Analytical Stability Analysis Research Articles

Analytical stochastic analysis of rock wedge stability using a JDRV method considering the residual factor as a random variable

The stability of a rock wedge is one of the main failure models of rock slopes. Although the strength parameters of the discontinuity planes are commonly deemed to be between the peak and residual strengths, the effect of a residual factor on the stability of a rock wedge still has not been considered. This paper presents an analytical stochastic method, considering the residual factor as a random variable, for determining the stability of a rock wedge. First, based on the Hoek–Bray method, this paper derives the formulas for evaluating the wedge stability in terms of the residual factor, and then, based on a jointly distributed random variables (JDRV) method, the calculation formulas for the probability distribution function of the safety factor is established. In the analytical method, the friction angle and cohesion are selected to satisfy the truncated normal distribution, and the residual factor is assumed to obey the generalized beta distribution. The study of a typical rock wedge shows that the results of the analytical method and the Monte Carlo simulation are almost consistent. Besides, it is also shown that the residual factor has a substantial influence on the estimation accuracy of the failure probability of the wedge.

Wedge Sliding Analysis of the Rock Slope Subjected to Uplift Forces and Surcharge Loads Conditions

The safety factor of the slope subjected to wedge failure under surcharge load and uplift forces was obtained using an analytical model. A detailed parametric study conducted for the physical slope involved examining the impacts of the surcharge loads and uplift forces acting along the joint planes due to water pressure on the safety factor of the slope subjected to wedge sliding. The study results indicated that the safety factor of the slope susceptible to wedge sliding decreases with an increase in the surcharge loads and the uplift forces acting along the joint planes due to water pressure. The reduction in the safety factor due to an increase in the surcharge load is independent of the uplift forces acting along the joint planes. The rate of deterioration in the stability of the rock slope susceptible to wedge sliding depends on the rate of increase in the surcharge loads and uplift forces acting along the joint planes due to water pressure. The proposed expression can be useful for analytical stability analyses of the rock slopes subjected to wedge sliding induced by the surcharge load and uplift forces.

Long-Term Stability of a 13.7 × 30.5-m (45 × 100-ft) Undercut Span Beneath Cemented Rockfill at the Turquoise Ridge Mine, Nevada.

In 2001, researchers from the National Institute for Occupational Safety and Health (NIOSH) installed instruments at the Turquoise Ridge Mine in cooperation with Placer Dome, Inc. to monitor the geomechanical behavior and stability of a cemented rockfill (CRF) sill and the surrounding host rock during test mining of a large undercut span beneath backfill. Six parallel, adjacent drifts were mined and backfilled to construct a CRF sill, approximately 22.9 m (75 ft) wide by 30.5 m (100 ft) long. The sill was then partially undercut, successfully creating a 13.7-m (45-ft) wide by 30.5-m (100-ft) long span beneath the CRF. Only small vertical displacements were measured in the overlying host rock during mining, with most of the movement occurring at shallow depths in the mine roof. Because the back above the CRF sill remained stable, the majority of the mining-induced stress was transferred to the host rock abutments rather than to the backfilled drifts. During retreat mining of the undercut span, the CRF sill and the mine roof remained stable. Most of the measured vertical displacement was caused by separation of the backfill from the overlying host rock, or deflection of the CRF sill, which was comparable to the deflection of a monolithic, elastic plate having similar dimensions, material properties, and undercut spans. The CRF sill moved in mass as a single unit rather than as individual drift segments, and the vertical cold joints between adjacent backfill drifts did not adversely affect their stability. Additional measurements collected from the instruments have shown that the backfill span is still intact and in stable condition more than 16 years after the completion of undercut mining. Displacements in the mine roof and abutments have stabilized, and vertical stress and deformation within the CRF have generally leveled off or decreased. Although only slight mining-induced loads were transferred to the backfilled drifts, the CRF has confined the abutment ribs and mine roof, thereby improving their long-term stability. Results of compressive and tensile strength tests conducted with CRF samples from the test site indicate that the long-term compressive strength gain for CRF is similar to that of concrete, and that the tensile-to-compressive strength ratio for CRF is about 1/6 rather than 1/10. Assuming the in-place CRF gained strength at the same rate as the lab samples, an analytical analysis of the flexural stability of the CRF undercut span shows that the Factor of Safety for the span should have logically increased over time. By providing a better understanding of the long-term strength properties and geomechanical behavior of CRF, these research findings help improve the methods that are used for designing stable, long-term undercut entries beneath cemented backfill.

Parametric subharmonic instability in a narrow-band wave spectrum

Parametric subharmonic instability arising in a narrow-band wave spectrum is investigated. Using a statistical equation that describes weakly nonlinear interactions in a random wave field, we perform analytical and numerical stability analyses for a modulating wave train. The analytically obtained growth rate $\unicode[STIX]{x1D706}=(-\unicode[STIX]{x1D707}+\sqrt{\unicode[STIX]{x1D707}^{2}+4CE_{B}})/2$ agrees favourably with the results from direct numerical experiments, where $\unicode[STIX]{x1D707}$ is the half-value width of the background wave frequency spectrum, $E_{B}$ is the background wave energy density, and $C$ is a constant. This expression has two asymptotic limits: $\unicode[STIX]{x1D706}\sim \sqrt{CE_{B}}$ for $\unicode[STIX]{x1D707}\ll \sqrt{CE_{B}}$ and $\unicode[STIX]{x1D706}\sim CE_{B}/\unicode[STIX]{x1D707}$ for $\unicode[STIX]{x1D707}\gg \sqrt{CE_{B}}$. In the terms of weak turbulence, these two growth rates correspond to the ones occurring in the dynamic and kinetic time scales. In this way, our formulation successfully unifies the two conventional types of parametric subharmonic instability and offers a new criterion to determine the applicability of the classical kinetic equation in three-wave systems.

Porcine Breed, Sex, and Production Stage Influence the Levels of Health Status Biomarkers in Saliva Samples.

In this study, the influence of several factors such as breed, sex, and production stage over the normal range values of salivary biomarkers of health status was evaluated in pigs. A total of 409 pigs of 2 different breeds (conventional Large White × Duroc and Iberian pigs) were included in the study. Animals were divided into different groups according to their sex (male or female) and the stage of the production cycle they were in (post-weaning, nursery, fattening, and finishing). The levels of an inflammatory marker, adenosine deaminase (ADA), and two acute phase proteins, C-reactive protein (CRP) and haptoglobin (Hp) were measured in saliva samples. Moreover, the total antioxidant capacity level (TAC) was quantified for the first time in porcine saliva; therefore, an analytical validation and stability analysis during storage at −80°C were also performed. Differences according to breed were observed for all the markers studied; thus, the influence of age and sex on the normal range values were studied separately for conventional and Iberian pigs. In Large White × Duroc pigs the overall median values of ADA, CRP, Hp and TAC were 282 U/L, 10.49 ng/mL, 0.88 μg/mL, and 21.73 μM Trolox equivalents, respectively. However, higher values of inflammatory marker and acute phase proteins were observed in males at the initial stages of the production cycle, while females presented higher values when they had reached sexual maturity. In Iberian pigs the overall median values observed were 585 U/L, 4.81 ng/mL, 0.63 μg/mL, and 21.21 μM Trolox equivalents for ADA, CRP, Hp, and TAC respectively with slight differences in the influence of the studied factors. Sex differences were not observed in the levels of acute phase proteins in Iberian pigs, probably due to the castration of males during the first days of life; however, ADA levels were found to be higher in male pigs at the end of the production cycle. It could be concluded that breed, sex, and production stage influence the range values of salivary markers of health status in pigs and should be taken into account to further establish reference intervals.

Analytical Design and Stability Analysis of the Universal Integral Regulator Applied in Flight Control

This paper considers the analytical design and stability analysis of an output feedback flight control problem for a rigid fighter aircraft which has a highly nonlinear dynamic. In this paper, a robust technique known as Universal Integral Regulator (UIR) has been chosen to solve the tracking problem due to the possibility to demonstrate the stability of the system and analytically compute the control parameters. The UIR is a combination of Continuous Sliding Mode Control (CSMC) and a Conditional Integrator (CI) which provides integral action only inside the boundary layer, enhancing the transient response of the system and providing an equilibrium point where the tracking error is zero. The general procedure consists firstly of rewriting the aircraft dynamics in the control-affine form, then the relative degree of the system is computed and the system is transformed to normal form. An output feedback controller using a CSMC controller is proposed, and a sliding surface considering a CI is designed. The controller parameters are designed analytically, taking into account two approaches. The first approach does not consider uncertain parameters and the second one treats a stability derivative as a parametric uncertainty. Simulations were performed in order to validate the design procedure of the control technique and to demonstrate the robustness of the UIR. Detailed step by step information about the computing of the controller parameters was done and an analytical analysis of stability was developed to demonstrate the convergence of the sliding surface, conditional integrator and tracking error dynamics.

Analytical structures and stability analysis of the simplest Takagi-Sugeno fuzzy two-term controllers

This paper deals with the simplest Takagi-Sugeno fuzzy two-term controllers. The analytical structures of the simplest fuzzy controllers were developed using a modified rule base. The number of tuning parameters of the controllers is reduced by introducing a novel rule base consisting of two rules. These controllers are termed as 'the simplest' since minimal (two) number of fuzzy sets is used for fuzzification. Algebraic product (AP)/minimum (min) triangular norm, bounded sum (BS)/maximum (max) triangular co-norm, different universes of discourses (UoDs) of inputs, and centre of gravity (CoG) defuzzification method are chosen to derive the mathematical models of the fuzzy controllers. The simplest fuzzy controller with modified rule base is equivalent to a (nonlinear) variable gain/structure PI/PD controller. The BIBO stability of the closed loop control system is investigated using the small gain theorem. The gain of the controller either varies or remains constant in different regions of the input plane. The gain variations and the computational burden of the fuzzy controllers are also studied. Two examples of nonlinear dynamical systems are considered to validate the developed models of the fuzzy controllers.

Analytical structures and stability analysis of the simplest Takagi-Sugeno fuzzy two-term controllers

Analytical structures and stability analysis of the simplest Takagi-Sugeno fuzzy two-term controllers

Vertically Sheared Horizontal Flow-Forming Instability in Stratified Turbulence: Analytical Linear Stability Analysis of Statistical State Dynamics Equilibria

Abstract Vertically banded zonal jets are frequently observed in weakly or nonrotating stratified turbulence, with the quasi-biennial oscillation in the equatorial stratosphere and the ocean’s equatorial deep jets being two examples. Explaining the formation of jets in stratified turbulence is a fundamental problem in geophysical fluid dynamics. Statistical state dynamics (SSD) provides powerful methods for analyzing turbulent systems exhibiting emergent organization, such as banded jets. In SSD, dynamical equations are written directly for the evolution of the turbulence statistics, enabling direct analysis of the statistical interactions between the incoherent component of turbulence and the coherent large-scale structure component that underlie jet formation. A second-order closure of SSD, known as S3T, has previously been applied to show that meridionally banded jets emerge in barotropic β-plane turbulence via a statistical instability referred to as the zonostrophic instability. Two-dimensional Boussinesq turbulence provides a simple model of nonrotating stratified turbulence analogous to the β-plane model of planetary turbulence. Jets known as vertically sheared horizontal flows (VSHFs) often emerge in simulations of Boussinesq turbulence, but their dynamics is not yet clearly understood. In this work S3T analysis of the zonostrophic instability is extended to study VSHF emergence in two-dimensional Boussinesq turbulence using an analytical formulation of S3T amenable to perturbation stability analysis. VSHFs are shown to form via an instability that is analogous in stratified turbulence to the zonostrophic instability in β-plane turbulence. This instability is shown to be strikingly similar to the zonostrophic instability, suggesting that jet emergence in both geostrophic and nonrotating stratified turbulence may be understood as instances of the same generic phenomenon.

Pilot-wave dynamics of two identical, in-phase bouncing droplets.

A droplet bouncing on the surface of a vibrating liquid bath can move horizontally guided by the wave it produces on impacting the bath. The wave itself is modified by the environment, and thus, the interactions of the moving droplet with the surroundings are mediated through the wave. This forms an example of a pilot-wave system. Taking the Oza-Rosales-Bush description for walking droplets as a theoretical pilot-wave model, we investigate the dynamics of two interacting identical, in-phase bouncing droplets theoretically and numerically. A remarkably rich range of behaviors is encountered as a function of the two system parameters, the ratio of inertia to drag, , and the ratio of wave forcing to drag, . The droplets typically travel together in a tightly bound pair, although they unbind when the wave forcing is large and inertia is small or inertia is moderately large and wave forcing is moderately small. Bound pairs can exhibit a range of trajectories depending on parameter values, including straight lines, sub-diffusive random walks, and closed loops. The droplets themselves may maintain their relative positions, oscillate toward and away from one another, or interchange positions regularly or chaotically as they travel. We explore these regimes and others and the bifurcations between them through analytic and numerical linear stability analyses and through fully nonlinear numerical simulation.

“On the fly” stabilization of the Coarse-Mesh Finite Difference acceleration for multidimensional discrete-ordinates transport calculations

In this paper, the Jacobian matrix of the Coarse-Mesh Finite Difference (CMFD) method is analyzed. Both the homogenization and the current preservation effects are studied in heterogeneous multidimensional configurations. Some bounding values of the spectral radius are also given. An analytical stability analysis is carried on an interface slab problem. This analysis leads to the computation of the stability parameter introduced in the Generalized Coarse Mesh Rebalancing method. A dynamical stabilization technique is proposed for multidimensional neutron lattice calculations. Numerical calculations show that the proposed technique dumps the unstable modes, in particular in optically thick configurations, where the classical CMFD method fails to converge.

Influence of Vertical Vibrations on the Stability of a Binary Mixture in a Horizontal Porous Layer Subjected to a Vertical Heat Flux

We present an analytical and numerical stability analysis of Soret-driven convection in a porous cavity saturated by a binary fluid mixture and subjected to vertical high-frequency and small-amplitude vibrations. Two configurations have been considered and compared: an infinite horizontal layer and a bounded domain with a large aspect ratio. In both cases, the initial temperature gradient is produced by a constant uniform heat flux applied on the horizontal boundaries. A formulation using time-averaged equations is used. The linear stability of the equilibrium solution is carried out for various Soret separation ratios $$\varphi $$ , vibrational Rayleigh numbers Rv, Lewis numbers Le and normalized porosity. For an infinite horizontal layer, the critical Rayleigh number $$\mathrm{Ra}_c$$ is determined analytically. For a steady bifurcation to a one-cell solution (the critical wavenumber is zero), we obtain $$\mathrm{Ra}_{c}={12}/{({\varphi }(\mathrm{Le}+1)+1)}$$ for all Rv. When the bifurcation is a Hopf bifurcation or when the critical wavenumber is not zero, we use a Galerkin method to compute the critical values. Our study is completed by a nonlinear analysis of the bifurcation to one-cell solutions in an infinite horizontal layer that is compared to numerical simulations in bounded horizontal domains with large aspect ratio.

An improved meshless method for solving two- and three-dimensional coupled Klein–Gordon–Schrödinger equations on scattered data of general-shaped domains

In present paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to obtain the solution of two- and three-dimensional coupled Klein–Gordon–Schrodinger (KGS) equations. First, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This approach is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. In the current work, the polyharmonic splines are used as the basis functions and to eliminate the nonlinearity, a simple predictor–corrector (P–C) scheme is performed. The aim of this paper is to show that the SMRPI procedure is suitable for the treatment of the KGS equations. The results of numerical experiments are compared with analytical solution and stability analysis is checked to confirm the accuracy and efficiency of the presented scheme.

Pivot point-based control for active rear-wheel steering in passenger vehicles

ABSTRACTThis paper presents a new application of active rear-wheel steering control to improve the lateral vehicle behaviour. In the state of the art, yaw or lateral velocity is used as control variable that means one degree of freedom being not directly controlled. A worse subjective impressions due to movements in the rear end of the vehicle during strong counter-steering are a consequence. To avoid this effect in urban surroundings, an innovative structure to control the pivot point distance of the vehicle is proposed. In this case the coupled elementary states yaw and lateral velocity can be influenced based on a higher level criteria. Analysis show that pivot point fixing provides a comprehensible reference behaviour. Solving the issue of singularity during disappearing yaw movement is the basis to design a performant modified feedforward input–output linearisation. An analytic stability analysis of the internal dynamics shows system immanent limitations which do not influence the target of improving the lateral vehicle dynamics in urban manoeuvres. Finally, the advantages of pivot-based control are highlighted by a comparison with state of the art rear axle control.

Analytical and numerical stability analysis of Soret-driven convection in a horizontal porous layer: The effect of vertical vibrations.

The authors studied the effect of vertical high-frequency and small-amplitude vibrations on the separation of a binary mixture saturating a porous cavity. The horizontal bottom plate was submitted to constant uniform heat flux and the top one was maintained at constant temperature while no mass flux was imposed. The influence of vertical vibrations on the onset of convection and on the stability of the unicellular flow was investigated for positive separation ratio [Formula: see text]. The case of high-frequency and small-amplitude vibrations was considered so that a formulation using time averaged equations could be used. For an infinite horizontal porous layer, the equilibrium solution was found to lose its stability via a stationary bifurcation leading to unicellular flow or multicellular one depending on the value of [Formula: see text]. The analytical solution of the unicellular flow was obtained and separation was expressed in terms of Lewis number, separation ratio and thermal Rayleigh number. The direct numerical simulations using the averaged governing equations and analytical stability analysis showed that the unicellular flow loses its stability via oscillatory bifurcation. The vibrations decrease the value of [Formula: see text], which allows species separation for a wide variety of binary mixtures. The vibrations can be used to maintain the unicellular flow and allow species separation over a wider range of Rayleigh numbers. The results of direct numerical simulations and analytical model are in good agreement.

Time-delayed model of RNA interference

RNA interference (RNAi) is a fundamental cellular process that inhibits gene expression through cleavage and destruction of target mRNA. It is responsible for a number of important intracellular functions, from being the first line of immune defence against pathogens to regulating development and morphogenesis. In this paper we consider a mathematical model of RNAi with particular emphasis on time delays associated with two aspects of primed amplification: binding of siRNA to aberrant RNA, and binding of siRNA to mRNA, both of which result in the expanded production of dsRNA responsible for RNA silencing. Analytical and numerical stability analyses are performed to identify regions of stability of different steady states and to determine conditions on parameters that lead to instability. Our results suggest that while the original model without time delays exhibits a bi-stability due to the presence of a hysteresis loop, under the influence of time delays, one of the two steady states with the high (default) or small (silenced) concentration of mRNA can actually lose its stability via a Hopf bifurcation. This leads to the co-existence of a stable steady state and a stable periodic orbit, which has a profound effect on the dynamics of the system.

Routes to chaos of natural convection flows in vertical channels

The aim of the present study is the analysis of the transition to turbulence of natural convection flows between two infinite vertical plates. For the study of the problem, a number of Direct Numerical Simulations (DNSs) have been performed. The continuity, momentum and energy equations, cast under the Boussinesq assumption, are tackled numerically by means of a pseudospectral method, through which the three-dimensional domain is decomposed with Chebychev polynomials in the wall-normal direction and with Fourier modes in the wall-parallel directions. For low Rayleigh number values, the predictions of the flow regimes are consistent with the classical analytical results and linear stability analyses. In particular, the first bifurcation (Ra ≈ 5800) from the so-called laminar conduction regime to steady convection is correctly captured. By increasing the Rayleigh number beyond a second critical value (Ra ≈ 10200), the flow regime becomes chaotic. This transition to chaos is found to be related with the amplification of spanwise instabilities occurring at scales larger than the channel gap, H. The study of the return of the system from the chaotic regime to the laminar base flow reveals a phenomenon of hysteresis, i.e. the chaotic regime persists even at Ra-values lower than the second critical value. From a numerical point of view, the predicted flow regimes appear to be extremely sensitive to the domain size, grid resolution and perturbation amplitude. These aspects are shown to be of crucial importance for the prediction of the heat transfer performance, and, hence, should be taken into consideration when numerical methods are used for the simulation of real-world problems.

Three-dimensional MHD flow over a shrinking sheet: Analytical solution and stability analysis

The magnetohydrodynamic (MHD) steady and unsteady axisymmetric flows of a viscous fluid over a two-dimensional shrinking sheet are addressed. The mathematical analysis is carried out in the presence of a large magnetic field. The steady state problem results in a singular perturbation problem having an infinite domain singularity. The secular term appearing in the solution is removed and a two-term uniformly valid solution is derived using the Lindstedt–Poincaré technique. This asymptotic solution is validated by comparing it with the numerical solution. The solution for the unsteady problem is also presented analytically in the asymptotic limit of large magnetic field. The results of velocity profile and skin friction are shown graphically to explore the physical features of the flow field. The stability analysis of the unsteady flow is made to validate the asymptotic solution.

Faraday instability of a two-layer liquid film with a free upper surface

The classic Faraday instability in a one-layer liquid film is extended to a two-layer film consisting of a lower liquid layer, supported by a vibrating solid plate, overlaid by a second layer of immiscible fluid with a free upper surface. A detailed analytical and numerical analysis of the linear stability of the film, and the interplay between the Faraday and the Rayleigh-Taylor instabilities, is presented.

Maxwell stress generated long wave instabilities in a thin aqueous film under time-dependent electro-osmotic flow

In this study the dynamics and stability of thin and electrically conductive aqueous films under the influence of a time-periodic electric field are explored. With the help of analytical linear stability analysis for long wavelength disturbances, the stability threshold of the system as a function of various electrochemical parameters and transport coefficients is presented. The contributions of parameters like surface tension, disjoining pressure, electric double layer (Debye length and interfacial zeta potential), and unsteady Maxwell and viscous stresses are highlighted with the help of appropriate dimensionless groups. The physical mechanisms affecting the stability of thin films are detailed with the above-mentioned forces and parametric dependence of stability trends is discussed.