Analytical Mechanics Research Articles

Understanding How In-Visualization Provenance Can Support Trade-off Analysis.

In domains such as agronomy or manufacturing, experts need to consider trade-offs when making decisions that involve several, often competing, objectives. Such analysis is complex and may be conducted over long periods of time, making it hard to revisit. In this paper, we consider the use of analytic provenance mechanisms to aid experts recall and keep track of trade-off analysis. We implemented VisProm, a web-based trade-off analysis system, that incorporates in-visualization provenance views, designed to help experts keep track of trade-offs and their objectives. We used VisProm as a technology probe to understand user needs and explore the potential role of provenance in this context. Through observation sessions with three groups of experts analyzing their own data, we make the following contributions. We first, identify eight high-level tasks that experts engaged in during trade-off analysis, such as locating and characterizing interest zones in the trade-off space, and show how these tasks can be supported by provenance visualization. Second, we refine findings from previous work on provenance purposes such as recall and reproduce, by identifying specific objects of these purposes related to trade-off analysis, such as interest zones, and exploration structure (e.g., exploration of alternatives and branches). Third, we discuss insights on how the identified provenance objects and our designs support these trade-off analysis tasks, both when revisiting past analysis and while actively exploring. And finally, we identify new opportunities for provenance-driven trade-off analysis, for example related to monitoring the coverage of the trade-off space, and tracking alternative trade-off scenarios.

Расчет системы амортизации шасси пассажирского самолета при ударном нагружении

The paper considers the shock absorption system of an advanced domestic mainline passenger aircraft, which includes oleo-pneumatic shock absorbers and wheels equipped with the pneumatic tires. Nonlinear mathematical models of single-active and two-active oleo-pneumatic shock absorbers of telescopic landing gear supports are proposed. The models take into account the process of polytropic gas compression, the forces of hydraulic resistance to the working fluid flow and the forces of dry friction arising in the system moving parts. To describe the vertical compression reaction of the pneumatic tires, the V.L. Biderman structural model was introduced, which parameters were determined by test results using the least squares method. Motion equations of the landing gear elements were obtained by the analytical mechanics methods involving the Ostrogradsky --- Hamilton variational principle of least action. Constraints imposed on the system were introduced using the penalty functions method. The problem statement of carrying out virtual computational experiments on the landing gear dvops was considered. In a wide range of the impact energies, models of shock absorbers of a passenger aircraft landing gear were validated, for which time realizations of the system state vector were determined, and areas of hysteresis loops of the shock absorber load characteristics were evaluated by the numerical integration. Satisfactory compliance of the results of simulated characteristics of the natural objects was shown. The obtained results could be used to improve the absorption quality criteria by numerical optimization methods, calculate the landing impacts and analyze the aircraft oscillations when moving on the runway

Mathematical Modelling and Designing of a Universal Conical Spreader for Granular Material

Abstract A universal conical spreader for uniform distribution of granular fertilizers and seeds of grain crops, grasses is proposed. The developed universal conical spreader is designed for subsurface application of above-mentioned granular materials using seed drills and can be used as a passive distributor in a wide variety of applications. The methods of theoretical mechanics and descriptive geometry, particularly the method of replacement of projection plane have been applied to arrive at the design of the cone – the shape of conical spreader. A parametric equation was derived for the truncated part of the cone, which is an ellipse. The unique design of the cone-shaped distributor has a truncating form in the direction transverse to the movement of the seeder. In order to determine the interval of variability of spreader parameters more precisely, an analysis was carried out using a Mathcad-15 software. The results of theoretical research were confirmed in the laboratory and field conditions. The width of seed distribution, uniformity, and stability of distribution were studied to arrive at the most effective cone distributor design. The smallest values of non-uniformity and instability of material distribution of experimental coulter are reached at the values of factors D = 117–119 mm, h = 17–18 mm, and H = 32–35 mm.

ДОСЛІДЖЕННЯ ПРОЦЕСУ СИЛОВОГО (БОРТОВОГО) ПОВОРОТУ МОСТОВИХ МАШИН

One of the ways to improve the maneuverability of wheeled axle machines when they move along the tracks of a constant technological track is the use of onboard turning. This simplifies the layout, increases the useful volume of the machine due to the release of niches in the machine body, necessary for placing the steering wheels when turning. However, the question of the efficiency of onboard turning of axle machines by the criterion of power input remains insufficiently studied. The aim of the research is to substantiate the criterion of the power required to implement on-board turns of an overhead machine. Theoretical research, synthesis of design schemes, parameters and modes of operation of bridge machines have been carried out by simulating on a personal computer the conditions of their functioning. The basics of theoretical mechanics and tractor theory were the basis of the research methods. As a result of the research, it has been established that power (onboard) turning is widely used for wheeled machines and has a prospect of application on bridge machines used in the track farming system. Approbation of the developed methodology of determining power inputs for power (onboard) turning of wheeled axle machines has shown that under the accepted conditions of turning the power required for turning of the axle machine is 28% of the engine power at a speed of 5 km⸱h–1. As the gauge of the bridge machine increases, the total power required for the power (onboard) turn increases exponentially.

Tensorial-formulation-based method for modeling of flight dynamics with generalized coordinates

For the sake of minimizing fuel consumption, a rocket typically keeps flying in or near a plane specified by the Earth's center, launch pad, and orbit-insertion point, where the plane is called the Reference Flight Plane (RFP) here. To facilitate trajectory analysis, a series of generalized position and velocity coordinates are defined based on the RFP. To develop the Ordinary Differential Equations (ODEs) of the generalized coordinates, a novel dynamics modeling method based on tensor theory is proposed. As the first step of the new method, the rotating Earth is taken as the reference frame and a flight dynamics model described by orthogonal coordinates is developed. Then, according to the definitions of the generalized coordinates, two special curvilinear coordinate systems are put forward, and the relationships between the curvilinear and orthogonal coordinate systems are found. Finally, skillfully using the relationships as well as tensor theory, the dynamics model for the orthogonal coordinates are transformed into that for the generalized coordinates. The benefit of the new method is that it can derive the ODEs of multiple coordinates simultaneously because it is entirely based on matrix operations. By contrast, the traditional modeling methods from Newtonian mechanics and analytical mechanics need to use the infinitesimal approach to develop the equation of each generalized coordinate one by one, which may result in a large amount of repeated derivation. Therefore, the new method has higher efficiency in developing flight dynamics model described by generalized coordinates, and helps to prevent mistakes in derivation.

The General Gauss Principle of Least Constraint

Abstract This paper develops a general form of Gauss’s Principle of Least Constraint, which deals with the manner in which Nature appears to orchestrate the motion of constrained mechanical systems. The theory of constrained motion has been at the heart of classical mechanics since the days of Lagrange, and it is used in various areas of science and engineering like analytical dynamics, quantum mechanics, statistical physics, and nonequilibrium thermodynamics. The new principle permits the constraints on any mechanical system to be inconsistent and shows that Nature handles these inconsistent constraints in the least squares sense. This broadening of Gauss’s original principle leads to two forms of the General Gauss Principle obtained in this paper. They explain why the motion that Nature generates is robust with respect to inaccuracies with which constraints are often specified in modeling naturally occurring and engineered systems since their specification in dynamical systems are often only approximate, and many physical systems may not exactly satisfy them at every instant of time. An important byproduct of the new principle is a refinement of the notion of what constitutes a virtual displacement, a foundational concept in all classical mechanics.

METHOD OF ANALYSIS OF THE STRESS-STRAIN STATE OF A BASE-CENTERING DEVICE AND A PIPELINE WHEN CARRYING OUT THE COMPLEX OF EMERGENCY AND RECOVERY REPAIRS

Link for citation: Rakhmatullina Yu.A., Alekseev V.A., Sultanmagomedov T.S., Khasanov R.R. Method of analysis of the stress-strain state of a base-centering device and a pipeline when carrying out the complex of emergency and recovery repairs. Bulletin of the Tomsk Polytechnic University. Geo Аssets Engineering, 2023, vol. 334, no. 7, рр.7-16. In Rus. The relevance of the research is caused by the need to carry out calculations of the stress-strain state of the support-centering device and the pipeline, when the device under consideration is exposed to it, when the axes are displaced due to the presence of residual stresses when carrying out a complex of emergency repair by cutting out a defective section of the pipeline. The purpose of the study is to assess the possibility of using the design of the device and to calculate the stress-strain state of the device when the axes of the boundary sections of the pipeline are displaced due to the release of prestresses during the emergency repair complex for cutting out the defective section of the pipeline. Methods: methods of classical theoretical mechanics, soil mechanics, finite element method. Results. Based on the patent, a preliminary 3D model of the support-centering device and a method for analyzing the boundary conditions and loads affecting the device and the pipeline during the emergency repair complex were developed. The calculation of SSS (stress-strain state) was carried out, where the following results were obtained: when the frame is displaced down/up by 100 mm in the device, respectively, along the length and displacement, stresses arise at the junctions of the enclosing ring and the frame, as well as on the «lugs», exceeding the yield strength of steel; the coefficient of friction μ for the considered diameter of 1020 mm should not exceed 0,15 to prevent the occurrence of areas of plastic deformation of the pipeline during the emergency repair complex for cutting out a defective area in swamp conditions.

Increase in performance of air-sieve grain-cleaning machines in separating heap of sunflower seeds

BACKGROUND: The performance of grain cleaning machines (GCM) of the air-sieve type and the quality of grain cleaning depend on the uniformity of the loading of the sieves, which is determined by the design of the supply-distribution device. A decrease in the inclination angle of the feeding device surface leads to a decrease in the intensity of supply of the heap closer to the axis, further reduction leads the heap to stop in this area. Therefore, the inclination angle reduction for the feeding device is below the friction coefficient that leads to a decrease in the loading distribution uniformity.
 AIMS: Determination the dependence of the sorting process parameter of a heap of sunflower seeds on the material for the feeding device surface in the MVU-1500 air-sieve GCM.
 METHODS: To determine the inclination angle of the feeding device in the process of sorting the Lakomka variety sunflower seeds, which affects the parameters of the pneumatic system of the MVU-1500 air-sieve GCM, an experimental installation was built to determine the heap movement velocity. The used methods were a privately developed methodology, methods of mathematical statistics and laws of theoretical mechanics.
 RESULTS: The movement velocity of the heap of sunflower seeds on the surface of the feeding device were considered. Using the feeding device, the experimental and theoretical movement velocities of the heap of sunflower seeds on its surface made of metal and fluoropolymer included in the standard and improved MVU-1500 air-sieve GCMs were determined.
 CONCLUSIONS: The practical value of the study lies in determining the theoretical movement velocity of the heap of sunflower seeds on the GCMs feeding device surface which depends on its inclination angle as well as on the optimal material for it.

Fracture process zone development and length assessment under the mixed-mode I/II load analysed by digital image correlation technique

In this paper, the fracture process zone (FPZ) of high-performance concrete (HPC) is investigated under mixed-mode I/II load conditions, and its formation is studied by applying digital image correlation (DIC). The experimental tests are performed on Brazilian disc specimens with central notch (BDCN). The traction-free crack and the FPZ extent ahead of the crack are localised. This is done by modification of the existing methodology for mode I cracks to account for various mixed-mode I/II loading conditions. Analytical and linear elastic fracture mechanics (LEFM) methods for the critical strain are used to find the FPZ extension. Lastly, this paper revisits the analytical formulas used in the prediction of mode I FPZ lengths. These formulas are adjusted to allow for the prediction of the FPZ length in the whole range of mixed-mode I/II load conditions. Experimental results show that the FPZ has a different size for various mixed-mode I/II load conditions.

System of pre-sowing seed inoculation

Pre-sowing inoculation of agricultural crops increases the parameters that affect the yield and quality of the crop, which has been proven by many Ukrainian and foreign researchers. The purpose of the study was to set the operating parameters of the injectors at different pressures, which allowed calculating the rate of discharge of the working solution during sowing to create a system of pre-sowing inoculation, which will simplify the process of seed treatment and increase the energy efficiency of farms. Methods of system and structural analysis, mathematical statistics, abstraction, and mathematical modelling based on the Euler-Lagrange equation, and using the foundations of theoretical mechanics, physics and machine theory were used for experimental research. The planning of the experiment met the current standards, and computer graphics systems and Microsoft Excel software suite were used for statistical data processing. Based on the conducted studies, the amount of liquid that one nozzle can pass in 1 min at pressures of 300, 400, and 500 kPA was determined. It was found that the lowest result was shown by a nozzle with a diameter of 0.1 mm with a pressure of 300 kPA – 10.493 ml/min, and the highest – a nozzle with a diameter of 0.8 mm and a pressure of 500 kPA – 146.379 ml/min. At the same time, injectors with a diameter of 0.4-0.6 mm used almost the same amount of liquid at pressures of 300 and 500 kPA. The amount of liquid that can be poured per 1 ha using a six-row seeding machine with a working width of 4.2 m per sowing unit of corn was calculated. The obtained data can be used to improve any systems that use similar nozzles, such as calculating the rate of discharge of agricultural sprayers or systems that allow cooling pig farms during the summer period

Dynamic sequential radar cross section properties of airborne corner reflector in array

AbstractWith the spread of airborne corner reflector (ACR) in the field of shipborne equipment, radar guided anti‐ship missile is facing new challenges. In order to achieve the mastery of the jamming principle of this apparatus, the paper studied its motion properties and sequential Radar Cross Section (RCS) properties. Firstly, the motion model of the ACR was obtained under certain constraints, by deriving the centroid dynamics equation and rotation dynamics equation according to the theoretical mechanics. Then, the dynamic sequential RCS model of ACR array was established by combining the motion model with the modified geometric optics method and coherent synthesis method. Through the simulation and analysis of the motion model, it was found that the flight process of the ACR can be divided into two stages: fast falling and steady falling. In the fast falling stage, the variables of the ACR system change rapidly, while the ACR rotates slowly and falls smoothly in the steady falling stage. From simulation of dynamic sequential RCS model, the results showed that the obvious depolarisation effect is occur in the fast falling stage. Further statistical analysis showed that, the omnidirectivity of single ACR is well from the dynamic perspective, meanwhile array placement can effectively enhance RCS amplitude and improve the probability density distribution.

Quaternion and Biquaternion Methods and Regular Models of Analytical Mechanics (Review)

The work is of a survey analytical nature. The first part of the work presents quaternion and biquaternion methods for describing motion, models of the theory of finite displacements and regular kinematics of a rigid body based on the use of four-dimensional real and dual Euler (Rodrigues–Hamilton) parameters. These models, in contrast to the classical models of kinematics in Euler–Krylov angles and their dual counterparts, do not have division-by-zero features and do not contain trigonometric functions, which increases the efficiency of analytical research and numerical solution of problems in mechanics, inertial navigation, and motion control. The problem of regularization of differential equations of the perturbed spatial two-body problem, which underlies celestial mechanics and space flight mechanics (astrodynamics), is discussed using the Euler parameters, four-dimensional Kustaanheimo–Stiefel variables, and Hamilton quaternions: the problem of eliminating singularities (division by zero), which are generated by the Newtonian gravitational forces acting on a celestial or cosmic body and which complicate the analytical and numerical study of the motion of a body near gravitating bodies or its motion along highly elongated orbits. The history of the regularization problem and the regular Kustaanheim–Stiefel equations, which have found wide application in celestial mechanics and astrodynamics, are presented. We present the quaternion methods of regularization, which have a number of advantages over Kustaanheimo–Stiefel matrix regularization, and various regular quaternion equations of the perturbed spatial two-body problem (for both absolute and relative motion). The results of a comparative study of the accuracy of numerical integration of various forms of regularized equations of celestial mechanics and astrodynamics in Kustaanheimo–Stiefel variables and Newtonian equations in Cartesian coordinates are presented, showing that the accuracy of numerical integration of regularized equations in Kustaanheimo–Stiefel variables is much higher (by several orders of magnitude) than the accuracy of numerical integration Newtonian equations.

Modeling of the Tillage Process by the Method of Stress Concentration in the Tilled Layer

Introduction. Tillage is the most important part of the agricultural crops cultivation technology. Currently, tillage accounts for 35‒40% of the energy consumption of the total volume of field work and up to 20% of the energy consumed in the agriculture sector. Tillage allows regulating the water-air regime and the intensity of biological processes, maintaining the necessary phytosanitary condition of the soil and crops. The energy intensity and quality of tillage depends on the degree of excellence of technical means. Mathematical modeling of the tillage process by various soil-cultivating working bodies is an urgent task to develop more advanced energy-efficient technical means. Aim of the Article. The purpose of the research is to develop mathematical models for determining and analyzing the indicators of tilling the soil by the bracket-shaped working body. Materials and Methods. In the process of theoretical studies of the bracket-shaped tillage working body for continuous surface tillage, there were used the methods of analysis, generalization and systematization, theoretical mechanics, mathematical statistics and modeling, probability theory, differential calculation, and the theory of interaction of working bodies with soil and their movement over the field surface. Results. The analytical models for determining the pressure, traction resistance, the total force of the soil layer compression, the time of passage of the working body, the required power and the quantity of energy expended on compression of the soil layer depending on the speed of the working body movement are obtained. Graphic dependencies of the investigated parameters of the bracket-shaped tillage working body are presented. The dynamics of parameter changes is identified. Discussion and Conclusion. The proposed mathematical models and the revealed regularities of changing the parameters of tilling the soil by the closed-cycle working body a will make it possible to develop new effective working bodies and optimize the design, technological parameters and modes of their operation.

Roles of sulfur-containing compounds in fermented beverages with 2-furfurylthiol as a case example.

Aroma is a critical component of the flavor and quality of beverages. Among the volatile chemicals responsible for fragrance perception, sulfur compounds are unique odorants due to their extremely low odor threshold. Although trace amounts of sulfur compounds can enhance the flavor profile of beverages, they can lead to off-odors. Sulfur compounds can be formed via Maillard reaction and microbial metabolism, imparting coffee aroma and altering the flavor of beverages. In order to increase the understanding of sulfur compounds in the field of food flavor, 2-furfurylthiol (FFT) was chosen as a representative to discuss the current status of their generation, sensory impact, enrichment, analytical methods, formation mechanisms, aroma deterioration, and aroma regulation. FFT is comprehensively reviewed, and the main beverages of interest are typically baijiu, beer, wine, and coffee. Challenges and recommendations for FFT are also discussed, including analytical methods and mechanisms of formation, interactions between FFT and other compounds, and the development of specific materials to extend the duration of aroma after release.

A Sprayer Movement Model for Treating Reeds in Rice Irrigation System

Introduction. Rice grawing is an important branch of crop sector in Kuban, which produces about 80% of Russian rice. One of the significant factors reducing the yield of this crop is the presence of high-stemmed weeds growing in the canals and on the rollers of the rice paddy fields. The problem is the lack of structural and technological scheme for the sprayer to protect the rice irrigation system from the reeds providing local treatment of their panicles. When treating the weeds that are on the perimeter of the rice paddy field, the unit is under the influence of disturbances that arise from the unevenness of the relief of the field. These disturbances cause oscillation of the boom in the vertical and horizontal directions, which has a negative impact on the quality of processing the reed panicles. Aim of the Article. The work is aimed at increasing the efficiency of rice paddy fields protection from the reeds by substantiating the design and technological scheme of the sprayer for local treatment of the reed panicles and the model of the sprayer movement according the relief of the rice cheek. Materials and Methods. In theoretical studies, the mathematics and theoretical mechanics methods were used. Experimental studies were conducted on rice irrigation systems of Krasnoarmeiskiy district of Krasnodar region using planning experiments. Results. The design and technological scheme of the sprayer with a right-handed boom, formed according the placement of the reed upper points y = 310,84x0,0955, located in the channel of the paddy field is proposed. Based on the sprayer movement model, the range of boom position variation differs from the regulated value for plowing – 3.1 %; rice stubble – 1.4 %; rape – 0.6 %; clean fallow – 2.3 %; winte r wheat – 1.4 %. Discussion and Conclusion. The obtained data show an insignificant difference in the variation of boom oscillation of the proposed sprayer from the adjustment value, which positively affects the quality and accuracy of treating reed panicles. The obtained results can be used for serial production of sprayers for treating high-stemmed weeds in rice irrigation system and high-growing crops.

Research on the Safety Assessment of Corbel Beams of Reservoir Radial Gates Based on Real States

The radial gate corbel beam is a kind of gate support structure more often used in large and medium-sized reservoirs, but the current corresponding structural design code does not give a review calculation method for it. Because there are obvious differences between a corbel and a corbel beam in structural form and force, if the corbel beam is just simplified in the calculation as a corbel structure, it can easily lead to misjudgment of insufficient bearing capacity. This misjudgment has a significant impact on later safety assessment, danger removal, and reinforcement. Currently, there are limited studies available on the internal stress distribution of the corbel beam. In this study, taking the danger removal and reinforcement of the radial gate beam of a medium reservoir in Beijing as an example, the concrete quality of the dam was tested by the core drilling method, and two safety review methods of the corbel beam for different types of reservoirs were proposed in combination with the Code for Design of Hydraulic Concrete Structures (SL191-2008). Then, three different types of calculation model were established by the method of theoretical mechanics calculation and finite element simulation. Combined with the safety test data, the stress state of the combined stress structure of the corbel beam and gate pier under the real state was analyzed and the safety evaluation was carried out. The calculation results of these two corbel beam safety review methods were respectively reduced by 32% and 47% compared with the current calculation method. Engineering practice has proved the rationality of the two safety evaluation methods proposed in this paper, which can provide a certain reference for similar engineering reinforcement.

Research on Compliant Controller of Underwater Mechanical Capture System Based on Impedance Theory

Due to the requirement of the exploitation of marine resources, the execution of specific underwater tasks by onboard manipulators has become one of the key research fields in domestic and all over the world. Based on the underwater capture system designed for the UUV recycling task, which consists of an underwater manipulator and a mechanical capture device, this paper first constructs the kinematics and dynamics model of the capture system through theoretical analysis such as theoretical mechanics and theory of mechanism. Then, combined with the requirements of the recycling task, through the theoretical basis of fluid mechanics such as Morrison equation, dynamic of the capture system in underwater environment is analysed, with a compliant controller designed for the capture system based on impedance theory in order to reduce the impact of underwater environment in the capture task. Moreover, as the capture system modelled in the Adams dynamics simulation platform, it is verified that the designed compliant controller can reduce the underwater environmental impact through simulation experiments in the Adams dynamic platform.

MATHEMATICAL MODEL OF STABLE EQUILIBRIUM OPERATION OF THE FLIGHT SIMULATOR BASED ON THE STEWART PLATFORM

This paper analyses the available mathematical models of flight simulators based on the Stewart platform. It was found that there is no model that describes the conditions for stable dynamic equilibrium operation of the Stewart platform as a function of a number of important motion parameters. In this context, a new physical model is proposed based on classical models of theoretical mechanics using the d’Alembert formalism, the concept of stable equilibrium of a mechanical system. This model mathematically separates the stable equilibrium of the flight simulator motion system from the general uniformly accelerated motion. The systems of equations obtained in the framework of the model connect the physical and geometrical parameters of the Stewart platform and make it possible to determine the reactions in the upper hinges of the platform support, the limit values of the position angles in the space of the base of the support of the Stewart platform, under which the condition of stable equilibrium operation of the Stewart platform is satisfied. The proposed physical model and the analytical relations obtained on its basis are of great practical importance: the operator controlling the operation of the Stewart platform-based flight simulator can control the range of parameters during training so as not to bring the flight simulator out of stable equilibrium.

Application of STEM Technologies on the Example of the Problem of a Thread with a Load

This paper demonstrates the application of STEM technologies using the example of the problem of a thread with a load. As a training problem, parameter finding of a flexible inextensible thread with a load sliding along it, as well as with a fixed load, is considered. A mathematical model of a thread with a load is created without using simplifying assumptions such as a negligibly small mass of the thread, a thread with a small sag or a parabolic approximation of the thread form. When solving the problem, methods of analysis from various branches of mathematics, theoretical mechanics, physics, history of mathematics, as well as various tools of computer mathematical packages (Mathcad), are used. The reasonability of considering such tasks from the perspective of teaching effectiveness for first-year students in engineering as well as in physics and mathematics is proved. Promising educational tasks based on the investigated problem are considered.

Value equilibrium analysis based on the New Theory of Value. I

The (traditional) labor theory of value, founded by Smith with the basic axiom that labor determines value, and supplemented and improved by Marx with the axiom that labor creates surplus-value. Neoclassical economics newly studied the utility theory of value, but it abandoned the traditional labor theory of value and theory of surplus-value. Under the basic axiomatic system analogous to theoretical mechanics, the above three theories of value has been included and developed by the new theory of value, which established the value complex variable function including three basic quantities of quantity, quality and time, as well as labor value, use-value and surplus-value. Starting from the value complex variable function, with the mathematical methods of Euler equation and Euler formula, this paper conduct a thorough and comprehensive analysis on the theory of value, under satisfactions of Marx’s first and second laws, focusing on value equilibrium by giving the mathematical explicit expressions of both value equilibrium function and price equilibrium function in various forms applicable to the general commodity economic movement process, so as to prove the existence and stability of the extremum solutions to the corresponding functions.