Methods of analytical calculation of a spatial console truss natural oscillations frequency

Introduction. Numerical methods are usually used to calculate the natural oscillation frequency of building structures. Methods for obtaining analytical solutions are known to estimate the oscillation frequency limits of simple statically determinate structures. If the structure is regular and has a periodic structure, the capabilities of analytical methods are expanded. The induction method adds an additional important parameter to the solution formula — the number of periodic structures of the structure, for example, the number of panels. The approximate Rayleigh method gives an upper estimate of the oscillation frequency, and the Dunkerley method gives a lower estimate. In this paper, a diagram of a cantilever spatial statically determinate truss with a regular structure is proposed and a formula for its first oscillation frequency is derived using three analytical methods.

Materials and methods. The truss consists of six flat trusses with a diagonal lattice connected along their long sides. The cantilever structure is fastened to the vertical base on six supports. To determine the rigidity of the truss, the Maxwell – Mohr formula and the Maple computer mathematics system are used. A formula is derived for the dependence of the first oscillation frequency based on the Dunkerley and Rayleigh methods, which are simplified due to summation. The dependence of the frequency on the number of panels is found by the inductive method of generalizing the results obtained for individual solutions in symbolic form to an arbitrary case.

Results. Analytical solutions are compared with the numerical one obtained for the first frequency from the frequency spectrum analysis. It is shown that the accuracy of the analytical solution depends non-monotonically on the number of panels and this dependence is different for the three methods used.

Conclusions. The modified Rayleigh method for a small number of panels showed the highest accuracy compared to the methods based on the Dunkerley approach. The error of all three methods depends significantly on the dimensions of the structure and the number of panels. The analytical form of the results allows using the found solutions in design optimization problems.

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