An efficient and accurate uncertainty propagation methodology for mechanics problems with random fields is developed in this paper. This methodology is based on the stochastic response surface method （SRSM） which ha...An efficient and accurate uncertainty propagation methodology for mechanics problems with random fields is developed in this paper. This methodology is based on the stochastic response surface method （SRSM） which has been previously proposed for problems dealing with random variables only. This paper extends SRSM to problems involving random fields or random processes fields. The favorable property of SRSM lies in that the deterministic computational model can be treated as a black box, as in the case of commercial finite element codes. Numerical examples are used to highlight the features of this technique and to demonstrate the accuracy and efficiency of the proposed method. A comparison with Monte Carlo simulation shows that the proposed method can achieve numerical results close to those from Monte Carlo simulation while dramatically reducing the number of deterministic finite element runs.展开更多
A convenient and universal residue calculus method is proposed to study the stochastic response behaviors of an axially moving viscoelastic beam with random noise excitations and fractional order constitutive relation...A convenient and universal residue calculus method is proposed to study the stochastic response behaviors of an axially moving viscoelastic beam with random noise excitations and fractional order constitutive relationship, where the random excitation can be decomposed as a nonstationary stochastic process, Mittag-Leffler internal noise, and external stationary noise excitation. Then, based on the Laplace transform approach, we derived the mean value function, variance function and covariance function through the Green＇s function technique and the residue calculus method, and obtained theoretical results. In some special case of fractional order derivative α , the Monte Carlo approach and error function results were applied to check the effectiveness of the analytical results, and good agreement was found. Finally in a general-purpose case, we also confirmed the analytical conclusion via the direct Monte Carlo simulation.展开更多
The dynamic response of offshore platforms is more serious in hostile sea environment than in shallow sea. In this paper, a hybrid solution combined with analytical and numerical method is proposed to compute the stoc...The dynamic response of offshore platforms is more serious in hostile sea environment than in shallow sea. In this paper, a hybrid solution combined with analytical and numerical method is proposed to compute the stochastic response of fixed offshore platforms to random waves, considering wave-structure interaction and non-linear drag force. The simulation program includes two steps: the first step is the eigenanalysis aspects associated the structure and the second step is response estimation based on spectral equations. The eigenanalysis could be done through conventional finite element method conveniently and its natural frequency and mode shapes obtained. In the second part of the process, the solution of the offshore structural response is obtained by iteration of a series of coupled spectral equations. Considering the third-order term in the drag force, the evaluation of the three-fold convolution should be demanded for nonlinear stochastic response analysis. To demonstrate this method, a numerical analysis is carried out for both linear and non-linear platform motions. The final response spectra have the typical two peaks in agreement with reality, indicating that the hybrid method is effective and can be applied to offshore engineering.展开更多
The joint probability density fimction （PDF） of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution ...The joint probability density fimction （PDF） of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution method, which is successfully developed to capture the instantaneous PDF of an arbitrary single response of interest, is extended to evaluate the joint PDF of any two responses. A two-dimensional partial differential equation in terms of the joint PDF is established. The strategy of selecting representative points via the number theoretical method and sieved by a hyper-ellipsoid is outlined. A two-dimensional difference scheme is developed. The free vibration of an SDOF system is examined to verify the proposed method, and a flame structure exhibiting hysteresis subjected to stochastic ground motion is investigated. It is pointed out that the correlation of different responses results from the fact that randomness of different responses comes from the same set of basic random parameters involved. In other words, the essence of the probabilistic correlation is a physical correlation.展开更多
We investigate the stochastic responses of a tumor–immune system competition model with environmental noise and periodic treatment. Firstly, a mathematical model describing the interaction between tumor cells and imm...We investigate the stochastic responses of a tumor–immune system competition model with environmental noise and periodic treatment. Firstly, a mathematical model describing the interaction between tumor cells and immune system under external fluctuations and periodic treatment is established based on the stochastic differential equation. Then, sufficient conditions for extinction and persistence of the tumor cells are derived by constructing Lyapunov functions and Ito＇s formula. Finally, numerical simulations are introduced to illustrate and verify the results. The results of this work provide the theoretical basis for designing more effective and precise therapeutic strategies to eliminate cancer cells, especially for combining the immunotherapy and the traditional tools.展开更多
In this paper, the analysis method of stochastic response of piled offshore platform excited by stationary filtered white noise is presented. With this method, the strong ground motion is considered as three direction...In this paper, the analysis method of stochastic response of piled offshore platform excited by stationary filtered white noise is presented. With this method, the strong ground motion is considered as three direction stationary filtered white noise process, the theoretic solutions of three special integration equations are derived with the residue theorem, and the expression of response nodal displacements and member forces of offshore platform excited by the stationary filtered white noise is put forward. The stochastic response of a piled offshore platform excited by the stationary filtered white noise, which is located 114.3 m in water depth, is computed. The results are compared with those obtained with the response spectrum analysis method and the stationary white noise model analysis method, and the corresponding conclusion is drawn.展开更多
The stochastic response of a noisy system with non-negative restoring force is investigated. The generalized cell mapping （GCM） method compute the transient and stationary probability density functions （PDFs） real...The stochastic response of a noisy system with non-negative restoring force is investigated. The generalized cell mapping （GCM） method compute the transient and stationary probability density functions （PDFs） real-power is used to Combined with the global properties of the noise-free system, the evolutionary process of the tran- sient PDFs is revealed. The results show that stochastic P-bifurcation occurs when the system parameter varies in the response analysis and the stationary PDF evolves from bimodal to unimodal along the unstable manifold during the bifurcation.展开更多
The approximate transient response of quasi in- tegrable Hamiltonian systems under Gaussian white noise excitations is investigated. First, the averaged It6 equa- tions for independent motion integrals and the associa...The approximate transient response of quasi in- tegrable Hamiltonian systems under Gaussian white noise excitations is investigated. First, the averaged It6 equa- tions for independent motion integrals and the associated Fokker-Planck-Kolmogorov （FPK） equation governing the transient probability density of independent motion integrals of the system are derived by applying the stochastic averag- ing method for quasi integrable Hamiltonian systems. Then, approximate solution of the transient probability density of independent motion integrals is obtained by applying the Galerkin method to solve the FPK equation. The approxi- mate transient solution is expressed as a series in terms of properly selected base functions with time-dependent coeffi- cients. The transient probability densities of displacements and velocities can be derived from that of independent mo- tion integrals. Three examples are given to illustrate the ap- plication of the proposed procedure. It is shown that the re- suits for the three examples obtained by using the proposed procedure agree well with those from Monte Carlo simula- tion of the original systems.展开更多
Great progress has been made in study on dynamic behavior of the damaged structures subject to deterministic excitation.The stochastic response analysis of the damaged structures,however,has not yet attracted people&#...Great progress has been made in study on dynamic behavior of the damaged structures subject to deterministic excitation.The stochastic response analysis of the damaged structures,however,has not yet attracted people's attention.Taking the damaged elastic beams for example,the analysis procedure for stochastic response of the damaged structures subject to stochastic excitations is investigated in this paper.First,the damage constitutive relations and the corresponding damage evolution equation of one-dimensional elastic structures are briefly discussed.Second,the stochastic dynamic equation with respect to transverse displacement of the damaged elastic beams is deduced.The finite difference method and Newmark method are adopted to solve the stochastic partially-differential equation and corresponding boundary conditions.The stochastic response characteristic,damage evolution law,the effect of noise intensity on damage evolution and the first-passage time of damage are discussed in detail.The present work extends the research field of damaged structures,and the proposed procedure can be generalized to analyze the dynamic behavior of more complex structures,such as damaged plates and shells.展开更多
Viscoelastic dampers,as spplementary energy dissipation devices,have been used in building structures un- der seismic excitation or wind loads.Different analytical models have been proposed to describe their dynamic f...Viscoelastic dampers,as spplementary energy dissipation devices,have been used in building structures un- der seismic excitation or wind loads.Different analytical models have been proposed to describe their dynamic force deform- ation characteristics.Among these analytieal models,the fractional derivative models have attracted more attention as they can capture the frequency dependence of the material stiffness and damping properties observed from tests very well.In this paper,a Fourier-transform-based technique is presented to obtain the fractional unit impulse function and the response of structures with added viscoelastic dampers whose foree-detormation relationship is described by a fractional derivative mod- el.Then,a Duhamel integral-type expression is suggested for the response analysis of a fractional damped dynamie system subjected to deterministic or random excitation.Through numerical verification,it is shown that viscoelastic dampers are ef- fective in reducing structural responses over a wide frequency range,and the proposed schmnes can be used to accurately predict the stochastic seismic response of structures with added viscoelastic dampers described by a Kelvin model wills frac- tional derivative.展开更多
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
Friction systems are a kind of typical non-linear dynamical systems in the actual engineering and often generate abundant dynamics phenomena.Because of non-smooth characteristics,it is difficult to handle these system...Friction systems are a kind of typical non-linear dynamical systems in the actual engineering and often generate abundant dynamics phenomena.Because of non-smooth characteristics,it is difficult to handle these systems by conventional analysis methods directly.At the same time,random perturbation often affects friction systems and makes these systems more complicated.In this context,we investigate the steady-state stochastic responses and stochastic P-bifurcation of friction systems under random excitations in this paper.And in order to retain the non-smooth of friction system,the generalized cell mapping(GCM)method is first used to the original stochastic friction systems without any approximate transformation.To verify the accuracy and validate the applicability of the suggested approach,we present two classical nonlinear friction systems,i.e.,Coulomb force model and Dahl force model as examples.Meanwhile,this method is in good agreement with the Monte Carlo simulation method and the computation time is greatly reduced.In addition,further discussion finds that the adjustable parameters can induce the stochastic P-bifurcation in the two examples,respectively.The stochastic P-bifurcation phenomena indicate that the stability of the friction system changes very sensitively with the parameters.Research of responses analysis and stochastic P-bifurcation has certain significances for the reliability and stability analysis of practical engineering.展开更多
A general method is developed for optimal application of dampers and actuators by installing them at optimal location on seismic-resistant structures.The study includes development of a statistical criterion,formulati...A general method is developed for optimal application of dampers and actuators by installing them at optimal location on seismic-resistant structures.The study includes development of a statistical criterion,formulation of a general optimization problem and establishment of a solution procedure.Numerical analysis of the seismic response in time-history of controlled structures is used to verify the proposed method for optimal device application and to demonstrate the effectiveness of seismic response control with optimal device location.This study shows that the proposed method for the optimal device application is simple and general,and that the optimally applied dampers and actuators are very efficient for seismic response reduction.展开更多
An isolated structure often possesses distinct non-proportional damping characteristics.However,traditional seismic calculation theory and methods are derived based on the assumption that damping is proportional.Based...An isolated structure often possesses distinct non-proportional damping characteristics.However,traditional seismic calculation theory and methods are derived based on the assumption that damping is proportional.Based on this drawback,a new,more efficient stochastic calculation method,an improvement on the pseudo-excitation method,is introduced.This method is then applied to the seismic analysis of an isolated structure.By comparing it with the forced decoupling,matrix inversion and iteration methods,it is shown that the presented method can produce accurate results while increasing the efficiency of the stochastic analysis.Moreover,the calculation process of the seismic response of an isolated structure is convergent.Based on the results of the example presented in this paper,the given method is applicable to the seismic analysis of an isolated structure and can be utilized in practice.展开更多
Reliability and optimization are two key elements for structural design. The reliability~ based topology optimization （RBTO） is a powerful and promising methodology for finding the optimum topologies with the uncert...Reliability and optimization are two key elements for structural design. The reliability~ based topology optimization （RBTO） is a powerful and promising methodology for finding the optimum topologies with the uncertainties being explicitly considered, typically manifested by the use of reliability constraints. Generally, a direct integration of reliability concept and topol- ogy optimization may lead to computational difficulties. In view of this fact, three methodologies have been presented in this study, including the double-loop approach （the performance measure approach, PMA） and the decoupled approaches （the so-called Hybrid method and the sequential optimization and reliability assessment, SORA）. For reliability analysis, the stochastic response surface method （SRSM） was applied, combining with the design of experiments generated by the sparse grid method, which has been proven as an effective and special discretization technique. The methodologies were investigated with three numerical examples considering the uncertainties including material properties and external loads. The optimal topologies obtained using the de- terministic, RBTOs were compared with one another; and useful conclusions regarding validity, accuracy and efficiency were drawn.展开更多
In this paper,stochastic dynamics of a single degree‐of‐freedom quasi‐linear system with multitime delays and Poisson white noises are investigated using an approximate procedure based on the stochastic averaging m...In this paper,stochastic dynamics of a single degree‐of‐freedom quasi‐linear system with multitime delays and Poisson white noises are investigated using an approximate procedure based on the stochastic averaging method.The simplified equations,including the averaged stochastic differential equation and the averaged generalized Fokker–Planck–Kolmogorov equation,are obtained to calculate the probability density functions(PDFs)to explore stationary responses.The expression of the Lyapunov exponent is presented to examine the asymptotic stochastic Lyapunov stability.An illustrative example of a quasi‐linear oscillator with two Poisson white noises controlled by two time‐delayed feedback forces is worked out to demonstrate the validity of the proposed method.The approximate stationary PDFs of stochastic responses and asymptotic stochastic stability are demonstrated numerically and theoretically.The results show that the Gaussian white noise has a stronger influence on the dynamics than the Poisson white noise with a small mean arrival rate.Moreover,the influence of the time delay and noise parameters on stochastic dynamics is investigated.It is found that the PDFs under the Poisson white noise approach those under Gaussian white noise as the mean arrival rate increases.The time delay can induce stochastic P‐bifurcation of the system.It is also found that the increase of time delay and the mean arrival rates of the Poisson white noises will broaden the unstable parameter region.The comparison between numerical and theoretical results shows the effectiveness of the proposed method.展开更多
This paper intends to study the stochastic response and reliability of the roll motion under the action of wind and wave excitation.The roll motion in random beam seas is described by a four-dimensional(4D)Markov dyna...This paper intends to study the stochastic response and reliability of the roll motion under the action of wind and wave excitation.The roll motion in random beam seas is described by a four-dimensional(4D)Markov dynamic system whose probabilistic properties are governed by the Fokker-Planck(FP)equation.The 4D path integration(PI)method,an efficient numerical technique based on the Markov property of the 4D system,is applied in order to solve the high dimensional FP equation and then the stochastic statistics of the roll motion are derived.Based on the obtained response statistics,the reliability evaluation of the ship stability is performed and the effect of wind action is studied.The accuracy of the 4D PI method and the reliability evaluation is assessed by the versatile Monte Carlo simulation(MCS)method.展开更多
基金The project supported by the National Natural Science Foundation of China(10602036)
文摘An efficient and accurate uncertainty propagation methodology for mechanics problems with random fields is developed in this paper. This methodology is based on the stochastic response surface method （SRSM） which has been previously proposed for problems dealing with random variables only. This paper extends SRSM to problems involving random fields or random processes fields. The favorable property of SRSM lies in that the deterministic computational model can be treated as a black box, as in the case of commercial finite element codes. Numerical examples are used to highlight the features of this technique and to demonstrate the accuracy and efficiency of the proposed method. A comparison with Monte Carlo simulation shows that the proposed method can achieve numerical results close to those from Monte Carlo simulation while dramatically reducing the number of deterministic finite element runs.
基金supported by the National Natural Science Foundation of China (11172233, 10932009 and 10972181)Program for New Century Excellent Talents in University+1 种基金the Shaanxi Project for Young New Star in Science & TechnologyNPU Foundation for Fundamental Research and New Faculties and Research Area Project
文摘A convenient and universal residue calculus method is proposed to study the stochastic response behaviors of an axially moving viscoelastic beam with random noise excitations and fractional order constitutive relationship, where the random excitation can be decomposed as a nonstationary stochastic process, Mittag-Leffler internal noise, and external stationary noise excitation. Then, based on the Laplace transform approach, we derived the mean value function, variance function and covariance function through the Green＇s function technique and the residue calculus method, and obtained theoretical results. In some special case of fractional order derivative α , the Monte Carlo approach and error function results were applied to check the effectiveness of the analytical results, and good agreement was found. Finally in a general-purpose case, we also confirmed the analytical conclusion via the direct Monte Carlo simulation.
基金National Natural Science Foundation of China(Grant No.59895410,59779002)
文摘The dynamic response of offshore platforms is more serious in hostile sea environment than in shallow sea. In this paper, a hybrid solution combined with analytical and numerical method is proposed to compute the stochastic response of fixed offshore platforms to random waves, considering wave-structure interaction and non-linear drag force. The simulation program includes two steps: the first step is the eigenanalysis aspects associated the structure and the second step is response estimation based on spectral equations. The eigenanalysis could be done through conventional finite element method conveniently and its natural frequency and mode shapes obtained. In the second part of the process, the solution of the offshore structural response is obtained by iteration of a series of coupled spectral equations. Considering the third-order term in the drag force, the evaluation of the three-fold convolution should be demanded for nonlinear stochastic response analysis. To demonstrate this method, a numerical analysis is carried out for both linear and non-linear platform motions. The final response spectra have the typical two peaks in agreement with reality, indicating that the hybrid method is effective and can be applied to offshore engineering.
基金the National Natural Science Foundation of Chinafor Innovative Research Groups Under Grant No.50621062the National Natural Science Foundation of China forYoung Scholars Under Grant No.10402030
文摘The joint probability density fimction （PDF） of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution method, which is successfully developed to capture the instantaneous PDF of an arbitrary single response of interest, is extended to evaluate the joint PDF of any two responses. A two-dimensional partial differential equation in terms of the joint PDF is established. The strategy of selecting representative points via the number theoretical method and sieved by a hyper-ellipsoid is outlined. A two-dimensional difference scheme is developed. The free vibration of an SDOF system is examined to verify the proposed method, and a flame structure exhibiting hysteresis subjected to stochastic ground motion is investigated. It is pointed out that the correlation of different responses results from the fact that randomness of different responses comes from the same set of basic random parameters involved. In other words, the essence of the probabilistic correlation is a physical correlation.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11402157 and 11571009)Shanxi Scholarship Council of China(Grant No.2015-032)+1 种基金Technological Innovation Programs of Higher Education Institutions in Shanxi,China(Grant No.2015121)Applied Basic Research Programs of Shanxi Province,China(Grant No.2016021013)
文摘We investigate the stochastic responses of a tumor–immune system competition model with environmental noise and periodic treatment. Firstly, a mathematical model describing the interaction between tumor cells and immune system under external fluctuations and periodic treatment is established based on the stochastic differential equation. Then, sufficient conditions for extinction and persistence of the tumor cells are derived by constructing Lyapunov functions and Ito＇s formula. Finally, numerical simulations are introduced to illustrate and verify the results. The results of this work provide the theoretical basis for designing more effective and precise therapeutic strategies to eliminate cancer cells, especially for combining the immunotherapy and the traditional tools.
文摘In this paper, the analysis method of stochastic response of piled offshore platform excited by stationary filtered white noise is presented. With this method, the strong ground motion is considered as three direction stationary filtered white noise process, the theoretic solutions of three special integration equations are derived with the residue theorem, and the expression of response nodal displacements and member forces of offshore platform excited by the stationary filtered white noise is put forward. The stochastic response of a piled offshore platform excited by the stationary filtered white noise, which is located 114.3 m in water depth, is computed. The results are compared with those obtained with the response spectrum analysis method and the stationary white noise model analysis method, and the corresponding conclusion is drawn.
基金Project supported by the National Natural Science Foundation of China(Nos.11172233,11302169,11302170,and 11472212)the Fundamental Research Funds for the Central Universities(No.3102014JCQ01079)
文摘The stochastic response of a noisy system with non-negative restoring force is investigated. The generalized cell mapping （GCM） method compute the transient and stationary probability density functions （PDFs） real-power is used to Combined with the global properties of the noise-free system, the evolutionary process of the tran- sient PDFs is revealed. The results show that stochastic P-bifurcation occurs when the system parameter varies in the response analysis and the stationary PDF evolves from bimodal to unimodal along the unstable manifold during the bifurcation.
基金supported by the National Natural Science Foundation of China(10902094,10932009,11072212 and 11272279)the Special Foundation for Young Scientists of Fujian Province of China(2008F3100)
文摘The approximate transient response of quasi in- tegrable Hamiltonian systems under Gaussian white noise excitations is investigated. First, the averaged It6 equa- tions for independent motion integrals and the associated Fokker-Planck-Kolmogorov （FPK） equation governing the transient probability density of independent motion integrals of the system are derived by applying the stochastic averag- ing method for quasi integrable Hamiltonian systems. Then, approximate solution of the transient probability density of independent motion integrals is obtained by applying the Galerkin method to solve the FPK equation. The approxi- mate transient solution is expressed as a series in terms of properly selected base functions with time-dependent coeffi- cients. The transient probability densities of displacements and velocities can be derived from that of independent mo- tion integrals. Three examples are given to illustrate the ap- plication of the proposed procedure. It is shown that the re- suits for the three examples obtained by using the proposed procedure agree well with those from Monte Carlo simula- tion of the original systems.
基金supported by the National Natural Science Foundation of China (Grant No. 11072076)
文摘Great progress has been made in study on dynamic behavior of the damaged structures subject to deterministic excitation.The stochastic response analysis of the damaged structures,however,has not yet attracted people's attention.Taking the damaged elastic beams for example,the analysis procedure for stochastic response of the damaged structures subject to stochastic excitations is investigated in this paper.First,the damage constitutive relations and the corresponding damage evolution equation of one-dimensional elastic structures are briefly discussed.Second,the stochastic dynamic equation with respect to transverse displacement of the damaged elastic beams is deduced.The finite difference method and Newmark method are adopted to solve the stochastic partially-differential equation and corresponding boundary conditions.The stochastic response characteristic,damage evolution law,the effect of noise intensity on damage evolution and the first-passage time of damage are discussed in detail.The present work extends the research field of damaged structures,and the proposed procedure can be generalized to analyze the dynamic behavior of more complex structures,such as damaged plates and shells.
文摘Viscoelastic dampers,as spplementary energy dissipation devices,have been used in building structures un- der seismic excitation or wind loads.Different analytical models have been proposed to describe their dynamic force deform- ation characteristics.Among these analytieal models,the fractional derivative models have attracted more attention as they can capture the frequency dependence of the material stiffness and damping properties observed from tests very well.In this paper,a Fourier-transform-based technique is presented to obtain the fractional unit impulse function and the response of structures with added viscoelastic dampers whose foree-detormation relationship is described by a fractional derivative mod- el.Then,a Duhamel integral-type expression is suggested for the response analysis of a fractional damped dynamie system subjected to deterministic or random excitation.Through numerical verification,it is shown that viscoelastic dampers are ef- fective in reducing structural responses over a wide frequency range,and the proposed schmnes can be used to accurately predict the stochastic seismic response of structures with added viscoelastic dampers described by a Kelvin model wills frac- tional derivative.
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the National Science Foundation of China through the Grants(11872306,11772256)the Central University Fundamental Research Fund(3102018zy043).
文摘Friction systems are a kind of typical non-linear dynamical systems in the actual engineering and often generate abundant dynamics phenomena.Because of non-smooth characteristics,it is difficult to handle these systems by conventional analysis methods directly.At the same time,random perturbation often affects friction systems and makes these systems more complicated.In this context,we investigate the steady-state stochastic responses and stochastic P-bifurcation of friction systems under random excitations in this paper.And in order to retain the non-smooth of friction system,the generalized cell mapping(GCM)method is first used to the original stochastic friction systems without any approximate transformation.To verify the accuracy and validate the applicability of the suggested approach,we present two classical nonlinear friction systems,i.e.,Coulomb force model and Dahl force model as examples.Meanwhile,this method is in good agreement with the Monte Carlo simulation method and the computation time is greatly reduced.In addition,further discussion finds that the adjustable parameters can induce the stochastic P-bifurcation in the two examples,respectively.The stochastic P-bifurcation phenomena indicate that the stability of the friction system changes very sensitively with the parameters.Research of responses analysis and stochastic P-bifurcation has certain significances for the reliability and stability analysis of practical engineering.
基金the National Science Foundation under grant CMS 9903136
文摘A general method is developed for optimal application of dampers and actuators by installing them at optimal location on seismic-resistant structures.The study includes development of a statistical criterion,formulation of a general optimization problem and establishment of a solution procedure.Numerical analysis of the seismic response in time-history of controlled structures is used to verify the proposed method for optimal device application and to demonstrate the effectiveness of seismic response control with optimal device location.This study shows that the proposed method for the optimal device application is simple and general,and that the optimally applied dampers and actuators are very efficient for seismic response reduction.
基金The authors gratefully acknowledge the financial support of this work,which was provided by the National Natural Science Foundation of China(Grant Nos.50938008,51108466)the National Science Foundation for Post-doctoral Scientists of China(Grant No.20110491277),the Science Foundation for Post-doctoral Scientists of Central South University.
文摘An isolated structure often possesses distinct non-proportional damping characteristics.However,traditional seismic calculation theory and methods are derived based on the assumption that damping is proportional.Based on this drawback,a new,more efficient stochastic calculation method,an improvement on the pseudo-excitation method,is introduced.This method is then applied to the seismic analysis of an isolated structure.By comparing it with the forced decoupling,matrix inversion and iteration methods,it is shown that the presented method can produce accurate results while increasing the efficiency of the stochastic analysis.Moreover,the calculation process of the seismic response of an isolated structure is convergent.Based on the results of the example presented in this paper,the given method is applicable to the seismic analysis of an isolated structure and can be utilized in practice.
基金Project supported by the National Natural Science Foundation of China(Nos.51275040 and 50905017)the Programme of Introducing Talents of Discipline to Universities(No.B12022)
文摘Reliability and optimization are two key elements for structural design. The reliability~ based topology optimization （RBTO） is a powerful and promising methodology for finding the optimum topologies with the uncertainties being explicitly considered, typically manifested by the use of reliability constraints. Generally, a direct integration of reliability concept and topol- ogy optimization may lead to computational difficulties. In view of this fact, three methodologies have been presented in this study, including the double-loop approach （the performance measure approach, PMA） and the decoupled approaches （the so-called Hybrid method and the sequential optimization and reliability assessment, SORA）. For reliability analysis, the stochastic response surface method （SRSM） was applied, combining with the design of experiments generated by the sparse grid method, which has been proven as an effective and special discretization technique. The methodologies were investigated with three numerical examples considering the uncertainties including material properties and external loads. The optimal topologies obtained using the de- terministic, RBTOs were compared with one another; and useful conclusions regarding validity, accuracy and efficiency were drawn.
基金This study was supported by the National Natural Science Foundation of China under Grant Nos.11872306,11702214,12072264the Natural Science Basic Research Plan in Shaanxi Province 2020JQ‐108+1 种基金Yong Xu was partially supported by the Key International(Regional)Cooperative Research Projects of the National Natural Science Foundation of China(Grant 12120101002)the Fundamental Research Funds for the Central Universities(Grant D5000220035).
文摘In this paper,stochastic dynamics of a single degree‐of‐freedom quasi‐linear system with multitime delays and Poisson white noises are investigated using an approximate procedure based on the stochastic averaging method.The simplified equations,including the averaged stochastic differential equation and the averaged generalized Fokker–Planck–Kolmogorov equation,are obtained to calculate the probability density functions(PDFs)to explore stationary responses.The expression of the Lyapunov exponent is presented to examine the asymptotic stochastic Lyapunov stability.An illustrative example of a quasi‐linear oscillator with two Poisson white noises controlled by two time‐delayed feedback forces is worked out to demonstrate the validity of the proposed method.The approximate stationary PDFs of stochastic responses and asymptotic stochastic stability are demonstrated numerically and theoretically.The results show that the Gaussian white noise has a stronger influence on the dynamics than the Poisson white noise with a small mean arrival rate.Moreover,the influence of the time delay and noise parameters on stochastic dynamics is investigated.It is found that the PDFs under the Poisson white noise approach those under Gaussian white noise as the mean arrival rate increases.The time delay can induce stochastic P‐bifurcation of the system.It is also found that the increase of time delay and the mean arrival rates of the Poisson white noises will broaden the unstable parameter region.The comparison between numerical and theoretical results shows the effectiveness of the proposed method.
文摘This paper intends to study the stochastic response and reliability of the roll motion under the action of wind and wave excitation.The roll motion in random beam seas is described by a four-dimensional(4D)Markov dynamic system whose probabilistic properties are governed by the Fokker-Planck(FP)equation.The 4D path integration(PI)method,an efficient numerical technique based on the Markov property of the 4D system,is applied in order to solve the high dimensional FP equation and then the stochastic statistics of the roll motion are derived.Based on the obtained response statistics,the reliability evaluation of the ship stability is performed and the effect of wind action is studied.The accuracy of the 4D PI method and the reliability evaluation is assessed by the versatile Monte Carlo simulation(MCS)method.
