The calculation of the bending frequencies and modes of space structures

Аuthors

Malykhina O. I.^*, Makaryants G. M.

Samara National Research University named after Academician S.P. Korolev, Moskovskoe shosse, 34, Samara, Russia

*e-mail: maloliya@ya.ru

Abstract

The article addresses selecting the optimal size for beam-type finite elements (FE) in dynamic analysis models for elastic structures of rocket and space technology. When designing a launch vehicle (LV), it is crucial to ensure strength reliability with minimal structural weight under intense dynamic loads. The accuracy of dynamic calculations depends on correctly identifying natural frequencies and mode shapes using FE models. However, uncertainty in choosing the FE size leads to difficulties in model verification. Using beam-type FEs with a lumped mass matrix (LMM) to model the LV structure, often related to liquid rocket fuel simulation, can reduce calculation accuracy. Meanwhile, standard verification through grid convergence studies by reducing FE size leads to unacceptable time costs in engineering practice. 
Therefore, this study aims to develop recommendations for the maximum allowable size of a beam-type FE with LMM for determining the modes and frequencies of an LV structure.
Initially, the problem of finding modes and frequencies was solved for a test case of a free-free beam with uniform mass and bending stiffness. This case has an analytical solution, which served as a reference for evaluating modeling accuracy. Two models were created: one with traditional LMM and another with coupled mass matrix (CMM). It was found that the rational number of elements using CMM is uniquely determined as a value equal to the increased by one the number of bending vibration modes within the frequency range of interest. Such a clear relationship between the number of elements and the vibration mode was not observed for the LMM case.
Subsequently, the developed recommendation for selecting beam-type FE size was used to create a reference dynamic model of a launch vehicle built with CMM. This reference model was used to verify the FE model with LMM for a real large-length tandem-type LV design, enabling the determination of an adequate number of elements for the LMM model.

Keywords:

finite element, beam model, launch vehicle, coupled mass matrix, lumped mass matrix.

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