Mathematical Model Of Oscillations Research Articles

Improving the damping of vehicle vibration by improving pneumatic springs

It is known that the air suspension of vehicles, in which diaphragm-type air springs are used as an elastic element, do not provide the necessary vibration damping. The reason for this is that such air springs have a relatively large passive part. As a result, a relatively small mass of compressed air crosses through the throttle installed between the air spring and the additional reservoir. This mass of air contains thermal energy, into which the energy of vibrations, which enters through the walls of the additional reservoir into the environment, has turned. This is interpreted as vibration damping, which is insufficient due to the low air mass. Therefore, hydraulic vibration dampers are installed parallel to the diaphragm air springs, which complicates and increases the cost of the vehicle. Increasing the damping properties of such air suspensions could eliminate these hydraulic vibration dampers, which would reduce costs and simplify operation. An air suspension with an improved air spring has been proposed, which has an increased effective area and a reduced "passive" capacity, an empirical formula has been built to determine its damping coefficient, as well as an expression for the stiffness coefficient. Mathematical modeling of oscillations of vehicles with different designs of pneumatic springs was carried out in order to improve their damping. The mathematical model takes into account the change in the parameters of the air spring during vibrations. The study was carried out for the diesel train DL-02. Using mathematical modeling, the effectiveness of the air suspension with an improved air spring has been proven: its damping index reaches 0.263, and the vibration damping coefficient is 45,859 kg/s, which corresponds to the values ​​​​recommended for vehicles.

Examining Beam Oscillations in the Space of Technogenic and Seismic Impact Parameters (Part I)

Free transverse oscillations of a compressed beam carrying two evenly distributed loads and two discrete masses in the span are studied. The mathematical model of oscillations is presented as a boundary value problem from the basic equation in hyperbolic partial derivatives of the fourth order in spatial coordinates, of second order in time, boundary conditions and block connection conditions. The technical theory of bending oscillations of rods based on Bernoulli’s hypothesis is used. We consider the spectral problem on the determination of eigenvalues, eigenmodes (Sturm-Liouville problem), which is necessary for the analysis of forced oscillations. It is argued that the solution by analytical methods is inexpedient in view of the large amount of transformations and calculations. The methods of separation of variables and finite differences are used. An algorithm of task solution has been developed and implemented in the Matlab software environment in the form of high precision graphoanalytic calculations. Practical conclusions have been made.

Моделювання динаміки гасника коливань із керованою частотою

Розроблено методику описання амплітудно-частотної характеристики динамічного гасника коливань, який є пружною консольною балкою із системою зосереджених мас. Математичною моделлю коливань такої системи є крайова задача із дискретною правою частиною. Використовуючи властивості системи власних функцій, які описують форми власних коливань вказаного тіла без зосереджених мас, методом регуляризації отримано аналітичні співвідношення, які описують амплітудо-частотну характеристику такого гасника коливань. Встановлено, зокрема, що його частота власних коливань приймає менші значення для: більших величин зосереджених мас, ближчого їх розміщення до кінця пружного тіла та більшої його довжини. Отримані співвідношення можуть бути базовими для налаштування вказаного типу гасників коливань з метою максимального виконання ними функціональних завдань. Ефективність застосування динамічних гасників коливань (ДГК) для гасіння коливань встановленого у транспортному засобі чутливого елемента залежить від багатьох чинників: способів і місця кріплення до підресореної частини транспортного засобу, його розмірів та ваги, матеріалу та його компоновки та ін. Сукупно зазначені чинники впливають на основні характеристики власних і вимушених його коливань, а відтак – на частину енергії, яку отримує ДГК від чутливого елемента, зумовлену рухом транспортного засобу вздовж пересіченої місцевості. Із фізичних міркувань остання значною мірою залежить від співвідношення між частотами власних коливань ДГК, чутливого елемента та підресореної частини. Отримано математичну модель ДГК, яка відповідає консольно закріпленій балці. Способом регуляризації дискретних зовнішніх сил отримано спектр власних частот ДГК, який враховує всі основні його характеристики: пружні властивості балки, її довжину, величину зосередженої маси. З використанням зазначеного вище отримано системи диференціальних рівнянь кутових коливань механічної системи підресореної частини транспортного засобу – чутливий елемент – ДГК. Програмна реалізація її дає змогу: визначити місце закріплення динамічних гасників коливань на турелі; визначити оптимальну масу динамічних гасників коливань; розрахувати оптимальні частоти власних коливань динамічних гасників коливань, закріплених на чутливому елементі, під час дії сили при навантаженні в русі транспортного засобу по пересіченій місцевості. Здійснено дослідження взаємодії турелі з динамічними гасниками коливань та обґрунтовано спосіб їх оптимального налаштування для уникнення явищ, близьких до резонансних.

Forced oscillations of beams with installation (Part II)

This article (Part II) is a continuation of Part I, published in the same edition, where the spectral problem is considered. Part II examines two types of transverse beam oscillations: forced harmonic and forced random. The beam carries its own distributed mass and discrete mass. The source of oscillations is operating equipment with an element moving during technological operations. The mathematical model of oscillations is presented as a boundary value problem from the main partial differential equation of the hyperbolic type of the fourth order in the spatial coordinate, the second order in time, the boundary conditions and the conditions of the beam sections‘ conjugation. The technical theory of the rods‘ bending oscillations is used, based on Bernoulli‘s hypothesis about the beam plane cross-sections‘ invariability. The methods of variables separation and finite differences are applied. The algorithms for solving the problems have been developed, implemented in the Matlab software environment. Verification of the proposed mathematical models is demonstrated using the specific examples. The particular examples have been carried out and the practical conclusions have been outlined.

Free oscillations of a beam with installation (1st part)

The transverse oscillations of a beam carrying a discrete mass in a span are studied. The mathematical model of oscillations is presented as a boundary value problem from the main partial differential equation of the fourth order hyperbolic type in the spatial coordinate, of the second order in time, the boundary conditions and the conditions for the sections‘ conjugation. The technical theory of the rods bending oscillations, based on Bernoulli‘s hypothesis, is used. The spectral problem of determining the eigenvalues and eigenmodes of oscillations (the Sturm-Liouville problem), necessary for analyzing the problems of forced oscillations, is considered. It is argued that the solution by analytical methods is impractical due to the large volume of transformations and cumbersome calculations. Methods of separation of variables and finite differences are used. An algorithm for solving the problem, implemented in the Matlab software environment in the form of high-precision graphic-analytical calculations, is developed. The conclusions for practical applications of the results are made.

Oscillations of the variable cross beams under seismic and technogenic influences (part II)

The forced transverse oscillations of the variable cross section compressed beams under seismic and technogenic impacts are considered. The source of oscillations is the kinematic perturbations of the beam ends in the form of a harmonic or random vector process. The mathematical model of oscillations is presented as a boundary value problem from the basic partial differential equation of the fourth order hyperbolic type in spatial coordinate and the second order in time, which is supplemented by the boundary conditions. The beam fluctuations are interpreted as the spatiotemporal random fields inhomogeneous in space and stationery in time. In the deterministic case, the amplitudes of the forced oscillations are determined, the influence of the kinematic disturbances’ frequency and the beam oscillations’ shift of their phases is studied. In the stochastic problem, the standard deviations for the deflections are calculated, the influence of the characteristic frequency and correlation of the components of the vector perturbation process on the deflections’ output function is studied. The problem is solved by the methods of the variables’ separation, finite differences and the theory of random processes. The examples are considered, the conclusions are drawn from the calculations results in the Matlab computing complex environment.

Oscillations of the variable cross beams under seismic and technogenic influences (part I)

Free compressed beam transverse vibrations of a variable cross section carrying a distributed mass are considered. The mathematical model of oscillations is presented as a boundary value problem from the basic partial differential equation of the fourth order hyperbolic type in spatial coordinate, the second order in time and boundary conditions. The technical theory of the rods’ bending vibrations, based on the Bernoulli hypothesis about the invariance of flat beam cross-sections, has been used. The task is to determine the eigenvalues, eigenmodes of oscillation and the attenuation coefficient (extended Sturm-Liouville problem). It has been argued that solving the problem by the analytical methods is impractical due to the large transformations and calculations volume. The variables separation methods, finite differences and computer technology have been used. An algorithm for solving the problem, implemented in the Matlab software environment in the form of high-precision graphical and analytical calculations, has been developed. On a concrete example of a beam, verification of the proposed mathematical model has been demonstrated. The practical conclusions have been made.

Suppression of Unstable Osclilations of Mechanical Structures

A natural phenomenon has been discovered—a natural property of nonlinear oscillations of elastic mechanical structures interacting with the environment. The indicated property, related in view of its objective reality to natural phenomena, is as follows: if the forces of excitation of structural vibrations from the environment exceed the strength of their damping, then the frequency of oscillations increases or decreases depending on whether the excitation of oscillations is associated with displacements or velocities of design points. On the basis of the natural property, a mathematical model of oscillations with a variable frequency was first constructed and the mechanisms of the implementation of three methods for suppressing unstable structural oscillations, allowing to switch from an unstable state to periodic structural oscillations, are presented.

Determination of Parameters Describing the Risk Areas of Ships Chaotic Rolling on the Example of LNG Carrier and OSV Vessel

One of the significant problems in the safe operation of vessels is the behavior of the ship on the wave. Of all degrees of freedom, the greatest threat to the safety of a ship is associated with excessive rolling. One of the best methods to improve the safety of a ship in this field is to carry out experiments on the ship model, performed at her design stage. The problem is that the model tests are costly. An alternative is to conduct simulation tests based on numerical models. The primary goal of the article is to present the results of the simulation regarding the determination of parameters describing the risk areas of chaotic rolling for the ship designed for transporting liquefied natural gas (LNG carrier) and offshore support vessel (OSV). The first discusses the state of knowledge on mathematical modeling of oscillations. Then, the theory of nonlinear differential equations is presented, and the mathematical model of ship rolling is described. This model is used to prepare and conduct a numerical simulation in the Mathematica package. The results of these studies and their discussion constitute the central part of the article. Finally, the conclusions are presented.

Mathematical Model of Oscillations of a Three-Layered Channel Wall Possessing a Compressible Core and Interacting with a Pulsating Viscous Liquid Layer

The paper deals with the formulation of a mathematical model to study a dynamics interaction of a three-layered channel wall with a pulsating viscous fluid layer in a channel. The narrow channel formed by two parallel walls was considered. The lower channel wall was a three-layered plate with a compressible core, and the upper one was absolutely rigid. The face sheets of the three-layered plate satisfied Kirchhoff's hypotheses. The plate core was considered rigid taking into account its compression in the transverse direction. Plate deformations were assumed to be small. The continuity conditions of displacements are satisfied at the layers' boundaries of the three-layered plate. The oscillations of the three-layered channel wall occurred under the action of a given law of pressure pulsation at the channel edges. The dynamics of the viscous incompressible fluid layer within the framework of a creeping motion was considered. The formulated mathematical model consisted of the dynamics equations of the three-layered plate with compressible core, Navier --- Stokes equations, and the continuity equation. The boundary conditions of the model were the conditions at the plate edges, the no-slip conditions at the channel walls and the conditions for pressure at the channel edges. The steady-state harmonic oscillations were investigated and longitudinal displacements and deflections of the plate face sheets were determined. Frequency-dependent distribution functions of amplitudes of plate layers displacements were introduced. These functions allow us to investigate the dynamic response of the channel wall and the fluid pressure change in the channel. The elaborated model can be used for the evolution of non-destructive testing of elastic three-layered elements contacting with a viscous fluid layer and being part of the lubrication, damping or cooling systems of modern instruments and units.

Mathematical modeling of oscillations during CO oxidation on Ni under reducing conditions

This paper presents the first mathematical model simulating the recently discovered oscillatory behavior during CO oxidation on a Ni foil in a continuous-flow catalytic reactor. These oscillations occur in the presence of excess CO; thus, the well-known Sales–Turner–Maple model, simulating oscillations in oxygen excess, cannot be applied. We suggest a modification of the Sales–Turner–Maple model by introducing the precursor-mediated adsorption of CO. This made it possible to simulate the oscillations under reducing conditions. The suggested microkinetic model coupled with a continuous-flow reactor model predicts that the oscillations proceed on the partially oxidized catalyst because of the periodic formation and reduction of the surface oxide. The introduction of oxygen diffusion into the subsurface layers allows the description of the experimentally observed variation in Ni color during the oscillatory cycle. The developed model successfully reproduces experimental data as the temperature range of the oscillations together with their shape and amplitudes.

Improvement of the bearing structure of the wagon-platform of the articulated type to ensure the reliability of the fixing on the deck of the railway ferry

In the article the results of studies of loading of the bearing structure of a wagon-platform of an articulated type, created on the basis of the existing construction, during transportation on the railway ferry by sea are given. In order to ensure the reliability of the fixing of the bearing structure of the wagon-platform of the articulated type relative to the deck of the railway ferry, it is proposed to equip it with knots for fixing the chain ties. To study the dynamic loading of a wagon-platform of an articulated type during transportation on a railway ferry by sea, constructed a mathematical model of oscillations with angular displacements relative to the longitudinal axis of the vessel (loll/heel). The maximal values of accelerations, as components of dynamic load, acting on the bearing structure of a wagon-platform are determined. The obtained acceleration values are taken into account when calculating the strength of the bearing structure using the finite element method implemented in the CosmosWorks software environment. It is established that the maximum equivalent tension do not exceed permissible values. The carried out researches will promote increase of efficiency of combined transportations in the direction of the international transport corridors.

Chemical Oscillations and Spatial Structures in Polymerisation Reactions

This article discusses the possibility of coupling a polymerization reaction to an oscillatory kinetic model, complemented by diffusion which can lead to spatial structure. We used three well–known mathematical models of oscillators: a variant of the Rossler multivibrator, a model proposed by Edelstein, and the Oregon Oscillator. One or some of the terms in the equations of these models come from a polymerization reaction, while the other terms of these equations will come from collateral processes. So, almost any reaction could become oscillatory and/or with dissipative structure, adding the adequate collateral processes. The propagation stages are considered as invariants and initiation reactions of order α = 0, 1, or 2, and termination reactions of order β = 1 or 2 are assumed. Except in the case α = 1, β = 2, all six remaining reactions combinations (α, β) can be coupled to least one, and often to several of the models. The effects of destabilization a stable homogeneous steady state by the presence of diffusion is also discussed, which is always be possible.

Математическая модель колебаний односекционной антенны

Математическая модель колебаний односекционной антенны

Mathematical modeling of wheelset oscillations as the basis of the method of acoustic control

This publication argues the need for onboard condition monitoring of the railway car wheelsets in operation. Mathematical modeling of oscillations of the wheelset with short isolated irregularities on the surface of the rolling wheels was offered in the article. To solve the problem was developed three-dimensional finite element model of the wheelset type RU1SH-957. Mathematical modeling conducted in MSC Nastran. In determining the natural of frequencies and waveform of the railway car wheelset, the method Lanczos was applied as the most effective for most designs. We obtained results that maximum vibrodisplacement take place on the first natural frequency of oscillations of the wheelset wile motion with short isolated roughness on wheel surface. Definitely the speed of the car in which the forced vibrations of the action of periodic forces overlap, and the duration of the shock interaction between wheel and rail is not dependent on the speed of the car and the size of the roughness on the surface of the rolling wheel. The results are used for building acoustic diagnostic model and as boundary conditions in the simulation of field of wheelset noise emissions in undercar space

Mathematical modeling of oscillations of a wheel tractor equipped with shock absorbers of front suspension

Vertical oscillations of front and back axles of a wheel tractor that has front axle suspension equipped with elastic springs and shock absorbers were considered. The mathematical model and calculation method for axles’ oscillations were proposed. The influence of front axles’ vertical oscillations on soil compaction was investigated.

Mathematical modeling of oscillations in a Lotka reaction on a catalyst surface

The article investigates a two-component heterogeneous-catalytic reaction on a catalyst surface with a regular square lattice. This reaction is an analog of the three-dimensional Lotka “predator–prey” reaction. The reaction kinetic scheme is applied to construct a hierarchical system of mathematical models, starting with a micro-level stochastic model and up to macro-level models. Two ODE systems are proposed with pair probabilities allowing for bounded diffusion rate of one of the reaction components. These systems display self-exciting oscillations under certain conditions. A parametric analysis of the pair probability models is carried out and self-exciting oscillation regions are determined in the parameter space. The solutions of these systems are compared with the dynamics of the microscopic stochastic model simulated by the dynamic Monte Carlo method. Conditions for the appearance of oscillations on the macro level are found.

Modeling of the Earth’s rotary—oscillatory motions in the three-body problem: Interpolation and prognosis

A mathematical model of oscillations of the Earth’s pole and uneven Earth’s rotation due to gravitational tidal forces produced by the Moon and the Sun is constructed using the dynamic Euler—Liouville equations. It is established that high-frequency lunar influences contribute appreciably when modeling the prognosis of the Earth’s pole oscillations. The developed theoretical models for Earth’s rotation parameters account for small-scale high-frequency influences of gravitational nature (short-periodic disturbances due to the Moon with combination frequencies) and geophysical nature. Statistically convincing theoretical interpolations and prognoses on long-term (from one year to several years) and short (15–25 days) time intervals compared to highly accurate observational and measuring data of the International Earth Rotation and Reference Systems Service (IERS) are provided.

Smoothing the discontinuous oscillations in the mathematical model of an oscillator with distributed parameters

We suggest a method of smoothing the discontinuous oscillations in mathematical models of oscillators with a long line segment in the feedback circuit. We show that under certain conditions the buffer phenomenon is realized in the corresponding boundary-value problem, that is, this problem can have a finite number of stable cycles (time-periodic solutions) as large as desired.

Mathematical model of oscillations in the human corneo-retinal potential.

Two mathematical models are developed describing the damped oscillations of the indirectly measured corneo-retinal potential in man. The variations of the potential are represented as feed-back systems between three structures, using a system of linear differential equations. The models were fitted to a typical electro-oculogram by means of an IBM 1410 digital computer. 97% of the observed variance can be explained by the models. The hypothesis is advanced that periodic oscillations of the indirectly measured corneo-retinal potential participate in a sensory homeostasis.