Methods Of Analytical Mechanics Research Articles

Deployment of a space tether in a centrifugal force field with alignment to the local vertical

This study is concerned with a small orbital tether of two bodies to be deployed from a spacecraft so that upon completion of the deployment it turns out to be aligned along the local vertical. The bodies of the tether have equal masses, and the thread connecting the bodies is supposed to be massless. The objective of the study is to build a program law of tether length control taking into account the variation of the angular momentum of the tether under the action of the gravitational torque from the central Newtonian field of forces. The deployment mode of the space tether in a centrifugal force field with its alignment at the conclusion of the deployment along the local vertical is studied. To produce centrifugal forces, the tether is pre-spinned about the orbit binormal. The study consists of two steps. The first step involves the construction of a tether length control law that would provide the planned deployment. At this step, use is made of the tether motion equations written in spherical coordinates for the special case of the tether motion in the orbital plane. A numerical simulation of the tether deployment dynamics is carried out at the second step using the constructed program law of tether length control. Hill-Clohessy-Wiltshire’s equations are used as a mathematical model of the tether. They describe the spatial motion of the tether bodies. These equations do not contain the tether length as a variable in explicit form. Therefore, these equations are modified. The tether tension force appearing in these equations is expressed in terms of the program law of tether length change and its two first time derivatives. The novelty of the study consists in the construction of a program control law that allows the tether to be deployed along the local vertical in a single stage. The study used methods of analytical mechanics, numerical methods, and methods developed by the authors. The obtained results make it possible to find the ranges of values of the deployment system parameters allowing a deployment of this type. The errors of the numerical simulation are estimated. The practical significance of the obtained results consists in the possibility of deploying small tethers in orbit with their alignment at the conclusion of the deployment along the local vertical in a single stage with controlling the tether length without the need for further dumping of libratory oscillations.

Numerical simulation of an external ballistic problem using analytical approach and atmosphere flow visualization by finite element method

Purpose. Analysis of the dynamic characteristics of the cargo in the presence of the speed of its carrier, taking into account the influence of the external environment and the determination of factors that affect the operational characteristics and parameters for the dynamic system control. Methodology. An approximate analytical approach to solving the nonlinear problem of external ballistics of a system with time-dependent parameters, which is implemented using the asymptotic perturbation method, and a numerical algorithm for modeling a dynamic process using a 3D software complex and the finite element method, which allows to visualize the nature of the flow around the object under wind load conditions. Findings. An analytical approach to solving the nonlinear problem of external ballistics of a system with time-varying parameters is proposed, as well as a numerical model, solution and visualization of a dynamic process that can be applied in problems of mathematical physics and engineering calculations. Originality. The use of an approximate analytical approach to solving the nonlinear problem of external ballistics, which includes a system of time-varying parameters, is a significant innovative step. For the first time, a three-dimensional distribution of the free fall parameters under atmospheric conditions was obtained depending on the time of free fall under the condition of an initial velocity different from zero. Obtaining the characteristics of the distribution, taking into account the aerodynamic quality of the object, made it possible to adapt the methods of analytical mechanics and differential equations with variable coefficients to the solution of the applied problem. The obtained characteristic three-dimensional surface allows to perform a study of the aerodynamic quality of the object under study using numerical methods and to visualize dynamic processes in three-dimensional space in order to obtain a qualitative picture of the perturbation of the object by the air flow. This integrated approach to the study of ballistic characteristics of cargo forms the scientific principle to perform engineering calculations to solve the problem of controlling dynamic systems. Practical value. An approximate analytical approach and a calculation model of the dynamic process of a mechatronic unmanned system make it possible to increase its efficiency in the presence of an initial speed and ensure the accuracy of the application of ballistic cargo delivery systems in real operating conditions.

Analytical mechanics methods in finite element analysis of multibody elastic system

The study of multibody systems with elastic elements involves at the moment the reevaluation of the classical methods of analysis offered by analytical mechanics. Modeling this system with the finite element method requires obtaining the motion equation for an element in the circumstances imposed by a multibody system. The paper aims to present the main analysis methods used by researchers, to make a comparative analysis, and to show the advantages or disadvantages offered by different methods. For the presentation of the main methods (namely Lagrange’s equations, Gibbs–Appell’s equations, Maggi’s formalism, Kane’s equations, and Hamilton’s equations) a unified notation is used. The paper provides a critical evaluation of the studied applications that involved some of these methods, highlighting the reason why it was decided to use them. Also, the paper identifies potential research areas to explore.

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

Modelling and calculation of stumulated oscillation for a crushing plant with vibration

The paper presents the results of the mathematical modelling of external stimuli to improve the efficiency of the removal of grain material from the grinding zone. Thus, the authors proposed to improve the evacuation conditions of the finished product from the working chamber of the crushing plant, as one of these solutions to improve the grinding process in crushers. As a result of creation of external vibrating influence. For this purpose, the calculation chart for modelling the proceeding process of the crushing plant body movement is created which takes into account weight and sizes of the crusher, weight, angular velocity of rotation and eccentric piece location, as well as rigidity of an elastic suspension on which the crusher is established. For the solution of the posed problem, we used methods of analytical mechanics, the Lagrange equation of the second kind, for the mechanical system with two degrees of freedom. As a result of calculations amplitude-frequency characteristics of mechanical system are determined. The conclusion about the choice of the angular velocity of the eccentric rotation for creation of necessary external stimulating influence on the crushing system and improvement of conditions of evacuation of a ready product from a working zone is given. The resonance frequencies of firstand second-order system vibrations and the vibrator frequency are determined to achieve resonance frequencies. Keywords: vibration, modelling, Lagrangian equations of the second kind, grinding, crushing plant.

Static and dynamic coefficient of friction in friction pairs of disc-pad brakes

Problem This scientific article is devoted to the value of static and dynamic coefficients of friction of disc-pad brakes. The issue of disc-pad brake wear is very acute because the safety of using the machine itself depends on its reliable operation. Therefore, the relevance of the chosen topic cannot be doubted. Goal. The aim is to establish patters of influence of the main operational parameters of friction pairs of a model disc-pad brake on the potential and kinematic coefficient of friction. Methodology. The methods of analytical mechanics and the theory of elasticity were used in the article. Results. Theoretical and experimental studies of the static (former rest) and dynamic coefficients of friction in the frictional interaction of materials FK-24A - steel 35KhNL were carried out on a model disk-shoe brake in a long-term braking mode. At the same time, it was established in the range of real impulse loads implemented during braking in the contact area of the friction lining with the brake disc, the stress-strain state of the contact according to the criterion of micro deformation in the contact corresponds to the elastic-plastic property. The elastic and plastic properties of materials appear when, in each individual case, the total stresses exceed the yield strength or reach its limit. With an increase in the impulse force of the normal pressing of the pad lining to the treadmill of the disc friction, the static coefficient of friction tends to increase, and with an increase in the static coefficient of mutual overlap kvz.s, it tends to decrease with the elastic-plastic property of materials. When choosing materials for friction pairs, the dynamic coefficient of friction must include the value of the static coefficient of friction and be at least 20% greater than the calculated value of the dynamic coefficient of friction. Practical value An increase in the static and dynamic friction coefficient in the friction pairs of a disk-pad brake leads to an increase in the plastic deformation of their microprotrusions. The data obtained can be used in the design of disc-shoe brakes

On the methods of analytical mechanics for mathematical modeling of the dynamics of non-free systems and some variants of their application to the dynamics of parallel manipulators

The level of development of actuator technology and the element base and software of control loops of modern automatic devices makes it possible to use in technical practice complex control algorithms based on the results of mathematical control theory, including with incomplete information about the state. The effectiveness of such algorithms is determined by the presence of an adequate mathematical model of the device under study. For such a widespread class of automatic devices as manipulators with parallel kinematics, the problem of developing methods for creating mathematical models that allow taking into account nonlinear effects, despite the efforts of numerous researchers, remains relevant. To a large extent, the complexity of modeling the dynamics of this class of automatic devices is associated with the presence of parallel kinematic chains, which leads to the need to apply the results of analytical mechanics of non-free systems. In this section of theoretical mechanics, several rigorous methods have been developed, as a result of which mathematical models of different dimensions are obtained with different levels of complexity of the algorithms for their derivation and study. This leads to the need for a critical analysis of both the methods themselves and their possible application in technical practice. In this paper, the main algorithms for obtaining mathematical models of the dynamics of systems with geometric constraints are presented in detail and some comparative analysis of their application to modeling the dynamics of manipulators with parallel kinematics is carried out.

INFLUENCE OF ROLLING STOCK VIBROACOUSTICAL PARAMETERS ON THE CHOICE OF RATIONAL VALUES OF LOCOMOTIVE RUNNING GEAR

Purpose.The success of the traffic on the railways of Ukraine depends on the number and the operational fleet of electric locomotives. Today, the locomotive depot exploit physically and morally outdated locomotives that have low reliability. Modernization of electric locomotives is not economically justified. The aim of this study is to improve the safety of the traction rolling stock by the frequency analysis of dynamical systems, which allows conducting the calculation of the natural (of resonant) frequencies of the design and related forms of vibrations.Methodology.The study was conducted by methods of analytical mechanics and mathematical modeling of operating loads of freight locomotive when driving at different speeds on the straight and curved track sections. The theoretical value of the work is the technique of choice of constructive schemes and rational parameters of perspective electric locomotive taking into account the electric inertia ratios and stiffness coefficients of Lagrange second-order equations.Findings. The problems of theoretical research and the development of a mathematical model of the spatial electric vibrations are solved. The theoretical studies of the effect of inertia ratios and stiffness coefficients on the dynamic values and the parameter values of electric locomotive undercarriages are presented.Originality.The set of developed regulations and obtained results is a practical solution to selecting rational parameters of bogies of the freight mainline locomotive for railways of Ukraine. A concept of choice of constructive scheme and rational parameters of perspective locomotive is formulated. It is developed the method of calculation of spatial electric locomotive oscillations to determine its dynamic performance. The software complex for processing the data of experimental studies of dynamic parameters of electric locomotive and comparing the results of the theoretical calculations with the data of full-scale tests is implemented.Practicalvalue.The theoretical study allows objectively assessing the impact of the partial frequency of electric locomotive elements on both the rational parameters of its underframe, and the performance and physiological condition of the locomotive crew. Thus, the practical value of the whole complex of theoretical research was to develop an algorithm and program, choosing rational parameters and design scheme of electric locomotive running gear. Using a software package for the study of spatial fluctuations of electric locomotive made it possible to evaluate the dynamic performance of the electric locomotive underframe with sufficient accuracy for practical problems.

MATHEMATICAL PENDULUM MODEL WITH MOBILE SUSPENSION POINT

Background. The new dynamic problem, which is posed and solved in this article, is a theoretical generalization of the well-known classical problem of free oscillations of a mathematical pendulum. In the proposed setting, it is new and relevant, and can be successfully used in such fields of technology as vibration protection, vibration isolation and seismic protection of high-rise flexible structures, long power lines, long-span bridges and other large-sized supporting objects. Objective. The aim of the work is to derive the equations of own oscillations of a new mathematical pendulum-absorber, to find a formula for determining the frequency of its small own oscillations and to establish those control parameters that allow you to tune the single-mass pendulum absorber to the frequency of the fundamental tone of the carrier object. Methods. To achieve this goal, we used the methods of analytical mechanics, namely, the Appel’s formalism, as well as the linearization of the obtained differential equations. Results. A mathematical model is constructed in the work that describes the own oscillations of a new-design mathematical pendulum with a movable (spring-loaded) suspension point with length L. The model is a system of differential equations obtained using the Appel’s formalism. Based on them, after linearization of nonlinear equations, a formula is established for the frequency of small own oscillations of a pendulum with a mobile suspension point. Conclusions. It is shown that the frequency of own oscillations of the new mathematical pendulum coincides with the frequency of own oscillations of the classical mathematical pendulum with an equivalent suspension length, which is equal to . In the case where the suspension point is fixed (k ® ¥), the frequency formula turns into a well-known formula for the frequency of small own oscillations of a classical mathematical pendulum . If the stiffness coefficient of elastic elements tends to zero (k ® 0), then the frequency w of the damper also tends to zero. An important structural feature of the proposed pendulum is noted, consisting in the fact that due to the appropriate choice of the three control parameters of the pendulum (k, L, m), its frequency in magnitude can be made any in the range from zero to .

Cross-country ability of tracked vehicles

Research aims to determine conditions and parameters under which the tracked vehicle overcomes stone fields and separate blocks (boulders). Research methodology included the methods of analytical mechanics and mathematical analysis. Results. Mining and geological conditions of deposits gradually get more complicated, while mining moves to remote northern regions, which is a characteristic feature of the Holocene epoch. Road, weather, and natural conditions in arctic and northern territories differ markedly from other regions. Ridges, mounds, lake plains, and marshland are the predominant landscape. Stone fields, marshes, permafrost, marshy hollows, and deep ravines impede the traffic. Due to these factors it is particularly difficult to access and explore in arctic and northern territories. There are particular technical specifications for the exploration equipment, namely, increased cross-country ability for weak soil (snow and swamp), ability to overcome stone fields, increased haulage and towing capacity with the account of maximum possible environmental regulatory compliance, etc. The conditions of ascent and overturning are derived from geometry, while the sliding condition is derived from statics. Summary. A mathematical model has been created, which makes it possible to describe the tracked vehicle’s travel across the uneven ground and, under known design values of the vehicle, determine the limitary obstacle clearance.

Kinematics of the Maneuver of the Multifunctional Machine for Metallurgical Furnaces

The article proposes a simulation model of a loading machine for metallurgical furnaces and builds a mathematical model that makes it possible to find optimal trajectories of motion and obtain laws of change in kinematic, dynamic and inertial characteristics for a full operation cycle. The description of the machine motion is made taking into account the nonholonomy of the imposed connections and provides for the possibility of considering arbitrary laws of motion along optimal trajectories. Compliance with optimal working conditions is also ensured by the choice of a special law of motion between the nodal points on the trajectory of movement, corresponding to the change in the direction of movement of the loading machine. Based on the research results, a design scheme and a simulation model with an analytical description of the mechanism operation, including kinematic graphs, are proposed. A simulation model of a loading machine for metallurgical furnaces and a description of its operation modes are made by methods of analytical mechanics.

覆冰输电导线舞动的Noether对称性和守恒量

To overcome the non-structure-preserving drawbacks in traditional numerical simulation of transmission-line nonlinear vibration responses, the Noether symmetry and conserved quantity of transmission lines’ 2-way galloping under ice and wind excitation were studied. Firstly, in view of the nonlinearity of the aerodynamic force and the line geometry, a 2-DOF galloping model of vertical and torsional vibrations was established based on the analytical mechanics method. Secondly, the group analysis theory was introduced, and the condition and the conserved quantity of the Noether symmetry were given according to the invariance principle. Finally, a conserved quantity-preserving discrete numerical algorithm was constructed. The dynamic characteristics of the nonlinear mechanical structure were studied with the Noether symmetry theory. The results show that, the proposed novel method is structure-preserving in a wide range of application, and is reliable and accurate.

The Lagrangian and Hamiltonian for the Two-Dimensional Mathews-Lakshmanan Oscillator

The purpose of this paper is to illustrate the theory and methods of analytical mechanics that can be effectively applied to the research of some nonlinear nonconservative systems through the case study of two-dimensionally coupled Mathews-Lakshmanan oscillator (abbreviated as M-L oscillator). (1) According to the inverse problem method of Lagrangian mechanics, the Lagrangian and Hamiltonian function in the form of rectangular coordinates of the two-dimensional M-L oscillator is directly constructed from an integral of the two-dimensional M-L oscillators. (2) The Lagrange and Hamiltonian function in the form of polar coordinate was rewritten by using coordinate transformation. (3) By introducing the vector form variables, the two-dimensional M-L oscillator motion differential equation, the first integral, and the Lagrange function are written. Therefore, the two-dimensional M-L oscillator is directly extended to the three-dimensional case, and it is proved that the three-dimensional M-L oscillator can be reduced to the two-dimensional case. (4) The two direct integration methods were provided to solve the two-dimensional M-L oscillator by using polar coordinate Lagrangian and pointed out that the one-dimensional M-L oscillator is a special case of the two-dimensional M-L oscillator.

Peg-on-hole: mathematical investigation of motion of a peg and of forces of its interaction with a vertically fixed hole during their alignment with a three-point contact

The task of the peg-in-hole assembly is a very common task in industry. Despite its intuitive comprehensibility and everyday experience, the industrial assembly faces specific problems. The naive approach of inserting with bare hands an industrial peg inside into an industrial hole can very certainly bring to the jamming and damaging because of a very small gap between them. Therefore, the initial alignment of the details before insertion, so-called peg-on-hole phase, or simply peg-on-hole, is of the special importance in industry to avoid jamming. This paper develops the description and analysis of this phase based on the methods of analytical mechanics. As a common approach, the details in this study are considered as cylindrical, and such that peg is supported at the edge of vertically fixed hole, and could freely move keeping a three-point contact with three degrees of freedom, namely (i) nutation angle or align describing planar motion of aligning parts, (ii) precession angle or slide describing rotational motion around hole axis, and (iii) self-rotation angle or slip describing rotational motion about its axis. The analytical approach implies the system of Lagrange Equations for this particular case of motion, which is called Dynamic Differential Equations (DDE). DDE describes interconnections between motion in three degrees of freedom on one side, reaction forces on another side, and the external forces and torques on the third side. DDE includes geometrical properties of the details, and full descriptions of velocities, normal reactions, and friction forces at the contact points. It is the most general description of the peg-on-hole case. As the example of application, the normal reactions were found for relative large alignment angles far from reduction into a two-point contact, and for a small alignment angles, when system transits to the two-point contact. It was shown that for both cases, reaction forces become larger during alignment because they have to balance the force of gravity. For the larger angles, slip reduces the reaction forces, whereas for small alignment angles, slip and slide increases them.

INVESTIGATION OF BAND LOOP FRAME ARTICULATED CONTEINER TRUCK STABILITY BY MOTION ON THECHNOLOGICAL ROADS

Mathematical model of disturbance motion for articulated container truck with band loop frame on pneumatic wheels by different conditions of cross stabilization has been worked out by analytical mechanics method with Lagrange equation of second type. Analytical expressions of critical parameters of system, which define stability of articulated container truck in cross plane has been received.

The application of analytical mechanics in a multimachine power system

Analytical mechanics methods play an important role in determining the energy function and Hamilton realization of a system. In this paper, the Hamilton function of the system is obtained directly from the system differential equation by means of analytical mechanics. In addition, the system Hamiltonization process requires that the system must be even order, and the commonly used generator model is a single-axis third-order model. For this reason, the odd-order power system model is extended to an even-order system. The Hamilton function of the system is deduced with coordinate transformation and the generalized force. Then the standard form of Hamilton realization for the multimachine power system is given. Based on the basic principle of nonconservative analytical mechanics and the state feedback control law of the power system, the stability control law of the multimachine power system is designed, which makes the system become asymptotically stable in the neighborhood of the equilibrium point. The control of the IEEE 14 is simulated with MATLAB utilizing the proposed controller. The control effect of the controller under the three-phase short-circuit fault is subsequently studied in the simulation, wherein the effectiveness of the proposed control strategy is verified.

Quasi-momentum theorem in Riemann-Cartan space

The geometric formulation of motion of the first-order linear homogenous scleronomous nonholonomic system subjected to active forces is studied with the nonholonomic mapping theory. The quasi-Newton law, the quasi-momentum theorem, and the second kind Lagrange equation of dynamical systems are obtained in the Riemann-Cartan configuration spaces. By the nonholonomic mapping, a Euclidean configuration space or a Riemann configuration space of a dynamical system can be mapped into a Riemann-Cartan configuration space with torsion. The differential equations of motion of the dynamical system can be obtained in its Riemann-Cartan configuration space by the quasi-Newton law or the quasi-momentum theorem. For a constrained system, the differential equations of motion in its Riemann-Cartan configuration space may be simpler than the equations in its Euclidean configuration space or its Riemann configuration space. Therefore, the nonholonomic mapping theory can solve some constrained problems, which are difficult to be solved by the traditional analytical mechanics method. Three examples are given to illustrate the effectiveness of the method.

Determining the parameters of oscillation dissipation in a column of sucker rods

In order to calculate a column of pumping rods as a mechanical system with concentrated masses, to determine dissipation coefficients, we applied a traditional method of analytical mechanics. A mathematical model is developed of longitudinal oscillations of a three-stage conditionally vertical column of sucker rods using the functions of displacement and load of its separate stages. Dependence for determining the dissipation coefficient of oscillations is derived from the solution of the system of equations. In accordance with the chosen set-up of a three-stage column of sucker rods, we examined a change in the dissipation coefficient of oscillations depending on the relationship between the rigidities of its stages. A change in the relationship between rigidities of the stages was carried out by changing their material. It was found that for the selected set-up of a column of sucker rods, the use of a fiberglass stage instead of that made of steel reduces its rigidity by approximately 4 times, and increases the dissipation of oscillations almost as much. Such approach makes it possible to prevent the phenomenon of resonance in the operation of a SR column during transition modes under the action of alternating load.

Flexoelectricity in low densification materials and its implication

Coupling between the strain gradient and the electric polarization, also named flexoelectricity, is a fundamental but often overlooked property in solid dielectrics. The past ten years has witnessed its great potential in sensing and actuating applications, especially in nanomaterials or nanostructures. The recently discovered giant flexoelectricity in barium strontium titanate (BST) perovskite above its Curie temperature has boosted the research interests on flexoelectricity to a new height. The possible interpretation of the enhanced flexoelectricity in BST was initially attributed to the non-crystalline polar-phases or polar nano-regions existing in the perovskites, then to the inner micro strain generated from the material densification process. In this paper, we analyzed the origin of flexoelectricity in this kind of materials by using both analytical mechanics method and finite element simulation method. Our results suggested the inner micro defect generated during the material densification process would weaken the flexoelectricity of the materials. The observed polarization in the studied materials was likely induced by other mechanisms rather than the pure flexoelectricity.

Methods of analytical mechanics in the constrained adaptive identification problems

The application of analytical mechanics methods to the problem of adaptive identification is discussed. The estimated constant parameters satisfy nonlinear equations considered as parametric constraints. Some modifications of identification algorithms are proposed to satisfy such constraints.