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National Advisory Committee for Aeronautics, Technical Notes - The Calculation of Modes and Frequencies of a Modified Structure from Those of the Unmodified Structure

A method is developed for the calculation of the natural coupled or
uncoupled frequencies and modes of a structure with modifications, such
as the addition of concentrated masses or springs, directly from the
known modes and frequencies of the unmodified structure. The modes of
the modified structure are expanded in terms of the modes of the unmodi-
fied structure. A characteristic equation and a frequency determinant,
the order of which is twice the number of modifications, are derived by
the use of the Galerkin method. Numerical examples are presented to show
the accuracy of the method and the number of modes and frequencies of the
unmodified structure necessary for agreement with exact solutions.

The calculation of the natural modes and frequencies of an airplane
structure is usually required for various loading conditions. The varia—
tion of conditions may be brought about by changes in pay load, changes
in the amount of fuel carried, the addition of tip tanks, and so forth.
These changes may be regarded as modifications to a primary structure.

In addition to weight changes, the addition of elastic restraints, such
as spring supports which may be used in ground vibration tests, may also
be considered as modifications. Thus, in this paper the basic or primary
structure is known as unmodified, and the structure after masses and
springs are added is known as modified.

Present methods of calculating modes and frequencies require a sepa—
rate and independent calculation for each modification to the primary
structure. In order to simplify the calculations of these modes and fre-
quencies, a method is developed in this paper that allows, with very
little extra work, the calculation of the modes and frequencies bf a modi—
fied structure directly from -the modes and frequencies of the unmodified
structure.

The paper presents both a theoretical analysis and numerical
examples. In the theoretical analysis a frequency determinant and the
modal functions are derived. The order of the frequency determinant is,
in general, twice the number of modifications to the primary structure.
Each element of the determinant is a series having terms that are func—
tions of the modal shapes and frequencies of the primary structure. The
numerical examples illustrate the convergence of the series and the accu-
racy with which the frequencies and modes of the modified structure can
be calculated.