Static Equilibrium Conditions Research Articles

Molten and glassy tellurium(IV) oxosulfato complexes in the TeO2 –K2S2O7 system studied by Raman spectroscopy: Stoichiometry, vibrational properties and molecular structure

Tellurium(IV) oxide dissolves in molten potassium pyrosulfate following a reaction that leads to TeIV oxosulfato complexes. High temperature Raman spectroscopy is used under static equilibrium conditions at temperatures of 550 – 740oC for inferring the structural properties of the complex(es) formed in TeO2-K2S2O7 molten mixtures (XTeO20 = 0 – 0.70) under O2 atmosphere. The composition effects on the Raman spectra together with the outcome of a quantitative exploitation of the relative Raman band intensities point to a 3:1 TeO2:K2S2O7 stoichiometry, which combined with the observed vibrational properties is suggestive of a (Te=O)3O2(SO4)22- structural formula for the simplest building unit formed. A 3-fold coordination around Te becomes evident comprising an oxide ligand and two oxygen atoms bridging to adjacent Te or S atoms. A gradual association of building units takes place with increasing XTeO20. Cooling results in formation of stable glasses with a concomitant change in the coordination number of TeIV from 3 to 4 by means of a structural mechanism suggestively involving a TeO2/2(=O)→ TeO4/2 transformation. Totally ten bands are attributed to the TeIV oxosulfato complex; the most characteristic ones pertain to the 1040 cm-1 band assigned to symmetric stretching unidentate sulphate and to the 850 cm-1 band assigned to Te=O terminal stretching, the latter band being attenuated on going from the molten to the glassy state due to the TeO2/2(=O)→ TeO4/2 interconversion.

An alternative method for computing hydrostatic performances of a floating body with arbitrary geometrical configurations

In the fields of naval architecture and ocean engineering applications, floating platforms or multi-hull vessels with very different geometrical appearance have been designed to meet their mission requirements. Many objects were built by composing a number of sub-objects in an arbitrary way because of the high flexibility available for the construction work. Because the composition of the sub-objects normally are not arranged along a certain direction in a fixed sequence, it could be sometimes troublesome for computing the hydrostatic data of such objects. In order to facilitate this computation more easily and flexibly, a method has been developed in this paper, which was derived from the simple principles of an exact pressure integration over triangles for getting the total buoyancy force vector and the static equilibrium condition between the buoyancy force and the weight of the floating object. The triangles thereby were generated by triangulation of the surfaces representing a whole floating object. Finally, applications on a high-speed trimaran hull and a floating Kuroshio current turbine were conducted for demonstrating the merit of this method. The placement of water compartments and free surface effects were further analysed to evaluate the changing ballast conditions for hydrostatic and transitional stabilities.

A nonlinear multi-point meshing model of spur gears for determining the face load factor

The face load factor, which takes into account the influence of the non-uniform distribution of the load over the face width on the resulting stresses, is a significant coefficient of the load capacity calculation of spur gears. In this study, a practical nonlinear multi-point meshing model of spur gears is proposed for the determination of the face load factor. Nonlinear gear meshing conditions over the face width can be simulated precisely and efficiently using the proposed nonlinear multi-point meshing model, which is composed of multiple nonlinear springs along the line of action and rigid bars connected with shaft beam element models. The static equilibrium condition of the entire spur gear system is obtained through an iterative calculation process, and both the load distribution and face load factor are obtained. The effectiveness and advantages of the proposed model are demonstrated through comparisons with different types of contact analysis models presented by previous studies.

Numerical simulation of drive axles considering the nonlinear couplings between gears and bearings

The drive axle is the core of the powertrain in automotive vehicles, and numerical simulation is a common and effective approach for designing and analysing drive axles. This article develops a new numerical simulation method for drive axles considering the nonlinear couplings between gears and bearings. Two types of models are proposed and integrated: a reduced model with nonlinear bearing elements and equivalent gear elements, which is efficient for nonlinear bearing simulation, and a gear contact analysis model with equivalent bearing stiffnesses, which can accurately reflect the gear mesh conditions and system deflection. The equivalent gear mesh parameters used in the reduced model are derived from the contact analysis model, and the linear equivalent bearing stiffness matrices used in the contact analysis model are derived from the reduced model. Therefore, the static equilibrium condition of the entire drive axle system is obtained through an iterative calculation process involving the two models under the same load condition. The good correlation between the numerical and experimental results of the gear-loaded contact pattern indicates that the proposed method is reliable.

Double Scaling in the Relaxation Time in the β-Fermi-Pasta-Ulam-Tsingou Model.

We consider the original β-Fermi-Pasta-Ulam-Tsingou system; numerical simulations and theoretical arguments suggest that, for a finite number of masses, a statistical equilibrium state is reached independently of the initial energy of the system. Using ensemble averages over initial conditions characterized by different Fourier random phases, we numerically estimate the time scale of equipartition and we find that for very small nonlinearity it matches the prediction based on exact wave-wave resonant interaction theory. We derive a simple formula for the nonlinear frequency broadening and show that when the phenomenon of overlap of frequencies takes place, a different scaling for the thermalization time scale is observed. Our result supports the idea that the Chirikov overlap criterion identifies a transition region between two different relaxation time scalings.

Dynamic equilibrium planform of embayed beaches: Part 2. Design procedure and engineering applications

The design of equilibrium beach systems employing beach nourishment techniques in conjunction with coastal support structures has received increased interest from coastal engineers in recent years. One such solution to remedy coastal erosion problems is to design an equilibrium pocket beach in order to stabilize the shoreline. Such coastal landforms may exist in nature in static or dynamic equilibrium states. The concept of the Static Equilibrium Planform (SEP) has received much more attention from coastal engineers than that paid to the Dynamic Equilibrium Planform (DEP). This paper presents a design procedure for embayed beaches in dynamic equilibrium which have a specific net littoral drift rate. The proposed methodology employs the net sediment transport rate passing through the bay, together with the time series of the wave climate impinging on the beach, in order to compute the angle (γd). This angle represents the difference in degrees between the direction of the mean wave energy flux at the diffraction point (θEF) and the orientation of the dynamic equilibrium beach (θDB), to be utilized in a newly derived DEP formula to predict the shoreline shape in the long term. The procedure was applied and validated using prototype bays in dynamic equilibrium with different net sediment transport rates along the Brazilian coast, producing good results. Furthermore, it was employed as a tool to investigate the influence of changes in the net annual littoral drift rate on the (DEP) shape in a case study along the Spanish coast. Taken together, these confirm its utility as a valuable tool for coastal management and engineering practice.

Research on limit equilibrium method of three-dimensional slope stability

The limit state design method for slope stability has many assumptions which are approximate solution and analyzing the safety state of soil. Thus, this method is not a strict calculation method satisfying all the static equilibrium conditions of a three-dimensional sliding body. Based on the stress analysis of a point on a space curved surface, the fundamental equation for slope stability analysis without any assumption was established. Actual engineering cases confirm that the results obtained using the method reported in the paper are reasonable and significant for broad applications.

Dynamic equilibrium planform of embayed beaches: Part 1. A new model and its verification

Equilibrium beach formulations are useful tools for diagnosing and managing coastal engineering problems, providing solutions for beach erosion problems. Headland Bay Beaches (HBBs) can be used as equilibrium coastal systems for stabilizing coastlines and mitigating erosion problems. These embayed beaches may exist in a state of static or dynamic equilibrium. Throughout the literature, several equations can be found for obtaining the Static Equilibrium Planform (SEP) of Headland Bay Beaches (HBBs) with almost negligible net littoral drift rates. However, the formulations used to define the Dynamic Equilibrium Planform (DEP) of embayed beaches with specific net sediment transport rates are scarce, and based on a limited number of studies. This paper proposes a new derived formula for obtaining the planform shape of HBBs in dynamic equilibrium conditions. The formula represents a general form of the Parabolic Bay Shape Equation (PBSE) with modified coefficients as a function of both the wave obliquity (β) and the net littoral drift rate passing through the bay (Q). The angular difference (γd) between the direction of the mean wave energy flux at the diffraction point of the headland and the beach orientation down-coast is also incorporated in the proposed formula. The model was verified against natural HBBs in dynamic equilibrium with different net littoral drift rates along the Brazilian coast, producing good results.

Nonquilibrium and Equilibrium Stationary States of Zwitterionic Surfactant Dynamic Adsorption on Limestone Cores at Oil-Reservoir Conditions

Dynamic adsorption of a zwitterionic-type surfactant cocamidopropyl hydroxysultaine on limestone cores at oil reservoir conditions, i.e., under 2000 psi pressure, 130 °C temperature, and in the presence of seawater with 30 800 ppm NaCl, has been studied through core-flood displacement tests and in the absence of oil, at an injection rate of 120 cm3·h–1 up to 60 pore volumes (PV). The amount of adsorbed surfactant has been determined from mass balance by calculating its concentration through high performance liquid chromatography (HPLC). The normalized surfactant concentrations (NSC) have been used in terms of the PV, and then they are obtained the corresponding dynamic adsorption curves. Dynamic adsorption isotherm curves of the zwitterionic surfactant diluted in seawater showed a steep rise between 1 and 3 pore volumes (PV), and the saturation conditions were kept up to 60 PV. An average surfactant adsorption of 0.1138 ± 0.0126 mg·g–1 was determined. Quantum theoretical studies show that the electronic energy gap of (104) the limestone is electrically neutral, preferably acquiring a negative charge tendency, these results imply that the calcite surface will preferentially interact with positive ions, so that the cations present in the solution interacting in the limestone, resulting in a positive charge in the rock that implies that the surfactant cocamidopropyl hydroxysultaine interacts in the rock by the anionic group.

Semi-analytical solutions for vertically and laterally loaded piles in multilayered soil deposits

In geotechnical practice, it is of considerable importance to assess the behavior of vertical–lateral coupled loading piles in multilayered soil deposits. This study deals with a semi-analytical formulation for the performance of a pile suffering from combined vertical and lateral loads. The emphasis is on quantifying the mobilization of the subgrade reaction provided by the layered soil stratums. In the proposed method, subgrade reactions, correlated with both the accumulative axial load transfer and side resistances depending upon the pile–soil interaction, are abstracted as a series of nonlinear springs in both vertical and lateral directions. On account of this, an alternative transfer matrix method is applied to characterize the pile reaction along the depth under the identified boundary conditions atop the pile; meanwhile, the condition of static equilibrium, between two specific pile segments located at the border of soil layers, is also essential. On this basis, validation of the solution is conducted by comparing with observations from experiments and predictions obtained from other existing methods. In addition, the influence of properties in shallow soil layer and the vertical load on the lateral response of the pile is also discussed. The results indicate a reduction in the lateral displacement and the maximum bending moment within the pile with the increase in the shallow soil stiffness, but a growth with the increase in vertical load due to the “P-Δ effect.”

Nonlinear Model Predictive Control of Position and Attitude in a Hexacopter with Three Failed Rotors

We applied nonlinear model predictive control (NMPC) to an unmanned aerial vehicle (a hexacopter) with three failed rotors out of the six rotors available. We considered three cases of failure: a case in which three rotors in alternate positions stop, a case in which two adjacent rotors and a separate rotor stop, and a case in which three adjacent rotors stop. Simulation results show that, by applying NMPC, the position and attitude of the hexacopter can be controlled from an initial state that significantly deviates from a state of equilibrium in consideration of constraints. Moreover, we simulated a case in which there was no static equilibrium state and succeeded in controlling the hexacopter’s position.

On a method for determination of Poisson’s ratio and Young modulus of a material

On the base of modernized NanoTest 600 Platform 3 indentation method is proposed to determine elastic parameters – Poisson’s ratio and Young’s modulus – of a material while loading in an elastic region. The experiment is based on procedure: lateral surface of indenter tip with the shape of parabolic cylinder penetrates into the specimen. NanoTest 600 was equipped by additional optics, backlight and device for spatial orientation of the specimen. This modernization allows to control the process of the indenter penetration both along its length and from the edges, so that one can observe and measure the width of the contact area and control the depth of the indentation area in a sample material. Mathematical modeling of the indentation process was conducted within the framework of plane theory of elasticity. This required solution of the contact problem on indentation of a rigid indenter with a parabolic shape into an elastic strip coupled with a non-deformable substrate. The fulfilment of condition of zeroing the contact stresses at the edges of the indenter with a known width of the contact area allows to determine the Poisson’s ratio, and condition of static equilibrium of the contact problem helps to find Young’s modulus of a strip material.

Recollimation shocks and radiative losses in extragalactic relativistic jets

We present the results of state-of-the-art simulations of recollimation shocks induced by the interaction of a relativistic jet with an external medium, including the effect of radiative losses of the shocked gas. Our simulations confirm that – as suggested by earlier semi-analytical models – the post-shock pressure loss induced by radiative losses may lead to a stationary equilibrium state characterized by a very strong focusing of the flow, with the formation of quite narrow nozzles, with cross-sectional radii as small as 10-3 times the length scale of the jet. We also study the time-dependent evolution of the jet structure induced by a density perturbation injected at the flow base. The set-up and the results of the simulations are particularly relevant for the interpretation of the observed rapid variability of the γ-ray emission associated to flat spectrum radio quasars. In particular, the combined effects of jet focusing and Doppler beaming on the observed radiation make it possible to explain the sub-hour flaring events such as that observed in the flat specrum radio quasar PKS 1222+216 by MAGIC.

Experimental and theoretical studies of ordinary Portland cement composites contains nano LSCO perovskite with Fokker-Planck and chemical reaction equations

In this study, LaSr0.5Co0.5O3 perovskite (LSCO), with 1, 2 and 4 wt% of LSCO nanoparticle inserted into the OPC (Ordinary Portland Cement) matrix via the sol-gel route. The sample nanocrystallite characteristics, surface topography, flexural strength, the influence of crystallite’s size, chemical interaction, and the correlation between carrier diffusion; surface roughness were studied using some related techniques and Master and Fokker-Planck approaches. The obtained results from LSCO/cement discrete cell structure show that the sample cement mortar containing 2 wt% LSCO nanocrystallites has a more stable mechanical structure and higher flexural strength due to the its permanent electric dipole, more occupation of the trap sites into the cement, and diffusion of particles (crystallites) through the cell boundary and inside the cells.For this purpose, microscopic parameters such as the trap and escaping rate in the LSCO+ cement discrete cells have been introduced and discussed with considering Master nonlinear Fokker–Planck and chemical reaction equations based on non-fluctuations of the LSCO particle position explored in the cells (which strongly correlated with random sequence). The results indicate an inhomogeneous nano structures characteristic, which is beyond the equilibrium stationary state with Gaussian fluctuations for the particle position.

Instabilities in the axisymmetric magnetoelastic deformation of a cylindrical membrane

We study the inflation of a weakly magnetizable isotropic incompressible cylindrical membrane and the effects of an external magnetic field generated by a current carrying wire placed along the axis of cylinder. A variational formulation based on magnetization is used and the computational results obtained by using four elastic constitutive models (neo-Hookean, Mooney–Rivlin, Ogden, and Arruda–Boyce) are studied and compared. Cylinders of various aspect ratios are studied in each case. Our study shows that the external magnetic field alters the elastic limit point, does not lead to equilibrium solutions below certain value of internal pressure, and can give rise to multiple equilibrium states for a given value of pressure. Presence of magnetic limit point, a phenomenon recently reported in the literature is reconfirmed. Magnetic limit point is a state where a further strengthening of the applied magnetic field at a given pressure does not yield any static equilibrium state. In this case it is detected when the cylindrical membrane deflates into the volume enclosed by itself. We also observe a quadratic relation between the defined magnetic energy parameter and the internal pressure at the magnetic limit point. Relaxed form of the strain energy density is used to account for wrinkling in this case of inward inflation. A finite difference method coupled with an arc-length technique is used for the computations and the stability of the solution is determined from the second variation.

Thermodynamic affinity in constrained free-energy systems

Affinity is the generic measure of the deviation of a state from stable equilibrium. Affinity, as introduced by de Donder, is a thermodynamic state property defined in terms of p, T, and system composition during the course of a chemical change. When incorporating reaction kinetic constraints to minimization of Gibbs energy of a multiphase system, affinity can be followed in terms of the extents of the constrained reactions. This property then becomes calculated in terms of the constraint potentials received as additional Lagrange multipliers in the minimization routine. Thus, received affinities are consistent with the respective values calculated from the chemical potentials of the reactants and products of the constrained reactions and their limiting behaviour corresponds to that defined for both stationary and stable equilibrium states. The intermediate affinities can be used in the respective reaction rate calculations, or as input parameters, to define the local chemical equilibrium set by known reaction kinetic constraints. Thus, they become a useful concept in modelling reactive processes.

Equilibration of energy in slow–fast systems

Ergodicity is a fundamental requirement for a dynamical system to reach a state of statistical equilibrium. However, in systems with several characteristic timescales, the ergodicity of the fast subsystem impedes the equilibration of the whole system because of the presence of an adiabatic invariant. In this paper, we show that violation of ergodicity in the fast dynamics can drive the whole system to equilibrium. To show this principle, we investigate the dynamics of springy billiards, which are mechanical systems composed of a small particle bouncing elastically in a bounded domain, where one of the boundary walls has finite mass and is attached to a linear spring. Numerical simulations show that the springy billiard systems approach equilibrium at an exponential rate. However, in the limit of vanishing particle-to-wall mass ratio, the equilibration rates remain strictly positive only when the fast particle dynamics reveal two or more ergodic components for a range of wall positions. For this case, we show that the slow dynamics of the moving wall can be modeled by a random process. Numerical simulations of the corresponding springy billiards and their random models show equilibration with similar positive rates.

Limit equilibrium analysis for stability of soil nailed slope and optimum design of soil nailing parameters

Reinforcement of slopes using soil nailing can effectively improve slope stability, and it has been widely used in upgrading cut slopes. Based on the assumptions of stresses on the slip surface, a new method for analyzing the stability of a slope reinforced with soil nails was established in the limit equilibrium theory framework, by considering that slope sliding occurs owing to shear failure of the slip surface, which subjects to Mohr–Coulomb (M–C) strength criterion. Meanwhile, in order to easily analyze the stability of a soil nailed slope in actual engineering and facilitate optimum design of parameters for soil nailing, factor of safety (FOS) contour curve charts were drawn on the basis of the established linear proportional relationship between the spacing of soil nails and slope height, and the length of soil nails and slope height. Then, by analyzing and verifying the results obtained from classic examples, some conclusions can be got as follows: 1) The results obtained from the current method are close to those obtained from the traditional limit equilibrium methods, and the current method can provide a strict solution for the slope FOS as it satisfies all the static equilibrium conditions of a sliding body, thus confirming the feasibility of the current method; 2) The slope FOS contour curve charts can be used not only to reliably analyze the stability of a soil nailed slope, but also to design optimally the parameters of soil nailing for the slope with a certain safety requirement.

The equilibrium alluvial river under variable flow and its channel‐forming discharge

AbstractWhen the water discharge, sediment supply, and base level vary around stable values, an alluvial river evolves toward a mean equilibrium or graded state with small fluctuations around this mean state (i.e., a dynamic or statistical equilibrium state). Here we present analytical relations describing the mean equilibrium geometry of an alluvial river under variable flow by linking channel slope, width, and bed surface texture. The solution holds in river normal flow zones (or outside both the hydrograph boundary layer and the backwater zone) and accounts for grain size selective transport and particle abrasion. We consider the variable flow rate as a series of continuously changing yet steady water discharges (here termed an alternating steady discharge). The analysis also provides a solution to the channel‐forming water discharge, which is here defined as the steady water discharge that, given the mean sediment supply, provides the same equilibrium channel slope as the natural long‐term hydrograph. The channel‐forming water discharge for the gravel load is larger than the one associated with the sand load. The analysis illustrates how the load is distributed over the range of water discharge in the river normal flow zone, which we term the “normal flow load distribution”. The fact that the distribution of the (imposed) sediment supply spatially adapts to this normal flow load distribution is the origin of the hydrograph boundary layer. The results quantify the findings by Wolman and Miller (1960) regarding the relevance of both magnitude and frequency of the flow rate with respect to channel geometry.

Theoretical calculation and experimental verification of the elastic angle of a cycloid ball planetary transmission based on the axial pretightening force

Mechanical models of deceleration meshing pair and equal-speed meshing pair are derived using the rigidity coefficient and static equilibrium conditions of plate (fixed plate, output shaft, or plan...