Vibration Science

1Introduction

1.1What Is Vibration

Vibration is the oscillatory motion of a body or structure about an equilibrium position. Every vibration can be described by three interdependent quantities: displacement (the distance from equilibrium), velocity (the rate of change of displacement), and acceleration (the rate of change of velocity). For a sinusoidal vibration at angular frequency $\omega$, if the displacement amplitude is $X$, then the velocity amplitude is $\omega X$ and the acceleration amplitude is $\omega^2 X$ [1, 2].

Vibrations are classified by their source characteristics. Free vibration occurs when a system is displaced and released, oscillating at its natural frequency until energy dissipates. Forced vibration occurs when a continuous external force drives the system, and the response depends on the relationship between the driving frequency and the natural frequency. Random vibration — the most common type in laboratory environments — contains energy distributed across a broad spectrum with no single dominant tone [1].

1.2Why Vibration Matters in Photonics

Precision optical experiments require positional stability at scales far below human perception. Visible light has wavelengths between 400 and 700 nm, and many experiments — interferometry, holography, confocal microscopy, nanopositioning — require stability at fractions of a wavelength. A vibration amplitude of even 100 nm can destroy an interference pattern, misalign a focused beam, or bury a signal in noise [3, 5].

The problem compounds with optical path length. A laser beam reflecting off multiple mirrors accumulates angular errors. If vibration tilts a mirror by angle $\theta$, the beam displacement at distance $L$ is $\Delta x = 2L\theta$ for each reflection. A system with 6 mirrors and a total path length of 2 meters can amplify a 0.1 µrad angular vibration into a 2.4 µm beam displacement — significant when the beam is focused to a few micrometers [5].

1.3Sources of Vibration

Vibration sources fall into three categories based on their coupling path to the optical system [5, 9]:

Seismic (floor) vibrations are ground-borne disturbances transmitted through the building structure. Sources include foot traffic (1–4 Hz), building HVAC systems (10–60 Hz), road traffic (5–25 Hz), and elevators. Typical laboratory floors exhibit vibration amplitudes of 1–10 µm in the 1–100 Hz range, with the spectrum generally falling with increasing frequency [5].

Acoustic vibrations are airborne sound waves coupling to optical surfaces. Sources include ventilation systems, vacuum pumps, conversation, and external noise. Acoustic coupling is most significant above 50 Hz and particularly problematic for large, lightweight optics with significant surface area [9].

On-table (direct) vibrations are mechanical disturbances from equipment mounted on the optical table. Sources include motorized stages, shutters, cooling fans, and vacuum pump tubing. These bypass the floor isolation system entirely and are often the most difficult to address [5, 9].

2Simple Harmonic Motion

2.1The Mass-Spring System

The simplest vibration model is a single mass $m$ connected to a linear spring with stiffness $k$. The spring exerts a restoring force $F = -kx$ proportional to displacement $x$ from equilibrium. Applying Newton’s second law yields the equation of motion [1, 2]:

The solution is a sinusoidal oscillation at the natural frequency:

where $X$ is the amplitude, $\phi$ is the phase angle set by initial conditions, and $\omega_0$ is the natural angular frequency.

2.2Natural Frequency

The natural frequency depends only on mass and stiffness [1, 2, 4]:

Increasing mass lowers $f_0$ (a heavier system oscillates more slowly), while increasing stiffness raises $f_0$ (a stiffer system oscillates faster). This relationship is fundamental to vibration isolation — lowering the natural frequency extends the range of frequencies over which attenuation occurs [4].

The natural frequency can also be expressed in terms of static deflection $\delta_{st} = mg/k$ under gravity:

This form is useful for pneumatic isolators, where static deflection is readily measurable. A typical pneumatic isolator with $\delta_{st} \approx 6$ cm has $f_0 \approx 2$ Hz [4, 5].

2.3Free Vibration Response

Free vibration begins when the system is displaced from equilibrium and released. For the undamped case, the motion continues indefinitely at constant amplitude and frequency. The total mechanical energy $E = \tfrac{1}{2}kX^2$ is conserved, oscillating between potential energy in the spring at maximum displacement and kinetic energy of the mass at zero displacement [1].

3Damped Vibration

3.1The Damped Oscillator

Real systems dissipate energy through viscous friction, internal material losses, air resistance, and structural damping. The standard model adds a viscous damper with damping coefficient $c$, producing a resistive force proportional to velocity [1, 2]:

3.2Damping Ratio

The damping ratio $\zeta$ (zeta) characterizes how much damping is present relative to the critical value [1, 2]:

Underdamped ($\zeta < 1$): The system oscillates with exponentially decaying amplitude at the damped natural frequency $\omega_d = \omega_0\sqrt{1-\zeta^2}$. This is the regime relevant to most isolation systems; typical pneumatic isolators have $\zeta$ = 0.05–0.2 [4, 5].

Critically damped ($\zeta = 1$): The system returns to equilibrium as fast as possible without oscillating — the theoretical boundary between oscillatory and non-oscillatory behavior.

Overdamped ($\zeta > 1$): The system returns to equilibrium without oscillating, but more slowly than the critically damped case.

3.3Logarithmic Decrement

The logarithmic decrement $\delta$ is a practical tool for measuring damping from experimental data. It is defined as the natural logarithm of the ratio of successive peak amplitudes in free decay [1, 2]:

For light damping ($\zeta \ll 1$), the approximation $\delta \approx 2\pi\zeta$ is accurate to within 1% for $\zeta < 0.1$ [1]. When multiple cycles are available, accuracy improves by measuring over $N$ cycles:

3.4Quality Factor

The quality factor $Q$ describes the sharpness of the resonance peak and is inversely related to damping [1, 2]:

High $Q$ means a sharp, tall resonance peak — the system stores energy efficiently but is difficult to damp. Low $Q$ means a broad, shallow peak. $Q$ also relates to the half-power bandwidth:

where $\Delta f$ is the frequency width at which the response falls to $1/\sqrt{2}$ of its peak value (the −3 dB points).

4Forced Vibration & Resonance

4.1Harmonic Forcing

When an external harmonic force $F(t) = F_0\cos(\omega t)$ acts on a damped mass-spring system, the equation of motion becomes [1, 2]:

After transients decay, the system settles into a steady-state response at the driving frequency: $x(t) = X\cos(\omega t - \phi)$, where $X$ is the steady-state amplitude and $\phi$ is the phase lag behind the driving force.

4.2Frequency Response

The magnification factor $M$ is the ratio of steady-state response amplitude to the static deflection $X_{st} = F_0/k$ [1, 2]:

where $r = \omega/\omega_0 = f/f_0$ is the frequency ratio.

4.3Resonance

Resonance occurs when the driving frequency approaches the natural frequency ($r \approx 1$). For a damped system, the peak magnification occurs at [1]:

Resonance is the primary concern for precision optics. A pneumatic isolator with $f_0 = 1.5$ Hz and $\zeta = 0.05$ has $Q = 10$, meaning floor vibrations at 1.5 Hz are amplified 10× before reaching the table surface [5].

4.4Phase Response

The phase angle between the driving force and the response is [1, 2]:

At low frequency ($r \ll 1$), $\phi \approx 0$ (response in phase with forcing). At resonance ($r = 1$), $\phi = 90°$ regardless of damping. At high frequency ($r \gg 1$), $\phi \to 180°$ (response anti-phase). The transition through 90° is sharp for low damping and gradual for high damping.

5Transmissibility

5.1Transmissibility Definition

Transmissibility $T$ is the ratio of output motion amplitude to input motion amplitude when a system is subjected to base excitation — when the floor moves and the question is how much motion reaches the table [1, 2, 4]:

This is the key performance metric for vibration isolation. For force transmissibility (transmitted force to applied force), the expression is numerically identical [1].

5.2Transmissibility Equation

For a single-degree-of-freedom system with viscous damping [1, 2, 4]:

where $r = f/f_0$ is the frequency ratio and $\zeta$ is the damping ratio.

5.3Three Regions

Static region ($r \ll 1$): $T \approx 1$. The input motion passes through unchanged. The spring is stiff enough relative to the driving frequency that the mass follows the base.

Resonance region ($r \approx 1$): $T > 1$. The input motion is amplified. Peak transmissibility is approximately $Q = 1/(2\zeta)$ for light damping.

Isolation region ($r > \sqrt{2}$): $T < 1$. The input motion is attenuated. Transmissibility decreases as $1/r^2$ for undamped systems — falling at 40 dB/decade [4, 5].

5.4The √2 Crossover

All transmissibility curves, regardless of damping, pass through $T = 1$ at $r = \sqrt{2} \approx 1.414$ [1, 2, 4]. This is exact — derived by setting $T = 1$ in the transmissibility equation. An isolation system can only attenuate vibrations at frequencies above $\sqrt{2} \times f_0$.

For a pneumatic isolator with $f_0 = 1.5$ Hz, isolation begins at $\sqrt{2} \times 1.5 = 2.12$ Hz. All floor vibrations below 2.12 Hz pass through or are amplified.

5.5The Damping Trade-Off

Damping creates a fundamental trade-off [4, 5]: at resonance, high damping reduces the amplification peak. In the isolation region, high damping degrades performance — an undamped system attenuates as $1/r^2$ (40 dB/decade), but a heavily damped system attenuates as only $1/r$ (20 dB/decade) at high frequency ratios [4].

The practical compromise is moderate damping ($\zeta$ = 0.05–0.15): low enough to maintain high-frequency isolation, high enough to limit the resonance peak and provide reasonable settling time. Active damping systems can selectively damp resonance without degrading high-frequency performance [5].

6Vibration Measurement & Representation

6.1Displacement, Velocity, Acceleration

For sinusoidal vibration at frequency $f$, the three quantities are interrelated by factors of $2\pi f$ [1, 2]:

Each quantity serves different purposes: displacement is relevant for positional accuracy (alignment, interferometry), velocity is used for vibration criteria because equipment sensitivity curves are approximately constant-velocity, and acceleration is what accelerometers directly measure [1, 7].

6.2Frequency Domain

Real environments contain energy across a broad frequency range. The power spectral density (PSD) decomposes a time-domain signal into frequency content. The PSD of acceleration, $\Phi_a(f)$, has units of (m/s²)²/Hz. The RMS acceleration in a frequency band is [1, 2]:

Conversions between PSDs follow the same frequency-scaling as the amplitudes:

6.3Decibel Representation

Vibration levels are frequently expressed in decibels [7, 8]:

Common reference values: velocity $v_{ref}$ = 1 µin/s = 25.4 nm/s (US convention), or 1 µm/s (SI convention). The VC curves use velocity in µm/s — for example, VC-A corresponds to 50 µm/s RMS [7].

6.4One-Third Octave Bands

Vibration criteria use one-third octave band analysis rather than narrowband spectra [7, 8, 10]. Each band spans a frequency ratio of $2^{1/3} \approx 1.26$, giving bandwidth proportional to center frequency. Standard center frequencies (Hz): 1, 1.25, 1.6, 2, 2.5, 3.15, 4, 5, 6.3, 8, 10, 12.5, 16, 20, 25, 31.5, 40, 50, 63, 80.

Proportional bandwidth is used because a structural resonance responds most strongly to excitation in a bandwidth proportional to its natural frequency — making constant-bandwidth velocity a natural metric for equipment sensitivity [7].

7Compliance

7.1Compliance Definition

Compliance $C(f)$ is the dynamic displacement response of a structure per unit applied force as a function of frequency [5, 6]:

Compliance is the inverse of dynamic stiffness. Lower compliance means a stiffer, more rigid structure — less deflection for a given force. Compliance is the primary specification for optical table performance because it directly determines how much the table surface deforms in response to vibration [5, 6].

7.2Static vs. Dynamic Compliance

A compliance curve plotted against frequency reveals structural behavior [5, 6]:

Rigid body region (below the first structural resonance): compliance decreases as $1/(m\omega^2)$ on a log-log plot — the “rigid body line.” All points on the surface move together; no relative motion occurs between components.

Resonant peaks (typically 100–500 Hz for optical tables): compliance peaks sharply. The table surface flexes, creating relative motion between components at different locations — the critical failure mode for optical experiments [5, 6].

High-frequency roll-off: above the resonant region, compliance generally decreases.

7.3Interpreting Compliance Curves

The compliance at resonant peaks determines relative motion across the table surface [5, 6]. Key considerations: peak-to-valley ratio matters more than absolute level; damping reduces peak compliance — broadband-damped tables show lower, wider peaks; a typical high-performance honeycomb table achieves compliance peaks of 0.005–0.02 µm/N at first resonance; and the first resonant frequency should be as high as possible (thicker tables have higher first resonance).

8Vibration Criteria

8.1Colin Gordon VC Curves

The Vibration Criterion (VC) curves are the standard framework for specifying acceptable vibration levels in sensitive facilities. Developed in the early 1980s by Eric Ungar and Colin Gordon, they define maximum RMS velocity in one-third octave bands from 4 to 80 Hz [7, 8]:

The VC curves are flat (constant velocity) from 4 to 80 Hz. Each successive curve is 6 dB below the previous — half the velocity. Below 4 Hz, the criteria are generally not applied because building resonances make achievement impractical [7, 8].

8.2NIST-A Criterion

The NIST-A criterion was developed for the NIST Advanced Measurement Laboratory for metrology and nanotechnology [8]. Above 20 Hz it is identical to VC-E (3.12 µm/s). Below 20 Hz it maintains constant displacement rather than constant velocity, making it more stringent at low frequencies. NIST-A is the standard target for the most demanding laboratory facilities.

8.3ISO Guidelines

The ISO 2631 standard provides human perception thresholds for vibration in buildings, serving as reference points on the VC chart: operating theatres (~200 µm/s), residential night (~100 µm/s), and offices (~400 µm/s). The contrast illustrates that equipment sensitivity requirements are 10–500× more stringent than human comfort thresholds.

8.4Matching Equipment to Criteria

9Vibration in Optical Systems

9.1Beam Deflection from Table Motion

Vibration affects optical systems through two mechanisms [5]. Translational motion moves the entire table rigidly — all components shift equally, with minimal effect on alignment. Angular (flexural) motion bends the table surface, creating angular displacement between components at different locations. This is far more destructive.

If two mounts separated by distance $d$ experience a relative angular tilt $\theta$, the beam deflection at a target distance $L$ from the second mount is [5]:

The factor of 2 applies to mirror reflection; for a transmissive element it is $\theta L$.

9.2Maximum Relative Motion

Maximum relative motion (MRM) is the peak-to-peak relative displacement between any two points on the table surface due to dynamic excitation. It is the specification most directly tied to optical performance — quantifying how much the surface bends under vibration [5, 6]. MRM is derived from the compliance curve integrated against the floor vibration spectrum. A well-designed system (broadband-damped table on pneumatic isolators) in a quiet lab achieves MRM below 0.01 µm.

9.3Thermal Effects

Temperature gradients cause slow structural deformation. A 1°C differential across a 1.5 m steel table (CTE ≈ 12 × 10⁻⁶/°C) creates a bending angle of approximately 8 µrad — orders of magnitude larger than vibration-induced angular motion [5]. Thermal effects are distinguished by timescale: drift over minutes to hours versus vibration at 1–100+ Hz. Mitigation uses enclosures and temperature control, not isolation.

9.4Coupling Paths

Vibration reaches optical components through multiple paths [5, 9]: through isolators (floor to table), through the table structure (on-table forces exciting resonances), through rigid pneumatic lines or cables that short-circuit isolation, and through acoustic coupling (airborne pressure fluctuations on optical surfaces).

Best practice: eliminate rigid connections between isolated and non-isolated equipment, use flexible tubing for all fluid/gas connections, suspend cables with slack loops, and enclose the optical system to reduce acoustic coupling.

10Vibration Mitigation Strategy

10.1Source Identification

Effective vibration control begins with characterizing the environment [5, 7]: perform a site survey with a low-noise accelerometer for at least 30 minutes during normal operation; plot the velocity spectrum in 1/3-octave bands against the relevant VC criterion; identify dominant sources (tonal peaks from rotating machinery, broadband from traffic); and address sources first — relocate vibrating machinery, repair unbalanced fans, add mass to lightweight floors. Source mitigation is always more effective than downstream isolation.

10.2The Isolation + Damping System

A complete vibration control system addresses two distinct problems [5, 6]:

Isolation (support legs): pneumatic or active isolators filter floor vibration before it reaches the table. Characterized by the transmissibility curve. Goal: lowest practical natural frequency for maximum isolation bandwidth.

Damping (table structure): the optical table damps structural resonances excited by forces that bypass the isolator. Characterized by the compliance curve. Goal: minimize compliance peaks at resonance frequencies.

These are complementary. An excellent isolator on a poorly damped table still suffers from on-table disturbances. A superbly damped table on rigid legs still transmits floor vibration.

10.3Selection Workflow

A step-by-step process for specifying a vibration control system:

1. Define the application: identify the vibration sensitivity and applicable VC criterion. 2. Measure the environment: perform a site vibration survey. 3. Determine isolation requirements: calculate required attenuation at each frequency. 4. Select isolator type: pneumatic (f₀ = 1–2 Hz, most common), active (f₀ < 1 Hz for demanding applications), or mechanical/elastomeric (f₀ = 5–20 Hz for lightweight platforms). 5. Select table specifications: size, thickness for first resonant frequency, and damping level. 6. Verify after installation: re-measure to confirm the system meets the criterion.

References

1. [1]S. S. Rao, Mechanical Vibrations, 6th ed. Pearson, 2017.
2. [2]D. J. Inman, Engineering Vibration, 4th ed. Pearson, 2013.
3. [3]E. Hecht, Optics, 5th ed. Pearson, 2017.
4. [4]Newport Corporation, “Fundamentals of Vibration,” Technical Note.
5. [5]Newport Corporation, “Vibration Control Fundamentals,” Technical Note.
6. [6]Newport Corporation, “Compliance and Transmissibility Curves,” Technical Note.
7. [7]C. G. Gordon, “Generic Vibration Criteria for Vibration-Sensitive Equipment,” SPIE Proc. Vol. 1619, pp. 71–85, 1991.
8. [8]H. Amick, M. Gendreau, T. Busch, and C. G. Gordon, “Evolving Criteria for Research Facilities: Vibration,” SPIE Proc. Vol. 5933, 2005.
9. [9]Thorlabs, “Optical Tables Tutorial,” Technical Resource.
10. [10]IEST-RP-CC024, “Measuring and Reporting Vibration in Microelectronics Facilities.”