RIGOROUS DERIVATION OF A SYSTEM OF LINEAR INTEGRO-DIFFERENTIAL EQUATIONS FOR VIBRATIONS OF CURVED RODS

In this article, the problem of deriving a system of linear integro-differential equations describing small oscillations of curvilinear rods is considered. With a curvature along the axis of the rod, the viscoelastic properties of the material are described by the Boltzmann-Volterra hereditary model. Based on kinematic perceptions and stress-strain relationships, the equations of motion are derived using the Hamilton principle or the virtual work method. The obtained results show that in special cases (straight rod, pure elastic material) they are reduced to classical equations.