Bearing Life Calculations

1. Understanding Bearing Life: It's a Game of Probabilities

Before we jump into equations, it's crucial to understand the philosophy behind bearing life calculation. Unlike calculating the stress in a beam, we cannot predict the exact life of a single bearing. Instead, we use a statistical approach to determine a reliable operational lifespan.

As the video explains, the standard for bearing life is the L10 life.

• L10 Life: Also known as the basic rating life, this is the number of revolutions that 90% of a group of identical bearings are expected to complete or exceed before fatigue failure. It's a measure of reliability.

To perform this validation, you must distinguish between the life you need and the life you can expect from a selected component.

Our goal today is to calculate the predicted L10 life of our bearings and check if it meets the required service life for our rotary mechanism project.

2. The L10 Life Calculation Formula

The standard formula for calculating L10 life, as established by ISO 281, is:

$L_{10}={\left(\frac{C}{P}\right)}^p$

Where:

• $L_{10}$ is the basic rating life in millions of revolutions.
• $C$ is the basic dynamic load rating of the bearing, found in the manufacturer's catalog. It represents the load a bearing can endure for one million revolutions.
• $P$ is the equivalent dynamic bearing load, which is the calculated load that represents the combined effect of all radial and axial forces on the bearing.
• $p$ is the life exponent: 3 for ball bearings and 10/3 (approx. 3.33) for roller bearings.

Once we have $L_{10}$ in revolutions, we can convert it to a more practical unit, hours, using the operating speed:

$L_{10h}=\frac{L_{10}\times 10^6}{n\times 60}$

Where:

• $L_{10h}$ is the rating life in operating hours.
• $n$ is the rotational speed in revolutions per minute (RPM).

3. Step-by-Step Calculation for Our Rotary Mechanism

Let's walk through this process for one of the bearings on our project's output shaft.

Step 1: Define Operating Conditions & Required Life

First, we must define the inputs. For our rotary mechanism, let's assume the following based on the power transmission requirements:

• Shaft Speed (n): 500 RPM
• Radial Load ($F_r$): 450 N (This would come from the tangential and radial forces of the gears on the shaft).
• Axial Load ($F_a$): 80 N (This could be from a helical gear or other components creating a thrust load).
• Shaft Diameter: 20 mm

Next, what is our required life? We'll use this chart to determine a reasonable target for our application.

Our mechanism falls under "General Industrial Machinery" intended for regular use (8-16 hours/day). Looking at the chart, a reasonable target for required life is in the range of 20,000 to 30,000 hours. Let's set our target at 25,000 hours.

Step 2: Initial Bearing Selection

Based on our 20 mm shaft diameter, we'll start with a standard deep groove ball bearing: a 6204. We look up its properties from a manufacturer catalog:

• Basic Dynamic Load Rating (C): 12.8 kN (or 12,800 N)
• Basic Static Load Rating ($C_0$): 6.55 kN (or 6,550 N)

Step 3: Calculate the Equivalent Dynamic Load (P)

Since we have both radial and axial loads, we can't just use $P=F_r$. We must calculate an equivalent load $P$ that accounts for both. The formula is:

$P=XV{F}_r+Y{F}_a$

The factors X (radial) and Y (thrust) depend on the bearing type and the ratio of axial to radial load. We find these from a table, like the one provided by NHBB.

To use this table for a deep groove ball bearing, we must follow a procedure:

1. Assume inner ring rotates: This is typical, so the rotation factor $V=1$.
2. Calculate $F_a/C_0$: $80\text{N}/6550\text{N}\approx 0.012$.
3. Find e from the table: Looking at the "Radial deep groove ball bearings" section, the value for $F_a/C_0$ is very small. The first row is for a different parameter, so we'd consult a more detailed catalog. For our purposes, let's use a common method shown in many tutorials where you find e based on a related factor. In a full SKF catalog, for a factor related to our $F_a/C_0$, the value of e is approximately 0.22.
4. Compare $F_a/(V{F}_r)$ with e:
   • $F_a/(V{F}_r)=80/(1\times 450)\approx 0.178$
   • Since $0.178\le e$ (i.e., $0.178\le 0.22$), the axial load is relatively small. In this case, the formula simplifies, and the factors are X = 1 and Y = 0.
   • This means the axial load is not significant enough to impact the fatigue life calculation for this specific bearing and load combination.

So, for this case:
$P=(1\times 1\times 450\text{N})+(0\times 80\text{N})=450\text{N}$

Note: If $F_a/(V{F}_r)$ had been greater than e, we would have used different X and Y factors from the table (e.g., X=0.56 and Y would be around 2.0), resulting in a much higher equivalent load P.

Step 4: Calculate the Bearing's L10 Life

Now we have all the pieces:

• C = 12,800 N
• P = 450 N
• p = 3 (for a ball bearing)

$L_{10}={\left(\frac{12,800}{450}\right)}^3\approx (28.44{)}^3\approx 23,015\text{ million revolutions}$

Now, convert this to hours at 500 RPM:
$L_{10h}=\frac{23,015\times 10^6}{500\times 60}=\frac{23,015\times 10^6}{30,000}\approx 767,167\text{ hours}$

Step 5: Compare and Conclude

• Calculated Life ($L_{10h}$): ~767,000 hours
• Required Life: 25,000 hours

Since the calculated life is far greater than the required life, the 6204 bearing is an excellent choice for this application from a fatigue life perspective. There is a very high margin of safety.

4. A Comprehensive Worked Example

The process of handling combined loads and iterating on the design can be complex. Watching a full, narrated example is one of the best ways to solidify your understanding.

The method shown in the video of calculating a "required C" is a powerful technique for quickly honing in on a suitable bearing without multiple rounds of trial-and-error.

Conclusion

Excellent work today. You've now learned how to perform one of the most common validation tasks in machine design: calculating bearing life. This skill allows you to move beyond just geometric fit and structural strength to ensure your designs are reliable and meet their intended service lifespan.

Key Takeaways:

• Bearing life is a statistical prediction (L10) based on fatigue.
• The process is iterative: define requirements, select a trial bearing, calculate the equivalent load ($P$), calculate the life ($L_{10h}$), and compare it to your requirement.
• The key parameters are the bearing's dynamic load rating ($C$), the equivalent load ($P$), and the operating speed ($n$).
• If a bearing's calculated life is insufficient, the primary solution is to select a bearing with a higher dynamic load rating ($C$).

Preview of the Next Lesson:
We've confirmed the housing is strong and the bearings will last. But there's one more critical component in that load path: the shaft itself. An overly flexible shaft can cause poor gear meshing and premature bearing failure, even if the bearings are theoretically rated for the load. In our next lesson, we will perform shaft deflection analysis to ensure stiffness requirements are met.