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.

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.

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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