Nonlinear Vibrations of a Thin Plate Under Simultaneous Internal and External Resonances

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
Usama H. Hegazy
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Thin Plate ◽  
Nonlinear Response ◽  
Phase Plane ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
State Response ◽  
The Stability ◽  
Parametric And External Excitations

The dynamic behavior of a rectangular thin plate under parametric and external excitations is investigated. The motion of the thin plate is modeled by coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought by applying the method of multiple scales. A reduced system of four first-order ordinary differential equations is determined to describe the time variation of the amplitudes and phases of the vibration in the horizontal and vertical directions. The steady-state response and the stability of the solutions for various parameters are studied numerically, using the frequency-response function and the phase-plane methods. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. Moreover, the chaotic motion of the thin plate is found by numerical simulation.

Dynamic Behavior of an AMB/Supported Rotor Subject to Parametric Excitation

2006 ◽  
Vol 128 (5) ◽  
pp. 646-652 ◽  
Author(s):  
Y. A. Amer ◽  
M. H. Eissa ◽  
U. H. Hegazy ◽  
A. S. Sabbah
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multiple Scales ◽  
Initial Conditions ◽  
Dynamical Behavior ◽  
Approximate Solutions ◽  
Sensitive Dependence ◽  
Response Function Method ◽  
The Stability ◽  
Gyroscopic Effects

The dynamical behavior of a parametrically excited simple rigid disk-rotor supported by active magnetic bearings (AMB) is investigated, without gyroscopic effects. The principal parametric resonance case is considered and studied. The motion of the rotor is modeled by a coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought applying the method of multiple scales. A reduced system of four first-order ordinary differential equations are determined to describe the time variation of the amplitudes and phases of the vibration in the horizontal and vertical directions. The steady-state response and the stability of the solutions for various parameters are studied numerically, using the frequency response function method. The numerical results show that the system behavior includes multiple solutions, jump phenomenon, and sensitive dependence on initial conditions. It is also shown that the system parameters have different effects on the nonlinear response of the rotor. Results are compared to previously published work.

scholarly journals Temperature Effects on Nonlinear Vibration Behaviors of Euler-Bernoulli Beams with Different Boundary Conditions

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Yaobing Zhao ◽  
Chaohui Huang
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Temperature Effects ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Different Boundary Conditions ◽  
The Stability ◽  
The Galerkin Method ◽  
Close Form

This paper is concerned with temperature effects on the modeling and vibration characteristics of Euler-Bernoulli beams with symmetric and nonsymmetric boundary conditions. It is assumed that in the considered model the temperature increases/decreases instantly, and the temperature variation is uniformly distributed along the length and the cross-section. By using the extended Hamilton’s principle, the mathematical model which takes into account thermal and mechanical loadings, represented by partial differential equations (PDEs), is established. The PDEs of the planar motion are discretized to a set of second-order ordinary differential equations by using the Galerkin method. As to three different boundary conditions, eigenvalue analyses are performed to obtain the close-form eigenvalue solutions. First four natural frequencies with thermal effects are investigated. By using the Lindstedt-Poincaré method and multiple scales method, the approximate solutions of the nonlinear free and forced vibrations (primary, super, and subharmonic resonances) are obtained. The influences of temperature variations on response amplitudes, the localisation of the resonance zones, and the stability of the steady-state solutions are investigated, through examining frequency response curves and excitation response curves. Numerical results show that response amplitudes, the number and the stability of nontrivial solutions, and the hardening-spring characteristics are all closely related to temperature changes. As to temperature effects on vibration behaviors of structures, different boundary conditions should be paid more attention.

scholarly journals On conjugate difference schemes: the midpoint scheme and the trapezoidal scheme

2021 ◽  
Vol 29 (1) ◽  
pp. 63-72
Author(s):  
Yu Ying ◽  
Mikhail D. Malykh
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Projective Geometry ◽  
Autonomous Systems ◽  
Approximate Solutions ◽  
Difference Schemes ◽  
Duality Principle ◽  
Integrals Of Motion ◽  
Quadratic Integral

The preservation of quadratic integrals on approximate solutions of autonomous systems of ordinary differential equations x=f(x), found by the trapezoidal scheme, is investigated. For this purpose, a relation has been established between the trapezoidal scheme and the midpoint scheme, which preserves all quadratic integrals of motion by virtue of Coopers theorem. This relation allows considering the trapezoidal scheme as dual to the midpoint scheme and to find a dual analogue for Coopers theorem by analogy with the duality principle in projective geometry. It is proved that on the approximate solution found by the trapezoidal scheme, not the quadratic integral itself is preserved, but a more complicated expression, which turns into an integral in the limit as t0.Thus the concept of conjugate difference schemes is investigated in pure algebraic way. The results are illustrated by examples of linear and elliptic oscillators. In both cases, expressions preserved by the trapezoidal scheme are presented explicitly.

Imprecise Solutions of Ordinary Differential Equations for Boundary Value Problems Using Metaheuristic Algorithms

2016 ◽  
pp. 401-421
Author(s):  
Ali Sadollah ◽  
Joong Hoon Kim
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Water Cycle ◽  
Error Function ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Metaheuristic Algorithms ◽  
General Strategy ◽  
Optimization Task

In this chapter, a general strategy is recommended to solve variety of linear and nonlinear ordinary differential equations (ODEs) with boundary value conditions. With the aid of certain fundamental concepts of mathematics, Fourier series expansion, and metaheuristic algorithms, ODEs can be represented as an optimization problem. The purpose is to reduce the weighted residual error (error function) of the ODEs. Boundary values of ODEs are considered as constraints for the optimization model. Inverted generational distance metric is utilized for evaluation and assessment of approximate solutions versus exact solutions. Four ODEs having different orders and features are approximately solved and compared with their exact solutions. The optimization task is carried out using different optimizers including the particle swarm optimization and the water cycle algorithm. The optimization results obtained show that the proposed method equipped with metaheuristic algorithms can be successfully applied for approximate solving of different types of ODEs.

Stability in terms of two measures for population growth models with impulsive perturbations

2020 ◽  
Vol 13 (06) ◽  
pp. 2050051
Author(s):  
Zhinan Xia ◽  
Qianlian Wu ◽  
Dingjiang Wang
Keyword(s):  
Differential Equations ◽  
Population Growth ◽  
Ordinary Differential Equations ◽  
Trivial Solution ◽  
Growth Models ◽  
Two Measures ◽  
Impulsive Perturbations ◽  
The Stability ◽  
Generalized Ordinary Differential Equations ◽  
Population Growth Models

In this paper, we establish some criteria for the stability of trivial solution of population growth models with impulsive perturbations. The working tools are based on the theory of generalized ordinary differential equations. Here, the conditions concerning the functions are more general than the classical ones.

Stability of Solutions of Ordinary Differential Equations with Respect to a Closed Set

1968 ◽  
Vol 20 ◽  
pp. 720-726
Author(s):  
T. G. Hallam ◽  
V. Komkov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Compact Set ◽  
Stability Of Solutions ◽  
Closed Set ◽  
The Stability ◽  
Stability Results

The stability of the solutions of an ordinary differential equation will be discussed here. The purpose of this note is to compare the stability results which are valid with respect to a compact set and the stability results valid with respect to an unbounded set. The stability of sets is a generalization of stability in the sense of Liapunov and has been discussed by LaSalle (5; 6), LaSalle and Lefschetz (7, p. 58), and Yoshizawa (8; 9; 10).

scholarly journals Ultimate Boundedness of the Systems Governed by Stochastic Differential Equations

1972 ◽  
Vol 47 ◽  
pp. 111-144 ◽  
Author(s):  
Yoshio Miyahara
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stochastic Differential Equations ◽  
Functional Differential Equations ◽  
Ultimate Boundedness ◽  
Functional Differential ◽  
The Stability

The stability of the systems given by ordinary differential equations or functional-differential equations has been studied by many mathematicians. The most powerful tool in this field seems to be the Liapunov’s second method (see, for example [6]).

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