Exponential Polynomial Closure Method Research Articles

The generalized EPC method for the non-stationary probabilistic response of nonlinear dynamical system

It is well known that the non-stationary response probability density function (PDF) plays an important role in the reliability and failure analysis. With current approximate or numerical methods, the non-stationary approximate PDF is generally obtained in terms of the ones at the discrete time instants. Repeated computation must be conducted for the ones at other time instants since the response PDF at only one time instant can be obtained after performing the solution procedure. Thus, the computational efficiency suffers a major setback. In this paper, the exponential polynomial closure (EPC) method is further improved and enhanced to obtain the completely non-stationary PDF solution, which is distributed continuously in the time domain. It takes the temporal base function into the PDF approximation. The unknown coefficients in the EPC solution are generalized to be explicit functions of the time parameter. With the least squares method, the explicit time functions can be determined based on the few simulated response PDF values. This implementation makes the continues PDF distribution in the time domain available, which greatly improves the computational efficiency without the repeating computation. Two typical nonlinear systems under stationary and non-stationary random excitations are taken as examples to illustrate the efficiency of the proposed method. Numerical results show that the results obtained by the proposed method agree well with the simulated results. In addition, the relationship between the explicit time function and the modulate function is discussed.

Random response analysis of nonlinear structures with inerter system

Most researches on inerter focused on the mechanism or deterministic vibration analysis, and some studies lied in the random vibration of linear structures. However, many structures exhibit the nonlinear characteristics under a large excitation, and the effects of inerter system on additional properties and probabilistic distribution of random response of nonlinear structure under random excitation have rarely been considered at present. Based on the enhancement effect of inerter system on nonlinear structure, the influence of various components of inerter system on the response suppression of nonlinear structure is investigated from the probability level, and a developed response analysis method suitable for inerter system is proposed in this paper. The stochastic differential equation of nonlinear structure with inerter is established in this work, in which both the nonlinear structure without hysteresis and the hysteretic structure are considered. The probability density function of structural response is obtained by the exponential polynomial closure (EPC) method combined with the state-space-split technique. In terms of nonstationary random excitation, a scheme of equivalent stationary excitation is proposed to split the state space vector. Meanwhile, the EPC method is developed through the reduction of undetermined coefficients in the probabilistic solution, which significantly improves the efficiency. Numerical results show that the response suppression effectiveness of inerter system is robust to nonlinear structures, and the inerter system mainly provides additional damping for the primary structure, thereby achieving the vibration mitigation. However, an excessively rigid inertia system may lead to a negative additional stiffness for the primary structure and inhibit the vibration mitigation.

Transient probabilistic analysis of nonlinear systems excited by correlated external and parametric Gaussian white noise

This paper explores using the exponential-polynomial-closure (EPC) method to analyze the transient probabilistic solutions to nonlinear stochastic oscillators excited by correlated external and parametric Gaussian white noise. The probabilistic solution to the transient responses of the nonlinear oscillator is governed by the Fokker–Planck–Kolmogorov (FPK) equation, in which the solution is denoted as an exponentially polynomial function with time-variant coefficients. These coefficients can be solved numerically by introducing a proper weighted function and making the projection of the residual error vanish. Meanwhile, three numerical examples are presented to verify the effectiveness and efficiency of the EPC method for analyzing the nonlinear oscillators through the comparisons with the Monte Carlo simulation (MCS) method. Moreover, the influence of the correlation coefficient between the excitations of external and parametric Gaussian white noise is investigated.

Investigation on the EPC method in analyzing the nonlinear oscillators under both harmonic and Gaussian white noise excitations

The objective of the current paper is to study and extend the exponential-polynomial-closure (EPC) method in analyzing the non-stationary response probabilistic solutions of nonlinear stochastic oscillators excited by combined deterministic harmonic and Gaussian white noise excitations. The probabilistic solution of the non-stationary responses of nonlinear stochastic oscillator is expressed as an exponential function of polynomial with time-variant coefficients and then the Fokker–Planck–Kolmogorov equation is solved approximately. Two validation examples are presented. The results obtained by Monte Carlo simulation (MCS) and EPC method are presented to show their good agreement, which confirms the effectiveness of the EPC method for the probabilistic solutions to nonlinear stochastic oscillators. The advantage of the EPC method for analyzing the oscillators with strong nonlinearities is investigated by comparison with Equivalent Linearization method. Furthermore, compared with MCS, the computational efficiency by using the EPC method is increased by about 56 times in the first example and 44 times in the second example.

Transient probabilistic solutions of stochastic oscillator with even nonlinearities by exponential polynomial closure method

The current work is devoted to analyze the transient probability density function solutions of stochastic oscillator with even nonlinearities under external excitation of Gaussian white noise by applying the extended exponential polynomial closure method. Specifically, the Fokker–Planck–Kolmogorov equation which governs the probability density function solutions of the nonlinear system is presented first. The residual error of the Fokker–Planck–Kolmogorov equation is then derived by assuming the probability density function solution as the type of exponential polynomial with time-dependent variables. Finally, by making the projection of the residual error vanish, a set of nonlinear ordinary differential equations is established and solved numerically. Numerical analysis show that the extended exponential polynomial closure method with polynomial order being six is both effective and efficient for solving the transient analysis of the stochastic oscillator with even nonlinearities by comparing the numerical results obtained by the proposed method with those obtained by Monte Carlo simulation method. Numerical results also show that the transient probability density function solutions of the system responses are not symmetric about their nonzero means due to the existence of even nonlinearities.

Probabilistic solutions of a variable-mass system under random excitations

The stationary probability density function (PDF) solution of a variable-mass system is calculated under Gaussian white noises and Poisson white noises, respectively. For small mass disturbance, the corresponding Fokker–Planck–Kolmogorov equation and Kolmogorov–Feller equation of the system are derived. The solution procedure based on the exponential–polynomial closure (EPC) method is formulated to obtain and study the probabilistic solutions of the strongly nonlinear variable-mass system subjected to Gaussian white noises and Poisson white noises. Both odd and even nonlinear variable-mass systems are considered. Compared with Monte Carlo simulation results, good agreement is achieved with the EPC method in the case of sixth-order polynomial. For large mass disturbance, the PDFs and logarithmic PDFs of displacement and velocity are numerically calculated via the fourth-order Runge–Kutta algorithm.

Stochastic responses of nonlinear systems to nonstationary non-Gaussian excitations

Some random excitations actually demonstrate a strong deviation from Gaussian. They are associated with strong non-Gaussian properties. In this paper, Poisson and filtered Poisson processes are utilized to describe such non-Gaussian excitations. Besides, nonstationarity has to be considered since excitation intensity is actually a varying process. It is modeled by multiplying stationary models with modulating functions. Based on the above, nonstationary responses of single-degree-of-freedom (SDOF) nonlinear systems under non-Gaussian excitations are investigated. Exponential polynomial closure (EPC) approximate method, which is previously proposed for analyzing stationary responses with Gaussian excitations, is further improved by taking time variable or additional state variables into account to determine nonstationary responses with non-Gaussian excitations. Examples of nonlinear systems under nonstationary Poisson and filtered Poisson excitations are analyzed to testify the improved solution procedure. Comparisons between EPC approximates and simulated results evidence that the EPC method is efficient. In addition, non-Gaussian and nonstationary properties of responses are analyzed. Nonationary effects with different modulating function are also discussed.

Transient influence of correlation between excitations on system responses

In this paper, transient influence of correlation between external and parametric excitations on system response is investigated. Time variable has been taken into account. This does not seem to have been studied previously. By describing correlation between excitations quantitively with correlated coefficient, general Fokker-Planck-Kolmogorov (FPK) equation is reformulated with two parts. The first part relates to the one of independent excitations, while the other part is caused by the correlated excitations. Then, exponential polynomial closure (EPC) method, by taking time variable into account, is further developed for transient responses of nonlinear oscillators under correlated excitations. With the improved EPC method, typical system of Duffing oscillator under correlated external and parametric Gaussian white noise excitations is investigated. Based on the results, it is found the influence of correlation between excitations on system response depends directly on the magnitude and the sign of the correlation coefficient. Besides, the influence seems stationary, and not affected by the time. In addition, the results obtained from the EQL method are not consistent with the actual ones with unsymmetrical PDFs and nonzero means. It is indicated that the EQL method is not applicable to such case.

Probabilistic Solutions of the In-Plane Nonlinear Random Vibrations of Shallow Cables Under Filtered Gaussian White Noise

The probabilistic solutions of the responses of shallow cable are studied when the cable is excited by filtered Gaussian white noise. The nonlinear multi-degree-of-freedom system is formulated which governs the random vibration of cable. The state-space-split (SSS) method and exponential polynomial closure (EPC) method are adopted to analyze the probabilistic solutions of cable systems in order to study the effectiveness and computational efficiency of SSS-EPC procedure in analyzing the probabilistic solutions of the cable systems under the excitation of filtered Gaussian white noise. Numerical results obtained by SSS-EPC method, Monte Carlo simulation, and equivalent linearization method are compared to examine the computational efficiency and numerical accuracy of SSS-EPC method in this case. Thereafter, the behaviors of probabilistic solutions of the cable systems are studied with different values of peak frequency and seismic intensity of excitation when the cable is excited by Kanai–Tajimi seismic force. Some observations and discussions are given by introducing a probabilistic quantity to show the influence of excitations on the probabilistic solutions.

Probabilistic Solutions of a Nonlinear Plate Excited by Gaussian White Noise Fully Correlated in Space

This paper addresses the nonlinear random vibration of a rectangular von Kármán plate excited by uniformly distributed Gaussian white noise which is fully correlated in space. The state-space-split method and exponential polynomial closure method are jointly utilized to analyze the probabilistic solutions of the plate. The computational efficiency and numerical accuracy of the methodology for analyzing the nonlinear random vibration of the plate are verified by comparing the computational effort and numerical results with those obtained by Monte Carlo simulation and equivalent linearization, respectively. Meanwhile, the convergence of the probabilistic solution in the sense of Galerkin’s approximation is examined by analyzing the plate modeled as single-degree-of-freedom and multi-degree-of-freedom systems. Some phenomena are discussed after numerically studying the behaviors of probabilistic solutions of the deflection at different locations of the plate.

Probabilistic solution of non-linear vibration energy harvesters driven by Poisson impulses

This paper develops a new solution procedure for obtaining the joint probability density function (PDF) of non-linear energy harvesters under Poisson impulses using the state-space-split method and the exponential-polynomial closure method. In the beginning, a generic electromechanical energy harvester is introduced and its governing equations are transformed into non-dimensional ones. According to the non-dimensional governing equations, a Duffing-type oscillator with piezoelectric conversion mechanism is further considered in this study. The proposed solution procedure includes three steps. First the joint PDF of this system is governed by the generalized Fokker-Plank-Kolmogorov (FPK) equation. The state-space-split method is used to reduce this generalized FPK equation to a low-dimensional one only about displacement and velocity. After that, the exponential-polynomial closure method is further adopted to solve the reduced FPK equation. Finally, the joint PDF of displacement, velocity and voltage can be approximated by the product of the obtained PDF and the conditional Gaussian PDF of voltage. Four cases are investigated considering the effects of non-linearity coefficient, impulse arrival rate and a bistable Duffing oscillator. The PDFs of these state variables and the harvested power are compared with the simulation results, respectively. The good agreement between the obtained PDFs and the simulated results is achieved. Compared with the results of the equivalent linearization method, the strong non-linearity in displacement and lower impulse rate lead the tail PDF of the harvested power to exhibiting hardening and softening behaviors, respectively. For the bistable Duffing oscillator, the PDF of the harvested power differs significantly from the result of the equivalent linearization method in the tail region.

Stationary Solution of Duffing Oscillator Driven by Additive and Multiplicative Colored Noise Excitations

A bistable Duffing oscillator subjected to additive and multiplicative Ornstein–Uhlenbeck (OU) colored excitations is examined. It is modeled through a set of four first-order stochastic differential equations by representing the OU excitations as filtered Gaussian white noise excitations. Enlargement in the state-space vector leads to four-dimensional (4D) Fokker–Planck–Kolmogorov (FPK) equation. The exponential-polynomial closure (EPC) method, proposed previously for the case of white noise excitations, is further improved and developed to solve colored noise case, resulting in much more polynomial terms included in the approximate solution. Numerical results show that approximate solutions from the EPC method compare well with the predictions obtained via Monte Carlo simulation (MCS) method. Investigation is also carried out to examine the influence of intensity level on the probability distribution solutions of system responses.

Influence of Nonzero Mean Impulse Amplitudes on the Response Statistics of Dynamical Systems

This paper investigates the influences of nonzero mean Poisson impulse amplitudes on the response statistics of dynamical systems. New correction terms of the extended Itô calculus, as a generalization of the Wong–Zakai correction terms in the case of normal excitations, are adopted to consider the non-normal property in the case of Poisson process. Due to these new correction terms, the corresponding drift and diffusion coefficients of Fokker–Planck–Kolmogorov (FPK) equation have to be modified and they become more complicated. Herein, the exponential–polynomial closure (EPC) method is employed to solve such a complex FPK equation. Since there are no exact solutions, the efficiency of the EPC method is numerically evaluated by the simulation results. Three examples of different excitation patterns are considered. Numerical results indicate that the influence of nonzero mean impulse amplitudes on system responses depends on the excitation patterns. It is negligible in the case of parametric excitation on displacement. On the contrary, the influence becomes significant in the cases of external excitation and parametric excitation on velocity.

Probabilistic solutions of nonlinear oscillators to subject random colored noise excitations

Random colored noise excitations, more close to real environmental excitations, are considered. In this paper, probabilistic solutions of nonlinear SDOF systems excited by colored noise are analyzed. With the aid of filtering method, the initial equation of motion is transferred to four coupled first-order SDEs. As a result, a Markov process of enlarged system responses comes into being, and traditional FPK equation method can be employed. However, this way makes the corresponding FPK equation become four-dimensional. In order to solve this problem, the developed and improved exponential polynomial closure (EPC) method is employed. In the solution procedure, an approximate exponential function is extended to include four state variables. Then additional unknown coefficients associated with extended state variables are produced, and the solution procedure becomes more complicated. On the other hand, since there is no exact solution, Monte Carlo simulation (MCS) is performed to verify the efficiency of the developed EPC method. Four examples associated with Gaussian and non-Gaussian colored noise excitations are considered. Numerical results show that the developed EPC method provides good agreement with the MCS method. Moreover, the effect of bandwidth for colored noise on system responses is taken into account.

Probabilistic solutions of nonlinear oscillators excited by combined colored and white noise excitations

In this paper, single-degree-of-freedom (SDOF) systems combined to Gaussian white noise and Gaussian/non-Gaussian colored noise excitations are investigated. By expressing colored noise excitation as a second-order filtered white noise process and introducing colored noise as an additional state variable, the equation of motion for SDOF system under colored noise is then transferred artificially to multi-degree-of-freedom (MDOF) system under white noise excitations with four-coupled first-order differential equations. As a consequence, corresponding Fokker-Planck-Kolmogorov (FPK) equation governing the joint probabilistic density function (PDF) of state variables increases to 4-dimension (4-D). Solution procedure and computer programme become much more sophisticated. The exponential-polynomial closure (EPC) method, widely applied for cases of SDOF systems under white noise excitations, is developed and improved for cases of systems under colored noise excitations and for solving the complex 4-D FPK equation. On the other hand, Monte Carlo simulation (MCS) method is performed to test the approximate EPC solutions. Two examples associated with Gaussian and non-Gaussian colored noise excitations are considered. Corresponding band-limited power spectral densities (PSDs) for colored noise excitations are separately given. Numerical studies show that the developed EPC method provides relatively accurate estimates of the stationary probabilistic solutions, especially the ones in the tail regions of the PDFs. Moreover, statistical parameter of mean-up crossing rate (MCR) is taken into account, which is important for reliability and failure analysis. Hopefully, our present work could provide insights into the investigation of structures under random loadings.

Stationary probabilistic solutions of the cables with small sag and modeled as MDOF systems excited by Gaussian white noise

Nonlinear random vibration of the cables with small sag-to-span ratio and excited by in-plane transverse uniformly distributed Gaussian white noise is studied by a nonlinear multi-degree-of-freedom system which is formulated with Galerkin’s method. The stationary probabilistic solutions of the nonlinear system are analyzed with the state-space-split method in conjunction with the exponential polynomial closure method. Effectiveness of this approach about the cable random vibration is examined through comparison with Monte Carlo simulation and equivalent linearization method. The probabilistic solutions of the cable random vibrations are also studied by modeling the cable as single-degree-of-freedom system and multi-degree-of-freedom system.

Stochastic response of a vibro-impact Duffing system under external Poisson impulses

This paper studies the stationary probability density function (PDF) of a vibro-impact Duffing system under external Poisson impulses. A one-sided constraint is located at the equilibrium position of the system, and the system collides with the constraint by instantaneous repetitive impacts. A recently proposed solution procedure is extended to the case of Poisson impulses including three steps. First, the Zhuravlev non-smooth coordinate transformation is utilized to make the original Duffing system and impact condition be integrated into one equation. An additional impulsive damping term is introduced in the new equation. Second, the PDF of the new system is obtained with the exponential–polynomial closure method by solving the generalized Fokker–Planck–Kolmogorov equation. Last, the PDF of the original system is established following the methodology on seeking the PDF of a function of random variables. In numerical analysis, different levels of nonlinearity degree and excitation intensity are considered in four illustrative examples to show the effectiveness of the proposed solution procedure. The numerical results show that when the polynomial order is taken as six in the proposed solution procedure, it can present a satisfactory PDF solution compared with the simulated result. The tail region of the PDF solution is also approximated well for both displacement and velocity.

Probabilistic solution of a multi-degree-of-freedom Duffing system under nonzero mean Poisson impulses

This paper investigates the probability density function (PDF) of multi-degree-of-freedom nonlinear systems excited by nonzero mean Poisson impulses. The PDF solution is governed by the Kolmogorov–Feller equation which is also called the generalized Fokker–Planck–Kolmogorov (FPK) equation. First the high-dimensional generalized FPK equation is reduced to a low-dimensional equation by a state-space-split method. The reduced FPK equation is further solved by an exponential-polynomial closure method. In numerical analysis, a ten-degree-of-freedom Duffing system is further investigated under nonzero mean Poisson impulses. The PDF distribution of impulse amplitudes is adopted with either a nonzero mean Gaussian distribution or a nonzero mean Rayleigh distribution. Comparison with the simulated results shows that the proposed solution procedure is effective to obtain a satisfactory PDF solution, especially in the tail region. The nonzero mean PDF of displacement is formulated due to nonzero mean Poisson impulses. The obtained PDF of displacement is not symmetrically distributed about its mean.

Stochastic response of a parametrically excited vibro-impact system with a nonzero offset constraint

In this paper, a recently proposed solution procedure is extended in a straightforward manner to obtain the probability density function (PDF) of the stochastic response of a vibro-impact Duffing system with a nonzero offset constraint. The Duffing system is simultaneously excited by external and parametric Gaussian white noises and it undergoes repetitive instantaneous impacts against the constraint. First, in terms of the Zhuravlev non-smooth coordinate transformation, the original equation of motion and the impact condition are combined into a new equation without any constraints by adding an additional impulsive damping term. Second, the PDF of the new system is governed by the associated Fokker–Planck equation which is solved by the exponential-polynomial closure method. Last, the PDF of the original vibro-impact Duffing system is formulated in terms of the methodology on seeking the PDF distribution of a function of random variables. In order to evaluate the effectiveness of the proposed solution procedure, four illustrative examples are studied considering different values of the parameters, namely nonzero offset, nonlinear stiffness and parametric excitation intensity. A comparison with the direct energy balance method is also presented. Comparison with the simulation result shows that the proposed solution procedure can provide a satisfactory PDF solution, even in the tail region for the examined examples. In addition, the nonzero offset constraint significantly affects the PDF distribution of the response.

The Probabilistic Solution of the Plate with Simple-Supported and Stretched Boundary and Uniform Load Being Gaussian White Noise

The nonlinear random vibration of the simply-supported rectangular isotropic von Kármán plate with in-plane stretched edges and excited by uniformly distributed Gaussian white noise is analyzed. The equation of motion of the plate with large deflection is a nonlinear partial differential equation in space and time. The multi-degree-of-freedom nonlinear stochastic dynamical system can be formulated by applying the Galerkin's method to the nonlinear partial differential equation. The probabilistic solution of the multi-degree-of-freedom nonlinear stochastic dynamical system is governed by the Fokker-Planck-Kolmogorov equation. The state-space-split method is used to make the Fokker-Planck-Kolmogorov equation in high dimensional space reduced to the Fokker-Planck-Kolmogorov equations in 2-dimensional space. Then the exponential polynomial closure method is used to solve the reduced Fokker-Planck-Kolmogorov equations in 2-dimensional space for the probability density function of the responses of the plate with moderately large deflection.