THE VIBRATIONS OF AN EXTENSIBLE FLEXIBLE THREAD WITH A SMALL SAG

The free vibrations of an extensible flexible thread with a small sag are considered. A new form of solution of the equations of equilibrium of an extensible flexible thread is obtained. The differential equati ons of small vibrations about the equilibrium position are derived. An asymptotic analysis of the vibrations out of the vertical plane is ca rried out. It is established that these vibrations are close to the vi brations of a string. An asymptotic analysis of the low-frequency and high-frequency vibrations in the vertical plane is carried out. It is established that the natural frequencies and forms of the low-frequenc y vibrations depend very much on two small parameters: the parameter e psilon, characterizing the sag value, and the parameter delta, charact erizing the degree of the thread stretching. It is proved that the low -frequency transverse vibrations when epsilon(2)/delta much less than 1 are close to the vibrations of a string, and when epsilon(2)/delta m uch greater than 1 they are close to vibrations of an inextensible thr ead. If the quantities epsilon(2) and delta have the same asymptotic o rder, the most representative asymptotic form is obtained. In the high -frequency region there are longwave longitudinal and short-wave trans verse vibrations, irrespective of the ratio of the two small parameter s. More complex forms of vibrations also sometimes arise, characterize d by interaction between the longitudinal and transverse motions. Asym ptotic expansions, which are uniformly valid over the whole frequency range, are obtained. (C) 1998 Elsevier Science Ltd. All rights reserve d.