scholarly journals MATHEMATICAL MODEL OF SPATIAL OSCILLATIONS OF A RAILWAY FOUR-AXLE AUTONOMOUS TRACTION MODULE

2021 ◽  
pp. 55-68
Author(s):  
František Bures
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Mechanical System ◽  
Traffic Safety ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Railway Track ◽  
Theoretical Studies ◽  
System Of Differential Equations ◽  
Spatial Oscillations

A description of the original mathematical model of spatial oscillations of a four-axle autonomous traction module during its movement along straight and curved sections of the railway track is proposed. In this case, the design of a four-axle autonomous traction module is presented as a complex mechanical system, and the track is considered as an elastic-viscous inertial system. The equations of motion were compiled using the Lagrange method of the ІІ kind. For each of the solids, the kinetic energy is determined by the Koenig theorem. The potential energy component is obtained by the Clapeyron theorem, as the sum of the energies accumulated in the elastic elements of the system during their deformations. The dissipative energy was also taken into account when compiling the equations of motion. Generalized forces that have no potential, in this case, include the forces of interaction between wheels and rails, which are determined using the creep hypothesis. It is important to note that the model takes into account the forces in the bonds between the bodies of the system and the spatial displacements of the rigid bodies of the mechanical system, taking into account possible restrictions. Moreover, the mathematical model developed by the author is a system of differential equations of the 100th order. This mathematical model is the base for further theoretical studies of the dynamics of railway four-axle autonomous traction modules in single motion or when moving as part of a train. To solve the resulting system of differential equations, the author develops special software that allows for complex theoretical studies of spatial oscillations of a four-axle autonomous tractionmodule to determine the indicators of its dynamic loading and traffic safety. 

scholarly journals Study of the dynamics of the working cell of the automatic mill of the pipe-rolling unit

2018 ◽  
pp. 74-80
Author(s):  
S.R. Rakhmanov ◽  
V.T Vyshinskyi
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Mechanical System ◽  
Equations Of Motion ◽  
Retention Mechanism ◽  
Operating Conditions ◽  
Dynamic Processes ◽  
Real Spectrum ◽  
Pipe Rolling ◽  
Automatic Mill

Purpose. Determining the real spectrum and the level of dynamic loads would make it possible to develop recommendations for improving the TPA-350 automatic mill, with the aim of expanding its technological capabilities, improving the reliability of operation and durability. Metodology. This work was carried out on the basis of the development of a mathematical model of dynamic processes in the mechanical system of an automatic TPA mill, where an attempt was made to identify the features of the functioning of a working stand with a rolled sleeve in the formulation of dynamic problems. Obviously, the proposed approach is more correct and convenient when studying complex dynamic phenomena in the elements of an automatic TPA mill. Findings. The results of the study of the dynamics of the working stand of the automatic mill of the pipe-rolling unit (TPA) are given The basic parameters of the functioning of the working stand, such as the automatic mill TPA 350, are established, and a mathematical model of the basic problem of the dynamics of a mechanical system is developed. The differential equations of motion for the selected model of the working stand and the elastic elements of the bed retention mechanism on the support nodes of the automatic TPA mill have been compiled. The corresponding patterns of oscillations of the working stand were obtained. Some features of the functioning of the working stand are identified by the example of an automatic mill TPA 350. The dynamic parameters of the working stand of the mill, affecting the difference in the thickness of the rolled sleeves, are established. Originality. The interrelation of the working stand dynamics with the operating conditions of an automatic TPA mill on the basis of a simplified mathematical model of dynamic processes is established. A mathematical model of the mechanical system of a working working stand of an automatic TPA mill has been compiled. In the most general form, the mechanical system of a working stand of a mill is described using differential equations describing the behavior of the selected design scheme of a system with two degrees of freedom. Practical value. A scheme has been proposed for modernizing the working stand and the mechanism for holding the working stand of the TPA-350 automatic mill. IL. 4. Biblography: 8 titles.

Mathematical model implementation in conditions of uncertainty and possible risks

2021 ◽  
pp. 137-145
Author(s):  
A. Kravtsov ◽  
◽  
D. Levkin ◽  
O. Makarov ◽  
◽  
...  
Keyword(s):  
Thermal Conductivity ◽  
Mathematical Model ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Layer Structure ◽  
Boundary Value ◽  
Goal Function ◽  
Cauchy Problems ◽  
System Of Differential Equations

The article presents the theoretical and methodological principles for forecasting and mathematical modeling of possible risks in technological and biotechnological systems. The authors investigated in details the possible approach to the calculation of the goal function and its parameters. Considerable attention is paid to substantiating the correctness of boundary value problems and Cauchy problems. In mechanics, engineering, and biology, Cauchy problems and boundary value problems of differential equations are used to model physical processes. It is important that differential equations have a single physically sound solution. The authors of this article investigate the specific features of boundary value problems and Cauchy problems with boundary conditions in a two-point medium, and determine the conditions for the correctness of such problems in the spaces of power growth functions. The theory of pseudo-differential operators in the space of generalized functions was used to prove the correctness of boundary value problems. The application of the obtained results will make it possible to guarantee the correctness of mathematical models built in conditions of uncertainty and possible risks. As an example of a computational mathematical model that describes the state of the studied object of non-standard shape, the authors considered the boundary value problem of the system of differential equations of thermal conductivity for the embryo under the action of a laser beam. For such a boundary value problem, it is impossible to guarantee the existence and uniqueness of the solution of the system of differential equations. To be sure of the existence of a single solution, it is necessary either not to take into account the three-layer structure of the microbiological object, or to determine the conditions for the correctness of the boundary value problem. Applying the results obtained by the authors, the correctness of the boundary value problem of systems of differential equations of thermal conductivity for the embryo is proved taking into account the three-layer structure of the microbiological object. This makes it possible to increase the accuracy and speed of its implementation on the computer. Key words: forecasting, risk, correctness, boundary value problems, conditions of uncertainty

scholarly journals Numerical simulation of oscillations of a nonlinear mechanical system with a spring-loaded viscous friction damper

2019 ◽  
Vol 265 ◽  
pp. 05030
Author(s):  
Viktor Nekhaev ◽  
Viktor Nikolaev ◽  
Marina Safronova
Keyword(s):  
Differential Equations ◽  
Mechanical System ◽  
Degrees Of Freedom ◽  
External Disturbance ◽  
Viscous Friction ◽  
Force Characteristic ◽  
Friction Damper ◽  
System Of Differential Equations ◽  
Nonlinear Force ◽  
Linear Spring

The dynamics of a mechanical system consisting of a parallel-connected main elastic element, an external disturbance compensator having a nonlinear force characteristic, and a viscous friction damper sprung by a linear spring are studied. The resulting system of differential equations describing the behavior of the system has one and a half degrees of freedom and has specific properties depending on the ratio of stiffness of the main spring and the spring suspension of a viscous friction damper. It is established that a single nonlinear system with one and a half degrees of freedom has either one or two harmonics. In the general solution of the system of differential equations, there are always two harmonics in the above-resonance zone, one of which is always equal to the disturbance frequency, and the second one is sufficiently close to the frequency k0. In the linear conservative case and the absence of suspension of the viscous friction damper, the natural frequency of the displacement of the system k0 =14.046 s-1.

Contributions to the Determination of the Equations of Motion for Multidegree of Freedom Systems

1971 ◽  
Vol 93 (1) ◽  
pp. 191-195 ◽  
Author(s):  
Desideriu Maros ◽  
Nicolae Orlandea
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
System Of Differential Equations ◽  
Original Contribution ◽  
Deductive Method ◽  
Method Of Solution ◽  
Further Development ◽  
Three Degrees Of Freedom

This paper is a further development of the kinematic problem presented in our 1967 paper [1] in which we have obtained the transmission functions for different orders of plane systems with many degrees of freedom. This paper establishes the corresponding system of differential equations of motion beginning with these functions. The purpose of this paper is to facilitate computer programming. Our study is based on the work of R. Beyer [2, 3] and is the first original addition to his papers. A second original contribution to Beyer’s theories is the deductive method of solution, from general to particular, which we have, incorporated in our work. Beyer concluded that the cases having two or three degrees of freedom can be considered as particular solutions to the results obtained.

scholarly journals Mathematical Model of Spatial Motion of the Controlled Parachute-Tether System of the Wind Kite Type

2021 ◽  
Vol 24 (4) ◽  
pp. 17-24
Author(s):  
V.M. Churkin ◽  
T.Yu. Churkina ◽  
A.M. Girin
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Solid Body ◽  
Equations Of Motion ◽  
Vertical Plane ◽  
Spatial Motion ◽  
Mathematical Task ◽  
Spatial Movement ◽  
Elastic Thread ◽  
The Mathematical Model

Mathematical modeling is created for the mathematical task of spatial motion of the controlled parachute-tether system of the “wind kite” type. The mathematical model parachute-tether system consists of a model of the main parachute and a model of the braking parachute. The parachutes are connected by the tether. The model of the main parachute is supposed to be the solid body. This solid body has two planes of symmetry. The braking parachute is the solid body with axial symmetry. The tether model is an absolutely flexible elastic thread. The tether is connected by ideal hinges with the main parachute and braking parachute. The control of the main parachute is carried out by changing the length of the control slings. Changing the length causes deformation of the dome. This is the reason for the change in its aerodynamics. Maneuvering of the main parachute occurs in the vertical plane, when the length of the control slings changes simultaneously. Maneuvering of the main parachute in space is carried out when the length of the control slings changes, when the slings are given a travel difference. The system of dynamic and kinematic equations is designed for calculating the controlled spatial movement of the main parachute, braking parachute and tether. The option exists when the mass of the tether and the forces applied to the tether cannot be neglected. The motion of the tether is represented by the equations of motion of an absolutely flexible elastic thread in projections on the axis of a natural trihedron. The mathematical model is represented by a system of ordinary differential equations and partial differential equations. The problem is solved using various numerical methods. The solution is possible with the help of an integrated numerical and analytical approach as well.

scholarly journals Investigation of the forced spatial vibrations of a wood shaper, caused by rotor’s unbalance of its electric motor and by cutting forces

2020 ◽  
Vol 317 ◽  
pp. 02001
Author(s):  
Valentin Slavov ◽  
Georgi Vukov
Keyword(s):  
Mechanical System ◽  
Analytical Solutions ◽  
Electric Motor ◽  
Degrees Of Freedom ◽  
Rigid Bodies ◽  
Damping Properties ◽  
System Of Differential Equations ◽  
Numerical Investigations ◽  
Matrix Modeling ◽  
Analysis And Synthesis

Mechanic-mathematical matrix modeling of the forced spatial vibrations of a wood shaper is performed in this study .The wood shaper is modeled as a mechanical system of three rigid bodies, which are connected by elastic and damping elements with each other and with the motionless floor. This mechanical system has 18 degrees of freedom. Formulas and algorithms are developed for computer calculating, analysis and synthesis of designing and investigating of this machine. This study renders an account the geometric, kinematic, mass, inertia, elastic and damping properties of the machine. A system of differential equations is derived. Analytical solutions are presented. The study presnts results of the numerical investigations of the forced spatial vibrations by using parameters of a particular machine. They allow to select parameters that reduce harmful vibrations for people and constructions.

Lagrangian descriptions of dissipative systems: a review

2020 ◽  
pp. 108128652097183
Author(s):  
Alberto Maria Bersani ◽  
Paolo Caressa
Keyword(s):  
Differential Equations ◽  
Equations Of Motion ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Dissipative Systems ◽  
Lagrangian Formalism ◽  
Lagrangian Description ◽  
System Of Differential Equations ◽  
Helmholtz Conditions ◽  
Necessary And Sufficient

In this paper, we review classical and recent results on the Lagrangian description of dissipative systems. After having recalled Rayleigh extension of Lagrangian formalism to equations of motion with dissipative forces, we describe Helmholtz conditions, which represent necessary and sufficient conditions for the existence of a Lagrangian function for a system of differential equations. These conditions are presented in different formalisms, some of them published in the last decades. In particular, we state the necessary and sufficient conditions in terms of multiplier factors, discussing the conditions for the existence of equivalent Lagrangians for the same system of differential equations. Some examples are discussed, to show the application of the techniques described in the theorems stated in this paper.

Identifying Sets of Constraint Forces by Inspection

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
Carlos M. Roithmayr ◽  
Dewey H. Hodges
Keyword(s):  
Mechanical System ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
The Body ◽  
Vector Form ◽  
Constraint Forces ◽  
Constraint Force ◽  
Symbolic Algebra ◽  
Constraint Equations ◽  
Angular Velocities

A mechanical system is often modeled as a set of particles and rigid bodies, some of which are constrained in one way or another. A concise method is proposed for identifying a set of constraint forces needed to ensure the restrictions are met. Identification consists of determining the direction of each constraint force and the point at which it must be applied, as well as the direction of the torque of each constraint force couple, together with the body on which the couple acts. This important information can be determined simply by inspecting constraint equations written in vector form. For the kinds of constraints commonly encountered, the constraint equations are expressed in terms of dot products involving velocities of the affected points or particles and angular velocities of the bodies concerned. The technique of expressing constraint equations in vector form and identifying constraint forces by inspection is useful when one is deriving explicit, analytical equations of motion by hand or with the aid of symbolic algebra software, as demonstrated with several examples.

scholarly journals CONSTRUCTION OF DYNAMIC INSTABILITY ZONES FOR HIGH STUCTURES UNDER SEISMIC IMPACT

2020 ◽  
Vol 2 (2) ◽  
pp. 42-50
Author(s):  
V Fomin ◽  
◽  
І Fomina ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Coordinate System ◽  
Mechanical System ◽  
Dynamic Instability ◽  
Vertical Component ◽  
System Of Differential Equations ◽  
High Rise ◽  
Concentrated Masses ◽  
Seismic Impact

Seismic impacts create the possibility of parametric resonances, i.e. the possibility of the appearance of intense transverse vibrations of structure elements (in particular, of high-rise structures) from the action of periodic longitudinal forces. As a design model of a high-rise structure, a model is used which adopted in the calculation of high-rise structures for seismic effects, - a weightless vertical rod (column) rigidly restrained at the base with a system of concentrated masses (loads) located on it (Fig. 1). By solving the differential equation of the curved axis influence function for a rod is constructed by means of which influence coefficients are determined for the rod points, in which the concentrated masses are situated. These coefficients are elements of the compliance matrix . Next, the elements of the stiffness matrix are determined by inverting the matrix . Using a diagonal matrix of the load masses and matrix a system of differential equations of free vibrations of a mechanical system, consisting of concentrated masses, is constructed, and the frequencies and forms of these vibrations are determined. From the vertical component of the seismic impact, its most significant part is picked out in the form of harmonic vibrations with the predominant frequency of the impact. Column vibrations are considered in a moving coordinate system, the origin of which is at the base of the column. The forces acting on the points of the mechanical system (concentrated masses) are added by the forces of inertia of their masses associated with the translational motion of the coordinate system. The forces of the load weights and forces of inertia create longitudinal forces in the column, periodically depending on time. Further, the integro-differential equation of the dynamic stability of the rod, proposed by V. V. Bolotin in [8], is written. The solution to this equation is sought in the form of a linear combination of free vibration forms with time-dependent factors. Substitution of this solution into the integro-differential equation of dynamic stability allows it to be reduced to a system of differential equations with respect to the mentioned above factors with coefficients that periodically depend on time. For some values of the vertical component parameters of the seismic action, namely the frequency and amplitude, the solutions of these equations are infinitely increasing functions, i.e. at these values of the indicated parameters, a parametric resonance arises. These values form regions in the parameter plane called regions of dynamic instability. Next, these regions are being constructed. A concrete example is considered.

scholarly journals THE RESEARCH RESULTS OF THE LINEAR ELECTROMAGNETIC MOTOR MATHEMATICAL MODEL

2010 ◽  
Vol 2 (1) ◽  
pp. 99-102
Author(s):  
Marijanas Molis
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Supply Voltage ◽  
Traction Force ◽  
Electromagnetic Properties ◽  
System Of Differential Equations ◽  
Transitional Processes ◽  
Linear Electromagnetic Motor ◽  
The Mathematical Model ◽  
Secondary Element

Development of the mathematical model of the linear electromagnetic motor and the dependencies of the inductance and traction force on the secondary element position expressed by mathematical equations, are presented in this research article. The dependency of the inductance on the secondary element position was obtained, approximating the inductance change diagram obtained experimentally. Also, using the theory of electromechanical energy transformation, mathematical expressions of the dependency of the traction force on the secondary element position were obtained. Mathematical model of the linear electromagnetic motor is composed of the system of differential equations. The Runge – Kutta calculation method was used to solve these equations. The transitional processes of the current, speed and secondary element position obtained with the solution of the system of differential equations at different supply voltage also the transitional processes of the dynamic traction force obtained at 24 V supply voltage of the motor. All obtained results of the dependencies and transitional processes of the mathematical model are presented in the graphic form. In accordance with the obtained results of the mathematical model the conclusions were formulated, specifying electromagnetic properties of the linear electromagnetic motor.

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