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Full text of "NASA Technical Reports Server (NTRS) 19730000384: Improved procedures for mass matrix-reductions in eigenvalue solutions"

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September 1973 




NASA Pasadena Office 

■ - - v — - - . •» ■- v ^ 

NASA Tech Briefs announce new technology derived from the U.S. space program. They are issued to encourage 
commercial application. Tech Briefs are available on a subscription basis from the National Technical Information 
Service, Springfield, Virginia 22151. Requests for individual copies or questions relating to the Tech Brief program may 
be directed to the Technology Utilization Office, NASA, Code KT, Washington, D.C. 20546. 

Improved Procedures for Mass Matrix-Reductions 
in Eigenvalue Solutions 

Determination of natural frequencies and mode 
shapes for structures with moderately large numbers 
of degrees of freedom (say, more than 200) requires 
solution of an eigenvalue problem for which mass 
matrix-reduction schemes are often employed to con- 
dense the order of the problem so that it is consistent 
with limitations of computer storage or available 
time. Prevalent mass matrix-condensation procedures 
can be considered in the context of the well-known 
Rayleigh-Ritz technique; that is, in order to solve a 
system of equations of high order, the order of the 
system is reduced by assuming that the solution can 
be expressed in terms of a much smaller subset of 
preselected displacement functions. These displace- 
ment functions are customarily derived to incorpo- 
rate unit displacements or unit external loads at 
selected “indicator” degrees of freedom. Straight- 
forward mathematical procedures are then applied 
to generate a reduced mass matrix with order equal 
to the number of displacement functions. Then, the 
order of the problem solution can be condensed to 
the order of the reduced mass matrix. In general, 
accuracy of the solution will depend to some extent 
upon validity of the selected functions. 

Analytical models of three structures (antenna 
pedestal and reflectors) were used to test mass matrix- 
reduction schemes using the Rayleigh-Ritz procedure. 
These models were of sufficiently small orders (165 to 
175 degrees of freedom) so that it was feasible to 
solve the eigenvalue problems without reduction for 
comparison with solutions obtained with reductions 
for various indicator modes. The accuracy of the four 
lowest mode shapes and frequencies was tracked 

through successive mass matrix-reductions with dim- 
inishing numbers of indicator degrees of freedom. 
These were chosen as carefully as would be expected 
in normal practice. Results for all three structures 
were approximately equivalent in accuracy and were 
consistently disappointing. The errors in natural fre- 
quency were higher than expected, and it was also 
found that retaining as many as 30 to 40 displacement 
functions (20% to 25% of the original orders) in the 
condensed problem did not always guarantee re- 
covery of the second (from the lowest) mode shape. 

Subsequently, two new procedures were developed 
and tested in an attempt to improve accuracy and 
regain validity for the previous results. The first pro- 
cedure was to augment an original set of displace- 
ment functions generated by indicators with six new 
displacement functions. These were developed by 
determining the six independent rigid body displace- 
ments of the structure for unit motions (3 translations 
and 3 rotations) at the foundation. Loading applied 
at each degree of freedom was equal to the product 
of the mass and rigid motion at the location, and the 
six sets of displacements for these loads were included 
as additional displacement functions. Accuracy of the 
solutions with indicator displacements was found to 
be greatly improved. 

The second procedure provided dramatic improve- 
ments for a given, and possibly poorly chosen, set of 
displacement functions. The procedure involves the 
use of loads, computed as the product of the full mass 
matrix and mode shapes found in the initial solution, 
to generate displacement functions that are the basis 
for a new solution; implementation is by the same 

(continued overleaf} 

This document was prepared under the sponsorship of the National 
Aeronautics and Space Administration. Neither the United States 
Government nor any person acting on behalf of the United States 

Government assumes any liability resulting from the use of the 
information contained in this document, or warrants that such use 
will be free from privately owned rights. 

procedure used to generate the rigid body loading 
displacements described previously. The remainder 
of the computation steps to complete the eigenvalue 
problem solution are identical to the initial steps. The 
additional computation time required, although pos- 
sibly insignificant in comparison with the solution 
time required without condensation, approaches the 
initial computation time. Some of the additional com- 
putation time could be saved by making -the number 
of mode shapes retained for the second cycle equal to 
only about two or three times the number of accurate 
mode shapes desired. Furthermore, although the re- 
sults clearly show a great improvement is achieved 
with only this second pass, additional improvements 
in accuracy can be obtained by continuing through 
several additional cycles. Indications are that with 
repeated cycling the results tend to converge toward 

the exact results for the number of mode shapes 


Requests for further information may be directed 

Technology Utilization Officer 
NASA Pasadena Office 
4800 Oak Grove Drive 
Pasadena, California 91103 
Reference: TSP '73-10384 

Source: Roy Levy of 
under contract to 
NASA Pasadena Office 


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