Mechanical vibrations for the PE Mechanical exam

Most exam vibration problems are a single degree of freedom — a mass on a spring, a machine on isolators, a beam treated as an equivalent spring. Get the natural frequency right (and use mass, not weight) and the rest is bookkeeping.

Natural frequency

For a spring-mass system:

ω_n = √(k/m) (rad/s), and f_n = ω_n / 2π (Hz).

The trap is units: use mass, not weight — if you're given a weight W, then m = W/g. And watch rad/s vs. Hz (the 2π). For a beam or other elastic element, replace k with the equivalent stiffness (load ÷ static deflection), so ω_n = √(g/δ_st) is a handy shortcut when you know the static deflection.

Damping

Real systems lose energy. The damping ratio ζ = c/c_c, where the critical damping c_c = 2√(k·m) = 2·m·ω_n:

• ζ < 1 underdamped (oscillates and decays — the usual case),
• ζ = 1 critically damped (fastest return without overshoot),
• ζ > 1 overdamped.

The damped natural frequency is slightly lower than the undamped: ω_d = ω_n·√(1 − ζ²).

Forced vibration, resonance, and isolation

Drive the system near its natural frequency and the response peaks — resonance at ω ≈ ω_n. For machine isolation, what matters is transmissibility (the fraction of force/motion passed to the foundation): it's governed by the frequency ratio r = ω/ω_n and the damping. The key design fact: you only get isolation (transmissibility < 1) when r > √2 — i.e., the forcing frequency is well above the natural frequency, so isolators are designed soft (low ω_n). Below √2, you can actually amplify.

Where people lose points

• Weight instead of mass — forgetting m = W/g in ω_n = √(k/m).
• Dropping the √ in the natural-frequency relation.
• ω_d vs. ω_n — the damped frequency is lower (×√(1 − ζ²)).
• Hz vs. rad/s — the 2π between f_n and ω_n.
• Isolation logic backwards — isolation needs the forcing frequency above √2·ω_n; stiffer isn't better for isolation.

Drill it

Practice these on real problems — the tutor walks any you miss, grounded in the worked solution. Where this lives: MD&M Module 5 — Components & Assemblies (this area); the stiffness term comes from Spring design.