Dimensional Mechanical Calibration

What is Required Dimensional Mechanical Calibration?

Required Dimensional Mechanical Calibration refers to the essential process of verifying and adjusting the accuracy of mechanical measuring instruments used to determine physical dimensions—such as length, thickness, diameter, angle, or flatness—to ensure they conform to national or international measurement standards.

🔧 Definition:

Dimensional Mechanical Calibration is the process of comparing the measurements made by a mechanical device (e.g., micrometer, caliper, gauge block, dial indicator) to a traceable standard of known accuracy. The goal is to detect, record, and correct measurement deviations.

📌 Why is it Required?

1. ✅ Accuracy Assurance: To ensure that equipment delivers correct measurements.
2. ✅ Regulatory Compliance: Required by ISO/IEC 17025, ISO 9001, and other quality standards.
3. ✅ Traceability: Links all measurements to national or international standards.
4. ✅ Product Quality: Directly impacts manufacturing precision and product tolerances.
5. ✅ Risk Reduction: Minimizes errors, rejects, and recalls.
6. ✅ Audit Preparedness: Demonstrates control over quality instruments during audits.

⚙️ Common Instruments That Require Dimensional Mechanical Calibration:

• Vernier calipers
• Micrometers (inside, outside, depth)
• Dial indicators
• Height gauges
• Bore gauges
• Feeler gauges
• Gauge blocks
• Thread plug and ring gauges
• Snap gauges

📅 When is Calibration Required?

• At scheduled intervals (typically every 6–12 months)
• After any mechanical shock or suspected damage
• After repair or modification
• Prior to critical measurements
• As per industry-specific calibration cycles (defined in SOPs)

🏭 Where is it Done?

• Accredited Calibration Laboratories
• In-house Metrology Labs with proper traceability
• On-site Calibration by certified calibration engineers (if portability is possible)

📋 How is it Done?

1. Pre-checks: Clean, visually inspect, and verify environmental conditions.
2. Reference Standards: Use of gauge blocks, certified standards, or master instruments.
3. Measurement: Take readings at defined intervals across the instrument’s range.
4. Error Analysis: Determine deviation between instrument reading and standard.
5. Adjustment/Correction: If within tolerance, record results; if outside, adjust or repair.
6. Calibration Certificate Issuance: Includes results, uncertainties, environmental conditions, traceability, and due date for next calibration.

📘 Case Study Example:

A leading aerospace component manufacturer found their micrometer used for turbine blade thickness measurements was miscalibrated by 0.04 mm. Post recalibration, defect rates in final assembly reduced by 8%, demonstrating how crucial proper dimensional mechanical calibration is to quality assurance.

📑 White Paper Tip:

“Precision Beyond Tolerance: A White Paper on Dimensional Mechanical Calibration in High-Reliability Industries” – explores calibration impact on Six Sigma, lean manufacturing, and aerospace quality control.

Who is Required Dimensional Mechanical Calibration?

Dimensional Mechanical Calibration is required by any individual, organization, or industry that relies on accurate mechanical measurements to ensure product quality, process control, or compliance with standards. Below is a breakdown of who specifically requires it:

🏭 1. Manufacturing & Engineering Industries

• Automotive
• Aerospace
• Defense
• Heavy Machinery
• Tool & Die Making
• Metal Fabrication
  These industries need highly accurate mechanical dimensions for parts and assemblies. Calibration ensures precision in components like shafts, gears, bolts, or casings.

🧪 2. Testing and Calibration Laboratories

• NABL-accredited labs (in India)
• ISO/IEC 17025-accredited metrology labs
  Labs are required to calibrate their reference standards and working instruments regularly to maintain traceability and credibility.

🏗️ 3. Quality Assurance & Inspection Departments

QA/QC departments across industries depend on calibrated instruments for:

• Incoming material inspection
• In-process inspection
• Final product validation

🏢 4. ISO-Certified Organizations

Organizations certified to:

• ISO 9001 (Quality Management Systems)
• ISO/TS 16949 (Automotive Quality)
• AS9100 (Aerospace Quality)
• ISO 13485 (Medical Devices)
  must ensure all dimensional measuring instruments are calibrated and traceable.

🔧 5. Tool Rooms & Maintenance Departments

Tool rooms that manage jigs, fixtures, molds, dies, and precision tools require calibration to avoid cumulative dimensional errors in production.

🧰 6. Equipment and Instrument Manufacturers

Manufacturers of:

• Vernier calipers
• Micrometers
• Gauges and fixtures
  must calibrate their products before dispatch to meet performance and accuracy specifications.

🧑‍🏫 7. Technical Training Institutes & Skill Development Centers

To maintain training standards, especially in:

• Mechanical engineering labs
• ITIs and polytechnics
• Skill India programs (e.g., NSDC certified courses)
  Institutes must use calibrated instruments for education and practical assessments.

📜 8. Government and Regulatory Bodies

Departments such as:

• Legal Metrology
• BIS (Bureau of Indian Standards)
• DRDO, ISRO, HAL (in India)
  need calibration for compliance, standardization, and national traceability.

⚙️ 9. Calibration & Metrology Service Providers

Any organization offering calibration services must itself undergo routine calibration of its master standards and working instruments.

👨‍🔧 10. Freelancers and Consultants in Quality & Engineering

Professionals involved in:

• Mechanical design
• Quality audits
• Metrology consulting
  require knowledge of calibrated tools for accurate assessments and reports.

When is Required Dimensional Mechanical Calibration?

Dimensional Mechanical Calibration is required at specific intervals or conditions to ensure the reliability and traceability of measurement instruments. Below are the key scenarios when calibration is mandatory:

📆 1. At Regular Intervals (Scheduled Calibration)

• Every 6–12 months, depending on:
  • Manufacturer’s recommendation
  • Instrument usage frequency
  • Customer/industry specifications
  • Quality system requirements (e.g., ISO 9001, ISO 17025)

Example: A micrometer used daily in a production line is typically calibrated every 6 months.

⚠️ 2. After Mechanical Shock or Damage

• If an instrument is dropped, mishandled, or exposed to vibration, it must be immediately recalibrated or verified before use.

Example: A dial indicator that fell from a bench must be recalibrated before use in inspection.

🛠️ 3. After Repair or Modification

• Any instrument that has been:
  • Repaired
  • Reassembled
  • Recalibrated internally
    must be validated and calibrated again to ensure accuracy.

📏 4. Before Critical Measurements

• Prior to:
  • Final inspection
  • Third-party audits
  • Customer acceptance testing
• Calibration is performed to ensure zero deviation in critical dimensions.

Example: Calibrating plug gauges before final acceptance of precision-machined engine components.

📋 5. As per Contractual or Customer Requirement

• Some clients or contracts specify:
  • Calibration frequency
  • Traceability documentation
  • Acceptance criteria
• Especially common in defense, aerospace, and medical device industries.

📉 6. When Measurement Deviations Are Suspected

• If inspection results become inconsistent or if products repeatedly fail dimensional checks, instruments must be checked and recalibrated.

🧪 7. During Internal or External Audits

• Calibration status is reviewed during audits. Any instrument without valid calibration may lead to:
  • Non-conformance reports
  • Audit failure
  • Loss of certification

📦 8. Before Placing Equipment into Service (Initial Calibration)

• New instruments must be calibrated before being used in production or testing environments to:
  • Validate accuracy
  • Create baseline data
  • Establish traceability

✅ Summary Table:

Where is Required Dimensional Mechanical Calibration?

Dimensional Mechanical Calibration is required wherever precise mechanical measurements are critical—from high-tech industries to everyday manufacturing environments. Below is a breakdown of where calibration is typically required and conducted:

🏭 1. Manufacturing Facilities

• Locations: Automotive plants, aerospace assembly lines, electronics manufacturing, metal workshops
• Purpose: Ensure parts meet dimensional tolerances and quality specifications during production.

🧪 2. NABL / ISO/IEC 17025 Accredited Calibration Laboratories

• These are specialized third-party labs equipped with traceable standards and environmental controls.
• Located in industrial zones, technology parks, or standalone facilities.
• Provide certified, traceable calibration services for gauges, calipers, micrometers, etc.

🏢 3. In-house Metrology or Quality Control Labs

• Large organizations maintain internal labs to:
  • Reduce downtime
  • Perform routine verifications
  • Maintain control over quality systems

Common in: Aerospace, automotive, defense, medical device companies

🧰 4. Tool Rooms and Maintenance Shops

• Calibration ensures tools, jigs, and fixtures used in production are dimensionally correct.
• Prevents tolerance stacking and wear-induced inaccuracies.

🏗️ 5. Construction and Infrastructure Sites

• Where precision instruments like laser distance meters, spirit levels, and dial gauges are used.
• Calibration ensures on-site measurement accuracy for alignment, leveling, and quality inspection.

📦 6. Incoming & Final Inspection Areas

• Instruments used to verify dimensions during goods receiving or before dispatch must be calibrated.
• Ensures acceptance criteria are met at both ends of the supply chain.

🏥 7. Medical Device and Pharmaceutical Industries

• Locations like cleanrooms, biotech labs, and QA departments use calibrated mechanical tools for verification of device components and packaging materials.

🎓 8. Educational and Skill Development Institutes

• Technical training institutes, ITIs, and polytechnics must ensure their measurement tools are calibrated to provide accurate practical training.

✈️ 9. Aerospace & Defense Facilities

• Where tight tolerances and regulatory compliance are mission-critical.
• Instruments in these facilities are calibrated with traceability to national/international standards (like NIST, NABL, UKAS).

🚛 10. Mobile Calibration Services / On-Site Calibration

• Offered by calibration service providers who visit your location with portable standards.
• Useful for:
  • Remote plants
  • Large-scale calibration jobs
  • Instruments that can’t be easily transported

✅ Summary Table:

How is Required Dimensional Mechanical Calibration?

Dimensional Mechanical Calibration is carried out through a systematic, standardized procedure that ensures measuring instruments provide accurate, traceable results. Here’s a step-by-step explanation of how the calibration process works:

🔍 Step-by-Step Calibration Procedure:

✅ 1. Preparation

• Clean the Instrument: Remove dust, oil, and debris.
• Stabilize Environment: Ensure controlled conditions—typically 20°C ± 1°C, low humidity, and no vibration.
• Visual Inspection: Check for damage, wear, rust, or deformation.

📏 2. Select Calibration Standards

• Use traceable reference standards such as:
  • Gauge blocks (for length)
  • Master rings and plugs
  • Certified micrometer standards
• Standards must be traceable to NIST, NABL, or other national/international metrology bodies.

🔧 3. Perform the Calibration

• Compare the instrument’s reading against the known value of the standard.
• Measure at multiple points across the instrument’s range (e.g., 0, 25, 50, 75, 100 mm for a 100 mm micrometer).
• Record the deviation (error) at each point.

Example:

📊 4. Calculate Uncertainty

• Determine measurement uncertainty, which includes:
  • Standard uncertainty of the reference
  • Environmental factors
  • Operator influence
  • Resolution of the instrument
• Express it typically with a confidence level of 95% (k=2).

🛠️ 5. Adjust or Repair (if needed)

• If the instrument is outside the allowable tolerance, it may be:
  • Adjusted using mechanical or digital methods
  • Repaired and then recalibrated
  • Rejected or tagged as non-conforming

🧾 6. Documentation and Certification

• Issue a Calibration Certificate that includes:
  • Instrument details (ID, make, serial number)
  • Calibration date and due date
  • Reference standards used
  • Results and uncertainties
  • Environmental conditions
  • Technician’s signature and lab accreditation

📌 7. Label the Instrument

• Apply a calibration sticker with:
  • Date of calibration
  • Next due date
  • Calibration status (Pass/Fail)

🔁 8. Maintain Calibration Records

• Keep records for audit and traceability
• Use a calibration management system (CMS) or logbook to track status and upcoming due dates

🧪 Types of Mechanical Instruments Calibrated This Way:

✅ Best Practices:

• Always calibrate before use in critical applications.
• Perform intermediate checks (e.g., zero setting, visual verification).
• Ensure all reference standards are within their calibration validity.
• Never use an instrument beyond its calibrated range.

Case Study on Dimensional Mechanical Calibration?

🏭 Industry: Precision Machining – Automotive Component Manufacturer

🧰 Application: Caliper and Micrometer Calibration

📍 Location: Pune, India

📆 Timeframe: January–March 2024

🎯 Problem Statement:

A Tier-1 supplier of engine pistons and connecting rods for a global automobile brand faced increasing rejections from the OEM during random dimensional audits. The defect rate rose from 0.8% to 4.6% within three months, with the primary issue being inconsistent measurements of piston diameters.

🔎 Initial Investigation:

• Piston diameter out-of-tolerance observed during OEM final inspection.
• In-process inspection team claimed conformity, citing results from calibrated vernier calipers and micrometers.
• Internal QA team suspected a measurement error, not a machining error.

⚙️ Root Cause Analysis:

1. Instruments in use:
   • 5 Vernier calipers (150 mm)
   • 3 Outside micrometers (0–50 mm)
2. Last calibration was over 14 months ago.
3. Two micrometers showed systematic deviation of +0.025 mm during re-verification.
4. Calipers had worn jaws, causing parallelism error up to +0.04 mm.

🧪 Action Taken – Dimensional Mechanical Calibration:

• All instruments sent to NABL-accredited calibration lab.
• Lab performed:
  • Gauge block comparison
  • Zero error check
  • Repeatability tests
• Instruments with excessive deviation were repaired and recalibrated.
• Calibrated tools were marked with:
  • Calibration date
  • Certificate number
  • Next due date

📈 Post-Calibration Results:

📋 Lessons Learned:

• Lack of timely calibration directly impacted product quality and client trust.
• Dimensional mechanical calibration improved measurement reliability, reduced false rejections, and restored compliance with OEM specifications.
• Company implemented:
  • 6-month calibration cycle
  • Internal instrument check every 30 days
  • Calibration tracking system using QR codes

🏁 Conclusion:

This case shows how neglecting regular dimensional mechanical calibration can cause costly quality failures, while timely calibration:

• Ensures product conformance
• Reduces scrap/rework
• Improves customer satisfaction
• Strengthens compliance with ISO 9001 & IATF 16949

White paper on Dimensional Mechanical Calibration?

Ensuring Measurement Accuracy in Modern Manufacturing

🏢 Published by: Six Sigma Labs / Deming Technologies

📅 Date: June 2025

📚 Keywords: Dimensional Calibration, Mechanical Metrology, ISO 17025, NABL, Measurement Uncertainty, Quality Assurance

🧭 Executive Summary

In today’s precision-driven industries, dimensional mechanical calibration plays a vital role in maintaining measurement integrity. From aerospace to automotive, even the slightest dimensional deviations can lead to failures, non-compliance, and increased costs. This white paper explores the methodology, standards, importance, and implementation of dimensional mechanical calibration, providing practical insights for quality managers, lab technicians, auditors, and engineers.

📌 1. Introduction

Dimensional mechanical calibration refers to the verification and adjustment of measuring instruments used to determine physical sizes—length, diameter, depth, and flatness. It ensures instruments produce accurate results that are traceable to national or international standards (e.g., NIST, NABL, UKAS).

⚙️ 2. Common Instruments Requiring Calibration

🔍 3. Why Dimensional Calibration is Required

• Compliance with Standards: ISO 9001, IATF 16949, ISO 13485, AS9100
• Product Quality Assurance: Eliminates dimensional variation
• Audit Preparedness: Required by certification bodies
• Customer Confidence: Ensures credibility in supply chains
• Regulatory Demand: Essential in defense, aerospace, and healthcare

🧪 4. Calibration Procedure

Step-by-Step:

1. Pre-inspection & Cleaning
   Remove contaminants; inspect for damage.
2. Environmental Conditioning
   Typically performed at 20°C ± 1°C and 45–55% humidity.
3. Reference Standards
   Use gauge blocks, master rings, or traceable artifacts with known uncertainties.
4. Measurement & Comparison
   Record readings at various points. Compare to standard values.
5. Adjustment (if needed)
   Instruments outside tolerance are adjusted or repaired.
6. Uncertainty Calculation
   Total uncertainty includes resolution, repeatability, environment, etc.
7. Certification
   Provide a calibration certificate with full traceability.
8. Labeling & Documentation
   Apply a calibration label and log the certificate in the database.

📊 5. Example Calibration Data (Micrometer)

📉 6. Risks of Non-Calibration

• Product rejections and recalls
• Legal and contractual non-compliance
• Inaccurate reporting of measurement data
• High costs due to rework or over-inspection
• Certification suspension or audit failure

🧱 7. Standards and Accreditation Bodies

🛠️ 8. Calibration Management System (CMS)

Modern industries use digital CMS platforms to:

• Track calibration status and due dates
• Store certificates digitally
• Generate alerts for expired instruments
• Integrate with ERP/QA systems (SAP, Oracle)

📈 9. Case Study Summary

A case study from an automotive supplier showed a 75% drop in dimensional rejections and zero audit non-conformities after implementing a 6-month dimensional calibration schedule using a NABL-accredited lab.

✅ 10. Conclusion & Recommendations

Dimensional mechanical calibration is not just a formality—it’s a foundational practice for quality, compliance, and precision manufacturing. Organizations should:

• Calibrate instruments periodically (every 6–12 months)
• Use only accredited labs
• Maintain traceable records
• Implement preventive maintenance and internal checks

📥 Appendix

• Sample Calibration Certificate Template
• Calibration Frequency Chart
• Calibration SOP (Standard Operating Procedure)
• List of Common Reference Standards

Industrial Application of Dimensional Mechanical Calibration?

Dimensional Mechanical Calibration plays a critical role across diverse industries where accuracy, consistency, and compliance with tolerances are non-negotiable. Here’s how it’s applied in key sectors:

🔩 1. Automotive Industry

📌 Use:

• Verification of piston diameters, gear tooth profiles, crankshaft lengths, and engine block bores.
• Calibration of tools like micrometers, bore gauges, and thread gauges.

🎯 Objective:

• Ensure part interchangeability
• Meet IATF 16949 and OEM quality requirements

✈️ 2. Aerospace and Defense

📌 Use:

• Calibrating height gauges, dial indicators, and surface plates for components like turbine blades, actuators, and fuselage parts.

🎯 Objective:

• Maintain extreme precision (±0.001 mm)
• Comply with AS9100 and military-grade tolerances

⚙️ 3. Precision Engineering & Tooling

📌 Use:

• Calibration of coordinate measuring machines (CMMs), snap gauges, plug/ring gauges, and optical comparators.

🎯 Objective:

• Tool room accuracy for dies, jigs, and fixtures
• Reduce cumulative error in assemblies

🏗️ 4. Construction & Civil Engineering

📌 Use:

• Verifying alignment tools, spirit levels, measuring tapes, and theodolites.

🎯 Objective:

• Maintain measurement integrity in layout, structural alignment, and leveling.

🧪 5. Pharmaceutical & Medical Device Industry

📌 Use:

• Calibration of gauges and calipers used for inspecting syringes, stents, surgical tools, and packaging dimensions.

🎯 Objective:

• Meet regulatory standards like ISO 13485, US FDA, and GMP.

🖨️ 6. 3D Printing & Additive Manufacturing

📌 Use:

• Calibration of measuring arms, surface roughness testers, and calipers used to verify print tolerances.

🎯 Objective:

• Validate printed parts against CAD model dimensions
• Minimize layer deviation and warping issues

⚡ 7. Electronics & Semiconductor Industry

📌 Use:

• Calibration of micrometers and dial indicators used to measure micro-components, PCB dimensions, and socket contacts.

🎯 Objective:

• Ensure miniaturized part conformance and IPC standards compliance

🍶 8. Food & Packaging Industry

📌 Use:

• Verifying fill depth, bottle neck dimensions, cap threading, and packaging alignment.

🎯 Objective:

• Maintain consistency in packaging dimensions
• Avoid mechanical jamming in automatic filling/sealing lines

⚗️ 9. Oil & Gas and Heavy Engineering

📌 Use:

• Calibration of tools for pipe diameters, valve seats, flange thickness, and gasket compression.

🎯 Objective:

• Ensure high-pressure component integrity
• Comply with ASME, API, and offshore standards

🏫 10. Training Institutes & Calibration Labs

📌 Use:

• Practical training using calibrated tools such as micrometers and vernier calipers.

🎯 Objective:

• Prepare students and technicians for industry-standard inspection tasks
• Demonstrate measurement uncertainty and standard procedures

📈 Summary Table:

References

1. ^ Jump up to:^a b Bolster, Diogo; Hershberger, Robert E.; Donnelly, Russell E. (September 2011). “Dynamic similarity, the dimensionless science”. Physics Today. 64 (9): 42–47. Bibcode:2011PhT….64i..42B. doi:10.1063/PT.3.1258.
2. ^ Jump up to:^a b BIPM (2019). “2.3.3 Dimensions of quantities”. SI Brochure: The International System of Units (SI) (PDF) (in English and French) (v. 1.08, 9th ed.). pp. 136–137. ISBN 978-92-822-2272-0. Retrieved 1 September 2021.
3. ^ Yalin, M. Selim (1971). “Principles of the Theory of Dimensions”. Theory of Hydraulic Models. pp. 1–34. doi:10.1007/978-1-349-00245-0_1. ISBN 978-1-349-00247-4.
4. ^ Jump up to:^a b Duff, M.J.; Okun, L.B.; Veneziano, G. (September 2002), “Trialogue on the number of fundamental constants”, Journal of High Energy Physics, 2002 (3): 023, arXiv:physics/0110060, Bibcode:2002JHEP…03..023D, doi:10.1088/1126-6708/2002/03/023, S2CID 15806354
5. ^ JCGM (2012), JCGM 200:2012 – International vocabulary of metrology – Basic and general concepts and associated terms (VIM) (PDF) (3rd ed.), archived from the original (PDF) on 23 September 2015, retrieved 2 June 2015
6. ^ Cimbala, John; Çengel, Yunus (2006). “§7-2 Dimensional homogeneity”. Essential of Fluid Mechanics: Fundamentals and Applications. McGraw-Hill. p. 203–. ISBN 9780073138350.
7. ^ de Jong, Frits J.; Quade, Wilhelm (1967). Dimensional analysis for economists. North Holland. p. 28.
8. ^ Waite, Lee; Fine, Jerry (2007). Applied Biofluid Mechanics. New York: McGraw-Hill. p. 260. ISBN 978-0-07-147217-3.
9. ^ Macagno, Enzo O. (1971). “Historico-critical review of dimensional analysis”. Journal of the Franklin Institute. 292 (6): 391–340. doi:10.1016/0016-0032(71)90160-8.
10. ^ Jump up to:^a b Martins, Roberto De A. (1981). “The origin of dimensional analysis”. Journal of the Franklin Institute. 311 (5): 331–337. doi:10.1016/0016-0032(81)90475-0.
11. ^ Martins, p. 403 in the Proceedings book containing his article
12. ^ Mason, Stephen Finney (1962), A history of the sciences, New York: Collier Books, p. 169, ISBN 978-0-02-093400-4
13. ^ Roche, John J (1998), The Mathematics of Measurement: A Critical History, Springer, p. 203, ISBN 978-0-387-91581-4, Beginning apparently with Maxwell, mass, length and time began to be interpreted as having a privileged fundamental character and all other quantities as derivative, not merely with respect to measurement, but with respect to their physical status as well.
14. ^ Mitchell, Daniel Jon (2017). “Making sense of absolute measurement: James Clerk Maxwell, William Thomson, Fleeming Jenkin, and the invention of the dimensional formula”. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 58: 63–79. doi:10.1016/j.shpsb.2016.08.004.
15. ^ Maxwell, James Clerk (1873), A Treatise on Electricity and Magnetism, p. 4
16. ^ Maxwell, James Clerk (1873), A Treatise on Electricity and Magnetism, Clarendon Press series, Oxford, p. 45, hdl:2027/uc1.l0065867749
17. ^ (Pesic 2005)
18. ^ Rayleigh, Baron John William Strutt (1877), The Theory of Sound, Macmillan
19. ^ Fourier (1822), p. 156.
20. ^ Maxwell, James Clerk (1873), A Treatise on Electricity and Magnetism, volume 1, p. 5
21. ^ Ramsay, Angus. “Dimensional Analysis and Numerical Experiments for a Rotating Disc”. Ramsay Maunder Associates. Retrieved 15 April 2017.
22. ^ Tao 2012, “With a bit of additional effort (and taking full advantage of the one-dimensionality of the vector spaces), one can also define spaces with fractional exponents …”.
23. ^ Tao 2012, “However, when working with vector-valued quantities in two and higher dimensions, there are representation-theoretic obstructions to taking arbitrary fractional powers of units …”.
24. ^ Tao 2012 “Similarly, one can define V^T−1 as the dual space to V^T …”
25. ^ Bridgman 1922, 2. Dimensional Formulas pp. 17–27
26. ^ Berberan-Santos, Mário N.; Pogliani, Lionello (1999). “Two alternative derivations of Bridgman’s theorem” (PDF). Journal of Mathematical Chemistry. 26: 255–261, See §5 General Results p. 259. doi:10.1023/A:1019102415633. S2CID 14833238.
27. ^ Berberan-Santos & Pogliani 1999, p. 256
28. ^ For a review of the different conventions in use see: Pisanty, E (17 September 2013). “Square bracket notation for dimensions and units: usage and conventions”. Physics Stack Exchange. Retrieved 15 July 2014.
29. ^ Jump up to:^a b Thompson, Ambler (November 2009). Guide for the Use of the International System of Units (SI): The Metric System (PDF). DIANE Publishing. ISBN 9781437915594.
30. ^ Duff, Michael James (July 2004). “Comment on time-variation of fundamental constants”. arXiv:hep-th/0208093v3.
31. ^ Woan, G. (2010), The Cambridge Handbook of Physics Formulas, Cambridge University Press, ISBN 978-0-521-57507-2
32. ^ Mosca, Gene; Tipler, Paul Allen (2007), Physics for Scientists and Engineers – with Modern Physics (6th ed.), San Francisco: W. H. Freeman, ISBN 978-0-7167-8964-2
33. ^ Martin, B.R.; Shaw, G.; Manchester Physics (2008), Particle Physics (2nd ed.), Wiley, ISBN 978-0-470-03294-7
34. ^ Gehani, N. (1977). “Units of measure as a data attribute”. Comput. Lang. 2 (3): 93–111. doi:10.1016/0096-0551(77)90010-8.
35. ^ Gehani, N. (June 1985). “Ada’s derived types and units of measure”. Software: Practice and Experience. 15 (6): 555–569. doi:10.1002/spe.4380150604. S2CID 40558757.
36. ^ Cmelik, R. F.; Gehani, N. H. (May 1988). “Dimensional analysis with C++”. IEEE Software. 5 (3): 21–27. doi:10.1109/52.2021. S2CID 22450087.
37. ^ Kennedy, Andrew J. (April 1996). Programming languages and dimensions (Phd). Vol. 391. University of Cambridge. ISSN 1476-2986. UCAM-CL-TR-391.
38. ^ Kennedy, A. (2010). “Types for Units-of-Measure: Theory and Practice”. In Horváth, Z.; Plasmeijer, R.; Zsók, V. (eds.). Central European Functional Programming School. CEFP 2009. Lecture Notes in Computer Science. Vol. 6299. Springer. pp. 268–305. CiteSeerX 10.1.1.174.6901. doi:10.1007/978-3-642-17685-2_8. ISBN 978-3-642-17684-5.
39. ^ Gundry, Adam (December 2015). “A typechecker plugin for units of measure: domain-specific constraint solving in GHC Haskell” (PDF). SIGPLAN Notices. 50 (12): 11–22. doi:10.1145/2887747.2804305. Archived (PDF) from the original on 10 August 2017.
40. ^ Garrigue, J.; Ly, D. (2017). “Des unités dans le typeur” (PDF). 28ièmes Journées Francophones des Langaeges Applicatifs, Jan 2017, Gourette, France (in French). hal-01503084. Archived (PDF) from the original on 10 November 2020.
41. ^ Teller, David (January 2020). “Units of Measure in Rust with Refinement Types”.
42. ^ Grecco, Hernan E. (2022). “Pint: makes units easy”.
43. ^ “CamFort: Specify, verify, and refactor Fortran code”. University of Cambridge; University of Kent. 2018.
44. ^ Bennich-Björkman, O.; McKeever, S. (2018). “The next 700 unit of measurement checkers”. Proceedings of the 11th ACM SIGPLAN International Conference on Software Language Engineering. pp. 121–132. doi:10.1145/3276604.3276613. ISBN 978-1-4503-6029-6. S2CID 53089559.
45. ^ Hart 1995
46. ^ Griffioen, P. (2019). A unit-aware matrix language and its application in control and auditing (PDF) (Thesis). University of Amsterdam. hdl:11245.1/fd7be191-700f-4468-a329-4c8ecd9007ba. Archived (PDF) from the original on 21 February 2020.
47. ^ McBride, Conor; Nordvall-Forsberg, Fredrik (2022). “Type systems for programs respecting dimensions” (PDF). Advanced Mathematical and Computational Tools in Metrology and Testing XII. Advances in Mathematics for Applied Sciences. World Scientific. pp. 331–345. doi:10.1142/9789811242380_0020. ISBN 9789811242380. S2CID 243831207. Archived (PDF) from the original on 17 May 2022.
48. ^ “NondimensionalizationTransform—Wolfram Language Documentation”. reference.wolfram.com. Retrieved 19 April 2023.
49. ^ “UnitDimensions—Wolfram Language Documentation”. reference.wolfram.com. Retrieved 19 April 2023.
50. ^ “DimensionalCombinations—Wolfram Language Documentation”. reference.wolfram.com. Retrieved 19 April 2023.
51. ^ Jump up to:^a b “UnityDimensions—Wolfram Language Documentation”. reference.wolfram.com. Retrieved 19 April 2023.
52. ^ “QuantityVariableDimensions—Wolfram Language Documentation”. reference.wolfram.com. Retrieved 19 April 2023.
53. ^ (Huntley 1967)
54. ^ Quincey, Paul (2021). “Angles in the SI: a detailed proposal for solving the problem”. Metrologia. 58 (5): 053002. arXiv:2108.05704. Bibcode:2021Metro..58e3002Q. doi:10.1088/1681-7575/ac023f.
55. ^ Siano (1985-I, 1985-II)
56.  JCGM 200:2008 International vocabulary of metrology Archived 2019-10-31 at the Wayback Machine — Basic and general concepts and associated terms (VIM)
57. ^ Jump up to:^a b ISO 9001: “Quality management systems — Requirements” (2008), section 7.6.
58. ^ Jump up to:^a b ISO 17025: “General requirements for the competence of testing and calibration laboratories” (2005), section 5.
59. ^ Faison, C. Douglas; Brickenkamp, Carroll S. (March 2004). “Calibration Laboratories: Technical Guide for Mechanical Measurements” (PDF). NIST Handbook 150-2G. NIST. Archived from the original (PDF) on 12 May 2015. Retrieved 14 June 2015.
60. ^ “Metrology, Pressure, Thermal & Eletrotechnical Measurement and Calibration”. Fluid Control Research Institute (FCRI), Ministry of Heavy Industries & Public Enterprises, Govt. of India. Archived from the original on 14 June 2015. Retrieved 14 June 2015.
61. ^ Haider, Syed Imtiaz; Asif, Syed Erfan (16 February 2011). Quality Control Training Manual: Comprehensive Training Guide for API, Finished Pharmaceutical and Biotechnologies Laboratories. CRC Press. p. 49. ISBN 978-1-4398-4994-1.
62. ^ Bare, Allen (2006). Simplified Calibration Interval Analysis (PDF). Aiken, SC: NCSL International Workshop and Symposium, under contract with the Office of Scientific and Technical Information, U.S. Department of Energy. pp. 1–2. Archived (PDF) from the original on 2007-04-18. Retrieved 28 November 2014.
63. ^ “ANSI/NCSL Z540.3-2006 (R2013)”. The National Conference of Standards Laboratories (NCSL) International. Archived from the original on 2014-11-20. Retrieved 28 November 2014.
64. ^ “Calibration Systems Requirements (Military Standard)” (PDF). Washington, DC: U.S. Department of Defense. 1 August 1998. Archived from the original (PDF) on 2005-10-30. Retrieved 28 November 2014.
65. ^ Jump up to:^a b Ligowski, M.; Jabłoński, Ryszard; Tabe, M. (2011), Jabłoński, Ryszard; Březina, Tomaš (eds.), Procedure for Calibrating Kelvin Probe Force Microscope, Mechatronics: Recent Technological and Scientific Advances, p. 227, doi:10.1007/978-3-642-23244-2, ISBN 978-3-642-23244-2, LCCN 2011935381
66. ^ Military Handbook: Evaluation of Contractor’s Calibration System (PDF). U.S. Department of Defense. 17 August 1984. p. 7. Archived (PDF) from the original on 2014-12-04. Retrieved 28 November 2014.
67. ^ Procedure for calibrating pressure gauges (USBR 1040) (PDF). U.S. Department of the Interior, Bureau of Reclamation. pp. 70–73. Archived (PDF) from the original on 2013-05-12. Retrieved 28 November 2014.
68. ^ “KNC Model 3666 Automatic Pressure Calibration System” (PDF). King Nutronics Corporation. Archived from the original (PDF) on 2014-12-04. Retrieved 28 November 2014.
69. ^ “the definition of calibrate”. Dictionary.com. Retrieved 18 March 2018.
70. ^ Baber, Zaheer (1996). The Science of Empire: Scientific Knowledge, Civilization, and Colonial Rule in India. SUNY Press. pp. 23–24. ISBN 978-0-7914-2919-8.
71. ^ Franceschini, Fiorenzo; Galetto, Maurizio; Maisano, Domenico; Mastrogiacomo, Luca; Pralio, Barbara (6 June 2011). Distributed Large-Scale Dimensional Metrology: New Insights. Springer Science & Business Media. pp. 117–118. ISBN 978-0-85729-543-9.
72. ^ Ackroyd, Peter (16 October 2012). Foundation: The History of England from Its Earliest Beginnings to the Tudors. St. Martin’s Press. pp. 133–134. ISBN 978-1-250-01367-5.
73. ^ Bland, Alfred Edward; Tawney, Richard Henry (1919). English Economic History: Select Documents. Macmillan Company. pp. 154–155.
74. ^ Tilford, Charles R (1992). “Pressure and vacuum measurements” (PDF). Physical Methods of Chemistry: 106–173. Archived from the original (PDF) on 2014-12-05. Retrieved 28 November 2014.
75. ^ Fridman, A. E.; Sabak, Andrew; Makinen, Paul (23 November 2011). The Quality of Measurements: A Metrological Reference. Springer Science & Business Media. pp. 10–11. ISBN 978-1-4614-1478-0.
76. ^ Cuscó, Laurence (1998). Guide to the Measurement of Pressure and Vacuum. London: The Institute of Measurement and Control. p. 5. ISBN 0-904457-29-X.