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National Advisory Committee for Aeronautics, Technical Notes - Calculation of Uncoupled Modes and Frequencies in Bending or Torsion of Nonuniform Beams

A procedure is presented for the calculation of frequencies and
modes of nonuniform beams in uncoupled bending. and torsional vibration.
Based on the principle of the Stodola method, the procedure consists
of solving the differential equation of equilibrium for vibration by
a method of successive approximations. Basic principles ofengineerins
beam theory are employed. in the method, and the integrations involved
are performed by improved. numerical methods. 'An effort has been made
to perform all calculations in a manner consistent with the accuracy
to which physical'constants in built-up beams are ordinarily known.

Higher modes are readily found by use of the orthogonality
relation between normal modes. The frequency is'found simply as the
square root of the proportionality factor existing between modal
deflection curves in successive approximations. All computations are
tabular in form and are performed mentally or with the aid of a slide
rule. Comparison made with available exact analytical solutions shows
that the method gives for practical purposes the. exact answer. Special
consideration has been given to the treatment of various boundary
conditions that are found in the vibration of aircraft structures.

The cantileVer bes'm, the free—free beam, beams with concentrated
masses, beams mounted on springs, beams elastically coupled to masses,
and so forth, are shown to be handled with practical simplicity. _ In
order to serve as a guide in the solution of practical problems, the
procedures for handling a mmber of different cases are illustrated by
a liberal use of examples.

In the dynamic analysis of aircraft structures, the determination
of the natural modes and frequencies is of basic importance. A
number of methods for calculating modes and frequencies have been
developed; each method has certain desirable features. The objective
of the present paper is to develop a procedure which is readily
learned by anyone familiar with engineering beam theory, is easy and .
quick to apply without the use of complicated computing devices, and
gives results within the range of accuracy with which the physical
properties of the structure (mass, stiffness distribution, etc.) can
ordinarily be determined;' An adaptation of-the successive- ,
approximation procedure of Stodola (see reference 1) fulfills these
requirements and has-the additional advantage that data necessary for
the analysis of stresses due to vibration are obtained during the
computations for the modes. In the present paper, this successive-
approximation or iteration method is employed to obtain solutions to
the differential equations of _equilibrium for bending and torsional
vibrations.