Act like a helpful tutor and exlain me :
Give step-by-step solution with explanation and final answer:For Problems 36-41, compute the bending stress number, s;,
using Equation (9-16). Assume that th

Question

Act like a helpful tutor and exlain me :
Give step-by-step solution with explanation and final answer:For Problems 36-41, compute the bending stress number, s;,
using Equation (9-16). Assume that the gear blank is solid
unless otherwise stated. (Note that the data in these problems
are used in later problems through Problem 59. You are advised
to keep solutions to earlier problems accessible so that you can
use data and results in later problems. The four problems that
are keyed to the same set of data require the analysis of bending
stress and contact stress and the corresponding specification of
suitable materials based on those stresses. Later design prob-
lems, 60-70, use the complete analysis within each problem.)
36. A pair of gears with 20°, full-depth, involute teeth trans-
mits 10.0 hp while the pinion rotates at 1750 rpm. The
diametral pitch is 12, and the quality number is A11. The
pinion has 18 teeth, and the gear has 85 teeth. The face
width is 1.25 in. The input power is from an clectric mo-
tor, and the drive is for an industrial conveyor. The drive
is a commercial enclosed gear unit.

VIPSolutions · Accepted Answer

# Step-by-Step Solution to Problem 36

Let's solve for the **bending stress number, $ S_t $** for the given gear problem using Equation (9–16) from the referenced text.

## **Given Data**

- **Transmitted Power, $ P $**: 10. hp
- **Pinion Speed, $ n_p $**: 175 rpm
- **Diametral Pitch, $ P_d $**: 12
- **Gear Quality Number, $ Q_v $**: 11
- **Pinion Teeth, $ N_p $**: 18
- **Gear Teeth, $ N_g $**: 85
- **Face Width, $ F $**: 1.25 in
- **Pressure Angle**: $ 20^\circ $ (full-depth)
- **Application**: Commercial enclosed gear unit (for service factor, etc.)

---

## **Step 1: Find Pitch Line Velocity ($ V $)**

Pitch line velocity (in ft/min):

$$
V = \pi d_p n_p / 12
$$

Where:
- $ d_p $ = pitch diameter of the pinion

First, find pitch diameter ($ d_p $):
$$
d_p = \frac{N_p}{P_d} = \frac{18}{12} = 1.5 \text{ in}
$$

Now, plug into the velocity formula:
$$
V = \frac{\pi \times 1.5 \times 175}{12} = \frac{3.1416 \times 1.5 \times 175}{12}
$$
$$
V \approx \frac{3.1416 \times 2625}{12} \approx \frac{8236.8}{12} \approx 686.4 \text{ ft/min}
$$

---

## **Step 2: Find Transmitted Load ($ W_t $)**

$$
\text{Horsepower} = \frac{W_t \times V}{33,000}
\Rightarrow W_t = \frac{\text{Horsepower} \times 33,000}{V}
$$
$$
W_t = \frac{10 \times 33,000}{686.4} \approx \frac{330,000}{686.4} \approx 481 \text{ lbf}
$$

---

## **Step 3: Lewis Bending Stress Equation (Equation 9-16)**

$$
S_t = \frac{W_t \cdot P_d}{F \cdot Y}
$$

Where:
- $ W_t $ = tangential load (lbf)
- $ P_d $ = diametral pitch
- $ F $ = face width (in)
- $ Y $ = Lewis form factor (depends on number of teeth and tooth form)

---

## **Step 4: Find Lewis Form Factor ($ Y $)**

For 20° full-depth involute, Lewis Y-factor for 18 teeth (from tables):

- For 18 teeth: $ Y \approx .30 $

---

## **Step 5: Calculate Bending Stress Number ($ S_t $)**

Now substitute all values:

$$
S_t = \frac{481 \times 12}{1.25 \times .30}
$$
$$
S_t = \frac{5772}{.375}
$$
$$
S_t \approx 15,392 \text{ psi}
$$

---

## **Final Answer**

$$
\boxed{S_t \approx 15,\!400~\text{psi}}
$$

---

### **Summary of Steps**
1. Found pitch diameter and pitch line velocity.
2. Calculated transmitted load using horsepower and velocity.
3. Used Lewis equation with the Lewis form factor for 18 teeth.
4. Solved for the bending stress number, $ S_t $.

If you need further steps (like dynamic factor, overload factor, or material selection), let me know!

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