A ball bearing is operating on a work cycle consisting of three parts:
 A radial load of 3000 N at 720 rpm for 30% of the cycle,
 A radial load of 7000 N at 1440 rpm for 40% of the cycle and
 A radial load of 5000 N at 900 rpm for remaining cycle.
 The basic dynamic capacity of the bearing is 307000 N. Calculate the rating life of bearing in
 Hours.

To calculate the rating life of a ball bearing, we can use the **equivalent dynamic load** and the bearing life formula based on the basic dynamic capacity.

The general formula for the life of a bearing in hours is:

\[
L_{10h} = \frac{(C/P)^p}{60 \times n}
\]

Where:
- \( L_{10h} \) = bearing life in hours
- \( C \) = basic dynamic capacity of the bearing (307000 N)
- \( P \) = equivalent dynamic load (N)
- \( n \) = speed (rpm)
- \( p \) = 3 for ball bearings (typical)

First, we need to calculate the **equivalent dynamic load**, \( P_e \), considering the loads and speeds for each part of the cycle. The formula for the equivalent dynamic load is:

\[
P_e = \left( P_1^3 \times t_1 + P_2^3 \times t_2 + P_3^3 \times t_3 \right)^{\frac{1}{3}}
\]

Where:
- \( P_1, P_2, P_3 \) are the radial loads (3000 N, 7000 N, 5000 N)
- \( t_1, t_2, t_3 \) are the time percentages for each load (30%, 40%, 30%)
  
Now, we will calculate \( P_e \):

\[
P_e = \left( (3000^3 \times 0.30) + (7000^3 \times 0.40) + (5000^3 \times 0.30) \right)^{\frac{1}{3}}
\]

After calculating the equivalent load, we can calculate the life in hours for each part of the cycle and sum the contributions.

Let's proceed step by step.The equivalent dynamic load, \( P_e \), is approximately 5675.34 N.

Now, we can calculate the bearing life in hours using the formula:

\[
L_{10h} = \frac{(C/P_e)^p}{60 \times n}
\]

For the entire cycle, we will use an average speed. Since each part of the cycle has a different speed, we calculate the **weighted average speed**, \( n_{avg} \), as follows:

\[
n_{avg} = (n_1 \times t_1) + (n_2 \times t_2) + (n_3 \times t_3)
\]

Where:
- \( n_1 = 720 \, \text{rpm} \)
- \( n_2 = 1440 \, \text{rpm} \)
- \( n_3 = 900 \, \text{rpm} \)

Let's calculate the average speed and then the rating life.The weighted average speed, \( n_{avg} \), is 1062 rpm.

Now we can calculate the bearing life in hours using the formula:

\[
L_{10h} = \frac{(C / P_e)^p}{60 \times n_{avg}}
\]

Where \( p = 3 \) (since it's a ball bearing). Let's calculate the bearing life in hours.The rating life of the bearing is approximately **2.48 hours**.

Let me know if you'd like further details or explanations on the process!

Here are five related questions you might find useful:
1. How would the bearing life change if the radial loads were different?
2. What impact does increasing the basic dynamic capacity have on the bearing life?
3. How do we calculate bearing life for a different type of bearing (e.g., roller bearings)?
4. How would a constant load and speed affect the bearing life calculation?
5. What other factors (e.g., lubrication or temperature) can influence bearing life?

**Tip:** Regular maintenance, such as proper lubrication, can significantly extend the life of a bearing.

A ball bearing is operating on a work cycle consisting of three parts: A radial load of 3000 N at 720 rpm for 30% of the cycle, A radial load of 7000 N at 1440 rpm for 40% of the cycle, and A radial load of 5000 N at 900 rpm for the remaining cycle. The basic dynamic capacity of the bearing is 307000 N. Calculate the rating life of the bearing in hours.

Learn how to calculate the rating life of a ball bearing operating under varying radial loads and speeds. This problem involves equivalent dynamic load, weighted average speed, and bearing life formulas. Suitable for university-level engineering students.

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