Mechanical Engineering Metrology and Measurements (MEMM)

Set of Jupyter notebooks, prepared by Prof. Alex Liberzon, School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University for the course that is called in many places as “Mechanical Measurements Lab 1” or “Theory and Design of Mechanical Measurements”, “Introduction to Measurements for Mechanical Engineers”, etc.

This book does not replace the course materials but rather organizes them in Jupyter and Markdown notebooks. We hope it is useful as an assistance learning material for undergraduate engineering laboratory courses. It is an open-source project, and any contribution is welcome ( contact on Github ).

Textbook and relevant books:¶

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

Metrology & Measurements — Course Introduction¶

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.

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¶

• Navigate chapters via the table of contents. Each chapter contains short explanatory pages and runnable notebooks for labs and examples.

• Do the notebooks interactively: create a local virtual environment, install requirements, and run the notebooks in Jupyter Lab/Notebook.

• Work through the “unsorted” and “archive” content only after core topics; many items are homework examples or experimental notes.

• Instructors: adopt notebooks as lab exercises, add assessment items, and redistribute with solutions for guided learning.

Quick environment notes¶

Recommended Python ecosystem: Python 3.9+, NumPy, SciPy, Matplotlib, pandas, jupyter-book, myst-nb. Add a pinned requirements.txt in the repo root for reproducible builds.

Start with the “theory”, “statistics”, and “a2d” chapters, then proceed to “signal_processing”, “dynamic_signals”, and “calibration” for lab work and examples.

Table of contents¶

• Mechanical Engineering Metrology and Measurements
• Metrology theory — Introduction and Learning Goals
  • Laboratory Notebook
  • Significant digits
  • Short summary of the ``Measurement good practice guide ‘’ by NPL
  • Standardization and Traceability
  • General measurement system diagram
  • Uncertainty 101
  • The Engineer’s 9-Step Uncertainty Analysis Checklist
  • Using simulations to explain uncertainty
  • Basic error analysis
  • Engineering Example: Uncertainty Analysis in Mechanical Measurements
  • How to estimate the uncertainty of a slope for static calibration or regression
  • Propagating uncertainty using Monte-Carlo simulations
  • Example: Volume Measurement Uncertainty Budget
  • 1. Problem Statement: Volume of a Cylinder
  • Full Uncertainty Budget of a Hot-Wire Anemometer for Isothermal Turbulent Air Flow in a Wind Tunnel
• Calibration — Introduction and Learning Goals
  • Simulation of calibration errors
  • Sensitivity estimate example
  • Introduction to linear regression
  • Lecture 6
  • Hysteresis and log-log calibration example
  • Linearity error example
  • Hysteresis example
  • Hysteresis and regression analysis example
  • Calibration of non-linear relations
  • Example of Weighing Scale Calibration Analysis
  • Static Calibration Errors: Engineering Testing & Measurements
  • Calibration and uncertainty analysis - virtual experiment
  • Simulated Static Calibration Data
  • LVDT example
  • Micrometer calibration using gage block
  • Measurement Uncertainty Budget Examples (SWGDRUG SD-3)
• Statistics — Introduction and Learning Goals
  • Very basic review of some statistics terms
  • Getting started with measurement uncertainty
  • Lecture 5 - probability and statistics
  • Probability Distributions and the Central Limit Theorem
  • Histogram To Distribution
  • Statistical distribution
  • “Student” t-distribution
  • t-test
  • Statistics example
  • Statistical parameters using probability density function
  • Outliers
  • Outliers
  • Outliers example 2
  • Outliers
  • Central limit theorem (or why errors often look Gaussian)
• Dynamic Signals — Introduction and Learning Goals
  • 1st order dynamic system
  • Step responses of dynamical systems
  • Log decrement method
  • Lab Worksheet: Measurement of Unknown Mass using Vibrating Beam
  • Example of pre-measurement design
  • Fast Fourier Transform (FFT) of a sum of two sine signals
  • Symbolic evaluation of Fourier series coefficients
  • Plot power spectrum of experimental data
  • Spectrum example
• Analog-to-Digital (A2D) — Introduction and Learning Goals
  • Mimic A/D conversion
  • Sampling, clipping and aliasing
  • Digital to Analog conversion using sinc
  • Analog to digital (A/D) and Digital to Analog (d/A) conversion example
  • Signal construction and helper functions for the frequency content class
• Signal Processing — Introduction and Learning Goals
  • Using FFT the right way to find the correct spectrum
  • Symbolic evaluation of Fourier coefficients
  • Fourier transforms
  • Digitizing the signal using pure Python
  • Fourier transforms with windowing
  • FFT demo of a real, periodic signal
  • Using FFT-based filter to reduce noise

Copyright Information¶

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