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Most multi-degree-of-freedom (MDOF) strongly nonlinear vibratory systems driven by Gaussian white noise can be modeled as quasi-non-integrable Hamiltonian systems. The prominent advantage of the stochastic averaging method of quasi-non-integrable Hamiltonian systems is that the averaged Ito stochastic differential equation (SDE) for Hamiltonian is only one-dimensional and the exact stationary solution of Fokker–Planck–Kolmogorov equation associated with the averaged Ito SDE can be readily obtained. However, for n-DOF quasi-non-integrable Hamiltonian systems with $$n>2$$ , one challenge to apply the stochastic averaging method for quasi-non-integrable Hamiltonian systems is how to evaluate ( $$2n-1$$ )-dimensional domain integrals in determining the averaged drift and diffusion coefficients. In the present paper, a novel method using two-step generalized elliptical coordinate transformations is proposed to convert the ( $$2n-1$$ )-dimensional domain integrals into ( $$n-1$$ )-fold repeated integrals and then n-fold repeated integrals. For quasi-non-integrable Hamiltonian systems with nonlinear momentum polynomial dampings and excitation coefficients, the first ( $$n-1$$ )-fold repeated integrals can be completed easily. Thus, the ( $$2n-1$$ )-dimensional domain integrals are actually reduced to much simple n-fold repeated integrals. A 5-DOF strongly nonlinear stochastic system is taken as an example to show the procedure and Monte Carlo simulation is used to show the validity of the proposed method.

More than 5000 landslides or potential landslides have been identified in the Three Gorges Reservoir (TGR) region since its impoundment in 2003. These hazards seriously threaten the continuation of reservoir operations and the safety of dams, waterways, and local residents. Understanding the deformation characteristics and kinematic evolution of these landslides can be helpful for assessing their stability and providing long-term landslide predictions. In this study, the deformation characteristics, influencing factors, sliding mechanism, and threshold reservoir level of the Muyubao landslide are revealed. Data from a professional monitoring network implemented in 2006 reveal that the deformation of this landslide is in a stage of creep characterized by overall movement, and the rate of creep is gradually slowing. The deformation mechanism is investigated by combining meteorological, hydrological, and reservoir level data with displacement measurements from surface cracks, manual and automatic GPS monitoring records, and field investigations and drilling surveys. The reservoir level is the main driving factor of deformation, whereas the filling-drawdown rate is not directly related to the deformation of the landslide. Furthermore, rainfall can promote deformation during periods with high reservoir levels. The deformation of the landslide follows a very distinctive pattern featuring large displacements after the reservoir level rises to a high level (from late October to early March of the following year) and a rapid transition to constant deformation as the reservoir level drops. The results indicate that the reservoir level of 172 m is the general threshold reservoir level of the Muyubao landslide. At present, the deformation rate of the Muyubao landslide is gradually decreasing, and the probability of large-scale sliding is low; however, the monitoring system must be strengthened and additional research must be performed on the deformation mechanism.

The reliability of quasi integrable and non-resonant Hamiltonian system under fractional Gaussian noise (fGn) excitation is studied. Noting rather flat fGn power spectral density (PSD) in most part of frequency band, the fGn is innovatively regarded as a wide-band process. Then, the stochastic averaging method for quasi integrable Hamiltonian systems under wide-band noise excitation is applied to reduce 2n-dimensional original system into n-dimensional averaged Ito stochastic differential equations (SDEs). Reliability function and mean first passage time are obtained by solving the associated backward Kolmogorov equation and Pontryagin equation. The validity of the proposed procedure is tested by applying it to an example and comparing the numerical results with those from Monte Carlo simulation.

The main difficulty of analyzing the response of nonlinear dynamical systems to fractional Gaussian noise (fGn) is the non-Markov property and non-usefulness of diffusion process theory. Currently, only numerical simulation can be applied to obtain the response of nonlinear systems to fGn. In the present paper, noting the rather flat property of the fGn power spectral density (PSD) in most part of frequency band, the stochastic averaging method for quasi-integrable Hamiltonian systems under wide-band noise excitation is applied to predict the response of quasi-integrable and non-resonant Hamiltonian systems to fGn. By using this method, the averaged Itô stochastic differential equations (SDEs) are established and the probability density function (PDF) of system response can be obtained from solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation. All of the statistics of system response are then obtained from the PDF analytically and verified through the comparison with the simulation results.

The major difficulty in studying the response of multi-degrees-of-freedom (MDOF) nonlinear dynamical systems driven by fractional Gaussian noise (fGn) is that the system response is not Markov process diffusion and thus the diffusion process theory cannot be applied. Although the stochastic averaging method (SAM) for quasi Hamiltonian systems driven by fGn has been developed, the response of the averaged systems still needs to be predicted by using Monte Carlo simulation. Later, noticing that fGn has rather flat power spectral density (PSD) in certain frequency band, the SAM for MDOF quasi-integrable and nonresonant Hamiltonian system driven by wideband random process has been applied to investigate the response of quasi-integrable and nonresonant Hamiltonian systems driven by fGn. The analytical solution for the response of an example was obtained and well agrees with Monte Carlo simulation. In the present paper, the SAM for quasi-integrable and resonant Hamiltonian systems is applied to investigate the response of quasi-integrable and resonant Hamiltonian system driven by fGn. Due to the resonance, the theoretical procedure and calculation will be more complicated than the nonresonant case. For an example, some analytical solutions for stationary probability density function (PDF) and stationary statistics are obtained. The Monte Carlo simulation results of original system confirmed the effectiveness of the analytical solutions under certain condition.

A stochastic averaging method of quasi integrable and non-resonant Hamiltonian systems under excitation of fractional Gaussian noise (fGn) with Hurst index 1/2

A stochastic averaging method for predicting the response of quasi partially integrable and non-resonant Hamiltonian systems to fractional Gaussian noise (fGn) with the Hurst index 1/2

A stochastic averaging method of quasi integrable and resonant Hamiltonian systems under excitation of fractional Gaussian noise (fGn) with the Hurst index 1/2 < H < 1 is proposed. First, the definition and the basic property of fGn and related fractional Brownian motion (fBm) are briefly introduced. Then, the averaged fractional stochastic differential equations (SDEs) for the first integrals and combinations of angle variables of the associated Hamiltonian systems are derived. The stationary probability density and statistics of the original systems are then obtained approximately by simulating the averaged SDEs numerically. An example is worked out to illustrate the proposed stochastic averaging method. It is shown that the results obtained by using the proposed stochastic averaging method and those from digital simulation of original system agree well.

On April 20th, 2013, a strong earthquake (Ms 7.0, depth of 13 km) occurred in LuShan County, Southwestern Sichuan province, China, followed by a series of aftershocks (Ms = 3.0∼5.4). Infrasonic waves associated with the earthquake were detected and recorded by a digital infrasound recording system in our lab located in Chengdu city. Fourier analysis was used to determine the amplitude and frequency characteristics of each infrasonic recording. A certain frequency distribution range was clearly identified in the spectrograms of all analyzed shocks. Furthermore, a logarithm regression relation is also found between aftershock magnitude and peak value of corresponding infrasound events in amplitude spectra based on analysis results of more than 30 observed events. This study presents a data set and an empirical relationship. This relationship requires additional testing with observations from other earthquake aftershock sequences. These results improve our understanding of local infrasound produced by the coupling of earth surface motion to the atmosphere.

Landslides and slope failures are very common phenomena in hilly regions, Southwestern China. These are hazardous because of the accompanying progressive movement of the slope-forming material. To minimize the landslide effects, slope failure analysis and stabilization require in-depth understanding of the process that governs the behavior of the slope. The present paper first briefly describes a three-dimensional numerical brittle creep model for rock. The model accounts for material heterogeneity, through a stochastic local failure stress field, and local material degradation using an exponential material softening law. Then a case study of the Jiweishan rockslide that occurred in China is numerically investigated considering the effect of the mining activity. Numerical simulations visualize the entire process of the Jiweishan rockslide from the fracture initiation, propagation and coalescence. The distribution and evolution of associated stress and deformation field during the slide are also presented. Numerical simulations show that the underground mining excavations have remarkably negative effect on the stability of the rock slope, which is one of the important triggering factors of the rockslide. Moreover, it is possible to take some precautions for the unstable failure of rock mass by monitoring acoustic emission (AE) events or microseismicities since the occurrence of clusters of AE events prior to the final unstable rockslide. The results are of general interest, because they can be applied to the investigation of time-dependent instability in rock masses, to the mitigation of associated rock hazards in rock engineering, and even to a better understanding of the seismic activities in geological and geophysical phenomena occurring in the earth’s crust.

With aim to master early deformation stages and motion laws of landslide, the raw displacement data of landslide are very important for the landslide researchers and environmental scientists. With the recent advances of Global Position System (GPS) and computer science, the differential GPS technology now can be used for the remote displacement observation of landslide. Therefore, the paper developed a landslide displacement remote observation system based on differential GPS technology to get the raw displacement data continually. The system is divided into hardware and software functional segments. A real landslide in southwest of China was monitored for several months through observation using this GPS-based system placed around and the observed results are presented in this paper. The practical application shows that this system has such advantages as being highly-precise, nearly real-time, remote, continuous and automatic.

Many physical systems can be modeled as quasi-Hamiltonian systems and the stochastic averaging method for quasi-Hamiltonian systems can be applied to yield reasonable approximate response statistics. In the present paper, the basic idea and procedure of the stochastic averaging method for quasi Hamiltonian systems are briefly introduced. The applications of the stochastic averaging method in studying the dynamics of active Brownian particles, the reaction rate theory, the dynamics of breathing and denaturation of DNA, and the Fermi resonance and its effect on the mean transition time are reviewed.

An important functioning mechanism of biological macromolecules is the transition between different conformed states due to thermal fluctuation. In the present paper, a biological macromolecule is modeled as two strands with side chains facing each other, and its stochastic dynamics including the statistics of stationary motion and the statistics of conformational transition is studied by using the stochastic averaging method for quasi Hamiltonian systems. The theoretical results are confirmed with the results from Monte Carlo simulation.

A nonlinear stochastic optimal control strategy for minimizing the first-passage failure of quasi integrable Hamiltonian systems (multi-degree-of-freedom integrable Hamiltonian systems subject to light dampings and weakly random excitations) is proposed. The equations of motion for a controlled quasi integrable Hamiltonian system are reduced to a set of averaged Ito stochastic differential equations by using the stochastic averaging method. Then, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximization of reliability and mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The final dynamical programming equations for these control problems are determined and their relationships to the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-passage time are separately established. The conditional reliability function and the mean first-passage time of the controlled system are obtained by solving the final dynamical programming equations or their equivalent Kolmogorov and Pontryagin equations. An example is presented to illustrate the application and effectiveness of the proposed control strategy.

In this paper two different control strategies designed to alleviate the response of quasi partially integrable Hamiltonian systems subjected to stochastic excitation are proposed. First, by using the stochastic averaging method for quasi partially integrable Hamiltonian systems, an n-DOF controlled quasi partially integrable Hamiltonian system with stochastic excitation is converted into a set of partially averaged Ito stochastic differential equations. Then, the dynamical programming equation associated with the partially averaged Ito equations is formulated by applying the stochastic dynamical programming principle. In the first control strategy, the optimal control law is derived from the dynamical programming equation and the control constraints without solving the dynamical programming equation. In the second control strategy, the optimal control law is obtained by solving the dynamical programming equation. Finally, both the responses of controlled and uncontrolled systems are predicted through solving the Fokker-Plank-Kolmogorov equation associated with fully averaged Ito equations. An example is worked out to illustrate the application and effectiveness of the two proposed control strategies.

A strategy for designing optimal bounded control to minimize theresponse of quasi non-integrable Hamiltonian systems is proposed basedon the stochastic averaging method for quasi non-integrable Hamiltoniansystems and the stochastic dynamical programming principle. Theequations of motion of a controlled quasi non-integrable Hamiltoniansystem are first reduced to an one-dimensional averaged Ito stochasticdifferential equation for the Hamiltonian by using the stochasticaveraging method for quasi non-integrable Hamiltonian systems. Then, thedynamical programming equation for the control problem of minimizing theresponse of the averaged system is formulated based on the dynamicalprogramming principle. The optimal control law is derived from thedynamical programming equation and control constraints without solvingthe equation. The response of optimally controlled systems is predictedthrough solving the Fokker–Planck–Kolmogrov (FPK) equation associatedwith completely averaged Ito equation. Finally, two examples are workedout in detail to illustrate the application and effectiveness of theproposed control strategy.

The optimal bounded control of quasi-integrable Hamiltonian systems with wide-band random excitation for minimizing their first-passage failure is investigated. First, a stochastic averaging method for multi-degrees-of-freedom (MDOF) strongly nonlinear quasi-integrable Hamiltonian systems with wide-band stationary random excitations using generalized harmonic functions is proposed. Then, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximizinig reliability and maximizing mean first-passage time are formulated based on the averaged It$\ddot{\rm o}$ equations by applying the dynamical programming principle. The optimal control law is derived from the dynamical programming equations and control constraints. The relationship between the dynamical programming equations and the backward Kolmogorov equation for the conditional reliability function and the Pontryagin equation for the conditional mean first-passage time of optimally controlled system is discussed. Finally, the conditional reliability function, the conditional probability density and mean of first-passage time of an optimally controlled system are obtained by solving the backward Kolmogorov equation and Pontryagin equation. The application of the proposed procedure and effectiveness of control strategy are illustrated with an example.

The changes of the stress state and strength of the slip zone in a landslide are determined by the material composition, micro-structural feature, and formation mechanism of the slip zone. It affects the activities of the landslide. In this paper, the mineral composition of the slip zone, which is composed of soft rock of the gray matter slate, is identified by DMAX-3C X-ray diffractometer, polarizing microscope, and scanning electron microscope (SEM). Results show that the content of calcite reaches up to 68 %. There are no changes in the slickensides composition of the sliding surface material. The microscopic structure has been analyzed by different experiments. The results show that the weak zone is composed of the banded calcareous tuffaceous slate, and its structure is characterized by gray black striped fine grains, heterogranular and irregular, and cryptocrystal texture. There is clay on these surfaces, and can be cut into pieces, so it is easy to track and accommodate the shear failure surfaces.

Single-file transportation of water across narrow nanochannels such as carbon nanotubes has attracted much attention in recent years. Such permeation can be greatly affected by the water-channel interactions; despite some progress, this issue has not been fully explored. Herein we use molecular dynamics simulations to investigate the effects of water-channel attractions on occupancy, translational (transportation) and orientational dynamics of water inside narrow single-walled carbon nanotubes (SWNTs). We use SWNTs as the model nanochannels and change the strength of water-nanotube attractions to mimic the changes in the hydrophobicity/polarity of the nanochannel. We investigate the dependence of water occupancy inside SWNTs on the water-channel attraction and identify the corresponding threshold values for drying states, wetting-drying transition states, and stably wetting states. As the strength of water-channel attractions increases, water flow increases rapidly first, and then decreases gradually; the maximal flow occurs in the case where the nanochannel is predominately filled with the 1D water wire but with a small fraction of 'empty states', indicating that appropriate empty-filling (drying-wetting) switching can promote water permeation. This maximal flow is unexpected, since in traditional view, the stable and tight hydrogen-bonding network of the water wire is the prerequisite for high permeability of water. The underlying mechanism is discussed from an energetic perspective. In addition, the effect of water-channel attractions on reorientational dynamics of the water wire is studied, and a negative correlation between the flipping frequency of water wire and the water-channel attraction is observed. The underlying mechanism is interpreted in term of the axial total dipole moment of inner water molecules. This work would help to better understand the effects of water-channel attractions on wetting properties of narrow nanochannels, and on single-file water permeation through nanochannels, which might be helpful in designs of high-flux nanochannels.