Grid Point Kinetic Energy (GPKE) is a simple and direct measure that can help determine the degrees of freedom that are important for various modes. GPKE identifies the DOF that have large mass and/or large amplitudes. Retaining degrees of freedom which have large kinetic energy will usually result in a good TAM.
Unfortunately, GPKE alone is often insufficient for determining which degrees of freedom to retain. One problem with the kinetic energy method is that it is not able to distinguish among a number of “equally important” DOF, since they will all have nearly equal GPKE. Another problem is that it tends to be mesh dependent. A portion of the structure with a fairly coarse mesh will have much more GPKE per DOF than a similar part of the structure with a finer mesh even though a similar number of DOF would be required in each region.
While the GPKE method alone may not clearly select the master DOF, the method can be a tremendous help to give a “candidate” selection of important DOF. In addition, Nastran provides GPKE information as a standard output option using the GPKE Case Control command.
The format of the GPKE command can be found in the Nastran Quick Reference Guide (QRG)[6]. The data is output using either the PRINT or PUNCH option, but is currently not written to an OUTPUT2 file for postprocessing in I-DEAS. The PUNCH option is recommended, since this is easily read into Matlab for sorting and reporting of results.
The GPKE in Nastran is calculated as:
(2.5.1-1)
where
= Mass matrix (G-set)
= Mode shapes (G-set)
Ä = Term-by-term matrix multiplication
The quantity has the property that the sum of the individual grid point kinetic energies for all DOF in the model will add up to total kinetic energy in the mode. Nastran normalizes the modes so that the sum of Grid Point Kinetic Energies for any mode is 100.0. For a diagonal mass matrix, the grid point kinetic energy is simply the mode shape coefficient squared times the mass at that DOF. For a fully populated mass matrix, the result is more complicated and involves mode shape coefficients at other DOF. In this case there is no guarantee that the individual grid point kinetic energies will be positive[7].
The GPKE can be processed directly from Nastran without the use of any special purpose alters. However, an alter called ‘write_gpke’ is supplied with TAMKIT. This performs two functions. The first is to write the GPKE in a DMIG format that includes only the nonzero terms. The second is to calculate GPKE based on A-set rather than G-set matrices. The calculation in (2.5.1-1) is on the G-set and will include DOF in the M-set. This can be useful in identifying important DOF, whether or not they are in the M-set, but it is not possible to put an M-set DOF in the A-set without first modifying the model. Because of this, it is often advantageous to do the GPKE calculation on the A-set[8]. This can be expressed as follows:
(2.5.1-2)
where
= Mass matrix (a-set)
= Mode shapes (a-set)
Another way to use GPKE to select DOF is to address
the problem as one of maximizing the frequencies of the problem. To do this one starts with a set of DOF and replaces
the ASET cards with SPC cards. The resulting modal calculation will be
the eigenvalues of
. To increase these eigenvalues
the user would examine the GPKE and SPC those DOF with high GPKE in the
lowest frequency modes. This process would be repeated until the lowest
frequency of
is above the frequency range
of interest.