A Symplectic Method For Vibration Of Functionally Graded Cylindrical Shell Conveying Fluid

Pipe conveying fluid plays a necessary role in the production and daily life. With the development of science and technology, the study of vibration of pipes conveying fluid is one of the focus of the academic research problem, and has a very broad sense. Under the action of hydrodynamic pressure, the necessary condition to the flow pipeline assumes as the beam model is the radius is much smaller than the length of the pipe. In order to optimize the pipe structure, the research of the shell model functionally graded materials (FGM) pipes vibration is becoming more meaningful. The main contents of this paper are as follows:(1) The mechanical model is the system of functionally graded cylindrical shell. Based on thin shell theory, obtained the Lagrange density equation of hollow flow pipeline. Through the Hamilton's principle, the Hamilton canonical vibration equation is obtained when the vibration equation is dimensionless and discrete. The general solution and the simplified solution under the simply supported boundary are obtained by symplectic method, the others, the simplified solutions under the clamped boundary conditions are obtained. The relationships between the natural frequencies versus materials volume faction, circumferential wave number and thickness-to-radius ratio are further analyzed.(2) Based on the free vibration of FGM cylindrical shell, research the fluid-solid mutual coupling vibration of the shell model functionally graded materials pipes conveying fluid.Through the thin shell theory, the dimensionless Lagrange density equation of the coupling vibration for the shell model FGM pipes is obtained. Through the Hamilton's principle, the vibration problems are studied in the symplectic system. It is a new method for the shell model functional graded materials pipes conveying fluid. The pipe inherent frequencies are obtained finally. And the changing regularity of the frequencies versus the flow velocity, material constituents and length-to-radius ratio are founded.(3) Using the Hamiltonian symplectic method to solve functionally graded cylindrical shell directly. The Lagrange function is derived in the frequency domain, and establishing the symplectic method in the Hamiltonian system to the cylindrical shell. In the Hamiltonian system, the natural frequencies and vibration modes of the pipes same as the symplectic eigenvalues and symplectic eigenvectors. It is convenient to solve the free vibration of the shell model pipes in different boundary conditions. And this method provides a broader option to the shell model FGM pipe conveying fluid.