Oscillations of finite-dimensional models of an extensible catenary

This article is devoted to the study of natural oscillation frequencies of finite-dimensional models of a stretchable flexible catenary. An analytical solution for two- and three-dumbbell models is presented, as well as the results of computer modeling of a twenty-dumbbell model of a stretchable catenary. In the case of an analytical approach, a coordinate solution method is used, in which the coordinates of the concentrated masses of dumbbell models in a deflected position are calculated. In the case of the numerical approach, the MSC.ADAMS software package is used, which allows analyzing the statics, kinematics, and dynamics of multibody systems. The results obtained for the considered stretchable catenary models are in good qualitative agreement with each other. Besides, when considering the limit transitions from the stretchable variant to the non-stretchable one, there is also a good consistency of the expected effects with the found results. For a finite-dimensional twenty-dumbbell model of a non-stretchable catenary with concentrated parameters, the first three dimensionless frequencies are compared with the frequencies of a continuous model, the values of which were found earlier. There is an excellent similarity of the results, confirming the applicability of the twenty-dumbbell scheme for describing the dynamics of catenary at low oscillation frequencies. In addition to determining the frequencies familiar to the classical non-stretchable catenary, an analysis of new ''migrating'' frequencies is carried out, which appear as a result of the emergence of additional degrees of freedom due to the consideration of stretchability. Frequency dependencies on the parameter characterizing the compliance of the catenary are constructed, which allows for estimating how quickly the ''migrating'' frequencies move from the high-frequency range to the low-frequency zone as the stiffness of the chain weakens. The formulas obtained and the models considered have both theoretical value and good applicability for applied tasks.

catenary, oscillation frequencies, stretchability, finite-dimensional model