Subharmonic Response of a Quasi-Isochronous Vibroimpact System to a Randomly Disordered Periodic Excitation

A quasi-isochronous vibroimpact system is considered, i.e. a linear system with a rigid one-sided barrier, which is slightly offset from the system's static equilibrium position. The system is excited by a sinusoidal force with disorder, or random phase modulation. The mean excitation frequency corresponds to a simple or subharmonic resonance, i.e. the value of its ratio to the natural frequency of the system without a barrier is close to some even integer. Influence of white-noise fluctuations of the instantaneous excitation frequency around its mean on the response is studied in this paper. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, or velocity jumps, thereby permitting the application of asymptotic averaging over the period for slowly varying inphase and quadrature responses. The averaged stochastic equations are solved exactly by the method of moments for the mean square response amplitude for the case of zero offset. A perturbation-based moment closure scheme is proposed for the case of nonzero offset and small random variations of amplitude. Therefore, the analytical results may be expected to be adequate for small values of excitation/system bandwidth ratio or for small intensities of the excitation frequency variations. However, at very large values of the parameter the results are approaching those predicted by a stochastic averaging method. Moreover, Monte-Carlo simulation has shown the moment closure results to be sufficiently accurate in general for any arbitrary bandwidth ratio. The basic conclusion, both of analytical and numerical simulation studies, is a sort of smearing of the amplitude frequency response curves owing to disorder, or random phase modulation: peak amplitudes may be strongly reduced, whereas somewhat increased response may be expected at large detunings, where response amplitudes to perfectly periodic excitation are relatively small.     

Dynamic Analysis of Prestressed Cables with Uncertain Pretension

This paper deals with finite element dynamic analysis of prestressed cables with uncertain pretension subjected to deterministic excitations. The theoretical model addressed for cable modeling is a two-dimensional finite-strain beam theory, which allows us to eliminate any restriction on the magnitude of displacements and rotations. The dynamic problem is formulated by referring the motion to the inertial frame, which leads to a simple uncoupled quadratic form for the kinetic energy. The effect of the externally applied stochastic pretension is approximately described by means of an uncertain axial component of stress resultant, which is assumed constant along the cable in its dead load configuration. The so-called improved perturbation approach is employed to solve this stochastic problem, obtaining two coupled systems of nonlinear deterministic ordinary differential equations, governing the mean value and deviation of response. An efficient and accurate iterative procedure is proposed to obtain the solution of these equations. In order to investigate the influence of random pretension on structural response, few numerical applications are presented and results are discussed.     

Identification of stiffness, dissipation and input parameters of randomly excited non-linear systems: Capability of restricted potential models (RPM)

A dynamic identification technique in the time domain for time invariant systems under random external forces is presented. This technique is based on the use of the class of restricted potential models (RPM), which are characterized by a non-linear stiffness and a special form of damping, that is a product of the input power spectral density (PSD) matrix and the velocity gradient of a non-linear function of the total mechanical energy. By applying stochastic differential calculus and by specific analytical manipulations, some algebraic equations, depending on the response statistics and on the mechanic parameters that characterize RPM, are obtained. These equations can be used for the dynamic identification of the above mechanic parameters once the response statistics of the system to be identified are evaluated. The proposed technique allows one to identify single-degree-of-freedom or multi-degrees-of-freedom systems in the case of unmeasurable input. Further, the probabilistic characteristics of the external forces can be completely estimated in terms of PSD matrix.     

Stochastic Response of Offshore Structures via Statistical Cubicization

This paper deals with the stochastic response of single-degree-of-freedom structures with polynomial restoring force excited by random Morisons forces with mean current. The problem is recast by expressing the excitation by means of a cubic polynomial of the wave elevation, which in turn is assumed as a stationary zero-mean Gaussian process, whose spectral density is given by the output of a cascade of two second order linear filters having a Gaussian white noise as primary excitation. Thus, Itôs stochastic differential calculus becomes applicable, and the solution is pursued with a moment equation approach by using a suitable closure scheme. The results of the applications compare well with digital simulation.     

Dynamic Response of Non-Linear Systems to Random Trains of Non-Overlapping Pulses

The stochastic excitation considered is a random train of rectangular, non-overlapping pulses, with random durations completed at latest at the next pulse arrival. For Erlang distributed interarrival times and for the actual distributions of pulse durations determined from the primitive Erlang distribution, the formulation of the problem in terms of a Markov chain allows to evaluate the mean value, the autocorrelation function and the characteristic function of the excitation process. However, the state vector of the dynamical system is a non-Markov process. The train of non-overlapping pulses with parameters , 1 and , 1 is then demonstrated to be a process governed by a stochastic equation driven by two independent Poisson processes, with parameters and , respectively. Hence, the state vector of the dynamical system augmented by this additional variable becomes a Markov process. The generalized Itôs differential rule is then used to derive the equations for the characteristic function and for moments of the response of a non-linear oscillator.     

Stochastic averaging of quasi-integrable and non-resonant Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A stochastic averaging method for predicting the response of quasi-integrable and non-resonant Hamiltonian systems to combined Gaussian and Poisson white noise excitations is proposed. First, the motion equations of a quasi-integrable and non-resonant Hamiltonian system subject to combined Gaussian and Poisson white noise excitations is transformed into stochastic integro-differential equations (SIDEs). Then $n$ -dimensional averaged SIDEs and generalized Fokker–Plank–Kolmogrov (GFPK) equations for the transition probability densities of $n$ action variables and $n$ - independent integrals of motion are derived by using stochastic jump–diffusion chain rule and stochastic averaging principle. The probability density of the stationary response is obtained by solving the averaged GFPK equation using the perturbation method. Finally, as an example, two coupled non-linear damping oscillators under both external and parametric excitations of combined Gaussian and Poisson white noises are worked out in detail to illustrate the application and validity of the proposed stochastic averaging method.     

Random Attractors for Degenerate Stochastic Partial Differential Equations

We prove the existence of random attractors for a large class of degenerate stochastic partial differential equations (SPDE) perturbed by joint additive Wiener noise and real, linear multiplicative Brownian noise, assuming only the standard assumptions of the variational approach to SPDE with compact embeddings in the associated Gelfand triple. This allows spatially much rougher noise than in known results. The approach is based on a construction of strictly stationary solutions to related strongly monotone SPDE. Applications include stochastic generalized porous media equations, stochastic generalized degenerate $p$ -Laplace equations and stochastic reaction diffusion equations. For perturbed, degenerate $p$ -Laplace equations we prove that the deterministic, $\infty $ -dimensional attractor collapses to a single random point if enough noise is added.     

Output feedback \varvec{\mathcal {H}}_{{\varvec{\infty }}} control of stochastic nonlinear time-delay systems with state and disturbance-dependent noises

This paper is concerned with the output feedback \(\mathcal {H}_\infty \) control problem for a class of stochastic nonlinear systems with time-varying state delays; the system dynamics is governed by the stochastic time-delay It \(\hat{o}\) -type differential equation with state and disturbance contaminated by white noises. The design of the output feedback \(\mathcal {H}_\infty \) control is based on the stochastic dissipative theory. By establishing the stochastic dissipation of the closed-loop system, the delay-dependent and delay-independent approaches are proposed for designing the output feedback \(\mathcal {H}_\infty \) controller. It is shown that the output feedback \(\mathcal {H}_\infty \) control problem for the stochastic nonlinear time-delay systems can be solved by two delay-involved Hamilton–Jacobi inequalities. A numerical example is provided to illustrate the effectiveness of the proposed methods.     

Model of a turbulent boundary layer with explicit identification of the coherent generation structure

Using the method of moments in the space of wavenumbers, a class of models of a developed turbulent flow of an incompressible fluid in a flatplate boundary layer is proposed. The models are based on an analysis of the Navier–Stokes equations that describe the behavior of dynamic coherent structures associated with vorticity generation and also the behavior of the stochastic component. A continuum analog of dynamic equations for a coherent structure is given in an explicit form. In the general case, the stochastic component should satisfy a system of equations of the kinetic type, which reduces to one equation under certain assumptions. It is also shown that the presence of coherent structures leads to generalization of the notion of statistical homogeneity.     

Comment on ‘fractal and superdiffusive transport and hydrodynamic dispersion in heterogeneous porous media’, by Muhammad Sahimi

Two fundamental questions regarding the application of percolation theory to transport in porous media are addressed. First, when critical path arguments (based on a sufficiently wide spread of microscopic transition rates) are invoked (in analogy to the case of transport in disordered semiconductors) to justify the application of percolation theory to the determination of relevant transport properties, then for long time scales (compared to the inverse of the critical percolation rate), the fractal structure of the critical path is relevant to transport, but not at short time scales. These results have been demonstrated concretely in the case of disordered semiconductors, and are in direct contradiction to the claims of the review. Second, the relevance of deterministic or stochastic methods to transport has been treated heretofore by most authors as a question of practicality. But, at least under some conditions, concrete criteria distinguish between the two types of transport. Percolative (deterministic) transport is temporally reproducible and spatially inhomogeneous while diffusive (stochastic) transport is temporally irreproducible, but homogeneous, and a cross-over from stochastic to percolative transport occurs when the spread of microscopic transition rates exceeds 4–5 orders of magnitude. It is likely that such conditions are frequently encountered in soil transport. Moreover, clear evidence for deterministic transport (although not necessarily percolative) exists in such phenomena as preferential flow. On the other hand, the physical limitation of transport to (fractally connected) pore spaces within soils (analogously to transport in metal-insulator composites) can make transport diffusive on a fractal structure, rather than percolative.Transport in Porous Media13 (1993), 3–40.     

Stochastic analysis of two-phase flow in porous media: I. Spectral/perturbation approach

Stochastic analysis of steady-state two-phase (water and oil) flow in heterogeneous porous media is performed using the perturbation theory and spectral representation techniques. The governing equations describing the flow are coupled and nonlinear. The key stochastic input variables are intrinsic permeability,k, and the soil and fluid dependent retention parameter, . Three different stochastic combinations of these two imput parameters were considered. The perturbation/spectral analysis was used to develop closed-form expressions that describe stochastic variability of key output processes, such as capillary and individual phase pressures and specific discharges. The analysis also included the estimation of the effective flow properties. The impact of the spatial variability ofk and on the variances of pressures, effective conductivities, and specific discharges was examined.     

Stability of elastic systems under a stochastic parametric excitation

An efficient method to investigate the stability of elastic systems subjected to the parametric force in the form of a random stationary colored noise is suggested. The method is based on the simulation of stochastic processes, numerical solution of differential equations, describing the perturbed motion of the system, and the calculation of top Liapunov exponents. The method results in the estimation of the almost sure stability and the stability with respect to statistical moments of different orders. Since the closed system of equations for moments of desired quantities y _j (t) cannot be obtained, the statistical data processing is applied. The estimation of moments at the instant t _n is obtained by statistical average of derived from the solution of equations for the large number of realizations. This approach allows us to evaluate the influence of different characteristics of random stationary loads on top Liapunov exponents and on the stability of system. The important point is that results found for filtered processes, are principally different from those corresponding to stochastic processes in the form of Gaussian white noises.     

Footing Settlement on a Consolidating Soil Layer with Stochastic Properties

Geotechnical engineering applications are characterized by various sources of uncertainties, most of them attributed to the stochastic nature of soil parameters and their properties. In particular, soil’s inherent random heterogeneity, inexact measurements and insufficient data necessitate numerical methods that incorporate the stochastic soil properties for a realistic representation of the soil behavior. In this paper, the process of consolidation of saturated soils is examined on the basis of the coupled u–p finite element formulation. A generalized Newmark implicit time integration scheme is implemented to treat the time integration of the coupled consolidation equations. A benchmark geotechnical engineering problem of a strip footing resting on a saturated soil layer is analyzed. The soil permeability coefficient k, as well as the elastic modulus E, are treated as lognormal random fields in two dimensions. The investigation of the effect of the spatial variability of the soil properties on the response of a footing–soil system is examined by means of the direct Monte Carlo simulation. The influence of the coefficient of variation and correlation length of the stochastic fields is quantified in terms of footing settlements, as well as excess soil water pore pressure. The effects of spatial variability of the permeability coefficient k and the elastic modulus E on the system response are demonstrated. It is shown that the footing differential settlement, along with generated excess pore pressures, is highly affected by the variation of the soil properties considered, as well as the correlation length of the underlying random fields.     

Stochastic averaging of quasi-generalized Hamiltonian systems

A stochastic averaging method for generalized Hamiltonian systems (GHS) subject to light dampings and weak stochastic excitations is proposed. First, the GHS are briefly reviewed and classified into five classes, i.e., non-integrable GHS, completely integrable and non-resonant GHS, completely integrable and resonant GHS, partially integrable and non-resonant GHS and partially integrable and resonant GHS. Then, the averaged and FPK equations and the drift and diffusion coefficients for the five classes of quasi-GHS are derived. Finally, the stochastic averaging for a nine-dimensional quasi-partially integrable GHS is given to illustrate the application of the proposed procedure, and the results are confirmed by using those from Monte Carlo simulation.     

Dynamics of stochastic non-Newtonian fluids driven by fractional Brownian motion with Hurst parameter H ∈ （1/4,1/2）

A two-dimensional (2D) stochastic incompressible non-Newtonian fluid driven by the genuine cylindrical fractional Brownian motion (FBM) is studied with the Hurst parameter $H \in \left( {\tfrac{1} {4},\tfrac{1} {2}} \right)$ under the Dirichlet boundary condition. The existence and regularity of the stochastic convolution corresponding to the stochastic non-Newtonian fluids are obtained by the estimate on the spectrum of the spatial differential operator and the identity of the infinite double series in the analytic number theory. The existence of the mild solution and the random attractor of a random dynamical system are then obtained for the stochastic non-Newtonian systems with $H \in \left( {\tfrac{1} {2},1} \right)$ without any additional restriction on the parameter H.     

Gaussian Closure of Transient Unsaturated Flow in Random Soils

The Gaussian closure approximation, previously used by the authors to solve steady state stochastic unsaturated flow problems in randomly heterogeneous soils, is extended here to transient flow. The method avoids linearizing the governing flow equations or the soil constitutive relations. It places no theoretical limit on the variance of constitutive parameters and applies to a broad class of soils with flow properties that scale according to a linearly separable model. Closure is obtained by treating the dimensionless pressure head as a multivariate Gaussian function. It yields a system of coupled nonlinear differential equations for the first and second moments of . We apply the Gaussian closure technique to the problem of transient infiltration into a randomly stratified soil. In each layer, hydraulic conductivity and water content vary exponentially with . Elsewhere we show that application of the technique to other constitutive relations is straightforward. Our solution for the mean and variance of in a one-dimensional layer with random conductivity compares well with Monte Carlo results over a wide range of parameters, provided that the spatial variability of the constitutive exponent is small. The solution provides considerable insight into the behavior of the transient unsaturated stochastic flow problem.     

Ergodic Properties of Weak Asymptotic Pseudotrajectories for Semiflows

It is known that various deterministic and stochastic processes such as asymptotically autonomous differential equations or stochastic approximation processes can be analyzed by relating them to an appropriately chosen semiflow. Here, we introduce the notion of a stochastic process X being a weak asymptotic pseudotrajectory for a semiflow and are interested in the limiting behavior of the empirical measures of X. The main results are as follows: (1) the weak* limit points of the empirical measures for X axe almost surely -invariant measures; (2) given any semiflow , there exists a weak asymptotic pseudotrajectory X of such that the set of weak* limit points of its empirical measures almost surely equal the set of all ergodic measures for ; and (3) if X is an asymptotic pseudotrajectory for a semiflow , then conditions on that ensure convergence of the empirical measures are derived.     

Asymptotic Behavior of Positive Solutions of Random and Stochastic Parabolic Equations of Fisher and Kolmogorov Types

We study the asymptotic behavior as t of positive solutions for random and stochastic parabolic equations of Fisher and Kolmogorov type. The following alternatives are established. Either (i) all positive solutions converge to one and the same trivial equilibrium, or (ii) every positive solution is neither bounded away from the trivial equilibria nor converges to them, or (iii) every positive solution is bounded away from the trivial equilibria. Moreover, for the random equation, we provide in case of alternative (iii) a fairly general condition under which every positive solution converges to uniformly positive equilibria. In the stochastic case, it is proved that there is no uniformly positive equilibrium, and under an appropriate condition, (iii) never occurs.     

High‐Frequency Changes in the Orientation of Nematic Liquid Crystals in Radio Frequency Crossed Electric Fields

The orientational dynamics of the director of a nematic liquid crystal located in radio frequency crossed electric fields were studied by numerical calculations and experimentally. This system is shown to be a physical object of nonlinear dynamics. Depending on the parameters of the problem, the following types of states of the director were observed: stationary (an analog of the nonthreshold Freedericksz transition), periodic, quasiperiodic (multimode), and stochastic of the strange attractor type. In the calculations, all states were obtained by solving a deterministic system of two timedependent nonlinear differential equations of the first order with no electrohydrodynamic terms. All types of solutions obtained, including stochastic ones, were observed experimentally.     

Pseudolinear vibroimpact systems: Non-white random excitation

Response analyses of vibroimpact systems to random excitation are greatly facilitated by using certain piecewise-linear transformations of state variables, which reduce the impact-type nonlinearities (with velocity jumps) to nonlinearities of the common type — without velocity jumps. This reduction permitted to obtain certain exact and approximate asymptotic solutions for stationary probability densities of the response for random vibration problems with white-noise excitation. Moreover, if a linear system with a single barrier has its static equilibrium position exactly at the barrier, then the transformed equation of free vibration is found to be perfectly linear in case of the elastic impact. The transformed excitation term contains a signature-type nonlinearity, which is found to be of no importance in case of a white-noise random excitation. Thus, an exact solution for the response spectral density had been obtained previously for such a vibroimpact system, which may be called pseudolinear, for the case of a white-noise excitation. This paper presents analysis of a lightly damped pseudolinear SDOF vibroimpact system under a non-white random excitation. Solution is based on Fourier series expansion of a signum function for narrow-band response. Formulae for mean square response are obtained for resonant case, where the (narrow-band) response is predominantly with frequencies, close to the system's natural frequency; and for non-resonant case, where frequencies of the narrow-band excitation dominate the response. The results obtained may be applied directly for studying response of moored bodies to ocean wave loading, and may also be used for establishing and verifying procedures for approximate analysis of general vibroimpact systems.