Non-linear free vibrations of the column loaded with a mass element and a local heat source

Abstract

In this paper, the problem of non-linear vibrations of an articulated column subjected to two loads was considered. The first load is axial compression associated with the action of the mass element. The second is the heat load in the form of a local heat source. In such a system, the problem analysis was divided into two parts. The first illustrates the influence of temperature on the physical and mechanical properties of the structure. The second - concerning non-linear vibrations of the system. The problem of heat flow in the column was formulated on the basis of the classical Fourier heat conduction equation and solved with the use of the Finite Element Method. Distribution of material properties (Young’s modulus) was obtained in subsequent moments of exposure of the column to the heat source. The obtained results were used in the problem of system vibrations, which was formulated on the basis of Hamilton’s principle. Due to the non-linearity, the perturbative small parameter method was used to finally derive the differential equations describing the behaviour of the system during vibrations. Numerical calculations were carried out on the basis of which the characteristic curves of the system (on the load - natural frequency plane) for various stages of heating the column were determined. Moreover, the influence of the heating time of the system on its non-linear natural frequency was presented. The significant influence of the longitudinal inertia of the mass element causing the load on the vibrations of the structure was shown. The mathematical model and the correctness of the performed numerical simulations were verified by experimental research.