1. Torque and RPM of Gears
𝑅𝑃𝑀 2 𝑅𝑃𝑀 1 = 𝑁 1 𝑁 2 = 𝑑 1 𝑑 2 = 𝑇 1 𝑇 2

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3. Torque
Torque is the moment applied about the longitudinal axis of a body (typically called a shaft).
Free Body Diagram of the Gear: x y
What is Rx for loading shown? P d r T
Rx = 0 Rx F Ry
What if another Load, P, was applied?
To keep the system in equilibrium, a resisting torque is applied to the shaft.
Rx = -P
Shaft torque constrains rotation
Bearing constrains translation

4. Gears
Toothed members that transmit rotary motion from one shaft to another.
The servos used with the Arduino contain a series of spur gears which transfer motion between parallel shafts.
Pitch diameter is the effective diameter of the gear.
If gear 1 has 30 teeth and gear 2 has 10 teeth, then the upper gear will turn ____ times when the lower gear turns ____ times.
If gear 1 turns one revolution per minute and gear 2 turns three revolutions per minute, can you develop a relationship between revolutions per minute and gear teeth? d1
GEAR 1
GEAR 2 1 d2 3
3 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠 𝑝𝑒𝑟 𝑚𝑖𝑛𝑢𝑡𝑒 1 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑝𝑒𝑟 𝑚𝑖𝑛𝑢𝑡𝑒 = 30 𝑡𝑒𝑒𝑡ℎ 10 𝑡𝑒𝑒𝑡ℎ
𝐺𝑒𝑎𝑟 2 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠 𝑝𝑒𝑟 𝑚𝑖𝑛𝑢𝑡𝑒 𝐺𝑒𝑎𝑟 1 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠 𝑝𝑒𝑟 𝑚𝑖𝑛𝑢𝑡𝑒 = 𝑇𝑒𝑒𝑡ℎ 𝑜𝑓 𝐺𝑒𝑎𝑟 1 𝑇𝑒𝑒𝑡ℎ 𝑜𝑓 𝐺𝑒𝑎𝑟 2
𝑅𝑃𝑀 2 𝑅𝑃𝑀 1 = # 𝑜𝑓 𝑡𝑒𝑒𝑡ℎ 𝑖𝑛 𝑔𝑒𝑎𝑟 1 # 𝑜𝑓 𝑡𝑒𝑒𝑡ℎ 𝑖𝑛 𝑔𝑒𝑎𝑟 2 = 𝑑 1 𝑑 2

5. Gear Analysis T2 T1 F F Rx2 Ry2 x y d1
Every force has an equal and opposite force (Newton’s 2nd Law) F d2 T2
Rx2
Ry2
Analysis of Gear 1:
Analysis of Gear 2:
𝐹 𝑥1 =𝐹+ 𝑅 𝑥1 =0
𝐹 𝑥2 =−𝐹+ 𝑅 𝑥2 =0
𝑅 𝑥1 =−𝐹
𝑅 𝑥2 =𝐹
𝐹 𝑦1 = 𝑅 𝑦1 =0
𝐹 𝑦1 = 𝑅 𝑦2 =0
𝑅 𝑦1 =0
𝑅 𝑦2 =0
𝑀 𝑐𝑒𝑛𝑡𝑒𝑟1 = −𝑇 1 +𝐹∙ 𝑑 1 2 =0
𝑀 𝑐𝑒𝑛𝑡𝑒𝑟2 = −𝑇 2 +𝐹∙ 𝑑 2 2 =0 + +
𝑇 1 =𝐹∙ 𝑑 1 2
𝑇 2 =𝐹∙ 𝑑 2 2

6. Gear Analysis T2 T1 F Rx2 Ry2 x y d1 d2 Gear 1: Gear 2: 𝑅 𝑥1 =−𝐹
𝑅 𝑥2 =𝐹
𝑅 𝑦1 =0
𝑅 𝑦2 =0
𝑇 1 =𝐹∙ 𝑑 1 2
𝑇 2 =𝐹∙ 𝑑 2 2
This is the same F.
Solve for F with both equation and set them equal to each other.
2∙ 𝑇 1 𝑑 1 =𝐹
2∙ 𝑇 2 𝑑 2 =𝐹
𝑅𝑃𝑀 2 𝑅𝑃𝑀 1 = 𝑁 1 𝑁 2 = 𝑑 1 𝑑 2 = 𝑇 1 𝑇 2
2∙ 𝑇 1 𝑑 1 = 2∙ 𝑇 2 𝑑 2
# of teeth is often represented with N.
𝑁 1 𝑁 2 is the same as # 𝑜𝑓 𝑡𝑒𝑒𝑡ℎ 𝑖𝑛 𝑔𝑒𝑎𝑟 1 # 𝑜𝑓 𝑡𝑒𝑒𝑡ℎ 𝑖𝑛 𝑔𝑒𝑎𝑟 2
𝑇 1 𝑇 2 = 𝑑 1 𝑑 2

7. Class Problem: Consider the gear train below
Class Problem: Consider the gear train below. Assume the gears are supported by bearings and that all gears have compatible teeth.
Find the RPM of gears 2, 3 and 4 (gears 2 and 3 are joined like many of your servo gears).
Find torque transmitted through gears 2, 3, and 4.
Gear
Teeth
RPM
Torque (in-lbs) 1 10
3600 2 30 3 8 4 40
𝑅𝑃𝑀 2 𝑅𝑃𝑀 1 = 𝑁 1 𝑁 2 = 𝑑 1 𝑑 2 = 𝑇 1 𝑇 2
1200 30
1200 30
240
150 a.
𝑅𝑃𝑀 2 𝑅𝑃𝑀 1 = 𝑁 1 𝑁 2 b.
𝑇 2 𝑇 1 = 𝑁 2 𝑁 1
𝑅𝑃𝑀 2 = 𝑁 1 𝑁 2 𝑅𝑃𝑀 1
𝑇 2 = 𝑁 2 𝑁 1 𝑇 1
𝑅𝑃𝑀 2 = =1200
=𝑅𝑃𝑀 3
𝑇 2 = 𝑖𝑛∙𝑙𝑏𝑠 =30 𝑖𝑛∙𝑙𝑏𝑠
=𝑇 3
Gear 2 and 3 are connected and so they must turn at the same speed
𝑇 4 = 𝑁 4 𝑁 3 𝑇 3 = 𝑖𝑛∙𝑙𝑏𝑠
𝑅𝑃𝑀 4 = 𝑁 3 𝑁 4 𝑅𝑃𝑀 3 =
=150𝑖𝑛∙𝑙𝑏𝑠
=240

8. Gear Summary Points to Remember:1
Points to Remember:
Bigger gear turns slower than smaller gear.
Bigger gear has more torque than smaller gear.
Both gears on a compound gear turn at the same rate and transmit the same torque.
𝑅𝑃𝑀 2 𝑅𝑃𝑀 1 = 𝑁 1 𝑁 2 = 𝑑 1 𝑑 2 = 𝑇 1 𝑇 2