Precision Motion & Vibration (`precision-motion`)

Precision Motion & Vibration

Flexures, isolation, thermal drift and optomechanical stiffness

Standards catalog

Validation: indicative · Method band: formula

Open calculator

Indicative method: Cantilever flexure and single-degree-of-freedom vibration screening

Assumptions

Linear elastic material behavior unless noted otherwise.
User is responsible for load combinations and load factors per the selected design code.
Design standard (US/EU/ISO) sets unit defaults and screening check labels — not a full code worksheet.

Limitations

Uses simple beam/flexure approximations; multi-axis flexures, controls, bearings and nonlinear motion errors are not included.

Engineering checks

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

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 $k = 3 E I / L^{3}$ for elastic modulus $E$ , second moment $I$ , and length $L$ . SDOF natural frequency $f_{n} = \frac{1}{2 π} k / m$ . Thermal drift $δ_{th} = α L Δ T$ from expansion coefficient $α$ .

Base-excitation transmissibility for damping ratio $ζ$ and frequency ratio $r = f_{exc} / f_{n}$ :

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

Values $T R < 1$ indicate isolation above resonance; near $r \approx 1$ , amplification occurs.

Advanced systems calculators use lumped-parameter screening models suitable for concept trade studies. Each calculator returns explicit assumptions and warnings arrays documenting model limits. Constants such as ( sigma ) (Stefan–Boltzmann), ( mu_0 ), and ( R ) (gas constant) use SI definitions from the solver source.

Results are not certified for regulatory submission without independent verification against detailed analysis or test data.

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 (advanced-systems/calculators). Resonance warning when $0.8 < r < 1.2$ .

Inputs

Parameter	Description
`elasticModulus`, `inertia`, `flexureLength`	Flexure geometry
`movingMass`	Payload mass
`alpha`, `referenceLength`, `deltaT`	Thermal drift
`excitationFrequency`, `dampingRatio`	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 modeled.
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.
PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Maintainer note: Flexure stiffness and SDOF vibration screening.

Precision Motion & Vibration

Assumptions

Limitations

Engineering checks

Precision Motion & Vibration (precision-motion)

Precision Motion & Vibration (`precision-motion`)