Precision Motion Calculator: Flexure Stiffness, Natural Frequency & Vibration Isolation

Q: What is transmissibility and when is it less than 1?

Transmissibility $TR$ is the ratio of response amplitude to base excitation amplitude. $TR < 1$ (isolation) occurs when the excitation frequency exceeds $\sqrt{2} \times fn$. Below that, the isolator amplifies vibration — worst at $r = 1$ (resonance).

Q: How do I choose between a stiff mount and a soft isolator?

Stiff mounts (high $fn$) work when disturbances are low-frequency and you need high static stiffness. Soft isolators (low $fn$) work when floor vibration is the dominant source and static load is handled by preload or gravity.

Q: What materials minimise thermal drift?

Invar ($\alpha \approx 1.2 \times 10^{-6}$), Super Invar ($\alpha \approx 0.3 \times 10^{-6}$), Zerodur ($\alpha \approx 0.05 \times 10^{-6}$), and carbon-fibre composites ($\alpha$ near zero along fibre). Cost and machinability trade against CTE.

Q: How accurate is the SDOF transmissibility model?

The SDOF model captures the dominant mode well for simple isolation systems. Multi-mode structures (granite-on-isolators, active tables) need frequency response function (FRF) measurement or multi-DOF models.

Q: What damping ratio is typical for precision isolators?

Passive rubber/elastomer: $\zeta = 0.05$–0.15. Air springs: $\zeta = 0.01$–0.05. Active systems with feedback: equivalent $\zeta = 0.2$–0.7.

Check	IND	US	EU	ISO
Flexure stiffness	implemented	—	—	—
Natural frequency	implemented	—	—	—
Isolation transmissibility	implemented	—	—	—

How engineers design precision motion systems

Precision instruments — coordinate measuring machines, lithography stages, optical mounts — demand sub-micrometre positioning accuracy. Three enemies threaten that accuracy: insufficient stiffness (compliance under load), vibration from the environment (floor, HVAC, adjacent equipment), and thermal drift from temperature changes. Engineers screen flexure stiffness, natural frequency, isolation transmissibility, and thermal drift to establish feasibility before detailed FEA and dynamic modelling.

This guide covers cantilever flexure mechanics, single-degree-of-freedom (SDOF) vibration isolation, and thermal dimensional stability.

Precision motion challenges and design levers

Challenge	Design lever	Key parameter
Compliance under load	Stiffer flexure or shorter span	$k = 3 E I / L^{3}$
Low natural frequency	Increase stiffness or reduce mass	$f_{n} = \frac{1}{2 π} k / m$
Floor vibration coupling	Isolator with low $f_{n}$	Transmissibility $T R$
Thermal drift	Low-CTE material, temperature control	$δ = α L Δ T$
Abbe error	Minimise offset from measurement axis	Geometry, not modelled here
Damping	Viscoelastic, constrained-layer, eddy-current	$ζ$

Engineering workflow

Define accuracy budget — total allowable error at the point of interest.
Allocate error sources — stiffness/load, thermal, vibration, Abbe, sensor.
Size flexures — compute stiffness and check that deflection under load is within budget.
Compute natural frequency — ensure it is well above (stiff mount) or well below (isolator) excitation frequencies.
Evaluate isolation — transmissibility at the dominant floor vibration frequency.
Estimate thermal drift — select low-CTE material or tighten temperature control.
Iterate — trade stiffness vs mass vs CTE vs cost until the error budget closes.

Key quantities and formulas

Cantilever flexure tip stiffness:

k = \frac{3 E I}{L ^{3}}

Single-degree-of-freedom natural frequency:

f_{n} = \frac{1}{2 π} \frac{k}{m}

Thermal drift:

δ_{th} = α L Δ T

Base-excitation transmissibility:

T R = \frac{1 + ( 2 ζ r ) ^{2}}{( 1 - r ^{2} ) ^{2} + ( 2 ζ r ) ^{2}}

where $r = f_{exc} / f_{n}$ is the frequency ratio and $ζ$ is the damping ratio.

Isolation condition: $T R < 1$ requires $r > 2$ , i.e., the excitation frequency must exceed $2 f_{n}$ .

Worked example

Given: Optical mount — cantilever flexure in Invar (E = 141 GPa, $α = 1.2 \times 1 0^{- 6}$ /°C). Flexure: 20 mm long, 5 mm wide, 0.5 mm thick. Moving mass 0.2 kg. Room temperature controlled to $\pm 0.5$ °C. Floor vibration at 15 Hz. Damping ratio $ζ = 0.02$ .

Inertia: $I = 5 \times 0. 5^{3} /12 = 0.052$ mm $^{4}$ = $5.2 \times 1 0^{- 14}$ m $^{4}$ .
Stiffness: $k = 3 \times 141 \times 1 0^{9} \times 5.2 \times 1 0^{- 14} / (0.02)^{3} = 2, 752$ N/m.
Natural frequency: $f_{n} = (1/2 π) 2752/0.2 = 18.7$ Hz.
Frequency ratio at 15 Hz: $r = 15/18.7 = 0.80$ . Transmissibility: $T R \approx 2.6$ — amplification, not isolation. The mount resonance is too close to the floor vibration.
Thermal drift over 100 mm reference length: $δ = 1.2 \times 1 0^{- 6} \times 0.1 \times 0.5 = 0.06$ $μ$ m — acceptable for micron-level work.

Fix: Lower the mount's natural frequency (add mass or soften the flexure) or raise it well above 15 Hz (stiffen the flexure and reduce mass). For $r > 2$ at 15 Hz, need $f_{n} < 10.6$ Hz — add an isolation pad.

Common mistakes and checks

Designing a flexure mount near the floor vibration frequency — creates resonant amplification instead of isolation.
Ignoring Abbe error — angular errors multiplied by offset distance dominate in many practical systems.
Using aluminium for thermal stability — its CTE ( $23 \times 1 0^{- 6}$ ) is 20× that of Invar or Zerodur.
Forgetting gravity sag — a horizontal cantilever deflects under its own weight, consuming error budget.
Assuming single-axis behaviour — real flexures have parasitic motions in secondary axes.
Neglecting creep in flexures — high-stress flexures near yield can exhibit time-dependent drift.

FAQ

What is transmissibility and when is it less than 1?

Transmissibility $T R$ is the ratio of response amplitude to base excitation amplitude. $T R < 1$ (isolation) occurs when the excitation frequency exceeds $2 \times f_{n}$ . Below that, the isolator amplifies vibration — worst at $r = 1$ (resonance).

How do I choose between a stiff mount and a soft isolator?

Stiff mounts (high $f_{n}$ ) work when disturbances are low-frequency and you need high static stiffness. Soft isolators (low $f_{n}$ ) work when floor vibration is the dominant source and static load is handled by preload or gravity.

What materials minimise thermal drift?

Invar ( $α \approx 1.2 \times 1 0^{- 6}$ ), Super Invar ( $α \approx 0.3 \times 1 0^{- 6}$ ), Zerodur ( $α \approx 0.05 \times 1 0^{- 6}$ ), and carbon-fibre composites ( $α$ near zero along fibre). Cost and machinability trade against CTE.

How accurate is the SDOF transmissibility model?

The SDOF model captures the dominant mode well for simple isolation systems. Multi-mode structures (granite-on-isolators, active tables) need frequency response function (FRF) measurement or multi-DOF models.

What damping ratio is typical for precision isolators?

Passive rubber/elastomer: $ζ = 0.05$ –0.15. Air springs: $ζ = 0.01$ –0.05. Active systems with feedback: equivalent $ζ = 0.2$ –0.7.

Use the PhyCalcPro calculator

Open the Precision motion calculator. Enter flexure geometry and material, moving mass, thermal parameters, and excitation frequency with damping ratio. Review flexure stiffness, natural frequency, thermal drift, frequency ratio, and transmissibility.

Purpose

Estimate flexure stiffness, natural frequency, thermal drift, and vibration isolation transmissibility for precision optomechanical and machine tool subsystems. Supports early-stage compliance and isolation design.

Physics & theory

Cantilever flexure tip stiffness $k = 3 E I / L^{3}$ . SDOF natural frequency $f_{n} = \frac{1}{2 π} k / m$ . Thermal drift $δ = α L Δ T$ . Base-excitation transmissibility for damping ratio $ζ$ and frequency ratio $r = f_{exc} / f_{n}$ : $T R < 1$ indicates isolation above $2 f_{n}$ ; near $r \approx 1$ , amplification occurs.

Governing equations

k = \frac{3 E I}{L ^{3}}, f_{n} = \frac{1}{2 π} \frac{k}{m}

δ_{th} = α L Δ T

T R = \frac{1 + ( 2 ζ r ) ^{2}}{( 1 - r ^{2} ) ^{2} + ( 2 ζ r ) ^{2}}

Numerical method

Closed-form flexure, thermal, and SDOF transmissibility. Resonance warning when $0.8 < r < 1.2$ .

Inputs

Parameter	Description
Elastic modulus, inertia, flexure length	Flexure geometry
Moving mass	Payload mass
CTE, reference length, temperature change	Thermal drift
Excitation frequency, damping ratio	Vibration isolation

Outputs

Flexure stiffness (N/m), natural frequency (Hz), thermal drift (m), frequency ratio, transmissibility.

Design codes & checks

Indicative: Stiffness, natural frequency, transmissibility screening
ISO: ISO 230 machine tool accuracy; ISO 20816 vibration context

Assumptions & limitations

Single cantilever flexure; multi-axis flexure systems not modelled.
SDOF isolation; no multi-mode or active control.
Linear elasticity; flexure stress limits not checked.
Abbe error and motion cross-coupling omitted.

References

Smith, S. T., & Chetwynd, D. G. Foundations of Ultraprecision Mechanism Design. Gordon and Breach.
Slocum, A. H. Precision Machine Design. SME.
ISO 230-1:2012. Test code for machine tools — Geometric accuracy.
Rao, S. S. Mechanical Vibrations, 6th ed., transmissibility chapter.

How Engineers Design Precision Motion Systems

Assumptions

Limitations

Engineering checks