Three-Dimensional Analysis of the Lower Frequencies of a Cantilevered Anisotropic Parallelepiped

Abstract

An approach to determining the lower frequencies of a cantilevered elastic parallelepiped using a three-dimensional problem statement and allowing for material anisotropy is developed. The approach combines the inverse-iteration and extended Kantorovich-Vlasov methods and is validated against the combination of the finite-element and Ritz methods. The influence of the anisotropy of the material on the lower frequencies of the parallelepiped is analyzed. It is shown that the variation in the frequencies of the parallelepiped with the boundary conditions being considered follows the variation in the predominant stiffness characteristics Keywords: elastic parallelepiped, cantilever, different types of anisotropy, three-dimensional problem statement, natural frequencies, extended Kantorovich-Vlasov methodIntroduction. Cantilevered elastic members of various shape and structure are widely used in various fields of engineering and construction. They are used to model wings of aircraft and blades of gas-turbine engines in mechanical engineering; elements of floors of buildings, balcony girders, and multistorey structures in construction; elements of scanning electron microscopes in nanoelectromechanical systems. No wonder that cantilever restraint is deemed to be a technological masterpiece. An analysis of the vibrations of such members is an important initial stage in their dynamic design, which may prevent emergencies under real service conditions. By now, the dynamic characteristics of cantilevered elastic members have been most extensively studied for beams and thin plates either of simple structure or with some complicating factors. For example, the influence of lamination of beams on their vibrations was studied in [9], the dependence of the lower frequencies on the elastic properties of composites was analyzed in [20], and discrete inclusions such as lumped masses were allowed for in [8]. We will use beam models, the classical Kirchhoff-Love model, and shear models of the first and higher orders.The vibrations of cantilevered bodies have been studied, using a three-dimensional problem statement, mainly for thick plates of various shapes such as an isosceles triangle, a rectangular parallelepiped, a prism with skew edge opposite to fixed one, etc. [18,19,21,22]. The three-dimensional problem statement in elasticity was compared in [18] with different two-dimensional problem statements for plates depending on their geometrical parameters. Particular attention was given to the boundary conditions on the lateral faces, and the frequency analysis was usually restricted to the case of an isotropic material [19]. The free vibrations of thick anisotropic plates were studied in few publications. The lower frequencies of square plates of different thickness made of materials with different types of anisotropy were analyzed in [3]. Various combinations of boundary conditions on the lateral faces, except for the cantilever case, were considered.To solve the relevant two-and three-dimensional eigenvalue problems, use is often...