Equilibrium Equations Research Articles

Reduced form of statistical equilibrium equations and radiative transfer equations

Abstract An extension of the vector algebra for irreducible spherical tensor operators (ITOs) is proposed, which involves coupling two ITOs into a third one (V (kv)= T (kt)× U (ku)*), with the additional condition that one of the operators is a complex conjugate and therefore is not an ITO. The correctness condition is considered for this extension. The new coupling rule treats the density matrices ρ(Kl) and all tensors related to light polarization (J (Kr) or Τ (K)) equally.
This approach has been applied to rewrite the fundamental statistical equilibrium equations and radiation transfer equations relevant to astrophysical polarimetry. The revised equations are more concise and eliminate the need for the nj symbols. Instead, they utilize vector and scalar products of ITOs. All rate coefficients in the equations consist of two independent factors, the reduced matrix element from dipole moments and the product (vector or scalar) of the tensors ρ(Kl) and J (Kr) (or Τ (K)).

New analytical model for multi-layered composite plates with imperfect interfaces under thermomechanical loading

AbstractA new analytical model for thermoelastic responses of a multi-layered composite plate with imperfect interfaces is developed. The composite plate contains an arbitrary number of layers of dissimilar materials and is subjected to general mechanical loads (both distributed internally and applied on edges for each layer) and temperature changes, which can vary from layer to layer and along two in-plane directions. Each layer is regarded as a Kirchhoff plate, and each imperfect interface is described using a spring-layer interface model, which can capture discontinuities in the displacement and stress fields across the interface. Unlike existing models, the governing equations and boundary conditions are simultaneously derived for each layer by using a variational procedure based on the first and second laws of thermodynamics, which are then combined to obtain the global equilibrium equations and boundary conditions for the multi-layered composite plate. A general analytical solution is developed for a symmetrically loaded composite square plate with an arbitrary number of layers and imperfect interfaces by using a new approach that first determines the interfacial normal and shear stress components on one interface. Closed-form solutions for two- and three-layer composite square plates are obtained as examples by directly applying the general analytical solution. Numerical results for two-, three- and five-layer composite plates under different loading and boundary conditions predicted by the current model are provided, which compare well with those obtained from finite element simulations using COMSOL, thereby validating the newly developed analytical model.

Solvation free energy in governing equations for DNA hybridization, protein-ligand binding, and protein folding.

This work examines the thermodynamics of model biomolecular interactions using a governing equation that accounts for the participation of bulk water in the equilibria. In the first example, the binding affinities of two DNA duplexes, one of nine and one of 10 base pairs in length, are measured and characterized by isothermal titration calorimetry (ITC) as a function of concentration. The results indicate that the change in solvation free energy that accompanies duplex formation (ΔGS) is large and unfavorable. The duplex with the larger number of G:C pairings yields the largest change in solvation free energy, ΔGS = +460 kcal·mol-1per base pair at 25 °C. A van't Hoff analysis of the data is complicated by the varying degree of intramolecular base stacking within each DNA strand as a function of temperature. A modeling study reveals how the solvation free energy alters the output of a typical ITC experiment and leads to a good, though misleading, fit to the classical equilibrium equation. The same thermodynamic framework is applied to a model protein-ligand interaction, the binding of ribonuclease A with the nucleotide inhibitor 3'-UMP, and to a conformational equilibrium, the change in tertiary structure of α-lactalbumin in molar guanidinium chloride solutions. The ribonuclease study yields a value of ΔGS = +160 kcal·mol-1, whereas the folding equilibrium yields ΔGS ≈ 0, an apparent characteristic of hydrophobic interactions. These examples provide insight on the role of solvation energy in binding equilibria and suggest a pivot in the fundamental application of thermodynamics to solution chemistry.

Optimistic value models of saddle point equilibrium control problems for uncertain jump systems

The saddle point equilibrium control problem is of great importance in differential game theory. In this paper, we investigated the optimistic value model for the saddle point equilibrium control problem based on multi-dimensional uncertain differential equations with jumps. The equilibrium equation for the proposed model is presented. Then a linear quadratic game optimistic value model and a bang-bang game optimistic value model are discussed, respectively. Finally, some applications are analysed by the results obtained in this paper.

Design and research of face gear central limited slip differential and analysis of its influence on vehicle performance

In this paper, a new design scheme of face gear central limited-slip differential (F-LSD) is proposed, which aims to unveil the technical veil of central limited-slip differential and fill the research blank of face gear application in the field of central limited-slip differential. Through the analysis of structural principle and motion characteristics, the dynamic equilibrium equation and locking coefficient model of F-LSD are established. The Simulink simulation model based on F-LSD dynamics and the torque distribution model of the whole transmission system are established. A 7-DOF vehicle dynamics model was established, and the performance of F-LSD vehicle in four conditions of acceleration, climbing, chassis twist sine wave and double line change was studied by using simulation software. At the same time, four groups of four-wheel drive vehicles with different central differential speeds were set as the control test group. The simulation results show that F-LSD system can effectively improve the vehicle's dynamic performance and extrication performance, and will not reduce the vehicle's handling stability, which provides a reference for the practical application of the face gear central limited slip differential.

Variable angle tow-steered fibres based rotating composite beam with rotary oscillations – Parametric excitation/instability analysis

This study is concerned with the stability analysis for various vibrational motions of tow-steered variable angle fibres based composite rotating beam with time varying speed. The formulation combines a higher order theory introducing sine function along with finite element methodology. It satisfies the plane stress condition in width direction of beam. The displacement kinematics are generic in the sense it includes various possible vibrational motions like chord and flap wise motions, torsional and axial vibration behaviours. The equilibrium equations for the proposed problem evolved adopting virtual dynamic work are then transformed into the equations of Mathieu-Hill type considering time varying harmonic rotational speed. Using Bolotin’s procedure coupled with C1 continuity-based beam element, the parametric resonance zones along with boundary limits are numerically evaluated. The eigenvalue type of solution methodology applied for the detailed study is validated for accuracy and computational efficiency against the time domain dynamic response analysis. Based on in-depth analysis, dynamic instability characteristics in terms of resonance range and origin of instability of composite beam subjected to the chord and flap wise, torsional and axial motions are studied considering curvilinear fibre path angles, beam cross section, hub radius, thickness ratio and mean rotary speed.

Mechanism design and mechanical analysis of pipeline inspection robot

Purpose This study aims to address the issues of limited pipe diameter adaptability and low inspection efficiency of current pipeline inspection robots, a new type of pipeline inspection robot capable of adapting to various pipe diameters was designed. Design/methodology/approach The diameter-changing mechanism uses a multilink elastic telescopic structure consisting of telescopic rods, connecting rods and wheel frames, driven by a single motor with a helical drive scheme. A geometric model of the position relationships of the hinge points was established based on the two extreme positions of the diameter-changing mechanism. Findings A pipeline inspection robot was designed using a simple linkage agency, which significantly reduced the weight of the robot and enhanced its adaptive pipe diameter ability. The analysis determined that the robot could accommodate pipe diameters ranging from 332 mm to 438 mm. A static equilibrium equation was established for the robot in the hovering state, and the minimum pressing force of the wheels against the pipe wall was determined to be 36.68 N. After experimental testing, the robots could successfully pass a height of 15 mm, demonstrating the good obstacle capacity of the robot. Practical implications This paper explores and proposes a new type of multilink elastic telescopic variable diameter pipeline inspection robot, which has the characteristics of strong adaptability and flexible operation, which makes it more competitive in the field of pipeline inspection robots and has great potential market value. Originality/value The robot is characterized by the innovative design of a multilink elastic telescopic structure and the use of a single motor to drive the wheel for spiral motion. On the basis of reducing the weight of the robot, it has good pipeline adaptability, climbing ability and obstacle-crossing ability.

Dissolution Mechanisms and Surface Charge of Clay Mineral Nanoparticles: Insights from Kinetic Monte Carlo Simulations

The widespread use of clay minerals and clays in environmental engineering, industry, medicine, and cosmetics largely stems from their adsorption properties and surface charge, as well as their ability to react with water. The dissolution and growth of minerals as a function of pH are closely related to acid–base reactions at their surface sites and their surface charge. The vivid tapestry of different types of surface sites across different types of clay minerals generates difficulties in experimental studies of structure–property relationships. The aim of this paper is to demonstrate how a mesoscale stochastic kinetic Monte Carlo (kMC) approach altogether with atomistic acid-base models and empirical data can be used for understanding the mechanisms of dissolution and surface charge behavior of clay minerals. The surface charge is modeled based on equilibrium equations for de/protonated site populations, which are defined by the pH and site-specific acidity constants (pKas). Lowered activation energy barriers for these sites in de/protonated states introduce pH-dependent effects into the dissolution kinetics. The V-shaped curve observed in laboratory experiments is reproduced with the new kMC model. A generic rate law for clay mineral dissolution as a function of pH is derived from this study. Thus, the kMC approach can be used as a hypothesis-testing tool for the verification of acid–base models for clay and other minerals and their influence on the kinetics of mineral dissolution and growth.

Mechanical Neural Networks with Explicit and Robust Neurons.

Mechanical computing provides an information processing method to realize sensing-analyzing-actuation integrated mechanical intelligence and, when combined with neural networks, can be more efficient for data-rich cognitive tasks. The requirement of solving implicit and usually nonlinear equilibrium equations of motion in training mechanical neural networks makes computation challenging and costly. Here, an explicit mechanical neuron is developed of which the response can be directly determined without the need of solving equilibrium equations. A training method is proposed to ensure the robustness of the neuron, i.e., insensitivity to defects and perturbations. The explicitness and robustness of the neurons facilitate the assembly of various network structures. Two exemplified networks, a robust mechanical convolutional neural network and a mechanical recurrent neural network with long short-term memory capabilities for associative learning, are experimentally demonstrated. The introduction of the explicit and robust mechanical neuron streamlines the design of mechanical neural networks fulfilling robotic matter with a level of intelligence.

Optimization of high-speed angular contact ball bearing for aircraft gearbox utilizing an evolutionary multi-objective algorithm, NSGA-III

We develop a program for predicting bearing performance, including the maximum contact stress, bearing power loss, minimum lubricating film thickness, static safety factor, and dynamic life, and an optimal design for high-speed angular contact ball bearings in aircraft gearboxes using optimization algorithms. A program is developed to predict bearing performance by calculating the loads and displacements of individual balls based on the bearing’s kinematic and static equilibrium equations. Next, an optimization program is developed with non-dominated sorting genetic algorithm III (NSGA-III), wherein calculated bearing performance indicators serve as constraints or objective functions. The performance of individuals within the final Pareto front is evaluated using the inter-criteria correlation (CRITIC) and weighted product methods. The optimization performances of NSGA-III and NSGA-II are compared based on their hypervolume indicators. A solution of the kinematic and static equilibrium equations under the load cases and duty cycles of real aircraft gearboxes is obtained, enhancing the prediction accuracy. Furthermore, optimization, including bearing performance metrics rarely considered in previous studies, is performed. Optimal specifications superior to reference bearings employed in aircraft gearboxes are obtained, enhancing all three objective functions or specific objective functions. The optimization performance of NSGA-III is confirmed to surpass that of conventional NSGA-II.

Elastodynamic behaviors of steady moving straight dislocation within thin nano film

The elastodynamic dislocation behaviors are of great interest for understanding the performances of structural alloys under intense dynamic loading conditions. The formation, propagations, and interactions of dislocations (such as injected dislocation, accelerating dislocation, steady moving dislocation at high constant speed) are quite different from static dislocations. For steady-moving dislocation within the isotropic infinite medium, the effects of surface and interface on steady-moving dislocations within limited space are still known. In this paper, we investigate the elastodynamic image stress simulation of steady moving dislocation within film of limited thickness at constant speed using Eigenstrain theory, Lorentz transformation, and steady dynamic equilibrium equations. We propose an efficient solution method that involves complex Fourier series, transforming partial differential equations into ordinary differential equations, and ultimately into a set of algebraic equations in spectral space. The effects of dislocation speed and position near the free surface on the image stress of steady-moving climbing and gliding dislocations within the thin film are examined. The results show that relativistic effects are significant for certain dislocation configurations and stress components, whereas other stress components are less sensitive to relativistic effects near the transonic speed region.

A Markovian description of the multi-level source function and its application to the Lyman series in the Sun

Aims. We introduce a new method to calculate and interpret indirect transition rates populating atomic levels using Markov chain theory. Indirect transition rates are essential to evaluate interlocking in a multi-level source function, which quantifies all the processes that add and remove photons from a spectral line. A better understanding of the multi-level source function is central to interpret optically thick spectral line formation in stellar atmospheres, especially outside local thermodynamical equilibrium (LTE). Methods. We compute the level populations from a hydrogen model atom in statistical equilibrium, using the solar FALC model, a 1D static atmosphere. From the transition rates, we reconstruct the multi-level source function using our new method and compare it with existing methods to build the source function. We focus on the Lyman series lines and analyze the different contributions to the source functions and synthetic spectra. Results. Absorbing Markov chains can represent the level-ratio solution of the statistical equilibrium equation and can therefore be used to calculate the indirect transition rates between the upper and lower levels of an atomic transition. Our description of the multi-level source function allows a more physical interpretation of its individual terms, particularly a quantitative view of interlocking. For the Lyman lines in the FALC atmosphere, we find that interlocking becomes increasingly important with order in the series, with Ly-α showing very little, but Ly-β nearly 50% and Ly-γ about 60% contribution coming from interlocking. In some cases, this view seems opposed to the conventional wisdom that these lines are mostly scattering, and we discuss the reasons why. Conclusions. Our formalism to describe the multi-level source function is general and can provide more physical insight into the processes that set the line source function in a multi-level atom. The effects of interlocking for lines formed in the solar chromosphere can be more important than previously thought, and our method provides the basis for further exploration.

Research on optimization of the vehicle handling stability considering flexibility of the front subframe

The elastic deformation of the front subframe during vehicle traveling affects the vehicle handling stability. The static equilibrium equations of the suspension and subframe system and the kinematics equations of the three-degree-of-freedom vehicle are established to illustrate the principle of the influence of the flexible front subframe on the handling stability. The front subframe is flexibly processed to establish a rigid-flexible coupled vehicle model. Simulations of the suspension kinematic and compliance alongside sine swept steering simulation of the vehicle are performed. The front subframe bushing stiffness, which has a greater impact on the handling stability, is selected as a design variable through parametric test design, and the NSGA-II algorithm is used for multi-objective optimization with the vehicle handling stability evaluation index as the optimization target. The simulation results show that the flexible front subframe will make the bushing force change to affect the suspension kinematic and compliance characteristics and then affect the vehicle handling stability, the gain of vehicle yaw rate and the delay time of lateral acceleration decrease, while the gain of roll angle increases. The optimization results show that the optimized vehicle exhibits improvements in the yaw rate gain, delay time of lateral acceleration, and roll angle gain.

First Failure Load of Rectangular Laminated and Sandwich Plates Using Isogeometric Analysis

This paper investigates the first failure load of simply supported and clamped laminate and sandwich plates loaded by a distributed normal traction on the top surface. A third-order shear and normal deformable plate theory, the isogeometric basis functions, and five failure criteria, namely, the maximum stress, the Tsai–Wu, the Tsai–Hill, the Hoffman, and the Hashin for laminated plates, and only the Tsai–Hill criterion for sandwich plates are used in this study. Of these, Hashin’s criteria distinguish between the fiber and the matrix failure. The in-plane stresses are found from the plate displacements and Hooke’s law, and the transverse stresses are recovered by integrating the equilibrium equations through the thickness. Effects of the plate aspect ratio, the fiber angle, the face sheet materials, and the core materials on the first failure load are identified. The computed results are found to agree well with those reported in the literature. Whereas the five failure criteria for laminated plates predict nearly the same value of the first failure load, they do not provide the same location of the failure initiation.

Spatio-temporal physics-informed neural networks to solve boundary value problems for classical and gradient-enhanced continua

Recent advances have prominently highlighted physics informed neural networks (PINNs) as an efficient methodology for solving partial differential equations (PDEs). The present paper proposes a proof of concept exploring the use of PINNs as an alternative to finite element (FE) solvers in both classical and gradient-enhanced solid mechanics. To this end, spatio-temporal PINNs are designed to represent continuous solutions of boundary value problems within spatio-temporal space. These PINNs directly incorporate the equilibrium and constitutive equations in their differential and rate forms, bypassing the requirement for incremental implementation. This simplifies application of PINNs to solve complex mechanical problems, particularly those involved in the context of gradient-enhanced continua. Moreover, traditional meshing is no longer required as it is replaced by a point cloud, making it possible to overcome meshing drawbacks. The results of this investigation prove the effectiveness of the proposed methodology, especially with regards to non-monotonic loading conditions and irreversible plastic deformation. Compared to classical FE approaches, the proposed spatio-temporal PINNs are more readily applied to complex problems, which are tackled in their raw form. This is especially true for gradient-enhanced continuum problems, where there is no need to introduce additional degrees of freedom as in classical FE approaches. However, PINNs training generally requires more computation time, a challenge that can be mitigated by employing the concept of transfer learning as shown in this paper. This concept, which is very useful when performing parametric studies, involves applying knowledge grained from solving one problem to another different but related one. The use of PINNs as mechanical solvers is shown to be highly promising in the forthcoming era, where advancements in GPU technology can further enhance their performance in terms of computation time.

Singularity analysis for V‐notches in a piezoelectric multimaterial and functionally graded piezoelectric materials

AbstractThis study develops a novel singularity analysis method that addresses the limitations posed by piezoelectric material quantity and the variation patterns of property functions in functionally graded piezoelectric materials (FGPMs). Given that piezoelectric composite materials consist of multiple materials, the material properties of FGPMs vary with respect to angle. By introducing the asymptotic assumption that governs the physical fields close to the notch vertex into the static equilibrium equations, a comprehensive set of characteristic equations is formulated. All singularity orders and corresponding characteristic angle functions of the notch are determined by numerically solving the established characteristic equations. The effects of the notch opening angle, piezoelectric material polarization direction, and boundary conditions on the electromechanical field singularity at the notch are assessed. The judicious selection of material variation patterns can alleviate notch singularity in FGPMs.

Numerical simulation of cable sheath damage detection based on torsional mode guided wave

In view of the cracking, sag, and damage of sheath caused by the load effect and external force impact of power cable, the echo parameters of cable sheath damage detection based on the characteristics of torsional guided wave propagation are studied in this work. According to the Navier displacement equilibrium equation, the dispersion curve of a magnetostrictive guided wave of the cable sheath was solved, and the T(0,1) mode with a group velocity of 1198.8 m/s and no dispersion was selected. Furthermore, while considering the excitation frequency, loss rate, and direction of the damaged section, the displacement field and the echo characteristic parameters of guided wave in the cable sheath were solved. Moreover, by analyzing the time-domain signals of damaged section echo, the cubic fitting function for the loss rate of the damaged section and the damaged section echo coefficient were obtained, which can effectively characterize the quantitative relationship between the damaged location, size, and guided wave echo of the cable sheath.

Equilibrium Solution of Two – Dimensional Non-Homogeneous Equations in the Theory of Elastic Mixtures

Abstract: The problem of plane elasticity for a doubly connected body with inner and outer boundaries in a regular polygonal form with common centre and parallel sides has been studied. The sides of the polygon were exposed to external forces. The nature of the force term was determined by application of complex variable theory. Kolosov’s method of solution was applied to obtain the biharmonic equation of the forcing term. The forces on the particle were studied under 2-dimensions from which the compatibility and equilibrium equations were derived. The compatibility and equilibrium equations were used to derive the force – stress relations. The results shows that there is a significant relationship between the angle of the force term on the plane of the particle and the stress state of the particle, which is in conformity with existing experimental results.

Morphological analysis of a novel scissor-type deployable cable dome structure

A novel deployable scissor-type strut cable dome structure was proposed, which can improve the lateral stiffness of the traditional cable structure and achieve the characteristics of fast unfolding and closing during construction. However, the geometric stability and optimal initial prestressing theory of traditional cable dome structures cannot be applied to the new structure. Based on the equilibrium matrix theory, the geometric stability analysis was conducted and the number of self-stress modes, mechanism displacement modes, and feasible prestress modes was calculated. Then, the formula for the initial prestress distribution of the structure was derived based on the node equilibrium equation. Furthermore, the geometric stability and initial prestress distribution of the structure with different rise-to-span ratios, number of circumferential struts, and hoop cables were studied. Finally, the dynamic properties of the structure are explored by finite element simulation. The analysis results show that the structure has excellent structural stability. The rise-to-span ratios, the number of circumferential struts, and hoop cables are the important factors affecting the geometric stability and the initial prestress distribution. The rise-to-span ratios have no effect on the number of structural modes, while the number of circumferential struts and cables will cause significant changes in the number of self-stress modes. The internal force of the structural members decreases with the rise-to-span ratio and the number of circumferential struts increases, and the number of circumferential struts has the greatest influence on its internal force. The structural natural frequency exhibits a dense distribution, which indicates that the structure has good stiffness. In summary, the research has provided an idea for the application of the deployable scissor-type strut cable dome structure as a new structural system in the field of structural engineering.

A new solution to force analysis including Coulomb friction in mechanism joints

The objective of this paper is to propose a novel approximate solution for determining the reactions of joints in mechanical systems, which involves the influence of Coulomb friction. It is widely acknowledged that if Coulomb friction is used in determining an exact solution to the equilibrium equations for a mechanism, then it will involve nonlinear systems of equations. With the abundance of computer tools now available, tasks of this kind, especially numerical computations, are not particularly challenging. However, new analytically tractable approximate methods are still valuable as a straightforward way of solving the problem. Several studies have been carried out in this field to find a simple solution. In this paper, a new approach based on friction circle concept and Babylonian algorithm is developed for various mechanisms, which is exceptionally well-suited for calculating and streamlining the solution process for mechanical systems by eliminating the necessity for iterative steps at each stage of the force analysis.