Oscillation Of Equations Research Articles

The Inverse Problem for the Nonhomogeneous Oscillation Equation on a Half-Line with a Boundary Condition of the Third Kind

The Inverse Problem for the Nonhomogeneous Oscillation Equation on a Half-Line with a Boundary Condition of the Third Kind

On Maxwell-Lorentz Equations in Dirac’s Symmetrisation and Their Analogs for Gravitation and Space-Time

In further discussion on the Maxwell-Lorentz equations in Dirac’s symmetrisation, I introduce the concept of magnetic monopole as an “act of electric current” in the 2nd equation (i.e. the analog of the “act of movement” in Classical Mechanics), I postulate a “magnetic displacement current” and a “magnetomotive force” in the 3rd and 4th equations, respectively (i.e. the analogs of the “electric displacement current” and of the “electromotive force” in the 4th and 3rd equations, respectively). As a consequence, I propose a generalised vision of the Electromagnetism in which inhomogeneous, microscopic, and relativistically linked equations describe the static and the oscillatory phenomena. Then, in the frame of Relativity, I propose analog microscopic equations to study the Gravitation and the Space-Time in terms of static and oscillatory phenomena: the static equations show the sources of newly defined vector fields (the generalised mass density as the source of the generalised mass field, the generalised time density as the source of the generalised space field, respectively), whereas the oscillatory equations describe the propagation of the gravitational waves and of the spatiotemporal waves, respectively. In other words, I propose to unify Electromagnetism, Gravitation, and Space-Time in terms of microscopic Maxwell-Lorentz-like equations in Dirac’s symmetrisation, where the unifying trait is c. Finally, using the concepts of the proposed generalised Electromagnetism, I discuss the conservation in Electromagnetism and the interaction between matter and electromagnetic waves.

TO THE QUESTION OF ANALYSIS OF EQUATIONS OF MOTION OF A RIGID BODY DURING THE MECHANICAL OSCILATIONS

Depending on the current position of the mass in different areas of the spring deformation during the oscillation process the values that determines the natural frequency of free continuous oscillations have opposite signs. It is defined by the change in the direction of acceleration of the mass in these areas, which makes it possible to determine a single inhomogeneous differential equation of the oscillation process in different areas of the movement of the mass. When the oscillation amplitude is much less than the static position of the mass, this inhomogeneous differential equation represents a homogeneous differential equation of free undamped oscillations.

Simulation of radial oscillations of a spring-loaded roll in a roll compactor

Carried out simulation of oscillations of a spring-loaded roll in a roll compactor when interacting the powder being compacted with the rolls. Considering the separation of the feed and compaction areas in the contact area of the roll with the material being compacted, we obtain the dependence of the force acting on the roll on the gap size between the rolls. It is shown that this dependence is non-linear, and it can be described with a sufficiently high accuracy degree by an exponential function with a negative exponent in the working range. The given numerical solution of the equation of free nonlinear oscillations of the spring-loaded roll has shown that considering the deformation of the material being compacted leads to a reduction of the natural frequency of the system by 20–25 % compared to the case, where the pressure force of the powder on the roll is assumed to be independent of the gap size. The nonlinearity of the dependence of the pressure force on the gap also leads to the increase by 10 % in the calculated values of the maximum displacements. The developed approach to the calculation of oscillations of the spring-loaded roll in the roll compactor enables to take into account the peculiarities of deformation of the powder being compacted during its interaction with the rolls. In addition, it allows estimating the frequencies and oscillation amplitudes and setting the optimum range of spring rate values, at which the occurrence of resonance in the machine is not possible.

Neutron star oscillations in pseudo-Newtonian gravity

ABSTRACT We investigate the oscillations of neutron stars using a purely Newtonian approach and three other pseudo-Newtonian formulations. Our work is motivated by the fact that pseudo-Newtonian formulations are commonly used in core-collapse supernova (CCSN) simulations. We derive and solve numerically the radial and non-radial perturbation equations for neutron star oscillations using different combinations of modified Newtonian hydrodynamics equations and gravitational potentials. We pay special attention to the formulation proposed recently by Zha et al. that implements the standard Case A effective potential in CCSN simulations with an additional lapse-function correction to the hydrodynamics equations. We find that this ‘Case A + lapse’ formulation can typically approximate the frequency of the fundamental radial mode of a 1.4-M⊙ neutron star computed in general relativity to about a few tens of per cent for our chosen EOS models. For the non-radial quadrupolar f mode, which is expected to contribute strongly to the gravitational waves emitted from a protoneutron star, the Case A + lapse formulation performs much better and can approximate the f-mode frequency to within about a few per cent even for the maximum-mass configuration in general relativity.

Simultaneous identification of mechanical properties of functionally graded plates under the Kirchhoff and Tymoshenko models

The paper presents models of deformation of functional-gradient round plates in the framework of the hypotheses of Kirchhoff and Timoshenko. Based on the equations of oscillations and boundary conditions obtained earlier using the Hamilton - Ostrogradsky variational principle, the formulations of problems in a cylindrical coordinate system are written, taking into account the variability of the functions of cylindrical stiffness and density along the radial coordinate. The plates was considered rigidly clamped along the edge. The case of steady-state vibrations caused by a load applied to the surface was considered. A scheme for solving direct problems calculating vibrations of plates based on the Galerkin method is constructed. An analysis of the influence of the functions of cylindrical stiffness and density on the amplitude-frequency characteristics (AFC, acoustic response) was carried out, which revealed that both functions significantly affect the AFC, and the greatest influence is observed in the vicinity of resonant frequencies. The results of the analysis made it possible to formulate new inverse problems of simultaneous identification of the functions of cylindrical stiffness and density of an inhomogeneous circular plate using additional information about the acoustic response for both hypotheses. To solve them, a special projection technique was built, based on the expansion of unknown functions of mechanical characteristics, as well as dynamic quantities (functions of deflection and angle of rotation of the normal) in terms of some systems of linearly independent functions that satisfy the boundary conditions. The coefficients of these expansions are determined from the solution of special systems of linear and nonlinear equations obtained from the formulated weak statements of both problems. As a result, it is possible to identify the desired characteristics in the given classes of functions. The identification results are illustrated with a set of computational experiments for various functions.

Compensatory bellows oscillations as a corrugated shell with liquid inside

The paper deals with a relevant problem of shipbuilding, i.e. calculation of free and forced vibrations of pipeline compensatory bellows. These devices are used to reduce the vibration load caused by ship power machines. When analyzing the vibrations of the compensatory bellows, it is necessary to take into account the liquid contained in the bellows. In this work, the design model of the bellows is represented by a corrugated elastic shell as a material surface with five degrees of freedom. A variant of the classical theory of shells, built on the basis of Lagrangian mechanics, is used. The influence of the liquid is taken into account by two models. First, the liquid is considered to be ideal and incompressible and is considered through the attached mass to the shell. The shell is replaced by a cylindrical surface with a radius in the middle line of the corrugation. To account for the influence of the frequency of bellows oscillations on the attached inertia of the liquid in the calculation we also used the acoustic approximation; and derived a formula for a generalized attached mass of the ideal compressible liquid. The equations of the bellows oscillations under the periodic loading are obtained. The problem has been solved by the finite difference method. The values of natural frequencies of free vibrations are obtained for the compensatory bellows from the corrosion-resistant heat-resistant steel. It is shown that by taking account of the liquid, we significantly change the natural frequencies of the bellows. With high-frequency vibrations it is necessary to take into account the compressibility of the liquid. The problem of the forced vibrations of the bellows caused by a displacement of its end face by the harmonic law is solved. The internal forces and moments are determined, as well as occurring stresses by Mises criterion in the bellows. We found the critical value of the end face displacement at a frequency of 50 Hz, at which the bellows goes into a plastic state.

Coupling between turbulence and solar-like oscillations: A combined Lagrangian PDF/SPH approach

Context. The development of space-borne missions such as CoRoT and Kepler now provides us with numerous and precise asteroseismic measurements that allow us to put better constraints on our theoretical knowledge of the physics of stellar interiors. In order to utilise the full potential of these measurements, however, we need a better theoretical understanding of the coupling between stellar oscillations and turbulent convection. Aims. The aim of this series of papers is to build a new formalism specifically tailored to study the impact of turbulence on the global modes of oscillation in solar-like stars. In building this formalism, we circumvent some fundamental limitations inherent to the more traditional approaches, in particular the need for separate equations for turbulence and oscillations, and the reduction of the turbulent cascade to a unique length and timescale. In this first paper we derive a linear wave equation that directly and consistently contains the turbulence as an input to the model, and therefore naturally contains the information on the coupling between the turbulence and the modes through the stochasticity of the equations. Methods. We use a Lagrangian stochastic model of turbulence based on probability density function methods to describe the evolution of the properties of individual fluid particles through stochastic differential equations. We then transcribe these stochastic differential equations from a Lagrangian frame to a Eulerian frame more adapted to the analysis of stellar oscillations. We combine this method with smoothed particle hydrodynamics, where all the mean fields appearing in the Lagrangian stochastic model are estimated directly from the set of fluid particles themselves, through the use of a weighting kernel function allowing to filter the particles present in any given vicinity. The resulting stochastic differential equations on Eulerian variables are then linearised. As a first step the gas is considered to follow a polytropic relation, and the turbulence is assumed anelastic. Results. We obtain a stochastic linear wave equation governing the time evolution of the relevant wave variables, while at the same time containing the effect of turbulence. The wave equation generalises the classical, unperturbed propagation of acoustic waves in a stratified medium (which reduces to the exact deterministic wave equation in the absence of turbulence) to a form that, by construction, accounts for the impact of turbulence on the mode in a consistent way. The effect of turbulence consists of a non-homogeneous forcing term, responsible for the stochastic driving of the mode, and a stochastic perturbation to the homogeneous part of the wave equation, responsible for both the damping of the mode and the modal surface effects. Conclusions. The stochastic wave equation obtained here represents our baseline framework to properly infer properties of turbulence-oscillation coupling, and can therefore be used to constrain the properties of the turbulence itself with the help of asteroseismic observations. This will be the subject of the rest of the papers in this series.

Action Principle and Ritz Method for Yarn Dynamics in Ring Spinning

AbstractRing spinning is the most relevant method of producing high quality staple yarns. The rotating yarn forms one or multiple balloon shapes governed by inertial, eyelet friction and drag forces. Traditionally, the stationary yarn path has been computed by direct integration of an equation of motion, and stability and natural oscillations have been assessed by application of Galerkin's method to these ([1], [4], and others). Recent technological advances by using frictionless superconducting magnetic bearings allow for increased process speeds and thus motivate a renewed interest in the dynamics of ring spinning.An integral formulation of the yarn dynamics problem is presented using Hamilton's principle for the one‐dimensional continuum and the magnetic bearing. While therefrom the known equations of motions can be deduced elegantly, here Ritz's method is employed to achieve direct discretization of the problem, with cases to be made for both local and global shape functions.While applicable to the classical study of stationary balloon shapes and their stability, the resulting models in particular enable the study of the instationary yarn path and natural and driven yarn oscillations, which both have received limited attention in literature, but are known to affect yarn tension and thus the process viability.

GENERALIZED MATHEMATICAL MODEL OF VIBRATIONS OF A VERTICAL AXIS ROTARY SYSTEM

The rotor’s response to dynamic actions depends on its physical parameters and the parameters of the supporting structure. The vibration measured on the bearings or on the rotor shaft, in addition to the physical properties of the rotor, also depends on the value of the unbalance, its distribution along the rotor axis, as well as on the rotational speed. These factors influence the choice of the balancing method. This article proposes a dynamic model of a rotary machine with a vertical axis of rotation and six degrees of freedom. Differential equations of free and forced oscillations of the rotor system are compiled. It is shown that when considering the general case of oscillations of the rotor system, there is a high connectivity of oscillations along all generalized coordinates. Based on the analysis of differential equations, ways of reducing vibration activity and connectivity of machine vibrations are shown, which provide for: reduction of the center of mass of the platform to the axis of rotation of the rotor; alignment of the center of stiffness of the elastic support system with the center of gravity of the platform, and the main axes of stiffness and constant viscous friction of dampers with the main central axes of inertia of the platform; the use of vertical suspensions for the oscillating system in the construction of the machine; replacement of the elastic sealing diaphragm of the rotor with a non-rigid one. The above measures to reduce the connectivity of vibrations and vibration activity do not completely eliminate the vibrations of the machine caused by stochastic, variable imbalances. Therefore, a further decrease in the vibration activity of the machine can be associated with the use of auto-ballasting devices, which make it possible to reduce the disturbing forces by reducing the magnitude of the stochastic imbalances of the rotary machine.

On the ubiquity of classical harmonic oscillators and a universal equation for the natural frequency of a perturbed system

A new perspective on the ubiquity of classical harmonic oscillators is presented based on the two-variable Taylor expansion of a perturbed system's total energy E(q,q̇), where q(t) is the system displacement as a function of time t and q̇(t)=dq/dt. This generalised approach permits derivation of the lossless oscillator equation from energy arguments only, yielding a universal equation for the oscillation frequency ω=(∂2E/∂q2)/(∂2E/∂q̇2) which may be applied to arbitrary systems without the need to form system-specific linearised models. As illustrated by a range of examples, this perspective gives a unifying explanation for the prevalence of harmonic oscillators in classical physics, can be extended to include damping effects and driving forces, and is a powerful tool for simplifying the analyses of perturbed systems.

DISSIPATIVE OSCILLATORS’ POWER CHARACTERISTIC NON-SYMMETRY DYNAMIC EFFECT

The features of motion of a non-linear oscillator under the instantaneous force pulse loading are studied. The elastic characteristic of the oscillator is given by a polygonal chain consisting of two linear segments. The focus of the paper is on the influence of the dissipative forces on the possibility of occurrence of the elastic characteristic non-symmetry dynamic effect, studied previously without taking into account the influence of these forces. Four types of drag forces are considered, namely linear viscous friction, Coulomb dry friction, position friction, and quadratic viscous resistance. For the cases of linear viscous friction and Coulomb dry friction the analytical solutions of the differential equation of oscillations are found by the fitting method and the formulae for computing the swings are derived. The conditions on the parameters of the problem are determined for which the elastic characteristic non-symmetry dynamic effect occurs in the system. The conditions for the effect to occur in the system with the position friction are derived from the energy relations without solving the differential equation of motion. In the case of quadratic viscous friction the first integral of the differential equation of motion is given by the Lambert function of either positive or negative argument depending on the value of the initial velocity. The elastic characteristic non-symmetry dynamic effect is shown to occur for small initial velocities, whereas it is absent from the system when the initial velocities are sufficiently large. The values of the Lambert function are proposed to be computed by either linear interpolation of the known data or approximation of the Lambert function by elementary functions using asymptotic formulae which approximation error is less than 1%. The theoretical study presented in the paper is followed up by computational examples. The results of the computations by the formulae proposed in the paper are shown to be in perfect agreement with the results of numerical integration of the differential equation of motion of the oscillator using a computer.

The Duffing Oscillator Equation and Its Applications in Physics

In this paper, we solve the Duffing equation for given initial conditions. We introduce the concept of the discriminant for the Duffing equation and we solve it in three cases depending on sign of the discriminant. We also show the way the Duffing equation is applied in soliton theory.

On Truly Nonlinear Oscillator Equations of Ermakov-Pinney Type

In this paper we present a general class of differential equations of Ermakov-Pinney type which may serve as truly nonlinear oscillators. We show the existence of periodic solutions by exact integration after the phase plane analysis. The related quadratic Lienard type equations are examined to show for the first time that the Jacobi elliptic functions may be solution of second-order autonomous non-polynomial differential equations.

Delay Identification in Thermoacoustics

Abstract The stability of thermoacoustic systems is often regulated by the time delay between acoustic perturbations and corresponding heat release fluctuations. An accurate estimate of this value is of great importance in applications since even small modifications can introduce significant changes in the system behavior. Different studies show that the nonlinear delayed dynamics typical of these systems can be well captured with low-order models. In this work, a method is introduced to estimate the most likely value of the time delay of a single thermoacoustic mode from a measured acoustic pressure signal. The mode of interest is modeled by an oscillator equation, with a nonlinear delayed forcing term modeling the deterministic flame contribution and an additive white Gaussian noise to embed the stochastic combustion noise. Additionally, other thermoacoustic relevant parameters are estimated. The model accounts for a flame gain, for a flame saturation coefficient, for linear acoustic damping, and for the background combustion noise intensity. The pressure data time series is statistically analyzed and the set of unknown parameters is identified. Validation is performed with respect to synthetically generated time series and low order model simulations, for which the underlying delay is known a priori. A discussion follows about the accuracy of the method, in particular, a comparison with existing methods is drawn.

Acoustic cavitation model based on a novel reduced order gas pressure law

The thermal behavior of a spherical gas bubble in a liquid excited by an acoustic pressure signal is investigated by constructing an iterative solution of the energy balance equations between the gas bubble and the surrounding liquid in the uniform pressure approximation. This iterative solution leads to hierarchy equations for the radial partial derivatives of the temperature at the bubble wall, which control the temporal rate of change of the gas pressure and gas temperature within the bubble. In particular, a closure relation for the hierarchy equations is introduced based on the ansatz that approximates the rapid change of state during the collapse of the bubble from almost isothermal to almost adiabatic behavior by time averaging the complex dynamics of change of state over a relatively short characteristic time. This, in turn, leads to the desired reduced order gas pressure law exhibiting power law dependence on the bubble wall temperature and on the bubble radius, with the polytropic index depending on the isentropic exponent of the gas and on a parameter that is a function of the Péclet number and a characteristic time scale. Results of the linear theory for gas bubbles are recovered by identifying this parameter as a function of the Péclet number based on the Minnaert frequency. The novel gas pressure law is then validated against the near-isothermal solution and against the results of the numerical simulations of the original energy balance equations for large amplitude oscillations using spectral methods. Consequently, an acoustic cavitation model that accounts for phase change but that neglects mass diffusion is constructed by employing the reduced order gas pressure law together with the Plesset–Zwick solution for the bubble wall temperature and the Keller–Miksis equation for spherical bubble dynamics. Results obtained using variable interface properties for acoustically driven cavitation bubbles in water show that the time variations of the bubble radius and the bubble wall temperature lie between those obtained by the isothermal and adiabatic laws depending on the value of the Péclet number and the characteristic time scale.

Comparison of Two Different Analytical Forms of Response for Fractional Oscillation Equation

The impulse response of the fractional oscillation equation was investigated, where the damping term was characterized by means of the Riemann–Liouville fractional derivative with the order α satisfying 0≤α≤2. Two different analytical forms of the response were obtained by using the two different methods of inverse Laplace transform. The first analytical form is a series composed of positive powers of t, which converges rapidly for a small t. The second form is a sum of a damped harmonic oscillation with negative exponential amplitude and a decayed function in the form of an infinite integral, where the infinite integral converges rapidly for a large t. Furthermore, the Gauss–Laguerre quadrature formula was used for numerical calculation of the infinite integral to generate an analytical approximation to the response. The asymptotic behaviours for a small t and large t were obtained from the two forms of response. The second form provides more details for the response and is applicable for a larger range of t. The results include that of the integer-order cases, α= 0, 1 and 2.

Study on free oscillations of a micromechanical gyroscope taking into account the nonorthogonality of the torsion axes

Introduction. The paper is devoted to the study on free oscillations of the sensing element of a micromechanical R-Rtype gyroscope of frame construction developed by the Kuznetsov Research Institute of Applied Mechanics, taking into account the nonorthogonality of the torsion axes. The influence of the instrumental manufacturing error on the accuracy of a gyroscope on a movable base in the case of free oscillations is studied. The work objective was to improve the device accuracy through developing a mathematical model of an R-R type micromechanical gyroscope, taking into account the nonorthogonality of the torsion axes, and to study the influence of this error on the device accuracy. The urgency of the problem of increasing the accuracy of micromechanical gyroscopes is associated with improving the accuracy of inertial navigation systems based on micromechanical sensors.Materials and Methods. A new mathematical model that describes the gyroscope dynamics, taking into account the instrumental error of manufacturing the device, and a formula for estimating the error of a gyroscope, are proposed. The dependences of the state variables obtained from the results of modeling and on the basis of the experiment are presented. Methods of theoretical mechanics and asymptotic methods, including the Lagrange formalism and the Krylov-Bogolyubov averaging method, were used in the research.Results. A new mathematical model of the gyroscope dynamics, taking into account the nonorthogonality of the torsion axes, is developed. The solution to the equations of small oscillations of the gyroscope sensing element and the estimate of the precession angle for the case of a movable base are obtained. A comparative analysis of the developed model and the experimental data obtained in the case of free oscillations of the gyroscope sensing element with a fixed base is carried out. The analysis has confirmed the adequacy of the constructed mathematical model. Analytical expressions are formed. They demonstrate the fact that the nonorthogonality of the torsion axes causes a cross-influence of the amplitudes of the primary vibrations on the amplitudes of the secondary vibrations of the sensing element, and the appearance of an additional error in the angular velocity readings when the gyroscope is operating in free mode.Discussion and Conclusions. The results obtained can be used to improve the device accuracy using the algorithm for analytical compensation of the gyroscope error and the method for identifying the mathematical model parameters.

Harmonic balance: simple and accurate estimation of nonlinear oscillator parameters

Harmonic balance (HB) analysis is applied to a number of nonlinear oscillator equations of motion; surprisingly accurate estimations of periodic solutions are obtained when the oscillator spectrum is narrowband, as here. Comparison with other approximation techniques shows its efficacy. HB is a useful tool for researchers, yet requires no advanced mathematics or specialist knowledge; as such it is appropriate for undergraduate teaching.

Initial–Boundary Value Problems for Equation of Oscillations of a Rectangular Plate

Initial–Boundary Value Problems for Equation of Oscillations of a Rectangular Plate