Periodic Ateb-functions Research Articles

Моделювання динаміки колісних транспортних засобів під час руху шляхом із нерівностями

The paper outlines the main provisions of the methodology for analytical research of the individual road irregularities impact on the longitudinal-angular oscillations of the wheeled vehicle sprung body. A flat system of three bodies (front, rear bridges, sprung part), whose relative motion is in the vertical plane, was chosen as the physical model for the research. Its peculiarity is that the sprung and unsprung parts interact with each other as elastic shock absorbers with non-linear characteristics of the restoring force. As for road irregularities, it is considered that they are described by smooth functions and the tires are in constant contact with the supporting surface during the wheeled vehicle movement. A mathematical model of the sprung part dynamics was built, which is a second-order nonlinear differential equation with the following feature: its right-hand side is a piecewise continuous function. Based on physically justified assumptions, the differential equation solution was constructed, which describes the relative longitudinal-angular oscillations of the sprung part. It is based on the idea of using: the special periodic Ateb-functions to construct the solution of differential equations with exponential nonlinearity; the construction of asymptotic approximations using the specified functions for new classes of differential equations. Taken together, the above made it possible to obtain differential equations in the standard form, which describe the amplitude-frequency characteristics of the sprung part oscillations. As for the individual irregularities influence on the sprung part dynamics, the amplitude of the longitudinal-angular oscillations of the exit from the irregularity is smaller: at higher vehicle speed; for suspension systems with a regressive change law of the restoring force of elastic shock absorbers; at smaller values of their static deformation for the progressive characteristic of the suspension system (and at larger values for the regressive one). Analytical dependences were obtained, calculations were made and dependences were constructed that describe the amplitude and frequency of these oscillations caused by the irregularities parameters and motion speed.The obtained results can be a basis not only for evaluating the quality of the suspension system, but also for developing algorithms for managing the suspension stiffness in order to improve operational characteristics.

Method of normal oscillations and substantiation of the choice of parameters for certain nonlinear systems with two degrees of freedom

On the example of the plane model of wheeled vehicle oscillations with adaptive power characteristic of the suspension system, the methodology for selecting its main parameters that would maximize the movement smoothness is developed. To solve this problem, the mathematical model of relative oscillations of the sprung part is constructed, provided that they are carried out in the vertical plane. The latter represents the system of two nonlinear differential equations describing the relative displacement of the center of mass of the sprung part and the angle of rotation of the latter around the transverse axis passing through the center of mass of the specified part. To construct the approximate analytical solution of this equations system, and thus to describe the main parameters that determine the relative position of the sprung part under reasonable assumptions, the method of normal oscillations of nonlinear systems with concentrated masses is used. This made it possible to obtain the system of ordinary differential equations of the first order that describe the amplitude–frequency characteristics of the sprung part vibrations. Due to the analysis of the latter it is determined that at a certain ratio between the parameters describing the power characteristics of the suspension system, it can perform isochronous vertical and longitudinal–angular oscillations, and thus it is possible to achieve maximum comfort in transporting passengers or dangerous cargo over rough terrain. The main obtained results can be used to create the software product for adaptive suspension, and their validity is confirmed by: a) in passing to the limit, obtaining results known from literary sources; b) generalization, based on the use of periodic Ateb-functions, of widely tested analytical methods for constructing solutions of differential equations with strong nonlinearity.

Mathematical model of dynamics of vibrating systems working environments

Using the apparatus of the special periodic Ateb-functions in combination with the asymptotic methods of nonlinear mechanics, the nonlinear mathematical models of motion of working environment of the oscillation system, which dependences take into account resilient and viscid making tensions from descriptions of the deformation state of environment, her physical and mechanical properties and features of co-operation of environment with the oscillation system, are worked out. The nonlinear model for describing the dynamics of the working environment of oscillating systems is more flexible, because the nonlinearity index, which depends on the type of working load, significantly affects the results of the oscillating loading process. It allows us to take into account the type of load, and, accordingly, increase the level of adequacy of the constructed analytical model of the oscillatory process that needs to be investigated. Taking into account this model, the study of various processes in oscillating systems can be carried out, in particular in different modes of vibration processing.

ON APPROXIMATION OF PERIODIC ATEB-FUNCTIONS

Two versions of approximation formulae for periodic Ateb-sine and Ateb-cosine in the first quarter of their common period are proposed. The first version is a Pade type approximation derived when constructing analytical solution of corresponding integral equation by iteration method with transforming the power series into a closed sum by Shanks’ formula. Two iteration approximations are considered. The first one is more concise but of worse approximation accuracy which deteriorates with increasing the argument value. To improve the approximation accuracy a hybrid approximation is proposed when the values of the Ateb-functions in the beginning (for the cosine) and in the end (for the sine) of the quarter period are computed by a separate formula obtained a priory by the asymptotic method. The comparison analysis of the approximate and exact values of the special functions indicates the error of the approximation proposed to be less than one per cent. The second variant of approximation is by replacing the periodic Ateb-functions by trigonometric functions of specific argument. The arguments are chosen so that the values of the special functions are exact at specific points of the quarter period. Five such collocation points are introduced in the paper. To implement this version of approximation a separate table of the values of the periodic Ateb-functions at the collocation points is compiled. The computational examples presented in the paper show the approximate values of the special functions obtained by the second version of approximation to have a good accuracy.

КОЛИВАННЯ ІМПУЛЬСНО НАВАНТАЖЕНОГО ОСЦИЛЯТОРА ЗІ СТЕПЕНЕВИМ ПОЗИЦІЙНИМ ТЕРТЯМ

Nonstationary oscillations of the oscillator with nonlinear positional friction caused by an instantaneous force pulse are described. The power dependence of the positional friction force on the displacement of the system, which generalizes the known models, is accepted. The corresponding dynamics problems were solved precisely by the method of adding and approximated by the method of energy balance. In the study, using periodic Ateb-functions, an exact analytical solution of the nonlinear differential equation of motion was constructed. Compact formulas for calculating oscillation ranges and half-cycle durations are derived. It is shown that the decrease in the amplitude of oscillations, as well as under the action of the force of linear viscous resistance, follows the law of geometric progression. The denominator of the progression is less than one and depends on the positional friction constants, in particular on the nonlinearity index. Thus, we have not only a decrease in the amplitude of oscillations, but also an increase in the durations of half-cycles, which is characteristic of nonlinear systems with a rigid force characteristic. Approximate displacement calculations use Pade-type approximations for periodic Ateb-functions. The error of these approximations is less than one percent. From the obtained analytical relations, as separate cases, the known dependences covered in the theory of oscillations for linear positional friction follow. It is shown that even in the case of nonlinear positional friction the process of oscillations caused by an instantaneous momentum has many oscillations and is not limited in time. In the case of power positional friction, the oscillation ranges of the pulse-loaded oscillator can be calculated by elementary formulas. The calculation of displacements in time is associated with the use of periodic Ateb-functions, the values of which are not difficult to determine by known asymptotic formulas. Calculations confirm that the obtained approximate formula does not give large errors. In order to verify the adequacy of the obtained analytical solutions, numerical computer integration of the original nonlinear differential equation of motion was performed. The results of the calculation, which lead to analytical and numerical solutions of the Cauchy problem, are well matched.

Influence of Impulse Disturbances on Oscillations of Nonlinearly Elastic Bodies

A method for studying the effect of impulse perturbation on the longitudinal oscillations of a homogeneous constant cross-section of the body and the elastic properties of a material which satisfies the essentially nonlinear law of elasticity has been developed. A mathematical model of the process is presented, which is an equation of hyperbolic type with a small parameter at the discrete right-hand side. The latter expresses the effect of impulse perturbation on the oscillatory process. As for the boundary conditions considered in the work, they are classic of the first, second and third genera. The methodology is based on: the principle of oscillation frequency in nonlinear systems with many degrees of freedom and distributed parameters; basic provisions of asymptotic methods of nonlinear mechanics; the idea of using special periodic Ateb-functions to construct solutions of some classes of nonlinear differential equations; properties of completeness and orthonormality of functions that describe the forms of oscillations of undisturbed motion. In general, the above allowed to obtain relations that describe for the first approximation the defining parameters of the oscillations of an elastic body. Their peculiarity is that even for undisturbed motion, the natural frequency of oscillations depends on the amplitude, and therefore, under the action of a periodic (over time) pulse force on the elastic body, both resonant and nonresonant processes are possible in the latter. It, in contrast to an elastic body with linear or quasilinear elastic properties of the body is determined not only by its basic physical and mechanical properties, but also by the amplitude of oscillations. As a special case, the oscillations of the body under the action of a constant periodic momentum perturbation are considered. It is shown that for the nonresonant case for the first approximation it does not affect the laws of change of amplitude and frequency of the process. As for the resonant is the amplitude of origin through the main resonance significantly depends not only on the speed but also on the points of action of the pulsed perturbation. Moreover, the closer the point of application of the pulsed force to the middle of the elastic body under boundary conditions of the first kind is greater (for boundary conditions of the second kind closer to the end).

Interpreting the Main Power Characteristics Choice of the Wheel Vehicles Guided Cushioning System

The method of investigation of the non-linear vibrations of the wheel vehicles with the guided cushioning system has been developed. It is based on the idea of the perturbation methods combined with the theory of special periodic Atebfunctions. All these made possible to obtain analytical relations, which describe the characteristic features of the cushioning area vibrations. These relations can be the basis for creation of the software product of the guided cushioning system with the purpose to minimize the dynamic loadings on the transported people and cargo and to increase the stability characteristics of the vehicle movement along the curvilinear areas of the bump road.

CESSATION OF FREE VIBRATIONS OF A NONLINEAR ELASTIC SYSTEM DUE TO VISCOUS RESISTANCE

The paper deals with free vibrations of a system with power-law nonlinear elasticity subjected to power-law viscous resistance. The relation between the nonlinearity indices is determined when the impact of the viscous resistance force causes the vibrations to die away. In this case the vibrations are limited in time i.e. consist of a finite number of cycles analogous to a system with Coulomb dry friction. The research exploits the energy balance method. The periodic Ateb-functions are used to obtain an approximate formula for the work of dissipative force over a semi-cycle of vibrations. A recursive power-law equation for the vibration swings is derived from the condition of equality of the work to the potential energy change. By analyzing the change of the coefficient in the equation, which is related to the change of the semi-cycle number as well as the vibration swings, the condition for the equation to have no positive root is determined, which means that the vibrations die away. The condition is formulated in the form of an inequality. It is shown to generalize the results previously known. The theoretical inferences are verified by numerical integration of the nonlinear differential equation of motion. It is shown that under the conditions proposed in the paper the free vibrations consist of a finite number of cycles even if dry friction is absent from the system. Special cases are highlighted, when the approximate energy balance method results into exact computational formulae. The length of the cycles increases during the motion since it depends on the swing of damped vibrations in the essentially nonlinear system with rigid force characteristics considered.

АНАЛІТИЧНИЙ МЕТОД ДОСЛІДЖЕННЯ ПОЗДОВЖНЬО-КУТОВИХ КОЛИВАНЬ КОЛІСНИХ ТРАНСПОРТНИХ ЗАСОБІВ ІЗ НЕЛІНІЙНОЮ СИЛОВОЮ ХАРАКТЕРИСТИКОЮ СИСТЕМИ ПІДРЕСОРЮВАННЯ

Influence of nonlinear restorative force of elastic elements and resistance force of damping devices of the suspension system on longitudinal-angular oscillations of the sprung part of wheeled vehicles is investigated. The basis of the research is the idea of asymptotic integration of equations with degree nonlinearity, which describe longitudinal-angular oscillations based on the use of special periodic Ateb-functions. The obtained analytical dependencies determine the law of basic parameters variation of the own damped oscillations of the sprung part as a function of the main power factors of the suspension system. The analysis of the obtained relations establishes that for the suspension system with the progressive law of restorative force variation of the elastic elements ( 0   ): ‒ the larger value of the oscillation amplitude the larger value of the own frequency (at the same values of static deformation of the elastic elements); ‒ the larger value of static deformation of the elastic elements (at the same amplitudes of the transverse angular oscillations and the parameter indicating the deviation of the elastic properties of the shock absorbers from the linear law –  ) the smaller value of the own frequency of oscillations; ‒ the overload, acting to the driver and the transported people, caused by the vibrations of the sprung part, is less than for the case of elastic elements with the linear law of restorative force (at the same values of static deformation of the sprung part) for small oscillation amplitudes. With regard to the suspension system with the regressive law of restorative force variation, the influence of oscillations of the sprung part is backward as for the progressive characteristic of the suspension system. As a result, it should be noted that the ergonomic requirements of operation are more satisfied by the progressive characteristic with static deformation of the suspension system м 2 , 0 at the change of the non-linearity parameter within 3 / 2 0   than the oscillation amplitude 16 , 0 025 . 0    a ; in the case of static deformation м 15 , 0 and 3 / 2 0   at the amplitude of transverse angular oscillations 2 , 0 05 , 0    a . As for the influence of the resistance forces of damping devices, they, at the real values of their parameters, slightly affect the law of time variation of the amplitude of the longitudinal-angular oscillations. The technique, based on which the above results were obtained, can be generalized to determine the response of the sprung part to the effect of single path irregularities, determine the critical speed of steady motion along curved sections of the way; the dependencies – during the design and modernization of suspension systems or the creation of a software product of the controlled suspension system.

ON THE POSSIBILITY OF STAGNATION OF FREE OSCILLATIONS OF A NONLINEARLY ELASTIC OSCILLATOR WITH A LINEAR VISCOUS RESISTANCE

The free oscillations of an oscillator with a power-nonlinear elasticity characteristic under the action of linear viscous resistance are considered. Using the energy balance method, which is widespread in mechanics, the calculation of the amplitudes of free damped oscillations is reduced to calculating the roots of an algebraic equation, which has an exact analytical solution only with linear elasticity of the oscillator. In the case of an arbitrary positive indicator of nonlinear elasticity, a numerical solution of the equation is required. For this, the Newton's iterative method was used in the work, which has fast convergence of iterations at an arbitrary initial approximation. According to the results of the analysis of the coefficients of the equation established, that in the case of a rigid characteristic of elasticity, when the nonlinearity is greater than unity, the oscillations are reduced to a finite number of decaying ranges, that is, they are limited in time, and in the case of a soft characteristic of elasticity, when the nonlinearity is less than unity, they continue to infinity, as linear dissipative oscillator. The research is given by the method of energy balance and numerical integration of the differential equation of oscillations on a computer. The work of the force of viscous resistance is calculated approximately using periodic Ateb functions that accurately describe free undamped oscillations in the absence of resistance. As a result, approximate iterative dependences are obtained for calculating the amplitudes of the ranges that decay during movement. The numerical results obtained using approximate formulas and numerical computer integration of the nonlinear Cauchy problem are compared. Their satisfactory agreement was noted. A satisfactory agreement was noted between the results for both hard and soft elastic characteristics, which confirmed the adequacy of approximate analytical solutions to the dynamics problem. The main advantage of the described approximate calculation method is that there is no need to build an analytical solution to the nonlinear differential equation of motion of the oscillator, which is a rather complicated mathematical problem. Furthermore, it made it possible to establish conditions under which the oscillator with a viscous and dry friction resistance have similar oscillation properties.

GENERALIZING GABOR FILTERS BASED ON ATEB-FUNCTIONS

Image filtering attempts to achieve greater resolution. There is a large number of filters that allows you to bring images with clear borders. In addition, noise is present when digitizing images. One of the most common types of filtering is the Gabor filter. It allows you to restore the image with the contour allocation at a certain frequency. Its core looks like elements of the Fourier basis, which is multiplied by Gaussian. The widespread use of Gabor filters for filtration is due to the fact that it gives a strong response at those points of the image where there is a component with local features of frequency in space and orientation. It is proposed to use the Ateb-Gabor filter, which greatly expands the well-known Gabor filter. The Ateb-Gabor filter combines all the properties of a harmonic function, which is multiplied by Gaussian. As a harmonic function, it is proposed to use the Ateb-functions that greatly extend the trigonometric effect. The developed filter is applied to the images. The Ateb-Gabor filter depends on the frequency and directions of the quasiperiodic structure of the image. Usually, to simplify the task, the average image frequency is calculated. It is unchanged at every point. Filtration of images is based on the generalized Ateb-Gabor filter. Influence of filtering parameters on images is investigated. The properties of periodic Ateb-functions are investigated. The value of the period from which the filtering results depend on is calculated. Ateb-Gabor filtering allowed for wider results than the classic Gabor filter. The one-dimensional Gabor filter based on the Ateb-functions gives the possibility to obtain more lenient or more convex forms of function at the maximum described in this study. In this way, filtration with a large spectrum of curves can be realized. This provides quick identification, since a more versatile kind of filtering has been developed.

УЗАГАЛЬНЕННЯ ФІЛЬТРІВ ГАБОРА НА ОСНОВІ ATEB-ФУНКЦІЙ

Image filtering attempts to achieve greater resolution. There is a large number of filters that allows you to bring images with clear borders. In addition, noise is present when digitizing images. One of the most common types of filtering is the Gabor filter. It allows you to restore the image with the contour allocation at a certain frequency. Its core looks like elements of the Fourier basis, which is multiplied by Gaussian. The widespread use of Gabor filters for filtration is due to the fact that it gives a strong response at those points of the image where there is a component with local features of frequency in space and orientation. It is proposed to use the Ateb-Gabor filter, which greatly expands the well-known Gabor filter. The Ateb-Gabor filter combines all the properties of a harmonic function, which is multiplied by Gaussian. As a harmonic function, it is proposed to use the Ateb-functions that greatly extend the trigonometric effect. The developed filter is applied to the images. The Ateb-Gabor filter depends on the frequency and directions of the quasiperiodic structure of the image. Usually, to simplify the task, the average image frequency is calculated. It is unchanged at every point. Filtration of images is based on the generalized Ateb-Gabor filter. Influence of filtering parameters on images is investigated. The properties of periodic Ateb-functions are investigated. The value of the period from which the filtering results depend on is calculated. Ateb-Gabor filtering allowed for wider results than the classic Gabor filter. The one-dimensional Gabor filter based on the Ateb-functions gives the possibility to obtain more lenient or more convex forms of function at the maximum described in this study. In this way, filtration with a large spectrum of curves can be realized. This provides quick identification, since a more versatile kind of filtering has been developed.

Basic principles for the approval of elective parameters of the adaptive surface of collected vehicles of special purpose

The use of wheeled vehicles of special purpose systems pidresoryuvannya relevant class production cars leads, at best, with output down some of its elements. Therefore, there is a need to create a system of sprinkling with such power characteristics, which on the one hand would as much as possible satisfy the ergonomic conditions, and on the other hand - provided special characteristics (stability of motion, permeability, controllability, high efficiency of driving east, maneuverability when changing the location of personal warehouse). To substantiate the choice of the main power parameters of wheeled vehicles with an adaptive sprinkler system, a mathematical model of vertical fluctuations of its submersible part during a movement along the path with an arbitrary placed system of inequalities is constructed. It represents non-autonomic nonlinear differential equations of the second order. For the latter, a method for constructing the first approximation of the asymptotic solution is developed. The method is based on the general provisions of perturbation methods and the idea of using special periodic Ateb-functions for constructing the solutions of some classes of nonlinear differential equations. In the aggregate, the above resulted in obtaining analytical relations that describe the determining parameters of oscillations of the submerged part of the vehicles during the movement by means of a system of randomly placed inequalities. By analyzing the latter, in particular, it was found that the amplitude of the output of single inequalities for larger lengths of the latter is greater, and for smaller inequalities at higher speeds of the KTZSP. The obtained analytical dependencies can serve as the basis for the choice of the main power parameters of an adaptive suspension, which would minimize the dynamic loads on people and goods transported.

Modeling the motion of an oscillator with a soft elastic characteristic

The free oscillations of a system with one degree of freedom are considered under the assumption that the elasticity of a spring is proportional to the cubic root of its deformation. Two forms of the analytical solution of the nonlinear differential equation of motion of the oscillator are obtained. In the first displacement of the oscillator in time is expressed in terms of incomplete elliptic integrals of the first and second kind. In the second form, the solution is expressed in terms of periodic Ateb-functions. The tables of the involved functions are made, which simplify the calculation. Formulas are also derived for calculating the oscillation periods when the oscillator is signaled or the initial deviation from the equilibrium position or the initial velocity (instantaneous pulse) in this position. The dependence of the oscillation period on the parameters of the oscillator and the initial conditions is established. Examples of calculations of oscillations are presented with the use of compiled tables of special functions and using the proposed approximations of the Ateb-functions. Comparison of numerical results obtained by different methods is made.

Construction of asymptotic solutions of certain boundary-value problems for the nonautonomous wave equation

Applying the periodic Ateb-functions we construct single-frequency asymptotic approximations of solutions of problems for the nonlinear nonautonomous wave equation.

On asymptotic approximation of a solution of a boundary-value problem for a nonlinear nonautonomous equation

On the basis of periodic Ateb functions, in the resonance and nonresonance cases, we construct the asymptotic approximation of one-frequency solutions of a boundary-value problem for a nonlinear nonautonomous equation.

Application of ateb-functions to the construction of solutions of some nonlinear partial differential equations

We construct asymptotic approximations of one-frequency solutions of some nonlinear partial differential equations by using periodic Ateb-functions.

On the construction of asymptotic approximations for a nonautonomous wave equation

For a nonautonomous wave equation with homogeneous boundary conditions, we construct one-frequency approximations of asymptotic solutions by using periodic Ateb-functions. Resonance and nonresonance cases are considered.

Construction of one-frequency solutions of boundary-value problems for a nonautonomous wave equation

By using special periodic Ateb-functions, we construct asymptotic representations of one-frequency solutions of boundary-value problems for a nonautonomous wave equation.

On a method for constructing one-frequency solutions of a nonlinear wave equation

A method for constructing one-frequency solutions of nonlinear wave equations is suggested. This approach is based on a modified representation of asymptotic expansions by using special periodic Atebfunctions. This method makes it possible to obtain approximate solution of the problem under consideration without difficulty.