# Calculating the Critical Speed of a Shaft Bearing with Mathcad

A few weeks ago, Jakov Kucan, the Development Director for Mathcad, asked me to help him make a few short videos for PlanetPTC Live (June 12-15 in Las Vegas, NV).  “Two to three minute videos highlighting the look and feel of Mathcad Prime 1.0, as well as some of its key capabilities,” he said.

Now I could have just stitched together some videos of individual features of Mathcad Prime 1.0, but I wanted to show how those individual features came together in a calculation that someone would make on typical day-to-day basis.

This led me to borrow (or “steal,” depending on how you look at it) the shaft bearing demo made by my fellow colleague, Thomas Devaraj, of the UK office.

In this demo, we are concerned about the critical speed of a shaft bearing.  As the machinery starts nearing this critical speed, the axle which the bearing supports will start to deflect significantly, or “whirl.”  (See http://www.roymech.co.uk/Useful_Tables/Drive/Shaft_Critical_Speed.html for a quick reference.)

Therefore, it is important to make sure that the shaft bearing you design is well above the maximum operating speed of the machinery.

In this worksheet, we start by defining the inputs – material property, shaft bearing dimensions, etc.  We can take advantage of Mathcad’s ability to handle units.

We then use the inputs to calculate the critical speed of the shaft bearing.  Mathcad’s natural math notation makes it easy for the reader to understand the underlying math.

This tells us the critical speed for this given set of input variables.  Suppose we want to test a range of input variables.  We can then use specification tables (a new feature in Mathcad Prime 1.0) to define an array as the inner and outer diameters – two input variables.

We now use Mathcad’s in-line programming capabilities to loop through all the combination of input and output diameters and compare it to the maximum operating speed.  If the critical speed is above the maximum operating speed – signifying that whirling will not happen during normal operation – the design passes.

Otherwise, it fails.  A pass/fail table lets us quickly check which designs (inner/outer diameters combinations) pass and which ones fail.

Finally, we can visualize the critical speed as a function of inner and outer diameters on a contour plot.

Here’s a video of these calculations:

I had to try very hard to fit everything into three minutes (okay, fine three minutes and seventeen seconds).  I barely took any breaths between sentences!  And notice how fast I was able to type all those equations!  As much as I would like to type that fast, I will admit that I had to use some video editing tricks to speed up the typing so it would sync up to the voiceover 😉

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## One thought on “Calculating the Critical Speed of a Shaft Bearing with Mathcad”

1. Balaji Sankar says:

I work in Foil bearings for high speed shafts. I usually code in c++ for my work and it’s surprising that so user friendly tools exist. Keep up the good work. Kudos:-)