Question 9.6: Analyzing Ferguson’s Paradox by the Tabular Method. Consider...
Analyzing Ferguson’s Paradox by the Tabular Method.
Consider Ferguson’s paradox train in Figure 9-39, which has the following tooth numbers and initial conditions:
Sun gear # 2 N_{2} = 100-tooth external gear
Sun gear # 3 N_{3} = 99-tooth external gear
Sun gear # 4 N_{4} = 101-tooth external gear
Planet gear N_{5} = 20-tooth external gear
Input to sun # 2 0 rpm
Input to arm 100 rpm counterclockwise
Sun gear 2 is fixed to the frame, thus providing one input (zero velocity) to the system. The arm is driven at 100 rpm counterclockwise as the second input. Find the angular velocities of the two outputs that are available from this compound train, one from gear 3 and one from gear 4, both of which are free to rotate on the main shaft.
1 The tabular solution for this train is set up in Figure 9-40 which shows the given data. Note that the row for gear 5 is repeated for clarity in applying the gear ratio between gears 5 and 4.
2 The known input values of velocity are the arm angular velocity and the zero absolute velocity of gear 2.
3 The gear ratios in this case are all negative because of the external gear sets, and their values reflect the direction of power flow from gear 2 to 5, then 5 to 3, and 5 to 4 in the second branch.
4 Figure 9-41 shows the calculated values added to the table. Note that for a counterclockwise 100 rpm input to the arm, we get a counterclockwise 1 rpm output from gear 4 and a clockwise 1 rpm output from gear 3, simultaneously.
