Calculating of a spatial cantilever truss natural vibration frequency with an arbitrary number of panels: analytical solution
The research object was a spatial cantilever statically determinate truss composed of three planar trusses with a triangular lattice. The spectrum of natural frequencies of the structure was analyzed. The same concentrated masses model the inertial properties of the truss at the nodes. The task was to obtain an analytical dependence of the lowest vibration frequency of a truss on the number of panels, mass, linear dimensions of the structure, and material properties. Method. The induction method and the Maple computer mathematics system operators were used to determine the forces in the rods and generalize the result to an arbitrary number of panels. The problem was solved using the Dunkerley approach, which gives a lower frequency estimate. Maxwell-Mohr's formula determines the rigidity of the structure. Homogeneous linear recurrent equations were compiled and solved to find the common members of the sequences of coefficients in the formula for the frequency. Results. The accuracy of the formula obtained by the Dunkerley method was estimated from comparison with a numerical calculation of the entire spectrum of natural frequencies. The comparison shows the good accuracy of the derived formula. As the number of panels increases, the accuracy of the lower estimate increases too. The same frequency for all trusses was found in the spectra of trusses with a different number of panels. This frequency is the spectral constant and depends only on the size of the system, the stiffness of the members, and the mass. The existence of spectral isolines with the property of asymptotically tending to a certain constant value was shown.