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Gears by the Numbers

Ratios, involute geometry, centre distance, backlash, strength and measurement for spur and helical gears

Work through one spur gear pair from ratio and geometry to contact ratio, backlash, tooth strength, contact stress and the data a gear drawing needs.

10 chapters · About 100 minutes · Free to read · Intermediate mechanical design

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BIZOMAT LibraryENGINEERINGGears by theNumbersBIZOMAT ORIGINAL BOOK10 CHAPTERS

About this book

Most machines buy their gears inside a gearbox, and most gearbox decisions are about catalogue ratings. Sooner or later, though, a designer has to draw a pair of gears, a machinist has to cut or check one, or a maintenance engineer has to decide whether a worn pair can be replaced with a different supplier’s parts. At that point, the numbers inside the gear matter: how module and tooth count set the size, why the centre distance must match, where backlash comes from, how strong the teeth are and how to measure what was made.

This book works through those numbers in a practical order. It follows one fictional design throughout: a single-stage spur reduction for a small conveyor drive, with a 3 kW motor turning at 1,440 rpm and a 20-tooth pinion driving a 60-tooth gear of module 3 mm with a 20° pressure angle. The drive, its loads and its results are invented for teaching. They are not BIZOMAT projects or design approvals.

Every calculation is shown in full and was checked independently. The strength checks in Chapters 8 and 9 are screening methods with explicit simplifications. They show how load, geometry and speed drive tooth stresses; they do not replace a gear rating to a recognised standard or a supplier’s application review for a real drive.

An original quantitative companion

The Library already covers selecting gearboxes and geared motors, bearings, shafts and couplings in the Machine Elements and Mechanical Design collection. This book is the geometry and calculation companion for anyone who needs to work inside the gear rather than choose the box around it.

The explanations, the conveyor case and all worked examples are original BIZOMAT writing, derived from standard involute relationships. Machinery’s Handbook’s gearing coverage helped identify the gap, and published references are cited for the strength and measurement methods. No tables, figures or text from any source are reproduced. Use your gear supplier’s data and the applicable standards for real designs.

Who this book is for

Mechanical designers, drafters, machinists, maintenance engineers and technical buyers who need to size, check, specify or measure a spur or helical gear pair rather than only select a packaged gearbox.

What you will learn

  • Calculate gear ratios, speeds and torques for simple and compound gear trains.
  • Size a gear pair from module and tooth numbers, and calculate pitch, base and tip diameters, centre distance and contact ratio.
  • Explain undercut, profile shift and backlash, and specify tooth thickness so a pair runs with controlled clearance.
  • Screen tooth bending and surface contact stress with the Lewis and Hertz relationships, knowing their limits.
  • Write the data block a gear drawing needs and check a gear with span measurement.

Inside this book

Chapters

  1. 01Ratios, speeds and gear trainsCalculate gear ratio, output speed and torque for simple and compound trains, and use tooth numbers to spread wear with a hunting tooth.
  2. 02The involute and the pressure angleSee why involute teeth keep a constant speed ratio, how the base circle and line of action are defined, and what the pressure angle does to tooth forces.
  3. 03Module, pitch and gear sizesUse module and tooth number to calculate pitch, tip and root diameters, circular pitch and tooth thickness, and avoid mixing module and diametral pitch gears.
  4. 04Centre distance and contact ratioCalculate the standard centre distance and the transverse contact ratio, and see how a centre distance error changes the pressure angle and overlap.
  5. 05Undercut and profile shiftSee why pinions with few teeth are undercut, calculate the minimum profile shift that prevents it, and find the new centre distance when shifts are added.
  6. 06Backlash and tooth thicknessSpecify backlash through tooth thickness and centre distance tolerances, and check that temperature and errors cannot close the clearance.
  7. 07Helical gears and axial thrustConvert between normal and transverse module, size a helical pair to a centre distance, and calculate the axial thrust and face overlap the helix creates.
  8. 08Tooth bending strengthScreen tooth root bending stress with the Lewis equation and a velocity factor, and see why modern rating methods add many more factors.
  9. 09Surface contact stressEstimate Hertzian contact stress between gear teeth, see how face width and size change it, and connect it to materials, hardening and lubrication.
  10. 10Measuring and specifying a gearCheck tooth thickness with span measurement, identify an unknown gear from its tooth count and diameter, and write the data a gear drawing needs.

Related Tools and Examples

References and source notes

The manuscript and worked examples are original BIZOMAT writing. References inform particular concepts; their authors do not endorse this book. Web references were consulted on 23 October 2026.

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