A scheme of a statically determinate planar truss is proposed and an analytical calculation of its deflection and displacement of the mobile support are obtained. The forces in the rods from the external load, uniformly distributed over the nodes of the lower or upper belt, are determined by the method of cutting out nodes using the computer mathematic system Maple. In the generalization of a number of solutions of trusses with a different number of panels to the general case, the general terms of the sequence of coefficients in the formulas are found from solutions of linear homogeneous recurrence equations. To compose and solve these equations, Maple operators were used. In the process of calculation it was revealed that for even numbers of panels in half the span, the determinant of the system of equations degenerates. This corresponds to the kinematic degeneracy of the structure. The corresponding scheme of possible speeds of the truss is given. The displacement was determined by the Maxwell-Mohr’s formula. The graphs of the obtained dependences have appreciable jumps, which in principle can be used in the selection of optimal design sizes.
A scheme of a statically definable truss with additional supports is proposed. Derive formulas for the dependence of the deflection of the truss against the number of panels for three types of symmetrical loads. It is shown that for definite numbers of panels the determinant of the system of equations for the equilibrium of nodes degenerates. This indicates an instant changeability of the structure. To generalize particular solutions to an arbitrary number of panels, the induction method is applied. For this purpose, in the computer mathematics system Maple linear recurrence equations are constructed for the terms of a sequence of coefficients from individual solutions. The graphs of the dependences obtained indicate a nonmonotonic character of the solutions found and the possibility of optimizing the design by choosing the number of panels.
Introduction. The article addresses a spatial model of a statically definable mast truss consisting of four identical planar trusses with a crosswise grid system and a base with four supports at the corners. The authors solve the problem of deriving the analytical dependence between the bottom vibration frequency of the mast truss and the number of panels, mass, linear dimensions of its construction and properties of the material. Materials and methods. To calculate the values of forces, arising in the rods of a mast truss with an arbitrary number of panels, and analyze the obtained results, the induction method and operators of the Maple computer system for mathematics were used. The problem of deriving the analytical dependence between the bottom frequency of vibrations of the mast truss and its parameters is solved using the Dunkerley method, which generates the bottom estimate of the natural frequency. The rigidity of the truss structure is calculated according to the Maxwell – Mohr formula. To calculate the common members of sequences of coefficients, homogeneous linear recurrent equations are derived and solved in the frequency formula. Results. A formula is obtained for estimating the first frequency of natural vibrations of a truss. The formula coefficients have the form of polynomials of no higher than the fourth order. The accuracy of the calculation formula, obtained using the Dunkerley method, is estimated by the comparison with the first frequency, obtained through the numerical calculation of the entire spectrum of natural frequencies. Conclusions. The analysis of the analytical results and their comparison with the numerical ones shows high accuracy of the derived formula. The authors have identified a dependence, whereby an increase in the number of mast truss panels boosts the accuracy of the bottom estimate of the natural frequency.
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