One of the most important prerequisites for scientific and technological progress in construction and mechanical engineering is the improvement of structural analysis methods for buildings and machinery. Among the tasks necessitated by the need to increase the reliability and cost-effectiveness of products, a primary focus is the development of methods for analyzing the dynamic loading of structural elements and parts. This involves using calculation schemes that most accurately reflect real operating conditions and the properties of the materials used.
When designing various construction and mechanical engineering structures, it is often necessary to calculate the steady-state vibrations of massive units and components in a three-dimensional formulation [1]. Recently, numerical methods based on potential method relations have been increasingly used to solve spatial problems. Their appeal for use in computer-aided design (CAD) systems stems from the fact that only the boundary of the computational domain needs to be discretized, leading to a reduction in the problem's dimensionality. At the same time, it should be noted that most existing boundary-element approaches are aimed at solving static problems. This is due to the complexity of dynamic fundamental solutions, the integration of which is the main and most labor-intensive part of the potential method algorithm. Such complexity increases significantly when it is necessary to account for the damping properties of the material, for example, when investigating vibrations with frequencies close to resonance [2]. In this work, the influence of internal friction on the parameters of the stress-strain state (SSS) is accounted for based on E.S. Sorokin’s theory, which utilizes a complex modulus of elasticity while maintaining the fundamental relationships of the linear theory of elasticity between SSS components.
These studies are dedicated to the development of an efficient numerical methodology for integrating the kernels of elastodynamic potentials in the problem of massive body vibrations, taking into account damping caused by internal friction in the material.
In the regular case—when the pole does not belong to the element over which integration is performed—this procedure usually does not cause complications. Depending on the location of the integration pole, two integration methods are used: 1) Gaussian quadrature formulas; 2) expansion of the exponent into a power series. Numerical experiments show that the closer the pole is to the integration domain, the more effective the second method becomes.
To calculate singular integrals, the following approach is proposed: divide the triangular element into two parts: the first part is a circle centered at the pole; the second is the remaining part of the triangle. Integrals over the circle exist in the sense of the Cauchy principal value and are calculated analytically. When integrating over the remaining part of the triangular fragment, a conformal mapping method is used, which allows transforming a curvilinear quadrilateral into a square, after which integration is performed using Gaussian quadrature formulas [1].
The problem of vibrations of an infinite layer under a uniformly distributed harmonic load applied to the upper surface was considered as a test case. An exact solution exists for this problem.
For numerical analysis, a computational fragment is extracted from the infinite layer in the form of a parallelepiped. A uniformly distributed harmonic load is specified on the upper face, the lower face is clamped, and symmetry conditions are specified on the side faces. The exact and numerical results were compared in the region of the first natural frequency. The problem was considered with mesh densities of forty and one hundred and sixty elements.
Analysis of the results indicates that as the mesh is refined, the approximate solution converges to the exact one. Thus, the effectiveness of the proposed numerical algorithm and the functionality of the computational complex are confirmed.
List of references
1. Bazhenov, V.A., Dekhtyariuk, Ye.S., Vorona, Yu.V. (2012). Dynamics of Structures. Kyiv: PAT "Vipol".2012 - 342 p.
2. Bazhenov, V.A., Perelmuter, A.V., Shishov, O.V. (2013). Structural Mechanics. Computer. Kyiv: PAT "Vipol".2013 - 896 p.
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