FINDING EIGENVALUES AND EIGENVECTOR OF A MATRIX WITH SMATH STUDIO

One of the limitations of Smath Studio is don't have a direct function to obtain de eigenvalues and eigenvectors of a Matrix, but it don't means that it is impossible to do it with the Smath Studio.

An important use of Eigenvalues and Eigenvectors is for identify the natural frequencies of a system with manu degrees of freedom, as rotor of turbines and several rotating machinery, as examples.

In "Introduction to the use of SMath Studio", Prepared by Gilberto E. Urroz, May 2010, it is showed many of functionalities of this freeware software, and the last item is about Eigenvalues and Eigenvectors. In the last Chapter of his work, is furnished a step-by-step how to obtain the eigenvalues and eigenvectors of any square matrix with real numbers as roots of the characteristic equation.

A - SQUARE MATRIX

λ - EIGENVALUES

I - IDENTITY MATRIX WITH THE SAME SIZE OF A.

Characteristic equation - "n" is the number of rows (or columns) of the matrix.

The following program steps is an updating of the work of professor Gilberto E. Urroz, for the 2021 version of the Smath Studio.

The "p" vector is the vector of the "p" factors of the characteristic equation. There is one "p" factor more than the number of rows (ou columns) of the matrix. In this case, in due of there is 4 (four) eingenvalues, it is showed 4 (four) eingenvectors.

The results of this example:

The Euclide's norm of each eigenvector is 1, and the product of the transposed vector by the each on vector is closely to 0, the eigenvalues are correct.

Solving the Problem 67 of "Mechanical Vibrations" - den Hartog:

For avoid the issue with units, it is working with off-dimension matrix.

The small difference is due to unity conversion and the Rotational Rigid modulus adopted.

In the highest critical speed, the vibration of the both discs are in opposites fase. In the lowest critical speed, both discs are in the same fase of vibration

REFERENCES

Introduction to the use of SMath Studio - Prepared by Gilberto E. Urroz, May 2010

Mechanical Vibrations - J. P. DEN HARTOG - DOVER PUBLICATIONS