Question 9.5: Epicyclic Gear Train Analysis by the Tabular Method. Conside...
Epicyclic Gear Train Analysis by the Tabular Method.
Consider the train in Figure 9-34, with the tooth numbers and initial conditions:
Sun gear N_{2} = 40-tooth external gear
Planet gear N_{3} = 20-tooth external gear
Ring gear N_{4} = 80-tooth internal gear
Input to arm 200 rpm clockwise
Input to sun 100 rpm clockwise
We wish to find the absolute output angular velocity of the ring gear.
1 The solution table is set up with a column for each term in equation 9.12 and a row for each gear in the train. It will be most convenient if we can arrange the table so that meshing gears occupy adjacent rows. The table for this method, prior to data entry, is shown in Figure 9-36.
\omega_{g e a r}=\omega_{a r m}+\omega_{g e a r / a r m} (9.12)
2 Note that the gear ratios are shown straddling the rows of gears to which they apply. The gear ratio column is placed next to the column containing the velocity differences \omega_{\text {gearlarm }} because the gear ratios only apply to the velocity difference. The gear ratios cannot be directly applied to the absolute velocities in the \omega_{\text {gear }} column.
3 The solution strategy is simple but is fraught with opportunities for careless errors. Note that we are solving a vector equation with scalar algebra and the signs of the terms denote the sense of the ω vectors which are all directed along the Z axis. Great care must be taken to get the signs of the input velocities and of the gear ratios correct in the table, or the answer will be wrong. Some gear ratios may be negative if they involve external gearsets, and others will be positive if they involve an internal gear. We have both types in this example.
4 The first step is to enter the known data as shown in Figure 9-37 which in this case are the arm velocity (in all rows) and the absolute velocity of gear 2 in column 1. The gear ratios can also be calculated and placed in their respective locations. Note that these ratios should be calculated for each gearset in a consistent manner, following the power flow through the train.
That is, starting at gear 2 as the driver, it drives gear 3 directly. This makes its ratio -N_{2} / N_{3} , or input over output, not the reciprocal. This ratio is negative because the gearset is external. Gear 3 in turn drives gear 4 so its ratio is +N_{3} / N_{4} . This is a positive ratio because of the internal gear.
5 Once any one row has two entries, the value for its remaining column can be calculated from equation 9.12, which is shown in the top row of Figures 9-37 and 9-38. Once any one value in the velocity difference column (column 3) is found, the gear ratios can be applied to calculate all other values in that column. Finally, the remaining rows can be calculated from equation 9.12 to yield the absolute velocities of all gears in column 1. These computations are shown in Figure 9-38 which completes the solution.
6 The overall train value for this example can be calculated from the table and is from arm to ring gear +1.25:1 and from sun gear to ring gear +2.5:1.
