Determination of vibration activity parameters of elastic bending rod systems with discrete-continuous mass distribution

Аuthors

Sobolev V. I.^1*, Zenkov E. V.^1**, Chernigovskay T. N.^{1, 2***}, Marhuk N. ^1****

1. Irkutsk National Research Technical University, 83, Lermontov str., Irkutsk, 664074, Russia
2. Irkutsk State Transport University (IrGUPS), 15, Chernyshevsky str., Irkutsk, 664074, Russia

*e-mail: sobolevvi@ex.istu.edu
**e-mail: jovanny1@yandex.ru
***e-mail: tannikch@gmail.com
****e-mail: fts07@mail.ru

Abstract

The proposed work is devoted to the issues of formation and vibration analysis of dynamic system models containing elastically bending elements with distributed and concentrated inertial and rigidity parameters based on the harmonic element method. Modeling of dynamic processes occurring in systems with irregular distribution of region boundaries and irregular boundary conditions - structures of various purposes is carried out, as a rule, on the basis of mass discretization. The use of such methods is associated with known difficulties - discrete models are a priori approximate with limited capabilities for error estimation; dynamic parameters of the model depend on its dimension, as well as on the transformation methods; numerical results with arrays and matrices of large dimension complicate the possibility of analysis and evaluation of the calculation results. Calculations of structures for stationary dynamic effects based on the use of elements with distributed and concentrated masses make it possible to avoid the listed consequences of full discretization. However, such discrete-continuous (hybrid) dynamic models are associated with the need to stitch together heterogeneous elements at the formation stage and the inevitable difficulties of solving such "combined" systems containing ordinary differential equations and partial differential equations. The listed problems are resolved using the author's harmonic element method, which performs nodal stitching of heterogeneous elements and allows obtaining solutions in the form of oscillation amplitudes of the nodes of the combined model in certain required directions. In this case, all the properties of the finite element method are preserved, allowing the solution to be implemented under various boundary conditions and irregular distribution of physical and geometric parameters and boundaries of the computational domains. Nodal stitching of model elements resembles the approach used in the formation of finite element models, but unlike the finite element method intended for solving dynamics problems and containing frequency and inertial parameters in the equations. These features allow us to distinguish the proposed method into a separate class called the harmonic element method.

Keywords:

bending elements, concentrated masses, Euler-Bernoulli equations, computational modeling, dynamic equations

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