Vibration Science — Abridged Guide

Quick-reference equations, tables, and rules of thumb for vibration fundamentals in precision optics. For full derivations, worked examples, and diagrams, see the Comprehensive Guide.

1.Overview

Vibration is oscillatory motion characterized by displacement, velocity, and acceleration. In photonics, vibration amplitudes as small as 100 nm can destroy interference patterns, misalign focused beams, and bury signals in noise.

Vibration sources divide into three coupling paths: seismic (floor), acoustic (air), and on-table (mechanical). Always identify the dominant source before selecting isolation equipment.

2.Simple Harmonic Motion

Natural Frequency

f_0 = \frac{1}{2\pi}\sqrt{\frac{k}{m}}

Also: f₀ = (1/2π)√(g/δ_st) where δ_st is static deflection under gravity.

Natural frequency depends only on mass and stiffness. Increasing mass or decreasing stiffness lowers f_0, extending the isolation bandwidth.

Quick estimate for pneumatic isolators: f₀ (Hz) ≈ 5/√(δ_st in cm). A 6 cm static deflection gives f₀ ≈ 2 Hz.

3.Damped Vibration

Damping Ratio

\zeta = \frac{c}{2\sqrt{km}} = \frac{c}{2m\omega_0}

ζ < 1 underdamped | ζ = 1 critical | ζ > 1 overdamped

Logarithmic Decrement

\delta = \frac{1}{N}\ln\frac{x_0}{x_N} \approx 2\pi\zeta \quad (\zeta < 0.1)

Quality Factor

Q = \frac{1}{2\zeta} = \frac{f_0}{\Delta f}

Q equals the resonance amplification factor for light damping.

The Q factor equals the resonance amplification. A pneumatic isolator with ζ = 0.05 amplifies vibrations 10× at its natural frequency.

ζ	Q	Regime	Typical System
0.01–0.05	10–50	Lightly damped	Pneumatic isolators
0.05–0.20	2.5–10	Moderately damped	Elastomeric mounts
0.20–1.0	0.5–2.5	Heavily damped	High-loss materials

4.Forced Vibration & Resonance

Magnification Factor

M = \frac{1}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}

r = f/f₀ is the frequency ratio. At resonance (r = 1), M ≈ Q = 1/(2ζ).

At resonance (r ≈ 1), the magnification factor equals Q = 1/(2ζ). Resonance amplification is the primary reason vibration analysis exists for precision optics.

Phase Angle

\phi = \arctan\frac{2\zeta r}{1-r^2}

ϕ = 0° at low f, 90° at resonance, 180° at high f.

5.Transmissibility

SDOF Transmissibility

T = \sqrt{\frac{1 + (2\zeta r)^2}{(1-r^2)^2 + (2\zeta r)^2}}

Isolation (T < 1) begins at r > √2 ≈ 1.414.

All transmissibility curves cross T = 1 at r = √2, regardless of damping. An isolator can only attenuate vibrations above √2 × f₀.

Damping creates a trade-off: more damping reduces the resonance peak but degrades high-frequency isolation. Moderate damping (ζ = 0.05–0.15) is the practical compromise.

r = f/f₀	T (ζ=0.05)	T (ζ=0.10)	T (ζ=0.20)	Region
0.5	1.33	1.32	1.29	Static
1.0	10.0	5.10	2.60	Resonance
√2	1.00	1.00	1.00	Crossover
3.0	0.128	0.135	0.163	Isolation
5.0	0.043	0.045	0.057	Isolation
10.0	0.010	0.011	0.015	Isolation

6.Vibration Measurement

Displacement–Velocity–Acceleration

|v| = 2\pi f\,|x|, \quad |a| = (2\pi f)^2\,|x|

Velocity is used for VC criteria; acceleration is what sensors measure.

Vibration criteria use RMS velocity in one-third octave bands because equipment sensitivity curves are approximately constant-velocity across frequency.

Decibels (Velocity)

L_v = 20\log_{10}\frac{v_{rms}}{v_{ref}} \quad \text{dB}

Common refs: v_ref = 1 µm/s (SI) or 1 µin/s = 25.4 nm/s (US/VC).

7.Compliance

Compliance

C(f) = \frac{X(f)}{F(f)} \quad \text{(m/N)}

Inverse of dynamic stiffness. Lower compliance = stiffer, more rigid structure.

Compliance measures how much a table surface deflects per unit force. Peak compliance at structural resonances determines the maximum relative motion across the table surface.

Look for tables where compliance peaks are broad and low (well-damped) rather than sharp and tall (undamped). Broadband-damped honeycomb tables outperform undamped granite at resonance.

8.Vibration Criteria

The VC curves (A through G) define maximum RMS velocity in one-third octave bands from 4–80 Hz. Each curve is 6 dB below the previous. Most photonics labs require VC-A to VC-C; interferometry and nanopositioning require VC-D or better.

Criterion	Max Velocity (µm/s)	Detail Size	Equipment
VC-A	50	8 µm	Optical microscopes (100×)
VC-B	25	3 µm	Microscopes (1000×), inspection
VC-C	12.5	1 µm	Lithography, SEMs
VC-D	6.25	0.3 µm	E-beam, sensitive SEMs
VC-E	3.12	0.1 µm	Long-path interferometry

9.Vibration in Optical Systems

Angular (flexural) table motion is far more damaging than rigid-body translation. A table bend angle θ produces beam deflection \Delta x = 2\theta L per mirror reflection at propagation distance L.

Eliminate rigid connections between isolated and non-isolated equipment. Flexible tubing, slack cable loops, and acoustic enclosures prevent vibration from bypassing the isolation system.

10.Vibration Mitigation Strategy

A complete vibration control system requires both isolation (pneumatic legs filter floor vibration) and damping (table structure attenuates resonances). These are complementary — neither alone is sufficient.

Always measure the vibration environment first: perform a site survey, compare to VC criteria, identify dominant sources, and address sources before specifying isolation equipment.

Formula	Expression	Use
Natural frequency	f₀ = (1/2π)√(k/m)	System resonance
Quality factor	Q = 1/(2ζ)	Resonance amplification
Transmissibility	T = √((1+(2ζr)²)/((1−r²)²+(2ζr)²))	Isolation performance
Isolation crossover	r = √2 ≈ 1.414	Where isolation begins
Disp–vel–accel	\|v\| = 2πf\|x\|, \|a\| = (2πf)²\|x\|	Unit conversion
Beam deflection	Δx = 2θL	Mirror vibration error

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Comprehensive Vibration Science Guide →Transmissibility Calculator →Vibration Unit Converter →Force & Mechanics (Abridged) →

The Comprehensive Guide includes 7 worked examples, 6 SVG diagrams, detailed derivations, and 10 cited references covering all formulas on this page.