A computer program facilitates the selection of the dominant modes of vibration of a structure - especially the dominant modes that interact with specific locations on the structure. This program is intended for use in conjunction with any of the various vibration-mode-analysis programs based on the finite-element method (FEM). The program was written to satisfy a need to accurately and quickly sort through enormous amounts of information about the vibrational modes of the developmental X-33 aerospace launch vehicle in order to select the dominant modes and thereby reduce the subsequent computational burden of analyzing the vibrations of the vehicle. In the X-33 application, the program has been used to determine which modes interact most with specific locations and pieces of equipment that affect and are affected by stability or instability with respect to (1) flight control and (2) a phenomenon, called "pogo" (because it is reminiscent of oscillations like those of a pogo stick), which involves deformations of liquid-oxygen plumbing combined with sloshing and/or fluctuations in pressure of the liquid oxygen inside the plumbing. The program could also be adapted for use in analyzing the vibrations of structures other than that of the X-33.
The system of dynamical equations that describe the undamped free vibrations of a structure yields a set of roots or eigenvalues, which in turn, determine a set of natural mode shapes or eigenvector coefficients. Because the equations are homogenous, there is not a unique solution for the eigenvector coefficients; one can obtain only ratios among the eigenvector coefficients. The shape of a given mode is defined by the ratios among the amplitudes of the motion that occurs at the various points on the structure when the structure is excited at the natural frequency of that mode. If one of the elements of the eigenvector is assigned a certain value, the rest of the elements of the same eigenvector are also fixed because the ratio between any two elements is constant.
The present computer program takes advantage of the facts presented in the preceding paragraph. The program is based partly on the use of orthonormal modes and partly on the root-sum-square (RSS) measure of the modes that affect each degree of freedom (DOF) in the FEM dynamical model of the structure. The displacements in all modes for all DOFs are normalized to the unity of the largest eigenvector displacement.
With the help of this program, one can make a direct comparison between modes for a particular location. The program computes magnitudes of the resultant displacement values for each mode at selected DOFs and sorts to locate modes with highest values. The modes with the highest RSS displacement values are deemed to have the highest overall motion and kinetic energy for the given location or node. Then by keying on the RSS displacement values for locations of interest, one can easily filter out thousands of modes that are not of interest, leaving only the modes of interest.
The program also sums the absolute displacement along the three coordinate axes for each mode. Again, the program sorts to locate modes with highest values. The result of this sorting process yields insight into the directional dynamic behavior of the modes. This aspect of the program can be used, for example, to identify the pogo modes as those for which the longitudinal (x) axis yield the highest values.
This program was written by Homero Hidalgo of Marshall Space Flight Center.