INFLUENCE OF DEGREES OF FREEDOM ON THE DETERMINATION OF NATURAL FREQUENCIES OF BRIDGE STRUCTURES

Abstract

The paper presents the results of a study of the influence of the number of degrees of freedom on the accuracy of determining the natural frequencies of bridge spans. The problem is related to approximate analytical modelling of the dynamic behaviour of bridge structures, where the continuously distributed mass of the span structure is replaced by a system of concentrated masses. This approach significantly simplifies the mathematical apparatus and allows analytical solutions to be obtained, but it is accompanied by errors, the magnitude of which depends on the number of degrees of freedom of the model. The aim of the study is to quantitatively assess the influence of the number of concentrated masses, which is equivalent to the number of degrees of freedom of the system, on the accuracy of determining the natural frequencies of bridge spans. To achieve this goal, analytical and numerical modelling of the dynamic characteristics of a beam span structure, which is considered as an elastic rod with constant stiffness and uniformly distributed mass, was performed. Within the analytical approach, the solution of the differential equation of beam vibrations is presented, taking into account the boundary conditions for a hinged bridge structure. Models with different numbers of concentrated masses (from one to three) were constructed, for which the natural frequencies of vibrations were determined. The obtained values are compared with the analytical results for a beam with distributed mass (a system with an infinite number of degrees of freedom). To verify the analytical results, a modal analysis was performed in the LIRA- FEM software package, which implements the finite element method. The analysis showed that when replacing the distributed mass of the system with one or two point masses, the error in the results is approximately 42% and 22%, respectively, compared to a beam with continuous mass. When the number of masses is increased to three, the analytical and numerical results are almost completely consistent.