Noisy and chaotic conditions of a pendulum motion

The effect of noise on the rotational mode of a pendulum which is excited kinematically in vertical direction has been analyzed. We have applied the multifractal analysis to distinguish chaotic and noisy solutions in transitions from the oscillations to rotations motion of a pendulum. During increasing the noisy disturbance of the system we analyzed the basic multifractals criteria of the system as correlation and complexity. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Brownian motion with a parabolic drift and airy functions

Summary Let {W(t): t } be two-sided Brownian motion, originating from zero, and let V(a) be defined by V(a)=sup}t : W(t)–(t–a)² is maximal}. Then {V(a): a } is a Markovian jump process, running through the locations of maxima of two-sided Brownian motion with respect to the parabolas f _a(t)=(t–a)². We give an analytic expression for the infinitesimal generators of the processes a , in terms of Airy functions in Theorem 4.1. This makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators (i.e. estimators derived under order restrictions). An example of this is given in Groeneboom (1985), where the asymptotic distribution of the (standardized) L ₁-distance between a decreasing density and the Grenander maximum likelihood estimator of this density is determined. On our way to Theorem 4.1 we derive some other results. For example, we give an analytic expression for the joint density of the maximum and the location of the maximum of the process {W(t)–ct ²: t }, where c is an aribrary positive constant. We also determine the Laplace transform of the integral over a Brownian excursion. These last results also have recently been derived by several other authors, using a variety of methods.This paper was awarded the Rollo Davidson prize 1985 (Cambridge, UK)     

Resistance of a circular cylinder due to a pair of vortices moving symmetrically

Zusammenfassung Es wird versucht, den veränderlichen Widerstand, welchen ein unter Bildung zweier Wirbel in eine gleichförmige Strömung gelegter, runder Zylinder erfährt, einer Berechnung zu unterziehen.     

Periodic solutions of a quasilinear wave equation

We study the properties of wave operators satisfying the periodicity condition with respect to time and homogeneous boundary conditions of the third kind and of Dirichlet type. We prove the existence of a nontrivial periodic (in time) sine-Gordon solution with homogeneous boundary conditions of the third kind and of Dirichlet type. We obtain theorems on the existence of periodic solutions of a quasilinear wave equation with variable (in x) coefficients and a boundary condition of the third kind.     

Non-existence of vortices in the small density region of a condensate

In this paper, we answer a question raised by Lev Pitaevskii and prove that the ground state of the Gross-Pitaevskii energy describing a Bose-Einstein condensate in a rotationally symmetric trap at low rotation does not have vortices in the low density region. Therefore, the first ground state with vortices has its vortices in the bulk. In fact we prove something stronger, which is that the ground state for the model at low and moderate rotations is equal to the ground state in a condensate with no rotation. This is obtained by proving that for small rotational velocities, the ground state is multiple of the ground state with zero rotation. We rely on sharp bounds of the decay of the wave function combined with weighted Jacobian estimates.     

Coupled motion of a rigid body and point vortices on a two-dimensional spherical surface

The paper is concerned with a class of problems which involves the dynamical interaction of a rigid body with point vortices on the surface of a two-dimensional sphere. The general approach to the 2D hydrodynamics is further developed. The problem of motion of a dynamically symmetric circular body interacting with a single vortex is shown to be integrable. Mass vortices on S ² are introduced and the related issues (such as equations of motion, integrability, partial solutions, etc.) are discussed. This paper is a natural progression of the author’s previous research on interaction of rigid bodies and point vortices in a plane.     

Periodic and chaotic dynamics in an asymmetric elastoplastic oscillator

We consider the dynamics of a harmonically forced oscillator with an asymmetric elastic–perfectly plastic stiffness function. The computed bifurcation diagrams for the oscillator show regions of periodic motion, hysteresis and large regions of chaotic motion. These different regions of dynamical behaviour are plotted in a two-dimensional parameter space consisting of forcing amplitude and forcing frequency. Examples of the chaotic motion encountered are shown using a discontinuity crossing map. Comparisons are made with the symmetric oscillator by computing a typical bifurcation diagram and considering previously published results for the symmetric system. From this we conclude that the asymmetric system is dominated by a large region of chaotic motion whereas in the symmetric oscillator period one motion and coexisting period three motion predominates.     

Resonant wave motion in a non-homogeneous system

A nonlinear theory of resonant wave motion in an inhomogeneous system (arising from different mechanical applications) is considered. Depending on the magnitude of the influence of inhomogeneity two different situations are encountered. The system may behave either as a nonlinear one-degree-of-freedom oscillator exhibiting a response curve with either soft or hard spring behaviour or for a sufficiently small influence of inhomogeneity, there are periodic shock waves in a certain frequency band about the linear resonance frequency, a phenomenon that is familiar from homogeneous systems like a gas filled tube being excited close to resonance.     

Impulsive rotatory motion of a circular disk in a viscous fluid

Zusammenfassung Es wird eine analytische Methode für die Bestimmung der Geschwindigkeit der Strömung und der Winkelgeschwindigkeit einer kreisförmigen Scheibe gegeben, die in einer zähen, inkompressiblen Flüssigkeit einen Drehstoss erhält.

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This research was sponsored by the US. Army under contract no. DA-11-022-ORD-2059 with the University of Wisconsin.     

Deflection of a circular plate due to heating of a concentric circular region

A circular plate kept zero initial temperature is heated for a certain interval of time over a concentric circular region of the upper face. Deflection is obtained by using Laplace transform technique. Taking the plate material to be aluminium, central deflection is obtained for several values of time and the results shown graphically.     

Diffraction of a plane wave by a circular disk

In this paper we solve the problem of diffraction of a normally incident plane wave by a circular disk. We treat both the hard and soft disk. In each case we obtain the solution as a series which converges when the product of the wave number and the radius of the disk is large. Our construction leads directly to asymptotic approximations to the solution for large wave number.     

Resistance on a circular cylinder due to any number of vortices lying in two rows

Zusammenfassung In dieser Arbeit wird der unterschiedliche Widerstand an einem Kreiszylinder berechnet, der in einem steten Strom angebracht ist und von einer Anzahl von Wirbelbewegungen in zwei Reihen begleitet ist.     

Stability of the Thomson vortex polygon with evenly many vortices outside a circular domain

We consider the stability problem for the stationary rotation of a regular point vortex n-gon lying outside a circular domain. After the article of Havelock (1931), the complete solution of the problem remains unclear in the case 2 ≤ n ≤ 6. We obtain the exhaustive results for evenly many vortices n = 2, 4, 6.     

Periodic and chaotic dynamics of motor-driven gear-pair systems with backlash

Dynamics of gear-pair systems driven by motors and presenting speed-dependent moment resistance is investigated. First, a suitable mechanical model is developed, taking into account the gear mesh backlash and static transmission error as well as the essential non-linearities due to the bearing clearance and contact characteristics. In comparison with earlier related studies, the new element of the present work is that the model developed determines the system response by simply specifying the external loads, rather than by assuming an a priori value for the constant mean angular velocity of the gear shafts. In fact, models employed in earlier research studies are shown to be obtained as special cases of the present model. The study is completed by numerical results. First, classical response diagrams are presented, illustrating the effect of the most important parameters on the system response. Finally, direct integration of the equations of motion is also performed, demonstrating the existence of quasi-periodic and chaotic long time response for selected combinations of the system parameters.