Reliability of Nonlinear Vibratory Systems Under Non-Gaussian Loads

2017 ◽  
Author(s):  
Vasileios Geroulas ◽  
Zissimos P. Mourelatos ◽  
Vasiliki Tsianika ◽  
Igor Baseski
Keyword(s):  
Duffing Oscillator ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Correlation Structure ◽  
Output Process ◽  
Quasi Monte Carlo ◽  
Standard Normal ◽  
Mc Simulation ◽  
Non Gaussian ◽  
First Four Moments

A general methodology is presented for time-dependent reliability and random vibrations of nonlinear vibratory systems with random parameters excited by non-Gaussian loads. The approach is based on Polynomial Chaos Expansion (PCE), Karhunen-Loeve (KL) expansion and Quasi Monte Carlo (QMC). The latter is used to estimate multi-dimensional integrals efficiently. The input random processes are first characterized using their first four moments (mean, standard deviation, skewness and kurtosis coefficients) and a correlation structure in order to generate sample realizations (trajectories). Characterization means the development of a stochastic metamodel. The input random variables and processes are expressed in terms of independent standard normal variables in N dimensions. The N-dimensional input space is space filled with M points. The system differential equations of motion are time integrated for each of the M points and QMC estimates the four moments and correlation structure of the output efficiently. The proposed PCE-KL-QMC approach is then used to characterize the output process. Finally, classical MC simulation estimates the time-dependent probability of failure using the developed stochastic metamodel of the output process. The proposed methodology is demonstrated with a Duffing oscillator example under non-Gaussian load.

Reliability Analysis of Nonlinear Vibratory Systems Under Non-Gaussian Loads

2017 ◽  
Vol 140 (2) ◽  
Author(s):  
Vasileios Geroulas ◽  
Zissimos P. Mourelatos ◽  
Vasiliki Tsianika ◽  
Igor Baseski
Keyword(s):  
Duffing Oscillator ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Correlation Structure ◽  
Output Process ◽  
Quasi Monte Carlo ◽  
Standard Normal ◽  
Non Gaussian ◽  
First Four Moments ◽  
Multidimensional Integrals

A general methodology is presented for time-dependent reliability and random vibrations of nonlinear vibratory systems with random parameters excited by non-Gaussian loads. The approach is based on polynomial chaos expansion (PCE), Karhunen–Loeve (KL) expansion, and quasi Monte Carlo (QMC). The latter is used to estimate multidimensional integrals efficiently. The input random processes are first characterized using their first four moments (mean, standard deviation, skewness, and kurtosis coefficients) and a correlation structure in order to generate sample realizations (trajectories). Characterization means the development of a stochastic metamodel. The input random variables and processes are expressed in terms of independent standard normal variables in N dimensions. The N-dimensional input space is space filled with M points. The system differential equations of motion (EOM) are time integrated for each of the M points, and QMC estimates the four moments and correlation structure of the output efficiently. The proposed PCE–KL–QMC approach is then used to characterize the output process. Finally, classical MC simulation estimates the time-dependent probability of failure using the developed stochastic metamodel of the output process. The proposed methodology is demonstrated with a Duffing oscillator example under non-Gaussian load.

Reliability Analysis and Random Vibration of Nonlinear Systems Using the Adjoint Method and Projected Differentiation

2020 ◽  
Vol 143 (6) ◽  
Author(s):  
Dimitrios Papadimitriou ◽  
Zissimos P. Mourelatos ◽  
Zhen Hu
Keyword(s):  
Reliability Analysis ◽  
Equations Of Motion ◽  
Random Variables ◽  
Random Processes ◽  
Time Dependent ◽  
Vibratory System ◽  
Standard Normal ◽  
Non Gaussian ◽  
Nonlinear Equations Of Motion ◽  
Adjoint Approach

Abstract This paper proposes a new methodology for time-dependent reliability and random vibrations of nonlinear vibratory systems using a combination of a time-dependent adjoint variable (AV) method and a projected differentiation (PD) method. The proposed approach is called AV-PD. The vibratory system is excited by stationary Gaussian or non-Gaussian input random processes. A Karhunen–Loeve (KL) expansion expresses each input random process in terms of standard normal random variables. The nonlinear equations of motion (EOM) are linearized using a Taylor expansion using the first-order derivatives of the output with respect to the input KL random variables. An adjoint approach obtains the output derivatives accurately and efficiently requiring the solution of as many sets of EOM as the number of outputs of interest, independently of the number of KL random variables. The proposed PD method then computes the autocorrelation function of each output process at an additional cost of solving as many sets of EOM as the number of outputs of interest, independently of the time horizon (simulation time). A time-dependent reliability analysis is finally performed using a KL expansion of the output processes and Monte Carlo simulation (MCS). The number of solutions of the EOM scales only with the number of output random processes which is commonly much smaller than the number of input KL random variables. The efficiency and accuracy of the proposed approach is demonstrated using a four degree-of-freedom (DOF) half-car vibratory problem.

Fatigue Life Analysis of Vehicle Systems under Non-Gaussian Excitation

2020 ◽  
Vol 17 (2) ◽  
pp. 7956-7966
Author(s):  
Z. Yang ◽  
D. Zhu ◽  
Q. Gao ◽  
L. Gong ◽  
D. Xu
Keyword(s):  
Monte Carlo ◽  
Fatigue Life ◽  
Equations Of Motion ◽  
Principal Component ◽  
Correlation Structure ◽  
Vehicle Systems ◽  
Life Analysis ◽  
Fatigue Life Analysis ◽  
Non Gaussian ◽  
First Four Moments

Fatigue life analysis is an important work in manufacturing of vehicle systems. The traditional method is to assume that stochastic loads are Gaussian type, then fatigue life is calculated by rain-flow counting, S-N curve and Miner linear damage rule. However, it is difficult to acquire accurate results by this means. In this paper, a numerical methodology is used to simulate non-Gaussian loads considering effects of skewness and kurtosis, as well as to estimate fatigue life under non-Gaussian stresses. Firstly, non-Gaussian inputs are represented by polynomial chaos expansion (PCE) and Karhunen-Loeve (KL) expansion when they are characterised using first four moments, i.e. mean, variance, skewness, kurtosis and a given correlation structure. During this process, we propose spectral decomposition to eliminate the influence of potential imaginary numbers, principal component analysis is also proposed to simplify calculating procedure in KL. Besides, original Monte Carlo sampling is replaced by quasi Monte-Carlo (QMC), which could greatly reduce the workload of numerical simulations. In order to get first four moments and correlation structure of outputs, differential equations of motion are numerically integrated by Runge-Kutta method. Meanwhile, response trajectories are represented based on PCE-KL-QMC approach. Eventually, the rain-flow counting is applied into these trajectories to obtain fatigue life variables, and a convenient formula about the saddlepoint approximations (SPA) represented by first four moments is proposed to provide fatigue life PDF. According to the above way, accurate and effective fatigue life estimation results can be presented

Reliability Analysis and Random Vibration of Nonlinear Systems Using the Adjoint Method and Projected Differentiation

2020 ◽  
Author(s):  
Dimitrios Papadimitriou ◽  
Zissimos P. Mourelatos ◽  
Zhen Hu
Keyword(s):  
Reliability Analysis ◽  
Equations Of Motion ◽  
Random Variables ◽  
Random Processes ◽  
Time Dependent ◽  
Vibratory System ◽  
Standard Normal ◽  
Non Gaussian ◽  
Nonlinear Equations Of Motion ◽  
Adjoint Approach

Abstract This paper proposes a new methodology for time-dependent reliability and random vibrations of nonlinear vibratory systems using a combination of a time-dependent adjoint variable (AV) method and a projected differentiation (PD) method. The proposed approach is called AV-PD. The vibratory system is excited by stationary Gaussian or non-Gaussian input random processes. A Karhunen-Loeve (KL) expansion expresses each input random process in terms of standard normal random variables. The nonlinear equations of motion (EOM) are linearized using a Taylor expansion using the first-order derivatives of the output with respect to the input KL random variables. An adjoint approach obtains the output derivatives accurately and efficiently requiring the solution of as many sets of EOM as the number of outputs of interest, independently of the number of KL random variables. The proposed PD method then computes the autocorrelation function of each output process at an additional cost of solving as many sets of EOM as the number of outputs of interest, independently of the time horizon (simulation time). A time-dependent reliability analysis is finally performed using a KL expansion of the output processes and Monte Carlo Simulation (MCS). The number of solutions of the EOM scales only with the number of output random processes which is commonly much smaller than the number of input KL random variables. The efficiency and accuracy of the proposed approach is demonstrated using a four degree-of-freedom (DOF) half-car vibratory problem.

Reliability Analysis of Nonlinear Vibratory Systems Under Non-Gaussian Loads Using a Sensitivity-Based Propagation of Moments

2020 ◽  
Vol 142 (6) ◽  
Author(s):  
Dimitrios Papadimitriou ◽  
Zissimos P. Mourelatos ◽  
Santosh Patil ◽  
Zhen Hu ◽  
Vasiliki Tsianika ◽  
...  
Keyword(s):  
Reliability Analysis ◽  
Autocorrelation Function ◽  
Computational Cost ◽  
Random Variables ◽  
Random Coefficients ◽  
Time Dependent ◽  
Vibratory System ◽  
First Order ◽  
Non Gaussian ◽  
First Four Moments

Abstract The paper proposes a new methodology for time-dependent reliability analysis of vibratory systems using a combination of a first-order, four-moment (FOFM) method and a non-Gaussian Karhunen–Loeve (NG-KL) expansion. The approach can also be used for random vibrations studies. The vibratory system is nonlinear and is excited by stationary non-Gaussian input random processes which are characterized by their first four marginal moments and autocorrelation function. The NG-KL expansion expresses each input non-Gaussian process as a linear combination of uncorrelated, non-Gaussian random variables and computes their first four moments. The FOFM method then uses the moments of the NG-KL variables to calculate the moments and autocorrelation function of the output processes based on a first-order Taylor expansion (linearization) of the system equations of motion. Using the output moments and autocorrelation function, another NG-KL expansion expresses the output processes in terms of uncorrelated non-Gaussian variables in the time domain, allowing the generation of output trajectories. The latter are used to estimate the time-dependent probability of failure using Monte Carlo simulation (MCS). The computational cost of the proposed approach is proportional to the number of NG-KL random variables and is significantly lower than that of other recently developed methodologies which are based on sampling. The accuracy and efficiency of the proposed methodology is demonstrated using a two-degree-of-freedom nonlinear vibratory system with random coefficients excited by a stationary non-Gaussian random process.

Reliability Analysis of Nonlinear Vibratory Systems Under Non-Gaussian Loads Using a Sensitivity-Based Propagation of Moments

2019 ◽  
Author(s):  
Dimitrios Papadimitriou ◽  
Zissimos P. Mourelatos ◽  
Santosh Patil ◽  
Zhen Hu ◽  
Vasiliki Tsianika ◽  
...  
Keyword(s):  
Reliability Analysis ◽  
Autocorrelation Function ◽  
Computational Cost ◽  
Random Variables ◽  
Random Coefficients ◽  
Time Dependent ◽  
Vibratory System ◽  
First Order ◽  
Non Gaussian ◽  
First Four Moments

Abstract This paper proposes a new methodology for time-dependent reliability analysis of vibratory systems using a combination of a First-Order, Four-Moment (FOFM) method and a Non-Gaussian Karhunen-Loeve (NG-KL) expansion. The vibratory system is nonlinear and it is excited by stationary non-Gaussian input random processes which are characterized by their first four marginal moments and autocorrelation function. The NG-KL expansion expresses each input non-Gaussian process as a linear combination of uncorrelated, non-Gaussian random variables and computes their first four moments. The FOFM method then uses the moments of the NG-KL variables to calculate the moments and autocorrelation function of the output processes based on a first-order Taylor expansion (linearization) of the system equations of motion. Using the output moments and autocorrelation function, another NG-KL expansion expresses the output processes in terms of uncorrelated non-Gaussian variables in the time domain, allowing the generation of output trajectories. The latter are used to estimate the time-dependent probability of failure using Monte Carlo Simulation (MCS). The computational cost of the proposed approach is proportional to the number of NG-KL random variables and is significantly lower than that of other recently developed methodologies which are based on sampling. The accuracy and efficiency of the proposed methodology is demonstrated using a two-degree of freedom nonlinear vibratory system with random coefficients excited by a stationary non-Gaussian random process.

Non-Linear Random Vibrations Using Second-Order Adjoint and Projected Differentiation Methods

2021 ◽  
Author(s):  
Dimitrios Papadimitriou ◽  
Zissimos P. Mourelatos ◽  
Zhen Hu
Keyword(s):  
Duffing Oscillator ◽  
Equations Of Motion ◽  
Random Variables ◽  
Random Excitation ◽  
Second Order ◽  
Output Process ◽  
Random Vibrations ◽  
Order Accuracy ◽  
Vibratory System ◽  
Non Linear

Abstract This paper proposes a new computationally efficient methodology for random vibrations of nonlinear vibratory systems using a time-dependent second-order adjoint variable (AV2) method, and a second-order projected differentiation (PD2) method. The proposed approach is called AV2-PD2. The vibratory system can be excited by stationary Gaussian or non-Gaussian random processes. A Karhunen-Loeve (KL) expansion expresses each input random process in terms of standard normal random variables. A second-order adjoint approach is used to obtain the required first and second-order output derivatives accurately by solving as many sets of equations of motion (EOMs) as the number of KL random variables. These derivatives are used to compute the marginal CDF of the output process with second-order accuracy. Then, a second-order projected differentiation method calculates the autocorrelation function of each output process with second-order accuracy, at an additional cost of solving as many sets of EOM as the number of outputs of interest, independently of the time horizon (simulation time). The total number of solutions of the EOM scales linearly with the number of input KL random variables and the number of output processes. The efficiency and accuracy of the proposed approach is demonstrated using a non-linear Duffing oscillator problem under a quadratic random excitation.

scholarly journals Information Length as a Useful Index to Understand Variability in the Global Circulation

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 299 ◽  
Author(s):  
Eun-jin Kim ◽  
James Heseltine ◽  
Hanli Liu
Keyword(s):  
Climate Model ◽  
Traditional Approach ◽  
Essential Feature ◽  
Complex Form ◽  
Time Dependent ◽  
Spatial Gradient ◽  
Mean Values ◽  
Global Circulation ◽  
New Perspective ◽  
Non Gaussian

With improved measurement and modelling technology, variability has emerged as an essential feature in non-equilibrium processes. While traditionally, mean values and variance have been heavily used, they are not appropriate in describing extreme events where a significant deviation from mean values often occurs. Furthermore, stationary Probability Density Functions (PDFs) miss crucial information about the dynamics associated with variability. It is thus critical to go beyond a traditional approach and deal with time-dependent PDFs. Here, we consider atmospheric data from the Whole Atmosphere Community Climate Model (WACCM) and calculate time-dependent PDFs and the information length from these PDFs, which is the total number of statistically different states that a system evolves through in time. Specifically, we consider the three cases of sampling data to investigate the distribution of information (information budget) along the altitude and longitude to gain a new perspective of understanding variabilities, correlation among different variables and regions. Time-dependent PDFs are shown to be non-Gaussian in general; the information length tends to increase with the altitude albeit in a complex form; this tendency is more robust for flows/shears than temperature. Much similarity among flows and shears in the information length is also found in comparison with the temperature. This means a strong correlation among flows/shears because of their coupling through gravity waves in this particular WACCM model. We also find the increase of the information length with the latitude and interesting hemispheric asymmetry for flows/shears/temperature, with the tendency of anti-correlation (correlation) between flows/shears and temperature at high (low) latitude. These results suggest the importance of high latitude/altitude in the information budget in the Earth’s atmosphere, the spatial gradient of the information length being a useful proxy for information flow.

scholarly journals Boussinesq modeling of surface waves due to underwater landslides

2013 ◽  
Vol 20 (3) ◽  
pp. 267-285 ◽  
Author(s):  
D. Dutykh ◽  
H. Kalisch
Keyword(s):  
Surface Waves ◽  
Shallow Water ◽  
Bottom Topography ◽  
Equations Of Motion ◽  
Boussinesq Equations ◽  
Time Dependent ◽  
Viscous Drag ◽  
Underwater Landslide ◽  
Landslide Mass ◽  
Run Up

Abstract. Consideration is given to the influence of an underwater landslide on waves at the surface of a shallow body of fluid. The equations of motion that govern the evolution of the barycenter of the landslide mass include various dissipative effects due to bottom friction, internal energy dissipation, and viscous drag. The surface waves are studied in the Boussinesq scaling, with time-dependent bathymetry. A numerical model for the Boussinesq equations is introduced that is able to handle time-dependent bottom topography, and the equations of motion for the landslide and surface waves are solved simultaneously. The numerical solver for the Boussinesq equations can also be restricted to implement a shallow-water solver, and the shallow-water and Boussinesq configurations are compared. A particular bathymetry is chosen to illustrate the general method, and it is found that the Boussinesq system predicts larger wave run-up than the shallow-water theory in the example treated in this paper. It is also found that the finite fluid domain has a significant impact on the behavior of the wave run-up.

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