A planetary gear system features a band brake that secures the ring gear in a fixed position. The sun gear rotates clockwise at 800 rpm with a torque of 16 N-m, and the armature is linked to a machine. The gears have a module (m) of 2 mm and a pressure angle of 20 degrees. There are 70 teeth on the ring gear and 20 on each planet gear. a) To calculate the circular pitch (p), you use the formula: \[ p = m \times \pi \] Plugging in m = 2 mm: \[ p = 2 \, \text{mm} \times \pi = 6.2832 \, \text{mm} \] b) The number of teeth on the sun gear (Z_s) can be calculated using the relationship between the number of teeth on the sun gear, the ring gear (Z_r), and the planet gears (Z_p) for a standard planetary gear set: \[ Z_r = 2 \times Z_p + Z_s \] Substituting Z_r = 70 and Z_p = 20, we find: \[ 70 = 2 \times 20 + Z_s \] \[ 70 = 40 + Z_s \] \[ Z_s = 70 - 40 \] \[ Z_s = 30 \] So, the sun gear has 30 teeth. c) To sketch the forces applied to the arm and gears, you need to visually represent the system and indicate the direction of rotation, torque application, and reaction forces at the interfaces of the gears, arm, and brakes. Remember to include the forces on the sun gear due to the input torque, the reaction force at the band brake holding the ring gear, and the forces between the planetary gears and both the sun and the ring gears. This is a physical illustration and needs to be provided separately. d) The output torque (T_out) on the arm (also called the carrier) can be determined by considering the gear ratio and efficiency of the system. Assuming the system is ideal (no losses), the torque is conserved when the ring gear is held stationary. With the given input torque (T_in) of 16 N-m and the gear ratio (GR), we can calculate T_out as: \[ GR = \frac{Z_r}{Z_s} \] \[ GR = \frac{70}{30} = \frac{7}{3} \] \[ T_out = T_in \times GR \] \[ T_out = 16 \, \text{N-m} \times \frac{7}{3} \] \[ T_out = \frac{112}{3} \, \text{N-m} \approx 37.33 \, \text{N-m} \] e) If the ring gear is to be kept stationary, the torque applied to the ring gear (T_ring) would be equal in magnitude but opposite in direction to the torque that the planetary gears would exert on the ring gear if it were free to rotate. Considering the system is in equilibrium (the ring gear is not rotating): \[ T_ring = - T_out \] Hence: \[ T_ring = -37.33 \, \text{N-m} \] The negative sign indicates the direction of the required torque on the ring gear is opposite to the output torque.

a) To calculate the circular pitch (p), you would use the formula with the given module (m) of 2 mm as follows: \[ p = m \times \pi \] \[ p = 2 \, \text{mm} \times \pi = 6.2832 \, \text{mm} \] So the circular pitch is 6.2832 mm.

b) The number of teeth on the sun gear (Z_s) can be found using the relationship between the number of teeth on the sun gear, the ring gear (Z_r), and the planet gears (Z_p): \[ Z_r = 2 \times Z_p + Z_s \] Substituting the given values (Z_r = 70 and Z_p = 20), we get: \[ 70 = 2 \times 20 + Z_s \] \[ 70 = 40 + Z_s \] \[ Z_s = 70 - 40 \] \[ Z_s = 30 \] Therefore, the sun gear has 30 teeth.

c) In order to sketch the forces applied to the arm and gears, one must visualize the gear system and carefully illustrate the applied torques, reaction forces, and how they may be transmitted through the gear connections. This would involve drawing the gears in proportion, showing the direction of the input rotational motion, and marking the points where forces are applied.

d) The output torque (T_out) on the arm can be found using the gear ratio (GR): \[ GR = \frac{Z_r}{Z_s} \] \[ GR = \frac{70}{30} = \frac{7}{3} \] Now, you can calculate the output torque if the system is ideal: \[ T_out = T_in \times GR \] \[ T_out = 16 \, \text{N-m} \times \frac{7}{3} \] \[ T_out = \frac{112}{3} \, \text{N-m} \approx 37.33 \, \text{N-m} \]

e) The torque on the ring gear (T_ring) that is required to keep it stationary when the system is in equilibrium can be determined by the output torque on the armature: \[ T_ring = - T_out \] \[ T_ring = -37.33 \, \text{N-m} \] The negative sign signifies that the direction of this torque is opposite to that of the output torque. This is the torque that must be countered by the band brake to fix the ring gear in place.

Extra: In a planetary gear system, you have several different gears interacting in a unique way. The sun gear is the central gear, which is surrounded by multiple planet gears that mesh with both the sun gear and the outer ring gear. The planet gears are mounted on a movable arm or carrier, which can also contribute to the output of the system.

The module (m) is a measure of the size of the gears, where each gear's pitch diameter is proportional to the number of teeth it has. The higher the module, the larger the teeth are.

The pressure angle (usually 20 degrees in many gear systems) affects the shape of the gear teeth and how they mesh together. A 20-degree pressure angle is commonly used for its balance between strength and how smooth the gear transitions are (less noise and vibration).

The circular pitch (p) is the distance from one point on a gear tooth to the corresponding point on the next gear tooth, measured along the pitch diameter.

In gears, the torque is transmitted from one gear to another. The gear ratio determines how the output speed and torque are related to the input speed and torque. In planetary gears, the ratios often result in higher torques compared to the input, making them useful for applications requiring significant power in a compact space, such as in automotive transmissions, industrial machinery, and robotics.