To calculation of frequencies and forms of beam vibrations with arbitrary number of elastically fixed bodies

This article presents a method for studying the natural vibrations of a beam with an arbitrary number of elastically fixed solids, which is based on Hamilton’s variational principle. In this case, the solution of the resulting hybrid system of differential equations, including both ordinary and partial differential equations, is understood in a generalized sense. The use of the concept of a generalized solution is caused by the presence in the equations of the Dirac delta function, which must be taken into account at the points of connection of bodies to the beam. This method is used to calculate the natural frequencies and vibration modes of the system under consideration and their numerical implementation. A comparative analysis of the calculations made with foreign studies is carried out, which showed excellent agreement. It should be noted that in the above foreign work, the usual technique is used, which consists in dividing a composite mechanical system into parts, the equations of motion of which are quite simple, and then eliminating the interaction reactions of these parts. In the methodology proposed in the article, these reactions do not need to be taken into account explicitly. If necessary, they can be easily calculated if you have a ready-made solution.