Mathematical modeling of flexural vibrations of a rod with an intermediate perfectly rigid massive insert

The paper considers a compound rod with an intermediate perfectly rigid insert of mass m and length s, which can be viewed as a simplified model of a gas turbine engine blade with a tip shroud. To calculate the bending vibration frequencies, the rod is divided into three coupled segments: two flexible segments and an intermediate perfectly rigid insert. Unlike traditional models that consider rods with point masses, this study takes into account the length of the rigid insert and its moment of inertia. This allows for a more accurate consideration of the influence of the translational and rotational motion of the rigid insert on the natural frequencies of the rod, which constitutes the scientific novelty of the study. In constructing the mathematical model, the continuity conditions for displacements and rotation angles of the rod's cross-sections are applied at the ends of the insert, along with the equilibrium conditions for the forces and moments acting on the insert. By applying the matching conditions and the boundary conditions at the ends of the flexible rod segments, a transcendental frequency equation is derived, the roots of which determine the natural frequency spectrum of the rod's free vibrations. Numerical analysis demonstrates that increasing the mass of the insert leads to a monotonic decrease in the natural frequencies, with the mass having the most significant effect on the lower natural frequencies. The model was verified by analyzing limiting cases (m → 0 and s → 0) and comparing the calculated results with known solutions for a homogeneous rod of constant cross-section. The calculation error for the natural frequencies does not exceed 1·10-4 %. The developed model is of practical value for the preliminary estimation of the natural frequencies of gas turbine engine blades with tip shrouds at the design stage, helping to prevent resonant operating conditions, which highlights the relevance of this study.

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