Hamiltonian structure for rigid body with flexible attachments in a circular orbit

Hamiltonian structure of a rigid body in a circular orbit is established in this paper. With the reduction technique, the Hamiltonian structure of a rigid body in a circular orbit is derived from Lie-Poisson structure of semidirect product, and Hamiltonian is derived from Jacobi's integral. The above method can be generalized to establish the Hamiltonian structure of a rigid body with a flexible attachment in a circular orbit. At last, an example of stability analysis is given. The project supported by National Natural Science Foundation of China and Aeronautic Science Foundation.     

On the stability of interconnected rigid bodies

Summary The aim of this paper is to determine conditions for equilibrium of an orbiting system of interconnected rigid bodies and to find the stability conditions for a given equilibrium configuration. The equations were expressed in a body mean frame and derived from the Roberson-Wittenburg formalism, the combination of the advantages of these methods leading to a rather simple stability analysis by use of the Liapunov technique.

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Übersicht Es werden die Bedingungen für das Gleichgewicht eines Systems von miteinander verbundenen starren Körpern aufgestellt, die sich auf einer Umlaufbahn befinden. Die Stabilitätsbedingungen für eine gegebene Gleichgewichtslage werden berechnet. Die Bewegungsgleichungen werden nach dem Vorbild von Roberson-Wittenburg abgeleitet und in einem körperfesten Zwischensystem ausgedrückt. Dank der Vorteile dieses Vorgehens kann die Stabilitätsanalyse verhältnismäßig einfach nach der Ljapunovschen Methode durchgeführt werden.

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Presented at the Symposium on Systemtheoretische Probleme der Mechanik, in the Mathematisches Forschungsinstitut, in Oberwolfach (6–12 August 1972), by P. Y. Willems.     

Modelling and simulation of rigid bodies transportation by means of rotating flexible rollers

The paper deals with a modelling and simulation approach intended to the solution of a problem of rigid bodies transportation using rotating rollers. It is motivated by problematic phenomenon of ceramic tiles transportation inside long kilns during firing. The row of tiles, which is initially straight at the kiln beginning, is coming out curved at the kiln end or after the passage through a certain part of the kiln. The main aim of the work is the development of the modelling and simulation method for the verification of the problems arising from the tile movement through very long kilns and for the understanding of the problem causation. The multibody dynamics approach is chosen for the development of a comprehensive methodology for the numerical simulation of the tiles and rollers movement. Two approaches to the modelling of the system of tiles and rollers using the SIMPACK simulation tool are shown in the paper. The more suitable one is based on the Polygonal Contact Model developed for the fast and efficient contact analysis between complexly shaped rigid bodies. It can be concluded that the problems of the curved row of tiles are influenced by the flexibility of bent rollers and also occur for ideally cylindrical rollers. The presented simulation methodology can be generally used for the investigation of similar transportation systems.     

The nonlinear stability of a heavy rigid plate supported by flexible columns

A heavy rigid platform is supported by thin elastic legs. The governing equations for large deformations are formulated and solved numerically by homotopy and quasi-Newton methods. Nonlinear phenomena such as nonuniqueness, catastrophe and hysteresis are found. A global critical load for nonlinear stability is introduced.     

Optimum trapezoidal frame with rigid members and flexible joints

A loaded trapezoidal frame with rigid members and flexible joints is studied. An exact solution method is used to find buckling and non-linear post buckling characteristics. In most cases a snap through phenomenon occurs. There exists an optimum trapezoidal shape for stability.     

Asymptotic stability for perturbed Hamiltonian systems

Communicated by J. Serrin     

Some general results on the stability and flow failure of rigid viscoplastic structures

The evolution of structures made of materials which obey a rigid viscoplastic constitutive law, conceived as a generalisation of the classical Bingham model, and submitted to several loading parameters, is examined here in connection with the limit analysis (or yield design) theorems. A minimum principle for the velocities is established, from which it is shown in particular that the structure remains motionless as far as the loading belongs to the domain of safe loadings, while flow failure is triggered as soon as it is loaded beyond its limit load. Such a property is illustrated on the example of a uniform layer of Bingham material flowing on an inclined plane, due to the combined action of gravity and surface shear loading, considered as two independent loading parameters.     

A finite element based stability test for equilibria of flexible structures in circular orbits

Large flexible structures could be one of the most challenging space technologies of the future. For example, the well-known space elevator may one day guide astronauts along a 36,000 km long tether from the earth's surface to a geostationary orbit. In this paper, we imagine an arbitrary space structure in a stationary circular motion around a celestial body.Usually such systems require two problems to be solved: the strength of the materials and an overall stability analysis. The latter task is a non-trivial one since in orbiting systems not the angular rate but the angular momentum must be kept constant when applying, for example, Dirichlet's criterion. In particular, this fact becomes important if the system's dimension is in the magnitude of the orbital radius. Therefore, this paper will focus on a general applicable method to determine the stability of flexible earth-orbiting structures.The analysis process is based on the reduced energy momentum method presented by J.C. Simo, T.A. Posbergh and J.E. Marsden, which has been customized for use in systems with cyclic coordinates by the author. In the presented approach the second variation of an amended potential is derived from a standard finite element analysis. An efficient stability test is introduced, which limits the computational effort to the evaluation of a few eigenvalues and eigenvectors of the system's tangent stiffness matrix. Finally, two examples are discussed to demonstrate the application of the method.     

Contact problems for elastic bodies with initial stresses (rigid punches)

Conclusions In this work, the results are presented of investigations into contact problems for prestressed solids with special reference to rigid punches. It should be mentioned that in the last 5–6 years further investigations into the contact problems for solids with inifial stresses with special reference to elastic punches using the formulation of the second approach, i. e., in the general united form for compressible and incompressible solids with elastic potentials of the arbitrary structure, were also carried out by the authors of this article.The authors believe that further progress in the development of the mechanics of contact interaction of prestressed solids (both for rigid and elastic punches) is determined by examination of more complicated problem groups (for elastic, viscoelastic, and plastic solids), and by carrying out experiments to determine the strength of the effect of the initial stresses on the main characteristics of contact interaction of the structural material. It is also evident that, from the practical viewpoint, it is interesting to carry out these investigations for nonhomogeneous initial states. However, this is associated with considerable mathematical problems, even in theoretical developments and, these problems are even greater in application in practice.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Khmel'nitsk Technological Institute of Household Maintenance. Translated from Prikladnaya Mekhanika, Vol. 25, No. 8, pp. 3–18, August, 1989.     

A highly redundant coordinate formulation for constrained rigid bodies

The paper presents a formulation for the dynamic analysis of rigid multibodies. An introductory part carries out the kinematic analysis and the definition of the highly redundant differential framework along with the choice of unknowns and equations. From the differential formulation the variational principles, either in Lagrangian or Hamiltonian form, are developed. The Hamiltonian formulation is then used to develop the numerical approximation by applying the finite element method in time. The application of the method in its multifield form is discussed and a solution algorithm is proposed. Some examples are finally presented in order to verify the effectiveness of the formulation.

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Sommario Il lavoro presenta una formulazione per lo studio della dinamica dei sistemi multicorpo rigidi. Nella parte introduttiva viene svolta l'analisi cinematica e si definisce il quadro differenziale con la scelta delle incognite e delle equazioni. Dalla formulazione differenziale vengono poi sviluppati dei principi variazionali nella forma Lagrangiana ed Hamiltoniana. La formulazione Hamiltoniana é quindi utilizzata per sviluppare l'approssimazione numerica col metodo degli elementi finiti di tempo. Viene discussa l'applicazione del metodo nella forma multi-campo e viene proposto un algoritmo di soluzione. Da ultimo, vengono discussi alcuni esempi per verificare la correttezza della formulazione.

     

The sliding interface crack with friction between elastic and rigid bodies

The paper analyzes the frictional sliding crack at the interface between a semi-infinite elastic body and a rigid one. It gives solutions in complex form for non-homogeneous loading at infinity and explicit solutions for polynomial loading at the interface. It is found that the singularities at the crack tips are different and that they are related to distinct kinematics at the crack tips. Firstly, we postulate that the geometry of the equilibrium crack with crack-tip positions b and a is determined by the conditions of square integrable stresses and continuous displacement at both crack tips. The crack geometry solution is not unique and is defined by any compatible pair (b,a) belonging to a quasi-elliptical curve. Then we prove that, for an equilibrium crack under given applied load, the “energy release rate” G_tip, defined at each crack tip by the J_ε-integral along a semi-circular path, centered at the crack tip, with vanishing radius ε, vanishes. For arbitrarily shaped paths embracing the whole crack, with end points on the unbroken zone, the J-integral is path-independent and has the significance of the rate, with respect to the crack length, of energy dissipated by friction on the crack.     

Existence and stability results for thermoelastic dipolar bodies with double porosity

This paper is concerned with the theory of thermoelastic dipolar bodies which have a double porosity structure. In contrast with previous papers dedicated to classical elastic bodies, in our context the double porosity structure of the body is influenced by the displacement field, which is consistent with real models. In this setting, we show instability of solution as the initial energy is negative while under an appropriated (and realistic) condition, we prove existence and uniqueness of solution using semi-group theory.     

On the principle of gyroscopic effect for self-excited rigid bodies

Summary A new method is described whereby the gyroscopic effect for self-excited rigid bodies is studied by means of a differential equation of the second order and the results of other Authors are generalized.

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Sommario Si descrive un metodo di calcolo, che permette di ricondurre la discussione sulla validità dell'effetto giroscopico per i solidi autoeccitati allo studio di una equazione differenziale del secondo ordine e di generalizzare i risultati ottenuti da altri Autori.

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This work has been done under the auspices of the Research Group no. 6 of C.N.R.     

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.     

Penetration of rigid bodies into loose and layered media

Penetration and motion of rigid bodies in ground media attracts the researchers’ attention because of various problems arising as the technology evolves. In fact, there are two independent directions of studies in this field: (1) the problem of earth excavation when a rigid body of a definite shape slowly moves along a given trajectory in the ground; (2) an impact of a rapidly flying free rigid or deformable body against the ground. In the latter case, to which the proposed studies pertain, it is sometimes of interest to study the medium behavior and the motion of the free body, which moves in the medium after the impact owing to the kinetic energy of itself. In this field, a majority of studies deal with collision and penetration of bodies of various shapes into clay media. An extensive survey of these studies is given in [1]. After this survey appeared, numerous paper dealing with complicated collision conditions have been published [2]. Penetration in loose media has been studied much more rarely. The direct collision with fractured rock was studied in connection with the expected landing of spacecraft on other planets [3, 4]. In this case, the influence of grain dimensions and the density of the filling and vacuum on the penetration was studied for the initial velocities in the range of 1.7–10 m/s. On the other hand, in [5], the results of investigating the penetration of conic bodies in sand at entry velocities of 700–900 m/s are given; these velocities significantly exceed the speed of sound in this medium, which lies in the range of 100–200 m/s for dry sand. Analyzing the experimental results, the author came to the conclusion that it is necessary to use different representations of the drag force in the supersonic and subsonic modes. In the present paper, we do not consider the influence of the grain distribution, sand density, and filling methods on the penetration. But, as follows from the experiments whose results are described in [6] and [7], to represent the results of penetration of rigid bodies at velocities up to several hundreds of meters per second, in addition to the characteristics listed above, it is also required to describe the technology of the experiment preparation, because such media have the property of shape “memory.”