ANALYTICAL EVALUATION OF THE FIRST NATURAL FREQUENCY OF A MULTI-SPAN TRUSS

The task is to obtain the dependence of the lower limit of the fundamental frequency of natural oscillations of a statically determinate truss on the number of spans. The upper belt of the truss is straight, the lower belt of spans is arched. The inertial properties of the truss are modeled by the same concentrated masses in the nodes. Each mass has two degrees of freedom. The forces in the rods and the reactions of the supports are found by cutting out the nodes in symbolic form from the solution of the general system of equations for the equilibrium of the truss nodes. All transformations are made in the Maple computer mathematics system. To find the structural stiffness matrix, the Maxwell – Mohr formula is used. The first natural frequency of the structure is found using the approximate Dunkerley method. The generalization of a series of individual solutions found for trusses with a different number of spans to an arbitrary number of spans is performed by induction using Maple system operators. For comparison, the lowest oscillation frequency of a system with many degrees of freedom obtained numerically is used. A formula is derived for the lower estimate of the first frequency depending on the size of the truss and the number of spans. Comparison of the analytical solution with the numerical one shows a good degree of approximation of the formula for the first frequency. It is shown that the error significantly depends on the height of the truss. Cases of kinematic variability of the structure with an even number of spans are revealed.