Analytical estimation of the first natural frequency and analysis of a planar regular truss oscillation spectrum

A statically determinate truss of a beam type with a triple lattice with short descending and long ascending braces is considered. The mass of a truss is modeled by concentrated loads at its nodes. For the first natural frequency, the Dunkerley’s method derives a formula for the dependence of its lower boundary on the number of panels. The calculation of the efforts in the truss required to obtain the stiffness value according to the Maxwell-Mohr formula is performed in the Maple computer mathematics system by cutting out the nodes. It is shown that for a certain number of panels the proposed scheme of the truss has the property of kinematic variability. For admissible numbers of panels, by induction, the sequence of solutions for trusses with different numbers of panels is generalized to an arbitrary case. The coefficients of the required dependence are obtained as solutions of linear homogeneous recurrent equations. To obtain and solve recurrent equations, the operators of the Maple system are used. The found solution is compared with the minimum frequency of the spectrum obtained numerically. It is shown that the accuracy of the analytical assessment monotonically increases with the increase in the number of panels. Multiple frequencies and the independence of several higher frequencies from the number of panels were found in the frequency spectrum.