Stability analysis of relative equilibria of mechanical systems with cyclic coordinates: a direct approach

Nonlinear stability of relative equilibria of mechanical systems has been investigated during the past two decades by notable authors and has resulted in the so-called energy momentum method. Although it has numerous important engineering applications, this theory involves subtle mathematical methods such as group theory with which engineers usually are not familiar. This paper develops a simple and natural approach to the problem for the case of cyclic coordinates in the Lagrangian since many practical examples can be easily formulated in terms of cyclic coordinates. Referring to standard algebraic operations, a stability criterion for relative equilibria is derived. As a computational benefit the presented approach does not require knowledge of a system's complete kinetic energy, either for formulating steady-state equations or for checking stability. The application of the method, which is closely related to Routh's method, will be demonstrated using the example of a dumbell satellite.     

On the dynamics of a tethered satellite system

The Hamiltonian structure for a fundamental model of a tethered satellite system is constructed. The model is composed of two point masses connected by a string with no restrictions on the motions of the two masses. A certain symmetry with respect to the special orthogonal group SO(3) for such a system is observed. The classical station-keeping mode for the tethered system is found to be nothing more than the relative equilibrium corresponding to the reduction of the system by the symmetry. The microgravity forces on the two point masses are responsible for the possible configurations of the string at the so-called radial relative equilibrium. A stability analysis is performed on the basis of the reduced energy-momentum method. Criteria for stability are derived, which could find potential applications in space technology.     

Dynamics of rigid and flexible polymer chains in confined geometries. part I: steady simple shear flow

This paper describes exact solutions to the response of both the elastic and rigid dumbbell models to a steady simple shear flow in a channel having a length scale comparable to the dumbbells themselves. Results are given for rheological properties over the entire range of the ratio of the channel width to the length of the dumbbell. It is found that both models lead to a decrease in viscosity as the channel is reduced in size with the elastic dumbbell predicting a stronger dependence on that parameter compared with the rigid dumbbell. The elastic dumbbell predicts shear independent rheological properties whereas the rigid dumbell predicts shear thinning as in the case of unbounded flows. The rate of shear thinning, however, decreases with decreasing channel width.     

The elastic energy-momentum tensor

The application to continuum mechanics of the general methods of the classical theory of fields is advocated and illustrated by the example of the static elastic field. The non-linear theory of elasticity is set up in the most convenient form (lagrangian coordinates and stress tensor). The appropriate energy-momentum tensor is derived, and it is shown that the integral of its normal component over a closed surface gives the force (as the term is used in the theory of solids) on defects and inhomogeneities within the surface. Other topics discussed are Günther's and related integrals, symmetrization of the energy-momentum tensor, and the Eulerian formulation. Some further extensions, existing and potential, are indicated.     

Synthetic interpretation of attitude dynamics of a spinning satellite under continuous and pulsed torques

Summary The problem of the dynamics of a rigid satellite in the case of no damping has been thoroughly treated in several papers[1]; less literature is available on the problem of damped satellites.Furthermore, the formulas describing the dynamics are quite complicated and so their interpretation is not straightforward.This study presents the results of one attempt to interpret the dynamics on the basis of relatively simple physical and cinematic considerations, which lend themselves to an easy interpretation of the performance and a simple determination of the most critical cases, for continuous and pulsed external torques both in the damped and undamped cases.

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Sommario Il problema della dinamica di un satellite rigido è stato trattato a fondo in vari articoli[1] in assenza di smorzamento. Esistono pochi articoli viceversa sul problema di satellite smorzato.Le formule che ne descrivono la dinamica sono assai complicate e di conseguenza la loro interpretazione non è immediata.Questo studio presenta i risultati di un tentativo di interpretazione della dinamica basato su considerazioni fisiche e cinematiche particolarmente semplici, che si prestano ad una facile interpretazione del funzionamento e a una chiara rappresentazione dei casi più critici sia in corrispondenza di eccitazione esterna a coppia costante che impulsiva.

     

Derivation of energy-momentum tensors in theories of micropolar hyperbolic thermoelasticity

In the framework of the classical field theory and using the theory of action variational symmetries, we consider the construction of canonical energy-momentum tensors for a coupled micropolar thermoelastic field taking account of the nonlocality of the Lagrangian density, which is typical of continuum micromechanics. We use the algorithms of group analysis to calculate the Noether currents and the energy-momentum tensors in three cases where the Lagrangian depends on the gradients of field variables of orders not exceeding 1, 2, and 3. In each of these cases, we present explicit formulas for the components of the canonical energy-momentum tensor. We construct the energy-momentum tensor for micropolar thermoelastic bodies in which the heat conduction process is characterized by a generalized heat equation of hyperbolic analytical type. In the equations of micropolar thermoelastic field, all possible restrictions on the microrotations are taken into account.     

Bilans d'energie et de quantité de mouvement pour un continu deformable,polarisable et magnétisable

In this paper, it is shown that the different electromagnetic energy-momentum tensors proposed by various authors for a continuum interacting with an electromagnetic field all lead to the same equations of balance for energy and momentum provided the definitions of stress and internal energy are suitably related. These various tensors come out from different partitions of the total energy-momentum tensor. From a particular partition, we derive an expression of the balance of energy suitable for application to continuum thermodynamics. In the classical approximation, the corresponding equation of balance of momentum gives rise to an expression for the electromagnetic force in a polarizable and magnetizable continuum.     

Lagrangian block diagonalization

Relative equilibria, i.e., steady motions associated to specified group motions, are an important class of steady motions of Hamiltonian and Lagrangian systems with symmetry. Relative equilibria can be identified by means of a variational principle on the tangent space of the configuration manifold. We show that relative equilibria can also be found by means of a variational principle on the configuration manifold itself. Formal stability of a relative equilibrium corresponds to definiteness of the second variation of the energymomentum functional, which is a specified combination of the total energy and the group momentum, on an appropriate subspace. We decompose this subspace into three subspaces by means of the Legendre transformation and the group action and show that the second variation block diagonalizes with respect to these subspaces. The techniques employed here are a generalization of the reduced energy-momentum method of Simoet al. (1991), which applies only to simple mechanical systems, to a more general class of conservative systems, including systems on which the symmetry group does not act freely. We briefly discuss a generalization of a result due to Patrick (1990) that provides conditions under which formal stability implies nonlinear orbital stability. Several simple examples, including natural mechanical systems, are used to illustrate the block diagonalization procedure.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅹ)--MASTER BALANCE LAW

Through a comparison between the expressions of master balance laws and the conservation laws derived by Noether's theorem, a unified master balance law and six physically possible balance equations for micropolar continuum mechanics are naturally deduced. Among them, by extending the well-known conventional concept of energymomentum tensor, the rather general conservation laws and balance equations named after energy-momentum, energy-angular momentum and energy-energy are obtained. It is clear that the forms of the physical field quantities in the master balance law for the last three cases could not be assumed directly by perceiving through the intuition. Finally,some existing results are reduced immediately as special cases.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅹ) —MASTER BALANCE LAW

Through a comparison between the expressions of master balance laws and the conservation laws derived by Noether's theorem, a unified master balance law and six physically possible balance equations for micropolar continuum mechanics are naturally deduced. Among them, by extending the well-known conventional concept of energymomentum tensor, the rather general conservation laws and balance equations named after energy-momentum, energy-angular momentum and energy-energy are obtained. It is clear that the forms of the physical field quantities in the master balance law for the last three cases could not be assumed directly by perceiving through the intuition. Finally, some existing results are reduced immediately as special cases.     

带偏心转子陀螺体的混沌运动

本文讨论了无力矩条件下带有质量偏心轴对称转子的非对称陀螺体的运动。利用动量变量列写动力学方程,并将系统化作受周期微扰作用下的Ｅｕｌｅｒ－Ｐｏｉｎｓｏｔ运动。应用Ｍｅｌｎｉｋｏｖ方法预测系统存在Ｓｍａｌｄ马蹄意义下的混沌运动,此结论与Ｐｏｉｎｃａｒｅ截面的数值计算相符。从Ｐｏｉｎｃａｒｅ截面的相图也可明显看出转子对于双自旋卫星的姿态稳定作用。     

Applications of an energy-momentum tensor in non-linear elastostatics

For equilibrium states of elastic materials some general formulae of conservation type have been established in recent papers by Knowles and Sternberg and by Green. It is shown that these results arise naturally from the application of standard integral identities to an energy-momentum tensor first introduced into elastostatics by Eshelby. A duality is exhibited between the energy-momentum tensor and the Cauchy stress which leads directly to inverse deformation relations for elastic solids due originally to Shield.     

Chaotic motion in a class of asymmetrical Kelvin type gyrostat satellite

This work is an investigation on the roots of chaotic attitudinal motion in a class of asymmetrical gyrostat satellites. The result shows that for a class of Kelvin type gyrostat satellite, there is an equivalent rigid spinning satellite with the same attitude dynamics. Finding some constants of motion and eliminating the cyclic coordinates, the rotational kinetic energy is changed to a quadratic form and using Jordan canonical form of the associated inertia tensor and transforming the coordinate system, the Hamiltonian has been changed to those of a rigid satellite. The Hamiltonian has been split into integrable and non-integrable parts. Using Deprit canonical transformation and Andoyer variables the integrable part has been reduced to a one-dimensional form. The reduced Hamiltonian shows that the regular dynamics of the satellite can be chaotic, under the influence of gravitational effects. To demonstrate various attitudinal dynamics of the satellite, a second-order Poincaré map is employed. This research shows firstly, that the attitudinal dynamics of Kelvin type gyrostat satellites and rigid satellites follow the same dynamical patterns, secondly, for non-linear analysis of dynamics of gyrostat satellite based on the perturbation methods, there is a preferable form for Hamiltonian of the system in the near-integrable fashion and thirdly the chaotic motion is originated from the gravitational field effects that can be suppressed by increasing the attitudinal energy of the satellite in comparison with the translational energy.     

Energy dissipation and contour integral characterizing fracture behavior of incremental plasticity

Jep -integral is derived for characterizing the frac- ture behavior of elastic-plastic materials. The J ep -integral differs from Rice’s J-integral in that the free energy density rather than the stress working density is employed to define energy-momentum tensor. The J ep -integral is proved to be path-dependent regardless of incremental plasticity and deformation plasticity. The J epintegral possesses clearly clear physical meaning: (1) the value J ep tip evaluated on the infinitely small contour surrounding the crack tip represents the crack tip energy dissipation; (2) when the global steadystate crack growth condition is approached, the value of J ep farss calculated along the boundary contour equals to the sum of crack tip dissipation and bulk dissipation of plastic zone. The theoretical results are verified by simulating mode I crack problems.     

A material force method for inelastic fracture mechanics

A material force method is proposed for evaluating the energy release rate and work rate of dissipation for fracture in inelastic materials. The inelastic material response is characterized by an internal variable model with an explicitly defined free energy density and dissipation potential. Expressions for the global material and dissipation forces are obtained from a global balance of energy-momentum that incorporates dissipation from inelastic material behavior. It is shown that in the special case of steady-state growth, the global dissipation force equals the work rate of dissipation, and the global material force and J-integral methods are equivalent. For implementation in finite element computations, an equivalent domain expression of the global material force is developed from the weak form of the energy-momentum balance. The method is applied to model problems of cohesive fracture in a remote K-field for viscoelasticity and elastoplasticity. The viscoelastic problem is used to compare various element discretizations in combination with different schemes for computing strain gradients. For the elastoplastic problem, the effects of cohesive and bulk properties on the plastic dissipation are examined using calculations of the global dissipation force.     

基于交叉降阶取样的改进响应面方法

针对经典响应面法和加权响应面法存在的计算精度不高和稳定性不强的问题,提出一种新的样本点选择策略,即交叉降阶思想,并基于此发展出一种新的改进响应面方法。该方法以均值点加上0.5倍的标准差作为初始迭代点。在Bucher试验设计的基础之上,将n维坐标系降阶为n个一维坐标系。然后,依据样本点的重要性,筛选出n+1个靠近真实失效面的优秀样本点,并用这些样本点去拟合线性响应面函数。最后依据样本中心间的距离与上一次迭代得到的设计点的范数之商作为其收敛准则。通过5个数值算例与工程算例,验证了交叉降阶响应面法具有较高的精度、效率、稳定性和一定的工程适用性。     

Estimating zero‐g flow rates in open channels having capillary pumping vanes

In vane‐type surface tension propellant management devices (PMD) commonly used in satellite fuel tanks, the propellant is transported along guiding vanes from a reservoir at the inlet of the device to a sump at the outlet from where it is pumped to the satellite engine. The pressure gradient driving this free‐surface flow under zero‐gravity (zero‐g) conditions is generated by surface tension and is related to the differential curvatures of the propellant‐gas interface at the inlet and outlet of the PMD. A new semi‐analytical procedure is prescribed for accurately calculating the extremely small fuel flow rates under reasonably idealized conditions. Convergence of the algorithm is demonstrated by detailed numerical calculations. Owing to the substantial cost and the technical hurdles involved in accurately estimating these minuscule flow rates by either direct numerical simulation or by experimental methods which simulate zero‐g conditions in the lab, it is expected that the proposed method will be an indispensable tool in the design and operation of satellite fuel tanks. Copyright © 2003 John Wiley & Sons, Ltd.     

Exploitation of the dissipation inequality in general relativistic continuum thermodynamics

The balance equations of energy-momentum and spin together with Einstein’s field equations are investigated by the Liu procedure to find constraints for the constitutive equations in such a way that the Second Law is satisfied. Special cases such as spinless systems and curvature insensitive materials are shortly discussed.     

Remarks on formulating an adhesion problem using Euler’s elastica (draft)

Three formulations for the problem of an elastica adhering to a rigid surface are discussed and compared. These include stationary principles, the surface integral of Eshelby’s energy-momentum tensor, and the material (configurational) force balance. The configuration at static equilibrium is predicted in closed form for a pair of structures that arise in nano- and microscale applications.     

二级多点近似方法在整星结构优化设计中的应用

以整星结构为研究对象,针对含有离散变量的混合变量优化问题,应用二级多点近似方法给出从卫星拓扑布局设计到尺寸的一体化优化求解过程。首先,对整星结构进行仿真和振动试验,设计满足总体指标要求且已得到在轨验证。其次,建立以极小化卫星质量为目标的多工况优化模型后利用分段多点逼近函数建立了显式近似问题。最后,采用遗传算法和变尺度法分别对离散/连续两类变量寻优,并以拓扑构型一致性和目标函数的下降程度作为收敛的判定依据。结果显示,在满足频率和静力的约束下,卫星质量降低了7.9%,且结构分析迭代次数仅为50~60次,从而验证了该方法在求解整星结构多变量多工况下的一体化设计问题时具有很高的计算效率和准确性。