Dynamics Analysis For Uncertain Structures

Stochastic model, fuzzy model and interval model are three kinds of models to solve uncertainty structures. In order to limit the growth of the result of interval arithmetic operation, an universal grey mathematics was applied to analyze the dynamic characteristics of structures with interval parameters, and then, an improved division operation rule for universal grey mathematics was put forward. A simple numerical example and engineering examples were used to illustrate the accuracy of the improved algorithm. Meanwhile, according to the fact that the deterministic dynamics modeling of flexible hub-beam system cannot well conform with actual operation, uncertain phenomenon existing in flexible hub-beam system was studied. Dynamic response and dynamic characteristic were investigated for three models, namely, plain flexible hub-beam model, plain flexible hub-beam model with additional mass, and spatial flexible hub-beam model. The results indicated that dynamic response and dynamic characterisitic of flexible hub-beam system had relation to the randomness of parameters. The main research works can be described as follows:The first part of the study focused on the calculation of Rayleigh quotient based on universal grey mathematics. The translation rules between universal grey number and interval number were introduced on the basis of the concept for universal grey number and interval number. It was showed that universal grey number-based approach can deal with some problems that interval number-based approach can’t. Then the solution to the interval eigenvalue equations was achieved based on universal grey number-based approach, which is based on the characteristic of Rayleigh quotient and the monotonicity of interval eigenvalues. Finally, two engineering examples were employed to verify that this method was simple, accurate and reliable.The second part of the study focused on analyzing the dynamic characteristics of uncertain chain structures based on universal grey number mathematics. On the basis of the analysis of four arithmetic operations of the universal grey numbers, an improved division operation for the universal grey number was presented, meanwhile, some drawbacks of the universal grey number-based approach for the interval analysis were pointed out. A nonlinear universal grey number-based equation of natural frequencies was obtained by using transfer matrix method and universal grey mathematics. Subsequently a forward and backward searching algorithm was presented to solve the equation. The feasibility and effectiveness of the improved division operation were confirmed through engineering examples.The third part of the study focused on the stochastic dynamic response of plane hub-beam system and plane hub-beam system with additional mass respectively. Using assumed modes method and Lagrange’s equations, the first-order approximate coupling stochastic dynamic equations of the planar system was derived.The polynomial chaos method with points of a regression-based collocation method and Galerkin method were applied to convert the completely implicit stochastic differential equations into a set of completely implicit pure differential equations. The results obtained through solving the deterministic equations were used to find the numerical characteristics of the response. As illustrating examples, dynamic model of the system with stochastic parameters was presented. Simulation results demonstrated that stochastic parameters had a significant effect on the dynamic response of the plane systems, and the dynamic modeling with stochastic parameters can reflect the dynamic behavior of the plane hub-beam systems.The forth part of the study focused on the stochastic dynamic response of a spetial flexible beam. Using assumed modes method and virtual work principle, the first-order approximate coupling stochastic dynamic equations of the spatial flexible beam was derived. The polynomial chaos method with points of monomial cubature rule and Galerkin method were applied to convert the completely implicit stochastic differential equations into a set of completely implicit pure differential equations. The deterministic equations was solved to find the numerical characteristics of the response. For illustration, the dynamic model of a spatial flexible beam with stochastic parameters was considered. The accuracy and efficiency of the method were verified by comparing the results with those obtained from the Monte Carlo simulation method.The fifth part of the study focused on stochastic dynamic characteristics of flexible hub-beam systems with probabilistic parameters. The stochastic analysises of the transversal bending natural frequencies for three systems, namely, plane hub-beam system, planar hub-beam system with tip mass and spatial flexible beam system with stochastic parameters, were studied based on the first order approximate coupling model. In the dynamical equations, several dimensionless parameters were used. Then, the influences of the dimensionless parameters on the randomness analysis of natural frequencies were studied for three models.