On the forced vibrations of three bodies elastically supported on a beam

In this article, the problem of forced vibrations of three bodies elastically supported on a beam is posed. To solve this problem, in contrast to the classical method, when the system is divided into parts for which equations of motion are compiled, their solutions are found, and then solutions are stitched together in the breakdown methods, Hamilton's variational principle is applied in this article, resulting in a system of differential equations, three of which are ordinary second-order differential equations with respect to time describing the motion of solids, and the partial differential equation of the second order with respect to time and the fourth order with respect to the longitudinal coordinate of the beam. The solution of the resulting system is sought in the form of the product of the amplitudes on the harmonic function of the frequency of the external disturbing disturbance. At the same time, for solids, the amplitudes are constant values, and for a beam, they are variable. Then, after some transformations of the resulting system of amplitude equations, the transmission coefficients are determined in the form of ratios of these amplitudes to the amplitude of disturbances. The above refers to the method of investigating the forced vibrations of three solids elastically fixed along the beam, which is based on the Hamilton variational principle. At the same time, the solution of the hybrid system of differential equations obtained as a result of the application of the variational principle, which includes both ordinary differential equations and partial derivatives, is understood in a generalized sense. The application 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 places where bodies are connected to the beam. According to the found transmission coefficients, resonant frequencies are determined, and they are compared with their own frequencies of the specified mechanical system.