Introduction. The first (lowest) frequency of natural vibrations of a structure is one of its most important dynamic characteristics. Analytical solutions supplement numerical ones; they can be efficiently used to perform a rapid assessment of properties of structures, to analyze and optimize constructions and to test numerical results. A space cantilever truss consisting of three planar trusses with a rectangular grid is considered in the article. The objective is to find the analytical dependence between the frequency of natural vibrations of a structure and the number of panels. It is assumed that the truss mass is distributed among the joints. Only the vertical mass displacement is taken into account. Materials and methods. Forces, arising in cantilever rods, are calculated by the Maple software as symbolic expressions, and the method of joint isolation is used here. The stiffness matrix is identified using the Mohr integral. Rods are assumed to be elastic, they have identical stiffness. The lower value of the vibration frequency is determined using the Dunkerley method. The final calculation formula used to identify the value of the vibration frequency is derived using the method of induction applied to a series of analytical solutions developed for trusses with a consistently increasing number of panels. When common members of sequences are found, genfunc operators of the Maple system are used. The analytical solution is compared with the numerical solution in terms of the first frequency using the analysis of the system spectrum featuring many degrees of freedom. The eigenvalues of the characteristic matrix are identified using the Eigenvalues operator from the Linear Algebra package. Results. The comparison between the analytical values and the numerical solution shows that the Dunkerley method ensures the accuracy varying from 20 % for a small number of panels to 3 % if the number of panels exceeds ten. The size of the structure, the weight and stiffness of rods have little effect on the accuracy of the obtained values. Conclusions. The lowest value obtained using the Dunkerley method in the form of a fairly compact formula has good accuracy, its application to a space structure with an arbitrary number of panels has a polynomial form equal to the number of panels, and it can be used in practical calculations.
The scheme of statically definable truss of spatial coverage is proposed. The formula for the dependence of the vibration frequency on the number of panels is derived. The Dunkerley lower bound and the induction method are used to generalize particular solutions to the case of an arbitrary number of panels. The calculation of the forces in the rods by cutting out the nodes and the analytical transformations to obtain the desired dependence are performed in the Maple computer mathematics system. The solution is compared with the numerical one obtained by solving the problem on the eigenvalues of the characteristic matrix for a system with many degrees of freedom. It is shown that the estimation accuracy depends on the number of panels.
The purpose of the study is to derive analytical expressions for estimating the lower vibration frequency of the power line support truss flat models. The forces in the bars of statically determinate structures are determined by the method of cutting out nodes in a program written in the Maple symbolic mathematics language. To find deformations, the Maxwell-Mohr's formula is used under the assumption that all rods are elastic, and that the supports are modeled by rigid rods. It is supposed that the hinges are ideal, and the mass of the structure in the form of point loads is distributed over the truss nodes, and only horizontal load displacements are considered. In comparison with similar problem statements with analytical forms of solution, the present study takes into account the masses at all nodes of the structure. For two-sided estimation of the fundamental frequency, the methods of Dunkerley and Rayleigh are used. The coefficients of the formulas in the solutions obtained for trusses with different numbers of panels form sequences, the common terms of which from the solution of linear recurrent equations give the final formula for the frequency dependence on the number of panels. As a result of the study, formulas for deflection and estimation of the fundamental frequency of truss natural vibration depending on the number of panels and dimensions of the structure have been derived. The obtained formulas can be used in carrying out engineering analyses of power transmission line supports. The formulas for the deflection and frequencies of the studied trusses have a form simple and convenient for use (in particular, for assessing the accuracy of numerical solutions). The frequency obtained by the Rayleigh method is much closer to the fundamental natural frequency than its value estimated using the Dunkerley method.
In this work, we investigate the static deformations of the spatial model of a statically determined truss of a power line support. The tetrahedral truss has a pyramidal extension at the base and a cross-shaped lattice. Brackets for attaching the supporting cables are located at the top of the truss. A spherical support hinge, a cylindrical one, and two vertical posts are located at the four corners of the structure base. We consider two types of loads: wind, and force. Horizontal forces applied to the nodes of one face model the wind load. The horizontal force is applied to the top of the structure. We aim to derive formulas for the dependence of the deflections of the truss on the number of its panels. We use the Maxwell-Mohr formula to determine the deflection. We find the efforts in the structural elements and the reactions of the supports from the general system of linear equations of equilibrium of all nodes of the truss. A series of solutions for trusses with different numbers of panels are summarized by the induction method in the Maple computer mathematics system. The sought formulas for the dependence of the vertical deflection of the console and the displacement of the top of the mast on the number of panels were obtained in the form of polynomials in the number of panels of degree not higher than the fourth. Some asymptotics of solutions is found in the work.
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