1. ME 302 DYNAMICS OF MACHINERY
GEAR FORCE ANALYSIS
Dr. Sadettin KAPUCU
© 2007 Sadettin Kapucu

2. GEAR FORCE ANALYSIS
Gears are used to transmit force and motion from one shaft to the other.

3. Spur Gear
In theory, one blank transmit force and motion via the friction force occurring at the pitch point. There must be no slip between the two blanks at the pitch points. Blanks roll but do not slip with respect to each other. Input /output relationship for circular gear is linear. So:

4. Spur Gear
Magnitudes of friction forces are generally small. So, if we want to transmit larger amounts of forces, friction becomes inadequate and slip occurs. To prevent slip, we make the joint between links 2 and 3 “form closed”. This is obtained by putting teeth around the periphery of the gear blanks. For fitting these details, we need some space. We simply separate the gear blanks apart a bit.
This separation causes the transmitted force to at an angle called “pressure angle”, denoted by Pressure angle is standardized;
In imperial system
In international system

5. Spur Gear
Characteristic dimension for the tooth is either the number of teeth on the blank or the length of the portion of the pitch circle within the tooth body.

6. Helical Gear
To improve the force carrying capacity of the gears the teeth are cut in a helix. This increase the tooth thickness, so helical gears are stronger. Also they operate with less noise.
Force acting normal to the tooth surface, hence it makes an angle of (helix angle) with the gear axis of rotation and with the common tangent.

7. Bevel Gear
Bevel gears are conical in shape and used to couple the shafts not parallel but intersecting. Point of intersection of shafts is called the apex. Gear force acts as distributed over the whole tooth thickness, but we can assume a resultant single force acting on the mid point of the tooth thickness.

8. Example 1
The gear train shown in the figure is composed of 6 diametral pitch spur gears and 20 degrees pressure angle. Link 2 is the driving gear, delivering 25 Hp at a CCW speed of 900 rpm. Gear 3 is an idler and gear 4 carries the external load. Draw freebody diagrams of the gears, show all the forces acting and calculate their magnitudes.
Given:

9. Example 1 t2 t4 t3 F2y F2x F2t x y + F2r F’3r F’3t F4r F3r F3x F4x F4y

10. Example 1 x y +
F2r
F2t
F2x
F2y t4
From second gear freebody diagram:

11. Example 1 t3 From third gear freebody diagram: x y + F’3t F’3r F3x F3y

12. Example 1 t4 y + From fourth gear freebody diagram: x
Radius of the gears can be calculated
following formulas:
F4t
F4r
F4x
F4y t4

13. Example 1 t4 F’3t F’3r F3x F3y F3t F3r F4t F4r F2r F2t F2x F2y F4x F4y+
Using the equations
unknowns become:
Gear forces of the second gear become:
Speed of the fourth gear is:
Then, gear forces of the fourth gear become: