Response Of Nonlinear Oscillators Research Articles

Stochastic response of nonlinear oscillators under non-homogeneous Poisson white noise excitations

Poisson white noise is a typical non-Gaussian process. It is generally utilized to model stationary loadings by the assumed constant impulse arrival rate. Much research work has been done on that. However, the actual impulse arrival times of some discrete loadings are not uniform in the given time duration. The resulting arrival rate is a time-varying function instead of a constant one. Corresponding Poisson process is non-homogeneous. Less work in the literature has been found. In this paper, a Poisson white noise process with a time-varying impulse arrival rate is considered in the excitation model. Under such excitation the response of nonlinear system is a Markov process. The resulting generalized time-dependent Fokker–Planck equation becomes more complicated. The perturbation method, combined with the exponential-polynomial closure (EPC) method, is utilized to solve this equation. Examples of typical nonlinear systems under non-homogeneous Poisson white noise excitations are investigated. The effect of time-varying impulse arrival rate on the response is illustrated. Numerical results show that the response statistical characteristics depend strongly on the impulse arrival rate. Large errors would generate if the arrival rate varies fast over time and is still treated as an average constant.

Stationary random response evaluation of nonlinear oscillators with Preisach hysteresis discretized by RNN model

This manuscript is devoted to evaluating random stationary responses of nonlinear oscillators including Preisach hysteresis, which is modeled by a powerful tool, viz. Recurrent Neural Networks (RNN). First, a generalized harmonic transformation is introduced by adopting a geometric viewpoint of RNN-described hysteresis and intentionally separating the hysteretic force into conservative and dissipative components, to circumvent direct mathematical calculations of amounts of difference equations. The hysteretic force could be equivalently mimicked by a force that only depends on the present system states. Then, the system dimension is reduced by stochastic averaging, which is derived from the corresponding low-dimensional Fokker–Planck–Kolmogorov (FPK) equation associated with the probability density of slow-varying process. Additionally, solving the FPK equation yields the stationary probability density as well as various-order response statistics. Finally, two numerical examples, i.e., Duffing oscillator and van der Pol oscillator, are investigated to verify the efficacy of the proposed procedure.

A Wiener path integral technique for determining the stochastic response of nonlinear oscillators with fractional derivative elements: A constrained variational formulation with free boundaries

A technique based on the concept of Wiener path integral (WPI) is developed for determining approximately the joint response probability density function (PDF) of nonlinear oscillators endowed with fractional derivative elements. Specifically, first, the dependence of the state of the system on its history due to the fractional derivative terms is accounted for, alternatively, by augmenting the response vector and by considering additional auxiliary state variables and equations. In this regard, the original single-degree-of-freedom (SDOF) nonlinear system with fractional derivative terms is cast, equivalently, into a multi-degree-of-freedom (MDOF) nonlinear system involving integer-order derivatives only. From a mathematics perspective, the equations of motion referring to the latter can be interpreted as constrained. Second, to circumvent the challenge of increased dimensionality of the problem due to the augmentation of the response vector, a WPI formulation with mixed fixed/free boundary conditions is developed for determining directly any lower-dimensional joint PDF corresponding to a subset only of the response vector components. This can be construed as an approximation-free dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. Thus, the original SDOF oscillator joint PDF corresponding to the response displacement and velocity is determined efficiently, while circumventing the computationally challenging task of treating directly equations of motion involving fractional derivatives. Two illustrative numerical examples are considered for demonstrating the reliability of the developed technique. These pertain to a nonlinear Duffing and a nonlinear vibro-impact oscillators with fractional derivative elements subjected to combined stochastic and deterministic periodic loading. Note that alternative standard approximate techniques, such as statistical linearization, need to be significantly modified and extended to account for such cases of combined loading. Remarkably, it is shown herein that the WPI technique exhibits the additional advantage of treating such types of excitation in a straightforward manner without the need for any ad hoc modifications. Comparisons with pertinent Monte Carlo simulation data are included as well.

Response of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitations: A Path Integral approach based on Laplace’s method of integration

In this paper, an approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional element and subjected to evolutionary excitations. This is achieved by a novel formulation of the Path Integral (PI) approach. Specifically, a stochastic averaging/linearization treatment of the original fractional order governing equation of motion yields a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is the Chapman–Kolmogorov (CK) equation governing the evolution in time of the non-stationary response amplitude PDF. Next, the PI technique is employed, which is based on a discretized version of the CK equation solved in short time steps. This is done relying on the Laplace’s method of integration which yields an approximate analytical solution of the integral involved in the CK equation. In this manner, the repetitive integrations generally required in the classical numerical implementation of the procedure are avoided. Thus, the non-stationary response amplitude PDF is approximately determined in closed-form in a computationally efficient manner. Notably, the technique can also account for arbitrary excitation evolutionary power spectrum forms, even of the non-separable kind. Applications to oscillators with Van der Pol and Duffing type nonlinear restoring force models, and Preisach hysteretic models, are presented. Appropriate comparisons with Monte Carlo simulation data are shown, demonstrating the efficiency and accuracy of the proposed approach.

A Time Marching Integration for Semianalytical Solutions of Nonlinear Oscillators Based on Synchronization

Abstract An efficient method is proposed in this study for solving the semi-analytical solutions of periodic responses of nonlinear oscillators. The basic ideas come from the fact that any periodic response can be described by Fourier series. By transforming the Fourier series into a system of harmonic oscillators, we thus establish an efficient numerical scheme for tracking the periodic responses, as long as a synchronized motion can be achieved between the system of harmonic oscillators and the nonlinear oscillators considered. The presented method can be implemented by conducting time marching integration only, but it is capable of providing semi-analytical solutions straightforwardly. Different from some widely used methods such as harmonic balance method and its improved forms, this method can solve solutions involving high order harmonics without incorporating any tedious derivations as it is totally a numerical scheme. Several typical oscillators with smooth as well as nonsmooth nonlinearities are taken as numerical examples to test the validity and efficiency.

Suppression of chaos in nonlinear oscillators using a linear vibration absorber

This paper proposes a nonfeedback control to suppress the chaotic response of nonlinear oscillators. A linear vibration absorber is used as nonfeedback method, whose key idea is to find conditions under which the nonlinear oscillator response converges to an equilibrium point or stay oscillating around it. Theoretical results show that if the ratio between the natural frequencies of the primary system $$(\omega _1)$$ and the undamped absorber $$(\omega _2)$$ , that is, $$\omega _r=\frac{\omega _2}{\omega _1}$$ is tuned to be equal to the excitation frequency $$(\Omega )$$ , then the chaotic behavior of a nonlinear oscillator is driven to a stable hyperbolic equilibrium point. Numerical results are presented for the Duffing oscillator shown that if $$\omega _r$$ is tuned close to excitation frequency, then the chaotic response of the Duffing oscillator is driven to periodic orbits. In the case of damped absorber, $$\omega _r$$ to be tuned close to the excitation frequency does not guarantee that the response of the primary system is periodic, it is also necessary to take into account the amplitude of the excitation.

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.

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.

Stochastic response and bifurcation of periodically driven nonlinear oscillators by the generalized cell mapping method

The stochastic response of nonlinear oscillators under periodic and Gaussian white noise excitations is studied with the generalized cell mapping based on short-time Gaussian approximation (GCM/STGA) method. The solutions of the transition probability density functions over a small fraction of the period are constructed by the STGA scheme in order to construct the GCM over one complete period. Both the transient and steady-state probability density functions (PDFs) of a smooth and discontinuous (SD) oscillator are computed to illustrate the application of the method. The accuracy of the results is verified by direct Monte Carlo simulations. The transient responses show the evolution of the PDFs from being Gaussian to non-Gaussian. The effect of a chaotic saddle on the stochastic response is also studied. The stochastic P-bifurcation in terms of the steady-state PDFs occurs with the decrease of the smoothness parameter, which corresponds to the deterministic pitchfork bifurcation.

Nonstationary response of nonlinear oscillators with optimal bounded control and broad-band noises

Nonstationary response of nonlinear oscillators with optimal bounded control and broad-band noise excitations is investigated. First, the stochastic averaging method is applied to obtain an averaged Itô stochastic differential equation for the amplitude process. Then, the dynamical programming equation is employed to minimize the system response and establish an optimal control law with a control constraint. The nonstationary probability density of the amplitude process can be solved from the corresponding Fokker–Planck–Kolmogorov equation by using the Galerkin method if only external excitations exist. In the case of parametric excitations are present, Monte Carlo simulations can be carried out for the simplified averaged system of the amplitude process with much less computational efforts. Two examples are given to illustrate the feasibility of the proposed procedure and the effectiveness of the optimal control strategy. The accuracy and efficiency of the proposed procedure are substantiated by those obtained from Monte Carlo simulation of the original system.

Multiple-peak probability density function of non-linear oscillators under Gaussian white noise

An exponential-polynomial closure (EPC) method is introduced to solve the Fokker–Planck equation for the stationary multiple-peak probability density function (PDF) of the response of non-linear oscillators under Gaussian white noise. The EPC method uses Gaussian PDFs as a part of the weighting functions in the solution procedure. The Gaussian PDFs can be obtained by a standard equivalent linearization (EQL) method. This study considers the case that the EQL method gives multiple Gaussian PDFs. An improvement is proposed to make the weighting functions be independent of the choice of the Gaussian PDFs. To assess the effectiveness of the proposed method, a non-linear oscillator with a three-peak PDF of displacement is examined. Both low-level excitation and high-level excitation are considered. Comparison with the exact PDF solution shows that the improved EPC method can present a unique exact stationary PDF solution to the Fokker–Planck equation no matter which Gaussian PDF is adopted from the EQL method.

Nonzero mean response of nonlinear oscillators excited by additive Poisson impulses

This paper investigates the nonzero mean probability density function (PDF) of nonlinear oscillators under additive Poisson impulses. The PDF is governed by the generalized Fokker–Planck–Kolmogorov (FPK) equation which is also called the Kolmogorov–Feller (KF) equation. An exponential-polynomial closure (EPC) method is adopted to solve the equation. Five examples are considered in numerical analysis to show the effectiveness of the EPC method. The nonzero mean response of nonlinear oscillators is formulated due to either nonlinearity type or nonzero mean amplitude of Poisson impulses. The analysis shows that the PDFs obtained with the EPC method agree with the simulated results when the polynomial order is 4 or 6. This agreement is also observed in the tail regions of the obtained PDFs. The comparison further shows that the nonzero mean PDF of displacement is nonsymmetrically distributed. Comparatively, the PDF of velocity still has a symmetrical distribution pattern when the nonlinearity only exists in displacement.

Noise-enhanced Response of Nonlinear Oscillators

Although often considered to be undesirable, noise can produce beneficial effects in a system. Here, the authors discuss two representative nonlinear systems and the influence of noise on the responses of these systems. One of these systems is a set of coupled monostable Duffing oscillators, while the second of these systems is a Rayleigh-Duffing system that has been considered in honor of Dr. Y. Ueda. For the coupled oscillators, it is shown that an appropriately chosen noise addition can be used to localize energy as well as shift energy localization locations. In the case of the Rayleigh-Duffing system, the authors illustrate how the addition of noise to a deterministic input can push the system from a periodic attractor in the case without noise to a “broken-egg attractor” in the case with noise. These representative examples serve to illustrate a range of possible noise-influenced responses, and it is expected that similar as well as a wider range of responses can be expected in other nonlinear systems.

Responses of Nonlinear Oscillators Excited by Nonzero-Mean Parametric Poisson Impulses on Displacement

AbstractThe nonzero-mean probability density function (PDF) solutions of nonlinear stochastic oscillators under the excitation of Poisson impulse process are obtained with exponential-polynomial closure (EPC) method. The excitations are assumed to be external Poisson impulse process and parametric Poisson impulse process on displacement. The PDF of the oscillator response is governed by the generalized Fokker-Planck-Kolmogorov (FPK) equation, which is solved with the EPC method. The nonlinear oscillator with external and parametric excitation on displacement is analyzed when the mean of oscillator response is nonzero. Different levels of oscillator nonlinearity and nonzero means of the impulse amplitude are considered in the analysis. The analytical results show that the PDF solutions given by the EPC method are in good agreement with the simulated results when the complete sixth-degree polynomial of state variables is taken in the EPC procedure. The good agreement is also observed in the tail regions of ...

Probabilistic solution of nonlinear oscillators excited by combined Gaussian and Poisson white noises

The stationary probability density function (PDF) solution of the stochastic response of nonlinear oscillators is investigated in this paper. The external excitation is assumed to be a combination of Gaussian and Poisson white noises. The PDF solution is governed by the generalized Kolmogorov equation which is solved by the exponential-polynomial closure (EPC) method. In order to evaluate the effectiveness of the EPC method, different nonlinear oscillators are considered in numerical analysis. Nonlinearity exists either in displacement or in velocity for these nonlinear oscillators. The impulse arrival rate, mono-modal PDF and bi-modal PDF are also considered in this study. Compared to the PDF given by Monte Carlo simulation, the EPC method presents good agreement with the simulated result, which can also be observed in the tail region of the PDF solution.

Probability density function for stochastic response of non-linear oscillation system under random excitation

Probability density function (PDF) of stochastic responses is a critical topic in uncertainty analysis. In this paper, orthogonal decomposition technique was extended to discuss non-stationary response of non-linear oscillator under random excitation. The PDF of stochastic reponses is represented by a set of standardized multivariable orthogonal polynomials. According to the Galerkin scheme, the original problem, which has to solve the Fokker–Planck–Kolmogorov (FPK) equation, was converted to a first-order linear ordinary differential equation, in terms of unknown time-dependent coefficients. Then, stationary and non-stationary PDFs of uncertainty responses were obtained. In numerical examples, first-order and second-order non-linear systems exposed to the Gaussian white noise were considered. Finally, the accuracy of the proposed method was demonstrated through appropriate comparisons to Monte-Carlo simulation and analytical results.

PDF solution of nonlinear oscillators subject to multiplicative Poisson pulse excitation on displacement

A solution procedure for the stationary probability density function (PDF) of the responses of nonlinear oscillators subjected to Poisson white noises is formulated with exponential-polynomial closure (EPC) method. The effectiveness of the solution procedure is investigated with nonlinear oscillators subjected to both external and multiplicative Poisson white noises at different levels of system nonlinearity, excitation intensity, and impulse arrival rates. Numerical results show that the PDFs obtained with the EPC procedure are in good agreement with those from Monte Carlo simulation.

Peak response of non-linear oscillators under stationary white noise

The use of the advanced censored closure (ACC) technique, recently proposed by the authors for predicting the peak response of linear structures vibrating under random processes, is extended to the case of non-linear oscillators driven by stationary white noise. The proposed approach requires the knowledge of mean upcrossing rate and spectral bandwidth of the response process, which in this paper are estimated through the stochastic averaging method. Numerical applications to oscillators with non-linear stiffness and damping are included, and the results are compared with those given by Monte Carlo simulation and by other approximate formulations available in the literature.

Response of nonlinear oscillators with random frequency of excitation, revisited

Periodically driven oscillators of low-frequency random excitations are analyzed. Computer simulation, which was carried out for the Duffing equation and forced vibrations of a pendulum, indicated that in these cases noise has a stabilizing effect. Computation of Lyapunov exponents showed that by adding noise to chaotic motion the largest Lyapunov exponent as a rule turns to negative and, consequently, the chaotic motion is annihilated.

On response of nonlinear oscillators with random frequency of excitation

Some types of nonlinear oscillators, for which the frequency of excitation is stochastic, are investigated. The paper consists of two parts. In the first part equations of motion are linearized. With the aid of stochastic averaging differential equations for the mean and variance of the process are obtained and integrated numerically. This approach is applicable for weakly nonlinear oscillators. The case of strong nonlinearity is considered in the second part. Making use of computer simulation, a number of stochastic realizations of the process are computed. The stochastic process is characterized by the mean and standard deviation of these realizations. Calculations have been carried out for the Duffing, Ueda and van der Pol equations and for forced vibrations of a pendulum. These calculations show that if attractors exist then the deterministic vibrations (which may be chaotic) turn regular by adding noise and the motion terminates in a stable fixed point or on a limit cycle.