(TI-58/59) Natural Frequencies & Mode Shapes of Multi-Degrees of Freedom Systems …

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An excerpt from RCA Review, Natural Frequencies and Mode Shapes of Multi-Degrees of Freedom Systems on a Programmable Calculator, December 1978, Volume 39 Number 4, ISSN 0033-6831, pages 604-617

Abstract - The Holzer tabulation method for determining the natural frequencies of multi -
              degree of freedom torsional systems is relatively easy to automate on a computer
              or a programmable calculator. The Holzer method has been extended to translational
              systems consisting of masses and springs configured so that the model
              starts with a mass and ends with a mass. For example, the method has been used
              to determine the natural frequencies of freight trains with an engine in the front
              and a caboose in the rear. The method presented here extends the basic Holzer
              theory further to accommodate lumped parameter structural models. A program
              is developed for a programmable calculator for determining the natural frequencies
              and mode shapes of multi-degree of freedom systems.

1. Holzer Tabulation Method
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2. Use of Programmable Calculator
It is obvious that a large number of simple calculations are necessary to
determine the natural frequencies and mode shapes of a multi-degree-
of-freedom structural model. Since the calculations are repetitive, it is
a simple job to program this problem for a computer or programmable
calculator.
  A program for a TI-58/59 programmable calculator has been developed.
The program assumes ω to be 10 radians and runs through the
Holzer tabulation calculations looking for a change in the sign of χ_N. If
χ_N changes sign (plus to minus) between 0 and 10 radians, the program
subtracts 5 radians from ω for averaging, divides by 2π, rounds the value
to the nearest whole number and displays the answer as 1 Hz. If χ_N does
not change sign in 10 radians, the program will add 10 radians to ω and
will repeat the above process. The angular frequency ω will be incremented
by 10 radians until χ_N changes sign. The calculator will then
compute the frequency and display the results in Hz. The displacement
between masses resides in the calculator memory and can be extracted
for developing mode shapes.
  The TI-58 contains enough memory to calculate the natural
frequencies and mode shapes of a system containing up to seven masses
and seven springs. The following description of the structural Holzer
program is presented here to enable the reader to use it without mastering
the art of programming calculators or computers …
…
3. Three-Mass, Three-Spring Structural Holzer Program
The details of a three-mass three-spring structural Holzer program for
a TI-58/59 are described below. …
…
The program for the calculator is as follows.
…
4. Example
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Once the calculator is programmed, the data is loaded as follows:
…
The natural frequencies are now computed with the following sequence:
…
  Before the mode shapes are determined, the calculator must be taken
out of the whole integer mode with the following key strokes: INV, 2nd,
and FIX. The mode shapes are determined by computing the deflections
between masses, normalizing to unit deflection, and plotting, as shown
in Fig. 7(a-c).

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