Euler Equations Of Dynamics Research Articles

On metallic-type asteroid rotation moving in magnetic field (introducing magnetic second-grade YORP effect)

A novel physical effect has been found to be illuminated of being physically self-consistent to take into account in further calculating the dynamics of magnetically-induced asteroid rotation, surface of which has mostly a conducting structure (whereas such mostly metallic-formed asteroid is assumed to be moving in an external magnetic field of planet but in the meantime to be orbiting preferably outside its sphere of effective attraction). The last condition of asteroid surface's having conducting structure means the existence of the applied external torques stemming from the physical interactions between external magnetic field and iron asteroid itself (including long-term effect due to torques stemming from arising eddy-currents on metallic surface of asteroid). Such eddy-currents should heat the conducting surface of asteroid with further non-uniformly re-directing the heat flow from surface of asteroid into outer space via fluxes of thermal photons which carry momentum (in orthogonal direction to the surface which is in most cases far from the ideal surface of sphere). This leads by taking into account the overall outcome of heat flow (from the non-ideal surface of asteroid) to a kind of long-term magnetic second-grade YORP effect. System of Euler equations for aforementioned dynamics of asteroid magnetic rotation has been investigated in regard to existence of semi-analytical solution. Various perturbations (such as collisions, YORP effect) may destabilize the rotation of asteroid deviating it from the current spin state, whereas the electric(eddy)current-induced dissipation of energy reduces kinetic one of asteroid spin. So, dynamics of the asteroid rotation should result in a spinning about maximal-inertia axis with the proper spin state corresponding to minimal energy with a fixed angular momentum.

Model of a microgravity platform based on the mechanical principle of operation

The paper considers a model of a mechanical microgravity platform. The purpose of the paper is to develop a model of a mechanical microgravity platform. The experience of creating small spacecraft for technological purposes for carrying out gravity-sensitive processes on board is analyzed. The principle of operation of a mechanical microgravity platform is described. The microgravity platform is equipped with an electrodynamic drive to compensate for micro-accelerations. The electrodynamic drive have three identical independent control circuits to compensate for microaccelerations along the three corresponding coordinate axes. The second Kirchhoff rule was used to describe the current strength in the closed circuit of the drive. Dynamic Euler equations were used to develop the control laws. The equations for estimating micro-accelerations inside the protected zone are obtained. The ways of meeting the requirements for microaccelerations for the implementation of gravity-sensitive processes on a spacecraft are shown. Recommendations are given on the use of the platform on board a small spacecraft for technological purposes.

Gravitational collapse of Bose-Einstein condensate dark matter halos with logarithmic nonlinearity

"If dark matter is composed of massive bosons, a Bose-Einstein Condensation process must have occurred during the cosmological evolution. Therefore galactic dark matter may be in a form of a condensate, characterized by a strong self-interaction. One of the interesting forms of the self-interaction potential of the condensate dark matter is the logarithmic form. In the present work we investigate one of the astrophysical implications of the condensate dark matter with logarithmic self-interaction, namely, its gravitational collapse. To describe the condensate dark matter we use the Gross-Pitaevskii equation, and the Thomas-Fermi approximation. By using the hydrodynamic representation of the Gross-Pitaevskii equation we obtain the equation of state of the condensate, which has the form of the ideal gas equation of state, with the pressure proportional to the dark matter density. In the Thomas-Fermi approximation, the evolution equations of the condensate reduce to the classical continuity, and Euler equations of fluid dynamics. We obtain the equations of motion of the condensate radius in spherical symmetry, by assuming certain particular forms for the velocity and density of the condensate. The collapse time required for the formation of a stable macroscopic astrophysical object is obtained in an integral form, and explicit numerical estimations for the formation of astrophysical objects with masses ranging from 106M⊙ to 1012M⊙ are presented."

The Role of Pickup Ions in the Interaction of the Solar Wind with the Local Interstellar Medium. I. Importance of Kinetic Processes at the Heliospheric Termination Shock

The role of pickup ions (PUIs) in the solar wind interaction with the local interstellar medium is investigated with 3D, multifluid simulations. The flow of the mixture of all charged particles is described by the ideal MHD equations, with the source terms responsible for charge exchange between ions and neutral atoms. The thermodynamically distinct populations of neutrals are governed by individual sets of gas dynamics Euler equations. PUIs are treated as a separate, comoving fluid. Because the anisotropic behavior of PUIs at the heliospheric termination shocks is not described by the standard conservation laws (a.k.a. the Rankine–Hugoniot relations), we derived boundary conditions for them, which are obtained from the dedicated kinetic simulations of collisionless shocks. It is demonstrated that this approach to treating PUIs makes the computation results more consistent with observational data. In particular, the PUI pressure in the inner heliosheath (IHS) becomes higher by ∼40%–50% in the new model, as compared with the solutions where no special boundary conditions are applied. Hotter PUIs eventually lead to charge-exchange-driven cooling of the IHS plasma, which reduces the IHS width by ∼15% (∼8–10 au) in the upwind direction, and even more in the other directions. The density of secondary neutral atoms born in the IHS decreases by ∼30%, while their temperature increases by ∼60%. Simulation results are validated with New Horizons data at distances between 11 and 47 au.

Computational Dynamics of Multi-Rigid-Body System in Screw Coordinate

This paper investigates the kinematics and dynamics of multi-rigid-body systems in screw form. The Newton–Euler dynamics equations are established in screw coordinates. All forces and torques of the multi-rigid-body system can be solved straightforwardly since they are explicit in the form of screw coordinates. The displacement and acceleration are unified in matrix form, which associates the kinematics and dynamics with variable of velocity. A one-step numerical algorithm only is needed to solve the displacements and accelerations. As a result, all absolute displacements, velocities, and accelerations are directly obtained by one kinematic equation. The kinematics and dynamics of Gough–Stewart platform validate this the method. In this paper, the kinematics and dynamics are carried out with the example of a Gough–Stewart platform, which represents the most complex multi-rigid-body system, to verify the computational dynamics method. The proposed algorithm is also fit for the kinematics and dynamics modeling of other multi-rigid-body systems.

On Control Moments Construction for Uniaxial Reorientation Problem of Rigid Body under Disturbances

The problem of uniaxial reorientation of an asymmetric rigid body is solved by means of external control moments. The given geometric restrictions are imposed on the control moments. External uncontrolled interference is taken into account, the statictical descriptions of which is not available. It is assumed that the control moments of external forces are "applied" to the main central axes of inertia connected with the body. The control process is modeled be a nonlinear conflict-controlled system of ordinary differential equations, including dynamic Euler equations and kinematic equations in Poisson variables. The control moments are formed according to the feedback principle as nonlinear functions (discontinuous) of the phase variables of the considered conflict-controlled system. The choise of such functions is determined by the following circumstances: 1) the solution of the original nonlinear reorientation problem can be reduced to the solution of linear antagonistic game problems (with an unfixed end time); 2) in the absence of interference, the control momets are time-suboptimal; 3) reorientation is achieved by one spatial turn without additional restrictions on the nature of the resulting movement (such as a flat turn, etc.). Solutions of the closed control system are understood in the Filippov sense. An estimate of admissible levels of external un controlled interference depending on the given restrictions on the control moments is indicated, the conclusion of which is based on the indicated interpretation of the solutions. This estimate is a sufficient condition under which a guaranteed solution of the reorientation problem is provided in finite time by means of proposed constructions of control moments. An iterative algorithm is given for finding the control moment parameters that determine the guaranteed reorientation time.

Improvements of the fifth-order WENO-JS-type scheme with normalized smoothing factor for gas dynamic Euler equations

In this paper, we developed a novel high-resolution fifth-order WENO-JS-type scheme that can achieve the optimal accuracy order at or near the arbitrary order critical points. To enhance the accuracy of the weight function of WENO-JS, we normalized the local smoothness factor to obtain a series of normalized variables. By introducing a mapping function that can increase the accuracy of the normalization variables, we developed a new normalized WENO-JS-type scheme called WENO-NSI. The WENO-NSI scheme can improve the shock capture ability by selecting the apposite parameter λ in the mapping function. Numerical experiments of one- and two-dimensional gas dynamic Euler equations for the present WENO-NSI scheme are compared with those of the WENO-JS/Z/M/D/IM and TENO-Z schemes. The results show that the present scheme can achieve theoretical accuracy at or near the arbitrary order critical points, and it has a better shock capturing ability than other schemes.

Classical solutions for the Euler equations of compressible fluid dynamics: A new topological approach

In this article we study a class of Euler equations of compressible fluid dynamics. We give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results we propose a new approach based upon recent theoretical results.

Quick-response attitude takeover control using multiple servicing spacecraft based on inertia properties identification

Multi-spacecraft is essential to on-orbit servicing (OOS) which leads to spacecraft life prolongation thus reducing space debris. However, the abrupt change of inertia parameters caused by the attachment of each servicing spacecraft is a critical obstacle to the stable and high-accuracy attitude takeover control of a target spacecraft during the entire servicing process using multiple servicing spacecraft. Therefore, a quick-response attitude takeover control method based on inertia properties identification is proposed in this paper. Combining the Euler dynamical equations of combined spacecraft with space environment moment model, a novel iterative identification equation integrating inertia matrix is established to eliminate ill-conditioned identification and realize the precise identification of inertia parameters using only single sampling data for each update. With the quickly and precisely estimated inertia parameters, a Lyapunov-based attitude controller and an optimal torque allocation method are designed to actualize the high-accuracy and globally stable attitude takeover control with the minimum energy consumption. The proposed method can actualize the quick-response and stable attitude takeover control with low computational cost, even at the moment of attachment of servicing spacecraft. Hence, it is greatly appropriate for the OOS missions with multiple servicing spacecraft. The ground experiments and numerical simulations were conducted for demonstrating the feasibility and effectiveness of the proposed method. The experiment result indicated that the trajectory tracking error could converge to ±0.02° and ±0.02°/s and the parameters identification error was less than 5.5%.

Maximal Kinematical Invariance Group of Fluid Dynamics and Applications

The maximal kinematical invariance group of the Euler equations of fluid dynamics for the standard polytropic exponent is larger than the Galilei group. Specifically, the inversion transformation (Σ:t→−1/t,x→→x→/t) leaves the Euler equation’s invariant. This duality has been used to explain the striking similarities observed in simulations of the supernova explosions and laboratory implosions induced in plasma by intense lasers. The inversion symmetry extends to discontinuous fluid flows as well. In this contribution, we provide a concise review of these ideas and discuss some applications. We also explicitly work out the implosion dual of the Sedov’s explosion solution.

Influence of Nonlinear Dynamics Behavior of the Roller Follower on the Contact Stress of Polydyne Cam Profile

The effect of the cam profile on the nonlinear dynamics phenomenon of the follower is studied at three involutes’ profiles for the cam. The value of the Lyapunov exponent parameter is calculated at different internal distances of the follower guide from inside and at different cam speeds. The effect of the Lyapunov exponent value on the contact stress is studied based on the clearance between the follower and its guides. The contact between the cam and the square grooving key and between the cam and the follower has been taken into consideration at different locations. The finite element method is used to calculate the contact stress numerically using the SolidWorks program. The nonlinear response of the follower is calculated analytically using the Newton–Euler equations of rigid body dynamics of translation and rotation motions while the follower position is tracked experimentally using a high-speed 3-D camera device. The contact stress is checked and verified using photo-elastic apparatus. The data of the contact stress against time are used in the Wolf algorithm code to extract the value of the largest Lyapunov exponent. A phase-plane diagram is drawn at different cam speeds and different internal distances of the follower guide from the inside. The periodic and non-periodic motions of the contact stress are examined at different contact locations for the follower with the cam profile. The higher the value of contact stress, the higher the value of the largest Lyapunov exponent parameter and the higher the value of bending deflection.

Computational Analysis of Strain-Induced Effects on the Dynamic Properties of C60 in Fullerite

A hybrid discrete-continuous physical and mathematical model is used to study what deformation characteristics cause the rolling effect of C60 fullerene in a fullerite crystal. The interaction of fullerene atoms with surrounding molecules is described using a centrally symmetric interaction potential, in which the surrounding molecules are considered as a spherical surface of uniformly distributed carbon atoms. The rotational motion of fullerene is described by the Euler dynamic equations. The results of a numerical study of the influence of the rate, magnitude, and direction of strain on the dynamic characteristics of the rotational and translational motion of C60 fullerene in a crystalline fragment are presented.

Относительная динамика оболочек бифуллеренового комплекса

In this work, mathematical modeling of relative dynamics of a bifullerene complex is carried out on the assumption that the inner shell does not form covalent bonds with an outer carbon skeleton. This fact enables free angular movements of the inner shell. In particular, the directed rotation of the inner fullerene can be provided. This, in turn, allows for accumulating of a significant fraction of kinetic energy at internal degrees of freedom of the complex under consideration. In this case, the direction of rotations is not related to temperature; the outer shell of the complex restrains the transfer of the stored energy into thermal vibrations. Therefore, calculations are performed to estimate the stability of the rotational motion of an encapsulated fullerene relative to translational displacements of the outer shell. The calculations are carried out using a separate description of the dynamics of closed carbon molecules in terms of translational and rotational displacements. Translational displacements are determined using the equations of motion for the centers of mass of molecules. Rotational displacements are found on the basis of the dynamic Euler equations. The power centers in the considered framework structures of the molecules are carbon atoms. Therefore, the strength characteristics of intermolecular interactions are obtained in accordance with an atom-atom approach. In this case, the interaction parameters of individual atoms correspond to the case when these atoms are located in a structure of the surface carbon crystal.

ДЕЯКІ ЗАДАЧІ ПРО РУХ ТВЕРДОГО ТІЛА У СЕРЕДОВИЩІ З ОПОРОМ

The dynamics of rotating rigid bodies is a classical topic of study in mechanics. In the eighteenth and nineteenth centuries, several aspects of a rotating rigid body motion were studied by famous mathematicians as Euler, Jacobi, Poinsot, Lagrange, and Kovalevskya. However, the study of the dynamics of rotating bodies of still important for aplications such as the dynamics of satellite-gyrostat, spacecraft, re-entry vehicles, theory of gyroscopes, modern technology, navigation, space engineering and many other areas. A number of studies are devoted to the dynamics of a rigid body in a resistive medium. The presence of the velocity of proper rotation of the rigid body leads to the apearance of dissipative torques causing the braking of the body rotation. These torques depend on the properties of resistant medium in which the rigid body motions occur, on the body shape, on the properties of the surface of the rigid body and the distribution of mass in the body and on the characters of the rigid body motion. Therefore, the dependence of the resistant torque on the orientation of the rigid body and its angular velocity can de quite complicated and requires consideration of the motion of the medium around the body in the general case. We confine ourselves in this paper to some simple relations that can qualitative describe the resistance to rigid body rotation at small angular velocities and are used in the literature. In setting up the equations of motion of a rigid body moving in viscous medium, we need to consider the nature of the resisting force generated by the motion of the rigid body. The evolution of rotations of a rigid body influenced by dissipative disturbing torques were studied in many papers and books. The problems of motion of a rigid body about fixed point in a resistive medium described by nonlinear dynamic Euler equations. An analytical solution of the problem when the torques of external resistance forces are proportional to the corresponding projections of the angular velocity of the rigid body is obtain in several works. The dependence of the dissipative torque of the resistant forces on the angular velocity vector of rotation of the rigid body is assumed to be linear. We consider dynamics of a rigid body with arbitrary moments of inertia subjected to external torques include small dissipative torques.

Numerical Simulation of Interaction between Kr+ Ion and Rotating C60 Fullerene towards for Nanoarchitectonics of Fullerene Materials

Dynamics of charged fullerene in a surface layer of fullerite is studied under the influence of neutral or charged particles of the gas phase surrounding the fullerite material. The translational displacements of the nodes of the crystal lattice structure are determined by the equations of motion of the centers of mass of fullerenes. Central fullerene, which is described as a discrete set of sixty carbon atoms, plays a special role in the presented mathematical model. Angular oscillations and rotations of the central fullerene are described by the dynamic Euler equations. All other fullerenes have a centrally symmetric field of the potential of interaction with the surrounding atoms and molecules. In this regard, we use the hybrid discrete–continuous mathematical model with four potentials that describe the interactions between the surrounding fullerenes, smoothed fullerene and an atom, a pair of atoms, and electric charges. The results of a numerical study of influence of the Coulomb interaction on the rotational and translational motion of the C60 fullerene are presented.

Entropy Minimization for Many-Body Quantum Systems

The problem considered here is motivated by a work by B. Nachtergaele and H.T. Yau where the Euler equations of fluid dynamics are derived from manybody quantum mechanics, see [10]. A crucial concept in their work is that of local quantum Gibbs states, which are quantum statistical equilibria with prescribed particle, current, and energy densities at each point of space (here R d , d $\ge$ 1). They assume that such local Gibbs states exist, and show that if the quantum system is initially in a local Gibbs state, then the system stays, in an appropriate asymptotic limit, in a Gibbs state with particle, current, and energy densities now solutions to the Euler equations. Our main contribution in this work is to prove that such local quantum Gibbs states can be constructed from prescribed densities under mild hypotheses, in both the fermionic and bosonic cases. The problem consists in minimizing the von Neumann entropy in the quantum grand canonical picture under constraints of local particle, current, and energy densities. The main mathematical difficulty is the lack of compactness of the minimizing sequences to pass to the limit in the constraints. The issue is solved by defining auxiliary constrained optimization problems, and by using some monotonicity properties of equilibrium entropies.

Gyroscopic effects in fullerite crystal upon deformation

In this paper, the effect of deformation of a fullerite crystal on the dynamic properties of a rotating C60 molecule located in the center of a selected fragment is analyzed. The spherical potential of pair interaction is used to determine the dynamic characteristics of surrounding molecules. The angular oscillations and rotation of the central C60 molecule are described using the dynamic Euler equations. The effect of the direction of deformation of a fragment of a fullerite crystal on the precessional displacement of a rotating fullerene molecule is investigated depending on the direction and magnitude of the initial angular velocity.

On the use of multibody dynamics techniques to simulate fluid dynamics and fluid–solid interaction problems

A multibody dynamics-based solution to the fluid dynamics problem is compared herein to two established Lagrangian-based techniques used by the computational fluid dynamics (CFD) community. The multibody dynamics-based solution has two salient attributes: it enforces the incompressibility condition through bilateral kinematic constraints, and it treats the coupling with the solid phase via unilateral kinematic constraints. The multibody dynamics-based solution, called herein the Kinematically Constrained Smoothed Particle Hydrodynamics (KCSPH) method, is a Lagrangian approach to solving the CFD problem. It relies on the Smoothed Particle Hydrodynamics (SPH) method to discretize the spatial differential operators in the Navier–Stokes equations, and on the Newton–Euler equations of multibody dynamics to convect the SPH particles forward in time. We show that the multibody dynamics-based approach is efficient and accurate by comparing its performance with the two most commonly used SPH algorithms in the CFD community: the weakly compressible SPH (WCSPH), and the implicit SPH (ISPH) methods. The comparison is carried out in conjunction with four tests: an incompressibility benchmark test, dam break, floating cylinder, and sloshing tank. We conclude that KCSPH is a robust alternative to conventional CFD approaches for fluid–solid interaction (FSI) problems with complex/moving boundaries. The solvers and models used herein are publicly available in an open-source software called Chrono; the implementations use GPU (for WCSPH and ISPH), and multicore CPU (for KCSPH) parallel computing.

Solving procedure for the Kelvin–Kirchhoff equations in case of non-stationary rotations of slim disc

We have presented in this communication a new solving procedure for Kelvin–Kirchhoff equations, considering the dynamics of the falling or ascending the rigid disc in an ideal incompressible fluid, assuming additionally the dynamical symmetry of rotation for the falling or ascending body, I1 = I2. Fundamental law of angular momentum conservation has been used for the aforementioned solving procedure. The system of Euler equations for dynamics of disc’s rotation has been explored in regard to the existence of an analytic way of presentation for the approximated solution (where we consider the case of laminar flow at slow regime of deceleration of the symmetric disc’s rotation). This scenario is associated with consideration of that the Stokes boundary layer phenomenon on the boundaries of the body has also been assumed at formulation of basic Kelvin-Kirchhoff equations. It allows us to use the results of Jeffery’s fundamental work, in which analytical solution for rotations of spheroid’s particle in Euler angles was obtained. The results of calculations for the components of angular velocity {Ωi} should then be used for solving momentum equation of Kelvin–Kirchhoff system.

A Nonlinear CFD/Multibody Incremental-Dynamic Model for A Constrained Mechanism

Numerical analysis of a multibody mechanism moving in the air is a complicated problem in computational fluid dynamics (CFD). Analyzing the motion of a multibody mechanism in a commercial CFD software, i.e., ANSYS Fluent®, is a challenging issue. This is because the components of a mechanism have to be constrained next to each other during the movement in the air to have a reliable numerical aerodynamics simulation. However, such constraints cannot be numerically modeled in a commercial CFD software, and needs to be separately incorporated into models through the programming environment, such as user-defined functions (UDF). This study proposes a nonlinear-incremental dynamic CFD/multibody method to simulate constrained multibody mechanisms in the air using UDF of ANSYS Fluent®. To testify the accuracy of the proposed method, Newton–Euler dynamic equations for a two-link mechanism are solved using Matlab® ordinary differential equations (ODEs), and the numerical results for the constrained mechanisms are compared. The UDF results of ANSYS Fluent® shows good agreement with Matlab®, and can be applied to constrained multibody mechanisms moving in the air. The proposed UDF of ANSYS Fluent® calculates the aerodynamic forces of a flying multibody mechanism in the air for a low simulation cost than the constraint force equation (CFE) method. The results could have implications in designing and analyzing flying robots to help human rescue teams, and nonlinear dynamic analyses of the aerodynamic forces applying on a moving object in the air, such as airplanes, birds, flies, etc.