proportional to displacement. F = -kx. Combine with F = ma and you get a differential equation:
Solution? Sines and cosines. The system oscillates forever (in the idealized, frictionless case):
A is amplitude (how far it swings). ω is angular frequency (how fast it oscillates). φ is phase (where it starts). This is simple harmonic motion (SHM)—the fundamental oscillatory behavior. Any linear restoring force produces it.
Stiffness k = 100 N/m, mass m = 4 kg
ω = √(k/m) = √(100/4) = 5 rad/s
Period T = 2π/ω = 1.26 seconds per cycle
SHM is everywhere: springs, pendulums, structural vibrations, electrical circuits (LC oscillators), even atomic bonds. Get comfortable with it. It's the foundation of vibration analysis.
Natural Frequency
Every system has a natural frequency—the rate it wants to oscillate when left alone. It's determined by physical properties: stiffness and inertia. Stiffer system → oscillates faster. More massive system → oscillates slower.
This isn't arbitrary. It falls directly out of the differential equation. The system doesn't "choose" to oscillate at ω_n. The physics forces it.
System A: m = 1 kg → ω_n = √(100/1) = 10 rad/s
System B: m = 4 kg → ω_n = √(100/4) = 5 rad/s
Quadruple the mass, halve the frequency. Heavier things oscillate more slowly.
Why This Matters Later
If you drive a system at its natural frequency, bad things happen. Small inputs cause large responses. That's resonance, and we'll get to it. For now: every system has a natural frequency. Know it. Avoid exciting it unless you mean to.
Damping
Ideal spring-mass system oscillates forever. Real systems don't. Friction, air resistance, internal material losses—energy dissipates. Oscillations decay. That's damping.
Add a damping term (proportional to velocity) to the equation of motion:
ζ (zeta) is the damping ratio. It tells you how much damping you have relative to critical damping. Three cases:
System still oscillates, but amplitude decays over time. Most real systems live here. Car suspension, buildings swaying, machinery vibrating then settling.
No oscillation. System returns to equilibrium as fast as possible without overshooting. Ideal for door closers, measuring instruments, anything where you want fast response without bouncing.
Still no oscillation, but slower return to equilibrium than critical damping. Too much damping. System sluggish.
Damping controls vibration amplitudes, noise, wear, and whether a system rings like a bell or settles quietly. Too little damping: things shake, resonate, break. Too much: slow, inefficient response. Engineering is finding the right amount.
Forced Vibrations and Resonance
Free vibration: displace the system once, let it oscillate. Forced vibration: apply a periodic external force. The system responds at the forcing frequency, not necessarily its natural frequency.
Equation of motion with external force F₀ cos(ωt):
Steady-state solution: system oscillates at the forcing frequency ω, with amplitude that depends on how close ω is to ω_n. Far from ω_n? Small response. Close to ω_n? Large response. Exactly at ω_n? That's resonance.
When forcing frequency matches natural frequency (ω ≈ ω_n), amplitude spikes. Without damping, amplitude would grow to infinity (in theory). With damping, amplitude is limited but still huge.
Response amplitude inversely proportional to damping ratio. Low damping → massive resonance peak. High damping → peak flattens out.
This is why you tune out of resonance or add damping. Tacoma Narrows Bridge: wind vortex shedding at bridge natural frequency, low damping, massive oscillations, structural failure. Washing machine on spin cycle hitting resonance: violent shaking. Engine mounts, suspension tuning, structural design—all about avoiding resonance or controlling it when unavoidable.
Resonance isn't always bad. Musical instruments rely on it. Sensors and filters use resonance for selectivity. But in structural/mechanical systems, unintended resonance is a common failure mode. Know your natural frequencies. Know your forcing frequencies. Keep them apart.
Waves
Vibrations are local oscillations. Waves are oscillations that travel. Pluck a guitar string at one end. Disturbance doesn't stay there—it propagates along the string. That's a wave.
Governing equation (one-dimensional wave equation):
u is displacement (or pressure, or voltage—depends on what's waving). c is wave speed. Partial derivatives because now you have spatial variation (x) and time variation (t) happening together.
Wavelength λ: distance between peaks
Frequency f: oscillations per second
Speed c: how fast the wave propagates
Related by: c = fλ
Sound in air: longitudinal waves (compression and rarefaction). Vibrations in beams and plates: transverse waves (bending motion). Stress waves in solids: elastic waves traveling at material-dependent speeds. When a hammer strikes metal, the impact doesn't instantly affect the whole part—stress wave propagates through it.
Waves reflect, refract, interfere, diffract. Standing waves form when boundary conditions constrain allowed wavelengths (guitar string fixed at both ends: only certain modes fit). Acoustic resonance in ducts, vibration modes in structures, signal transmission in mechanical systems—all wave phenomena.
Hit a machine component with a hammer. The impact doesn't stay local—stress waves radiate out, reflect off boundaries, interfere with each other, set up vibration patterns. Understanding waves means understanding how energy and disturbances move through real, connected systems instead of idealized point masses.
Worked Example: Identifying System Properties from the Equation of Motion
Find the natural frequency, write the general solution, and explain what the coefficient 25 represents physically.
Standard SHM form is d²x/dt² + ω²x = 0. Match patterns: ω² = 25, so ω = 5 rad/s. That's the natural frequency. System completes one cycle every 2π/5 ≈ 1.26 seconds.
Natural frequency: ω = 5 rad/s
Second-order linear ODE with constant coefficients → sinusoidal solution. Two arbitrary constants (A and B) because it's second-order. Initial position and initial velocity determine A and B.
Equivalently: x(t) = C cos(5t + φ), where C is amplitude and φ is phase. Same information, different packaging.
It's ω² = k/m if this came from a spring-mass system. Larger value → faster oscillations, stiffer system or lighter mass. Smaller value → slower oscillations. The coefficient completely determines the time scale of motion. Change 25 to 100? Frequency doubles (ω = 10 rad/s). Change to 6.25? Frequency halves (ω = 2.5 rad/s).
Physical meaning: It's the ratio of restoring stiffness to inertia. That ratio sets natural frequency for any oscillatory system—springs, pendulums, electrical circuits, structural modes. Same math, different physics.
Core lesson: The differential equation tells you everything about system behavior. Read the coefficients, extract natural frequency, solve for motion. No coefficients? No restoring force, no oscillation. Wrong coefficients? Wrong predictions, wrong design. Getting the equation of motion right is half the battle.
Ready for the Next Level?
Once you understand vibrations and wave phenomena, you're ready to explore thermodynamics and fluid mechanics.
Continue to Level 5: Thermal-Fluid Systems →