Force & Mechanics — Abridged Guide

Essential quick reference for force, torque, deflection, clamping, and material properties in optical system design. For the full treatment with worked examples and diagrams, see the Comprehensive Guide.

1.Overview

Optical performance depends on mechanical stability. Every lens, mirror, and detector is held by a mechanical structure. Deflection, vibration, and inadequate clamping degrade alignment and image quality. This guide covers the mechanics you need for confident payload calculations and hardware selection.

The most common mechanical failure in optics is not breakage — it is deflection. Structures that are perfectly strong can still deflect enough to misalign a beam. Always check deflection before checking stress.

2.Force Fundamentals

Newton's Second Law

F = m \cdot a

F in newtons (N), m in kg, a in m/s². Weight: F = mg where g = 9.81 m/s²

A 1 kg mass weighs 9.81 N. Force is a vector — direction matters. For a system in static equilibrium, all forces and all torques sum to zero.

Static Equilibrium Conditions

\sum F = 0 \qquad \sum \tau = 0

Stage acceleration force: To accelerate a payload of mass m to velocity v in time t, the required force is F = mv/t. In practice, allow 2–3× this value for friction and cable drag.

3.Weight & Payload

Weight

W = m \cdot g

Check whether catalog specs give payload in kg (mass) or N (force)

Component	Mass	Weight (N)
25 mm mirror (N-BK7, 6 mm)	~7 g	0.07
50 mm lens (N-BK7, 10 mm)	~46 g	0.45
Kinematic mirror mount (50 mm)	~200 g	1.96
Post (12.7 mm × 75 mm, steel)	~75 g	0.74
Post + holder + mount + optic	~350 g	3.43
Cage system assembly (4-plate)	~500 g	4.9

Center of Gravity (1D)

\bar{x} = \frac{\sum m_i \, x_i}{\sum m_i}

Keep CG low and centered over the base for stability

Payload ratings assume vertical (axial) loading. Cantilevered (offset) loads drastically reduce effective capacity because of the bending moment on the support structure.

4.Torque & Moments

Torque

\tau = F \times d

τ in N·m. F = applied force, d = perpendicular distance to pivot

Torque arises whenever a load's CG is offset from its support axis — virtually every cantilevered mount, right-angle bracket, and off-axis holder. Minimizing lever arms is the single most effective way to improve mechanical stability.

Quick torque check: Mass (kg) × 9.81 × offset (m) = torque (N·m). Example: 232 g at 45 mm offset → 0.232 × 9.81 × 0.045 = 0.10 N·m.

Every mm of unnecessary offset multiplies the bending load on the structure. When possible, center the optic directly over the post axis to eliminate torque entirely.

5.Stress, Strain & Elasticity

Normal Stress

\sigma = \frac{F}{A}

σ in Pa (= N/m²). Practical units: MPa = 10⁶ Pa, GPa = 10⁹ Pa

Hooke's Law

\sigma = E \cdot \varepsilon

E = Young's modulus (stiffness). ε = ΔL/L (strain, dimensionless)

Safety Factor

\text{SF} = \frac{\sigma_y}{\sigma_{\text{applied}}}

Typical SF: 3–5 for static loads, 5–10 for dynamic/shock loads in optics

Material	E (GPa)	σ_y (MPa)	Notes
Aluminium 6061-T6	69	276	Most common optomechanical material
Stainless Steel 303	193	240	Standard posts and hardware
Invar 36	141	276	Ultra-low CTE for thermal stability
Titanium Ti-6Al-4V	114	880	High strength-to-weight ratio

Axial compression is never the limiting factor for posts. A 12.7 mm steel post with 2 kg load has a safety factor of ~1,500 against compression. The limit is always bending deflection.

6.Beam & Cantilever Deflection

Cantilever End Deflection — Point Load

\delta = \frac{F \cdot L^3}{3\,E\,I}

F = force (N), L = length (m), E = Young's modulus (Pa), I = second moment of area (m⁴)

Cantilever End Deflection — End Moment

\delta = \frac{M \cdot L^2}{2\,E\,I}

M = applied moment (N·m). Used for cantilevered offset loads on posts

Second Moment of Area — Solid Circle

I = \frac{\pi\,d^4}{64}

Second Moment of Area — Rectangle

I = \frac{b\,h^3}{12}

b = width, h = height in bending direction

Post Diameter	I (mm⁴)	Relative Stiffness
6 mm	63.6	1×
12.7 mm (½″)	1,277	20×
25.4 mm (1″)	20,430	321×
38.1 mm (1.5″)	103,400	1,626×

Deflection scales with L³ (length cubed). Doubling the post height increases deflection by 8×. Stiffness scales with d⁴ (diameter to the fourth power). A 1″ post is 16× stiffer than a ½″ post. Always use the shortest post and largest diameter practical.

Simply Supported Beam — Center Point Load

\delta = \frac{F \cdot L^3}{48\,E\,I}

16× stiffer than a cantilever of the same length and load

7.Static vs. Dynamic Loading

Static loads (gravity) are constant. Dynamic loads (vibration, stage acceleration, shock) vary with time and can exceed static loads by large factors.

Dynamic Load Factor (Suddenly Applied Load)

\delta_{\text{dynamic}} = 2 \times \delta_{\text{static}}

DLF = 2.0 for a load released from rest. Higher for drops and impacts

Natural Frequency — Cantilever with End Mass

f_n = \frac{1}{2\pi}\sqrt{\frac{3\,E\,I}{m\,L^3}}

Push f_n above 100 Hz — well above building vibration (1–50 Hz)

Resonance amplifies vibration 10–100×. If a structural natural frequency matches an environmental vibration frequency, the response is amplified dramatically. Design for high natural frequency: stiffer, shorter, lighter.

Gentle handling matters. Place optical assemblies onto supports gradually — a suddenly applied load produces twice the static deflection. Tighten clamps progressively, not abruptly.

8.Clamping Force & Friction

Static Friction

f_s = \mu_s \cdot N

μ_s = static friction coefficient, N = normal (clamping) force

Contact Pair	μₛ (Static)
Steel on steel (dry)	0.6 – 0.8
Aluminium on steel	0.45 – 0.65
Glass on metal	0.5 – 0.7
Nylon on metal	0.15 – 0.25
PTFE on metal	0.04 – 0.10

Minimum Clamping Force

F_{\text{clamp}} \geq \frac{m \cdot a}{\mu_s}

a = expected acceleration (shock spec, e.g. 10g). Apply SF = 2

Bolt Preload from Torque

F = \frac{T}{K \cdot d}

T = torque (N·m), d = bolt diameter (m), K = nut factor (0.15–0.25)

Clamp firmly, not excessively. Enough force to resist the specified shock environment (with SF = 2) — no more. Over-clamping deforms optics, inducing wavefront error and stress birefringence. Use spring-loaded retainers for controlled, repeatable force.

9.Optomechanical Materials

Material	E (GPa)	ρ (g/cm³)	CTE (ppm/°C)	Best For
Aluminium 6061-T6	69	2.70	23.6	General purpose, lightweight, easy to machine
Stainless Steel 303	193	8.03	17.2	Posts, precision hardware, non-magnetic (316L)
Invar 36	141	8.05	1.3	Thermally critical mounts, metrological frames
Super Invar	144	8.15	0.3	Interferometric-grade thermal stability
Titanium Ti-6Al-4V	114	4.43	8.6	Lightweight, high strength, low CTE
Brass C36000	97	8.49	20.5	Adjusters, thumbscrews, low friction
Fused Silica	73	2.20	0.55	Ultra-low CTE optical substrates
Zerodur	91	2.53	0.05	Near-zero CTE mirror substrates

Specific Stiffness

\text{Specific stiffness} = \frac{E}{\rho}

Higher is better for lightweight, stiff structures. Al, Ti, SiC are leaders

Material selection rule of thumb: General lab → Al 6061. Thermal stability required → Invar. Weight-critical → Titanium. Maximum stiffness → Stainless steel or SiC.

10.Practical Load Checks

A 5-step workflow for verifying optomechanical payload adequacy:

Sum all masses

Optic + mount + adapter + cables. Multiply by 9.81 for weight in N.

Measure offset distances

Distance from the CG of each load to its support axis. Calculate torque τ = Wd.

Check post deflection

Use δ = ML²/2EI for cantilevered loads. Compare to your alignment tolerance.

Verify payload ratings

Check each stage, post, and mount against its rated capacity. For off-center loads, calculate the applied moment (M = Fd) and compare to the manufacturer's moment capacity spec.

Check clamping force

F_clamp ≥ 2 × ma/μ for the expected acceleration environment.

Applied Moment from Off-Center Load

M_A = F \times d_{\text{offset}}

Compare M_A to the stage's maximum moment capacity (MMC) from the datasheet

Off-center loads create moments that derate stage capacity. Crossed roller bearings handle moments better than ball bearings or dovetail slides. Air bearings are least sensitive. Always check the manufacturer's moment capacity specification rather than using rules of thumb.

For multi-stage stacks: work from top down. Each stage carries all stages above it plus the payload. The bottom stage bears the cumulative load. Always verify capacity at every level.

Post selection shortcut: Payload <200 g, no offset → 12.7 mm post. Payload <1 kg or any significant offset → 25.4 mm post. Payload > 2 kg → 38.1 mm post or pedestal.

Continue Learning

Comprehensive Force & Mechanics Guide →Post Deflection Calculator →Clamping Force Calculator →Stage Payload Budget →

The Comprehensive Guide includes 7 worked examples, SVG diagrams, and full mathematical derivations for every formula on this page.