Why machines shake, and when it turns dangerous

Pull a mass and release it and it vibrates on its own; stiffer or lighter is faster, ωₙ = √(k/m)

Still at the ends but the spring is full, fastest at the center; ½kx²+½mv²=½kA² stays constant

A damper bleeds energy each cycle so amplitude decays by a ratio; ζ=c/(2√(km)) splits under-, critical, overdamped

When the drive matches ωₙ the magnification M=1/√((1−r²)²+(2ζr)²) explodes; damping (≈Q) sets the peak

Two coupled masses have in-phase (ω₁) and out-of-phase (ω₂) modes; any motion is their sum, N masses → N modes (TMD)

Let the masses go to infinity and you get a string or beam; the modes are integer-multiple standing waves fₙ = n·f₁ (harmonics), pitch set by f₁ = (1/2L)√(T/μ)

A beam vibrates by bending stiffness (EI), not tension; modes are not sines and frequencies not integer multiples (cantilever 1:6.27:17.5), f₁ ∝ √(EI/ρA)/L²

When the base shakes, the response ratio is the transmissibility TR=X/Y; isolation (TR<1) needs r>√2, so lower ωₙ (soft mount) — and damping is double-edged

An eccentric mass makes F₀=m·e·ω² (grows with speed²); response M·X/(m·e)=r²/√((1−r²)²+(2ζr)²), at high speed X→m·e/M, fixed by balancing

Relative displacement of a caged mass z=x−y, Z/Y=r²/√((1−r²)²+(2ζr)²); r≫1 (low ωₙ)=seismometer (displacement), r≪1 (high ωₙ)=accelerometer (Z≈a/ωₙ²)