Mathematical Modeling of the Dynamic Deformation of a Rod-Strip Fixed on a Double-Sided Support Element through Elastic Interlayers - V. N. Paimushin; V. M. Shishkin; S. F. Chumakova (Учёные записки Казанского университета: Серия Физико-математические науки)
The simplest transformation model of the dynamic deformation along the length of a rod-strip consisting of two segments was constructed. The model is based on the classical geometrically linear Kirchhoﬀ–Love model for the unﬁxed segment, while the ﬁxed segment of ﬁnite length is assumed to be connected to a rigid and immovable support element through elastic interlayers. On the ﬁxed segment, the deﬂections of the rod and interlayers were considered zero. For axial displacements within the thicknesses of the rod and interlayers, approximations were adopted according to S.P. Timoshenko’s shear model, subject to the conditions of continuity at the points where they connect to each other and immobility at the points where the interlayers connect to the support element. The conditions for kinematic coupling of the unﬁxed and ﬁxed segments of the rod were formulated. Taking them into account and using the D’Alembert–Lagrange variational principle, the equations of motion and boundary conditions for the considered segments were derived, and the conditions for force coupling of the segments were obtained. With the help of the derived equations, exact analytical solutions of the problems of free and forced harmonic vibrations of the rod of the studied type were found. These solutions were employed in the numerical experiments to determine the natural modes and frequencies of bending vibrations, as well as the dynamic response during the resonant vibrations of the rod-strip made of a unidirectional ﬁbrous composite based on ELUR-P carbon tape and XT-118 binder. The ﬁndings show a signiﬁcant transformation of transverse shear stresses when passing through the boundary from the unﬁxed segment of the rod to the ﬁxed one, as well as their pronounced localization in the region of the ﬁxed segment near this boundary.