Mode Participation Factor in Dynamic Analysis

Hello Engineers,

Hope you have enjoyed my post on Mass Stiffness relationship

As a Dynamic / FEA Engineer, you would have performed modal analysis for determining the natural frequency and mode shape pattern of the structure.

Have you ever noticed the solution statistics once after executing the Modal dynamics problem?

We have a Participation factor calculation table for 6 directions, which includes translation and rotation of mass in X,Y & Z directions.

In this post, I will be explaining how the solver computes the mass participation factor and the components present in the above mentioned table.

Before going into the calculation part, It will be highly significant to know what exactly the mode participation is!!!

For describing the calculation, I am using a cantilevered bar (dimension is not much important here) made of steel material which has a mass of 294 g as shown in the figure:

The dynamics problem is executed in ANSYS mechanical. We do not want to worry much about the software tool provided that computations is going to be the same for all solvers.

I have solved the problem by specifying the number of Eigen frequencies to 12.

The reason for specifying this value is that the solver should not miss any of the important modes of our interest.

A basic thumb rule behind is that 👍

The ratio of effective mass to the total mass must be greater than 0.90 

Here is the participation table along Y direction excitation, 

Tap to Zoom the Participation table

In the above table only Natural frequency and Participation factors are calculated by the solver based on mass and stiffness matrices.  In addition we require only the final ratio value (i.e.,) 0.933 to derive all the parameters in the table.

Let us dive into the calculation part 👇

Effective Mass = (Participation factor)²

So Effective Mass for first mode = (13.410)² = 179.8 g

Effective mass for second mode = (-7.4537)² = 55.57 g     

Ratio is calculated based on the participation factor values 

Ratio for first mode = Participation for first mode / dominant mode

                                             = 13.410/13.410 = 1

        Ratio for second mode  = 7.4537/13.410 = 0.5558

Cumulative mass fraction  is calculated based on Cumulative Effective Mass / Effective Mass.

Cumulative effective mass for mode 1 : 179.817 + 0 = 179.817 g

For mode 2 it is :179.817 + 55.5574 = 235.3744 g

For mode 3 it is :235.3744 + 0.431263 *(10-7) = 235.3744 g

As mentioned previously we can derive the mass participating in the Y direction by the below method:

Effective Mass participating in Y direction  = Ratio of Total to Effective mass * Mass of the cantilevered bar considered for simulation

Effective mass = 0.933 * 294 g = 274.302 g

Cumulative mass fraction for first frequency     

= 179.8 / 274.65 = 0.654

Cumulative mass fraction for second frequency 

    = 235.3744/274.657 = 0.855

Cumulative mass fraction for third frequency   

    = 235.3744/274.657 = 0.855

In this way the solver computes the values in the participation table.

I hope that the participation factor table will not be a myth for you in future ☺

Please let me know your comments and suggestions in the below section.

Keep visiting my blog for more concepts!!!