Calculating Natural Frequencies With Extended Tuplin’s Method

Abstract

In this paper an efficient numerical procedure is described which yields the eigenvalue of a lumped mass torsional vibration directly from the frequency equation of the system. Special characteristics of Tuplin’s frequency equation allow all eigenvalues to be easily located and accurately evaluated from the frequency polynomial. In contrast to the general belief that extracting roots of polynomials is less efficient than matrix reduction methods, this paper demonstrates that the direct solution competes favorably with the modern eigenvalue routines such as QR and tridiagonal methods [15] [16] [17] [21] in torsional vibration problems. A BASIC program FUNG has been written based on the numerical concepts of this paper. The current version is able to solve multiple branch systems of many degrees of freedom subject to the restriction that no branch shall exceed 4 rotors and 4 shafts. The program has been tested for various examples and the output compared with the known results. Within the above range of applicability, this method beats the modern tridiagonal eigenvalue subroutines [16] [21] by a comfortable margin which ranges from 15 times to 120 times faster. The comparisons were made on the basis of solving the same problems on the same computer. FUNG assumes a tight tolerance of convergence for iteration (correct to approximately 14 significant digits).