scholarly journals Nonlinear Analysis of Tropical Waves and Cyclogenesis Excited by Pressure Disturbance in Atmosphere

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3038
Author(s):  
Zi-Liang Li ◽  
Jin-Qing Liu
Keyword(s):  
Solitary Waves ◽  
Direct Method ◽  
Equations Of Motion ◽  
Nonlinear Process ◽  
Analytic Solutions ◽  
External Pressure ◽  
Homogeneous Fluid ◽  
Pressure Disturbance ◽  
Cyclone Genesis ◽  
Fkdv Equation

The horizontal equations of motion for an inviscid homogeneous fluid under the influence of pressure disturbance and waves are applied to investigate the nonlinear process of solitary waves and cyclone genesis forced by a moving pressure disturbance in atmosphere. Based on the reductive perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies the Korteweg–de Vries equation with a forcing term (fKdV equation for short), which describes the physics of a shallow layer of fluid subject to external pressure forcing. Then, with the help of Hirota’s direct method, the analytic solutions of the fKdV equation are studied and some exact vortex solutions are given as examples, from which one can see that the solitary waves and vortex multi-pole structures can be excited by external pressure forcing in atmosphere, such as pressure perturbation and waves. It is worthy to point out that cyclone and waves can be excited by different type of moving atmospheric pressure forcing source.

scholarly journals Generation of upstream advancing solitons by moving disturbances

1987 ◽  
Vol 184 ◽  
pp. 75-99 ◽  
Author(s):  
T. Yao-Tsu Wu
Keyword(s):  
Solitary Waves ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Theoretical Models ◽  
Computer Code ◽  
Basic Mechanism ◽  
Stationary Solutions ◽  
Boussinesq Model ◽  
Dispersive Waves ◽  
Fkdv Equation

This study investigates the recently identified phenomenon whereby a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, periodically, a succession of solitary waves, advancing upstream of the disturbance in procession, while a train of weakly nonlinear and weakly dispersive waves develops downstream of a region of depressed water surface trailing just behind the disturbance. This phenomenon was numerically discovered by Wu & Wu (1982) based on the generalized Boussinesq model for describing two-dimensional long waves generated by moving surface pressure or topography. In a joint theoretical and experimental study, Lee (1985) found a broad agreement between the experiment and two theoretical models, the generalized Boussinesq and the forced Korteweg-de Vries (fKdV) equations, both containing forcing functions. The fKdV model is applied in the present study to explore the basic mechanism underlying the phenomenon.To facilitate the analysis of the stability of solutions of the initial-boundary-value problem of the fKdV equation, a family of forced steady solitary waves is found. Any such solution, if once established, will remain permanent in form in accordance with the uniqueness theorem shown here. One of the simplest of the stationary solutions, which is a one-parameter family and can be scaled into a universal similarity form, is chosen for stability calculations. As a test of the computer code, the initially established stationary solution is found to be numerically permanent in form with fractional uncertainties of less than 2% after the wave has traversed, under forcing, the distance of 600 water depths. The other numerical results show that when the wave is initially so disturbed as to have to rise from the rest state, which is taken as the initial value, the same phenomenon of the generation of upstream-advancing solitons is found to appear, with a definite time period of generation. The result for this similarity family shows that the period of generation, Ts, and the scaled amplitude α of the solitons so generated are related by the formula Ts = const α−3/2. This relation is further found to be in good agreement with the first-principle prediction derived here based on mass, momentum and energy considerations of the fKdV equation.

Multi-Parameter Optimization of Damped Linear Continuous Systems

1972 ◽  
Vol 94 (1) ◽  
pp. 1-7 ◽  
Author(s):  
O. B. Dale ◽  
R. Cohen
Keyword(s):  
Optimal Parameter ◽  
Equations Of Motion ◽  
Analytic Solutions ◽  
System Response ◽  
Continuous Systems ◽  
Frequency Interval ◽  
State Variables ◽  
Initial Value ◽  
Unknown Parameters ◽  
One Dimensional

A method is presented for obtaining and optimizing the frequency response of one-dimensional damped linear continuous systems. The systems considered are assumed to contain unknown constant parameters in the boundary conditions and equations of motion which the designer can vary to obtain a minimum resonant response in some selected frequency interval. The unknown parameters need not be strictly dissipative nor unconstrained. No analytic solutions, either exact or approximate, are required for the system response and only initial value numerical integrations of the state and adjoint differential equations are required to obtain the optimal parameter set. The combinations of state variables comprising the response and the response locations are arbitrary.

scholarly journals Resonant Stellar Orbits in Spiral Galaxies

1975 ◽  
Vol 69 ◽  
pp. 237-244
Author(s):  
P. O. Vandervoort
Keyword(s):  
Excellent Agreement ◽  
Spiral Galaxy ◽  
Harmonic Balance ◽  
Spiral Galaxies ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Analytic Solutions ◽  
Stellar Orbits ◽  
Method Of Harmonic Balance ◽  
Resonance Phenomena

This paper reviews a series of investigations of the orbits of stars in the regions of the Lindblad resonances of a spiral galaxy. The analysis is formulated in an epicyclic approximation. Analytic solutions of the epicyclic equations of motion are obtained by the method of harmonic balance of Bogoliubov and Mitropolsky. These solutions represent the resonance phenomena exhibited by the orbits in generally excellent agreement with numerical solutions.

scholarly journals Vibration Suppression Control of a Flexible Gantry Crane System with Varying Rope Length

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Tung Lam Nguyen ◽  
Trong Hieu Do ◽  
Hong Quang Nguyen
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Vibration Suppression ◽  
Direct Method ◽  
Equations Of Motion ◽  
Gantry Crane ◽  
Partial Differential ◽  
Control Approach ◽  
Lyapunov’S Direct Method

The paper presents a control approach to a flexible gantry crane system. From Hamilton’s extended principle the equations of motion that characterized coupled transverse-transverse motions with varying rope length of the gantry is obtained. The equations of motion consist of a system of ordinary and partial differential equations. Lyapunov’s direct method is used to derive the control located at the trolley end that can precisely position the gantry payload and minimize vibrations. The designed control is verified through extensive numerical simulations.

Coherent manipulations of atoms using laser light

2008 ◽  
Vol 58 (3) ◽  
Author(s):  
Bruce Shore
Keyword(s):  
Hilbert Space ◽  
Pulse Propagation ◽  
Laser Light ◽  
Equations Of Motion ◽  
Pulse Sequences ◽  
Three Dimensional ◽  
Analytic Solutions ◽  
Quantum States ◽  
Excitation Energies ◽  
Mathematical Concepts

Coherent manipulations of atoms using laser lightThe internal structure of a particle - an atom or other quantum system in which the excitation energies are discrete - undergoes change when exposed to pulses of near-resonant laser light. This tutorial review presents basic concepts of quantum states, of laser radiation and of the Hilbert-space statevector that provides the theoretical portrait of probability amplitudes - the tools for quantifying quantum properties not only of individual atoms and molecules but also of artificial atoms and other quantum systems. It discusses the equations of motion that describe the laser-induced changes (coherent excitation), and gives examples of laser-pulse effects, with particular emphasis on two-state and three-state adiabatic time evolution within the rotating-wave approximation. It provides pictorial descriptions of excitation based on the Bloch equations that allow visualization of two-state excitation as motion of a three-dimensional vector (the Bloch vector). Other visualization techniques allow portrayal of more elaborate systems, particularly the Hilbert-space motion of adiabatic states subject to various pulse sequences. Various more general multilevel systems receive treatment that includes degeneracies, chains and loop linkages. The concluding sections discuss techniques for creating arbitrary pre-assigned quantum states, for manipulating them into alternative coherent superpositions and for analyzing an unknown superposition. Appendices review some basic mathematical concepts and provide further details of the theoretical formalism, including photons, pulse propagation, statistical averages, analytic solutions to the equations of motion, exact solutions of periodic Hamiltonians, and population-trapping "dark" states.

Characteristics of solitary waves, quasiperiodic solutions, homoclinic breather solutions and rogue waves in the generalized variable-coefficient forced Kadomtsev–Petviashvili equation

2017 ◽  
Vol 31 (36) ◽  
pp. 1750350 ◽  
Author(s):  
Xue-Wei Yan ◽  
Shou-Fu Tian ◽  
Min-Jie Dong ◽  
Li Zou
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Variable Coefficient ◽  
Wave Equations ◽  
Direct Method ◽  
Dynamical Behavior ◽  
Rogue Waves ◽  
Nonlinear Wave Equations ◽  
Quasiperiodic Solutions ◽  
Petviashvili Equation

In this paper, the generalized variable-coefficient forced Kadomtsev–Petviashvili (gvcfKP) equation is investigated, which can be used to characterize the water waves of long wavelength relating to nonlinear restoring forces. Using a dependent variable transformation and combining the Bell’s polynomials, we accurately derive the bilinear expression for the gvcfKP equation. By virtue of bilinear expression, its solitary waves are computed in a very direct method. By using the Riemann theta function, we derive the quasiperiodic solutions for the equation under some limitation factors. Besides, an effective way can be used to calculate its homoclinic breather waves and rogue waves, respectively, by using an extended homoclinic test function. We hope that our results can help enrich the dynamical behavior of the nonlinear wave equations with variable-coefficient.

Lyapunov-Based Attitude Tracking Control of a Quadrotor UAV

2012 ◽  
Author(s):  
Mohammad A. Ayoubi ◽  
Chokri Sendi
Keyword(s):  
Tracking Control ◽  
Direct Method ◽  
Equations Of Motion ◽  
Attitude Motion ◽  
Slow Time ◽  
Modified Rodrigues Parameters ◽  
Attitude Tracking ◽  
Aerial Vehicle ◽  
Attitude Tracking Control ◽  
Globally Stable

In this paper, we use the Newton-Euler formulation to derive the equations of motion of a quadrotor unmanned aerial vehicle. We use the Modified Rodrigues Parameters to describe the attitude motion of a quadrotor for large attitude angles. Then, a globally stable feedback law for the problem of attitude tracking control of the vehicle was derived based on the Lyapunov’s direct method. Simulation results confirm that the proposed controller can track a reference attitude signal in the presence of parameter uncertainty, time delay, and slow time-varying external moments.

Carl Størmer’s auroral discoveries 1This article is part of a Special Issue that honours the work of Dr. Donald M. Hunten FRSC who passed away in December 2010 after a very illustrious career.

2012 ◽  
Vol 90 (8) ◽  
pp. 785-793
Author(s):  
Alv Egeland ◽  
William J. Burke
Keyword(s):  
Large Scale ◽  
Equations Of Motion ◽  
Theoretical Work ◽  
Analytic Solutions ◽  
Auroral Zone ◽  
Space Physics ◽  
Natural Phenomenon ◽  
Southern Norway ◽  
Scale Motion ◽  
Trajectory Problem

In the first half of the 20th century Professor Carl Størmer took aurora research to new heights, devoting all of his energy to solving the riddle of this fascinating natural phenomenon. He began his pioneering research by calculating the trajectories allowed to energetic charged particles. Because the equations of motion did not have analytic solutions, he was forced to invent new numerical methods to follow each particle’s path, step by step. Through a series of treatises he presented now classic solutions to the trajectory problem. To explain the large scale motion of the auroral zone, he was first to introduce the concept of a ring current in 1911. His theoretical work also provided the basis for understanding later discoveries of cosmic rays and the radiation belts. Størmer contributed many important scientific achievements to space physics. In 1909 he constructed the first useful auroral camera needed to make precise space–time mappings of auroral characteristics. Over the course of four sunspot cycles he took more than 100 000 auroral photographs with his network of stations spread across southern Norway. These parallactic auroral photographs gave not only the heights of individual auroral features, but also their occurrence rates, locations, and orientations. He classified the different auroral forms by publishing the first Auroral Atlas (Størmer. Photographic Atlas of Auroral Forms. Brøggers Boktrykkeri, Oslo. 1930.). Among Størmer’s other fascinating discoveries was his identification of sunlit aurorae and descriptions of their remarkable properties.

Analytical Solutions for a Spinning Rigid Body Subject to Time-Varying Body-Fixed Torques, Part I: Constant Axial Torque

1993 ◽  
Vol 60 (4) ◽  
pp. 970-975 ◽  
Author(s):  
J. M. Longuski ◽  
P. Tsiotras
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Analytic Solutions ◽  
Axisymmetric Body ◽  
Attitude Motion ◽  
Time Varying ◽  
Torque Vector ◽  
Constant Torque ◽  
Symmetric Rigid Body ◽  
Body Subject

Analytic solutions are derived for the general attitude motion of a near-symmetric rigid body subject to time-varying torques in terms of certain integrals. A methodology is presented for evaluating these integrals in closed form. We consider the case of constant torque about the spin axis and of transverse torques expressed in terms of polynomial functions of time. For an axisymmetric body with constant axial torque, the resulting solutions of Euler’s equations of motion are exact. The analytic solutions for the Eulerian angles are approximate owing to a small angle assumption, but these apply to a wide variety of practical problems. The case when all three components of the external torque vector vary simultaneously with time is much more difficult and is treated in Part II.

Traveling and solitary waves in monodisperse and dimer granular chains

2017 ◽  
Vol 31 (10) ◽  
pp. 1742001 ◽  
Author(s):  
Yuli Starosvetsky ◽  
K. R. Jayaprakash ◽  
Alexander F. Vakakis
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Traveling Waves ◽  
Granular Media ◽  
Equations Of Motion ◽  
Continuum Approximation ◽  
Granular Chains ◽  
Solitary Pulse ◽  
Countably Infinite ◽  
Dimer Granular Chains

We provide a review of propagating traveling waves and solitary pulses in uncompressed one-dimensional ([Formula: see text]) ordered granular media. The first such solution in homogeneous granular media was discovered by Nesterenko in the form of a single-hump solitary pulse with energy-dependent profile and velocity. Considering directly the discrete, strongly nonlinear governing equations of motion of these media (i.e., without resorting to continuum approximation or homogenization), we show the existence of countably infinite families of stable multi-hump propagating traveling waves with arbitrary wavelengths. A semi-analytical approach is used to study the dependence of these waves on spatial periodicity (wavenumber) and energy, and to show that in a certain asymptotic limit, these families converge to the single-hump Nesterenko solitary wave. Then the study is extended in dimer granular chains composed of alternating “heavy” and “light” beads. For a set of specific mass ratios between the light and heavy beads, we show the existence of multi-hump solitary waves that propagate faster than the Nesterenko solitary wave in the corresponding homogeneous granular chain composed of only heavy beads. The existence of these waves has interesting implications in energy transmission in ordered granular chains.

Sign in / Sign up

Export Citation Format

Share Document