Gaussian White Noise Excitation Research Articles

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.

Time-varying system identification by enhanced Empirical Wavelet Transform based on Synchroextracting Transform

In this paper, an enhanced Empirical Wavelet Transform (EWT) approach based on Synchroextracting Transform (SET) is proposed for time-varying system identification. When a structure of time-varying physical properties, i.e. mass, stiffness or damping, is under external excitations, structural dynamic responses are usually non-stationary because the system has time-varying dynamic vibration characteristics. Under this circumstance, it would be difficult to determine the number of Intrinsic Mode Functions (IMFs) included in structural dynamic responses by using Fourier spectrum. Considering that the filtering boundaries of traditional EWT method are defined based on the segmental Fourier Spectrum of a processed signal, directly using it for non-stationary signal decomposition may not be effective and accurate. To apply the EWT method for time-varying system identification, in this study, time-frequency analysis based on SET is first performed to determine the frequency components of a non-stationary vibration signal instead of using Fourier spectrum. The filtering boundaries for EWT analysis are determined based on the time-frequency representation. Then, the IMFs are extracted from the non-stationary vibration signals by using EWT with the above defined filtering boundaries. When the IMFs are accurately obtained, the instantaneous frequencies of IMFs are identified by using Hilbert Transform (HT). In numerical simulations, a simulated signal with a high level noise is analyzed to verify the feasibility of using SET to define the filtering boundaries. Then the proposed approach is used to identify the instantaneous frequencies of a time-varying two-storey shear type building under earthquake and Gaussian white noise excitations, respectively. Experimental investigations on a time-varying bridge-vehicle system are conducted to verify the effectiveness of the proposed approach. The results in both numerical simulations and experimental validations demonstrate that the enhanced EWT approach can effectively and reliably identify the instantaneous frequencies of time-varying systems.

Stochastic P-bifurcation in a tri-stable Van der Pol system with fractional derivative under Gaussian white noise

In this paper, we study the tri-stable stochastic P-bifurcation problem of a generalized Van der Pol system with fractional derivative under Gaussian white noise excitation. Firstly, using the principle for minimal mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force, so that the original system can be transformed into an equivalent integer order system. Secondly, we obtain the stationary Probability Density Function (PDF) of the system’s amplitude by the stochastic averaging, and using the singularity theory, we find the critical parametric conditions for stochastic P-bifurcation of amplitude of the system, which can make the system switch among the three steady states. Finally, we analyze different types of the stationary PDF curves of the system amplitude qualitatively by choosing parameters corresponding to each region divided by the transition set curves, and the system response can be maintained at the small amplitude near the equilibrium by selecting the appropriate unfolding parameters. We verify the theoretical analysis and calculation of the transition set by showing the consistency of the numerical results obtained by Monte Carlo simulation with the analytical results. The method used in this paper directly guides the design of the fractional order controller to adjust the response of the system.

Stationary distribution simulation of rare events under colored Gaussian noise

Forward flux sampling (FFS) has provided a convenient and efficient way to simulate rare events in equilibrium as well as non-equilibrium stochastic systems. In the present paper, the FFS scheme is applied to systems driven by colored Gaussian noise through enlarging the dimension to deal with the non-Markovian property. Besides, the parameters of the FFS scheme have to be reconsidered. Interestingly, by analyzing the effect of colored Gaussian noise on stationary distributions, some results are found which are clearly different from the case of Gaussian white noise excitation. We mainly found that the probability of the occurrence of rare events is inversely proportional to the correlation time. Comparing to the case of Gaussian white noise with the same intensity, the presence of colored Gaussian noise exerts a hindrance to the occurrence of rare events. Meanwhile, the FFS results show a good agreement with those from Monte Carlo simulations, even for the colored Gaussian noise case. This provides a potential insight into rare events of systems under non-white Gaussian noise via the FFS scheme.

Stochastic response and bifurcations of a dry friction oscillator with periodic excitation based on a modified short-time Gaussian approximation scheme

In this paper, a modified short-time Gaussian approximation (STGA) scheme is incorporated into the generalized cell mapping (GCM) method to study the stochastic response and bifurcations of a nonlinear dry friction oscillator with both periodic and Gaussian white noise excitations. Firstly, the Fokker–Planck–Kolmogorov equation and the moment equations with Gaussian closure are derived. Because of the periodic force and the singularity of moment equations in the initial condition being a Dirac delta function, a set of novel STGA solutions of the conditional probability density functions (PDFs) over each small fraction of one period is then constructed in order to form the solution over the whole period. At last, the transient and steady-state response PDFs are computed by the GCM method. The accuracy of the results is demonstrated by the direct Monte Carlo simulations. The transient PDFs are presented when evolving from a Gaussian initial distribution to a non-Gaussian steady-state one. The effects of periodic excitation and dry friction damping on the steady-state stochastic response are particularly discussed. When the amplitude of periodic excitation is changed, both stochastic P-bifurcation and D-bifurcation are detected. It is also found that the increase in the dry friction damping coefficient makes the chaotic response disappear gradually, inducing stochastic D-bifurcation but no stochastic P-bifurcation.

Non-stationary response of MDOF dynamical systems under combined Gaussian and Poisson white noises by the generalized cell mapping method

A block matrix analysis procedure of the generalized cell mapping (GCM) method is proposed in this paper. The proposed method solves the storage problem in the response analysis of nonlinear stochastic dynamical system with the GCM method, and makes it possible to compute the non-stationary and stationary probability density functions (PDFs) of multi-degree-of-freedom (MDOF) nonlinear systems without using supercomputing. Two examples of two-degree-of-freedom systems under external or parametric excitations of combined Gaussian and Poisson white noises are presented to demonstrate the efficiency of the proposed method. Monte Carlo (MC) simulations are used to verify the accuracy of the solutions obtained with the proposed method.

Bifurcation Analysis of a Vibro-Impact Viscoelastic Oscillator with Fractional Derivative Element

To the best of authors’ knowledge, little work has been focused on the noisy vibro-impact systems with fractional derivative element. In this paper, stochastic bifurcation of a vibro-impact oscillator with fractional derivative element and a viscoelastic term under Gaussian white noise excitation is investigated. First, the viscoelastic force is approximately replaced by damping force and stiffness force. Thus the original oscillator is converted to an equivalent oscillator without a viscoelastic term. Second, the nonsmooth transformation is introduced to remove the discontinuity of the vibro-impact oscillator. Third, the stochastic averaging method is utilized to obtain analytical solutions of which the effectiveness can be verified by numerical solutions. We also find that the viscoelastic parameters, fractional coefficient and fractional derivative order can induce stochastic bifurcation.

Path integral solution of vibratory energy harvesting systems

A transition Fokker-Planck-Kolmogorov (FPK) equation describes the procedure of the probability density evolution whereby the dynamic response and reliability evaluation of mechanical systems could be carried out. The transition FPK equation of vibratory energy harvesting systems is a four-dimensional nonlinear partial differential equation. Therefore, it is often very challenging to obtain an exact probability density. This paper aims to investigate the stochastic response of vibration energy harvesters (VEHs) under the Gaussian white noise excitation. The numerical path integration method is applied to different types of nonlinear VEHs. The probability density function (PDF) from the transition FPK equation of energy harvesting systems is calculated using the path integration method. The path integration process is introduced by using the Gauss-Legendre integration scheme, and the short-time transition PDF is formulated with the short-time Gaussian approximation. The stationary probability densities of the transition FPK equation for vibratory energy harvesters are determined. The procedure is applied to three different types of nonlinear VEHs under Gaussian white excitations. The approximately numerical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulation (MCS).

Resonance response interaction without internal resonance in vibratory energy harvesting

Abstract Resonance response interaction in nonlinear multiple degrees of freedom vibration systems has always involved internal resonance. In this article, bubble shaped response curve is uncovered without pre-designated resonances relationship that is a necessary condition for internal resonance. The objective of this article is twofold: first to explore the connection between the resonance response interaction and bubble shaped response curve that may appear in the forced response of a nonlinear magnetoelectric coupled system; and, second, to exploit resonance response interaction to enhance the vibratory energy harvesting bandwidth. A nonlinear magnetoelectric oscillator coupled linearly to an additional oscillator is used to demonstrate the phenomena and the enhanced performance. The lateral springs could produce the geometrical nonlinear stiffness and its stiffness to adjust the two resonance responses such that the bubble shaped response curve appears in the power-frequency response plot. The method of harmonic balance is well established method for the analysis of such the power-frequency response. The results are also validated by some numerical work. The power-frequency response curves are generated for distinct harvesting mass, manifesting the resonance response interaction could increase the bandwidth and output power. At last, the resonance response interaction designed energy harvesting from Gaussian white noise excitation is numerically addressed to illustrate positive effects of geometrical nonlinearity.

Stochastic P-Bifurcation of a Bistable Viscoelastic Beam with Fractional Constitutive Relation under Gaussian White Noise

In this paper, we study the stochastic P-bifurcation problem for axially moving of a bistable viscoelastic beam with fractional derivatives of high order nonlinear terms under Gaussian white noise excitation. First, using the principle for minimum mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force, so that the original system can be simplified to an equivalent system. Second, we obtain the stationary Probability Density Function (PDF) of the system’s amplitude by stochastic averaging, and using singularity theory, we find the critical parametric condition for stochastic P-bifurcation of amplitude of the system. Finally, we analyze the types of the stationary PDF curves of the system qualitatively by choosing parameters corresponding to each region within the transition set curve. We verify the theoretical analysis and calculation of the transition set by showing the consistency of the numerical results obtained by Monte Carlo simulation with the analytical results. The method used in this paper directly guides the design of the fractional order viscoelastic material model to adjust the response of the system.

The closed-form stationary probability distribution of the stochastically excited vibro-impact oscillators

The response of vibro-impact oscillators under random excitations has been studied using various techniques in the last three decades. However, the results available were smooth approximations of the intrinsically non-smooth response of vibroimpact oscillators. This paper proposes a new procedure for the closed-form stationary probability density function (PDF) of the response of single-degree-of-freedom (SDOF) vibro-impact system under Gaussian white noise excitation. First, the Zhuravlev-Ivanov transformation is adopted to convert a vibro-impact oscillator with one-side barrier into an oscillator without barrier. The probabilistic description of the system is subsequently defined through the corresponding Fokker-Planck-Kolmogorov (FPK) equation. The closed-form stationary PDF of the response is obtained by solving the reduced FPK equation using the iterative method of weighted residue together with the concepts of the circulatory probability flow and the potential probability flow. Two examples are examined to illustrate the effectiveness of the proposed procedure. Good agreements are found between the analytical solutions and the simulated results of the PDFs and logarithmic PDFs of the examples. The reported studies also show that the proposed procedure can deal with case of small restitution factor. In particular, the closed-form solution for the case of small restitution factor is firstly obtained in literature, and can be used as the benchmark problem to examine the other method in random vibration. Furthermore, the approximate solutions obtained with the proposed procedure are piece-wise form, reflecting the true nature of the intrinsically discontinuous vibro-impact oscillators.

Dynamical reliability of internally resonant or non-resonant strongly nonlinear system under random excitations

Abstract The dynamical reliability of multi-degrees-of-freedom (MDOF) strongly nonlinear system under Gaussian white noise excitations is studied, including resonance and non-resonance. Firstly, the equations of motion of the original system with or without internal resonance are reduced to a set of Ito stochastic differential equations after stochastic averaging. Then, the backward Kolmogorov equation and the Pontryagin equation associated with the resonantly or non-resonantly averaged Ito stochastic differential equations, which determine the conditional reliability function and the mean first-passage time of the original random system, are constructed under appropriate boundary and (or) initial conditions, respectively. In particular, if the non-resonantly averaged system is completely decoupled, the conditional reliability function and the mean first-passage time of the original non-resonant system can be obtained by solving a set of simplified backward Kolmogorov equations. A system comprising two weakly coupled and strongly nonlinear mechanical oscillators is given as a concrete example to show the application of the proposed method. The 1:1 internal resonance or non-resonance is discussed. The corresponding high-dimensional backward Kolmogorov equation and Pontryagin equation are established and solved numerically. All theoretical results are validated by a Monte Carlo digital simulation.

Approximate Solution of the Fokker–Planck Equation for a Multidegree of Freedom Frictionally Damped Bladed Disk Under Random Excitation

Bladed disks are subjected to different types of excitations, which cannot, in any, case be described in a deterministic manner. Fuzzy factors, such as slightly varying airflow or density fluctuation, can lead to an uncertain excitation in terms of amplitude and frequency, which has to be described by random variables. The computation of frictionally damped blades under random excitation becomes highly complex due to the presence of nonlinearities. Only a few publications are dedicated to this particular problem. Most of these deal with systems of only one or two degrees-of-freedom (DOFs) and use computational expensive methods, like finite element method or finite differences method (FDM), to solve the determining differential equation. The stochastic stationary response of a mechanical system is characterized by the joint probability density function (JPDF), which is driven by the Fokker–Planck equation (FPE). Exact stationary solutions of the FPE only exist for a few classes of mechanical systems. This paper presents the application of a semi-analytical Galerkin-type method to a frictionally damped bladed disk under influence of Gaussian white noise (GWN) excitation in order to calculate its stationary response. One of the main difficulties is the selection of a proper initial approximate solution, which is applicable as a weighting function. Comparing the presented results with those from the FDM, Monte–Carlo simulation (MCS) as well as analytical solutions proves the applicability of the methodology.

Wiener Path Integral based response determination of nonlinear systems subject to non-white, non-Gaussian, and non-stationary stochastic excitation

The recently developed Wiener Path Integral (WPI) technique for determining the joint response probability density function of nonlinear systems subject to Gaussian white noise excitation is generalized herein to account for non-white, non-Gaussian, and non-stationary excitation processes. Specifically, modeling the excitation process as the output of a filter equation with Gaussian white noise as its input, it is possible to define an augmented response vector process to be considered in the WPI solution technique. A significant advantage relates to the fact that the technique is still applicable even for arbitrary excitation power spectrum forms. In such cases, it is shown that the use of a filter approximation facilitates the implementation of the WPI technique in a straightforward manner, without compromising its accuracy necessarily. Further, in addition to dynamical systems subject to stochastic excitation, the technique can also account for a special class of engineering mechanics problems where the media properties are modeled as stochastic fields. Several numerical examples pertaining to both single- and multi-degree-of-freedom systems are considered, including a marine structural system exposed to flow-induced non-white excitation, as well as a bending beam with a non-Gaussian and non-homogeneous Young's modulus. Comparisons with Monte Carlo simulation data demonstrate the accuracy of the technique.

Stochastic P-bifurcation of fractional derivative Van der Pol system excited by Gaussian white noise

This paper aimed to investigate the stochastic P-bifurcation of Van der Pol oscillator with a fractional derivative damping term driven by Gaussian white noise excitation. Firstly, based on the method of stochastic averaging method and Stratonovich–Khasminskii theorem, the corresponding Fokker–Plank–Kolmogorov (FPK) equation is deduced. To describe the P-bifurcation of system, the stationary probability densities of amplitude can be obtained by solving the FPK equation. Then, the effects of the fractional order, the fractional coefficient, and the intensity of Gaussian white noise on the fractional systems are discussed in detail. The results show that increasing order α will change obviously the number and the height of peaks under certain parameter conditions. Finally, comparing the analytical and numerical results, a very satisfactory agreement can be found.

Hybrid electromagnetic and piezoelectric vibration energy harvester with Gaussian white noise excitation

In this work, we investigated the dynamical behavior of the hybrid energy harvester under Gaussian white noise using probabilistic approach. We find that under the influence of this kind of noise, the dynamics of the nonlinear electromechanical system exhibited the stochastic bifurcation which is characterized by a qualitative change of the stationary probability distribution. A stochastic averaging method is applied in this system in the aim to build the Itô Stochastic differential equations. From these equations, the Fokker–Planck Equations (FPE) of the electromechanical system is constructed whose the solution at the stationary state is a probability density. By combining the harmonic excitation to the random force, the harvested energy is improved. We also provided the optimization rate of the hybrid system with respect to the piezoelectric system. Besides, the comparison between the power obtained by the hybrid model, piezoelectric and electromagnetic energy harvester shows the interest to build the hybrid model. The analytical results agree very well with numerical simulation.

Stochastic analysis of monostable vibration energy harvesters with fractional derivative damping under Gaussian white noise excitation

To the best of authors’ knowledge, the dynamical behaviors of vibration energy harvesters with fractional derivative damping have not been discussed by researchers with the help of the stochastic averaging method. As the fractional-order models are more accurate than the classical integer-order models, so it is necessary to investigate the dynamical behaviors of fractional vibration energy harvesters. This paper aims to investigate the stochastic response of monostable vibration energy harvesters with fractional derivative damping under Gaussian white noise excitation. First, we can get the equivalent stochastic system with the help of variable transformation. Then, the approximately analytical solutions of the equivalent stochastic system can be obtained by the stochastic averaging method. Third, the numerical results are considered as the benchmark to prove the effectiveness of the proposed method. The results indicate that the proposed method has a satisfactory level of accuracy. We also discuss the effect of system parameters on the mean square voltage.

Response of non-linear oscillator driven by fractional derivative term under Gaussian white noise

This paper aimed to investigate the stationary response of non-linear system with fractional derivative damping term under Gaussian white noise excitation. The corresponding Fokker–Plank–Kolmogorov (FPK) equation can be deduced by utilizing the stochastic averaging method and Stratonovich–Khasminskii theorem in the first place. And then we can solve the FPK equation to obtain the stationary probability densities (SPDs) of amplitude, which in fact can be used to describe the response of system. Furthermore, the analytical results coincide with the Monte Carlo results. Finally, one found that reducing fractional derivative order is able to enhance the response of system and increasing fractional coefficient can weaken the response of system. So the fractional derivative damping term has a great effect on the response of Duffing-Van der Pol oscillator. In addition, the response can also be influenced by other system parameters.

Reliability Estimation of Stochastic Dynamical Systems with Fractional Order PID Controller

In this paper, the reliability of stochastic dynamical systems under Gaussian white noise excitations with fractional order proportional–inegral–derivative (FOPID) controller is estimated. First, the FOPID controller is approximated by a set of combination of displacement and velocity based on the generalized van der Pol transformation. Then, the stochastic averaging method of energy envelope is applied to obtain a diffusive differential equation, from which the Backward Kolmogorov equation, governing the conditional reliability function, and the Generalized Pontryagin equation, governing the statistical moments of first-passage time, are derived from the averaged equation and solved numerically. Finally, in the two examples, the critical parameters in the FOPID controller are shown to be capable of improving the reliability of the stochastic dynamical system apparently, and all numerical results are verified to be efficient and correct by the Monte Carlo simulation.

Probabilistic response of nonsmooth nonlinear systems under Gaussian white noise excitations

This paper proposes an approach to obtain the steady state probability density function (PDF) of nonsmooth nonlinear dynamical systems driven by the Gaussian white noise. The steady state PDF solution is assumed to contain two probability flows obtained by the detailed balance method. The weighted mean squared residuals of the error of the FPK equation are minimized with respect to the unknown coefficients in the assumed solution. The integrability of the weighted mean squared residuals is found to provide a means to deal with the singularities in the steady state PDF that arise from the nonsmooth dynamics of the nonlinear stochastic system. The accuracy of the proposed approach is estimated by the Monte Carlo simulations via two examples.