This paper introduces an orthogonal expansion method for general stochastic processes. In the method, a normalized orthogonal function of time variable t is first introduced to carry out the decomposition of a stochas...This paper introduces an orthogonal expansion method for general stochastic processes. In the method, a normalized orthogonal function of time variable t is first introduced to carry out the decomposition of a stochastic process and then a correlated matrix decomposition technique, which transforms a correlated random vector into a vector of standard uncorrelated random variables, is used to complete a double orthogonal decomposition of the stochastic processes. Considering the relationship between the Hartley transform and Fourier transform of a real-valued function, it is suggested that the first orthogonal expansion in the above process is carried out using the Hartley basis function instead of the trigonometric basis function in practical applications. The seismic ground motion is investigated using the above method. In order to capture the main probabilistic characteristics of the seismic ground motion, it is proposed to directly carry out the orthogonal expansion of the seismic displacements. The case study shows that the proposed method is feasible to represent the seismic ground motion with only a few random variables. In the second part of the paper, the probability density evolution method （PDEM） is employed to study the stochastic response of nonlinear structures subjected to earthquake excitations. In the PDEM, a completely uncoupled one-dimensional partial differential equation, the generalized density evolution equation, plays a central role in governing the stochastic seismic responses of the nonlinear structure. The solution to this equation will yield the instantaneous probability density function of the responses. Computational algorithms to solve the probability density evolution equation are described. An example, which deals with a nonlinear frame structure subjected to stochastic ground motions, is illustrated to validate the above approach.展开更多
Due to the current trend towards lightweight design in automotive industry,hollow stepped gear shafts for automobile and its radial forging process are widely investigated.Utilizing coupled finite element thermo-mecha...Due to the current trend towards lightweight design in automotive industry,hollow stepped gear shafts for automobile and its radial forging process are widely investigated.Utilizing coupled finite element thermo-mechanical model,radial forging process of a hollow stepped gear shaft for automobile was simulated.The optimal combination of three process parameters including initial temperature,rotation rate and radial reduction was also selected using orthogonal design method.To examine the strain inhomogeneity of the forging workpiece,the strain inhomogeneity factor was introduced.The results reveal that the maximum effective strain and the minimum effective strain appeared in the outermost and innermost zones of different cross sections for the hollow stepped gear shaft,respectively.Optimal forging parameters are determined as a combination of initial temperature of 780°C,rotation rate of 21°/stroke and radial reduction of 3 mm.展开更多
A method of equivalent simplification,using equivalent-plate models(EPMs),is developed.It is to achieve goals of rapid modeling and effective analysis in structural dynamics and flutter analysis of complex wing struct...A method of equivalent simplification,using equivalent-plate models(EPMs),is developed.It is to achieve goals of rapid modeling and effective analysis in structural dynamics and flutter analysis of complex wing structures.It is on the assumption that the wing structures discussed are composed of skin,beams and ribs,and the different plate units(such as skin,beam web,rib web)are not distinguished in modeling,which is to avoid the complex pre-processing and make it more generalized.Taking the effect of transverse shear deformation into consideration,the equivalence is based on the first-order shear deformation theory,and it can import the model files of MSC/NASTRAN and process the information to accomplish the equivalent modeling.The Ritz method is applied with the Legendre polynomials,which is used to define the geometry,structure and displacements of the wing.Particularly,the selection of Legendre polynomials as trial functions brings good accuracy to the modeling and can avoid the ill-conditions.This is in contrast to the EPM method based on the classical plate theory.Through vibration and flutter analysis,the results obtained by using EPM agree well with those obtained by the finite element method,which indicates the accuracy and effectiveness in vibration and flutter analysis of the EPM method.展开更多
Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet ...Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.展开更多
基金National Natural Science Foundation of China for Innovative Research Groups Under Grant No.50321803 & 50621062National Natural Science Foundation of China Under Grant No.50808113 & 10872148
文摘This paper introduces an orthogonal expansion method for general stochastic processes. In the method, a normalized orthogonal function of time variable t is first introduced to carry out the decomposition of a stochastic process and then a correlated matrix decomposition technique, which transforms a correlated random vector into a vector of standard uncorrelated random variables, is used to complete a double orthogonal decomposition of the stochastic processes. Considering the relationship between the Hartley transform and Fourier transform of a real-valued function, it is suggested that the first orthogonal expansion in the above process is carried out using the Hartley basis function instead of the trigonometric basis function in practical applications. The seismic ground motion is investigated using the above method. In order to capture the main probabilistic characteristics of the seismic ground motion, it is proposed to directly carry out the orthogonal expansion of the seismic displacements. The case study shows that the proposed method is feasible to represent the seismic ground motion with only a few random variables. In the second part of the paper, the probability density evolution method （PDEM） is employed to study the stochastic response of nonlinear structures subjected to earthquake excitations. In the PDEM, a completely uncoupled one-dimensional partial differential equation, the generalized density evolution equation, plays a central role in governing the stochastic seismic responses of the nonlinear structure. The solution to this equation will yield the instantaneous probability density function of the responses. Computational algorithms to solve the probability density evolution equation are described. An example, which deals with a nonlinear frame structure subjected to stochastic ground motions, is illustrated to validate the above approach.
基金Projects(51774054,51974050)supported by the National Natural Science Foundation of China。
文摘Due to the current trend towards lightweight design in automotive industry,hollow stepped gear shafts for automobile and its radial forging process are widely investigated.Utilizing coupled finite element thermo-mechanical model,radial forging process of a hollow stepped gear shaft for automobile was simulated.The optimal combination of three process parameters including initial temperature,rotation rate and radial reduction was also selected using orthogonal design method.To examine the strain inhomogeneity of the forging workpiece,the strain inhomogeneity factor was introduced.The results reveal that the maximum effective strain and the minimum effective strain appeared in the outermost and innermost zones of different cross sections for the hollow stepped gear shaft,respectively.Optimal forging parameters are determined as a combination of initial temperature of 780°C,rotation rate of 21°/stroke and radial reduction of 3 mm.
文摘A method of equivalent simplification,using equivalent-plate models(EPMs),is developed.It is to achieve goals of rapid modeling and effective analysis in structural dynamics and flutter analysis of complex wing structures.It is on the assumption that the wing structures discussed are composed of skin,beams and ribs,and the different plate units(such as skin,beam web,rib web)are not distinguished in modeling,which is to avoid the complex pre-processing and make it more generalized.Taking the effect of transverse shear deformation into consideration,the equivalence is based on the first-order shear deformation theory,and it can import the model files of MSC/NASTRAN and process the information to accomplish the equivalent modeling.The Ritz method is applied with the Legendre polynomials,which is used to define the geometry,structure and displacements of the wing.Particularly,the selection of Legendre polynomials as trial functions brings good accuracy to the modeling and can avoid the ill-conditions.This is in contrast to the EPM method based on the classical plate theory.Through vibration and flutter analysis,the results obtained by using EPM agree well with those obtained by the finite element method,which indicates the accuracy and effectiveness in vibration and flutter analysis of the EPM method.
基金supported by the Foundation for Innovation Research Groups of the National Natural Science Foundation of China(No.51621092)National Natural Science Foundation of China(Nos.51339003 and 51409186)
基金the National Natural Science Foundation of China (Nos.11571238,11601332,91130014,11471312 and 91430216).
文摘Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.
