To analyze the simple epicyclic gear train as described, we can apply fundamental gear principles and some equations related to epicyclic gear systems.
Given Data:
- Sun Gear (C): 16 teeth
- Planetary Gear (E): 20 teeth
- Gear D: 24 teeth
- Ring Gear (G): 60 teeth
- Input speed of Sun Gear (C): \( N_C = 120 \) r/min (clockwise)
1. Speed and Direction of Rotating Annulus G when Arm (A) is Fixed
When the arm is fixed, the motion of the gears must be analyzed with respect to the fixed arm. We can utilize the relation of the gears:
Gear Relation
The relationship between the number of teeth and the rotational speeds of gears in contact can be given by the formula:
\[ \frac{N_C}{N_G} = -\frac{T_G}{T_C} \]
Where:
- \( N_C \) = Speed of Sun (input) gear C
- \( N_G \) = Speed of ring gear G (unknown, which we want to find)
- \( T_G \) = Number of teeth of Ring Gear (G) = 60
- \( T_C \) = Number of teeth of Sun Gear (C) = 16
Rearranging the formula gives us:
\[ N_G = -N_C \cdot \frac{T_G}{T_C} \]
Substituting in the known values:
\[ N_G = -120 \cdot \frac{60}{16} \]
Calculating this:
\[ N_G = -120 \cdot 3.75 = -450 \text{ r/min} \]
The negative sign indicates that the ring gear is rotating in the opposite direction to the sun gear. Thus, the ring gear G rotates at 450 r/min counterclockwise.
2. Output Shaft Speed when Ring Gear G is Fixed
In this case, we must analyze the speed of the sun gear (C) and the planet gears (E and D) with the ring gear fixed. The fixed ring gear means that any rotation at the sun gear must cause the planet gears to rotate around their center.
Relationship of Speeds with Fixed G:
Given \( N_G = 0 \) (fixed ring gear), the relationship can be written as:
\[ \frac{N_E}{N_C} = -\frac{T_C}{T_E} \]
Where:
- \( N_E \) = Speed of Planet Gear E
- \( T_E \) = Number of teeth on Planet Gear E = 20
Using the same way as above,
\[ N_E = -N_C \cdot \frac{T_E}{T_C} \]
Substituting the known values:
\[ N_E = -120 \cdot \frac{20}{16} = -120 \cdot 1.25 = -150 \text{ r/min} \]
Now considering that gears E and D are mounted on the same axis, the speed of Gear D will be affected likewise. We can use the same relationship for Gear D:
\[ N_D = -N_E \cdot \frac{T_E}{T_D} \]
Where \( T_D \) = Number of teeth on Gear D = 24.
Using E's speed:
\[ N_D = -N_E \cdot \frac{T_E}{T_D} \]
Using \( N_E = -150 \):
\[ N_D = -(-150) \cdot \frac{20}{24} = 150 \cdot \frac{5}{6} = 125 \text{ r/min} \]
Output Shaft Speed (B):
The output speed is determined by the speed of the planet gears as they rotate around the axis of the sun gear.
- The average speed of the planet gears contributes to the output shaft speed:
Since two planet gears rotating in opposite directions around a fixed point (the axis of the arm) will average their speeds,
Therefore, the output speed at arm B can be considered as an average of the speeds of the two planet gears.
Since the vibrations and detailed geometries might introduce error margins, we can say:
The average speed at output (on arm) is:
\[ N_{out} = \frac{N_E + N_D}{2} = \frac{-150 + 125}{2} = \frac{-25}{2} = -12.5 \text{ r/min} \]
The negative indicates the output direction is contrary to input.
Thus, the answers are summarized:
- The speed of the rotating ring gear G is 450 r/min counterclockwise.
- The output shaft speed when the ring gear is fixed is 12.5 r/min counterclockwise.