Akademik Veri Yönetim Sistemi
Asia Pasific 9th International Modern Sciences Congress, Baku, Azerbaycan, 16 - 18 Ocak 2024, ss.287-298, (Tam Metin Bildiri)
Cylindrical structures are used in various fields of engineering. Solid cylinders serve as shafts in the powertrains of land, sea, and air vehicles, as rods in machinery, as well as pistons in hydraulic and pneumatic forming systems. They also act as structural support elements in architecture and civil engineering. These cylindrical elements, designed to fulfill specific roles or bear loads, operate under various working conditions. Examining their response to vibrations is a crucial engineering concern.Conducting experimental studies for such investigations is not always feasible due to high costs. Therefore, a mathematical examination of the vibration of cylindrical elements provides valuable technical data for researchers and engineers, enabling significant improvements in subsequent designs. This study focuses on the axisymmetric motion of a solid cylinder made of homogeneous, isotropic, linear elastic material. Navier-Lame equations, derived using the generalized Hooke's law, which establishes the relationship between stresses and strains of a linear elastic material, along with the dynamic equilibrium equations derived from Newton's second law of motion, are employed for this purpose. The Navier-Lame equations are suitable for problems with boundary conditions specified in terms of displacements. The study solves these equations using two different methods: the potential method and the Bessel method. In the potential method, the displacement vector is represented as a combination of a vector potential function and a scalar potential function. Substituting these representations into the vector form of the Navier-Lame equations yields a system of partial differential equations. On the other hand, in the Bessel method, the Navier-Lame equations are expressed in terms of radial and axial displacements in cylindrical coordinates. Displacements are initially assumed in terms of Bessel functions. Both methods consider time-dependent harmonic wave motion in the axial direction of the solid material. The model assumes no stress on the outer surface of the cylinder. Solving the problem involves finding a non-zero solution for a set of homogeneous equations, considering the boundary conditions. The characteristic equation, giving the eigenvalue angular frequency, is found by setting the determinant of the 2x2 dimensional coefficient matrix equal to zero. As an analytical solution was not possible, a numerical solution was implemented. Graphs depicting the variation of angular frequency with respect to dimensionless wave number are provided for both methods. The results indicate compatibility between the two solution methods.
Keywords: Cylinder Vibration, Bessel, Potential Solution, Numerical Solution.