Analytical calculation of the fundamental oscillation frequency of a multi-dome roof

Строительная механика

The object of the study is a spatial model of a statically determinate truss of a regular type, consisting of separate domed hexagonal trusses connected in a row. Truss eigenfrequencies are investigated. Method. The lower bound of the first natural frequency is sought in analytical form. The Dunkerley partial frequency method is used. The stiffness of the structure is calculated using the Maxwell – Mohr formula, assuming that the stiffnesses of the elements are the same. The partial frequencies of the masses concentrated in the truss nodes are calculated from separate oscillation equations. Each mass is assumed to have three degrees of freedom. Several solutions for trusses with a successively increasing number of panels are generalized by induction to an arbitrary number of panels. For analytical transformations and calculation of common members of sequences of coefficients of the desired formula, operators of the Maple computer mathematics system are used. In numerical form, the natural frequency spectra of a family of regular trusses are searched for and analyzed. Result. The obtained dependence of the fundamental frequency on the number of panels has coefficients in the form of polynomials of order not higher than the fourth. Comparison of the analytical result with the first frequency of the entire frequency spectrum obtained numerically shows the high accuracy of the found formula. As the number of panels increases, the accuracy of the proposed solution increases. In the frequency spectrum, it is found that the highest frequency of natural vibrations does not depend on the order of the truss.