Steady motions of an axisymmetric satellite: an atlas of their bifurcations

The steady motions of an axially symmetric rigid satellite orbiting a fixed spherically symmetric rigid body are considered in this paper. Based on a model for the dynamics of the satellite which incorporates the classical approximation for the gravitational potential and includes the attitude-orbit coupling, the bifurcations and non-linear stability of the steady motions of the satellite are discussed. These results extend earlier works on this problem by Likins, Pringle, Rumyantsev, Stepanov, and Thomson, by showing how the motions found by these authors are interrelated, and how several of them are physically unrealistic because their orbital radii are too small.     

用水下卡片间隙试验法测定炸药冲击波感度

将炸药的水下爆炸能量测试方法与冲击波感度测试方法（小隔板法）相结合,研究了一种测试炸药冲击波感度的方法水下卡片间隙试验法,并用该方法得到了6种炸药冲击波感度的相对排序（从高到低）:炸药泰安、黑索金、8701、钝化黑索金、梯恩梯、膨化硝酸铵的冲击波感度依次降低,该试验结果与文献值、传统隔板试验结果一致,表明水下卡片间隙试验法的测试结果是可信的,水下卡片间隙试验法用于测定炸药冲击波感度是可行的,在一定程度上可代替传统的隔板试验法,此方法还可用于研究炸药装药密度对其冲击波感度的影响。     

On steady motions of a rigid body bearing three-degree-of-freedom control momentum gyroscopes and their stability

Equations of motion are obtained for a rigid body bearing N three-degree-of-freedom control momentum gyroscopes in gimbals and the entire set of steady motions in a homogeneous external field is determined. The steady motion dependence on the magnitude of the system angular momentum is studied and a detailed analysis of the secular stability is performed.     

Free surface wave motions and interactions

This paper investigates the problems involved in the numerical simulation of free surface wave motions and surface wave effects on marine structures. Various approaches that might be taken in meeting these problems are discussed, and their relative advantages and disadvantages are considered. One particular approach combines a Lagrangian formulation of the governing equations, a triangular grid and a finite-difference solution procedure. Since this approach has some distinct advantages in the numerical calculation of fluid flows which include a free surface, it formed the basis for the development of one particular computer code, SPLISH. Sufficient progress has been made with the SPLISH code to demonstrate the attractiveness of numerical calculations for wave flow problems. Recent computational results demonstrate that realistic time-varying local flow fields, pressures and forces on and near structures such as a half-cyclinder on the ocean floor can be determined from numerical calculations for certain conditions. Good agreement is found in comparison of the numerical results from SPLISH, recent linear wave Green's function and fifth-order asymptotic solutions for wave motion over a bottom seated half-cylinder, and an experimental simulation in a wave channel.     

Nonholonomic motions of gyroscopic and wheeled systems

An approach to modeling the motion of mechanical systems with small generalized velocities is discussed. The motion of a gyroscope in a gimbal suspension and the lateral movements of a railway wheelset are discussed as examples for the situations when the solutions to the corresponding problems are evolved near the manifold defined by the primary Dirac constraints.     

The Hele-Shaw problem and area-preserving curve-shortening motions

We prove existence, locally in time, of a solution of the following Hele-Shaw problem: Given a simply connected curve contained in a smooth bounded domain, find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve in and whose boundary value on the curve equals its curvature. We show that this motion is a curve-shortening motion which does not change the area of the region enclosed by the curve. In case is the whole plane ², we also show that if the initial curve is close to an equilibrium curve, i.e., to a circle, then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity.     

Self-similarity of nonstationary boundary-layer motions

In this paper we show that problems concerning the development of a boundary layer on a semi-infinite plate when the outer flow speed is of the form U = (1 + ct)^b a, and on a cylinder when the outer flow speed has the forms U = ctx^m and U = (1 + ct)^b ax^m, are self-similar. We present the results of numerical calculations for various values of , b, and m. We consider the problem of a stepwise nonstationary heating of a plate, impulsively set into motion in an incompressible fluid; we show that this problem is self-similar and obtain its solution numerically.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 122–125, July–August, 1975.     

On periodic motions in magnetohydrodynamics

Summary A uniqueness theorem and an existence theorem for motions asymptotically stable in the mean and periodically depending on time, are demonstrated in magnetohydrodynamics, on the hypothesis that a motion sufficiently regular and asymptotically stable in the mean exists. In particular, similar theorems for steady motions are deduced. The procedure used is also a simple method for constructing periodic and steady motions.

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Sommario Si dimostra un teorema di unicità ed un teorema di esistenza per i moti magnetoidrodinamici periodicî nel tempo ed asintoticamente stabili in media, subordinatamente all'ipotesi che esista un moto asintoticamente stabile in media sufficientemente regolare, e si deducono, in particolare, analoghi teoremi per i moti stazionari. Il procedimento adoperato costituisce anche un semplice metodo per la effettiva costruzione dei moti periodici e dei moti stazionari.

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This work was done in the sphere of activity of the C.N.R. groups for mathematical research and, in Italian, has already been presented for publication in Rend. Acc. Sc. Fis. Mat., Napoli.     

Particle motions in non-Newtonian media

Summary An experimental study of the behaviour of rigid and deformable particles suspended in pseudoplastic and elasticoviscous liquids undergoing slowCouette flow was undertaken. The velocity profiles deviated slightly from those obtained forNewtonian fluids, but the measured angular velocities of rigid spheres showed that the rotation of the field was equal to half the velocity gradient. While the measured angular velocities of rods and discs were in accord with theory applicable toNewtonian liquids, in both non-Newtonian media there was a steady drift in the orbit towards an asymptotic value corresponding to minimum energy dissipation in the flow. Furthermore, discs in elasticoviscous solutions of polyacrylamide at higher shear stresses aligned themselves in the direction of the flow and ceased to rotate.Migration of rigid particles across the planes of shear in the annul us of theCouette was also observed. In pseudoplastic liquids, the migration was towards the region of higher shear, whereas the opposite was true in elasticoviscous liquids.The deformation, orientation and burst of pseudoplastic drops inNewtonian liquids and that ofNewtonian drops in pseudoplastic fluids were similar to those previously in completelyNewtonian systems. With elasticoviscous drops, however, the deformation was smaller than given by theory.As in elasticoviscous fluids, two-body collisions of rigid uniform spheres in the pseudoplastic liquids were unsymmetrical and irreversible, thus differing from collisions inNewtonian systems where complete reversibility is observed.While some of the observed phenomena in elasticoviscous suspensions could be qualitatively interpreted, particle behaviour in the pseudoplastic liquids could not be explained in terms of the known rheological properties of the fluids.

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Zusammenfassung Es wurde experimentell das Verhalten von festen und deformierbaren Teilchen untersucht, die bei der Suspension in strukturviskosen und viskoelastischen Flüssigkeiten einer langsamenCouette-Strömung ausgesetzt sind. Die Geschwindigkeitsprofile zeigten gewisse Abweichungen von denenNewtonscher Flüssigkeiten, aber die gemessenen Winkelgeschwindigkeiten der festen Kügelchen ergaben, daß die Drehung des Feldes gleich dem halben Geschwindigkeitsgradienten war. Die gemessenen Winkelgeschwindigkeiten der Stäbchen und Scheiben stimmten mit der Theorie, die auf Newtonsche Flüssigkeiten zutrifft, überein. In beiden nicht-Newtonschen Flüssigkeiten verschob sich jedoch die Kreisbahn stetig zu einem asymptotischen Wert, der einem Minimum der Dissipationsenergie der Strömung entsprach. Scheibchen in viskoelastischen Lösungen von Polyacrylamid richteten sich bei höherer Scherspannung in Strömungsrichtung aus und zeigten keine Drehung mehr.Es wurden auch Wanderungen von festen Teilchen über die Scherebene im Spalt derCouette-Anordnung beobachtet. In strukturviskosen Flüssigkeiten erfolgte die Wanderung in Richtung der höheren Scherung, während auf elastische Flüssigkeiten das Gegenteil zutraf.Die Deformation, Orientierung und das Aufbrechen strukturviskoser Tröpfchen inNewtonschen Flüssigkeiten und das Verhalten von Newtonschen Tröpfchen in strukturviskosen Flüssigkeiten waren den früher in rein-Newtonschen Systemen beobachteten Phänomenen ähnlich. Die Deformation der viskoelastischen Tröpfchen war jedoch kleiner als die von der Theorie vorhergesagt worden war.Zweikörper-Zusammenstöße zwischen festen gleichförmigen Kügelchen in strukturviskosen Flüssigkeiten waren unsymmetritch und irreversibel. Darin unterschieden sie sich von Zusammenstößen inNewtonschen Flüssigkeiten, in denen völlige Umkehrbarkeit beobachtet worden war.Während einige der beobachteten Phänomene in viskoelastischen Suspensionen qualitativ gedeutet werden konnten, ließ sich das Teilchenverhalten in strukturviskosen Flüssigkeiten nicht anhand der bekannten Theologischen Eigenschaften der Flüssigkeiten erklären.

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This work was supported by the Defence Research Board of Canada (DRB Grant 9530-47).     

Nonlinear motions of beam-mass structure

We study the planar dynamic response of a flexible L-shaped beam-mass structure with a two-to-one internal resonance to a primary resonance. The structure is subjected to low excitation (mili g) levels and the resulting nonlinear motions are examined. The Lagrangian for weakly nonlinear motions of the undamped structure is formulated and time averaged over the period of the primary oscillation, leading to an autonomous system of equations governing the amplitudes and phases of the modes involved in the internal resonance. Later, modal damping is assumed and modal-damping coefficients, determined from experiments, are included in the analytical model. The locations of the saddle-node and Hopf bifurcations predicted by the analysis are in good agreement, respectively, with the jumps and transitions from periodic to quasi-periodic motions observed in the experiments. The current study is relevant to the dynamics and modeling of other structural systems as well.     

Method of analyzing irregular motions

This article examines problems in identifying vibrations with regular and chaotic attractors in certain oscillating systems. A new approach is developed on the basis of the dynamical principle of symmetry, the construction of regions of aperiodic solutions, and analysis of the characteristic indices of quasistatic solutions for a circular trajectory (in polar variables). S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 8, pp. 87–93, August, 1999.     

Particle motions in non-newtonian media

Summary The rotation of rigid rods and discs, and the deformation and burst of liquid drops were observed in elasticoviscous solutions of polyacrylamide undergoingCouette flow at shear rates from 0.1 to ~ 10 sec^–1. The study extended previous work (1) to cylindrical particles having axis ratios closer to unity and to liquid drops in systems having zero interfacial tension.The period of rotation of discs was appreciably longer than predicted by theory applicable toNewtonian fluids. Depending on the axis ratio and shear rate, the particles drifted into orbits in which the conventional orbit constantC varied periodically throughout each orbit. Above a given shear rate, rods aligned themselves in the direction of flow without rotating, in a metastable equilibrium position; in the case of discs at higher shear rates the position of alignment was stable.The oscillating deformation and orientation of drops in a system of two solubleNewtonian silicone oils was in fair accord with theory. With pairs of elasticoviscous polyacrylamide solutions, deformation and burst of viscous drops occurred along the vorticity axis, in contrast to the behaviour of elasticoviscous drops of polyisobutylene in Decalin having a finite interfacial tension, which deformed and burst in a manner similar toNewtonian drops.

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Zusammenfassung Die Rotation starrer Stäbchen und Scheibchen sowie die Verformung und das Aufbrechen flüssiger Tröpfchen wurde in viskoelastischen Polyacrylamid-Lösungen untersucht, die in einerCouette-Apparatur mit Schergeschwindigkeiten zwischen 0,1 und ca. 10 sec^–1 geschert wurden. Diese Untersuchung ergänzt eine frühere Arbeit (1), insofern zylindrische Teilchen mit einem Achsenverhältnis näher bei eins und Tröpfchen in Systemen mit verschwindender Grenzflächenspannung berücksichtigt werden.Die Umdrehungsperiode der Scheibchen kam erheblich langsamer heraus, als sie auf Grund der fürnewtonsche Flüssigkeiten gültigen Theorie vorausgesagt wird. Die Teilchen trieben, abhängig von ihren Achsenverhältnissen und der Schergeschwindigkeit, in Kreisbahnen hinein, für welche die konventionelle KreisbahnkonstanteC sich periodisch änderte. Oberhalb einer bestimmten Schergeschwindigkeit orientierten sich die Stäbchen in Richtung der Strömung in einer metastabilen Gleichgewichtseinstellung; für Scheibchen war bei höheren Schergeschwindigkeiten diese Einstellung stabil.Oszillationsverformung und Orientierung der Tröpfchen in einem System zweier ineinander lösbarernewtonscher Siliconöle stimmte zufriedenstellend mit der Theorie überein. Für Paare viskoelastischer Polyacrylamid-Lösungen wurden die zähen Tröpfchen entlang der Richtung der Rotationsachse verformt und aufgebrochen im Gegensatz zu viskoelastischen, eine endliche Grenzflächenspannung aufweisenden Tröpfchen von Polyisobutylen in Dekalin, die ähnlich wienewtonsche Tröpfchen verformt und aufgebrochen wurden.

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This paper is, in effect, an addendum to Reference (1). To avoid unnecessary repetition, identical symbols have been used (with the addition of andD which are defined in the text), and frequent use is made of figures and equations from references (1) and (2).

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This work was supported by the Defence Research Board of Canada. (DRB Grant 9530-47).

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With 3 figures and 2 tables     

Stability of Cosserat fluid motions

We obtain criteria of stability of the unsteady motion of an incompressible Cosserat fluid in an arbitrary time-dependent domain, employing a general energy method due to J. Serrin. It is shown that the original motion is stable if R _e ² 80 + 12800 C ₀ or if R _e80 + 6400 C ₀. The quantities R _e and C ₀ are the Reynolds number and Cosserat number, respectively, and - is the lower bound for the eigenvalues of the strain rate tensor D _ij.The theorems established for the stability criteria are universal in the sense that they do not depend either on the shape of the domain or on the distribution of the basic field variables. Finally an experimental scheme is proposed to determine the upper bound of the Cosserat number and consequently the characteristic length of a Cosserat fluid.     

Unsteady motions with steady streamlines

Communicated by K. R. Rajagopal