Let's solve the problem step by step using the **tabular method** for compound planetary gear trains. ### **Given Data** - Number of teeth: - \( N_2 = 30 \) (Sun 1) - \( N_3 = 25 \) (Planet 1) - \( N_4 = 45 \) (Planet 2) - \( N_5 = 50 \) (Ring 1) - \( N_6 = 200 \) (Ring 2) - Angular velocities: - \( \omega_2 = 50 \) rpm (Sun 1) - \( \omega_6 = 0 \) rpm (Ring 2 is fixed) - **Find:** \( \omega_{\text{arm}} \) (Arm angular velocity) --- ## **Step 1: Tabular Approach Setup** ### **Compound Planetary Gear Train** - The arm (carrier) holds planets 3 and 4. - Sun gear 2 meshes with planet 3. - Planet 4 meshes with ring 6 (fixed). - Planets 3 and 4 are rigidly attached and rotate together. --- ## **Step 2: Relative Motion Table** We use the principle: - **Step 1:** Lock the arm (arm = 0), rotate sun gear 2 by +1 rev. - **Step 2:** Add +\( x \) revs to all elements (move everything by \( x \) revs). Let: - \( x \) = Arm motion (to be determined) - \( y \) = Relative motion (when arm is locked, sun 2 rotates +1 rev) | Step | Arm (\(\omega_{\text{arm}}\)) | Sun 2 (\(\omega_2\)) | Ring 6 (\(\omega_6\)) | |---------------------|------------------|----------------|-----------------| | 1. Arm locked, +1 rev at Sun 2 | 0 | +1 | ? | | 2. Add \(x\) to all | \(x\) | \(x + y\) | \(x + z\) | We'll focus on Sun 2, Arm, and Ring 6 (since that's where we have the velocities). --- ## **Step 3: Find the Ratio for Locked Arm** **With the arm locked**: - Sun 2 (\(N_2\)) meshes with planet 3 (\(N_3\)), which is rigid to planet 4 (\(N_4\)), which meshes with ring 6 (\(N_6\)). - This forms a compound gear train from Sun 2 to Ring 6. **Gear Ratio:** \[ \text{Overall ratio} = \frac{N_2 \cdot N_4}{N_3 \cdot N_6} \] The movement of ring 6 when sun 2 rotates +1 rev (arm locked): - From compound train: (since planet arms are locked, it’s like a simple train) \[ \text{If sun 2 rotates by } +1 \text{ rev, ring 6 rotates by:} \] \[ \frac{\omega_6'}{\omega_2'} = -\frac{N_2 \cdot N_4}{N_3 \cdot N_6} \] \[ \omega_6' = -\frac{N_2 \cdot N_4}{N_3 \cdot N_6} \cdot 1 \] Plug in the numbers: \[ \omega_6' = -\frac{30 \cdot 45}{25 \cdot 200} = -\frac{1350}{5000} = -0.27 \] So, when the arm is locked and sun 2 rotates +1 rev, ring 6 rotates \(-0.27\) rev. --- ## **Step 4: Write the General Equation** Let \( x = \omega_{\text{arm}} \), and let the motion we just found be \( y \) times the actual sun rotation (here, \( y = \omega_2 - x \)). The actual motions are: - Sun 2: \( \omega_2 \) - Ring 6: \( \omega_6 \) - Arm: \( \omega_{\text{arm}} \) The tabular equation: \[ \omega_{\text{element}} = \omega_{\text{arm}} + (\text{relative motion with respect to arm}) \cdot (\omega_2 - \omega_{\text{arm}}) \] For ring 6: \[ \omega_6 = \omega_{\text{arm}} + (\text{ratio}) \cdot (\omega_2 - \omega_{\text{arm}}) \] Here, ratio is \(-0.27\) (from above). Given: \( \omega_6 = 0 \) \[ 0 = \omega_{\text{arm}} + (-0.27) \cdot (\omega_2 - \omega_{\text{arm}}) \] --- ## **Step 5: Solve for \(\omega_{\text{arm}}\)** \[ 0 = \omega_{\text{arm}} - 0.27 (\omega_2 - \omega_{\text{arm}}) \] \[ 0 = \omega_{\text{arm}} - 0.27 \omega_2 + 0.27 \omega_{\text{arm}} \] \[ 0 = (1 + 0.27)\omega_{\text{arm}} - 0.27 \omega_2 \] \[ 1.27\omega_{\text{arm}} = 0.27 \omega_2 \] \[ \omega_{\text{arm}} = \frac{0.27}{1.27} \omega_2 \] Plug in \( \omega_2 = 50 \) rpm: \[ \omega_{\text{arm}} = \frac{0.27}{1.27} \times 50 \] \[ \omega_{\text{arm}} = \frac{13.5}{1.27} \approx 10.63 \text{ rpm} \] --- ## **Step 6: Final Answer** \[ \boxed{\omega_{\text{arm}} \approx 10.63 \text{ rpm}} \] --- ### **Summary Table** | Gear | Number of Teeth | RPM (Given/Found) | |--------------|:--------------:|:-----------------:| | Sun 2 | 30 | 50 | | Planet 3 | 25 | - | | Planet 4 | 45 | - | | Ring 5 | 50 | - | | Ring 6 | 200 | 0 | | Arm | - | **10.63 (Ans)** | --- ## **Explanation** 1. **Locked Arm:** Find the ratio of sun's motion to the ring (with arm locked). 2. **Tabular Equation:** Express actual velocities in terms of arm and relative motion. 3. **Solve:** Use given velocities and solve for the unknown arm velocity. --- **Final Answer:** \[ \boxed{\omega_{\text{arm}} \approx 10.63 \text{ rpm}} \]