Study On The Nonlinear Mode Interaction And Resonance Behaviour Of Functionally Graded Circular Cylindrical Thin Shells

This dissertation is devoted to research on the nonlinear dynamic characteristic of thin-walled circular cylindrical shells made of functionally graded material (FGM). The emphasis is placed mainly on the nonlinear mode interaction and resonance of functionally graded cylindrical thin shells. Based on the Hamiltonian dynamics theory and perturbation method, the mode interaction and energy exchange existing in the two-mode free vibration of functionally graded cylindrical thin shells are analysed. And the two-mode primary resonance behaviour of functionally graded cylindrical thin shells subjected to steady temperature distribution and distributed loading is studied via the Lagrangian dynamics and multi-scale method. The nonlinear dynamic phenomena in the vibration of functionally graded cylindrical shells are displayed.First of all, the volume fraction variation of constituent materials of FGM along the thickness direction is described as a power-law distribution, and the equivalent material properties are given by a simple mixture law based on the Voigt's uniform-strain modal. The dynamic equations for linear and nonlinear vibration of functionally graded cylindrical thin shells are established according to the Donnell's simplified cylindrical shell theory and the von Karman's large deflection theory. The frequence characteristic for linear free vibration of functionally graded cylindrical thin shells with simply supported boundary condition is analysed, which validate the present theory.The kinetic energy and potential energy formulation are obtained. An axi-symmetric mode and a non-axisymmetric mode are selected as approximate transverse deflection function; the kinetic and potential energy expressions for functionally graded cylindrical thin shells are gained. The Hamiltonian dynamics theory is introduced into the analysis of nonlinear two-mode free vibration of cylindrical shells. The averaged equation describing the nonlinear two-mode free vibration of functionally graded cylindrical thin shells is obtained using perturbation method; and the phase trajectories of system at exact1:2internal resonance and near resonance conditions are discovered, the phenomena of mode interaction and energy exchange are displayed comprehensively. Moreover, the energy bifurcation point which triggers the ineraction between the non-symmetric mode and axi-symmetric mode is obtained. The influence of grade exponent of FGM on the internal resonance condition and bifurcation character is discussed. Through numerical integration the chaotic motion of the nonlinear two-mode resonance of functionally graded cylindrical thin shells at the high energy level is revealed.The differential equations of motion for two-mode forced vibration of infinitely long functionally graded cylindrical thin shells subjected to transverse loading are obtained using Lagrangian method. By the application of multi-scale method, the dynamic behaviour of system when the high-frequence primary resonance and low-frequence primary resonance occuring is studied in detail. The amplitude-frequency curves of response and the bifurcation behaviour of system are analysed using numerical continuation method, and the path leading the system to chaos is revealed. The influence of grade exponent of FGM on the amplitude response of system is discussed. It is found that, at high-frequence primary resonance the system can appear quasi-periodic motion, and turn into chaotic motion through period doubling bifurcation of quasi-periodic tori; moreover, the quasi-periodic tori and chaotic attractor possess symmetry, and the attractors will hit each other, which results in a single symmetric tori or chaotic attractor. Similarly, the system at the low-frequence primary resonance condition will also exhibit quasi-periodic and chaotic motion, which, however, is asymmetric.The vibration behaviour of functionally graded cylindrical thin shells in thermal environment is investigated. The multi-scale method is used again to analyse the dynamic behaviour of system at high-frequence and low-frequence primary resonance condition when the system possesses steady temperature distribution. It is found that the steady temperature distribution will not affect the qualitative dynamic behaviour of primary resonance of functionally graded cylindrical shells, while it will change the relation of the amplitude response of system with grade exponent of FGM.