Mathematical model of the damped variable cross section beam in transportation

The paper concerns the problem of vibrations of beamlike system with variable cross section. The beam is treated as the movable system in transportation. The considered problem focuses on modelling and dynamic analysis of geometrically nonlinear beam systems in rotational motion within the context of damping. The major scientific purpose of the paper is to elaborate the mathematical model of such a system. Additionally, the main motion impact on the local vibrations due to the mathematical sense is determined. Moreover, it is necessary to remember the interactions between damping forces of the above mentioned mechanisms and the transportation effect. The main motion of the system is treated as transportation, whereas the vibrations of the system are treated as relative motion. There are two types of systems considered: simple vibrating longitudinally and simple vibrating transversally in the plane transportation. The most interesting elements of the analysis determine the dynamic state of the system and present the mutual coupling of vibration amplitudes, natural frequency, and transportation velocity. Analysis of systems moving with low velocities or vibrating only locally treats the systems as already known models in literature. There are many scientific articles where the forms of vibrations of these systems have been described. Due to the obtained results it will be possible to confront mathematical models with the known stationary and non-stationary systems. As regards complex and simple systems running at high speed, the resonance phenomenon can be noticed, and depending on the amplitude and frequency of vibrations, we consider the following cases: when the amplitude reaches theoretical infinity leading in practice to permanent damage of the mechanism or when the amplitude of vibration reaches a certain speed which can cause the decrease of durability of the whole system. The adequate practical usage of the above mentioned researches is justified by its wide range of applications. In the majority of technical cases, further analysis of the systems is considered to be far too much simplified when we ignore the elements of flexibility, damping, or the nonlinear geometry of the beam. All the mentioned influences are presented in the derived mathematical model in form of equations of motion.