Question

Two meshing spur gears 1 and 2 with diametral pitch of 8 teeth per mm and an angular velocity ratio \(|\omega_2|/|\omega_1| = 1/4\), have their centers 30 mm apart. The number of teeth on the driver (gear 1) is ................ (Answer in integer) 

Hint:

Be careful with gear terminology and units. In the metric system, the {module} (\(m\), in mm) is standard. In the imperial system, {Diametral Pitch} (\(P_d\), in teeth/inch) is standard. They are related by \(m \cdot P_d = 25.4\). If a problem gives \(P_d\) in metric units like "teeth/mm", it's highly likely that they mean it's the reciprocal of the module (\(m = 1/P_d\)).

Answer 1

Step 1: Understanding the Concept:
This problem involves the fundamental kinematic and geometric relationships of a pair of meshing spur gears. We will use the relationships between velocity ratio, number of teeth, pitch circle diameters, and center distance. The unit "diametral pitch" is given in an unusual metric form (teeth per mm), which is the reciprocal of the module.
Step 2: Key Formula or Approach:
1. Velocity Ratio: For meshing gears, the ratio of angular velocities is inversely proportional to the ratio of their numbers of teeth (\(T\)) and their pitch circle diameters (\(d\)). \[ \frac{|\omega_2|}{|\omega_1|} = \frac{T_1}{T_2} = \frac{d_1}{d_2} \] 2. Center Distance (External Meshing): The distance between the centers of two externally meshing gears is the sum of their radii. \[ C = r_1 + r_2 = \frac{d_1 + d_2}{2} \] 3. Diametral Pitch and Module: Module (\(m\)) is defined as \(m = d/T\). Diametral pitch (\(P_d\)) is typically \(T/d\). Given \(P_d = 8\) teeth/mm, the module is the reciprocal: \[ m = \frac{1}{P_d} = \frac{1}{8} \text{ mm/tooth} \] Step 3: Detailed Calculation:
Given:
- Velocity ratio \(\frac{|\omega_2|}{|\omega_1|} = \frac{1}{4}\)
- Center distance \(C = 30\) mm
- Module \(m = 1/8\) mm/tooth
From the velocity ratio, we have: \[ \frac{T_1}{T_2} = \frac{1}{4} \implies T_2 = 4T_1 \] Also, since \(d = m \cdot T\): \[ \frac{d_1}{d_2} = \frac{m T_1}{m T_2} = \frac{T_1}{T_2} = \frac{1}{4} \implies d_2 = 4d_1 \] Now use the center distance formula: \[ C = \frac{d_1 + d_2}{2} \] \[ 30 = \frac{d_1 + 4d_1}{2} = \frac{5d_1}{2} \] Solve for the pitch circle diameter of gear 1 (\(d_1\)): \[ d_1 = \frac{30 \times 2}{5} = \frac{60}{5} = 12 \text{ mm} \] Finally, calculate the number of teeth on gear 1 (\(T_1\)) using the module formula: \[ m = \frac{d_1}{T_1} \implies T_1 = \frac{d_1}{m} \] \[ T_1 = \frac{12 \text{ mm}}{1/8 \text{ mm/tooth}} = 12 \times 8 = 96 \] Step 4: Final Answer:
The number of teeth on the driver (gear 1) is 96.
Step 5: Why This is Correct:
The solution systematically uses the standard gear formulas. The key was correctly interpreting the unusual unit for diametral pitch as the reciprocal of the module. The resulting number of teeth is an integer, which is a necessary condition for a physical gear.