13.3 Mathematical Anomalies

In this section artifacts of the mathematical methods used by Mech are discussed.

13.3.1 Zero Unit Vectors

If the concept of a zero-length unit vector seems a little strange, rest assured that Mech also finds it difficult. Unit vectors are used internally by Mech to generate an applied force vector when the magnitude of the force is specified, and the direction of the force is given as a vector based on mechanism geometry. The applied force vector, in this case, is the supplied direction vector divided by its own magnitude, times the specified force magnitude. If the magnitude of the supplied direction vector becomes zero at some point in the mechanism motion, a singularity results.
The most common occurrence of this problem is in the application of frictional forces. If the negation of the velocity vector of a point is used to specify the direction of a force, the velocity of the point may not go to zero at any point in time when a solution is sought. If it does, Mathematica evaluates 0/0 and produces an error message.
While this is to be expected since a zero vector makes no sense as a direction specification, it can be quite difficult to formulate a workaround. Setting the magnitude of the force equal to zero for all points in time at which the direction vector becomes zero may not work, because this still results in 0*0/0, which is no better than 0/0.

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The simplest workaround is to use the CutOff option for Force and Moment.

A fix for problems with zero-length force vectors.

The following example shows the effect that CutOff has on the force vector generated by a Mech Force function. The force is defined with a constant magnitude, but the direction vector of the force is a function of a. Clearly, the force is indeterminate if a goes to zero.

Here is a force that fails if a = 0.

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Here is the same force using the CutOff option.

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The value of CutOff should be extremely small so that it has a negligible effect on the magnitude of the load in the operating range of the variable a.