Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

Vibrations of Square and Hexagonal Cylinders in a Liquid

1981 ◽  
Vol 103 (3) ◽  
pp. 233-239 ◽  
Author(s):  
Y. Shinohara ◽  
T. Shimogo
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fuel Assembly ◽  
Dynamic Behavior ◽  
Nuclear Fuel ◽  
Equations Of Motion ◽  
Hydrodynamic Forces ◽  
Partial Differential ◽  
Nuclear Fuel Assembly

A mathematical model is proposed to describe the dynamic behavior of square and hexagonal cylinder bundles immersed in a liquid. First, the hydrodynamic forces associated with cylinder motions are examined, and then equations of motion of the spring-mounted cylinders including liquid coupling are derived. When the number of cylinders is very large, these equations are replaced by partial differential equations on the assumption that the cylinder bundles form a continuum. The results of this study have application in the modeling of vibration of a nuclear fuel assembly under the excitation of earthquakes.

scholarly journals A New Numerical Procedure for Vibration Analysis of Beam under Impulse and Multiharmonics Piezoelectric Actuators

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Yassin Belkourchia ◽  
Lahcen Azrar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Methodological Approach ◽  
Numerical Procedure ◽  
Piezoelectric Actuators ◽  
Partial Differential ◽  
Simple Method ◽  
Regularization Procedure ◽  
Dirac Delta

The dynamic behavior of structures with piezoelectric patches is governed by partial differential equations with strong singularities. To directly deal with these equations, well adapted numerical procedures are required. In this work, the differential quadrature method (DQM) combined with a regularization procedure for space and implicit scheme for time discretization is used. The DQM is a simple method that can be implemented with few grid points and can give results with a good accuracy. However, the DQM presents some difficulties when applied to partial differential equations involving strong singularities. This is due to the fact that the subsidiaries of the singular functions cannot be straightforwardly discretized by the DQM. A methodological approach based on the regularization procedure is used here to overcome this difficulty and the derivatives of the Dirac-delta function are replaced by regularized smooth functions. Thanks to this regularization, the resulting differential equations can be directly discretized using the DQM. The efficiency and applicability of the proposed approach are demonstrated in the computation of the dynamic behavior of beams for various boundary conditions and excited by impulse and Multiharmonics piezoelectric actuators. The obtained numerical results are well compared to the developed analytical solution.

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.

XIV.—On General Dynamics:—I. Hamilton's Partial Differential Equations and the Determination of their Complete Integrals

1913 ◽  
Vol 32 ◽  
pp. 164-174
Author(s):  
A. Gray
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Equations ◽  
Equations Of Motion ◽  
Partial Differential ◽  
General Dynamics ◽  
Elementary Manner ◽  
Derivatives Of

The present paper contains the first part of a series of notes on general dynamics which, if it is found worth while, may be continued. In § 1 I have shown how the first Hamiltonian differential equation is led up to in a natural and elementary manner from the canonical equations of motion for the most general case, that in which the time t appears explicitly in the function usually denoted by H. The condition of constancy of energy is therefore not assumed. In § 2 it is proved that the partial derivatives of the complete integral of Hamilton's equation with respect to the constants which enter into the specification of that integral do not vary with the time, so that these derivatives equated to constants are the integral equations of motion of the system.*

CHAOTIC BEHAVIOR AND CHAOS CONTROL FOR A CLASS OF COMPLEX PARTIAL DIFFERENTIAL EQUATIONS

2001 ◽  
Vol 12 (06) ◽  
pp. 889-899 ◽  
Author(s):  
G. M. MAHMOUD ◽  
H. A. ABDUSALAM ◽  
A. A. M. FARGHALY
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fixed Points ◽  
Chaos Control ◽  
Chaotic Behavior ◽  
Point Of View ◽  
Partial Differential ◽  
Ginzburg Landau ◽  
Practical Point ◽  
Nagumo Equations

Systems of complex partial differential equations, which include the famous nonlinear Schrödinger, complex Ginzburg–Landau and Nagumo equations, as examples, are important from a practical point of view. These equations appear in many important fields of physics. The goal of this paper is to concentrate on this class of complex partial differential equations and study the fixed points and their stability analytically, the chaotic behavior and chaos control of their unstable periodic solutions. The presence of chaotic behavior in this class is verified by the existence of positive maximal Lyapunov exponent.The problem of chaos control is treated by applying the method of Pyragas. Some conditions on the parameters of the systems are obtained analytically under which the fixed points are stable (or unstable).

On the Parametrically Excited Response of the Quadratically Nonlinear Model of a Rotating Ring on an Elastic Hub

1993 ◽  
Author(s):  
Rick I. Zadoks ◽  
Charles M. Krousgrill
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Parametric Excitation ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Excitation Frequency ◽  
Normal Modes ◽  
Point Load ◽  
Tangential Displacement ◽  
Partial Differential

Abstract As a first approximation, a steel-belted radial tire can be modeled as a one dimensional rotating ring connected elastically to a moving hub. This ring can be modeled mathematically using a set of three nonlinear partial differential equations, where the three degrees of freedom are a radial displacement, a tangential displacement and a section rotation. In this study, only quadratic geometric nonlinearities are considered. The system is excited by a temporally harmonic point load f^(t) and a temporally harmonic hub motion z^(t) that have the same harmonic frequency. The point load f^(t) appears in the equations of motion as a single in-homogeneous term, while the hub motion z^(t) appears in inhomogeneous and parametric excitation terms. To simplifying the ensuing analysis, the rotation rate of the hub is assumed to be constant. The partial differential equations of motion are reduced to a set of four second-order ordinary differential equations by using two linear normal modes to approximate the spatial distribution of the displacements. A region of the parameter space, as defined by ranges of values of the excitation amplitude z and the excitation frequency ω (or detuning parameter σ), is identified, from a Strutt diagram, where the parametric excitation is expected to be dominant. In this region σ is varied to locate a secondary Hopf bifurcation that leads to a set of complex steady-state quasi-periodic solutions. These solutions contain two families of frequency components where the fundamental frequencies of these families are non-commensurate, and they are characterized by Poincaré sections with closed or nearly closed “orbits” as opposed to the distinct points displayed by periodic responses and the strange attractor sections displayed by chaotic solutions.

Longitudinal Vibration Modeling and Control of a Flexible Transporter System with Arbitrarily Varying Cable Lengths

2005 ◽  
Vol 11 (3) ◽  
pp. 431-456 ◽  
Author(s):  
Yuhong Zhang ◽  
Sunil K. Agrawal ◽  
Peter Hagedorn
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Longitudinal Vibration ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Governing Equations ◽  
Partial Differential ◽  
Modeling And Control ◽  
Assumed Mode Method ◽  
Mode Method

We present a systematic procedure for deriving the model of a cable transporter system with arbitrarily varying cable lengths. The Hamilton principle is applied to derive the governing equations of motion. The derived governing equations are nonlinear partial differential equations. The results are verified using the Newton law. The assumed mode method is used to obtain an approximate numerical solution of the governing equations by transforming the infinite-dimensional partial differential equations into a finite-dimensional discretized system. We propose a Lyapunov controller, based directly on the governing partial differential equations, which can both dissipate the vibratory energy during the motion of the transporter and guarantee the attainment of the desired goal point. The validity of the proposed controller is verified by numerical simulation.

scholarly journals Phason dynamics of quasicrystals

2014 ◽  
Vol 70 (a1) ◽  
pp. C86-C86
Author(s):  
Eleni Agiasofitou ◽  
Markus Lazar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Mass Density ◽  
Equations Of Motion ◽  
Type Equation ◽  
Unified Theory ◽  
Partial Differential ◽  
Diffusion Type ◽  
Average Mass ◽  
Proposed Model

Phason dynamics constitutes a challenging and interesting subject in the study of quasicrystals, since there is not a unique model in the literature for the description of the dynamics of the phason fields. Here, we introduce the elastodynamic model of wave-telegraph type for the description of dynamics of quasicrystals [1, 2]. Phonons are represented by waves, and phasons by waves damped in time and propagating with finite velocity; that means the equations of motion for the phonons are partial differential equations of wave type, and for the phasons partial differential equations of telegraph type. The proposed model constitutes a unified theory in the sense that already established models in the literature can be recovered as asymptotic cases of it. Several noteworthy features characterize the proposed model. The influence of the damping in the dynamic behavior of the phasons is expressed by the tensor of phason friction coefficients, which gives the possibility to take into account that the phason waves can be damped anisotropically. In terms of the phason friction coefficient and the average mass density of the material an important quantity, the characteristic time of damping, can been defined. Another important advantage of the model is that it provides a theory valid in the whole regime of possible wavelengths for the phasons. In addition, with the telegraph type equation there is no longer the drawback of the infinite propagation velocity that exists with the equation of diffusion type.

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