On the Chaotic Instability of a Nonsliding Liquid-Filled Top with a Small Spheroidal Base via Melnikov-Holmes-Marsden Integrals

Chaotic orientations of a top containing a fluid filled cavity are investigated analytically and numerically under small perturbations. The top spins and rolls in nonsliding contact with a rough horizontal plane and the fluid in the ellipsoidal shaped cavity is considered to be ideal and describable by finite degrees of freedom. A Hamiltonian structure is established to facilitate the application of Melnikov-Holmes-Marsden (MHM) integrals. In particular, chaotic motion of the liquid-filled top is identified to be arisen from the transversal intersections between the stable and unstable manifolds of an approximated, disturbed flow of the liquid-filled top via the MHM integrals. The developed analytical criteria are crosschecked with numerical simulations via the 4th Runge-Kutta algorithms with adaptive time steps.     

Chaotic attitude motion of gyrostat statellites in a central force field

The nonlinear attitude motion of gyrostat satellites in a central force field is investigated, with particular emphasis on their long-time dynamic behavior for a wide range of parameters. The numbers of equilibrium solutions, as well as their stability, vary with the rotor speed, and bifurcation diagrams have been obtained. Various dynamic behaviors of gyrostat satcllites, e.g. periodic, quasiperiodic, and chaotic, are studied via the Poincaré map technique. It is shown that the rotor speed has a significant effect on the dynamic behavior of gyrostat satellites.     

Parametric identification of a chaotic base-excited double pendulum experiment

The parametric identification of a chaotic system was investigated for a double pendulum. From recorded experimental response data, the unstable periodic orbits (UPOs) were extracted and then used in a harmonic balance identification process. By applying digital filtering, digital differentiation and linear regression techniques for optimization, the results were improved. Verification of the related simulation system and linearized system also corroborated the success of the identification algorithm.     

On the existence of intermediate magnetohydrodynamic shock waves

We consider the questions related to the structure of shock waves for a system of magnetohydrodynamic equations. Using Conley's connection matrix, we recover and extend earlier results due to C. Conley and J. Smoller. In particular, we give a simpler proof of the existence of fast and slow shocks with structure. We also demonstrate that for some viscosity parameters intermediate shocks occur. Furthermore, under an assumption of transversality, we show that there exist multi-parameter families of these intermediate shocks.This research was done while both authors were visiting the Lefschetz Center for Dynamical Systems at Brown University.Supported in part by the NSF under Grant DMS-8507056.Supported in part by AFOSR 87-0347.     

About the basic integral variants of holonomic nonconservative dynamical systems

In this paper, we prove that for holonomic nonconservative dynamical system the Poincaré and Poincaré-Cartan integral invariants do not exist. Instead of them, we introduce the integral variants of Poincaré-Cartan's type and of Poincare's type for holonomic nonconservative dynamical systems, and use these variants to solve the problem of nonlinear vibration. We also prove that the integral invariants introduced in references [1] and [2] are merely the basic integral variants given by this paper. Project supported by National Natural Science Foundation of China     

Numerical modeling of the rotation of a solid body with a liquid-filled cavity having radial ribs

The boundary-value problem of unsteady vortex flow of a viscous incompressible fluid in a cylindrical vessel with radial ribs rotating at a variable angular velocity is solved using a finite-difference method. The results of the solution are used to calculate the motion of a system of a solid body and a cavity filled with a liquid. The results are compared with available experimental data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 135–139, March–April, 2007.     

On the uniqueness of boundary integral equation for the exterior Helmholtz problem

From the point of view of energy analysis,the cause that the uniqueness of theboundary integral equation induced from the exterior Helmholtz problem does not hold isinvestigated in this paper.It is proved that the Sommerfeld’s condition at the infinity ischanged so that it is suitable not only for the radiative wave but also for the absorptive wavewhen we use the boundary integral equation to describe the exterior Helmholtz problem.Therefore,the total energy of the system is conservative.The mathematical dealings toguarantee the uniqueness are discussed based upon this explanation     

On the rotational motion of a gyrostat about a fixed point with mass distribution

The perturbed rotational motion of a gyrostat about a fixed point with mass distribution near to Lagrange’s case is investigated. The gyrostat is subjected under the influence of a variable restoring moment vector, a perturbing moment vector, and a third component of a gyrostatic moment vector. It is assumed that the angular velocity of the gyrostat is sufficiently large, its direction is close to the axis of dynamic symmetry, and the perturbing moments are small as compared to the restoring ones. These assumptions permit us to introduce a small parameter. Averaged systems of the equations of motion in the first and second approximations are obtained. Also, the evolution of the precession angle up to the second approximation is determined. The graphical representations of the nutation and precession angles are presented to describe the motion at any time.     

On the regularization method of the first kind of Fredholm integral equation with a complex kernel and its application

The regularized integrodifferential equation for the first kind of Fredholm integral equation with a complex kernel is derived by generalizing the Tikhonov regularization method and the convergence of approximate regularized solutions is discussed. As an application of the method, an inverse problem in the two-demensional wave-making problem of a flat plate is solved numerically, and a practical approach of choosing optimal regularization parameter is given. Project supported by the National Natural Science Foundation, of China     

三维体内含垂直于双材料界面混合型裂纹的超奇异积分解法

利用双材料位移基本解和Somigliana公式,将三维体内含垂直于双材料界面混合型裂纹问题归结为求解一组超奇异积分方程。使用主部分析法,通过对裂纹前沿应力奇性的分析,得到用裂纹面位移间断表示的应力强度因子的计算公式,进而利用超奇异积分方程未知解的理论分析结果和有限部积分理论,给出了超奇异积分方程的数值求解方法。最后,对典型算例的应力强度因子做了计算,并讨论了应力强度因子数值结果的收敛性及其随各参数变化的规律。     

On the computation of the near-tip stress intensities for three-dimensional wedges via two-state M-integral

Singular stress fields around three-dimensional wedges are examined, and the near-tip intensity is calculated via the two-state M-integral with the aid of the domain integral representation. Two numerical examples demonstrate the effectiveness and accuracy of the present scheme for computing the stress intensities of singular stresses near the generic three-dimensional wedges.     

Solution of the laminar duct flow problem by a boundary integral equation method

Summary A boundary integral equation method is proposed for approximate numerical and exact analytical solutions to fully developed incompressible laminar flow in straight ducts of multiply or simply connected cross-section. It is based on a direct reduction of the problem to the solution of a singular integral equation for the vorticity field in the cross section of the duct. For the numerical solution of the singular integral equation, a simple discretization of it along the cross-section boundary is used. It leads to satisfactory rapid convergency and to accurate results. The concept of hydrodynamic moment of inertia is introduced in order to easily calculate the flow rate, the main velocity, and the fRe-factor. As an example, the exact analytical and, comparatively, the approximate numerical solution of the problem of a circular pipe with two circular rods are presented. In the literature, this is the first non-trivial exact analytical solution of the problem for triply connected cross section domains. The solution to the Saint-Venant torsion problem, as a special case of the laminar duct-flow problem, is herein entirely incorporated.     

On the Chaotic Behavior of a Third-Order Log-Domain Filter

Operation of log-domain filters revolves around the large-signal forward-active mode exponential current-voltage relationship of the bipolar junction transistor, which is used to map the input currents to the logarithmic domain, where the analog processing takes place, and to convert the filtered voltage waveforms back to the linear domain. Sometimes, the inherent internal nonlinearity of such circuits is the cause of unwanted externally-nonlinear effects. In this paper a differential third-order log-domain Chebyshev low-pass filter is designed by applying the method of operational simulation of doubly-terminated LC ladders. When the six grounded shunt capacitors of this filter are properly replaced with three half-sized floating capacitors, as is common practice in internally-linear fully-differential capacitively-loaded circuits, under particular conditions the resulting system exhibits limit-cycle oscillations, period-doubling bifurcations and chaotic behavior for zero-input. The presence of chaos is confirmed by extracting the spectrum of Lyapunov exponents of the system from the time series of the output voltage.     

On the dynamics of free and excited oscillations of a simple portal frame foundation

In this paper we search for the dynamics of a simple portal structure in the free and in the periodic excitation cases. By using the Center Manifold approach and Averaging Method, we obtain results on both stability and bifurcation of equilibrium points and periodic orbits.     

A domain independent integral expression for the crack extension force of a curved crack in three dimensions

An integral expression that is domain independent in curvilinear coordinates and compatible with zero divergence of Eshelby's (Phil. Trans. Roy. Soc. (London) 244 (1951) 87.) energy momentum tensor was obtained from the principle of virtual work. By applying Eshelby's definition of the force on a material defect a general expression of the crack extension force for a curved crack in three dimensions, here called the F-integral, was derived from the domain independent integral expression. The F-integral is given explicitly for a number of curved cracks and found to be in agreement with previously known solutions, when available. The influence of crack surface and crack front curvature upon the various forms of the F-integral is discussed. The F-integral presented in this work is a generalisation of the J-integral (Rice, J. Appl. Mech. 35 (1968) 379.) to curved cracks in orthogonal curvilinear coordinates.     

Reissner型板表面裂纹应力强度因子的线弹簧-不连续位移边界积分方程解法

本文由Reissner型板的不连续位移基本解，根据Betti互换定理，导出了Reissuer型板的不连续位移边界积分方程；结合平面问题的不连续位移边界积分方程─—边界元方法和线弹簧模型，给出了Rrissner型板表面裂纹应力强度因子的线弹簧-不连续位移边界积分方程解法。     

The integral as a function of the upper limit and depending on a parameter to solve drawing through idling rolls

The velocity and strain-rate fields which are different from Avitzur’s have been established in Cartesian coordinates. Using the integral as a function of the upper limit and integration depending on a parameter, an analytical upper-bound solution to drawing stress through idling rolls has been obtained in this paper.     

On partial sliding along a planar crack: the case of a circular sliding zone

Summary A problem of partial sliding along a planar crack with a local drop in frictional resistance is investigated. A sliding zone initiates in the area of reduced friction, and then propagates as the applied shear load is monotonously increased. The problem is formulated in general terms, and then solved for the case when sliding spreads as a penny-shaped zone. Conditions under which the front of the zone stays circular during sliding are analyzed. It is observed that the axisymmetry of the profile of frictional resistance does not necessarily guarantee uniform propagation of sliding in the radial direction. The circular shape becomes the most favorable growth condition only if the shear modes are related in a certain way. The problem is studied based on the criterion of propagation that stress intensity factors(SIFs) for II and III modes vanish on the boundary of the sliding zone. The singular integrals in expressions for the SIFs are reduced to non-singular ones. Analytical solutions are derived for a number of special cases where the radius of the sliding zone is related to the applied shear load.This work is supported by the National Science Foundation through grant DGE-0209543 to the University of New Mexico.     

On the asymptotical behaviour of solutions of a class of nonlinear control systems

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