Averaged Equations Of Motion Research Articles

A Second-Order Closed-Form J_2 Model for the Draper Semi-Analytical Satellite Theory

A second-order closed-form semi-analytical solution of the main problem of the artificial satellite theory (J_2 contribution) consistent with the Draper Semi-analytic Satellite Theory (DSST) is presented. This paper aims to improve the computational speed of the numerical-based approach, which is only available in the GTDS-DSST version. The short-period terms are removed by means of an extension of the Lie-Deprit method using Delaunay variables. The averaged equations of motion are given explicitly and transformed to the non-singular equinoctial elements. Finally, the second-order terms in the equations of motion are included in the C/C++ version of the DSST orbit propagator.

Features of the relativistic motion of a single electron entering a waveguide

Properties of the motion of an electron with relativistic energy entering an arbitrary Hmn-mode wave in a rectangular waveguide are considered. The derivation of the averaged motion is performed using the Krylov–Bogolyubov averaging method. The ratio of the amplitude of the quiver velocity of the particle in the wavefield to the speed of light is considered as the small parameter. The averaged equations of motion and the periodic additions to the smoothed variables are obtained up to second-order expansions over the small parameter. It is shown that the averaged (ponderomotive) force along the longitudinal axis of the waveguide is absent independent of the wave mode. Numerical solutions of both exact and averaged systems of equations are obtained, demonstrating an excellent agreement of the models. The importance of the correct setting of the initial conditions for the averaged equations based on the values of the periodic additions is emphasized. The conditions, under which the effect of reflection or refraction of the electron by the waveguide field takes place, are established in the case when the electron is injected transversely.

Mathematical Modeling of Evolution of Swirling Turbulent Jet in Coflowing Stream

A numerical modeling of a swirling turbulent jet in a coflowing stream was carried out. The flow description involved two second-order mathematical models. The first one includes the averaged equations of motion and the differential equations for transfer of normal Reynolds stresses and dissipation rate in the thin shear layer approximation. The second model relies on the far wake approximation. The distances from the source of the jet in the calculations reached very large values. At small distances, the calculated profiles of the averaged velocity components agree well with the known experimental data from Lavrent’ev Institute of Hydrodynamics SB RAS. At large distances from the source, the flow becomes close to the self-similar one, with degeneration laws and normalized profiles consistent with the known theoretical concepts of the dynamics of swirling turbulent jets in a coflowing stream. The problem of asymptotic behavior of a nonswirling turbulent jet in a coflowing stream was also considered. A self-similar solution based on numerical experiments was obtained.

The Averaged Equations of Motion in the Presence of an Inverse-Square Perturbing Acceleration

The motion of a zero-mass point under the action of a gravitational force toward a central body $${\cal S}$$ and a perturbing acceleration P′ whose magnitude is inversely proportional to the square of the distance to $${\cal S}$$ is considered. The direction of P′ is constant in one of the three coordinate systems most widely used in astronomy: the main inertial system $${\cal O}$$ and two orbiting systems $${{\cal O}_s}$$ with their x axes along the radius vector for s = 1 and along the velocity vector for s = 2.The ratio of |P′|to the main acceleration due to the gravitation of the central body is taken to be small. An averaging transformation in a first approximation in a small parameter of the problem is applied to the equations of motion in the osculating elements. Closed expressions are obtained for the right-hand sides of the equations of motion in the mean elements. These are expressed in terms of elementary functions in the systems $${\cal O}$$ and $${{\cal O}_1}$$ ; complete elliptical integrals arise in the system $${{\cal O}_2}$$ . Closed expressions are obtained for the change-of-variable functions. All the functions encountered in the systems $${\cal O}$$ and $${{\cal O}_1}$$ are elementary functions, apart from those determining the variations of the mean anomaly. The latter is given by an integral of an elementary function, as well as a series in powers of the eccentricity that converges absolutely and uniformly when 0 ⩽ e ⩽ 1. All functions in the system $${{\cal O}_2}$$ apart from those determining the variations of the mean anomaly can be expressed in terms of incomplete elliptical integrals. The variations of the mean anomaly are calculated using a Fourier series in the mean anomaly. Integration of the averaged equations of motion will be considered in future papers. Possible applications of this model problem include the motion of an asteroid taking into account the Yarkovsky-Radziewski effect, and the motion of a spacecraft with a solar sail, when the perturbing action is inversely proportional to the square of the distance from the Sun. It stands to reason that determining the components of the vector P′ requires knowledge of the thermal-physical characteristics of the body in question and the parameters of its rotational motion in the former case, and of the orientation of the solar sail in the latter case.

Strong tidal energy dissipation in Saturn at Titan’s frequency as an explanation for Iapetus orbit

Context. Natural satellite systems present a large variety of orbital configurations in the solar system. While some are clearly the result of known processes, others still have largely unexplained eccentricity and inclination values. Iapetus, the furthest of Saturn’s main satellites, has a still unexplained 3% orbital eccentricity and its orbital plane is tilted with respect to its local Laplace plane (8° of free inclination). On the other hand, astrometric measurements of saturnian moons have revealed high tidal migration rates, corresponding to a quality factor Q of Saturn of around 1600 for the mid-sized icy moons. Aims. We show how a past crossing of the 5:1 mean motion resonance between Titan and Iapetus may be a plausible scenario to explain Iapetus’ orbit. Methods. We have carried out numerical simulations of the resonance crossing using an N-body code as well as using averaged equations of motion. A large span of migration rates were explored for Titan and Iapetus was started on its local Laplace plane (15° with respect to the equatorial plane) with a circular orbit. Results. The resonance crossing can trigger a chaotic evolution of the eccentricity and the inclination of Iapetus. The outcome of the resonance is highly dependent on the migration rate (or equivalently on Q). For a quality factor Q of over around 2000, the chaotic evolution of Iapetus in the resonance leads in most cases to its ejection, while simulations with a quality factor between 100 and 2000 show a departure from the resonance with post-resonant eccentricities spanning from 0 up to 15%, and free inclinations capable of reaching 11°. Usually high inclinations come with high eccentricities but some simulations (less than 1%) show elements compatible with Iapetus’ current orbit Conclusions. In the context of high tidal energy dissipation in Saturn, a quality factor between 100 and 2000 at the frequency of Titan would bring Titan and Iapetus into a 5:1 resonance, which would perturb Iapetus’ eccentricity and inclination to values observed today. Such rapid tidal migration would have avoided Iapetus’ ejection around 40–800 million years ago.

Engineered bio-inspired coating for passive flow control

Flow separation and vortex shedding are some of the most common phenomena experienced by bluff bodies under relative motion with the surrounding medium. They often result in a recirculation bubble in regions with adverse pressure gradient, which typically reduces efficiency in vehicles and increases loading on structures. Here, the ability of an engineered coating to manipulate the large-scale recirculation region was tested in a separated flow at moderate momentum thickness Reynolds number, [Formula: see text] We show that the coating, composed of uniformly distributed cylindrical pillars with diverging tips, successfully reduces the size of, and shifts downstream, the separation bubble. Despite the so-called roughness parameter, [Formula: see text], falling within the hydrodynamic smooth regime, the coating is able to modulate the large-scale recirculating motion. Remarkably, this modulation does not induce noticeable changes in the near-wall turbulence levels. Supported with experimental data and theoretical arguments based on the averaged equations of motion, we suggest that the inherent mechanism responsible for the bubble modulation is essentially unsteady suction and blowing controlled by the increasing cross-section of the tips. The coating can be easily fabricated and installed and works under dry and wet conditions, increasing its potential impact on a diverse range of applications.

Analytic propagation of near-circular satellite orbits in the atmosphere of an oblate planet

In the present paper, an averaging perturbation technique leads to the determination of a time-explicit analytic approximate solution for the motion of a low-Earth-orbiting satellite . The two dominant perturbations are taken into account: the Earth oblateness and the atmospheric drag. The proposed orbit propagation algorithm comprises the Brouwer–Lyddane transformation (direct and inverse), coupled with the analytic solution of the averaged equations of motion. This solution, based on equinoctial elements, is singularity-free, and therefore it stands for low inclinations and small eccentricities as well. The simplifying assumption of a constant atmospheric density is made, which is reasonable for near-circular orbits and short-time orbit propagation. Two sets of time-explicit equations are provided, for moderate and small eccentricities ( $$\mathcal {O} ( e^{4}) =0$$ and $$\mathcal {O}( e^{2}) =0,$$ respectively), and they are obtained by performing (1) a regularization of the original averaged differential equations of motion for the vectorial orbital elements, and (2) Taylor series expansions of the aforementioned equations with respect to the eccentricity. The numerical simulations show that the errors due to the use of the proposed analytic model in the presence of drag are almost the same as the errors of the Brouwer first-order approximation in the absence of drag.

The motion of a two-body limbless locomotor along a straight line in a resistive medium

The behavior of a two-body self-propelling locomotion system in a resistive environment is studied. The motion of the system is excited and sustained by means of a periodic change in the distance between the bodies. A complete analysis of the motion of the system is performed for the case where the resistance forces applied by the environment to the bodies of the system are represented by linear functions of the velocities of these bodies relative to the environment. For the case where the resistance forces are nonlinear functions of the velocities of the bodies, a model based on the averaged equation of motion is used. This model assumes the forces of friction acting in the system to be small in comparison with the excitation force. The motion of the system along a horizontal straight line in an isotropic dry friction environment is investigated in detail for two particular types of excitation modes. The calculated results are compared with the experimental data.

Self-induced vibrations in a string-bow system

The vibrations of a thin stretched string is studied in the case when a bow slides on it with a constant velocity orthogonal to the string. The interaction between the bow and the string is governed by a smooth nonlinear law of friction with a falling segment of the characteristic. The motion of this mechanical system is described by an infinite coupled system of nonlinear ordinary differential equations. Some averaged equations of motion are derived in terms of the action-angle variables. The stationary points corresponding to self-vibration modes are found. The stability of these modes is analyzed.

Connection between three-body configuration and four-body configuration of the Sitnikov problem when one of the masses approaches zero: circular case

In this manuscript we have established averaged equation of motion of the Sitnikov restricted three- body and four-body problem when all the primaries are point masses, by applying the Van der Pol transformation and averaging technique of J. Guckenheimer and P. Holmes (in Nonlinear Oscillations, Dynamical System Bifurcations of Vector Fields, Springer, Berlin, 1983). In addition to the resonance criterion at the 3/2 commensurability we have chosen ω=2n/3,n=4, ω is the angular velocity of the coordinate system. Further we established the Series solution of the three-body and four-body problem in the sense of Sitnikov. Lastly the periodicities of the solutions have been examined by the Poincare section and four-body and three-body problem have been compared by different comparative graphs and surfaces.

Crossing the dividing surface of transition state theory. II. Recrossing times for the atom–diatom interaction

We consider a triatomic system with zero total angular momentum and demonstrate that, no matter how complicated the anharmonic part of the potential energy function, classical dynamics in the vicinity of a saddle point is constrained by symmetry properties. At short times and at not too high energies, recrossing dynamics is largely determined by elementary local structural parameters and thus can be described in configuration space only. Conditions for recrossing are given in the form of inequalities involving structural parameters only. Explicit expressions for recrossing times, valid for microcanonical ensembles, are shown to obey interesting regularities. In a forward reaction, when the transition state is nonlinear and tight enough, one-fourth of the trajectories are expected to recross the plane R = R* (where R* denotes the position of the saddle point) within a short time. Another fourth of them are expected to have previously recrossed at a short negative time, i.e., close to the saddle point. These trajectories do not contribute to the reaction rate. The reactive trajectories that obey the transition state model are to be found in the remaining half. However, no conclusion can be derived for them, except that if recrossings occur, then they must either take place in the distant future or already have taken place in the remote past, i.e., far away from the saddle point. Trajectories that all cross the plane R = R* at time t = 0, with the same positive translational momentum P(R*) can be partitioned into two sets, distinguished by the parity of their initial conditions; both sets have the same average equation of motion up to and including terms cubic in time. Coordination is excellent in the vicinity of the saddle point but fades out at long (positive or negative) times, i.e., far away from the transition state.

A way to generate a model of secular perturbations for planetary orbits

The reciprocal distance between two material points that rotate around a central body in nonintersecting orbits is expanded and the results are presented. The expansion is obtained accurate to the tenth order with respect to small parameters: the eccentricities and sine of the orbital inclination angle. The result is the basis of the averaging operation of the perturbation function in the system of eight major planets in the solar system, and of the numerical integration of the averaged equations of motion. The averaged Hamiltonian contains the terms whose period of variation is greater than 200 years. Forty eight equation of first order are numerically integrated with increments of 100 years for two intervals from the beginning of the Christian era: 25 million years forward and 25 million years backward over time. To present the results of calculation, the website (URL: http://vadimchazov.narod.ru/secequat.htm) was developed, where the initial codes, executable program modules, the results of calculations presented in graphical form, text files with initial conditions, tables for expanding the reciprocal distance between two material points, and the tables with the results of expansion of the perturbation function for eight major planets of the solar system are presented.

Flow of power-law fluids in fixed beds of cylinders or spheres

AbstractAn ensemble average of the equations of motion for a Newtonian fluid over particle configurations in a dilute fixed bed of spheres or cylinders yields Brinkman’s equations of motion, where the disturbance velocity produced by a test particle is influenced by the Newtonian fluid stress and a body force representing the linear drag on the surrounding particles. We consider a similar analysis for a power-law fluid where the stress $\boldsymbol{\tau} $ is related to the rate of strain $ \mathbisf{e} $ by $\boldsymbol{\tau} = 2m \mathop{ \vert \mathbisf{e} \vert }\nolimits ^{n\ensuremath{-} 1} \mathbisf{e} $, where $m$ and $n$ are constants. In this case, the ensemble-averaged momentum equation includes a body force resulting from the nonlinear drag exerted on the surrounding particles, a power-law stress associated with the disturbance velocity of the test particle, and a stress term that is linear with respect to the test particle’s disturbance velocity. The latter term results from the interaction of the test particle’s velocity disturbance with the random straining motions produced by the neighbouring particles and is important only in shear-thickening fluids where the velocity disturbances of the particles are long-ranged. The solutions to these equations using scaling analyses for dilute beds and numerical simulations using the finite element method are presented. We show that the drag force acting on a particle in a fixed bed can be written as a function of a particle-concentration-dependent length scale at which the fluid velocity disturbance produced by a particle is modified by hydrodynamic interactions with its neighbours. This is also true of the drag on a particle in a periodic array where the length scale is the lattice spacing. The effects of particle interactions on the drag in dilute arrays (periodic or random) of cylinders and spheres in shear-thickening fluids is dramatic, where it arrests the algebraic growth of the disturbance velocity with radial position when $n\geq 1$ for cylinders and $n\geq 2$ for spheres. For concentrated random arrays of particles, we adopt an effective medium theory in which the drag force per unit volume in the medium surrounding a test particle is assumed to be proportional to the local volume fraction of the neighbouring particles, which is derived from the hard-particle packing. The predictions of the averaged equations of motion are validated by comparison with simulations of randomly distributed hydrodynamically interacting cylinders.

An Exact Thickness-Weighted Average Formulation of the Boussinesq Equations

Abstract The author shows that a systematic application of thickness-weighted averaging to the Boussinesq equations of motion results in averaged equations of motion written entirely in terms of the thickness-weighted velocity; that is, the unweighted average velocity and the eddy-induced velocity do not appear in the averaged equations of motion. This thickness-weighted average (TWA) formulation is identical to the unaveraged equations, apart from eddy forcing by the divergence of three-dimensional Eliassen–Palm (EP) vectors in the two horizontal momentum equations. These EP vectors are second order in eddy amplitude and, moreover, the EP divergences can be expressed in terms of the eddy flux of the Rossby–Ertel potential vorticity derived from the TWA equations of motion. That is, there is a fully nonlinear and three-dimensional generalization of the one- and two-dimensional identities found by Taylor and Bretherton. The only assumption required to obtain this exact TWA formulation is that the buoyancy field is stacked vertically; that is, that the buoyancy frequency is never zero. Thus, the TWA formulation applies to nonrotating stably stratified turbulent flows, as well as to large-scale rapidly rotating flows. Though the TWA formulation is obtained by working on the equations of motion in buoyancy coordinates, the averaged equations of motion can then be translated into Cartesian coordinates, which is the most useful representation for many purposes.

Self-oscillations of two bodies in the case of nonlinear friction

The motion of two bodies is studied. One of them is attached to a fixed base by a spring, whereas the other one is attached to a body moving at a constant velocity. The interaction of the bodies is described by a nonlinear friction law with a falling section of the characteristic. The averaged equations of motion are obtained in terms of the canonical action-angle variables. The stationary points corresponding to self-oscillation modes are found and their stability is studied.

Bifurcations and chaos of an inclined cable

The nonlinear behavior of an inclined cable subjected to a harmonic excitation is investigated in this paper. The Galerkin’s method is applied to the partial differential governing equations to obtain a two-degree-of-freedom nonlinear system subjected to harmonic excitation. The nonlinear systems in the presence of both external and 1:1 internal resonances are transformed to the averaged equations by using the method of averaging. The averaged equations are numerically examined to obtain the steady-state responses and chaotic solutions. Five cascades of period-doubling bifurcations leading to chaotic solutions, 3-periodic solutions leading to chaotic solution, boundary crisis phenomena, as well as the Shilnikov mechanism for chaos, are observed. In order to study the global dynamics of an inclined cable, after determining the averaged equations of motion in a suitable form, a new global perturbation technique developed by Kovacic and Wiggins is used. This technique provides analytical results for the critical parameter values at which the dynamical system, through the Shilnikov type homoclinic orbits, possesses a Smale horseshoe type of chaos.

The Effect of Imperfect Contact on the Homogenization of a Micro-periodic Helix

Under study are the equations governing the elastodynamics of a micro-periodic helix, i.e. a helix made of a sequence of unit cells, each containing a thin imperfect interphase embedded within a finite number of other phases. An averaged equation of motion, along with its effective constitutive coefficients, is determined via an asymptotic homogenization method. The results are valid in the case of wavelengths much longer than the length of the unit cell. Formulae for shorter wavelengths can be derived by admitting higher order terms in the expansion.

Homogenization of a Micro-Periodic Helix with Parabolic or Hyperbolic Heat Conduction

ABSTRACT Based on a recent model for vibration of an elastic helix [6], a thermoelastic heterogeneous helix is studied by the asymptotic homogenization method. The objective of the study is the determination of the averaged equation of motion and of the effective coefficients of a one-dimensional micro-periodic thermoelastic helix. The results are valid in the case of waves much longer than the length of the periodic unit cell, and for any finite number of phases for within that cell. Also perfect contact conditions between phases are considered. Generally, the constitutive coefficients are harmonic averages, while the mass density and polar moment of inertia are arithmetic averages.

On the Hydrodynamic Thickness of Cellular Detonations

The characterization of the detonation dynamic parameters (detonability limits, direct initiation energy, critical tube diameter, etc.) requires a characteristic length scale for the detonation wave in the direction of propagation. However, most detonations are unstable, their reaction zones are turbulent, and their structure departs significantly from the idealized one-dimensional Zel'dovich-Von Neumann-Doring model. It is argued that the most suitable length scale to characterize a turbulent detonation wave is the location of the sonic surface, which separates the statistically stationary flow of the reaction zone structure from the unsteady expansions behind the wave. Previous real and numerical experiments are reviewed in order to determine the relation between the global location of the mean sonic surface and the chemical, mechanical, and thermodynamic relaxation processes occurring in the detonation wave structure. Based on the experimental evidence, we postulate that the structure of turbulent detonations can be modeled in the one-dimensional Zel'dovich-Neumann-Doring framework, with the turbulence effects as source terms in the momentum and energy equations. These source terms involve the relaxation rates for the mechanical fluctuations, thermal fluctuations and the chemical exothermicity towards equilibrium. In the framework of the idealized one-dimensional structure with source terms, the sonic surface location is governed by the balance between the competing source terms satisfying the generalized Chapman-Jouguet criterion. We recommend that future work in detonation research should be focused at: 1) acquiring a large experimental database for the mean detonation properties (detonation velocity, location of sonic surface and mean reaction zone profiles); 2) the development of the appropriate source terms involving the turbulent fluctuations in the averaged equations of motion.

Geodesics on vibrating surfaces and curvature of the normal family

In this note we study the motion of a particle confined to a moving surface, in other words, the geodesic motion where the surface is allowed to vary. We show that in the case of a rapidly vibrating surface, the differential geometry of a certain family of normal curves plays a role. Certain curvature terms appear in the averaged equations of motion.