Multiplicative Random Excitations Research Articles

Stochastic Response of an Impact-Hybrid Energy Harvesting System with Viscoelastic Damping Under Additive and Multiplicative Random Excitations

Stochastic Response of an Impact-Hybrid Energy Harvesting System with Viscoelastic Damping Under Additive and Multiplicative Random Excitations

A novel methodology for dynamic response maximisation in multi-axis accelerated random fatigue testing

This paper investigates the effects caused by simultaneous multiple random excitations on the fatigue-life of the specimen under test. In multi-axis accelerated fatigue testing, the test specifications are usually provided in terms of PSDs only. However, different combinations of the test specifications can significantly affect the fatigue behaviour of the specimen, resulting in altered failure modes and test durations. In this context, the original contribution of this paper is to provide a novel method for combining the PSD test specifications, which is able to recreate in the laboratory the most severe and damaging vibration environment possible. The aim of the present methodology is to get out the extreme dynamic response of the specimen by fully exploiting the total energy offered by the test specifications. This method avoids the risk of underestimating the fatigue damage to undergo the specimen during laboratory testing. The paper offers the mathematical implementation of the method and its experimental validation achieved throughout an intense test campaign. Fatigue tests have been performed on specially designed specimens, by exploiting a three-axial electro-dynamic shaker.

Stochastic response and stability of system with friction and a rigid barrier

• A stochastic system with friction, impact and viscoelastic force is analyzed. • Responses and P-bifurcation are discussed by stochastic averaging method. • Asymptotic stability of probability one is investigated. • Effectiveness of analytical results is validated by numerical simulations. In the present paper, considering the significant nonsmooth factors, friction and impact, the responses and asymptotic stability with probability one of a system are investigated for the cases of additive and multiplicative Gaussian white noise random excitations. First, the original system which contains impact and viscoelastic terms can be approximated as an equivalent system by using nonsmooth transformation and converting the viscoelastic force to the sum of stiffness and damping terms. Then, the stationary probability density functions are analytically obtained by utilizing the stochastic averaging method. And the equation of the Top Lyapunov Exponent is also obtained with the help of combining the stochastic averaging method and Khasminskii method, which is used to discuss the effects of various parameters on the system’s stability. The validity of the analytical results derived from the proposed procedures is verified by comparing with directly numerical simulation. In particular, under certain conditions, there are examples about how to control the change of stability of the system by adjusting the variation of the restitution coefficient and the friction from the mathematical standpoint.

Research on the reliability of friction system under combined additive and multiplicative random excitations

In this paper, the reliability of a non-linearly damped friction oscillator under combined additive and multiplicative Gaussian white noise excitations is investigated. The stochastic averaging method, which is usually applied to the research of smooth system, has been extended to the study of the reliability of non-smooth friction system. The results indicate that the reliability of friction system can be improved by Coulomb friction and reduced by random excitations. In particular, the effect of the external random excitation on the reliability is larger than the effect of the parametric random excitation. The validity of the analytical results is verified by the numerical results.

Stationary response analysis of vibro-impact system with a unilateral nonzero offset barrier and viscoelastic damping under random excitations

The stationary response analysis of stochastic vibro-impact (VI) dynamical system with a unilateral nonzero offset barrier and viscoelastic damping under additive and multiplicative random excitations is explored. First, the conversion of the original stochastic VI system coupled with viscoelastic damping into an equivalent stochastic VI system without viscoelastic damping terms is performed by adding the equivalent stiffness and damping. Then, the equivalent stochastic VI system is studied according to the level of system energy instead of non-smooth transformation of the previous literature; thus, description on effects of impacts draws support from the loss of the system energy. The energy loss at each impact is found to be related to the restitution factor and the energy level before impact. Furthermore, the probability density functions of the equivalent VI system based on assumption of the weak damping and random perturbation is produced by the application of the stochastic averaging of energy envelope. The effectiveness of the suggested method is verified by comparing the analytical results with those from Monte Carlo simulations.

Stochastic responses of a viscoelastic-impact system under additive and multiplicative random excitations

This paper deals with the stochastic responses of a viscoelastic-impact system under additive and multiplicative random excitations. The viscoelastic force is replaced by a combination of stiffness and damping terms. The non-smooth transformation of the state variables is utilized to transform the original system to a new system without the impact term. The stochastic averaging method is applied to yield the stationary probability density functions. The validity of the analytical method is verified by comparing the analytical results with the numerical results. It is invaluable to note that the restitution coefficient, the viscoelastic parameters and the damping coefficients can induce the occurrence of stochastic P-bifurcation. Furthermore, the joint stationary probability density functions with three peaks are explored.

Global Response Sensitivity Analysis of Randomly Excited Dynamic Structures

The response of structural dynamical systems excited by multiple random excitations is considered. Two new procedures for evaluating global response sensitivity measures with respect to the excitation components are proposed. The first procedure is valid for stationary response of linear systems under stationary random excitations and is based on the notion of Hellinger's metric of distance between two power spectral density functions. The second procedure is more generally valid and is based on the l2 norm based distance measure between two probability density functions. Specific cases which admit exact solutions are presented, and solution procedures based on Monte Carlo simulations for more general class of problems are outlined. Illustrations include studies on a parametrically excited linear system and a nonlinear random vibration problem involving moving oscillator-beam system that considers excitations attributable to random support motions and guide-way unevenness. (C) 2015 American Society of Civil Engineers.

Finite element method analysis of Fokker–Plank equation in stationary and evolutionary versions

The problems that often arise in stochastic dynamics can be investigated using the Fokker–Planck (FP) equation. The response of a such systems being subjected to additive and/or multiplicative random noise is represented by probability density function (PDF) that gives the full information about a response random character. Various analytic and semi-analytic solution methods have been developed for various systems to obtain results requested. However numerical approaches offer a powerful alternative. In particular the Finite Element Method (FEM) seems to be very effective. A couple of single dynamic linear/non-linear (Duffing and Van Der Pol type) systems under additive and multiplicative random excitations are discussed using FEM as a solution tool of the FP equation. The resulting PDFs are analyzed and if the analytic results exist mutually compared.

Experimental investigation of a vibration isolation system using negative stiffness structure

The purpose of this study is to investigate and propose a vibration isolation system with a negative stiffness structure (NSS) for driver seats in low frequency vibration conditions. In this paper, the nonlinear stiffness characteristic of the system is presented. The relationship between the configurative parameters and the stiffness of the elastic element, for which the dimensionless nonlinear stiffness of the proposed system ranges from zero to one and the range of the displacement of the isolation equipment (mass) is the maximum without exceeding a desired value of stiffness, will be obtained. An experimental apparatus is subsequently configured to examine the characteristic of the vibration transmissibility of the system according to various values of the configurative parameters. Next, the time responses to the multiple frequency and random base excitations are also investigated. In addition, the dynamic responses of the system with and without the negative stiffness structure are considered. The results show that the proposed system has a wide frequency range of isolation, especially since the resonance phenomenon almost does not occur.

Stochastic dynamical analysis of a kind of vibro-impact system under multiple harmonic and random excitations

There have been many contributions concerned with non-smooth dynamics. The purpose of this study is focused on the global stochastic dynamics of a kind of vibro-impact oscillator under the multiple harmonic and bounded noisy excitations. The well-known cell-to-cell mapping method is firstly developed to investigate the incursive fractal boundaries between the attracting domains of different random attractors, and a specific Poincaré map is then set up to explore the noise-contaminated dynamical transitions in the system. Lastly, the leading Lyapunov exponents and the surrogate tests are used to identify the noise-contaminated dynamics. It is shown that several random attractors will coexist in the phase space of the randomly driven system by adjusting the parameters’ values, and fractal boundaries may also arise between the attracting domains of different random attractors. Under the joint action of the harmonic excitation and the weak bounded noise excitation, the noisy period-doubling process, similar to a deterministic one, can appear in the Poincaré’s global cross-section by increasing the strength of the bounded noisy excitation. Moreover, the noisy periodic, the noisy chaotic, and the random-dominant dynamics are also distinguished from the noise-contaminated signals.

Stochastic responses of Duffing-Van der Pol vibro-impact system under additive and multiplicative random excitations

The paper is devoted to an averaging approach to study the responses of Duffing-Van der Pol vibro-impact system excited by additive and multiplicative Gaussian noises. The response probability density functions (PDFs) are formulated analytically by the stochastic averaging method. Meanwhile, the results are validated numerically. In addition, stochastic bifurcations are also explored.

Response of nonlinear systems in probability domain using stochastic averaging

Determination of probability density function (PDF) of the response for strongly nonlinear single-degree-of-freedom system subjected to both multiplicative and additive random excitations using stochastic averaging technique is faced with few difficulties. Firstly, the size of excitations should be small such that the response of the system converges weakly to a Markov process. Secondly, the excitations should preferably be broad banded so that the analytical results are reasonably accurate and finally, the nonlinear functions should be integrable if closed-formed expressions for the result are desired. The above issues are examined in the paper with a view to show (a) limiting values of the size parameter of excitations for which stochastic averaging technique can be applied to obtain reasonable estimates of PDF and mean square value of response, (b) the effect of the nature of the frequency contents of excitation on the response and (c) the use of the stochastic averaging method employing generalized harmonic functions for nonintegrable functions representing the dynamic system. It is shown that the stochastic averaging procedure provides results which compare very well with those obtained from simulation analysis not only for wide band excitations, but also for narrow band excitations for a wide range of the size parameter of excitation. Further, the procedure can be easily implemented for highly irregular functions by employing a numerical scheme with FFT.

Stochastic response of nonlinear system in probability domain

A stochastic averaging procedure for obtaining the probability density function (PDF) of the response for a strongly nonlinear single-degree-of-freedom system, subjected to both multiplicative and additive random excitations is presented. The procedure uses random Van Der Pol transformation, Ito’s equation of limiting diffusion process and stochastic averaging technique as outlined by Zhu and others. However, the equations are rederived in generalized form and arranged in such a way that the procedure lends itself to a numerical computational scheme using FFT. The main objective of the modification is to consider highly irregular nonlinear functions which cannot be integrated in closed form and also to solve problems where analytical expressions for probability density function cannot be obtained. The procedure is applied to obtain the PDF of the response of Duffing oscillator subjected to additive and multiplicative random excitations represented by rational power spectral density functions (PSDFs). The results are verified by digital simulation. It is shown that the procedure provides results which compare very well with those obtained from simulation analysis not only for wide-band excitations but also for very narrow-band excitations, which are weak (when normalized with respect to mass of the system.)

Seismic spatial effects on long-span bridge response in nonstationary inhomogeneous random fields

The long-span bridge response to nonstationary multiple seismic random excitations is investigated using the PEM (pseudo excitation method). This method transforms the nonstationary random response analysis into ordinary direct dynamic analysis, and therefore, the analysis can be solved conveniently using the Newmark, Wilson-θ schemes or the precise integration method. Numerical results of the seismic response for an actual long-span bridge using the proposed PEM are given and compared with the results based on the conventional stationary analysis. From the numerical comparisons, it was found that both the seismic spatial effect and the nonstationary effect are quite important, and that both stationary and nonstationary seismic analysis should pay special attention to the wave passage effect.

Stochastic bifurcation in Duffing–Van der Pol oscillators

Global analysis of bifurcation in the Duffing–van der Pol oscillators under both additive and multiplicative random excitations is explored in detail by the generalized cell mapping method using digraph. System parameters are chosen in the range that two co-exist attractors and a saddles. As an alternative definition, stochastic bifurcation may be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value. It is found that under certain conditions stochastic bifurcation always occurs when a stochastic attractor collides with a stochastic saddle. Our study reveals that the generalized cell mapping method with digraph is also a powerful tool for global analysis of stochastic bifurcation. By this global analysis the mechanism of development, occurrence, and evolution of stochastic bifurcation can be explored clearly and vividly.

Non-linear systems of multiple degrees of freedom under both additive and multiplicative random excitations

A quasi-linearization procedure is developed for non-linear systems under both additive and multiplicative white-noise excitations to obtain response statistical properties, including moments, correlation functions, and spectral densities. In the proposed procedure, only the system properties are linearized, while all excitations are kept unchanged. In particular, the basic characteristics associated with the multiplicative excitations are preserved, which is the key factor for the accuracy of the approach. Numerical examples are given, and the accuracy of the procedure is substantiated by comparing the analytical results with those obtained from Monte Carlo simulations.

Transmissibility Matrix in Harmonic and Random Processes

The transmissibility concept may be generalized to multi-degree-of-freedom systems with multiple random excitations. This generalization involves the definition of a transmissibility matrix, relating two sets of responses when the structure is subjected to excitation at a given set of coordinates. Applying such a concept to an experimental example is the easiest way to validate this method.

Response probability estimation for randomly excited quasi-linear systems using a neural network approach

A quasi-linear system is referred to as a system linear in properties and subjected to multiplicative random excitations appearing also in the linear terms. It is known that exact solutions for the stationary moments can be obtained analytically for such a quasi-linear system if the excitations are Gaussian white noises. However, the exact response probability, which is non-Gaussian, is not obtainable analytically. In this paper, a neural network approach is proposed to evaluate the stationary response probability for quasi-linear systems under both additive and multiplicative excitations of Gaussian white noises based on the obtained exact statistical moments. Numerical examples show that the procedure yields accurate results if an appropriate form is assumed for the probability density function. The accuracy of the results is substantiated by comparing them with those obtained from Monte Carlo simulations.

Global analysis of stochastic bifurcation in a Duffing-van der Pol system

Stochastic bifurcation of the Duffing-van der Pol oscillators under both additive and multiplicative random excitations is studied in detail by the generalized cell mapping method using digraph.As an alternative definition,stochastic bifurcation may be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value.It is found that under certain conditions stochastic bifurcation mostly occurs when a stochastic attractor collides with a stochastic saddle.Our study reveals that the generalized cell mapping method with digraph is also a powerful tool for global analysis of stochastic bifurcation.By this global analysis ,the mechanism of development,occurrence,and evolution of a stochastic bifurcation can be explored clearly and vividly.

Exponential closure method for some randomly excited non-linear systems

The probability density function (PDF) of the responses of non-linear stochastic system excited by white noise is approximated with the exponential function of polynomial in state variables. Special measure is taken to satisfy FPK equation in the weak sense of integration with the assumed PDF. Gaussian closure method is a special case of the proposed method. Examples are given to show the application of the method to the systems with additive random excitations and those with both additive and multiplicative random excitations. The PDFs obtained with the proposed method and conventional Gaussian closure method are compared with obtainable exact ones. Numerical results showed that the PDFs obtained with the proposed method can be very close to the exact ones regardless of the degree of system non-linearity. In some cases, even exact solution can be obtained with the proposed method.