Isolation Principles & Techniques

From single-degree-of-freedom theory through passive and active isolation methods to system selection — the complete engineering framework for protecting sensitive optical and photonics equipment from environmental vibration.

Comprehensive Guide

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1Introduction

Vibration isolation is the practice of decoupling a sensitive payload from its environment so that ground-borne, structure-borne, or acoustically coupled disturbances are attenuated before they reach the equipment. In photonics laboratories, the equipment in question is almost always an optical table or breadboard carrying interferometers, microscopes, lithography stages, or laser cavities — systems where nanometre-level stability determines whether an experiment succeeds or an instrument meets its specification.

This guide presents the engineering principles behind vibration isolation: the single-degree-of-freedom (SDOF) model that underpins all isolator design, the transmissibility function that quantifies performance, the extension to multi-DOF systems, and the practical implementation of passive and active isolation techniques. The goal is to give the reader a working command of the theory and the vocabulary needed to specify, evaluate, and troubleshoot isolation systems in real optical environments.

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1.1Scope & Context

This guide focuses on isolation — the reduction of transmitted vibration between a source and a receiver. It does not cover damping of structural resonances (treated in the companion Damping guide) or the design of optical tables themselves (treated in the Optical Tables guide), though both topics are closely related and cross-referenced where relevant. The emphasis is on principles and techniques rather than specific commercial products, giving the reader the tools to evaluate any isolation system from first principles.

1.2Relationship to Vibration Science

The Vibration Science guide establishes the fundamentals: simple harmonic motion, damped and forced oscillation, frequency-domain representation, and vibration criteria. This Isolation Principles guide builds directly on that foundation. Readers who are unfamiliar with concepts such as natural frequency, damping ratio, transmissibility, and power spectral density should review the Vibration Science guide first. Here, we take those concepts as given and apply them to the specific engineering problem of vibration isolation.

1.3Who This Guide Is For

This guide is written for optical engineers, photonics researchers, and laboratory managers who need to specify or troubleshoot vibration isolation systems. It assumes familiarity with basic mechanics and vibration concepts at the level covered in the Vibration Science guide. No advanced control theory or finite-element analysis is required — the mathematics is kept to closed-form expressions and back-of-the-envelope calculations that can be performed with a calculator or spreadsheet.

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2Isolation Theory — The SDOF Model

Every vibration isolation system, no matter how complex, can be understood as an elaboration of the single-degree-of-freedom (SDOF) model: a rigid mass supported on a spring and a dashpot. The mass represents the payload (optical table plus instruments), the spring represents the isolator stiffness, and the dashpot represents energy dissipation within the isolator. This section develops the SDOF model from first principles and derives the key parameters that govern isolation performance.

Figure 2.1 — Single-degree-of-freedom isolation model showing a rigid payload mass m supported on a spring of stiffness k and a viscous dashpot with damping coefficient c, driven by base displacement x_b.

2.1Equation of Motion

Consider a payload of mass

m

resting on an isolator with stiffness

k

and viscous damping coefficient

c

. The base (floor) undergoes displacement

x_b(t)

and the payload displacement is

x_p(t)

. Applying Newton's second law to the payload, the equation of motion in terms of the relative displacement

z = x_p - x_b

is:

Equation of Motion (SDOF Isolation Model)

m\,\ddot{x}_p + c\,(\dot{x}_p - \dot{x}_b) + k\,(x_p - x_b) = 0

This can be rewritten in standard form by dividing through by the mass:

Standard Form

\ddot{x}_p + 2\zeta\omega_n\,(\dot{x}_p - \dot{x}_b) + \omega_n^2\,(x_p - x_b) = 0

where

\omega_n

is the undamped natural frequency and

\zeta

is the damping ratio. These two parameters completely characterize the SDOF system and determine its isolation performance at every frequency.

2.2Natural Frequency & Static Deflection

The undamped natural frequency of the SDOF system is:

Natural Frequency

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

A particularly useful relationship links the natural frequency to the static deflection of the isolator under the payload weight. If the isolator compresses by

\delta_{st}

under the gravitational load

mg

, then

k = mg / \delta_{st}

, and:

Natural Frequency from Static Deflection

f_n = \frac{1}{2\pi}\sqrt{\frac{g}{\delta_{st}}}

This equation is enormously practical: it means the natural frequency of any spring-based isolator can be estimated simply by measuring (or calculating) how far the spring compresses under the payload weight. A static deflection of 25 mm corresponds to about 3.1 Hz; a deflection of 8 mm corresponds to about 5.6 Hz.

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Worked Example: WE 1 — SDOF Natural Frequency from Static Deflection

Problem: A pneumatic isolator supporting a 500 kg optical table compresses 6 mm under the payload weight. Estimate the vertical natural frequency.

Solution:

\delta_{st} = 6\;\text{mm} = 0.006\;\text{m}

f_n = \frac{1}{2\pi}\sqrt{\frac{9.81}{0.006}} = \frac{1}{2\pi}\sqrt{1635} = \frac{40.4}{6.283} \approx 6.4\;\text{Hz}

The vertical natural frequency is approximately 6.4 Hz. For a pneumatic isolator, this is somewhat high — most pneumatic systems target 1.5–3 Hz — suggesting the air spring pressure may be set too high or the effective volume is too small.

2.3Damping Ratio

The damping ratio

\zeta

is the ratio of the actual damping coefficient to the critical damping coefficient:

Damping Ratio

\zeta = \frac{c}{2\sqrt{km}} = \frac{c}{c_{cr}}

where

c_{cr} = 2\sqrt{km} = 2m\omega_n

is the critical damping coefficient. For vibration isolation, the damping ratio plays a dual role: higher damping reduces the resonant peak (beneficial near the natural frequency) but degrades high-frequency isolation (detrimental above

\sqrt{2}\,f_n

). Typical isolation systems operate with

\zeta

in the range 0.05–0.20, representing a deliberate compromise between resonance control and high-frequency performance.

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3Transmissibility

Transmissibility is the central metric of isolation performance. It describes the ratio of the payload response to the base excitation as a function of frequency. A transmissibility less than unity means the isolator is attenuating vibration; greater than unity means it is amplifying it. Understanding the transmissibility curve — its shape, its dependence on damping, and its asymptotic behavior — is essential for specifying and evaluating any isolation system.

Figure 3.1 — Transmissibility curves for the SDOF model at several damping ratios, showing the resonant peak, the crossover at r = sqrt(2), and the isolation region above crossover.

3.1The Transmissibility Equation

For harmonic base excitation at frequency

f

, the displacement transmissibility of the SDOF model is:

Displacement Transmissibility

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

where

r = f / f_n

is the frequency ratio. At the resonant peak (approximately

r \approx 1

for light damping), the transmissibility reaches its maximum:

Peak Transmissibility (approximate)

T_{\text{peak}} \approx \frac{1}{2\zeta} \quad (\zeta \ll 1)

All transmissibility curves pass through

T = 1

r = \sqrt{2}

, regardless of the damping ratio. Below this crossover, the isolator amplifies; above it, the isolator attenuates. This is the fundamental design rule: the driving frequency must be at least

\sqrt{2}

times the natural frequency before any isolation is achieved.

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3.2Decibel Transmissibility

Transmissibility is frequently expressed in decibels:

Transmissibility in Decibels

T_{\text{dB}} = 20 \log_{10} T

A transmissibility of 0.1 (90% attenuation) corresponds to −20 dB; a transmissibility of 0.01 (99% attenuation) corresponds to −40 dB. Conversely, a resonant peak with

T = 10

corresponds to +20 dB of amplification. The decibel scale makes it easier to read isolation performance from logarithmic plots and to compare isolators that differ by orders of magnitude in performance.

3.3High-Frequency Rolloff

Well above the natural frequency (

r \gg 1

), the transmissibility simplifies to asymptotic forms that depend on whether the damping is viscous or structural:

High-Frequency Rolloff — Viscous Damping

T \approx \frac{2\zeta}{r} \quad (r \gg 1, \;\text{viscous damping dominant})

High-Frequency Rolloff — Low Damping

T \approx \frac{1}{r^2} = \left(\frac{f_n}{f}\right)^2 \quad (r \gg 1, \;\zeta \ll 1)

For lightly damped systems, transmissibility rolls off as

1/r^2

(−40 dB/decade), giving 12 dB of additional isolation for every octave above the natural frequency. For systems with significant viscous damping, the rolloff degrades to

1/r

(−20 dB/decade) at high frequencies because the damper acts as a rigid link at high velocities, transmitting force directly. This is the fundamental damping trade-off in isolation design: more damping reduces the resonant peak but sacrifices high-frequency attenuation.

3.4Isolation Efficiency

Isolation efficiency expresses the fraction of vibration that is removed:

Isolation Efficiency

\eta = (1 - T) \times 100\%

An efficiency of 90% means the payload sees only 10% of the base vibration amplitude at that frequency. Isolation efficiency is only meaningful for T < 1 (i.e., above the crossover frequency). At resonance, the “efficiency” is negative, indicating amplification rather than attenuation.

Worked Example: WE 2 — Transmissibility at 10 Hz

Problem: An isolator has a natural frequency of 3 Hz and a damping ratio of 0.10. Calculate the displacement transmissibility at 10 Hz and express it in decibels.

Solution:

r = \frac{f}{f_n} = \frac{10}{3} = 3.33

1 + (2 \times 0.10 \times 3.33)^2 = 1 + 0.444 = 1.444

(1 - 3.33^2)^2 + (2 \times 0.10 \times 3.33)^2 = (1 - 11.09)^2 + 0.444 = 101.8 + 0.444 = 102.2

T = \sqrt{\frac{1.444}{102.2}} = \sqrt{0.01413} = 0.119

T_{\text{dB}} = 20 \log_{10}(0.119) = 20 \times (-0.925) = -18.5\;\text{dB}

At 10 Hz, the isolator transmits about 12% of the base vibration amplitude, corresponding to 18.5 dB of attenuation. The isolation efficiency is approximately 88%.

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4Multi-DOF Systems

Real isolation systems do not have a single degree of freedom. An optical table on four isolators has six degrees of freedom — three translational (vertical, lateral X, lateral Y) and three rotational (roll, pitch, yaw). Each degree of freedom has its own natural frequency, and poor design can result in low-frequency rocking modes that are more problematic than the vertical bounce mode addressed by the SDOF model.

Figure 4.1 — Six degrees of freedom for an isolated optical table: three translations (X, Y, Z) and three rotations (roll, pitch, yaw).

4.1Six Degrees of Freedom

For a symmetric rectangular payload on four identical isolators at the corners, the vertical (heave) natural frequency is determined by the total vertical stiffness

k_z = 4k_{\text{leg}}

and the total payload mass. The horizontal natural frequencies depend on the lateral stiffness of each isolator leg, which is typically different from the vertical stiffness. In pneumatic isolators, horizontal stiffness is provided by a pendulum mechanism or by the diaphragm geometry, and it is common for horizontal natural frequencies to be somewhat lower than the vertical natural frequency.

4.2Rocking Modes

Rocking (pitch and roll) modes are rotational oscillations about horizontal axes. For a rectangular table of length

L

and width

W

on four isolators spaced at

L_s \times W_s

, the pitch rocking frequency is approximately:

Rocking Mode Frequency (pitch)

f_{\text{rock}} = \frac{1}{2\pi}\sqrt{\frac{k_z \, L_s^2}{2\,I_{\text{pitch}}}}

where

I_{\text{pitch}}

is the mass moment of inertia of the payload about the pitch axis. Rocking frequencies are typically 1.5–3 times the vertical natural frequency, depending on the geometry. If the center of gravity is above the plane of the isolator attachment points, the effective rocking frequency decreases — a condition that can lead to instability if the CG is too high.

Worked Example: WE 3 — Rocking Mode Frequency

Problem: A 1200 mm × 600 mm optical table (mass 200 kg) sits on four isolators spaced 1000 mm × 400 mm. Each isolator has a vertical stiffness of 15,000 N/m. The table is modeled as a uniform rectangular slab. Estimate the pitch rocking frequency.

Solution:

I_{\text{pitch}} = \frac{1}{12}\,m\,L^2 = \frac{1}{12} \times 200 \times 1.2^2 = 24.0\;\text{kg·m}^2

L_s = 1.0\;\text{m}

k_z = 2 \times 15{,}000 = 30{,}000\;\text{N/m (per side)}

f_{\text{rock}} = \frac{1}{2\pi}\sqrt{\frac{30{,}000 \times 1.0^2}{2 \times 24.0}} = \frac{1}{2\pi}\sqrt{625} = \frac{25.0}{6.283} \approx 4.0\;\text{Hz}

The pitch rocking frequency is approximately 4.0 Hz. For comparison, the vertical bounce frequency with total stiffness

4 \times 15{,}000 = 60{,}000

N/m and 200 kg mass is about 2.8 Hz. The rocking mode is about 1.4 times the bounce frequency, which is typical for a table of this aspect ratio.

4.3Center-of-Gravity Effects

When the payload center of gravity (CG) is offset from the geometric center of the isolator support points, the static load is distributed unevenly across the isolators. In pneumatic systems, each leg can be independently pressurized to level the table, but the dynamic behavior changes: the rocking modes become coupled with translational modes, and asymmetric loading can introduce cross-axis coupling where vertical excitation produces horizontal payload motion.

A high CG is particularly problematic. If the CG is above the isolator attachment plane, gravity creates a destabilizing moment during rocking: the gravitational torque acts in the same direction as the angular displacement, effectively reducing the rotational stiffness and lowering the rocking frequency. In extreme cases, this can reduce the rocking frequency to zero — the system becomes statically unstable and the table topples. As a rule of thumb, the CG height above the isolator plane should be less than one-quarter of the smallest isolator spacing dimension.

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5Damping Strategies

Damping in an isolation system serves two purposes: it limits the resonant amplification at the natural frequency, and it controls transient settling time after impulsive disturbances. However, as the transmissibility equation shows, viscous damping in the isolator degrades high-frequency isolation. Two distinct strategies address this trade-off: broadband damping, which accepts the compromise, and tuned damping, which targets specific resonances without affecting the isolation bandwidth.

Figure 5.1 — Comparison of broadband damping versus tuned mass damper response, showing how a TMD can reduce a specific resonance without degrading high-frequency rolloff.

5.1Broadband Damping

Broadband damping is applied directly in the isolation spring, typically through the inherent hysteresis of elastomeric materials or through viscous fluid dampers in parallel with the spring. It affects all frequencies simultaneously. For isolation systems with moderate damping ratios (

\zeta = 0.05

0.15

), the resonant peak is controlled to a factor of 3–10 while maintaining reasonable high-frequency rolloff. This is the approach used in most commercial elastomeric and pneumatic isolators.

5.2Tuned Mass Dampers

A tuned mass damper (TMD), also called a dynamic vibration absorber, is a small mass-spring-damper subsystem attached to the payload and tuned to the problematic resonant frequency. When properly tuned, the TMD absorbs energy at the target frequency, splitting the single resonant peak into two smaller peaks and dramatically reducing the maximum transmissibility. Crucially, the TMD does not add damping to the main isolation spring, so the high-frequency rolloff is preserved.

5.3TMD Design Equations

The key design parameters for a tuned mass damper are the mass ratio, the tuning ratio, and the damper damping ratio. For optimal equal-peak design (Den Hartog criterion), the relationships are:

Mass Ratio

\mu = \frac{m_d}{m_p}

Optimal Tuning Ratio

q_{\text{opt}} = \frac{f_d}{f_n} = \frac{1}{1 + \mu}

Optimal Damping Ratio for TMD

\zeta_{d,\text{opt}} = \sqrt{\frac{3\mu}{8(1 + \mu)^3}}

where

m_d

is the TMD mass,

m_p

is the primary (payload) mass,

f_d

is the TMD natural frequency, and

f_n

is the primary system natural frequency. A mass ratio of 1–5% is typical; larger mass ratios give wider suppression bandwidth but add dead weight to the payload.

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Worked Example: WE 4 — TMD Sizing

Problem: A 500 kg optical table on pneumatic isolators has a vertical natural frequency of 2.0 Hz with insufficient damping (resonant transmissibility of 15). Design a tuned mass damper using a 3% mass ratio.

Solution:

m_d = \mu \times m_p = 0.03 \times 500 = 15\;\text{kg}

q_{\text{opt}} = \frac{1}{1 + 0.03} = 0.971

f_d = q_{\text{opt}} \times f_n = 0.971 \times 2.0 = 1.94\;\text{Hz}

k_d = m_d\,(2\pi f_d)^2 = 15 \times (2\pi \times 1.94)^2 = 15 \times 148.6 = 2{,}229\;\text{N/m}

\zeta_{d,\text{opt}} = \sqrt{\frac{3 \times 0.03}{8 \times (1.03)^3}} = \sqrt{\frac{0.09}{8.74}} = \sqrt{0.0103} = 0.101

c_d = 2\,\zeta_{d,\text{opt}}\,\sqrt{k_d\,m_d} = 2 \times 0.101 \times \sqrt{2229 \times 15} = 0.202 \times 182.9 = 36.9\;\text{N·s/m}

The TMD consists of a 15 kg mass on a spring of 2,229 N/m with a dashpot of 36.9 N·s/m. This will split the 2 Hz resonance into two peaks of roughly equal height, each substantially lower than the original peak of 15.

5.4Broadband vs. Tuned Comparison

Parameter	Broadband Damping	Tuned Mass Damper
Mechanism	Energy dissipation in isolator spring/dashpot	Auxiliary mass-spring-damper tuned to resonance
Frequency range	All frequencies affected	Narrow band around target frequency
Resonant peak reduction	Moderate (factor of 3–10)	Large (factor of 10–50 possible)
High-frequency isolation	Degraded (rolloff reduced to −20 dB/decade)	Preserved (−40 dB/decade maintained)
Added weight	None (inherent to isolator)	1–5% of payload mass
Tuning required	No	Yes — must match target frequency precisely
Sensitivity to payload changes	Low — performance degrades gradually	High — detuned TMD is ineffective
Typical application	General-purpose isolation	Specific problematic resonance on an otherwise well-isolated system

Table 5.1 — Comparison of broadband damping and tuned mass damper strategies.

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6Passive Isolation Methods

Passive isolators require no external power, control system, or sensors. They achieve isolation purely through the mechanical properties of springs, elastomers, or air volumes. Passive systems are the workhorses of vibration isolation — reliable, maintenance-free, and cost-effective for the vast majority of photonics applications.

Figure 6.1 — Common passive isolator types: elastomeric mounts, coil spring isolators, pneumatic (air spring) isolators, and pendulum/negative-stiffness mechanisms.

6.1Elastomeric Isolators

Elastomeric (rubber) isolators are the simplest and most widely used passive isolators. A block or pad of elastomeric material provides both stiffness and damping in a single element. The damping is hysteretic (frequency-independent) rather than viscous, which partially avoids the high-frequency isolation penalty of viscous damping. Typical elastomeric isolators achieve natural frequencies of 5–25 Hz, making them suitable for isolating machinery vibration but generally insufficient for precision optical work where sub-5 Hz natural frequencies are required.

Elastomeric materials are available in a range of durometers (Shore A hardness), allowing the stiffness to be tailored to the payload weight. Softer compounds (Shore 30–40A) give lower natural frequencies but have lower load capacity and greater creep. Harder compounds (Shore 60–80A) support heavier loads but provide less isolation. Environmental factors — temperature, ozone, oils, and UV exposure — can degrade elastomeric performance over time.

6.2Mechanical Spring Isolators

Coil spring isolators provide lower natural frequencies than elastomers — typically 2–5 Hz — because metal springs can achieve greater static deflection without creep or fatigue. However, metal springs have very low inherent damping (ζ < 0.01), so a separate damping element (usually a viscous dashpot or constrained-layer pad) must be added in parallel. Spring isolators are common in building mechanical systems and heavy equipment isolation but less common in precision optics, where pneumatic isolators offer even lower natural frequencies with better damping characteristics.

6.3Pneumatic Isolators

Pneumatic (air spring) isolators are the standard choice for optical table isolation. An air volume enclosed by a flexible diaphragm acts as the spring; the stiffness is determined by the air pressure and the effective piston area. The key advantage of pneumatic isolators is that the natural frequency is nearly independent of the payload mass — adding more weight increases the pressure proportionally, maintaining the same stiffness-to-mass ratio.

Pneumatic Isolator Stiffness

k_{\text{air}} = \frac{n\,P\,A_e^2}{V}

where

n

is the polytropic index (1.0 for isothermal, 1.4 for adiabatic),

P

is the absolute pressure,

A_e

is the effective piston area, and

V

is the air volume. The natural frequency is typically 1.5–3 Hz for bench-top systems and can be reduced below 1 Hz with large external reservoirs. Damping is provided by orifices connecting the main chamber to auxiliary volumes, providing velocity- dependent energy dissipation with damping ratios of 0.05–0.15.

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Worked Example: WE 5 — Pneumatic Isolator Sizing

Problem: A pneumatic isolator has an effective piston area of 0.01 m², an air volume of 2 liters, and operates at an absolute pressure of 200 kPa. Assuming adiabatic conditions (n = 1.4), calculate the vertical stiffness and the natural frequency for a 100 kg payload.

Solution:

k_{\text{air}} = \frac{n\,P\,A_e^2}{V} = \frac{1.4 \times 200{,}000 \times (0.01)^2}{0.002} = \frac{1.4 \times 200{,}000 \times 10^{-4}}{0.002} = \frac{28.0}{0.002} = 14{,}000\;\text{N/m}

f_n = \frac{1}{2\pi}\sqrt{\frac{k}{m}} = \frac{1}{2\pi}\sqrt{\frac{14{,}000}{100}} = \frac{1}{2\pi}\sqrt{140} = \frac{11.83}{6.283} = 1.88\;\text{Hz}

The pneumatic isolator provides a natural frequency of about 1.9 Hz, which is within the typical 1.5–3 Hz range for precision optical table isolation. Good isolation (−20 dB) begins at approximately 6 Hz, and excellent isolation (−40 dB) at approximately 19 Hz.

6.4Pendulum & Negative-Stiffness Isolators

For applications requiring sub-hertz natural frequencies — gravitational-wave detectors, scanning probe microscopes, and the most demanding interferometric measurements — conventional springs cannot provide sufficient static deflection. Two passive techniques address this: pendulum isolation and negative-stiffness mechanisms.

A simple pendulum provides horizontal isolation with a natural frequency determined solely by its length:

Pendulum Frequency

f_{\text{pend}} = \frac{1}{2\pi}\sqrt{\frac{g}{L}}

A 250 mm pendulum gives a horizontal natural frequency of about 1 Hz; a 1 m pendulum gives about 0.5 Hz. Many pneumatic isolators incorporate a pendulum mechanism for horizontal isolation while using the air spring for vertical isolation.

Negative-stiffness mechanisms use a combination of compressed springs or flexures arranged so that the net restoring force is very small — the negative-stiffness element partially cancels the positive stiffness of the load-bearing spring, yielding an extremely low effective stiffness and a natural frequency of 0.5 Hz or below. These systems achieve the lowest natural frequencies of any passive isolator but require careful setup and are sensitive to payload changes.

Isolator Type	Typical f_n (Hz)	Damping Ratio	Load Range	Advantages	Limitations
Elastomeric	5–25	0.05–0.15	1–10,000 kg	Simple, compact, no maintenance	High f_n limits isolation; creep; temperature sensitive
Coil spring	2–5	< 0.01 (needs added damper)	10–50,000 kg	Wide load range; linear; no creep	Low damping; large static deflection; resonance if undamped
Pneumatic	1.5–3	0.05–0.15	20–5,000 kg per leg	f_n nearly independent of mass; self-leveling available	Requires air supply; more complex; higher cost
Pendulum	0.3–1.0	0.01–0.05	Horizontal axis only	Very low f_n; simple concept	Horizontal only; long pendulum required for very low f_n
Negative-stiffness	0.3–0.5	0.01–0.05	5–2,000 kg	Lowest passive f_n; no air supply needed	Sensitive to payload changes; complex setup; expensive

Table 6.1 — Comparison of passive isolator types.

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7Active Isolation

Active isolation systems use sensors, actuators, and feedback control to achieve performance beyond what passive elements alone can provide. Where a passive isolator is limited by the fundamental trade-off between resonance control and high-frequency isolation, an active system can electronically suppress the resonant peak while maintaining or exceeding the passive high-frequency rolloff. The cost is complexity, power consumption, and the potential for control-system instabilities.

Figure 7.1 — Active feedback isolation architecture: floor sensor measures base vibration, controller drives actuator to cancel transmitted force, payload sensor verifies performance.

7.1Feedback Control Architecture

The most common active isolation architecture uses inertial feedback: a sensor (geophone or accelerometer) on the payload measures absolute payload velocity or acceleration, and the controller drives an actuator (electromagnetic voice coil or piezoelectric) to generate a force that opposes the measured motion. The effect is equivalent to adding a very large electronic damping coefficient at low frequencies — suppressing the resonant peak — while allowing the passive spring to provide the high-frequency isolation.

Feedforward architectures are also used, particularly for periodic disturbances: a sensor on the floor (or on the vibration source) measures the incoming disturbance, and the controller drives the actuator to cancel the disturbance before it reaches the payload. Feedforward is effective for tonal vibrations (machinery, HVAC) but less effective for broadband random vibration, which requires high-bandwidth feedback.

7.2Sensor Technologies

Geophones are the most common payload sensors for active isolation in the 0.5–100 Hz range. A geophone is a velocity transducer: a coil suspended on a spring inside a magnetic field produces a voltage proportional to the relative velocity between the coil and the housing. Below the geophone's own natural frequency (typically 1–4.5 Hz), the sensitivity falls off, limiting the low-frequency bandwidth of the active system. Accelerometers (piezoelectric or MEMS) extend the bandwidth to higher frequencies but have higher noise floors at low frequencies. For the most demanding applications, seismometers (broadband force-balance instruments) provide sub-hertz sensing with extremely low noise.

7.3Actuator Technologies

Voice-coil (electromagnetic) actuators are the standard for active optical table isolation. They provide linear force output proportional to the drive current, with bandwidth from DC to several hundred hertz. The force range is typically 10–200 N per actuator, sufficient to control payloads from tens to thousands of kilograms. Piezoelectric actuators offer higher bandwidth (to several kilohertz) and sub-nanometre resolution, but have limited stroke (typically tens of micrometres) and require high-voltage amplifiers. Hybrid systems use voice coils for low-frequency control and piezoelectric stacks for high-frequency fine correction.

7.4Hybrid Active-Passive Systems

The majority of commercial active isolation systems are hybrid: a passive pneumatic isolator provides the basic isolation (typically with a natural frequency of 1.5–3 Hz) and an active feedback loop electronically damps the resonance and extends the isolation bandwidth to lower frequencies. The passive element carries the static load and provides failsafe support if the active system loses power. The active element provides 10–20 dB of additional isolation in the 0.7–10 Hz range compared to the passive-only performance.

7.5Limitations of Active Isolation

Active systems are not without drawbacks. They require continuous electrical power and a compressed-air supply (for hybrid systems). The control electronics introduce noise, and sensor noise sets a floor below which vibration cannot be reduced regardless of loop gain. The feedback loop has finite bandwidth and phase margins, and instability can result from structural resonances within the loop bandwidth. Active systems are also more expensive — typically 3–10 times the cost of equivalent passive-only systems — and require periodic calibration and maintenance. For many photonics applications, a well-specified passive pneumatic system is sufficient, and the added complexity of active control is justified only when the passive performance margin is genuinely inadequate.

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8Design Parameters

Specifying an isolation system requires quantifying several key parameters before selecting hardware. Omitting any of these parameters risks either over-specifying (and overspending) or under-specifying (and discovering inadequate performance after installation). This section outlines the essential design inputs.

8.1Payload Weight & Center of Gravity

The total payload weight determines the required load capacity of each isolator leg and, for pneumatic systems, the operating pressure. The payload includes the optical table itself plus all instruments, optics, mounts, and cabling that will be placed on the table. It is important to estimate the maximum loaded weight, not just the bare table weight. The center of gravity (CG) must be within the support polygon formed by the isolator positions, with reasonable margin — ideally the CG is within the central 50% of the support polygon in each axis to avoid excessive rocking-mode coupling and uneven isolator loading.

8.2Disturbance Spectrum

The floor vibration spectrum at the installation site is the single most important environmental input. It is measured using a triaxial accelerometer or geophone placed on the floor at the intended equipment location, typically recorded as a one-third-octave velocity spectrum in dB re 1 micro-m/s or micro-inches/s. Without this measurement, isolation system selection is guesswork. Many isolator manufacturers offer site survey services, and portable vibration measurement kits are available for in-house characterization.

8.3Required Isolation Frequency

The natural frequency of the isolation system must be chosen so that effective isolation begins below the lowest frequency of concern. As a rule of thumb, the natural frequency should be at least 3 times below the lowest frequency at which isolation is needed — this ensures at least 20 dB of attenuation at that frequency. For optical tables in typical laboratory environments, the critical vibration band is 5–100 Hz, leading to natural frequency requirements of 1–3 Hz. For scanning probe microscopes and interferometers sensitive to sub-hertz vibration, even lower natural frequencies may be required.

8.4Allowable Motion

The maximum acceptable payload motion is determined by the application. For general optical table work (beam alignment, component testing), allowable motion of 1–10 micrometres is typical. For interferometry and lithography, the requirement may be 10–100 nanometres. For scanning probe microscopy, sub-nanometre stability is often required. The allowable motion specification should include the frequency bandwidth (e.g., 1–100 Hz RMS) and the direction (vertical, horizontal, or both) to be meaningful.

8.5Settling Time

Settling time is the time required for the payload to return to within a specified tolerance of its equilibrium position after an impulsive disturbance (e.g., a user bumping the table or a sample stage moving). For lightly damped systems, settling time can be several seconds or more. Active systems and systems with tuned mass dampers typically settle in under one second. In automated fabrication or inspection tools, where throughput depends on step-and-settle speed, settling time can be the dominant performance specification, more important than steady-state isolation.

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9Environmental Considerations

Isolation system performance depends not only on the hardware but also on the environment in which it operates. Floor vibration, acoustic noise, thermal drift, and seismic activity all influence the achievable payload stability. A well-specified isolator installed in a poorly characterized environment will underperform; conversely, a modest isolator in a quiet environment may exceed expectations.

9.1Floor Vibration Environments

Floor vibration levels vary enormously between sites. A basement laboratory in a rural research campus may see floor velocities of 1–3 micro-m/s RMS in the 4–80 Hz band, while a ground-floor laboratory adjacent to a mechanical room in an urban building may see 10–50 micro-m/s RMS. Upper floors amplify low-frequency building sway (typically 0.5–2 Hz) and can introduce resonances in the 5–15 Hz range from floor panel bending modes. Knowing the floor vibration spectrum is essential for selecting the right isolation approach and for predicting the residual vibration on the isolated payload.

9.2Vibration Criteria (VC Curves)

The Colin Gordon vibration criteria (VC curves) are the industry-standard benchmark for specifying floor vibration environments and equipment sensitivity. The VC curves define maximum allowable one-third-octave RMS velocity levels as a function of frequency. The criteria range from VC-A (least stringent, suitable for general optical microscopy) through VC-G (most stringent, suitable for the most demanding electron-beam and scanning-probe instruments).

Figure 9.1 — Colin Gordon VC curves showing one-third-octave velocity limits from VC-A through VC-G.

Criterion	Velocity Limit (micro-m/s)	Typical Application
VC-A	50	Optical microscopes (400×), microbalances, optical comparators
VC-B	25	Optical microscopes (1000×), inspection and lithography equipment (line widths > 3 micro-m)
VC-C	12.5	Lithography and inspection (1–3 micro-m line widths), most laser/optical research
VC-D	6.25	Lithography and inspection (sub-micron line widths), long-path interferometry
VC-E	3.12	Electron-beam lithography, scanning electron microscopes at high magnification
VC-F	1.56	Demanding SEM and TEM work, sensitive interferometers
VC-G	0.78	Most demanding scanning probe microscopes, gravitational wave detectors

Table 9.1 — Colin Gordon VC curves and typical application sensitivities.

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9.3Acoustic Excitation

Airborne acoustic noise can excite the payload directly, bypassing the isolation system entirely. Sound pressure levels above 65–70 dBA can produce measurable vibration on an optical table, particularly at low frequencies where acoustic wavelengths are comparable to the table dimensions. Acoustic enclosures, curtains, and baffles are often necessary supplements to vibration isolation in noisy environments. HVAC systems, fume hoods, and even human conversation can be significant acoustic disturbance sources.

9.4Thermal Effects on Isolation

Temperature changes affect isolation systems in several ways. Elastomeric materials stiffen at low temperatures and soften at high temperatures, shifting the natural frequency and damping ratio. Pneumatic isolators are less temperature-sensitive, but the air pressure (and hence the leveling height) changes with temperature unless an auto-leveling valve is present. Thermal gradients across the optical table can produce slow drift that is not vibration in the traditional sense but is equally damaging to optical alignment. Temperature stability of ±0.5°C or better is recommended for precision optical environments.

9.5Seismic Considerations

In seismically active regions, isolation systems must survive earthquake loading without damage. Soft isolators with large travel ranges (pneumatic, negative-stiffness) can bottom out or overtavel during seismic events, potentially damaging the payload. Travel-limiting snubbers, seismic restraints, and automatic lockout systems are common provisions. It is important to distinguish between the vibration isolation function (normal operation, micro-vibration) and the seismic protection function (rare event, large displacement) — the two may require different design features within the same isolation system.

Worked Example: WE 6 — VC Compliance Check

Problem: A floor vibration measurement shows a one-third-octave velocity level of 30 micro-m/s at 8 Hz. The isolation system has a natural frequency of 2 Hz and a damping ratio of 0.10. Does the isolated payload meet VC-C (12.5 micro-m/s limit)?

Solution:

r = \frac{8}{2} = 4.0

T = \sqrt{\frac{1 + (2 \times 0.10 \times 4)^2}{(1 - 16)^2 + (2 \times 0.10 \times 4)^2}} = \sqrt{\frac{1 + 0.64}{225 + 0.64}} = \sqrt{\frac{1.64}{225.64}} = 0.0853

v_{\text{payload}} = T \times v_{\text{floor}} = 0.0853 \times 30 = 2.56\;\text{micro-m/s}

The isolated payload sees 2.56 micro-m/s at 8 Hz, which is well below the VC-C limit of 12.5 micro-m/s. In fact, it meets VC-F (1.56 micro-m/s limit would require checking, and 2.56 exceeds it). The system meets VC-C and VC-D and VC-E, but does not quite meet VC-F at this frequency.

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10Performance Metrics

Isolation system performance is characterized by a set of frequency-domain and statistical metrics that describe how well the system attenuates vibration across the spectrum. These metrics allow quantitative comparison between different isolation approaches and provide the basis for specification and acceptance testing.

10.1Compliance Transfer Function

Compliance is the ratio of displacement to force as a function of frequency. For an isolated payload, the compliance describes how much the payload displaces in response to an applied force (such as a researcher leaning on the table):

Compliance

C(f) = \frac{X(f)}{F(f)} = \frac{1}{k}\,\frac{1}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}}

Low compliance at the frequencies of interest means the payload resists on-board disturbances effectively. Compliance is specified in units of micrometres per newton (micro-m/N) and is typically plotted on a log-log scale. For optical tables, the table stiffness (not the isolator) dominates compliance above 50–100 Hz, where the table ceases to behave as a rigid body.

10.2RMS Displacement from PSD

The root-mean-square (RMS) displacement of the payload is computed by integrating the power spectral density (PSD) of the displacement over the frequency band of interest:

RMS Displacement from PSD

x_{\text{RMS}} = \sqrt{\int_{f_1}^{f_2} S_x(f)\,df}

where

S_x(f)

is the displacement PSD in units of m²/Hz. If the floor vibration PSD is

S_{\text{floor}}(f)

and the transmissibility is

T(f)

, then

S_x(f) = T^2(f) \cdot S_{\text{floor}}(f)

. This integral is the definitive measure of isolation performance: it accounts for both the isolator transfer function and the actual disturbance environment across all frequencies.

10.3Power Spectral Density

The PSD represents the distribution of vibration energy across frequency. It is computed from measured time-domain data using the fast Fourier transform (FFT), typically with windowing and averaging to reduce statistical variance. The PSD is the preferred representation for random vibration because it normalizes for measurement bandwidth: the PSD value at a given frequency does not change with the FFT resolution bandwidth, making it a true spectral density. Acceleration PSD (in g²/Hz or (m/s²)²/Hz), velocity PSD (in (m/s)²/Hz), and displacement PSD (in m²/Hz) are related by factors of

(2\pi f)^2

and can be converted freely.

10.4Coherence & Cross-Coupling

Coherence measures the linear correlation between two signals as a function of frequency and ranges from 0 (no correlation) to 1 (perfect linear correlation). In isolation testing, coherence between the floor input and the payload output should be high (close to 1) at frequencies where the transmissibility is being measured; low coherence indicates that the payload motion is dominated by sources other than the floor (e.g., acoustic excitation or on-board disturbances), and the measured transmissibility is unreliable. Cross-coupling coherence between orthogonal axes reveals whether the isolation system introduces unwanted coupling — for example, vertical floor vibration producing horizontal payload motion through asymmetric isolator mounting or CG offset.

Worked Example: WE 7 — RMS Displacement from PSD

Problem: A floor vibration measurement shows a flat displacement PSD of

S_{\text{floor}} = 1 \times 10^{-18}\;\text{m}^2/\text{Hz}

from 1 to 100 Hz. The isolation system has a natural frequency of 2 Hz and a damping ratio of 0.10. Estimate the RMS payload displacement from 1 to 100 Hz, assuming the transmissibility at each frequency can be approximated by the SDOF formula.

Solution:

This requires integrating

T^2(f) \cdot S_{\text{floor}}

from 1 to 100 Hz. For a flat input PSD, the integral is dominated by the resonant peak near 2 Hz. A useful approximation is to compute the equivalent noise bandwidth of the resonance:

\Delta f_{\text{eq}} = \frac{\pi f_n}{2} \cdot \frac{1}{2\zeta} = \frac{\pi \times 2}{2} \times \frac{1}{0.20} = 3.14 \times 5.0 = 15.7\;\text{Hz}

T_{\text{peak}}^2 \approx \left(\frac{1}{2\zeta}\right)^2 = 25

\int T^2 \cdot S_{\text{floor}}\,df \approx T_{\text{peak}}^2 \times S_{\text{floor}} \times \Delta f_{\text{eq}} = 25 \times 10^{-18} \times 15.7 = 3.93 \times 10^{-16}\;\text{m}^2

x_{\text{RMS}} = \sqrt{3.93 \times 10^{-16}} = 1.98 \times 10^{-8}\;\text{m} \approx 20\;\text{nm RMS}

The estimated RMS payload displacement is approximately 20 nm, dominated by the resonant amplification near 2 Hz. This illustrates why even modest damping is important: reducing the resonant peak directly reduces the broadband RMS.

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11Selection Workflow

Selecting an isolation system is a structured engineering process, not a catalog browsing exercise. The following workflow distills the key steps into a practical sequence that ensures all critical parameters are addressed before hardware is specified.

11.1Define the Requirement

Start with the application. What is the sensitive instrument or process? What is the maximum allowable vibration level, in what frequency band, and in which axes? Express the requirement quantitatively — as a VC curve, as an RMS displacement or velocity, or as a maximum amplitude at a specific frequency. If the equipment manufacturer specifies a vibration sensitivity, use that as the starting point. If no specification exists, measure the equipment's sensitivity by introducing known vibration and observing the effect on performance (image blur, noise floor, yield loss, etc.).

11.2Characterize the Environment

Measure the floor vibration spectrum at the intended installation site. Use a calibrated triaxial sensor and record data for at least 30 minutes, capturing both baseline conditions and worst-case activity (HVAC cycling, foot traffic, nearby elevator operation). Plot the data as one-third-octave velocity spectra and compare to the VC curves. This measurement establishes the input to the isolation system and determines how much attenuation is needed. Also assess the acoustic environment: measure the sound pressure level and identify dominant noise sources.

11.3Choose Isolation Approach

Compare the measured floor spectrum to the application requirement. The gap between the two — in decibels, frequency by frequency — defines the required isolation performance. If the gap is less than 20 dB across the critical frequency band, passive pneumatic isolation is usually sufficient. If the gap is 20–40 dB and extends to frequencies below 5 Hz, active or hybrid active-passive isolation is warranted. If the gap exceeds 40 dB, consider site relocation, structural modification, or a combination of isolation with on-board vibration control. For the horizontal axes, check whether a pendulum mechanism is needed for low-frequency lateral isolation.

11.4Size & Specify

Calculate the total payload weight (table + instruments + margin) and the center of gravity. Select isolator legs with adequate load capacity and the desired natural frequency. Verify that the rocking-mode frequencies are acceptable. For pneumatic systems, specify the air supply requirements (pressure, flow, cleanliness). For active systems, specify the sensor type, control bandwidth, and power requirements. Request transmissibility data from the manufacturer at the actual payload weight, not just the catalog nominal weight. Compute the predicted payload vibration by multiplying the measured floor PSD by the isolator transmissibility squared and integrating — this is the definitive performance prediction.

11.5Verify & Iterate

After installation, measure the actual payload vibration and compare it to the prediction and the requirement. If the performance is inadequate, diagnose the cause: is the floor vibration higher than expected? Is the isolator natural frequency different from the specification (wrong payload weight, incorrect pressure setting)? Is there acoustic coupling or on-board disturbance? Is the isolator properly leveled and free of mechanical short circuits (cables, hoses, or other rigid connections bridging the isolator)? Iterative measurement and adjustment is the norm — few isolation systems deliver optimal performance out of the box without site-specific tuning.

References

[]Harris, C.M. and Piersol, A.G., Harris' Shock and Vibration Handbook, 5th ed., McGraw-Hill, 2002. The comprehensive reference for vibration isolation theory and practice.
[]Rivin, E.I., Passive Vibration Isolation, ASME Press, 2003. Detailed treatment of passive isolation methods including pneumatic, elastomeric, and negative-stiffness systems.
[]Ungar, E.E., Sturz, D.H., and Amick, H., "Vibration Control Design of High Technology Facilities", Sound and Vibration, Vol. 24, No. 7, pp. 20–27, 1990. Introduction of the VC curve framework.
[]Gordon, C.G., "Generic Vibration Criteria for Vibration-Sensitive Equipment", Proceedings of SPIE, Vol. 1619, pp. 71–85, 1991. Original publication of the VC-A through VC-E criteria.
[]Den Hartog, J.P., Mechanical Vibrations, 4th ed., Dover, 1985. Classic text including the theory of tuned mass dampers (dynamic vibration absorbers).
[]Platus, D.L., "Negative-Stiffness-Mechanism Vibration Isolation Systems", Proceedings of SPIE, Vol. 1619, pp. 44–54, 1991. Theory and implementation of negative-stiffness passive isolation.
[]Preumont, A., Vibration Control of Active Structures: An Introduction, 3rd ed., Springer, 2011. Authoritative treatment of active vibration control theory and implementation.
[]Amick, H., Gendreau, M., Busch, T., and Gordon, C.G., "Evolving Criteria for Research Facilities: Vibration", Proceedings of SPIE, Vol. 5933, 2005. Updated VC criteria including VC-F and VC-G.
[]ISO 10811-1:2000, "Mechanical vibration and shock — Vibration and shock in buildings with sensitive equipment — Part 1: Measurement and evaluation". International standard for building vibration assessment.
[]Newport Corporation, Vibration Control Technical Notes. Practical application notes covering pneumatic isolator selection, optical table compliance, and active system integration. Available at www.newport.com.

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