Study On Nonlinear Dynamic Behavior Of Non-Conservative FGM Rods And Plates

Functionally graded material (FGM) is a new type of composite materials arisen in material science field in recent twenty years, and has wider application foreground in many fields such as spaceflight, biomedicine, and nuclear industry. Therefore analyses of the dynamic behavior of non-conservative FGM structure have important significance. On the basis of the existing research, the vibration and stability of non-conservative rod with various elastic supports, the post-buckling behavior of non-conservative FGM rod, the post-buckling behavior of FGM rod subjected to thermal loads, and the linear and nonlinear vibration behavior of non-conservative rectangular plate made of FGM are analyzed respectively in this paper. The main research work is as follows.(1) The dynamic behaviors and stability of Kelvin's viscoelastic rods with interior point elastic supports subjected to uniformly distributed tangential follower forces are investigated by integral equation theory. In solving the complex characteristic problem of the differential equation, firstly, nucleus function is obtained by using superposition principle, and the differential equation of mode shape is reduced to an integral equation. Then, based on the integral equation, complex characteristic equation is derived in accidence with the properties of degenerative nucleus. At last, the effect of the elastic constant of the elastic point supports, location of point support and dimensionless delay time of Kelvin's materials on complex frequencies and stability of non-conservative rods are analyzed by the examples. It is shown that the proposed method is effective and practical in solving the complex eigenvalue problem of differential equation with generalizedδ-function and complex variable coefficients.(2) The dynamic behaviors and stability of simply supported elastic rods with point elastic supports subjected to uniformly distributed tangential follower force are investigated. For the case of point elastic supports, it is described by the differential equations of mode shape in piecewise and continuous condition as well as boundary conditions. The recurrence formula is obtained by using finite difference method, and the effect of elastic constant of the linear elastic supports, location of elastic support and moment of inertia on vibration frequencies and stability of non-conservative rods are analyzed.(3) The stability of pipes conveying fluid with removable elastic support is studied. A differential equation of the pipe in piecewise, continuous conditions at movable elastic support and boundary conditions are established. The recurrence formulas are obtained by using the finite difference method. The effect of elastic constant of the elastic support, location of elastic support on vibration frequencies and stability of elastic pipe conveying fluid are analyzed, and the stability regional diagram (divergence region and flutter region) are plotted.(4) Based on the large deformation theory of extensible elastic rod, nonlinear governing differential equations of FGM rod subjected to a uniform distributed tangential load along the central axis are established. By using Runge-Kutta method and shooting method, the post-buckling behaviors of movable hinge and fixed hinge FGM rod made of metal and ceramic are analyzed,and compared with the properties of conservative FGM rod. The post-buckling characteristics curves of FGM rod under the different gradient index are plotted, and compared with the properties of metal and ceramic material rods. The influence of ratio of the length of the rod to the height of rectangular cross-section on the post-buckling behaviors of FGM rod is discussed.(5) The non-linear governing differential equations of fixed hinge /clamped FGM rod subjected to thermal loads were derived. The thermal post-buckling behaviors of FGM rod made of ZrO₂ and Ti-6Al-4Vwere analyzed by shooting method. Firstly, the thermal post-buckling equilibrium paths of the FGM rod with different gradient index in the uniform temperature field were plotted, and compared with the behaviors of the homogeneous rods made of ZrO₂ and Ti-6A1-4V materials respectively. For given value of end rotation angles, the influence of gradient index on the thermal post-buckling behaviors of FGM rod was discussed. Secondly, the thermal post-buckling characteristics of the FGM rod were analyzed when the temperature difference parameter is changing while the bottom temperature parameter remains constant, and when the bottom temperature parameter is changing while the temperature difference parameter remains constant, and compared with the characteristics of the two homogeneous material rods.(6) The coupled differential equations of motion of FGM rectangular plates under the action of uniformly distributed tangential follower forces are derived, which are expressed by introducing stress function and deflection function. The stability of simply supported FGM rectangular plates subjected to non-conservative forces is analyzed by using Fourier series method. The variation curves of vibration frequency and divergence load for different aspect ratio and gradient index are plotted, and the effects of gradient index on vibration frequency and divergence load are investigated.(7) Nonlinear vibration of simply supported FGM rectangular plates subjected to uniformly distributed tangential follower forces is presented. The material properties of a FGM plate were graded continuously in the direction of thickness, according to a simply power-law distribution of the volume fraction of the constituents. The governing nonlinear partial differential equations which expressed by stress function and deflection function are obtained using the von Karman theory. Then the nonlinear partial differential equations are transformed into nonlinear ordinary differential equation using Galerkin method. For simply supported ceramic/metal FGM plates under the action of uniformly distributed tangential follower forces, the effect of follower force, gradient index and aspect ratio on the dynamic behavior of the plates is discussed. The relation between central amplitude and nonlinear fundamental frequency for different parameter are derived.