On boundedness of the motions of weakly connected nonlinear systems

Conclusions A general theorem on the boundedness of motions of weakly connected systems is established based on the Lyapunov function method and a comparison principle. Sufficient conditions for uniform boundedness contain the condition of sign-determinacy of a special matrix. A problem on the uniform ultimate-boundedness of motion of a nonlinear system with weak linear bonds is solved as an example.Institute of Mathematics, Academy of Sciences of the Ukraine, Kiev. Kiev University, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 29, No. 3, pp. 68–73, March, 1993.     

刚体运动学的几何讨论

从几何的角度描述刚体的运动学. 刚体的几何描述对应着一个正交变换. 刚体的定轴转动对应着一个特征值为1的正交变换,相应的特征向量为刚体转动的定轴. 通过刚体定点转动的描述,简单地证明了有限转轴定理,并且给出了自由刚体运动的速度和位移公式.     

Multimode laws of control of rigid body motions

The problem of control of motion of a rigid body (aircraft, satellite, or landing module) is studied. A multimode control law that can stabilize not only one prescribed motion of the body but any motion in a sufficiently broad class is constructed. The problem is solved in nonlinear setting under indeterminacy conditions on the basis of Lyapunov functions. The control law is constructed in the class of bounded discontinuous laws, special zero-overshoot response modes are constructed, and the specific character of mechanical systems is used.     

结合刚体基本运动分析三刚片体系几何构造

为探寻体系几何构造分析技巧,总结了三刚片连接的所有可能形式,在简化模型分析基础上,结合刚体的基本运动特征,针对三刚片连接形式中有无穷远瞬铰出现的特殊情况进行详细分析,获取体系构造分析中的一般规律,为学习体系几何构造分析提供帮助.     

ON UNILATERALLY CONSTRAINED MOTIONS OF RIGID BODIES SYSTEMS

ＯＮＵＮＩＬＡＴＥＲＡＬＬＹＣＯＮＳＴＲＡＩＮＥＤＭＯＴＩＯＮＳＯＦＲＩＧＩＤＢＯＤＩＥＳＳＹＳＴＥＭＳＬｉＨｏｎｇｂｏ（李洪波）（ＭＭＲＣ，ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｓＳｃｉｅｎｃｅ，ＡｃａｄｅｍｉａＳｉｎｉｃａ，Ｂｅｉｊｉｎｇ１０００８０...     

GENERALIZATION OF STEINER FORMULAFOR THE HOMOTHETIC MOTIONS ONTHE PLANAR KINEMATICS

The Steiner formula and the mixture area formula given by M(U|¨)ller were expressed under the one-parameter closed planar homothetic motions in the complex sense . Also, using the generalization of Steiner formula, the result of Holditch theorem for homothetic motions is got. In the case of the homothetic scale h≡1 the results given by M(U|¨)ller are obtained as a special case.     

Chaotic motions of a rigid rotor in short journal bearings

In the present paper the conditions that give rise to chaotic motions in a rigid rotor on short journal bearings are investigated and determined. A suitable symmetry was given to the rotor, to the supporting system, to the acting system of forces and to the system of initial conditions, in order to restrict the motions of the rotor to translatory whirl. For an assigned distance between the supports, the ratio between the transverse and the polar mass moments of the rotor was selected conveniently small, with the aim of avoiding conical instability. Since the theoretical analysis of a system's chaotic motions can only be carried out by means of numerical investigation, the procedure here adopted by the authors consists of numerical integration of the rotor's equations of motion, with trial and error regarding the three parameters that characterise the theoretical model of the system: m, the half non-dimensional mass of the rotor, , the modified Sommerfeld number relating to the lubricated bearings, and , the dimensionless value of rotor unbalance. In the rotor's equations of motion, the forces due to the lubricating film are written under the assumption of isothermal and laminar flow in short bearings. The number of numerical trials needed to find the system's chaotic responses has been greatly reduced by recognition of the fact that chaotic motions become possible when the value of the dimensionless static eccentricity % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabew7aLnaaBaaaleaacaWGZbaabeaaaaa!4046!\[\varepsilon _s \] is greater than 0.4. In these conditions, non-periodic motions can be obtained even when rotor unbalance values are not particularly high (=0.05), whereas higher values (>0.4) make the rotor motion periodic and synchronous with the driving rotation. The present investigation has also identified the route that leads an assigned rotor to chaos when its angular speed is varied with prefixed values of the dimensionless unbalance . The theoretical results obtained have then been compared with experimental data. Both the theoretical and the experimental data have pointed out that in the circumstances investigated chaotic motions deserve more attention, from a technical point of view, than is normally ascribed to behaviours of this sort. This is mainly because such behaviours are usually considered of scarce practical significance owing to the typically bounded nature of chaotic evolution. The present analysis has shown that when the rotor exhibits chaotic motions, the centres of the journals describe orbits that alternate between small and large in an unpredictable and disordered manner. In these conditions the thickness of the lubricating film can assume values that are extremely low and such as to compromise the efficiency of the bearings, whereas the rotor is affected by inertia forces that are so high as to determine severe vibrations of the supports.Nomenclature C radial clearance of bearing (m) - D diameter of bearing (m) - e dimensional eccentricity of journal (m) - e _s value of e corresponding to the static position of the journal - E dimensional static unbalance of rotor (m) - f _x, f _y =F _x/(P), F _y/(P): non-dimensional components of fluid film force - F _x, F _y dimensional components of fluid film force (N) - g acceleration of gravity (m/s²) - L axial length of bearing (m) - m % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabg2da9maalaaabaGaeqyYdC3aaWbaaSqa% beaacaaIYaaaaaGcbaGaeqyYdC3aa0baaSqaaGabciaa-bdaaeaaca% WFYaaaaaaakiabg2da9maalaaabaGaeqyYdC3aaWbaaSqabeaacaaI% YaaaaOGaam4qaaqaaiabeo8aZjaadEgaaaaaaa!4C14!\[ = \frac{{\omega ^2 }}{{\omega _0^2 }} = \frac{{\omega ^2 C}}{{\sigma g}}\]: half non-dimensional mass of rotor - M half mass of rotor (kg) - n angular speed of rotor (in r.p.m.=60/2) - t time     

Slow motions of a rigid sphere in incompressible viscoelastic fluids

Problems concerned with the force of resistance and the moment of forces acting from the side of a viscoelastic fluid on a sphere moving with acceleration are considered in a linear formulation. Fairly simple relations are obtained for a fluid with a single relaxation time or a single after-action time. A discussion of the asymptotic expressions is given for a fluid with a large number of times.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 9–16, March–April, 1976.     

Isolated large amplitude periodic motions of towed rigid wheels

This study investigates a low degree-of-freedom (DoF) mechanical model of shimmying wheels. The model is studied using bifurcation theory and numerical continuation. Self-excited vibrations, that is, stable and unstable periodic motions of the wheel, are detected with the help of Hopf bifurcation calculations. These oscillations are then followed over a large parameter range for different damping values by means of the software package AUTO97. For certain parameter regions, the branches representing large-amplitude stable and unstable periodic motions become isolated following an isola birth. These regions are extremely dangerous from an engineering point of view if they are not identified and avoided at the design stage. Research Group on Dynamics of Machines and Vehicles, Hungarian Academy of Sciences.     

Stability problem for pendulum-type motions of a rigid body in the Goryachev-Chaplygin case

We consider the motion of a rigid body with a single fixed point in a homogeneous gravity field. The body mass geometry and the initial conditions for its motion correspond to the case of Goryachev—Chaplygin integrability. We study the orbital stability problem for periodic motions corresponding to vibrations and rotations of the rigid body rotating about the equatorial axis of the inertia ellipsoid.In [1], it was proved that these periodic motions are orbitally unstable in the linear approximation. It was also shown that, to solve the stability problem in the nonlinear setting, it does not suffice to analyze terms up to the fourth order in the expansion of the Hamiltonian function in the canonical variables.The present paper shows that in this problem one deals with a special case where standard methods for stability analysis based on the coefficients in the normal form of the Hamiltonian of the perturbed equations of motion do not apply. We use Chetaev’s theorem to prove the orbital instability of these periodic motions in the rigorous nonlinear statement of the problem. The proof uses the additional first integral of the Goryachev—Chaplygin problem in an essential way.     

Rheological equations of state of weakly concentrated suspensions of rigid ellipsoidal particles

Rheological equations of state of dilute suspensions of rigid ellipsoidal particles (ellipsoids of revolution) are derived [1–4] from the vantage point of the structural-continuum approach, with attention given both to rotational Brownian motion of particles and to their inertia and the outer force fields. Interaction between particles is ignored in those treatments given the low concentration of the suspended particles. In this paper, the earlier findings [1–4] are generalized to higher concentrations. The effect of hydrodynamical interaction between particles on the rheological behavior of the suspension is treated in the light of the Simha approach [5].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 141–145, January–February, 1973.     

Load-induced stress singularities in the bending of cosserat plates

Summary Presented in this paper are the solutions to several plate bending problems as governed by a recent theory developed by Green and Naghdi, into which couple-stress is incorporated. Specifically, each problem considered is subjected to boundary conditions emanating from a singular load distribution acting on the free edge of a semi-infinite plate. The method of integral transforms is applied in the solutions.In general, it is found that the singularities in the shear and moment resultants are of the same order as those given in Reissner's plate theory, however the detailed structures of these singular functions are altered. The present theory also suggests that, in most cases, the maximum magnitudes of the moment resultants are diminished as compared to the corresponding results given in Reissner's theory.Also discussed is the exact relationship between the Green-Naghdi theory and Reissner's theory.

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Sommario In questa memoria sono presentate le soluzioni di alcuni problemi relativi a piastre inflesse studiati in base ad una teoria recentemente sviluppata da Green e Naghdi, in cui si considerano distribuzioni di coppie di volume.In ogni problema considerato le condizioni al contorno presentano una singolarità nella distribuzione di carico agente su un bordo libero di una piastra semi-indefinita. Per la risoluzione è applicato il metodo delle trasformate integrali.In generale risulta che le singolarità nel taglio e nel momento sono dello stesso ordine di quelle date dalla teoria delle piastre di Reissner, benchè gli andamenti di queste funzioni singolari siano diversi. La teoria sviluppata porta inoltre a valutare, nella maggior parte dei casi, momenti massimi più piccoli di quelli corrispondenti ottenuti in base alla teoria di Reissner.Si discute inoltre l'esatta relazione tra le teorie di Green-Naghdi e di Reissner.

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This research is sponsored by the U. S. Navy under Contract Nonr-610(06) with the Office of Naval Research in Washington, D. C.     

On the rheology of a dilute suspension of rigid spheres in a weakly viscoelastic matrix fluid

The complete solution for the pressure and the velocity field up to O(De) of a dilute suspension of neutrally buoyant, non-Brownian rigid spheres suspended in an unbounded, weakly viscoelastic matrix fluid, where is the solid volume fraction and De is the Deborah number of the matrix fluid, is presented. The spheres are subjected to an arbitrary linear velocity profile at infinity. The analytical solution is used for the prediction of the bulk stress, and specifically for the calculation of the first and the second normal stress differences in simple shear and uniaxial elongational flows. A comparison of the results with available values reported in the literature is also offered. The final expressions for the bulk normal stress differences in shear and uniaxial elongational flow fully agree with those reported earlier by Greco et al., J. Non-Newton. Fluid Mech., 147 (2007) 1–10.     

Characterization of nonlinear and chaotic motions by the cepstral analysis

A technique based on the power cepstrum has been developed to analyze and characterize data of various nonlinear and chaotic motions. For repeatability and ready availability, nonlinear response data of a Duffing oscillator and van der Pol oscillator, generated numerically by the fourth order Runge-Kutta algorithm, were used in the investigation. Results obtained by the proposed technique using spectrum, cepstrum, and specepstrum which is defined as the spectrum of the logarithmic cepstrum, indicate that it is superior to methods previously available in the literature.     

Particle motions in sheared suspensions XXI: Interactions of rigid spheres (theoretical)

Summary Based on a method developed byFaxén, approximate expressions were developed for the rotational and translational velocities of a single rigid sphere near a plane rigid wall bounding Couette flow, and near another sphere of either different or the same size in Couette flow. The solutions to the latter problem provide equations for the orbits of each sphere about the other, some of which are closed. In many respects the results are similar to the corresponding two-dimensional problems for cylinders.

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Zusammenfassung Angenäherte Gleichungen wurden aufgrund einer vonFaxén entwickelten Methode für die Rotations- und Translations-Geschwindigkeiten einer einzelnen, starren Kugel in der Nähe einer ebenen, starren Wand, die einenCouette- Fluß begrenzte, aufgestellt, und außerdem für eine zweite Kugel von gleicher oder verschiedener Größe imCouette-Fluß. Die Lösungen für dieses zweite Problem geben Gleichungen für die Bahnen jeder Kugel um die andere und einige davon stellen geschlossene Ausdrücke dar. In vieler Hinsicht sind die Resultate änlich wie für die entsprechenden zweidimensionalen Zylinder-Probleme.

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This work was supported in part by the National Heart Institute of the United States Public Health Service (Grant HE-05911).     

Particle motions in sheared suspensions XXII: Interactions of rigid spheres (experimental)

Summary The behaviour of rigid spheres suspended in viscous liquids undergoing laminar shear flow was compared experimentally with the approximate theory developed earlier. Good agreement was obtained for a sphere near a wall and for equatorial interactions between two spheres of both equal and unequal diameters. In the region close to but not touching the wall, the theory byGoldman et al. accurately described the motion of a sphere. The prediction that a sphere touching the wall should stick to it could not be confirmed. A viewing device, designed to study motions in two planes simultaneously was used to make several interesting observations on permanent and separating doublet. Some similarities between collisions in the equatorial plane and two-body interactions of vertically oriented rigid circular cylinders were noted. The asymmetry and reversibility of triple collisions as well as the ease of formation of permanent doublets in flowing suspensions were established.

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Zusammenfassung Das Verhalten von starren, in viskosen Flüssigkeiten suspendierten Teilchen unter dem Einfluß von laminarem Scherfluß wurde experimentell mit der früher entwickelten, genäherten Theorie verglichen. Gute Übereinstimmung wurde erhalten für eine Kugel in Wandnähe und für äquatoriale Einflüsse zwischen zwei Kugeln von entweder gleichem oder verschiedenem Durchmesser. Die Theorie vonGoldmann et al. gab eine genaue Beschreibung der Bewegung einer Kugel in Wandnähe, aber ohne Wandberührung. Die Voraussage, daß eine Kugel an einer berührenden Wand kleben würde, konnte nicht bestätigt werden. Eine Beobachtungs-Einrichtung, die für das Studium der gleichzeitigen Bewegung in zwei Ebenen konstruiert war, wurde für verschiedene interessante Beobachtungen an permanenten und an sich trennenden Dubletts benutzt. Einige Änlichkeiten zwischen Zusammenstößen in der Äquatorial-Ebene und Zweikörperbeinflussungen von vertikal orientierten, starren kreisförmigen Zylindern wurde festgestellt. Die Asymmetrie und Reversibilität von dreifachen Stößen wie auch die leichte Bildung von permanenten Dubletts wurde im Fluß von Suspensionen festgestellt.

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This work was also supported by the National Heart Institute of the United States Public Health Service (Grant HE-05911).