Complex Response Study On Flow-Induced Vibration Of Nonlinear Plate-Type Structures

Plate-type structures are widely used in the integrative designs of fuel elements in some research, power nuclear reactors and other structural engineering designs. The vibration and complex response of plate-type structure in incompressible, inviscid and viscous fluids are systematically investigated in this dissertation. The bifurcation and stability of plate-type beams in incompressible, inviscid fluids are studied by utilizing an algebraic criterion of Hopf bifurcation. Based on incompressible viscous fluid theory, the Semi-analytical fluid pressures acting on plates are deduced, appropriate fluid-solid coupled dynamic models of nonlinear plate-type beams vibrating in axial incompressible viscous fluids are founded, and further the dynamic behaviors close by the bifurcation points of these systems with different nonlinear situations are analyzed to determine periodicity and stability. The phenomena of plate-type flow-induced system bifurcating from simply movements to complex behaviors are revealed. The main contributions of this dissertation are as the follows:1 By assuming that all plates having the same deflections at any instant, the dynamic model of plate-type beams with cubic nonlinear stiffness in incompressible inviscid fluids by considering the interaction between fluid and solid is established. An algebraic criterion of Hopf bifurcation is proposed to analyze the flow-induced vibration, instability and bifurcation of these systems. Usually, to a multi-freedom-degree system, it is difficult to judge the cross condition of the bifurcation points while using the classical Hopf theory directly. In this dissertation, with Hopf bifurcation criterion used, it's proved that Hopf bifurcation will not occur for nonlinear simply-supported plate-type beams vibrating in an incompressible inviscid fluid. The analytic expression of critical fluid velocity of nonlinear cantilevered plate-type beams in an incompressible inviscid fluid is deduced and thus avoids the hard task of judging the cross condition technically.2 For the first time, the semi-analytical fluid pressures acting on the plates of a parallel flat plate-type structure in axial flow are deduced by theoretical analysis.Incompressible viscous flow theory is used to describe the interaction between fluid and plates. First, based on the assumption that plates doing small harmonic vibration, the semi-analytical fluid pressures of a typical plate-fluid-plate sub-structure are gotten. And with the results extending to the whole structures, the semi-analytical fluid pressures acting on the plates with fluid flowing along the axial in a rigid water trough and in a rigid rectangular tube are obtained respectively, which will benefit much for further analytical analysis of plate-type fluid-solid coupled structure. As an example, the semi-analytical fluid forces acting on the plates in a rigid rectangular tube are gotten, from where the added mass coefficients, added damping and added stiffness coefficients of fluids are determined, and the changing tendency with structure's parameters are discussed in detail. The results show that viscous effects play a significant role. The damping coefficient has an inverse ratio with the cubic value of gap, which is consistent to the results of a hexagonal cylinder and a parallel flat plate-type structure in still water done before. Affected by velocity of flow, the added stiffness coefficient has an inverse ratio with the cubic value of gap and it becomes zero while the velocity of flow reduced to zero, which satisfies well with the results obtained in still water. 3 The assumption mode method is proposed to establish differential equation of a four-span plate-type-beam vibrating in incompressible viscous fluid flowing along the axial in a rigid water trough. The dynamic characteristics and free vibration stability of coupled system are analyzed systemically. It reveals that, the relative error of the 1st order frequency between computational value and the test one of beam vibrating in air is 1.7%, which satisfies the needs of engineering. The added mass coefficient and added stiffness coefficients have significant influence on natural frequencies of system. The frequencies of the structure in flow decrease compared with the corresponding ones in air and with the increase of fluid velocities, the frequencies decrease directly to zero in the end. This changing tendency is quite similar to the results of circular cylinder shells in annular axial flow gotten by potential flow theory. On the other hand, the stability analysis shows that the simply-supported four-span plate-type-beam flutters while conveying free vibration in a two-dimensional axial incompressible viscous flow. And the flutter critical velocity decreases with the increase of gap while increases with the increase of beam's thickness.4 Reasonable nonlinear flow-induced dynamic model of a simply-supported plate-type beam in axial incompressible viscous flow with cubic stiffness for rigid rectangular tube boundary condition is established. The equivalent linearization method is introduced into the analysis of limit cycle flutter of plate-type structure, and the stability of the limit cycle flutter is judged via analysis of corresponding coupling figures. The fourth order Runge-Kutta method is utilized to analyze the bifurcation forms of the original nonlinear system. Numerical integrations show that, the response is stable while the fluid velocity is on the small side. As the increase of fluid velocity, the system loses its stability by flexure within a certain range of fluid velocities, and along with more accretion of fluid velocity, system loses stability by stable limit cycle flutter. Numerical integrations verify that it is feasible using equivalent linearization method for the limit cycle flutter analysis of plate-type flow-induced system.5 By nonlinear dynamic model of a simply-supported plate-type beam with asymmetric piecewise linear stiffness vibrating in axial incompressible viscous flow in a rigid rectangular tube established, the emphasis focuses on the complex responses of nonlinear system. The region of fluid velocity of limit cycle flutter is presented by using equivalent linearization method for nonlinear flutter and stable limit cycle of plate-type system is determined qualitatively. Complex dynamic behaviors close by the bifurcation points of nonlinear system are posted by numerical integrations. The results show that, for plate-type structure with asymmetric piecewise linear nonlinearity, there are many new and complex phenomena which is the first time observed as existing literatures at the present time. That is, the system has two or more stable limit cycles in the same bifurcation velocity of flow. With the increase of fluid velocity, the system undertakes bifurcation through the path of periodic 1 - periodic 2 - periodic 4 -periodic 8. In some certain parameter regions, another stable limit cycle of period one loses stability and period doubling response is observed.