A solar system model with dissipation

A problem of motion for an arbitrary number of planets is discussed with consideration of the forces of gravitational interaction according to the law of universal gravitation. The planets are assumed to be homogeneous viscoelastic spheres. In the process of motion, the planets are deformed and the dissipation of energy takes place due to internal viscous forces. On the basis of the motion separation method, an approximate system of equations is obtained to describe the motion of planet centers of mass and the variation of planet angular momenta with respect to the centers of mass. The equations of motion contain small conservative corrections to the law of universal gravitation and small dissipative forces whose influence causes a decrease of the total mechanical energy. The motion under consideration admits the following first integral: the law of angular momentum conservation for the system with respect to the centers of mass. When the system executes the steady motion corresponding to its rotation with a constant angular velocity as a rigid body, the dissipative forces do not perform work, since the deformed planets have no time-dependent deformations.     

Melnikov's method for rigid bodies subject to small perturbation torques

Summary In this paper, the global motion of rigid bodies subjected to small perturbation torques, either conservative or dissipative, is investigated by means of Melnikov's method. Deprit's variables are introduced to transform the equations of motion into a standard form which is rendered suitable for the application of Melnikov's method. The Melnikov method is used to predict the transversal intersections of stable and unstable manifolds for the pertubed rigid-body motion. The chosen examples are a self-excited rigid body subject to a small periodic torque in a viscous medium, and the heavy rigid body. It is shown in both cases that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.     

Dynamics of a rigid cylinder near a plane boundary in the radiation field of an acoustic wave

An approach is described for investigation of the interaction between a rigid body and a viscous fluid boundary under acoustic wave propagation. The influence of the liquid on the rigid body is determined as a mean force, which is a constant in the time component of the hydrodynamic force. This enables the use of a previously developed technique for calculation of pressure in a compressible viscous liquid. The technique takes into account the second-order terms with respect to the wave field parameters and is based on investigation of a system of initially nonlinear hydromechanics equations that can be simplified with respect to the wave motion parameters of the liquid. It has proven possible to retain the second-order terms for determination of stresses in the liquid without having to solve the system of nonlinear equations. The stresses can be expressed in terms of parameters found in the solution of the linearized equations of the compressible viscous liquid. In this way, the solution of linearized equations is expressed in terms of a scalar and vector potentials. The problem statement is derived for a rigid cylinder located near a rigid flat wall under the effects of a wave propagating perpendicular to the wall. The solution for this particular example is obtained.     

The Motion of a Fluid–Rigid Disc System at the Zero Limit of the Rigid Disc Radius

We consider the two-dimensional motion of the coupled system of a viscous incompressible fluid and a rigid disc moving with the fluid, in the whole plane. The fluid motion is described by the Navier–Stokes equations and the motion of the rigid body by conservation laws of linear and angular momentum. We show that, assuming that the rigid disc is not allowed to rotate, as the radius of the disc goes to zero, the solution of this system converges, in an appropriate sense, to the solution of the Navier–Stokes equations describing the motion of only fluid in the whole plane. We also prove that the trajectory of the centre of the disc, at the zero limit of its radius, coincides with a fluid particle trajectory.     

Oscillations of a vessel containing a viscous fluid

The oscillations of a rigid body having a cavity partially filled with an ideal fluid have been studied in numerous reports, for example, [1–6]. Certain analogous problems in the case of a viscous fluid for particular shapes of the cavity were considered in [6, 7]. The general equations of motion of a rigid body having a cavity partially filled with a viscous liquid were derived in [8]. These equations were obtained for a cavity of arbitrary form under the following assumptions: 1) the body and the liquid perform small oscillations (linear approximation applicable); 2) the Reynolds number is large (viscosity is small). In the case of an ideal liquid the equations of [8] become the previously known equations of [2–6]. In the present paper, on the basis of the equations of [8], we study the free and the forced oscillations of a body with a cavity (vessel) which is partially filled with a viscous liquid. For simplicity we consider translational oscillations of a body with a liquid, since even in this case the characteristic mechanical properties of the system resulting from the viscosity of the liquid and the presence of a free surface manifest themselves.The solutions are obtained for a cavity of arbitrary shape. We then consider some specific cavity shapes.     

Numerical simulation of plate autorotation in a viscous fluid flow

A general formulation of the plane coupled dynamical and aerodynamical problem of the motion of a rigid body with a rotational degree of freedom in a viscous incompressible fluid flow is given. A computation technique for solving the Navier-Stokes equations based on the meshless viscous vortex domain method is used. The autorotation of a single plate and a pair of plates is investigated. The effect of the reduced moment of inertia and the Reynolds number on the angular rotation velocity is determined. The time dependences of the hydrodynamic loads are compared with the corresponding instantaneous flow patterns. The increased the autorotation velocity of two plates in tandem is detected.     

The Method of Averaging for Euler's Equations of Rigid Body Motion

We formulate the method of averaging for perturbations of Euler's equations of rotational motion. Euler's equations are three strongly nonlinear coupled differential equations that can be viewed as a three dimensional oscillator. The method of averaging is used to determine the long-term influence of perturbation terms on the motion by averaging about the nominal rigid body motion. The treatment is applicable to a large class of motions including precession with large nutation – it is not restricted to small motions about simple spins or nearly axi-symmetric bodies. Three examples are shown that demonstrate the accuracy of the method's predictions.     

A discrete model of a rope with bending stiffness or viscous damping

A discrete model of a rope is developed and used to simulate the plane motion of the rope fixed at one end.Actually,two systems are presented,whose members are rigid but non-ideal joints involve elasticity or dissipation.The dissipation is reflected simply by viscous damping model, whereas the bending stiffness conception is based on the classical curvature-bending moment relationship for beams and simple geometrical formulas.Equations of motion are derived and their complexity is discussed from the computational point of view.Since modified extended backward differentiation formulas(MEBDF)of Cash are implemented to solve the resulting initial value problems,the technique scheme is outlined.Numerical experiments are performed and influences of the elasticity and damping on behaviour of the model are analyzed.Basic energy principles are used to verify the obtained results.     

Control structure interaction in the nonlinear analysis of flexible mechanical systems

The effect of the control structure interaction on the feedforward control law as well as the dynamics of flexible mechanical systems is examined in this investigation. An inverse dynamics procedure is developed for the analysis of the dynamic motion of interconnected rigid and flexible bodies. This method is used to examine the effect of the elastic deformation on the driving forces in flexible mechanical systems. The driving forces are expressed in terms of the specified motion trajectories and the deformations of the elastic members. The system equations of motion are formulated using Lagrange's equation. A finite element discretization of the flexible bodies is used to define the deformation degrees of freedom. The algebraic constraint equations that describe the motion trajectories and joint constraints between adjacent bodies are adjoined to the system differential equations of motion using the vector of Lagrange multipliers. A unique displacement field is then identified by imposing an appropriate set of reference conditions. The effect of the nonlinear centrifugal and Coriolis forces that depend on the body displacements and velocities are taken into consideration. A direct numerical integration method coupled with a Newton-Raphson algorithm is used to solve the resulting nonlinear differential and algebraic equations of motion. The formulation obtained for the flexible mechanical system is compared with the rigid body dynamic formulation. The effect of the sampling time, number of vibration modes, the viscous damping, and the selection of the constrained modes are examined. The results presented in this numerical study demonstrate that the use of the driving forees obtained using the rigid body analysis can lead to a significant error when these forces are used as the feedforward control law for the flexible mechanical system. The analysis presented in this investigation differs significantly from previously published work in many ways. It includes the effect of the structural flexibility on the centrifugal and Coriolis forces, it accounts for all inertia nonlinearities resulting from the coupling between the rigid body and elastic displacements, it uses a precise definition of the equipollent systems of forces in flexible body dynamics, it demonstrates the use of general purpose multibody computer codes in the feedforward control of flexible mechanical systems, and it demonstrates numerically the effect of the selected set of constrained modes on the feedforward control law.     

New solutions of classical problems in rigid body dynamics

The equations of motion of a rigid body acted upon by general conservative potential and gyroscopic forces were reduced by Yehia to a single second-order differential equation. The reduced equation was used successfully in the study of stability of certain simple motions of the body. In the present work we use the reduced equation to construct a new particular solution of the dynamics of a rigid body about a fixed point in the approximate field of a far Newtonian centre of attraction. Using a transformation to a rotating frame we also construct a new solution of the problem of motion of a multiconnected rigid body in an ideal incompressible fluid. It turns out that the solutions obtained generalize a known solution of the simplest problem of motion of a heavy rigid body about a fixed point due to Dokshevich.     

A quasi-static approach to the stability of the path of heavy bodies falling within a viscous fluid

We consider the gravity-driven motion of a heavy two-dimensional rigid body freely falling in a viscous fluid. We introduce a quasi-static linear model of the forces and torques induced by the possible changes in the body velocity, or by the occurrence of a nonzero incidence angle or a spanwise rotation of the body. The coefficients involved in this model are accurately computed over a full range of Reynolds number by numerically resolving the Navier–Stokes equations, considering three elementary situations where the motion of the body is prescribed. The falling body is found to exhibit three distinct eigenmodes which are always damped in the case of a thin plate with uniform mass loading or a circular cylinder, but may be amplified for other geometries, such as in the case of a square cylinder.     

Sur une méthode de pénalisation tensorielle pour la résolution des équations de Navier–Stokes

An original reformulation of the viscous stress tensor is proposed for the motion equations dedicated to an incompressible fluid. Four different tensors appear in this decomposition, associated with viscosities of compression, elongation, shearing and rotation. This new model allows us to build a numerical solver of the Navier–Stokes equations based on a technique of tensorial penalization which is generalized with all the stresses acting on a flow. The processing of incompressibility and the rotation of a rigid body in a flow are described thanks to the model. Several numerical applications are proposed to illustrate the abilities of the new penalization method.     

Rocking response of rectangular rigid blocks under random noise base excitations

The response of a rigid rectangular block resting on a rigid foundation and acted upon simultaneously by a horizontal and a vertical random white-noise excitation is considered. In the equation of motion, the energy dissipation is modeled through a viscous damping term. Under the assumption that the body does not topple, the steady-state joint probability density function of the rotation and the rotational velocity is obtained using the Fokker-Planck equation approach. Closed form solution is obtained for a specific combination of system parameters. A more general but approximate solution to the joint probability density function based on the method of equivalent non-linearization is also presented. Further, the problem of overturning of the block is approached in the framework of the diffusion methods for first passage failure studies. The overturning of the block is deemed incipient when the response trajectories in the phase plane cross the separatrix of the conservative unforced system. Expressions for the moments of first passage time are obtained via a series solution to the governing generalized Pontriagin-Vitt equations. Numerical results illustra- tive of the theoretical solutions are presented and their validity is examined through limited amount of digital simulations.     

Global Weak Solutions¶for the Two-Dimensional Motion¶of Several Rigid Bodies¶in an Incompressible Viscous Fluid

We consider the two-dimensional motion of several non-homogeneous rigid bodies immersed in an incompressible non-homogeneous viscous fluid. The fluid, and the rigid bodies are contained in a fixed open bounded set of ?². The motion of the fluid is governed by the Navier-Stokes equations for incompressible fluids and the standard conservation laws of linear and angular momentum rule the dynamics of the rigid bodies. The time variation of the fluid domain (due to the motion of the rigid bodies) is not known a priori, so we deal with a free boundary value problem. The main novelty here is thedemonstration of the global existence of weak solutions for this problem. More precisely, the global character of the solutions we obtain is due to the fact that we do not need any assumption concerning the lack of collisions between several rigid bodies or between a rigid body and the boundary. We give estimates of the velocity of the bodies when their mutual distance or the distance to the boundary tends to zero.     

Evolution of a heavy rigid body rotation under the action of unsteady restoring and perturbation torques

The paper develops an approximate solution to the system of Euler’s equations with additional perturbation term for dynamically symmetric rotating rigid body. The perturbed motions of a rigid body, close to Lagrange’s case, under the action of restoring and perturbation torques that are slowly varying in time are investigated. We describe an averaging procedure for slow variables of a rigid body perturbed motion, similar to Lagrange top. Conditions for the possibility of averaging the equations of motion with respect to the nutation phase angle are presented. The averaging technique reduces the system order from 6 to 3 and does not contain fast oscillations. An example of motion of the body using linearly dissipative torques is worked out to demonstrate the use of general equations. The numerical integration of the averaged system of equations is conducted of the body motion. The graphical presentations of the solutions are represented and discussed. A new class of rotations of a dynamically symmetric rigid body about a fixed point with account for a nonstationary perturbation torque, as well as for a restoring torque that slowly varies with time, is studied. The main objective of this paper is to extend the previous results for problem of the dynamic motion of a symmetric rigid body subjected to perturbation and restoring torques. The proposed averaging method is implemented to receive the averaging system of equations of motion. The graphical representations of the solutions are presented and discussed. The attained results are a generalization of our former works where µ and M_i are independent of the slow time τ and M_i depend on the slow time only.

     

一类刚-梁耦合系统平面稳态运动的稳定性

研究了中心力场中的一类刚-弹耦合系统的平面运动动力学，模型是带有一悬臂 梁的刚体. 综合考虑了系统轨道运动与姿态运动，在Lagrange力学体系下给出了系统的运 动方程，在保守系统和考虑梁的材料黏滞阻尼两种情况下，利用能量-动量方法给出了一类 相对平衡点稳定性的充分条件.     

带约束平面多刚体系统动力学方程的隐式算法

在多体系统动力学正则方程的基础上建立了平面多体系统正则方程的隐式数值算法。利用平面运动的特性，对正则方程进行了简化，导出了该方程的Ｊａｃｏｂｉ矩阵的一般表达式，给出了Ｒｕｎｇｅ－Ｋｕｔａ多体系统动力学方程隐式数值计算方法。算例表明，该方法是一种计算速度和精度均理想的数值方法。     

Self-induced jumping of a rigid body of revolution on a smooth horizontal surface

The article is devoted to the study of the motion of a rigid body of revolution on a rigid and perfectly smooth horizontal surface under the influence of the uniform gravitational field. Basic equations are listed and their solutions are given. The unilateral contact between the body and the plane at non-steady motion is investigated and the procedure of calculation of threshold values of the body energy above which the contact is broken is given. In contrast to Shimomura et al. [Dynamics of an axisymmetric body spinning on a horizontal surface. II. Self-induced jumping. Proc. R. Soc. A 461 (2005) 1775-1809], who assumed sliding friction in their analysis, it is found that the self-induced jumping can also occur in the absence of friction at the very beginning of the motion. The free motion after the contact is lost and impact of the body when it again makes contact with the plane is discussed. The motion of a spheroid and a disk which illustrate the results of the general theory are discussed in some detail.     

Global Existence of Weak Solutions for Viscous Incompressible Flows around a Moving Rigid Body in Three Dimensions

We study the motion of a rigid body of arbitrary shape immersed in a viscous incompressible fluid in a bounded, three-dimensional domain. The motion of the rigid body is caused by the action of given forces exerted on the fluid and on the rigid body. For this problem, we prove the global existence of weak solutions.