# Calculating the Probability of Failure for Anti-Friction Bearings in Mathcad Prime 2.0

*Guest blog by Dirk Jordan, PTC Mathcad Senior Technical Sales Specialist*

For a German tier one automotive supplier, calculating the probability of failure for anti-friction bearings is a critical task. These bearings are used in high-pressure pumps and other components within all types of vehicles. Calculations are based on and must comply with ISO 281:2007.

From the ISO website:

ISO 281:2007 specifies methods of calculating the basic dynamic load rating of rolling bearings within the size ranges shown in the relevant ISO publications, manufactured from contemporary, commonly used, high quality hardened bearing steel, in accordance with good manufacturing practice and basically of conventional design as regards the shape of rolling contact surfaces.

ISO 281:2007 also specifies methods of calculating the basic rating life, which is the life associated with 90% reliability, with commonly used high quality material, good manufacturing quality and with conventional operating conditions. In addition, it specifies methods of calculating the modified rating life, in which various reliabilities, lubrication condition, contaminated lubricant and fatigue load of the bearing are taken into account.

Rolling bearings are used in many different automotive applications and components.  Take a look at the bearings found in high-pressure pumps, like the ones used as fuel pumps for internal combustion engines. These bearings’ nominal lifespans are calculated at different loads, operating hours, and temperatures (100C or 120C).  They can be used in a variety of engines, and in turn used in different trucks (12-ton or 28-ton).  Typically, these bearings are designed to exceed 300,000 km of operation.

Probability of failure is one aspect that is carefully considered in the design of the pump.  Bearings need to be appropriately spec’ed out early in the development process to avoid serious changes late in the cycle.  Calculations need to be made for the range of temperatures and loads that the bearings are used in.

So why do these types of calculations in PTC Mathcad? One can leverage PTC Mathcad’s standard math notation to easily and clearly represent the calculations specified in the standard. You can put all the input parameters into a matrix, and then use the SUBMATRIX function to extract the relevant values used in various calculations.

If you understand the lifespan of rolling bearings to be limited by metal fatigue, then the lifespan distribution can be described by a Weibull distribution. One can use PTC Mathcad Prime 2.0’s solve block functionality to determine the parameters governing the Weibull distribution.

Using solve blocks to determine the parameters governing the Weibull distribution

You can then calculate the failure probability as a percentage of the specified operating hours for different storage temperatures. We do this by symbolically solving the equations with the various input parameters.  Below, we can see from the results (for this made up data set) that the probability of failure is high, especially at the higher 120C operating temperature (second line of calculation).

Standard Equation for Weibull Analysis in Mathcad Prime 2.0

Try using PTC Mathcad Express to solve symbolically, to use matrices, and to ensure your calculations are compliant, no matter what engineering discipline you fall in. Or, see how engineers of all disciplines use PTC Mathcad through customer presentations at Mathcad Engage 2012, PTC Mathcad’s virtual event.

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## 3 thoughts on “Calculating the Probability of Failure for Anti-Friction Bearings in Mathcad Prime 2.0”

1. Alan Stevens says:

The solve block in your first image looks very strange!! b doesn’t appear in the Constraints anywhere, yet a solution is still found for it (different from the value given to it in the Guess Values section!).

Alan

2. Hi, Alan. You had me second-guessing myself for a minute! But if you look carefully, there’s a ‘1/b’ in the exponent.

1. Alan Stevens says:

So there is! Guess I need stronger glasses!