# Approximate calculation of natural frequencies of oscillations of the plate with variable cross-section of the discrete-continuous inter-resonance vibrating table

2022;
: 41-50

Revised: May 10, 2022
Accepted: May 30, 2022
1
Department of Robotics and Integrated Mechanical Engineering Technologies, Lviv Polytechnic National University
2
Lviv Polytechnic National University
3
Department of Designing and Operation of Machines
4
Department of Strength of Materials and Structural Mechanics, Lviv Polytechnic National University,

Problem statement.  To ensure highly efficient inter-resonance modes of operation of vibrating equipment, the oscillating masses of the system must have certain inertia-rigid parameters, as well as a certain frequency of natural oscillations. The disadvantage of highly efficient inter-resonance oscillatory systems is that the third reactive mass must be small, and therefore the use of complex and large structures is impossible. Therefore, it is best to use the reactive mass as a continuous section. The continuous  section,  which  is  a  flexible  body,  optimally  combines  inertial  and  rigid  parameters. Scientific works have already considered the design of the vibrating table, in which the continuous section  is  an  ordinary  rectangular  plate  hinged  in  the  intermediate  mass.  This  decision  looks  quite promising.  However,  likely,  the  rectangular  shape  of  the  plate  is  not  the  best  option  to  ensure maximum  energy  efficiency.  Purpose.  Extend  the  method  of  calculating  the  natural  frequency  of oscillations  of  the  plates  by  the  approximate  Rayleigh-Ritz  method  using  the  general  hyperboloid equation to plates with variable cross-section for the proposed types of plates and check the results with the calculation in Ansys software. Methodology.  The calculations of the plates were performed using the basic principles of the theory of oscillations, in particular the Rayleigh-Ritz method in the software product MathCAD. Findings (results) and originality (novelty). Two types of elastic plates with  variable  cross-sections  are  considered.  In  the  first  case,  the  shape  of  the  plate  was  given  by quadratic functions, in the second case, it was described by trigonometric functions of cosine. In both cases, the same conditions of attachment in the intermediate mass were observed. The calculation of the first natural frequency of oscillations of the considered plates was performed using the approximate Rayleigh-Ritz method with the assumption that the deflection of the plates occurs on the surface of the hyperboloid.  The  reliability  of  the  obtained  results  was  verified  by  numerical  calculation  in  the software product Ansys. Practical value. It is assumed that the proposed types of plates can increase the dynamic potential of the vibrating machine. Scopes of further investigations. For further study of the considered types of plates as a continuous section of the inter-resonance vibrating machine, it is necessary to calculate their deflections at forced oscillations.

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