Mechanical Engineering Metrology and Measurements (MEMM)¶

TLDR; Uncertainty¶

Every measurement has doubt. Error is not doubt. Repeat the test to get Type A. Use certificates and limits to get Type B. Combine them. Report the result with the doubt and the confidence.

Do not start a measurement if you do not have a plan to quantify the uncertainty. Without a plan, the result has no meaning. See steps 1 through 3 below.

Report every result in this form:

\[ L = \bar{L} \pm U \;\; \mathrm{(units, \; 95\% \; confidence)} \]

Example:

\[ L = 21.49 \pm 0.05 \; \mathrm{mm\;(k=2,\;95\%)} \]

Where the terms are:

\(\bar{L}\) is the mean, the best estimate
\(U\) is the expanded uncertainty, the range of doubt
\(k=2\) is the coverage factor for 95% confidence

Get Type A from repeated readings:

\[ s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}}, \qquad u_A = \frac{s}{\sqrt{n}} \]

Example: \(n=24\), \(\bar{x}=21.493\;\mathrm{mm}\), \(u_A = 0.021\;\mathrm{mm}\).

Get Type B from other information, for rectangular limits \(\pm a\):

\[ u_B = \frac{a}{\sqrt{3}} \]

Example: \(a = 0.02\;\mathrm{mm}\), \(u_B = 0.012\;\mathrm{mm}\).

Combine them, uncorrelated:

\[ u_c = \sqrt{u_A^2 + u_B^2} \]

Example: \(u_c = \sqrt{0.021^2 + 0.012^2} = 0.024\;\mathrm{mm}\).

Expand to 95% confidence:

\[ U = k \cdot u_c, \qquad k=2 \]

Example: \(U = 2 \times 0.024 \approx 0.05\;\mathrm{mm}\).

For a result \(y = f(x_1, x_2, \dots)\):

\[ u_{c}(y) = \sqrt{\sum_{i=1}^{N}\left(u_{x_i}\frac{\partial y}{\partial x_i}\right)^2} \]

Example: \(Q = V/t\), \(V = 1.15 \pm 0.05\;\mathrm{L}\), \(t = 33.0 \pm 0.1\;\mathrm{s}\), then \(Q = 2.09 \pm 0.09\;\mathrm{L/min\;(95\%)}\). Volume causes almost all the doubt. Improve volume first.

Rules:

Do not start a measurement if you do not have a plan to quantify the uncertainty. Without a plan, the result has no meaning.
Record the date, the name, the instrument, and the temperature.
Repeat the measurement.
Calculate Type A from data.
Get Type B from certificates and limits.
Combine the values.
Report the result, the uncertainty, \(k\), and the confidence.

An open-source book of interactive notebooks and worked examples, prepared by Prof. Alex Liberzon, School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, for the course called in many places "Mechanical Measurements Lab 1", "Theory and Design of Mechanical Measurements", or "Introduction to Measurements for Mechanical Engineers".

This book does not replace the course materials but organizes them into a readable sequence. We hope it is useful as learning material for undergraduate engineering laboratory courses. It is an open-source project, and any contribution is welcome (contact on GitHub).

How this book is organised¶

The chapter sequence and the topic grouping follow a widely used structure for undergraduate instrumentation courses. We are grateful to Adam W. Duran for permitting the reuse of that structure and for the reference material that shaped it:

MEGN 300: Instrumentation & Automation course materials by Adam W. Duran, Colorado School of Mines, used under CC BY 4.0. Source: https://github.com/professor-duran/MEGN300

All content, code, and worked examples in this book are our own; the attribution above covers the course structure and sequencing only.

Textbook and relevant books¶

This course follows the (Figliola & Beasley, 2020). It is also recommended to consult (Dunn & Davis, 2017) and (Wheeler & Ganji, 2009).

This book collects practical notebooks and concise explanations to teach core concepts in mechanical engineering metrology and measurements. Content emphasizes hands-on examples, reproducible analyses, and problem-solving skills appropriate for undergraduate laboratory and lecture use.

The Central Theme: Understanding and Managing Measurement Uncertainty¶

Every measurement has uncertainty — whether you quantify it or not. This book teaches you to:

Understand where uncertainty comes from — in your instruments, your data collection, your analysis
Quantify it rigorously — using statistics, error budgets, and propagation methods
Reduce it where possible — through careful calibration and experimental design
Report it correctly — so others know how much to trust your results

This is the journey of a metrologist: Metrology and Uncertainty → Statistics → Calibration → Real-world Systems → Uncertainty Budget → Informed Reporting.

How This Book Is Organized: A Learning Pathway¶

Each chapter builds on the previous one, all focused on the central goal of managing measurement uncertainty:

Chapter	Purpose	Contributes to Understanding
Metrology and Uncertainty	Foundational concepts: where errors come from, how uncertainty is defined	What is uncertainty?
Statistics	Tools to quantify data variability, detect outliers, compute confidence intervals	How do I measure uncertainty from data?
Calibration	Use statistics to characterize instruments, reduce systematic errors, build calibration curves	How do I reduce and control uncertainty in my instruments?
Dynamic Signals	Understand how measurement systems respond to changes; recognize when you're measuring the system, not the quantity	What affects the fidelity of my measurement?
Signal Processing	Separate true signals from noise using frequency-domain tools; understand sampling and filtering trade-offs	How do I extract clean measurements from noisy data?
Analog vs Digital	Understand how analog signals are digitized; manage aliasing and quantization effects	How do I capture measurements without introducing new errors?

Key Insights You'll Build¶

After Metrology and Uncertainty: You understand what uncertainty is, where it comes from, and that every measurement has a "confidence zone" around it
After Statistics: You can quantify uncertainty from real data using Type A (statistical) methods and recognize when your sample size is adequate
After Calibration: You can characterize and reduce systematic errors in instruments, and you understand Type B (systematic) uncertainty sources
After Dynamic Signals & Signal Processing: You understand that your measurement system itself affects the result — bandwidth, frequency response, and noise filtering all matter
After Analog vs Digital: You know how to capture signals digitally without losing information or introducing aliasing artifacts
Capstone: You can plan a complete measurement workflow: identify uncertainty sources → design an experiment → collect data → analyze with appropriate statistical rigor → propagate and report uncertainty

Learning objectives¶

By the end of this course/readings, students will be able to: - Explain fundamental measurement concepts: accuracy, precision, resolution, and uncertainty. - Apply statistical tools to analyze measurement data (distributions, confidence intervals, t‑tests, outlier detection). - Perform calibration and regression analysis for common sensors and instruments. - Analyze dynamic signals using time‑domain and frequency‑domain methods (FFT, windowing, spectral interpretation). - Understand sampling, aliasing, and basic reconstruction for A/D systems. - Model simple measurement systems (first and second order) and interpret step/impulse responses. - Propagate measurement uncertainty (analytical and Monte Carlo) and report results following good practice. - Implement reproducible experiments and analyses using Python and Jupyter notebooks.

Recommended prerequisites¶

Students should be comfortable with: - Calculus and basic differential equations - Linear algebra (vectors, matrices) - Introductory probability and statistics - Basics of signals and systems (sinusoids, frequency, convolution helpful but not required) - Basic Python programming (variables, functions, NumPy arrays) - Familiarity with Jupyter notebooks and command-line usage is helpful

How to use this book¶

Reading order: work through the chapters in the order they appear in the sidebar. Each chapter opens with a page explaining what it covers, what you should already know, and the reasoning behind its sequence — start there rather than jumping into a notebook.

Every chapter begins with an introduction page: learning objectives, key concepts, prerequisites, a suggested reading path, and a note on what the chapter enables you to do next.
Pages are numbered notebooks with prose between them. Read the prose, then run the notebook; the figures are regenerated when the book is built, so what you see is what the code actually produces.
Many pages carry Open in molab, View on GitHub, and Download .py buttons, so you can pick up any example as a live, editable notebook.
Interactive pages have a slider or switch you can move; the figure updates as you drag. These are worth exploring rather than reading.
If a concept feels under-explained, the "Ordered reading" list at the end of each chapter introduction points to the pages that develop it in depth.

Quick environment notes¶

Python 3.11 or newer, with NumPy, SciPy, Matplotlib, pandas, and SymPy. See the README for installation and for how to build the book locally.

Table of contents¶

Copyright and attribution¶

Content in this book is released under CC0 1.0 Universal.

To the extent possible under law, the person who associated CC0 with this work has waived all copyright and related or neighboring rights to this work.

The chapter structure and topic sequencing follow MEGN 300: Instrumentation & Automation course materials by Adam W. Duran, Colorado School of Mines, used under CC BY 4.0.

References¶

Dunn, P., & Davis, M. (2017). Measurement and Data Analysis for Engineering and Science. CRC Press.
Figliola, R., & Beasley, D. (2020). Theory and Design for Mechanical Measurements. Wiley.
Wheeler, A., & Ganji, A. (2009). Introduction to Engineering Experimentation. Prentice Hall.