System Of Second-order Differential Equations Research Articles

МАТЕМАТИЧЕСКАЯ МОДЕЛЬ КОЛЕБАНИЙ В ЦЕПОЧКЕ ФИЗИЧЕСКИХ ОСЦИЛЛЯТОРОВ

The analysis of the oscillatory process in a linear system of spring oscillators from several spherical bodies is carried out. The case is considered when one end of the system is fixed, and force acts on the opposite end. The cause of oscillations is the initial displacement of one or more bodies from the equilibrium state or an external load. It is shown that at a sufficiently large amplitude of the oscillations, neighboring bodies can come into contact with each other, as a result of which their velocities will change stepwise in accordance with the laws of conservation of kinetic energy and conservation of momenta. The mathematical model of the process is presented in the form of a system of second-order differential equations. The solution to the problem was obtained using the numerical Rung-Kutta method. For calculations, a program was developed in the C ++ algorithmic language, which allows you to build phase portraits taking into account the collision of neighboring elements, as well as determine the time dependences of the deviation and the speed of this deviation of each body. Variant calculations showed that the possibility of using a mathematical or physical model of the oscillatory process is determined by the ratio of the maximum amplitude to the diameter of the balls. If this ratio is much less than unity, you can use the model of mathematical oscillators. Otherwise, it is necessary to apply the model of physical oscillators. The adequacy of the physical model is maintained under any initial conditions and for any external load. The magnitude of the jumps usually increases in the direction from the first to the last oscillator, to which the force is applied. Fracture on the graph of the time dependence of the deviations of the balls from the equilibrium position at low load is usually manifested to a small extent. In this case, you should refer to the analysis of the phase portrait. Key words: oscillatory process, mathematical model, oscillator, phase portrait, physical oscillator.

The features of behavior of the Lyapunov's exponents from the symmetry of thermodynamic potential described by lifshitz invariant

The influence of the thermodynamic potential ( n ) symmetry on the behavior of the Lyapunov exponents is investigated. The calculation of the system of second-order differential equations was performed by a multi-step variable-order BDF method. The Lyapunov coefficients were calculated using Python software using Skipy, JiTCODE libraries. It is established that for values of n ≥ 5, the dynamics of the incommensurate superstructure is determined by the spatial change in the amplitude of the order parameter (the first integral of the motion of the incommensurate superstructure). For n < 5, the dynamics of a incommensurate superstructure is determined by the spatial change of the phase of the order parameter and its perturbation. It is shown that the dynamics of the superstructure is not affected by the ferroelectric or ferroelastic nature of the incommensurate superstructure. In order to obtain a detailed picture of the dynamics of the incommensurate superstructure, phase portraits were investigated in the coordinates R , R `, and φ at different values of the anisotropic interaction of the superstructure and the initial phase values of the order parameter φ 0 . It is established that the dynamics of this system is determined by the cascades of transition to chaos and areas of self-organization of the superstructure. It is shown that at the final stage of the transition to chaos, the system is characterized by the emergence of a chaotic attractor, which has a more complex phase portrait structure. The projection of such an attractor consists of two symmetrical parts with respect to the axis of the abscissa. The movement of a typical trajectory of a chaotic attractor can be divided into two phases. In the first phase of the trajectory, chaotic movements are made along the turns of the upper (lower) part of the attractor from time to time, passing to the boundary of the localization region where the stability of the upper (lower) symmetric boundary cycles is lost. At an unpredictable time, the trajectory moves from the upper (lower) part of the attractor to its upper (lower) part. This process is repeated infinitely many times. Thus, the transition to chaos has features characteristic of the Feigenbaum scenario (infinite cascade of bifurcations of doubling cycles), and for mixing (unpredictable mixing between the upper and lower parts of the emerging chaotic attractor). The self-organization process can be associated with the localization of a wave vector of incommensurate superstructure at a commensurate higher order value. It is shown that the distortion of the order parameter phase at the edge of the crystal can be related to boundary conditions in the first approximation. According to the authors, such cascading transitions to chaos are related to the modes of existence of a incommensurate superstructure. The emergence of chaos in the third stage is caused by the transition of a incommensurate superstructure to its stochastic mode of existence. In a stochastic mode of existence of a incommensurate superstructure, a chaotic, possibly incommensurate phase arises, which is a prototype of this chaotic state. It should be noted that the initial conditions (boundary conditions) stabilize the system, thereby leading to the removal of the degeneracy of the system, and the transition of a incommensurate superstructure to a state characterized by the existence of long-period proportional phases. Key words: incommensurate superstructures, phase portraits, Lyapunov’s exponents.

НАПРЯЖЕННОЕ СОСТОЯНИЕ ДВУХСЛОЙНОЙ ПРЯМОУГОЛЬНОЙ ПЛАСТИНКИ ПРИ СДВИГЕ. УПРОЩЕННАЯ ДВУМЕРНАЯ МОДЕЛЬ

A considerable attention is given to studying the stressed state of adhesive joints due to the structure’s bearing capacity, which is usually determined by the strength of the connections, where the stressed state is irregular. Most of the existing mathematical models of joints are one-dimensional, i.e. they imply an even distribution of stresses across the width of joints. However, there are constructions for which the classical models are not applicable. In order to calculate the stressed state of such joints it’s necessary to take into consideration the irregularity of stresses not only in length but also in width of joints. To solve such problems, a simplified two-dimensional model of the overlapping adhesive joint of the rectangular plates is proposed. The simplification consists in the fact that the displacements of the layers are considered only along one of the axes. The model is actually a two-dimensional generalization of the Volkersen classical connection model. The stresses are considered even distributed over the layer thickness and the adhesive layer works only on a shift. These simplifications allowed us to obtain an analytical solution to the studied problem. The problem of the stressed state of the adhesive jointing of the two rectangular plates is solved, one of which is rigidly fixed along one side, and the second one is loaded with an uneven shear load on the opposite side. The problem is reduced to a system of second-order differential equations in partial derivatives with respect to longitudinal displacements of the two carrier (outer) layers. The solution is constructed using the method of separating the variables, in the form of a functional series consisting of the Eigen functions of the spectral problem. The boundary conditions at the unloaded ends are accurately satisfied. The satisfaction of the boundary conditions on the sides leads to a system of linear equations for the unknown coefficients of the functional series. The convergence of the obtained solution is proved. The model problem is solved and the numerical results are compared with the results of the calculations performed using the finite element method. It is shown that the proposed approach has a sufficient accuracy for engineering tasks.

Investigation of seismic wave reaction of spatial structure

The stochastic behavior of the spatial structure under seismic action was studied on the basis of theories and methods of nonlinear mechanics, finite elements, traveling wave and wavelet analysis. The spatial concrete structure in the form of a square plate, which rests on four columns rigidly fixed in the foundation was presented. A probabilistic simulation of acceleration of seismic action with different magnitudes using the statistical method of Ruiz and Penzien was performed. The effect of horizontal seismic displacement in the soil on the design was taken into account with the help of simulation of a transverse bending traveling wave in the form of an initial imperfection of the four columns of the structure. Mathematical models of non-stationary stochastic vibrations structure in the form of a system of second-order differential equations in generalized coordinates were formed on the basis of the Dahlberger-Lagrange method. The influence of the traveling waves and surface pressure on the static characteristics structure was estimated. The nonlinear static problem by the Newton-Ruffson method and the stability problem by Lanczos method were solved. A modal analysis of the spatial structure without and with traveling waves and surface pressure by the Lanczos method was carried. Realizations of dynamic characteristics of the structure: acceleration, velocity and displacement were obtained by the direct method of numerical integration of Runge-Kutta of the fourth order. The wavelet analysis of the seismic acceleration and structure reactions to the seismic action of different magnitudes was performed using discrete orthogonal (Dobesh4) and continuous nonorthogonal (Morle) one-dimensional complex wavelet functions. Wavelet-spectrograms and Fourier-images of the seismic acceleration and of the structure reactions were presented. The expediency of accounting for the flexural traveling waves in a spatial structure in the study of its seismic behavior was estimated.

A Trajectory-Based Method for Constrained Nonlinear Optimization Problems

A trajectory-based method for solving constrained nonlinear optimization problems is proposed. The method is an extension of a trajectory-based method for unconstrained optimization. The optimization problem is transformed into a system of second-order differential equations with the aid of the augmented Lagrangian. Several novel contributions are made, including a new penalty parameter updating strategy, an adaptive step size routine for numerical integration and a scaling mechanism. A new criterion is suggested for the adjustment of the penalty parameter. Global convergence properties of the method are established.

DYNAMIC LOADS INFLUENCING ON ELEMENTS OF MULTI-MOTOR HYDRAULIC DRIVE OF CCM COOLER

Based on dynamic model of multi-motor drive of continuous casting machine (CCM) coolers, mathematical model is formed and evaluation of low- frequency dynamic processes occurring in the drive is performed. Model of the coolers drive is an eight-mass dynamic system with elastic connections between masses. Description of links takes into account presence of gaps and damping properties in them. Active load in dynamic model is described by mechanical characteristic of hydraulic drive. The cooler moving beams and cooling metal mass are regarded as reactive loads. Motion of all masses of the model is described through the system of second-order differential equations. Integration of differential equations was carried out by Runge-Kutta method. To analyze influence of various­ factors on loads that arise in cooler drive, a software program was written. Evaluation of dynamic processes in the CCM cooler drive shows that dynamic component of load in the elements of machine for vertical displacement of mobile beams is of a significant value. Dynamic response factor K d in the drive elements reaches 2.2 – 2.3. Analysis of the factors influencing dynamic processes in the cooler drive reveals that the optimum ratio of masses of metal located in cooler and moving beams of the cooler is close to 1.6 – 1.7. The minimum dynamic coefficient is close to 1.5. The nature of change in drive dynamics due to viscosity of working fluid of hydraulic drive is smoothly decreasing linear dependence with a minimum corresponding to fluid viscosity value of 4·10–5 m2/s. Speed of vertical displacement of the moving beams of the coolers also affects dynamic processes that arise in their drive. At the same time, the dynamic response factor is higher with the higher speed. Within the limits of change in speed of movement of the moving beams of the cooler referred to in the paper, K d varies from 2 to 2.2. Dynamic model of multi-motor hydraulic drive of cooler of a walking-type used in the work makes it possible to analyze low-frequency dynamic processes occurring in hydraulic drive. As a result, it be comes possible to identify degree of influence of various design and power conditions of operation of hydraulic drive on low-frequency oscillations in it and, to develop optimal design solutions in terms of its operability using calculating tool.

Asymptotic representation of intermediate solutions to a cyclic systems of second-order difference equations with regularly varying coefficients

Asymptotic representation of intermediate solutions to a cyclic systems of second-order difference equations with regularly varying coefficients

On modified TDRKN methods for second-order systems of differential equations

ABSTRACTThis paper concerns modified Two-Derivative Runge–Kutta–Nyström (TDRKN) methods for solving second-order initial value problems. Compromised with the deduced order and symmetry criteria, two implicit and forth-order two-stage TDRKN schemes are derived through a mixed collocation approach. Phase and periodic stability features are examined. Numerical experiments are carried out to illustrate the effectiveness and competence of our new methods. Comparisons with existing highly accurate and efficient numerical methods are given.

Solvability of a class of boundary value problems in the space of convergent sequences

We consider two types of infinite systems of second-order differential equations with boundary conditions in the Banach sequence space c. For each system, sufficient conditions are provided for the existence of solutions. Our approach is based on the use of measure of noncompactness concept and the application of Darbo’s theorem for condensing operators. Some examples are presented to illustrate the obtained results.

Exponential matrix method for the solution of exact 3D equilibrium equations for free vibrations of functionally graded plates and shells

The present paper analyzes the convergence of the exponential matrix method in the solution of three-dimensional equilibrium equations for the free vibration analysis of functionally graded material structures. The three-dimensional equilibrium equations are written in general orthogonal curvilinear coordinates for one-layered and sandwich plates and shells embedding functionally graded material layers. The resulting system of second-order differential equations is reduced to a system of first-order differential equations redoubling the variables. This system is exactly solved using the exponential matrix method and harmonic displacement components. In the case of functionally graded material plates, the differential equations have variable coefficients because of the material properties which depend on the thickness coordinate z. For functionally graded material shells, the differential equations have variable coefficients because of both changing material properties and curvature terms. Several mathematical layers M can be introduced to approximate the curvature terms and the variable functionally graded material properties to obtain differential equations with constant coefficients. The exponential matrix is applied to solve the resulting system of partial differential equations with constant coefficients, where the used expansion has a very fast convergence ratio. The present work investigates the convergence of the proposed method related to the order N used for the expansion of the exponential matrix and to the number of mathematical layers M used for the approximation of curvature shell terms and variable functionally graded material properties. Both N and M values are analyzed for different geometries, thickness ratios, materials, functionally graded material laws, lamination sequences, imposed half-wave numbers, frequency orders, and vibration modes.

The multiplicity of solutions for a system of second-order differential equations

The multiplicity of solutions for a system of second-order differential equations

Dynamic Behavior of Axially Functionally Graded Pipes Conveying Fluid

Dynamic behavior of axially functionally graded (FG) pipes conveying fluid was investigated numerically by using the generalized integral transform technique (GITT). The transverse vibration equation was integral transformed into a coupled system of second-order differential equations in the temporal variable. TheMathematica’s built-in function, NDSolve, was employed to numerically solve the resulting transformed ODE system. Excellent convergence of the proposed eigenfunction expansions was demonstrated for calculating the transverse displacement at various points of axially FG pipes conveying fluid. The proposed approach was verified by comparing the obtained results with the available solutions reported in the literature. Moreover, parametric studies were performed to analyze the effects of Young’s modulus variation, material distribution, and flow velocity on the dynamic behavior of axially FG pipes conveying fluid.

The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

In this paper, we will focus on the dynamical behavior of a rigid body suspended on an elastic spring as a pendulum model with three degrees of freedom. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity. The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the fourth-order Runge-Kutta algorithms through Matlab packages. These solutions are represented graphically in order to describe and discuss the behavior of the body at any instant for different values of the physical parameters of the body. The obtained results have been discussed and compared with some previous published works. Some concluding remarks have been presented at the end of this work. The importance of this work is due to its numerous applications in life such as the vibrations that occur in buildings and structures.

Application of measures of noncompactness to the infinite system of second-order differential equations in ℓ p $\ell_{p}$ spaces

In this article, we use the technique based upon measures of noncompactness in conjunction with a Darbo-type fixed point theorem with a view to studying the existence of solutions of infinite systems of second-order differential equations in the Banach sequence space $\ell_{p}$ . An illustrative example is also given in support of our existence result.

Structure preserving model order reduction of large sparse second-order index-1 systems and application to a mechatronics model

ABSTRACTNowadays, mechanical engineers heavily depend on mathematical models for simulation, optimization and controller design. In either of these tasks, reduced dimensional formulations are obligatory in order to achieve fast and accurate results. Usually, the structural mechanical systems of machine tools are described by systems of second-order differential equations. However, they become descriptor systems when extra constraints are imposed on the systems. This article discusses efficient techniques of Gramian-based model-order reduction for second-order index-1 descriptor systems. Unlike, our previous work, here we mainly focus on a second-order to second-order reduction technique for such systems, where the stability of the system is guaranteed to be preserved in contrast to the previous approaches. We show that a special choice of the first-order reformulation of the system allows us to solve only one Lyapuov equation instead of two. We also discuss improvements of the technique to solve the Lyapunov equation using low-rank alternating direction implicit methods, which further reduces the computational cost as well as memory requirement. The proposed technique is applied to a structural finite element method model of a micro-mechanical piezo-actuators-based adaptive spindle support. Numerical results illustrate the increased efficiency of the adapted method.

Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method

The reproducing kernel method is a numerical as well as analytical technique for solving a large variety of ordinary and partial differential equations associated to different kind of boundary conditions, and usually provides the solutions in term of rapidly convergent series in the appropriate Hilbert spaces with components that can be elegantly computed. The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions for systems of second-order differential equations with periodic boundary conditions. A reproducing kernel space is constructed in which the periodic conditions of the systems are satisfied, whilst, the smooth kernel functions are used throughout the evolution of the method to obtain the required grid points. An efficient construction is given to obtain the approximate solutions for the systems together with an existence proof of the exact solutions is proposed based upon the reproducing kernel theory. Convergence analysis and error behavior of the presented method are also discussed. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed.

Lie and Noether point symmetries of a class of quasilinear systems of second-order differential equations

We study the Lie and Noether point symmetries of a class of systems of second-order differential equations with $n$ independent and $m$ dependent variables ($n\times m$ systems). We solve the symmetry conditions in a geometric way and determine the general form of the symmetry vector and of the Noetherian conservation laws. We prove that the point symmetries are generated by the collineations of two (pseudo)metrics, which are defined in the spaces of independent and dependent variables. We demonstrate the general results in two special cases (a) a system of $m$ coupled Laplace equations and (b) the Klein-Gordon equation of a particle in the context of Generalized Uncertainty Principle. In the second case we determine the complete invariant group of point transformations, and we apply the Lie invariants in order to find invariant solutions of the wave function for a spin-$0$ particle in the two dimensional hyperbolic space.

Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations

A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints and admitting a given Lie algebra as Vessiot-Guldberg-Lie algebra. In particular, well-known types, such as the Milne-Pinney or Kummer-Schwarz equations, are recovered as special cases of this classification. The analogous problem for systems of second-order differential equations in the real plane is considered for a special case that enlarges the generalized Ermakov systems.

Positive Solutions for a System of Second-Order Differential Equations with Integral Boundary Conditions

Positive Solutions for a System of Second-Order Differential Equations with Integral Boundary Conditions

Precise Estimates of the Walk Speed of Solutions of Second-Order Linear Systems

We consider some classes of nonautonomous second-order systems of differential equations whose coefficients are bounded by a given constant M, in particular, diagonal systems, triangular systems, and systems corresponding to a single equation. It is shown that the walk speeds of solutions of various systems from these classes fill up a certain interval, and precise estimates are obtained for the length of that interval in terms of the value of M.