Lever Analogy for Simpson gearset (30 points)
Given a Simpson gearset, its planetary gearset 1’s sun gear tooth number Ns1 = 30, and annulus (ring) gear’s tooth number Na1 = 71; and its planetary gearset 2’s sun gear tooth number Ns1 = 35, and annulus (ring) gear’s tooth number Na1 = 82. The annulus gear of planetary gearset 1 is rigidly connected to The annulus gear of planetary gearset 2, and the carrier of planetary gearset 1 is rigidly connected to the carrier of planetary gearset 2. The relationship is shown in the following figure. Description: A1
A2
71
82
CA
C2
30
S1
35
S2
output
input
Convert the Simpson gearset into a lever
The Correct Answer and Explanation is :
A Simpson gearset, composed of two planetary gearsets, can be analyzed and converted into a lever analogy by recognizing the various gears’ roles and relationships. In this case, the problem describes a Simpson gearset with two planetary gearsets. The lever analogy simplifies the understanding of how gears interact by using a lever-based system of input and output forces.
Gearset Parameters:
- Planetary Gearset 1:
- Sun Gear ( N_{s1} = 30 )
- Annulus Gear ( N_{a1} = 71 )
- Planetary Gearset 2:
- Sun Gear ( N_{s2} = 35 )
- Annulus Gear ( N_{a2} = 82 )
Key Configurations:
- The annulus gear of planetary gearset 1 is rigidly connected to the annulus gear of planetary gearset 2.
- The carrier of planetary gearset 1 is rigidly connected to the carrier of planetary gearset 2.
These rigid connections create an integrated system where the gears of the two planetary gearsets operate in unison.
Conversion to Lever Analogy:
To convert the Simpson gearset into a lever analogy, consider each gear as a point of force application on a lever. The following relationships can help establish the analogy:
- Input and Output Forces:
- The input torque is applied to the sun gear of the first planetary gearset (planetary gearset 1).
- The output torque is extracted from the sun gear of the second planetary gearset (planetary gearset 2).
- Lever Arms:
- The lever arms can be visualized as the distances between the axis of rotation (center of the gears) and the point of application of the force (the radius of the gears).
- The input arm corresponds to the distance from the center of the first sun gear, and the output arm corresponds to the distance from the center of the second sun gear.
Gear Ratios:
The gear ratios in the system will affect how the input force translates to the output. The gear ratio ( G_1 ) between the two planetary gearsets depends on the number of teeth on the sun and annulus gears:
[
G_1 = \frac{N_{a1}}{N_{s1}} = \frac{71}{30} = 2.37
]
[
G_2 = \frac{N_{a2}}{N_{s2}} = \frac{82}{35} = 2.34
]
These ratios indicate how much the input torque is multiplied or reduced by the system, and thus how the levers’ lengths are adjusted to reflect the same ratio of torque transformation.
Final Explanation:
In the lever analogy, the rigid connection between the two planetary gearsets forces them to act as a single system. The sun gears of the planetary gearsets are analogous to the applied forces at two points on the lever, with the output torque resulting from the combined effect of these forces. The gear ratios ( G_1 ) and ( G_2 ) dictate the effectiveness of the input force in generating the output force.