Bifurcation and Modal Interaction in a Simplified Model of Bending-Torsion Vibrations of the Thin Elastica

1995 ◽  
Vol 117 (1) ◽  
pp. 30-42 ◽  
Author(s):  
J. P. Cusumano ◽  
D.-C. Lin
Keyword(s):  
Differential Equations ◽  
Floquet Theory ◽  
Numerical Study ◽  
Normal Modes ◽  
Elastic Beam ◽  
Pitchfork Bifurcation ◽  
Statistical Technique ◽  
Simplified Model ◽  
Modal Interaction ◽  
Amplitude Parameter

This paper presents a numerical study of bifurcation and modal interaction in a system of partial differential equations first proposed as a simplified model for bending-torsion vibrations of a thin elastic beam. A system of seven ordinary differential equations obtained using the first six bending and first torsional normal modes is studied, and Floquet theory is used to locate regions in the forcing frequency, forcing amplitude parameter plane where “planar” (i.e., zero torsion) motions are unstable. Numerical branch following and symmetry considerations show that the initial instability arises from a subcritical pitchfork bifurcation. The subsequent nonplanar chaotic attractor is part of a branch of 2-frequency quasiperiodic orbits which undergoes torus-doubling bifurcations. A new statistical technique which identifies interacting modes and the average stability properties of the associated subspaces is presented. The technique employs the Lyapunov vectors used in the calculation of the Lyapunov exponents. We show how this method can be used to split the modes into active and passive sets: active modes interact to contain the attractor, whereas passive modes behave like isolated driven oscillators. In particular, large amplitude modes may simply serve as conduits through which energy is supplied to the active modes.

Numerical Study of the Combined Free-Forced Convection and Mass Transfer Flow Past a Vertical Porous Plate in a Porous Medium with Heat Generation and Thermal Diffusion

2006 ◽  
Vol 11 (4) ◽  
pp. 331-343 ◽  
Author(s):  
M. S. Alam ◽  
M. M. Rahman ◽  
M. A. Samad
Keyword(s):  
Mass Transfer ◽  
Flow Field ◽  
Differential Equations ◽  
Forced Convection ◽  
Heat Generation ◽  
Numerical Study ◽  
Porous Plate ◽  
Integration Scheme ◽  
Suction Parameter ◽  
Transfer Flow

The problem of combined free-forced convection and mass transfer flow over a vertical porous flat plate, in presence of heat generation and thermaldiffusion, is studied numerically. The non-linear partial differential equations and their boundary conditions, describing the problem under consideration, are transformed into a system of ordinary differential equations by using usual similarity transformations. This system is solved numerically by applying Nachtsheim-Swigert shooting iteration technique together with Runge-Kutta sixth order integration scheme. The effects of suction parameter, heat generation parameter and Soret number are examined on the flow field of a hydrogen-air mixture as a non-chemical reacting fluid pair. The analysis of the obtained results showed that the flow field is significantly influenced by these parameters.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

scholarly journals Numerical study of time-fractional fourth-order differential equations with variable coefficients

2011 ◽  
Vol 23 (1) ◽  
pp. 91-98 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Nasir-Uddin Khan ◽  
Muhammad Ayaz ◽  
Amir Mahmood ◽  
Noor Fatima
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Numerical Study ◽  
Variable Coefficients

scholarly journals Normal modes of a waveguide as eigenvectors of a self-adjoint operator pencil

2021 ◽  
Vol 29 (1) ◽  
pp. 14-21
Author(s):  
Mikhail D. Malykh
Keyword(s):  
Electromagnetic Field ◽  
Differential Equations ◽  
Wave Equations ◽  
Adjoint Operator ◽  
Normal Modes ◽  
Operator Pencil ◽  
Homogeneous Case ◽  
Permittivity And Permeability ◽  
Simply Connected ◽  
Magnetic Potentials

A waveguide with a constant, simply connected section S is considered under the condition that the substance filling the waveguide is characterized by permittivity and permeability that vary smoothly over the section S, but are constant along the waveguide axis. Ideal conductivity conditions are assumed on the walls of the waveguide. On the basis of the previously found representation of the electromagnetic field in such a waveguide using 4 scalar functions, namely, two electric and two magnetic potentials, Maxwells equations are rewritten with respect to the potentials and longitudinal components of the field. It appears possible to exclude potentials from this system and arrive at a pair of integro-differential equations for longitudinal components alone that split into two uncoupled wave equations in the optically homogeneous case. In an optically inhomogeneous case, this approach reduces the problem of finding the normal modes of a waveguide to studying the spectrum of a quadratic self-adjoint operator pencil.

scholarly journals Numerical Exploration via Least Squares Estimation on Three Dimensional MHD Yield Exhibiting Nanofluid Model with Porous Stretching Boundaries

2021 ◽  
Vol 5 (4) ◽  
pp. 167
Author(s):  
Tamour Zubair ◽  
Muhammad Usman ◽  
Umar Nazir ◽  
Poom Kumam ◽  
Muhammad Sohail
Keyword(s):  
Differential Equations ◽  
Numerical Study ◽  
Nonlinear Partial Differential Equations ◽  
Three Dimensional ◽  
Least Square Method ◽  
Least Square ◽  
Comparison Study ◽  
Nano Fluid ◽  
Complicated Geometry ◽  
Maple Code

The numerical study of a three-dimensional magneto-hydrodynamic (MHD) Casson nano-fluid with porous and stretchy boundaries is the focus of this paper. Radiation impacts are also supposed. A feasible similarity variable may convert a verbalized set of nonlinear “partial” differential equations (PDEs) into a system of nonlinear “ordinary” differential equations (ODEs). To investigate the solutions of the resulting dimensionless model, the least-square method is suggested and extended. Maple code is created for the expanded technique of determining model behaviour. Several simulations were run, and graphs were used to provide a thorough explanation of the important parameters on velocities, skin friction, local Nusselt number, and temperature. The comparison study attests that the suggested method is well-matched, trustworthy, and accurate for investigating the governing model’s answers. This method may be expanded to solve additional physical issues with complicated geometry.

Model Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes

1999 ◽  
Author(s):  
E. Pesheck ◽  
C. Pierre ◽  
S. W. Shaw
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Single Equation ◽  
Rotating Beam ◽  
Model Size ◽  
Nonlinear Normal

Abstract Equations of motion are developed for a rotating beam which is constrained to deform in the transverse (flapping) and axial directions. This process results in two coupled nonlinear partial differential equations which govern the attendant dynamics. These equations may be discretized through utilization of the classical normal modes of the nonrotating system in both the transverse and extensional directions. The resultant system may then be diagonalized to linear order and truncated to N nonlinear ordinary differential equations. Several methods are used to determine the model size necessary to ensure accuracy. Once the model size (N degrees of freedom) has been determined, nonlinear normal mode (NNM) theory is applied to reduce the system to a single equation, or a small set of equations, which accurately represent the dynamics of a mode, or set of modes, of interest. Results are presented which detail the convergence of the discretized model and compare its dynamics with those of the NNM-reduced model, as well as other reduced models. The results indicate a considerable improvement over other common reduction techniques, enabling the capture of many salient response features with the simulation of very few degrees of freedom.

Numerical Solution for Boundary Layer Flow of a Dusty Micropolar Fluid Due to a Stretching Sheet with Constant Wall Temperature

2021 ◽  
Vol 87 (1) ◽  
pp. 30-40
Author(s):  
Anisah Dasman ◽  
Abdul Rahman Mohd Kasim ◽  
Iskandar Waini ◽  
Najiyah Safwa Khashi’ie
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Wall Temperature ◽  
Micropolar Fluid ◽  
Stretching Sheet ◽  
Numerical Solutions ◽  
Numerical Study ◽  
Particle Interaction ◽  
Dust Particles ◽  
Constant Wall Temperature

This paper aims to present the numerical study of a dusty micropolar fluid due to a stretching sheet with constant wall temperature. Using the suitable similarity transformation, the governing partial differential equations for two-phase flows of the fluid and the dust particles are reduced to the form of ordinary differential equations. The ordinary differential equations are then numerically analysed using the bvp4c function in the Matlab software. The validity of present numerical results was checked by comparing them with the previous study. The results graphically show the numerical solutions of velocity, temperature and microrotation distributions for several values of the material parameter K, fluid-particle interaction parameter and Prandtl number for both fluid and dust phase. The effect of microrotation is investigated and analysed as well. It is found that the distributions are significantly influenced by the investigated parameters for both phases.

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