Modal mass and kinetic energy in FEM modal analysis

[Arjan82]

TL;DR Summary

What is the formula for modal mass and kinetic energy of a modal analysis FEM computation? I do *not* mean effective modal mass.

So, I use Ansys (well known FEM software) and get the next output for a modal analysis toy problem (If you happen to know Ansys that's a pre, but I promise it shouldn't matter). The problem is a simple beam, clamped at one end. I used 160 20-node brick elements to solve it (so no Timoshenko beams or something like that).

 The modes requested are mass normalized (Nrmkey on MODOPT).  However, 
 the modal masses and kinetic energies below are calculated with unit 
 normalized modes.                                                     

        ***** MODAL MASSES, KINETIC ENERGIES, AND TRANSLATIONAL EFFECTIVE MASSES SUMMARY *****

                                                                         EFFECTIVE MASS
  MODE  FREQUENCY   MODAL MASS     KENE      |      X-DIR      RATIO%   Y-DIR      RATIO%   Z-DIR      RATIO%
     1   81.73       39.42      0.5199E+07   |     0.000        0.00   95.85       61.05   0.000        0.00
     2   159.3       40.53      0.2030E+08   |     0.000        0.00   0.000        0.00   96.07       61.19
     3   490.2       41.77      0.1981E+09   |     0.000        0.00   30.22       19.25   0.000        0.00
     4   593.4       31.73      0.2206E+09   |     0.000        0.00   0.000        0.00   0.000        0.00
     5   859.8       48.92      0.7138E+09   |     0.000        0.00   0.000        0.00   31.92       20.33
     6   1268.       77.93      0.2472E+10   |     126.6       80.61   0.000        0.00   0.000        0.00

So I know exactly how to get the effective mass, which is dependent on direction. And by exactly I mean exactly. I extract the mass and stiffness matrix from Ansys, compute the eigenvectors of this problem (with Matlab):

$$
\left[ K \right] \left\{ d \right\} = w \left[ M \right] \left\{ d \right\}
$$

with $\left[ K \right]$ the stiffness matrix, $\left[ M \right]$ the mass matrix, $\left\{ d \right\}$ an eigenvector and $w = \omega^2$ the eigenvalue. All is 'mass normalized' such that $\left\{ d \right\}^T \left[ M \right] \left\{ d \right\} = 1$. And now we can compute the participation factor for the x-direction (assuming mass normalization):

$$
L_x = \left\{ d \right\}^T \left[ M \right] \left\{ r_x \right\}
$$

with $\left\{ r_x \right\}$ the influence vector, or just a vector with 1's at all degrees of freedom of the x-direction and 0 everywhere else. And now the effective modal mass in the x-direction is simply $L_x^2$. If I do this with e.g. Matlab I get exactly the same results as Ansys does (all digits are the same, except maybe the last). So, no problem there.

But the modal mass and kinetic energy (KENE) on the left side of this table are a mystery to me. I need the formula for that but cannot seem to find it, not in the documentation of Ansys, not on the internet (for which the results get swamped by explanations of effective mass) and not in any book about the subject I own. Who knows how to compute those numbers?

 

[Arjan82]

Allright, apparently the documentation of Ansys is also a mystery for me, the answer is just right in there 😆. Also, I'm overthinking things, as usual 🤔

The modal mass is simply

$$
m = \left\{ d \right\}^T \left[ M \right] \left\{ d \right\}
$$

And the kinetic energy

$$
KE = \frac{1}{2} m \omega^2
$$

 

[Arjan82]

Oh, and crucially, the $\left\{d\right\}$ vector is not mass normalized, but unit normalized (i.e. the max absolute value of $\left\{d\right\}$ is equal to 1), otherwise $m$ would just be 1.