Resonant Oscillations of a Beam-Pendulum System

1963 ◽  
Vol 30 (2) ◽  
pp. 181-188 ◽  
Author(s):  
R. A. Struble ◽  
J. H. Heinbockel
Keyword(s):  
Energy Transfer ◽  
Differential Equations ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Asymptotic Methods ◽  
Physical Parameters ◽  
Pendulum System ◽  
Long Period ◽  
Resonance Phenomena

The resonance phenomena and energy transfer associated with a set of coupled nonlinear differential equations are analyzed using asymptotic methods. The equations describe the motion of a beam-pendulum system which exhibits autoparametric excitation. Precise conditions under which resonant or nonresonant oscillations arise, are obtained. These conditions not only depend upon the physical parameters of the system but also upon the energy of the oscillations (i.e., initial conditions). In general, the envelopes of the resonant oscillations are periodic of long period and there is intermittent energy transfer between the pendulum and beam modes.

Numerical exploration of mixed convection heat transfer features within a copper-water nanofluidic medium occupied a square geometrical cavity

2021 ◽  
Vol 8 (4) ◽  
pp. 807-820
Author(s):  
M. Zaydan ◽  
◽  
A. Wakif ◽  
E. Essaghir ◽  
R. Sehaqui ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Mixed Convection ◽  
Nonlinear Differential Equations ◽  
Convection Heat Transfer ◽  
Local Heat Transfer ◽  
Physical Parameters ◽  
Convection Heat ◽  
Linear Temperature ◽  
Mixed Convection Heat Transfer

The phenomenon of mixed convection heat transfer in a homogeneous mixture is deliberated thoroughly in this study for cooper-water nanofluids flowing inside a lid-driven square cavity. By adopting the Oberbeck-Boussinesq approximation and using the single-phase nanofluid model, the governing partial differential equations modeling the present flow are stated mathematically based on the Navier--Stokes and thermal balance formulations, where the important features of the scrutinized medium are presumed to remain constant at the cold temperature. Note here that the density quantity in the buoyancy body force is a linear temperature-dependent function. The characteristic quantities are computed realistically via the commonly used phenomenological laws and the more accurate experimental correlations. A feasible non-dimensionalization procedure has been employed to derive the dimensionless conservation equations. The resulting nonlinear differential equations are solved numerically for realistic boundary conditions by employing the fourth-order compact finite-difference method (FOCFDM). After performing extensive validations with the previously published findings, the dynamical and thermal features of the studied convective nanofluid flow are revealed to be in good agreement for sundry values of the involved physical parameters. Besides, the present numerical outcomes are discussed graphically and tabularly with the help of streamlines, isotherms, velocity fields, temperature distributions, and local heat transfer rate profiles.

Multiple large-scale coherent mode interactions in a developing round jet

1993 ◽  
Vol 248 ◽  
pp. 383-401 ◽  
Author(s):  
Sang Soo Lee ◽  
J. T. C. Liu
Keyword(s):  
Differential Equations ◽  
Energy Method ◽  
Large Scale ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Wave Mode ◽  
Mean Flow ◽  
Round Jet ◽  
The Mean ◽  
Integral Energy

The integral energy method has been used to study the nonlinear interactions of the large-scale coherent structure in a spatially developing round jet. The streamwise development of a jet is obtained in terms of the mean flow shear-layer momentum thickness, the wave-mode kinetic energy and the wave-mode phase angle. With the energy method, a system of partial differential equations is reduced to a system of ordinary differential equations. The nonlinear differential equations are solved with initial conditions which are given at the nozzle exit. It is shown that the initial wave-mode energy densities as well as the initial phase angles play a significant role in the streamwise evolution of the large-scale coherent wave modes and the mean flow.

CALCULATION OF THE INFLUENCE OF TEMPERATURE FIELDS ON THE QUALITY OF THE RESULTING COMPOSITE COATINGS

2021 ◽  
Vol 2 (143) ◽  
pp. 130-134
Author(s):  
Sergey Yu. Zhachkin ◽  
◽  
Marina N. Krasnova ◽  
Aleksandr V. Biryukov ◽  
Nikita A. Pen’kov ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Plasma Spraying ◽  
Performance Indicators ◽  
Plasma Torch ◽  
Initial Conditions ◽  
Composite Coatings ◽  
Nonlinear Differential Equations ◽  
Research Purpose ◽  
Mathematical Tool

Increased attention is currently being paid to improving the reliability and efficiency of power units. The use of materials with special coatings that have the necessary set of characteristics allows you to increase the service life of the restored components and assemblies. One of the ways to obtain such coatings is by plasma spraying. (Research purpose) The research purpose is in predicting the performance indicators of composite coatings using a mathematical tool that adequately describes the process of coating build-up during plasma spraying. (Materials and methods) Authors used a universal UPU-3 installation with modifications to produce coatings, which allowed controlling the heat transfer process with high accuracy. Authors used small-sized cooling water temperature sensors located at the connection point of the current-carrying hoses to the sprayer to record the enthalpy of the jet. During the coating process, the restored part remained stationary, and a plasma torch carried out the movement along the sprayed part. The advantage of this method is the independence of the mass of the recovered parts from the drive power of the plasma torch. (Results and discussion) The article presents the mathematical model, which is a system of nonlinear partial differential equations describing the process of heat transfer during the application of plasma-sputtered coatings, taking into account the initial conditions determined using the UPU-3 installation. To solve the presented system of nonlinear differential equations, the method of perturbation theory with the use of thermal potentials is used. (Conclusions) The article proposes the ways to solve the problem of heat transfer during the application of plasma-coated coatings, which allow us to predict the performance indicators of the restored parts.

scholarly journals Deterministic Chaos Theory: Basic Concepts

2016 ◽  
Vol 39 (1) ◽  
Author(s):  
Mauro Cattani ◽  
Iberê Luiz Caldas ◽  
Silvio Luiz de Souza ◽  
Kelly Cristiane Iarosz
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Chaos Theory ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Deterministic Chaos ◽  
Chaotic Motions ◽  
And Mathematics ◽  
Irregular Behavior ◽  
Mathematics And Physics

This article was written to students of mathematics, physics and engineering. In general, the word chaos may refer to any state of confusion or disorder and it may also refer to mythology or philosophy. In science and mathematics it is understood as irregular behavior sensitive to initial conditions. In this article we analyze the deterministic chaos theory, a branch of mathematics and physics that deals with dynamical systems (nonlinear differential equations or mappings) with very peculiar properties. Fundamental concepts of the deterministic chaos theory are briefly analyzed and some illustrative examples of conservative and dissipative chaotic motions are introduced. Complementarily, we studied in details the chaotic motion of some dynamical systems described by differential equations and mappings. Relations between chaotic, stochastic and turbulent phenomena are also commented.

scholarly journals Investigation of iceberg towing processes based on finite-dimensional models of continuous media

2020 ◽  
pp. 82-88
Author(s):  
В.А. Коршунов ◽  
Р.С. Мудрик ◽  
Д.А. Пономарев ◽  
А.А. Родионов
Keyword(s):  
Numerical Modeling ◽  
Differential Equations ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Mathematical Formulation ◽  
Element Model ◽  
Dynamic Factor ◽  
System Of Differential Equations ◽  
Finite Dimensional ◽  
Simplified Algorithm

В работе рассмотрена проблема управления ледовой обстановкой при освоении участков на морской шельфе. Отмечено, что наиболее эффективным способом управления движения айсбергов и их осколков является использование буксировочных систем. В зависимости от размера айсберга в качестве буксировочной системы может использоваться либо одиночный канат, либо ледовая сетка. Описана технология осуществления буксировки. Рассмотрено два варианта математической формулировки задачи буксировки. Инженерный подход основан на решении системы нелинейных дифференциальных уравнений динамики с известными начальными условиями. Наиболее точным является аппарат механики сплошных сред, который опирается на фундаментальные законы сохранения. Он позволяет построить связанную нелинейную систему дифференциальных уравнений с минимальным количеством допущений. Использование данного подхода возможно при численном моделировании процесса буксировки. В работе создана конечно-элементная модель взаимодействия айсберга с буксировочной системой. Разработан упрощенный алгоритм учета жидкости. Получены кинематические и динамические характеристики процесса буксировки. Определен коэффициент динамичности усилий. The paper deals with the problem of ice management during the development of the sea shelf. It is noted that the most effective way to control the movement of icebergs and their fragments is to use towing systems. Depending on the size of the iceberg, either a single rope or an ice net can be used as a towing system. The technology of towing is described. Various of the mathematical formulation of the towing problem are considered. The engineering approach is based on solving a system of nonlinear differential equations of dynamics with initial conditions. More accurate approach is the computational continuum mechanics, which is based on fundamental conservation laws. It allows to build a nonlinear system of differential equations with a minimum of assumptions. This approach can be used for numerical modeling of the towing process. In the paper, a finite element model of the interaction of an iceberg with a towing system is created. To account liquid the simplified algorithm is prescribed. The kinematic and dynamic characteristics of the towing process are obtained. The dynamic factor of axial force is determined.

Physical aspects of magnetized Jeffrey nanomaterial flow with irreversibility analysis

2021 ◽  
Author(s):  
Fazal Haq ◽  
Muhammad Ijaz Khan ◽  
Sami Ullah Khan ◽  
Khadijah M. Abualnaja ◽  
M. A. El-Shorbagy
Keyword(s):  
Differential Equations ◽  
Entropy Generation ◽  
Nonlinear Differential Equations ◽  
Transportation Systems ◽  
Physical Parameters ◽  
Bejan Number ◽  
Convective Boundary ◽  
Homotopy Analysis ◽  
Permeability Parameter ◽  
Thermal Features

Abstract This analysis presents the applications of entropy generation phenomenon in incompressible flow of Jeffrey nanofluid in presence of distinct thermal features. The novel aspects of various features like Joule heating, porous medium, dissipation features and radiative mechanism is addressed. In order to improve the thermal transportation systems based on nanomaterials, the convective boundary conditions are introduced. The thermal viscoelastic nanofluid model is expressed in term of differential equations. The problem is presented via nonlinear differential equations for which analytical expressions are obtained by using homotopy analysis method(HAM). The accuracy of solution is ensured. The effective outcomes of all physical parameters associated with the flow model are carefully examined and underlined through various curves. The observations summarized from current analysis reveal that presence of permeability parameter offers resistance to the flow. A monotonic decrement in local Nusselt number is noted with Hartmann number and Prandtl number. Moreover, entropy generation and Bejan number increases with radiation parameter and fluid parameter.

Application of New Homotopy Perturbation Method for Solving Partial Differential Equations

2018 ◽  
Vol 15 (2) ◽  
pp. 500-508 ◽  
Author(s):  
Musa R. Gad-Allah ◽  
Tarig M. Elzaki
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Novel Technique ◽  
Homotopy Perturbation ◽  
Linear And Nonlinear ◽  
Linear Differential

In this paper, a novel technique, that is to read, the New Homotopy Perturbation Method (NHPM) is utilized for solving a linear and non-linear differential equations and integral equations. The two most important steps in the application of the new homotopy perturbation method are to invent a suitable homotopy equation and to choose a suitable initial conditions. Comparing between the effects of the method (NHPM), is given exact solution, and the method (HPM), is given approximate solution, in this paper, we make some instances are provided to prove the ability of the method (NHPM). Show that the method (NHPM) is valid and effective, easy and accurate in solving linear and nonlinear differential equations, compared with the Homotopy Perturbation Method (HPM).

Laplace–Pade Parametric Homotopy Perturbation Method for Uncertain Nonlinear Oscillators

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
S. Chakraverty ◽  
N. R. Mahato
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Nonlinear Oscillator ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Parametric Form ◽  
Homotopy Perturbation

Nonlinear oscillators have wide applicability in science and engineering problems. In this paper, nonlinear oscillator having initial conditions varying over fuzzy numbers has been initially taken into consideration. Here, the fuzziness in the uncertain nonlinear oscillators has been handled using parametric form. Using parametric form in terms of r-cut, the nonlinear uncertain differential equations are reduced to parametric differential equations. Then, based on classical homotopy perturbation method (HPM), a parametric homotopy perturbation method (PHPM) is proposed to compute solution enclosure of such uncertain nonlinear differential equations. A sufficient convergence condition of parametric solution obtained using PHPM is also proved. Further, a parametric Laplace–Pade approximation is incorporated in PHPM for retaining the periodic characteristic of nonlinear oscillators throughout the domain. The efficiency of Laplace–Pade PHPM has been verified for uncertain Duffing oscillator. Finally, Laplace–Pade PHPM is also applied to solve other uncertain nonlinear oscillator, viz., Rayleigh oscillator, with respect to fuzzy parameters.

scholarly journals Vibration of the Biomass Boiler Tube Excited with Impact of the Cleaning Device

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1519
Author(s):  
Dragan Cveticanin ◽  
Nicolae Herisanu ◽  
Istvan Biro ◽  
Miodrag Zukovic ◽  
Livija Cveticanin
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Mode Shapes ◽  
Straight Beam ◽  
Special Cases ◽  
Cleaning Device ◽  
The Impact

In boilers with biomass fuel, a significant problem is caused due to the slag layer formed from the unburned particles during combustion. In the paper, a tube cleaning method from slag is proposed. The method is based on the impact effect of the end of the tube with the aim to produce vibration for slag elimination. The tube is modeled as a clamped-free nonlinear oscillatory system. The initial impact of the tube causes vibrations. The mathematical model of the system is a nonlinear partial differential equation with zero initial deflection. To obtain the ordinary differential equations, the Galerkin method is applied. By discretizing the equation into a finite degree of freedom system, using the undamped linear mode shapes of the straight beam as basic functions, the reduced order model, consisting of ordinary differential equations in time, is obtained. The ordinary time equations are analytically solved by adopting the Krylov–Bogoliubov procedure. Special cases of nonlinear differential equations are investigated. In the paper, the influence of the nonlinear parameters and initial conditions on the vibration properties of the tube is obtained. We use the procedure developed in the paper and the analytical results for computation of the impact parameters of the cleaning device.

scholarly journals Effects of Geopotential and Atmospheric Drag Effects on Frozen Orbits Using Nonsingular Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Paula Cristiane Pinto Mesquita Pardal ◽  
Rodolpho Vilhena de Moraes ◽  
Helio Koiti Kuga
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Orbital Evolution ◽  
Atmospheric Drag ◽  
Orbit Design ◽  
Frozen Orbits ◽  
Long Period ◽  
Frozen Orbit ◽  
And Control ◽  
Analytical Expressions

The concept of frozen orbit has been applied in space missions mainly for orbital tracking and control purposes. This type of orbit is important for orbit design because it is characterized by keeping the argument of perigee and eccentricity constant on average, so that, for a given latitude, the satellite always passes at the same altitude, benefiting the users through this regularity. Here, the system of nonlinear differential equations describing the motion is studied, and the effects of geopotential and atmospheric drag perturbations on frozen orbits are taken into account. Explicit analytical expressions for secular and long period perturbations terms are obtained for the eccentricity and the argument of perigee. The classical equations of Brouwer and Brouwer and Hori theories are used. Nonsingular variables approach is used, which allows obtaining more precise previsions for CBERS (China Brazil Earth Resources Satellite) satellites family and similar satellites (SPOT, Landsat, ERS, and IRS) orbital evolution.

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