What is the magnitude of the deflections from Modal Analysis?

Question

What is the magnitude of the deflections from Modal Analysis?

Answer

The deflections produced by Modal analysis are not “real”.  They are essentially arbitrary.  For each mode, they describe the shape of the mode, but this shape can have any size (amplitude) and this is not known.

In order to derive a real amplitude and hence deflections, the energy input into a structural system must be specified in some manner.  This would require:

1. A definition of the energy - commonly in the form of a time vs force or acceleration function - AND
2. A more complex form of analysis such as for example "Time History" or "Time step" analysis. 

While such complex forms of analysis exist, this is NOT what Tekla Structural Designer's Modal Analysis is.  For more technical information about what it is (and is not) see the topic Modal Analysis.

So, the Modal analysis deflections have no real magnitude or unit.  Having said this, what then determines the modal deflection values we see?  See the next section.

Mode Shape Scaling

The basic theory for modal normalization, or scaling, is as follows:

Two common forms of mode shape normalization (i.e. ways of reporting) are Mass scaling and Unity scaling.  Until more recently, the shapes reported (in Tekla Structural Designer) were always normalized to Mass not unity.  Release 2022 SP4 introduced a new option to report mode shapes normalized to Mass or to Unity.  For more about this see the 2022 SP4 Release Notes.

Unity Normalization

Normalizing to unity means the largest (resultant) displacement of the shape is set = 1.0 and all the other nodal displacements are scaled accordingly, such that the shape is unchanged.  The unity scaled mode shape is commonly used to calculate Modal Mass.  For more more about this see the topic Modal Analysis.

Mass Normalization

Scaling in this manner generally means the deflection values are very small numbers.  Furthermore, the larger the structure, the larger the mass and so the smaller the modal deflection values.  Scaling to mass satisfies the following equation:
 

Image

Where:

Image

  is the mode shape and 

Image

is the mass matrix.

Note that this is the equation for what is termed Modal Mass, which you will thus appreciate can have any value, depending on the method of modal scaling being used.

Any good reference on Vibration analysis theory will give more detail on this topic should you need it.  We can recommend the following:

• Anil K. Chopra, Dynamics of Structures: Theory and Applications to Earthquake Engineering (4nd Edition), Pearson, 2017.