Study On Vibration Of Structural Systems With Parametric Uncertainties

In this paper,the statistical characteristics of vibration of the structural system with uncertain factors are mainly studied.The properties of orthogonal polynomial function system and methods such as dimension decomposition of multivariate function are used to solve the stochastic problems in the field of practical engineering and science.And the statistical characteristics of random responses are compared with those obtained by the direct Monte Carlo simulation method.Firstly,the basic concepts and properties of orthogonal polynomials such as Hermite,Legendre and so on are introduced,they are a useful tool for approximation of response functions.Be based on the dimensional decomposition algorithm of any continuous differentiable multivariate function,and under the condition of the uncertainty factors of system obeying the respective and independent Gauss distribution,the stochastic responses of the structural system are analyzed by embedding the local Monte Carlo simulation,which emply the Fourier-Hermite polynomial expansion,generalized ROM and multi-dimensional Gauss-Hermite quadrature to determine each expansion coefficients and to obtain the explicit orthogonal polynomial function of them.The errors of the first four order moments of them and the direct Monte Carlo simulation method is obtained by the error formula to compare them in a more intuitive way.Secondly,in the practical structural dynamic system,as one of the structural dynamic characteristic parameters,natural frequency is a key parameter of system design,structure analysis and stability,sensitivity analysis.And when considering the uncertainties of the system,it has the characteristics of randomness.The numerical simulation of the natural frequencies of a spring mass system with three degrees of freedom and neglecting damping is carried out,and the results show that the statistical results of the natural frequencies of the "black box" method are better than that of the implicit function expression structure method;With the increase of the number of parameter variables considering the uncertainty of the system,the statistical characteristics of the natural frequencies of the system will be more in line with the results of the direct Monte Carlo simulation.However,the cost of computing will also be in a polynomial growth,we should choose the appropriate number of random variables according to the error analysis in order to meet precision as much as possible to reduce the computational workload.Finally,the plate and shell structures which is widely used in practical engineering structures is parameterized modeling.Under the boundary conditions of bilateral freedom,fixed on the left side and supported using spring on the right side,the orgin vibration response of the plate structure is excited by single point harmonic force.Based on the above analysis method,the vibration displacement response of the z direction at the origin of the steady state is analyzed to obtain the probabilistic characteristics of the random response of the system.The grid of plate structure is refined gradually by the refinement criteria of grid element,and it can be used to analyze the effect of the changeable elastic support random boundary conditions on the random displacement response.The simulation results show that the proposed method can obtain the same results as the direct Monte Carlo simulation,and obtain the statistical characteristics of the vibration response of the random plate structure.Through the mesh refinement according to the finite element method on the boundary of the discrete stiffness,the statistical characteristics of the vibration response of the random plate structure with the continuous stiffness boundary is tended to a certain distribution.