Editing Openai/68fceef0-7248-8013-bf5d-581461dec5bc (section)

=== Assistant: Great—here’s a tight, source-backed checklist matching every calc you asked for. ===
Great—here’s a tight, source-backed checklist matching every calc you asked for. I’m using full URLs (for your records).

1) DP / Module sizing → pinion pitch diameter & radius
* DP definition and relation DP=z/d\text{DP} = z/dDP=z/d (inch); radius from pitch dia. https://khkgears.net/new/gear_knowledge/gear-nomenclature/diametral-pitch.html KHK Gears<ref>{{cite web|title=KHK Gears|url=https://khkgears.net/new/gear_knowledge/gear-nomenclature/diametral-pitch.html|publisher=KHK Gears|access-date=2025-11-15}}</ref>
* “Module” fundamentals and standard equations (incl. circular pitch ties). https://khkgears.net/new/gear_knowledge/abcs_of_gears-b/basic_gear_terminology_calculation.html KHK Gears<ref>{{cite web|title=KHK Gears|url=https://khkgears.net/new/gear_knowledge/abcs_of_gears-b/basic_gear_terminology_calculation.html|publisher=KHK Gears|access-date=2025-11-15}}</ref>
* DP ↔ Module conversion and p=πmp=\pi mp=πm statement (metric). https://sdp-si.com/resources/elements-of-metric-gear-technology/index.php sdp-si.com<ref>{{cite web|title=sdp-si.com|url=https://sdp-si.com/resources/elements-of-metric-gear-technology/index.php|publisher=sdp-si.com|access-date=2025-11-15}}</ref>

2) Circular pitch and rack tooth count over a length
* Circular pitch definition; use Nrack=⌊L/p⌋N_{\text{rack}}=\lfloor L/p\rfloorNrack=⌊L/p⌋ since tooth spacing = circular pitch. https://shop.sdp-si.com/products/gears-differentials-pinions-racks/spur-gear-technical-information.html shop.sdp-si.com<ref>{{cite web|title=shop.sdp-si.com|url=https://shop.sdp-si.com/products/gears-differentials-pinions-racks/spur-gear-technical-information.html|publisher=shop.sdp-si.com|access-date=2025-11-15}}</ref>

3) Torque ↔ linear force at a drum/pinion (use F=τ/rF = \tau/rF=τ/r)
* Standard torque relationship (torque as r×Fr\times Fr×F); rearrange to F=τ/rF=\tau/rF=τ/r. https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_7B_-_General_Physics/7%3A_Momentum/7.5%3A_Torque Physics LibreTexts<ref>{{cite web|title=Physics LibreTexts|url=https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_7B_-_General_Physics/7%3A_Momentum/7.5%3A_Torque|publisher=Physics LibreTexts|access-date=2025-11-15}}</ref> https://en.wikipedia.org/wiki/Torque Wikipedia<ref>{{cite web|title=Wikipedia|url=https://en.wikipedia.org/wiki/Torque|publisher=en.wikipedia.org|access-date=2025-11-15}}</ref>

4) Static breakaway vs running friction (need a starting torque threshold)
* SKF guidance on starting torque and running friction/power loss in bearings. https://www.skf.com/group/products/rolling-bearings/principles-of-rolling-bearing-selection/bearing-selection-process/operating-temperature-and-speed/bearing-friction-power-loss-and-starting-torque?Information=true SKF<ref>{{cite web|title=SKF|url=https://www.skf.com/group/products/rolling-bearings/principles-of-rolling-bearing-selection/bearing-selection-process/operating-temperature-and-speed/bearing-friction-power-loss-and-starting-torque?Information=true|publisher=skf.com|access-date=2025-11-15}}</ref>
* SKF frictional moment model components (what contributes to τs\tau_sτs). https://cdn.skfmediahub.skf.com/api/public/0901d1968065e9e7/pdf_preview_medium/0901d1968065e9e7_pdf_preview_medium.pdf SKF Media Hub<ref>{{cite web|title=SKF Media Hub|url=https://cdn.skfmediahub.skf.com/api/public/0901d1968065e9e7/pdf_preview_medium/0901d1968065e9e7_pdf_preview_medium.pdf|publisher=SKF Media Hub|access-date=2025-11-15}}</ref>

5) Angle / projection “loss” (% based on cosine)
* Component of a force along an axis uses cosine of the deviation angle (our 1−cos⁡θ1-\cos\theta1−cosθ presentation is a readable way to show the penalty). https://www.physicsclassroom.com/class/vectors/Lesson-3/Inclined-Planes The Physics Classroom<ref>{{cite web|title=The Physics Classroom|url=https://www.physicsclassroom.com/class/vectors/Lesson-3/Inclined-Planes|publisher=The Physics Classroom|access-date=2025-11-15}}</ref> https://openstax.org/books/physics/pages/5-4-inclined-planes OpenStax<ref>{{cite web|title=OpenStax|url=https://openstax.org/books/physics/pages/5-4-inclined-planes|publisher=openstax.org|access-date=2025-11-15}}</ref>

6) Running efficiency as a simple multiplicative factor
* Backed by bearing/friction losses compounding through stages (we collapse to ηtopηbot\eta_{top}\eta_{bot}ηtopηbot). https://www.skf.com/group/products/rolling-bearings/principles-of-rolling-bearing-selection/bearing-selection-process/operating-temperature-and-speed/bearing-friction-power-loss-and-starting-torque?Information=true SKF<ref>{{cite web|title=SKF|url=https://www.skf.com/group/products/rolling-bearings/principles-of-rolling-bearing-selection/bearing-selection-process/operating-temperature-and-speed/bearing-friction-power-loss-and-starting-torque?Information=true|publisher=skf.com|access-date=2025-11-15}}</ref>

If you want, I can paste a tiny “references” block (these URLs) for the site’s “Verify” links exactly next to each result section.