Nonlinear Dynamics of Axially Moving Plates With Rotational Springs

2013 ◽  
Author(s):  
Mergen H. Ghayesh ◽  
Marco Amabili
Keyword(s):  
Nonlinear Dynamics ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Plate Theory ◽  
Lagrange Equation ◽  
Global Dynamics ◽  
Nonlinear Ordinary Differential Equations ◽  
Moving Plate ◽  
Time Histories ◽  
Axially Moving

In this paper, the in-plane and out-of-plane nonlinear dynamics of an axially moving plate with distributed rotational springs at boundaries is examined numerically. The Von Kármán plate theory along with the Kirchhoff’s hypothesis are employed to construct the kinetic and potential energies of the system. The Lagrange equation is used so as to obtain the equations of motion which are in the form of a set of second-order nonlinear ordinary differential equations. This set is recast into a set of first-order nonlinear ordinary differential equations with coupled terms. Gear’s backward-differentiation-formula is employed to integrate this set of equations numerically, yielding the generalized coordinates of the system as a function of time. The bifurcation diagrams of Poincaré maps are then constructed by sectioning these time histories in every period of the external excitation force. The results are shown in the form of time histories, phase-plane portraits, and Poincaré sections. The effect of the stiffness of the rotational springs on the global dynamics of the system is also investigated.

The Nonlinear Dynamics of Long, Slender Cylinders

1984 ◽  
Vol 106 (2) ◽  
pp. 250-256 ◽  
Author(s):  
Y. C. Kim ◽  
M. S. Triantafyllou
Keyword(s):  
Nonlinear Dynamics ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Geometric Nonlinearity ◽  
Linear Problem ◽  
Large Deformations ◽  
Nonlinear Ordinary Differential Equations ◽  
Wkb Method ◽  
Fluid Drag ◽  
Nonlinear Fluid

The nonlinear dynamics of long, slender cylinders for moderately large deformations are studied by projecting the solution along the set of eigenmodes of the linear problem. The resulting set of nonlinear ordinary differential equations is truncated on the basis of bandlimited response. The efficiency of the method is due to the derivation of asymptotic solutions for the linear problem in its general form, by using the WKB method. Applications for the dynamics of risers, including the effects of nonlinear fluid drag and geometric nonlinearity demonstrate the features of the method.

scholarly journals Nonlinear Parametric Vibration and Chaotic Behaviors of an Axially Accelerating Moving Membrane

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Mingyue Shao ◽  
Jimei Wu ◽  
Yan Wang ◽  
Qiumin Wu
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibration ◽  
Plate Theory ◽  
Velocity Variation ◽  
Mean Velocity ◽  
Thin Plate Theory ◽  
Time Histories ◽  
Axially Moving ◽  
The Stability ◽  
The Galerkin Method

Nonlinear vibration characteristics of a moving membrane with variable velocity have been examined. The velocity is presumed as harmonic change that takes place over uniform average speed, and the nonlinear vibration equation of the axially moving membrane is inferred according to the D’Alembert principle and the von Kármán nonlinear thin plate theory. The Galerkin method is employed for discretizing the vibration partial differential equations. However, the solutions concerning to differential equations are determined through the 4th order Runge–Kutta technique. The results of mean velocity, velocity variation amplitude, and aspect ratio on nonlinear vibration of moving membranes are emphasized. The phase-plane diagrams, time histories, bifurcation graphs, and Poincaré maps are obtained; besides that, the stability regions and chaotic regions of membranes are also obtained. This paper gives a theoretical foundation for enhancing the dynamic behavior and stability of moving membranes.

BIFURCATIONS AND CHAOS OF AN AXIALLY MOVING PLATE UNDER EXTERNAL AND PARAMETRIC EXCITATIONS

2012 ◽  
Vol 12 (04) ◽  
pp. 1250023 ◽  
Author(s):  
J. T. LIU ◽  
X. D. YANG ◽  
L. Q. CHEN
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Linear Instability ◽  
Transverse Motion ◽  
Largest Lyapunov Exponent ◽  
Moving Plate ◽  
Instability Range ◽  
Axially Moving ◽  
Deflection Theory

The chaos and bifurcations in transverse motion of an axially moving thin plate under external and parametric excitations are studied herein. The geometric nonlinearity is introduced by using the von Karman large deflection theory. The coupled partial differential equations of transverse deflection and stress are truncated into a set of ordinary differential equations. By using the Poincaré map and the largest Lyapunov exponent, the dynamical behaviors including chaos are identified based on numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented for different parameters, such as axially moving velocity, damping, external and parametric excitation amplitudes. The chaos is detected in both cases of external and parametric excitations. The interesting relevance between onset of chaos with the corresponding linear instability range are indicated in the external and parametric responses.

A NOVEL ANALYTICAL METHOD TO INVESTIGATE EFFECT OF RADIATION ON FLOW OF A MAGNETO-MICROPOLAR FLUID PAST A CONTINUOUSLY MOVING PLATE WITH SUCTION AND BLOWING

2010 ◽  
Vol 01 (02) ◽  
pp. 219-238
Author(s):  
MOHAMMAD MEHDI RASHIDI ◽  
ESMAEEL ERFANI
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Analytical Method ◽  
Velocity Profiles ◽  
Nonlinear Ordinary Differential Equations ◽  
Moving Plate ◽  
Transform Method ◽  
Surface Mass ◽  
Mass Transfer Parameter ◽  
Suction And Blowing

In this article, Differential Transform Method (DTM) and Padé approximants (DTM-Padé), are considered for finding analytical solutions of a magneto-micropolar flow past a continuously moving plate with suction and blowing and radiation effect. This technique is extended to give solutions for system of nonlinear ordinary differential equations with boundary conditions at infinity. The analytic solutions of the system of nonlinear ordinary differential equations are constructed in the ratio of two polynomials. Graphical results are presented to investigate influence of the radiation parameter, magnetic field parameter, Prandtl number, coupling constant parameter and the surface mass transfer parameter on velocity profiles, angular velocity profiles and temperature profiles. In addition, the numerical method is used to investigate the validity of this analytical method, an excellent agreement is observed between the solutions obtained from the DTM-Padé and numerical results. The results reveal that the DTM-Padé is very effective and convenient for solving engineering problems especially for boundary-layer problems.

scholarly journals Impact of double-diffusive convection and motile gyrotactic microorganisms on magnetohydrodynamics bioconvection tangent hyperbolic nanofluid

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 74-88 ◽  
Author(s):  
Tanveer Sajid ◽  
Muhammad Sagheer ◽  
Shafqat Hussain ◽  
Faisal Shahzad
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Transfer Rate ◽  
Mass Transfer Rate ◽  
Physical Nature ◽  
Working Fluid ◽  
Nonlinear Ordinary Differential Equations ◽  
Gyrotactic Microorganisms ◽  
Motile Gyrotactic Microorganisms ◽  
Double Diffusive

AbstractThe double-diffusive tangent hyperbolic nanofluid containing motile gyrotactic microorganisms and magnetohydrodynamics past a stretching sheet is examined. By adopting the scaling group of transformation, the governing equations of motion are transformed into a system of nonlinear ordinary differential equations. The Keller box scheme, a finite difference method, has been employed for the solution of the nonlinear ordinary differential equations. The behaviour of the working fluid against various parameters of physical nature has been analyzed through graphs and tables. The behaviour of different physical quantities of interest such as heat transfer rate, density of the motile gyrotactic microorganisms and mass transfer rate is also discussed in the form of tables and graphs. It is found that the modified Dufour parameter has an increasing effect on the temperature profile. The solute profile is observed to decay as a result of an augmentation in the nanofluid Lewis number.

A Multiple Interval Chebyshev-Gauss-Lobatto Collocation Method for Ordinary Differential Equations

2016 ◽  
Vol 9 (4) ◽  
pp. 619-639 ◽  
Author(s):  
Zhong-Qing Wang ◽  
Jun Mu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
High Order Accuracy ◽  
Nonlinear Ordinary Differential Equations ◽  
Initial Value ◽  
Order Accuracy ◽  
Long Time

AbstractWe introduce a multiple interval Chebyshev-Gauss-Lobatto spectral collocation method for the initial value problems of the nonlinear ordinary differential equations (ODES). This method is easy to implement and possesses the high order accuracy. In addition, it is very stable and suitable for long time calculations. We also obtain thehp-version bound on the numerical error of the multiple interval collocation method underH1-norm. Numerical experiments confirm the theoretical expectations.

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