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

[SlideRule]

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
…
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
…
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).

BEST!
SlideRule

[Maximilian Hohmann]

Hello,

this made me curious because it coincides with the era when I used my Ti-59 at university for similar calculations. But I have never heard of "Holzers method" or Holzer himself before, nor has Wikipedia. In other papers I have found references to a 1907 shipbuilding texbook - so at least we know how old this method is!

Next I asked my new friend ChatGPT who told me quite a lot about it, not uninteresting:

Holzer's method is a procedure used in the field of mechanical engineering, particularly in the analysis of torsional vibration in multi-degree-of-freedom (MDOF) systems such as shafts and rotors. It is a tabular method that helps in determining the natural frequencies and the mode shapes of the torsional system. Here’s a detailed overview:
Basic Concept

Holzer's method involves:

Defining the System: The mechanical system is modeled as a series of discrete masses (inertia) connected by massless, elastic shafts (springs). The torsional stiffness and moments of inertia for each segment are identified.

Formulating the Equations: The method uses the fact that the sum of torques (due to inertia and restoring torques from the shaft stiffness) at each mass must be zero for a free vibration condition.

Assuming a Trial Frequency: A trial value for the angular frequency (ωω) is assumed.

Tabulating Calculations: The assumed frequency is used to compute the angular displacements and torques throughout the system in a step-by-step manner. The computations are typically organized in a table, starting from one end of the system and progressing to the other.

Checking for Compatibility: The end conditions are checked. If the trial frequency leads to a compatible solution (e.g., the torque at the last mass is zero), it is considered a natural frequency. If not, the process is repeated with a new trial frequency.

Steps in Holzer’s Method

Initialization: Start at one end of the system, typically setting the angular displacement of the first mass to an arbitrary value (often zero for simplicity).

Calculations:
For each mass, calculate the torsional deflection using the assumed frequency and the torsional stiffness of the connecting shafts.
Compute the corresponding torques due to the inertia and restoring torques.
Propagate these calculations to the next mass.

Iteration:
Continue this process to the last mass.
If the last computed torque does not satisfy the boundary condition (typically zero torque for free ends), adjust the trial frequency and repeat.

Natural Frequencies: The correct natural frequencies are those for which the computed torques and displacements satisfy the boundary conditions at both ends of the system.

Example

Consider a system with three disks connected by two shafts. The procedure would be:

Assume an initial angular frequency.
Start with an initial displacement for the first disk.
Compute the angular displacement for the second disk using the torsional stiffness of the first shaft and the assumed frequency.
Compute the torque in the second disk due to its inertia and the restoring torque from the first shaft.
Propagate this to the third disk.
Check if the final computed torque at the last disk matches the boundary condition.
Adjust the frequency and repeat until a satisfactory solution is found.

Applications

Holzer’s method is widely used for:

Design and analysis of drive shafts and rotor systems: To ensure that the operating speeds do not coincide with the natural frequencies, which can lead to resonance and potential failure.
Vibration analysis of mechanical systems: To identify and mitigate sources of harmful vibrations.

Advantages

Simplicity: It is straightforward and does not require advanced computational tools.
Applicability: Effective for both simple and moderately complex systems.

Limitations

Manual Iteration: Can be tedious for systems with many degrees of freedom.
Approximation: The method is an iterative approximation, and finding the exact natural frequencies can be time-consuming.

Holzer's method remains a fundamental technique for torsional vibration analysis, providing engineers with a reliable way to predict and mitigate vibration issues in mechanical systems.

And about Holzer himself (shortened):

Holzer's method is named after Heinrich Holzer, an Austrian engineer who developed it in the early 20th century. Heinrich Holzer made significant contributions to the field of mechanical vibrations, particularly in the analysis of torsional vibrations in rotating machinery.
Heinrich Holzer's Contribution

Development of the Method: Heinrich Holzer

developed the method to provide a systematic way to determine the natural frequencies of torsional systems, which was crucial for ensuring the reliability and safety of mechanical systems such as drive shafts and rotors.

Application in Engineering: Holzer's method became widely adopted in mechanical engineering due to its straightforward approach and practical applicability, especially before the advent of modern computational tools.

Regards
Max