Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: the case of arbitrary smooth body shape

We present a (noncanonical) Hamiltonian model for the interaction of a neutrally buoyant, arbitrarily shaped smooth rigid body with N thin closed vortex filaments of arbitrary shape in an infinite ideal fluid in Euclidean three-space. The rings are modeled without cores and, as geometrical objects, viewed as N smooth closed curves in space. The velocity field associated with each ring in the absence of the body is given by the Biot–Savart law with the infinite self-induced velocity assumed to be regularized in some appropriate way. In the presence of the moving rigid body, the velocity field of each ring is modified by the addition of potential fields associated with the image vorticity and with the irrotational flow induced by the motion of the body. The equations of motion for this dynamically coupled body-rings model are obtained using conservation of linear and angular momenta. These equations are shown to possess a Hamiltonian structure when written on an appropriately defined Poisson product manifold equipped with a Poisson bracket which is the sum of the Lie–Poisson bracket from rigid body mechanics and the canonical bracket on the phase space of the vortex filaments. The Hamiltonian function is the total kinetic energy of the system with the self-induced kinetic energy regularized. The Hamiltonian structure is independent of the shape of the body, (and hence) the explicit form of the image field, and the method of regularization, provided the self-induced velocity and kinetic energy are regularized in way that satisfies certain reasonable consistency conditions.      

The stochastic energy-Casimir methodLa méthode d'énergie-Casimir stochastique

In this paper, we extend the energy-Casimir stability method for deterministic Lie–Poisson Hamiltonian systems to provide sufficient conditions for stability in probability of stochastic dynamical systems with symmetries. We illustrate this theory with classical examples of coadjoint motion, including the rigid body, the heavy top, and the compressible Euler equation in two dimensions. The main result is that stable deterministic equilibria remain stable in probability up to a certain stopping time that depends on the amplitude of the noise for finite-dimensional systems and on the amplitude of the spatial derivative of the noise for infinite-dimensional systems.     

Floating rigid bodies: a note on the conservativeness of the hydrostatic effects

Within the framework of Lagrangian mechanics, the conservativeness of the hydrostatic forces acting on a floating rigid body is proved. The representation of the associated hydrostatic potential is explicitly worked out. The invariance of the resulting Lagrangian with respect to surge, sway and yaw motions is used in connection with the Routh procedure in order to convert the original dynamical problem into a reduced one, in three independent variables. This allows to put on rational grounds the study of hydrostatic equilibrium, introducing the concept of pseudo-stability, meant as stability with respect to the reduced problem. The small oscillations of the system around a pseudo-stable equilibrium configuration are discussed.     

Stochastic stability of quasi-partially integrable and non-resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises

The asymptotic Lyapunov stability with probability one of multi-degree-of freedom quasi-partially integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises is studied. First, the averaged stochastic differential equations for quasi partially integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are derived by means of the stochastic averaging method and the stochastic jump-diffusion chain rule. Then, the expression of the largest Lyapunov exponent of the averaged system is obtained by using a procedure similar to that due to Khasminskii and the properties of stochastic integro-differential equations. Finally, the stochastic stability of the original quasi-partially integrable and non-resonant Hamiltonian systems is determined approximately by using the largest Lyapunov exponent. An example is worked out in detail to illustrate the application of the proposed method. The good agreement between the analytical results and those from digital simulation show that the proposed method is effective.     

Hamiltonian structure and relative equilibria of an axisymmetric rigid body in a second degree and order gravity field: cylindrical and generalized hyperbolic equilibria

The full dynamics of an axisymmetric rigid body in a uniformly rotating second degree and order gravity field are investigated, where orbit and attitude motions of the body are coupled through the gravity. Compared with the classical orbital dynamics with the body considered as a point mass, the full dynamics is a higher-precision model in close proximity of the central body where the gravitational orbit–attitude coupling is significant, such as a spacecraft about a small asteroid or an irregular-shaped natural satellite about a planet. The full dynamics are modeled by using the non-canonical Hamiltonian structure, in terms of variables expressed in the frame fixed with the central body. A Poisson reduction is carried out by means of the axial symmetry of the body, and a reduced system with lower dimension, as well as its non-canonical Hamiltonian structure and equations of motion, is obtained through the reduction process. With the second-order potential, three types of relative equilibria are found to be possible: cylindrical equilibria, generalized hyperbolic equilibria, and conic equilibria, which are counterparts to cylindrical equilibria, hyperbolic equilibria, and conic equilibria of an axisymmetric rigid body in a spherical gravity field, respectively. The geometrical properties and existence of the cylindrical equilibria and generalized hyperbolic equilibria are investigated in detail. It has been found that compared with the classical results in a spherical gravity field, the relative equilibria in this study are more complicated and diverse. The most significant difference is that the non-spherical gravity field enables the existence of non-Lagrangian hyperbolic equilibria, called generalized hyperbolic, which cannot exist in a spherical gravity.     

Rigid rotor dynamic stability using Floquet theory

The dynamic behaviour of a rigid rotor elastically connected by a constant speed joint is investigated. The effects of inertial inequalities and stiffness inequalities are evaluated and the stability characteristics are worked out. The Floquet theory is extensively applied to this system in order to obtain the stability limit curves. Comparisons with experimental results obtained by other authors, available in the literature, are reported in order to verify the procedure. Results concerning the unsymmetrical rotors are widely shown through parameters values restricted to realistic cases.     

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.

     

SYMPLECTIC STRUCTURE OF POISSON SYSTEM

When the Poisson matrix of Poisson system is non-constant, classical symplectic methods, such as symplectic Runge-Kutta method, generating function method, cannot preserve the Poisson structure. The non-constant Poisson structure was transformed into the symplectic structure by the nonlinear transform.Arbitrary order symplectic method was applied to the transformed Poisson system. The Euler equation of the free rigid body problem was transformed into the symplectic structure and computed by the midpoint scheme. Numerical results show the effectiveness of the nonlinear transform.     

On the stability of the rotational motion of a rigid body having a liquid filled cavity under finite initial disturbance

In this paper the problem of the stability of rotational motion of a rigid body which has a liquid filled cavity and a fixed point is investigated without any approximation. Criteria of stability and instability under finite disturbance are obtained. The region of stability is found out explicitly.     

非线弹性平面杆系的应力应变分析

以指数函数近似表示非线弹性材料的应力－应交关系,推导出了非线弹性材料平面杆系结构应力应变计算的普遍表达式,编制了通用程序,使这一类问题有了一个通用的解题方法．     

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.     

线弹性幂强化材料平面杆系弹塑性分析的数值解

各杆任意铰接在一个刚体上的平面杆系是一种比较复杂的杆系结构,某些其它类型的平面杆系常常可以看作是它的特例。本文将材料的本构关系描述为线性幂强化形式,推导出了该类平面杆系结构弹塑性分析的普遍表达式,编制了通用程序,使这一类问题有了一个通用的解题方法。     

基于机械振动理论的垂直侵彻弹靶作用模型

为了给侵彻引信抗高过载优化设计提供准确的力学输入,将机械振动理论引入侵彻过程建模领域,提出了一种侵彻战斗部刚体运动与一阶轴向振动相结合的垂直侵彻弹靶作用模型。在垂直侵彻过程受力分析的基础上,基于牛顿第二定律建立了战斗部刚体运动模型,基于单自由度弹簧-质量-阻尼系统建立了战斗部一阶轴向振动模型,并采用数值积分的方法获得了垂直侵彻过程中各物理量的变化规律。和火炮试验实测加速度信号的对比分析结果表明:考虑战斗部一阶轴向振动后的垂直侵彻弹靶作用模型能更准确地描述侵彻过程,能更有效地指导侵彻引信的抗高过载优化设计。     

Planar isotropic structures with negative Poisson’s ratio

A new design principle is suggested for constructing auxetic structures – the structures that exhibit negative Poisson’s ratio (NPR) at macroscopic level. We propose 2D assemblies of identical units made of a flexible frame with a sufficiently rigid reinforcing core at the centre. The core increases the frame resistance to the tangential movement thus ensuring high shear stiffness, whereas the normal stiffness is low being controlled by the local bending response of the frame. The structures considered have hexagonal symmetry, which delivers macroscopically isotropic elastic properties in the plane perpendicular to the axis of the symmetry. We determine the macroscopic Poisson’s ratio as a ratio of corresponding relative displacements computed using the direct microstructural approach. It is demonstrated that the proposed design can produce a macroscopically isotropic system with NPR close to the lower bound of ?1. We also developed a 2D elastic Cosserat continuum model, which represents the microstructure as a regular assembly of rigid particles connected by elastic springs. The comparison of values of NPRs computed using both structural models and the continuum approach shows that the continuum model gives a healthy balance between the simplicity and accuracy and can be used as a simple tool for design of auxetics.     

带充液腔重刚体的自旋稳定性

本文讨论带任意个充液腔的重刚体的自旋稳定性。以平均涡量作为液体的离散化变量,导出解析形式稳定性判据,并用于讨论充液自旋弹丸。对腔内隔板增强自旋稳定性的实际效果进行了估计。     

On the problem of free deceleration of a rigid body in a resisting medium

A mathematical model of the influence of a medium on a rigid body with some part of its external surface being flat is considered with due allowance for an additional dependence of the moment of the medium action force on the angular velocity of the body. A full system of equations of motion is given under quasi-steady conditions; the dynamic part of this system forms an independent third-order system, and an independent second-order subsystem is split from the full system. A new family of phase portraits on a phase cylinder of quasi-velocities is obtained. It is demonstrated that the results obtained allow one to design hollow circular cylinders (“shell cases”), which can ensure necessary stability in conducting additional full-scale experiments.     

中心舱体与薄壁梁刚-柔耦合系统的热弹性-结构动力学分析

空间柔性结构受太阳热流冲击而诱发的振动是导致航天器失效的典型模式之一,准确预测结构热致振动的响应及稳定性是卫星设计的基础.针对常见的中心舱体与附属薄壁杆件组成的空间结构,提出了考虑刚-柔耦合、耦合热弹性和耦合热-结构三重耦合效应的热致振动分析理论模型.其中,刚-柔耦合是指舱体姿态角、顶端集中质量转动与柔性附件运动的耦合...     

Squeeze flow of a Bingham-type fluid with elastic core

The squeeze flow of a Bingham-type material between finite circular disks is considered. The material is modelled assuming that the unyielded region behaves like a linear elastic core. A lubrication approximation is considered. It is shown that no paradox can arise, such as that has been pointed out for many years by various authors when the unyielded region in the fluid is supposed to be perfectly rigid. The unyielded region is shown to be always detached from the axis of symmetry. Some numerical simulations are worked out for different squeezing rates.     

Eulerian equilibria of a triaxial gyrostat in the three body problem: rotational Poisson dynamics in Eulerian equilibria

We consider the noncanonical Hamiltonian dynamics of a triaxial gyrostat in the three body problem. By means of geometric-mechanics methods, we will study the approximated dynamics that arises when we develop the potential in series of Legendre and truncate the series to the second harmonics. Working in the reduced problem, we will study the existence of equilibria that we will denominate of Euler in analogy with classic results on the topic. In this way, we generalize the classical results on equilibria of the three-body problem and many of those obtained by other authors using more classic techniques for the case of rigid bodies. The instability of Eulerian equilibria is proven in this approximate dynamics if the gyrostat is close to the sphere. The rotational Poisson dynamics of the gyrostat placed at an Eulerian equilibrium and the study of the nonlinear stability of some equilibria is considered. The analysis is done in vectorial form avoiding the use of canonical variables and the tedious expressions associated with them.     

Motion stability dynamics for spacecraft coupled with partially filled liquid container

This paper presents a canonical Hamiltonian model of liquid sloshing for the container coupled with spacecraft. Elliptical shape of rigid body is considered as spacecraft structure. Hamiltonian system is an important form of mechanical system. It mostly used to stabilize the potential shaping of dynamical system. Free surface movement of liquid inside the container is called sloshing. If there is uncontrolled resonance between the motion of tank and liquid-frequency inside the tank then such sloshing can be a reason of attitude disturbance or structural damage of spacecraft. Equivalent mechanical model of simple pendulum or mass attached with spring for sloshing is used by many researchers. Mass attached with spring is used as an equivalent model of sloshing to derive the mathematical equations in terms of Hamiltonian model. Analytical method of Lyapunov function with Casimir energy function is used to find the stability for spacecraft dynamics. Vertical axial rotation is taken as the major axial steady rotation for the moving rigid body.