Deformations and Natural Frequency of a Triangular Truss: Analytical Solutions

The object of research. A new scheme of a lattice externally statically indeterminate truss in the form of a triangle with a lower chord supported by vertical posts and a fixed hinge is considered. When looking for the forces in critical bars and deflection, a vertical load is considered, which is evenly distributed over all external and internal nodes of the truss. The dependence of the deflection of the truss top on the load, dimensions and number of truss panels is given. A formula is derived for the lower estimate of the first frequency of natural oscillations. Method. The calculation of forces is carried out by cutting out all the nodes of the structure. The number of unknowns of the system of linear equilibrium equations in the projection on the coordinate axes includes both forces and reactions of supports. The deflection is calculated in analytical form using the Maxwell-Mohr formula and is generalized by induction from solving a number of problems for trusses with a different number of panels to an arbitrary order of a regular truss. To find an analytical estimate of the first frequency of natural oscillations of nodes endowed with masses, each of which has two degrees of freedom, the Dunkerley lower estimate method is used. Results. The formulas obtained for the forces in the rods, deflection and the first frequency have a compact form, which can be used to obtain simple evaluation solutions. The lower analytical estimate of the first oscillation frequency is in good agreement with the numerical solution for the entire spectrum of structure oscillations. All necessary transformations are performed in the Maple symbolic mathematics system. Linear asymptotics of solutions for deflection and forces are found.