الفهرس 
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Abstract
There are many practical flexible structures, which have beam -like shapes and under large overall motions as ’rods, satellite antenna, rotating shafts of the twisting spindles, the spinning spindles, the shafts of the electric spindles, the shafts of the centrifugal pot...etc.
A flexible rotating shaft is a part of a great interest for engineers in the industry as the Textile industry; this industry is a very vital industry for the economy of Egypt. The spinning and twisting machines are the bottleneck of the mill. Therefore, owing to the necessity of increasing the productivity, it is necessary to increase the speeds of the shafts.
The main problem that stands against the increase of the rotating speeds is the increase of the mechanical vibrations; thus it is very important to choose the working speeds of the shafts carefully so that, we can eliminate or, at least decrease the mechanical vibrations to a reasonable limit. Thus, it is very important to investigate the dynamics of the spindle i.e. to evaluate the natural frequencies, critical speeds and the corresponding deflection along the shaft’s length.
The dynamical investigations of any flexible structure as the spindle of the ring twisting machines are mainly based on either the SIMPLE BEAM THEORY (by EULLER- BERNOULLI [3]), or, THE TIMOSHINKO’S BEAM THEORY (by TlMOSHfNKO [4].)
Many research