Stability Charts Research Articles

Evaluating Road-Cut Slope Stability Using Newly Proposed Stability Charts and Rock Microstructure: An Example from Dharasu-Uttarkashi Roadway, Lesser Himalayas, India

Evaluating Road-Cut Slope Stability Using Newly Proposed Stability Charts and Rock Microstructure: An Example from Dharasu-Uttarkashi Roadway, Lesser Himalayas, India

On the stability of thin-walled circular cylindrical shells under static and periodic radial loading

In this paper, we study the dynamic behavior of the cylindrical shells subjected to static and time-varying (periodic) radial loading. Flügge-Luré-Byrne shell theory has been used to derive dynamic equations in the form of nonlinear partial differential equations (PDEs). Linearization of PDEs about the steady-state periodic solutions, followed by Galerkin projection, yields a system of ordinary differential equations (ODEs) with time-varying periodic coefficients. Those are similar to the higher-order Mathieu-Hill equations and their stability analyses have been ascertained using Floquet theory. Dynamic instabilities of the cylindrical shells have been presented through stability charts in the plane of forcing parameters, namely, the static component of the pressure, the amplitude, and frequency of the fluctuating component of the pressure. Also, possible sub and super harmonic resonances have been explored through the stability charts. Four different boundary conditions, viz., simply supported, clamped–free, pinned–free (axial constraint at the pinned end) and pinned–free (axial constraint at the free end) have been considered in the present analysis. Presented charts can be very useful during the design of the shell structures subjected to static/dynamic loading.

Stability Analysis of 3D Deep Tunnel Face under Steady Seepage Flow Condition

The instabilities of tunnel face may occur while shields advance through the rivers or region rich in water. It is of significance to attach the importance to the estimation of face stability during the tunnel excavation. Thus, this paper adopted a three-dimensional (3D) rotational mechanism to analyze the face stability under steady seepage flow field with the aid of the kinematic method in the context of limit analysis. The hydraulic head that derived by fitting the results of trial and error is employed herein to portray the steady seepage flow field. With the aid of a double integration method, the work rate induced by seepage forces acting on the soil skeleton is calculated and accordingly a closed-form solution of support pressure is derived based on the work balance equation. By comparing with the literature, one can validate the rationality and correctness of the present results. The research shows that the proposed method based on rotational mechanism provides a better upper bound estimation, when under seepage condition, the influence of internal frictional angle on the stability of the excavation face is not always monotonic and the influence of head difference on the support pressure is significant. Finally, the normalized stability charts are provided for a quick assessment for practical use during tunnel excavation in water-rich stratum.

Stability Charts for Closely Spaced Strip Footings on Hoek–Brown Rock Mass

Footings are constructed closely to each other in mountainous rock areas and the determination of closely spaced bearing capacity footings on rock masses has become an important topic of recent research. The primary aim of this paper is to determine an efficiency factor, just like in the design of a pile group, that can be used to evaluate the bearing capacity of closely spaced footings on Hoek–Brown mass. The advanced finite element limit analysis of upper and lower bound theorems is used to study the stability of two and multiple interfering footings using the Hoek–Brown parameters such as GSI (geological strength index) and mi (Hoek–Brown yield parameter). This study found that the efficiency factor is influenced by three dimensionless parameters including GSI, mi, and S/B. Failure mechanisms of the problems are also investigated, and stability charts and tables produced for practical uses. Overall, these results broadly support the common understanding of the two key parameters GSI and mi in the HB failure criterion. As a result of these investigations, suggestions were identified for future research.

Slope Stability Classification under Seismic Conditions Using Several Tree-Based Intelligent Techniques

Slope stability analysis allows engineers to pinpoint risky areas, study trigger mechanisms for slope failures, and design slopes with optimal safety and reliability. Before the widespread usage of computers, slope stability analysis was conducted through semi analytical methods, or stability charts. Presently, engineers have developed many computational tools to perform slope stability analysis more efficiently. The challenge associated with furthering slope stability methods is to create a reliable design solution to perform reliable estimations involving a number of geometric and mechanical variables. The objective of this study was to investigate the application of tree-based models, including decision tree (DT), random forest (RF), and AdaBoost, in slope stability classification under seismic loading conditions. The input variables used in the modelling were slope height, slope inclination, cohesion, friction angle, and peak ground acceleration to classify safe slopes and unsafe slopes. The training data for the developed computational intelligence models resulted from a series of slope stability analyses performed using a standard geotechnical engineering software commonly used in geotechnical engineering practice. Upon construction of the tree-based models, the model assessment was performed through the use and calculation of accuracy, F1-score, recall, and precision indices. All tree-based models could efficiently classify the slope stability status, with the AdaBoost model providing the highest performance for the classification of slope stability for both model development and model assessment parts. The proposed AdaBoost model can be used as a screening tool during the stage of feasibility studies of related infrastructure projects, to classify slopes according to their expected status of stability under seismic loading conditions.

Integral Resonant Controller to Suppress the Nonlinear Oscillations of a Two-Degree-of-Freedom Rotor Active Magnetic Bearing System

Within this article, the nonlinear vibration control of the rotor active magnetic bearings system is tackled utilizing the integral resonant controller for the first time. Two integral resonant controllers are proposed to mitigate the system lateral oscillations in the horizontal and vertical directions. Based on the suggested control technique, the whole system dynamical model is derived as a two-degree-of-freedom nonlinear system (i.e., rotor system) coupled linearly to two first-order filters (i.e., the integral resonant controllers). The nonlinear autonomous system that governs the oscillation amplitudes of the controlled system as a function of the control parameters is extracted by applying perturbation analysis. The obtained autonomous system showed that the linear damping coefficients of the rotor system are functions of the control gains, feedback gains, and internal loop feedback gains of the coupled controller. Accordingly, the sensitivity of the rotor oscillation amplitudes to the different control parameters is explored. The stability margins and the optimal control gains are reported via plotting the different stability charts in two-dimensional space. The main acquired results demonstrated that the vibration suppression efficiency of the proposed controller is proportional to the product of both the control and feedback signal gains, and inversely proportional to the square of the internal loop feedback gains. In addition, the analytical investigations confirmed that the proposed integral resonant control method can force the rotor system to respond as a linear one with a single periodic attractor when the control parameters are designed properly. Finally, numerical simulations are performed that have illustrated the excellent correspondence with the obtained analytical results.

Development and application of a numerical analysis method for investigating hydro static and hydrodynamic responses of pocket bearing rotor systems

A numerical tool is presented for simulating the dynamics of a vertical rotor attached with pocket bearing assembly immersed in a liquid pool. The journal of the bearing has a heterogeneous geometry with the presence of pockets and grooves. The fluid pressure supplied to the pockets is a quadratic function of shaft speed. The radial, circumferential and axial flow paths constitute a complex fluid network in the coupled system. The network has been conceived as an assemblage of straight and curved fluid flow paths for the generic applications. An efficient solution strategy has been employed involving both analytical and numerical techniques.A Case study was performed on a representative pocket bearing assembly that is commonly used in the coolant pumps of sodium cooled fast reactor. The study has brought out some specific dynamic characteristics of such bearings. The flow field exhibits both hydrostatic and hydrodynamic effects, thereby hybrid bearing characteristics. The dynamic characteristics are strong function of spinning velocity, eccentricity and angular orientation of the rotor in fluid annulus. Based on an elaborate parametric study, a rotor dynamic stability chart has been established for the practical applications. Interestingly, the dynamic behavior within a limited operating regime exhibits the characteristics of a large eccentric homogeneous rotor rotating in fluid annulus with moderate gaps, predicted by Antunes et al. (1996). Moreover, it is found that the pocket type bearing has better stability control, mainly due to lower added fluid mass. The numerical tool is experimentally validated by comparing the displacements measured on a shaft coupled with pocket bearing assembly, rotating in a water pool.

The Consolidated Mathews Stability Graph for Open Stope Design

The stability graph method for open stope design is one of the most widely used approaches for predicting the stability of stopes in underground metalliferous mines. The primary purpose of this work is to propose a new stability chart, which includes all relevant case histories, and to exclude parameters with uncertainties for determining stability number. The modified stability number was used to achieve this goal, and the Extended Mathews database was recalculated and compared with the new stability graph. In this study, a new refined Consolidated stability graph was developed by excluding the entry mining methods data from the Extended graph data, and only the non-entry methods data was used. The applicability of the proposed Consolidated stability chart was demonstrated by an open stope example. The probabilities of stability for each stope surface were determined using the logistic regression model and the developed Consolidated stability chart. Comparing the stability analysis results with that of other published works of the same example shows that the determined Consolidated chart, in which the entry-method data is excluded, produces a more conservative and safer design. In conclusion, the size and quality of the dataset dictate the reliability of this approach.

Semi-Discretization for Fractional-Order Feedback Systems with Input Delay

In this paper, we apply the semi-discretization method to controllable single-input linear systems with input delay and fractional-order feedback. This requires the fractional derivative to be discretized in a way that the resulting discrete map is linear and time-invariant. To this end, three different techniques are investigated, namely, the short-memory principle, the application of adaptive time steps, and the exponential approximation of the derivative weights. For all three approaches, we construct a linear map that describes the dynamics of the approximate semi-discrete system. The matrices of these linear maps can be used to investigate the stability of the system. As an example, stability charts are determined for the fractional-order proportional-derivative delayed control of the inverted pendulum, and the numerically obtained stability charts are compared with the exact stability boundaries.

Stability Analysis of a Hierarchical Lane-keeping Controller with Feedback Delay

A hierarchical lane-keeping controller of passenger cars is analyzed, with the consideration of feedback delay in the control loop. The higher-level controller generates the desired steering angle based on the delayed feedback of the lateral position and yaw angle of the vehicle. The feedback delay of the lateral position and the yaw angle are treated separately in the model, therefore more general sensor setups and estimation algorithms can be handled. A lower-level proportional-integral-derivative power steering controller ensures that the desired steering angle signal is followed properly. The linear stability analysis of the resulting closed-loop system is performed, with a focus on the controller parameters and the delay values. The optimal control gains, leading to the fastest decay of the solution are also investigated in detail. The results are showcased with the help of stability charts and numerical simulations.

Stability and local bifurcation analyses of two-wheeled trailers considering the nonlinear coupling between lateral and vertical motions

The nonlinear dynamics of two-wheeled trailers is investigated using a spatial 4-DoF mechanical model. The non-smooth characteristics of the tire forces caused by the detachment of the tires from the ground and other geometrical nonlinearities are taken into account. Beyond the linear stability analysis, the nonlinear vibrations are analyzed with special attention to the nonlinear coupling between the vertical and lateral motions of the trailer. The center manifold reduction is performed leading to a normal form up to third degree terms. The nature of the emerging periodic solutions, and, thus, the sense of the Hopf bifurcations are verified semi-analytically and numerically. Simplified models of the trailer are also used in order to point out the practical relevance of the study. It is shown that the linearly independent pitch motion affects the sense of the Hopf bifurcations at the linear stability boundary. Namely, the constructed spatial trailer model provides subcritical bifurcations for higher center of gravity positions, while the commonly used simplified mechanical models explore the less dangerous supercritical bifurcations only. Domains of loss of contact of tires are also detected and shown in the stability charts highlighting the presence of unsafe zones. Experiments are carried out on a small-scale trailer to validate the theoretical results. A good agreement can be observed between the measured and numerically determined critical speeds and vibration amplitudes.

Investigations of Silty Soil Slopes under Unsaturated Conditions Based on Strength Reduction Finite Element and Limit Analysis

Matric suction plays a key role in slope stability by conferring an apparent cohesion component to the unsaturated portion of the soil. This paper adopts the total cohesion method to investigate the contribution of apparent cohesion on the stability of silty slopes under hydrostatic conditions. Phase2 and Optum G2 numerical programs, based on strength reduction finite element analysis and finite element limit analysis methods, respectively, are used for numerical analysis. Generally, Phase2 and Optum G2 results are in good agreement with each other. Optum G2 yields slightly higher factor of safety results than Phase2, particularly for steep slopes β≥ 30°. The results are presented in form of stability charts which are validated with a case from a previous study. Notably, the contribution of apparent cohesion to unsaturated shear strength is most pronounced when varying the water table. An examination of the slope failure mechanism reveals that the toe failure mechanism is the dominant failure mechanism. The depth of the failure surface is most sensitive to changes in the slope angle, cohesion and water table position. The influence of the air-entry value on the depth of the failure surface is contingent upon the location of the water table.

FPGA implementation of nonlinear equations with delay

The study of dynamic systems has been the subject of growing interest in recent decades. Its purpose is to reveal the phenomena or behaviors of the systems, and even those hidden, for given initial conditions and parameter values. Their implementations allow them to be exploited in various fields of application. The present work deals with the study and FPGA implementation of nonlinear time-delay equations. These are the Johnson and Moon equation and the Mackey-Glass equation. The stability analysis of these two models allowed to plot the stability chart in the parameter space of each system, thus delimiting the areas of stability and instability. Numerical simulation for different values of the control parameter revealed a damped, periodic and even chaotic oscillation for the two systems. An architecture based on the fourth-order Runge-Kutta method (RK4) taking into account the delay, in which the calculations are performed according to the 32-bit IEEE-754–1985 floating-point numbers standard, has been proposed for the hardware description of the systems. The corresponding VHDL code in each case was synthesized on the Xilinx Virtex-6 family XCV6VLX130T-3FF484 FPGA chip and the maximum operating frequencies obtained are of the order of 431.727 MHz and 432.280 MHz respectively for the Johnson and Moon and Mackey-Glass equation. The results obtained from the Modelsim simulation of the FPGA implementation of the models are identical to those of the numerical simulation. This contribution highlighted a new approach for the FPGA implementation of time-delay systems and can be exploited for other time-delay systems.

Limit equilibrium analysis on the stability of rock wedges with linear and nonlinear strength criteria

The stability of a wedge is an important research topic for rock slopes. A rock wedge is formed by two sets of intersecting discontinuous structural surfaces distributed in a rock mass and cutting the rock slope. Hence, its stability is related to the occurrence of structural surfaces. To evaluate the stability of rock wedges, the limit equilibrium (LE) methods, which do not fully consider the shear forces on the slip surface of both sides of the rock wedge, are usually adopted. As a result, the calculated rock wedge factor of safety (FOS) may be overestimated. Here, a spatial model of a rock wedge is established from the occurrence parameters of the structural planes and slope surfaces. Moreover, reasonable assumptions of internal forces on both sides of a micro-unit body are adopted with the micro-unit body in the rock wedge as the analysis object. Subsequently, the analytical formulas for the normal and shear forces of the rock wedge with full consideration of the shear forces on the slip surface of both sides are derived by applying mechanical equilibrium conditions . Furthermore, the solution of the normal stress on the slip surface of a rock wedge under the assumption of a linear distribution is given. Afterwards, the formulas for calculating the FOS of a rock wedge under the linear Mohr–Coulomb strength criterion and nonlinear generalised Hoek–Brown strength criterion are derived. By comparing and analysing some rock slope examples, the feasibility of the proposed method is verified, and it is shown that the conventional LE methods overestimate the stability of rock wedges. In addition, stability charts of rock wedges are proposed to illustrate the influence trend of dips and dip directions of discontinuous structural planes on the rock wedge FOS when the structural planes have certain discreteness and distribution ranges. • A rigorous LE model is established for analysing the stability of rock wedges. • The FOS of a rock wedge under the nonlinear GHB strength criterion is calculated. • Stability charts of rock wedges are proposed and plotted.

Control Performance, Stability Conditions, and Bifurcation Analysis of the Twelve-Pole Active Magnetic Bearings System

The active magnetic bearings system plays a vital role in high-speed rotors technology, where many research articles have discussed the nonlinear dynamics of different categories of this system such as the four-pole, six-pole, eight-pole, and sixteen-pole systems. Although the twelve-pole system has many advantages over the eight-pole one (such as a negligible cross-coupling effect, low power consumption, better suspension behaviors, and high dynamic stiffness), the twelve-pole system oscillatory behaviors have not been studied before. Therefore, this article is assigned to explore the effect of the magneto-electro-mechanical nonlinearities on the oscillatory motion of the twelve-pole system controlled via a proportional derivative controller for the first time. The normalized equations of motion that govern the system vibrations are established by means of classical mechanics. Then, the averaging equations are extracted utilizing the asymptotic analysis. The influence of all system parameters on the steady-state oscillation amplitudes is explored. Stability charts in a two-dimensional space are constructed. The stable margin of both the system and control parameters is determined. The obtained investigations reveal that proportional gain plays a dominant role in reshaping the dynamics and motion bifurcation of the twelve-pole systems. In addition, it is found that stability charts of the system can be controlled by simply utilizing both the proportional and derivative gains. Moreover, the numerical simulations showed that the twelve-poles system can exhibit both quasiperiodic and chaotic oscillations besides the periodic motion depending on the control parameters’ magnitude.

Dynamic instability of fiber composite cylindrical shell with metal liner subjected to internal pulse loading

Thin-walled cylindrical shells are commonly used structures in explosion containment vessels. When subjected to internal blast loading, these cylindrical shells are prone to dynamic buckling failure.In this work the dynamic stability of cylindrical fiber composite shells with metal liner subjected to uniform internal pressure pulse was investigated through both finite element simulations and theoretical modelling. In particular, pulse buckling of the inner metal liner and vibrational buckling of the outer fiber composite shell were observed and studied. Dynamic, Implicit analysis in ABAQUS Standard considering the initial geometric imperfections were conducted to study the mechanisms for the dynamic buckling of the inner metal liner and outer fiber composite shell. The simulated buckled shape using a simplified two-dimensional model could successfully reproduce the previous experimental results. The dynamic pulse buckling of the metal liner was found to occur during the plastic compression process. The effect of the buckling amplitude of the inner metal liner on the dynamic stability of the outer fiber composite shell was also revealed using numerical simulations. Instead of preventing the outer fiber composite shell from buckling instability, severely buckled metal liner with large buckling amplitude may even accelerate the growth of unstable mode of the fiber composite shell. The theoretical solution for the radial deformation of the metal liner was established by modelling the growth of the buckling mode as the amplification of the initial geometric or velocity imperfection. The most amplified mode and corresponding amplification were determined by the derived amplification function. The solutions for the radial deformation of the outer fiber composite shell were represented by a set of Mathieu differential equations. The dynamic instability of the fiber composite shells was determined from the Mathieu stability charts. The dependence of the dynamic instability of the bilayer composite shells on various parameters such as composite layup, thickness-to-radius ratio and impulse of pressure pulse loading was theoretically predicted and compared with the finite element simulation results.

Limit analysis for the seismic stability of three-dimensional rock slopes using the generalized Hoek-Brown criterion

The parameters that influence slope stability and their criteria of failure are fairly understood but over-conservative design approaches are often preferred, which can result in excessive overburden removal that may jeopardize profitability in the context of open pit mining. Numerical methods such as finite element and discrete element modelling are instrumental to identify specific zones of stability, but they remain approximate and do not pinpoint the critical factors that influence stability without extensive parametric studies. A large number of degrees of freedom and input parameters may make the outcome of numerical modelling insufficient compared to analytical solutions. Existing analytical approaches have not tackled the stability of slopes using non-linear plasticity criteria and three-dimensional failure mechanisms. This paper bridges this gap by using the yield design theory and the Hoek-Brown criterion. Moreover, the proposed model includes the effect of seismic forces, which are not always taken into account in slope stability analyses. The results are presented in the form of rigorous mathematical expressions and stability charts involving the loading conditions and the rock mass properties emanating from the plasticity criterion.

Modelling and stability analysis of a longitudinal wheel dynamics control loop with feedback delay

Vehicle safety is an important field of today's engineering practice and the scientific research as well. One of the basics of the modern vehicle system dynamics control systems is braking or driving torque control. Ensuring proper connection between the tyre and the road surface is essential for safety and good handling behaviour. Fundamental questions may arise about the details of the model, such as feedback delay, sampling effect, numerical differentiation in the control algorithm or the dynamical characteristics of the tyre model. In this article, a connected dynamical system is presented which contains a wheel model and a PID type traction controller. Steady state and dynamical brush type tyre models are presented and compared. Two different ways are described to model the feedback delay. The fully continuous time model contains constant time delay, while there is another case, where the sampling effect and the numerical differentiation methods of the control algorithm are presented. Finally, stability analysis of the delayed system is performed. The stability and performance characteristics and different bifurcations are presented using stability charts. The type of the tyre model has significant effect on the stability characteristics.

Assessing the Impact of Reactive Power Droop on Inverter Based Microgrid Stability

Droop control is the most common approach for controlling inverter-based micro-grids. The active power droop gain has always been considered as the main parameter for identifying the micro-grid stability margin. Increasing this margin improves the transient performance and provides robustness to the micro-grid for a wide range of operations. Previous work on droop control focused on the active power droop gain, which is required for accurate power sharing as well as for micro-grid stability assessment. This paper utilizes small-signal stability analysis to analyze the impact of the reactive power droop gain on micro-grid stability, which is ignored in previous work. Consequently, a micro-grid domain of stability chart is proposed and defined in the mp <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> -nq plane, which represents the zone within which the micro-grid will maintain stable operation. The proposed domain of stability chart is utilized to assess and compare the impact of the conventional and proportional derivative (PD) reactive power droop controller on the micro-grid stability margin. The results show that there exists a reactive power droop gain at which the stability margin is minimum. Furthermore, it has been shown, through the domain of stability chart, that the PD reactive power droop controller is capable and sufficient to significantly increase the micro-grid stability margin while maintaining equal load sharing. Further, the domain of stability chart can serve as a useful tool for defining the micro-grid droop gain operational boundaries and for assessing and comparing inverter-based micro-grid control schemes.

Generalized parametric resonance in a spin-1 Bose-Einstein condensate

We propose a generalized Mathieu equation (GME) which describes well the dynamics for two different models in spin-1 Bose-Einstein condensates. The stability chart of this GME differs significantly from that of Mathieu's equation and the unstable dynamics under this GME is called generalized parametric resonance. A typical region of $\epsilon \gtrsim 1$ and $\delta \approx 0.25$ can be used to distinguish these two equations. The GME we propose not only explains the experimental results of Hoang et al. [Nat. Commun. 7, 11233 (2016)] in nematic space with a small driving strength, but predicts the behavior in the regime of large driving strength. In addition, the model in spin space we propose, whose dynamics also obeys this GME, can be well-tuned such that it is easily implemented in experiments.