Page 384 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

12

Classical

Methods

The exact analysis for the vibration of systems of many degrees of freedom is
generally difficult and its associated calculations are laborious. Even with high­
speed digital computers that can solve equations of many DOF, the results beyond
the first few normal modes are often unreliable and meaningless. In many cases, all
the normal modes of the system are not required, and an estimate of the
fundamental and a few of the lower modes is sufficient. For this purpose,
Rayleigh’s method and Dunkerley’s equation are of great value and importance.
In many vibrational systems, we can consider the mass to be lumped. A shaft
transmitting torque with several pulleys along its length is an example. Holzer
devised a simple procedure for the calculation of the natural frequencies of such a
system. Holzer’s method was extended to beam vibration by Myklestad and both
methods have been matricized into a transfer matrbc procedure by Pestel. Many of
these procedures were developed in the early years and can be considered as
classical methods. They are now routinely processed by digital computer; however,
a basic understanding of each these methods is essential.

12.1 RAYLEIGH METHOD

The fundamental frequency of multi-DOF systems is often of greater interest than
its higher natural frequencies because its forced response in many cases is the
largest. In Chapter 2, under the energy method, Rayleigh’s method was introduced
to obtain a better estimate of the fundamental frequency of systems that contained
flexible elements such as springs and beams. In this section, we examine the
Rayleigh method in light of the matrix techniques presented in previous chapters
and show that the Rayleigh frequency approaches the fundamental frequency from
the high side.

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