## Analytical Modeling of Equipment

1Q. What does modeling of machinery mean?

1A. Analytical modeling is when the operation of a piece of equipment is represented by free body diagrams or sketches and analytical expressions are used which describe the essence of the function you are interested in. For example, consider the case of a shaft failed in a groove on a large motor shaft with multi-belt drive. Your model would contain as a minimum, a free body diagram of the rotor shaft and bearings, and equations which contain belt loading, force, moment and stress. Since you are concerned with too high of a fatigue stress which causes failures you will want to see the effect of belt tension or shaft/groove geometry in the equations. Instead of guessing what to modify the solution of the equations indicate exactly what and how much the parameters should be changed. Changes are made on paper and evaluated before any expensive field modifications are performed. You can even determine how long the component can be expected to function until another failure occurs.

2Q. What is the difference between deterministic modeling and probabilistic modeling?

2A. If you do a tensile test on a round bar of diameter d and pull on it with force F, the stress can be represented as σ = F / (0.785 d 2) and the minimum ultimate tensile stress σtensile would be that of the bar.

We would normally use the largest force F and smallest diameter d to calculate the stress and compare this with the minimum tensile strength. When σ / σtensile is greater than 1 we usually say the piece has failed. This would produce one answer, and would be conservative. It is a deterministic solution.

Now suppose instead of taking one F maximum value and d minimum, and σtensile minimum measurement we had a frequency distribution of many tests for each. From these distributions we could randomly pull samples (i.e. 100,000+) from each distribution using a technique called The Monte Carlo Method and build another frequency distribution. From this new distribution we could now look at the probability of σ / σtensile being greater than 1 rather than looking at a single result. In a simplistic way this is called probabilistic modeling. It still requires you to determine what risk you are willing to take, but you can make statements now like there is an 85% chance that the rod will not fail in tension.

3Q. We send our failures to an outside laboratory for a metallurgical analysis, why bother with a mechanical analysis?

3A. Please see Failure Investigations 2A.

4Q. What do you mean when you say the same equations used in design aren't the same used in troubleshooting?

4A. When designing, you need to consider all modes of possible failure. The design equations, for example in shaft design, have safety design factors built into the equation for impact, stress risers, etc. to determine the diameter size. When such a shaft fails, the troubleshooter has to determine the cause. The troubleshooter may have to determine the effect of the material properties, stress concentrations or impact on the failure. For example, if yielding of the shaft has occurred then the loads causing it to yield will need to be determined. This is difficult to do with a general design equation. 