scholarly journals Discussion: “Flexural Wave Solution of Coupled Equations Representing the More Exact Theory of Bending” (Miklowitz, J., 1953, ASME J. Appl. Mech., 20, pp. 511–514)

1954 ◽  
Vol 21 (2) ◽  
pp. 204
Author(s):  
M. A. Dengler
Keyword(s):  
Wave Solution ◽  
Flexural Wave ◽  
Coupled Equations ◽  
Exact Theory

Flexural Wave Solutions of Coupled Equations Representing the More Exact Theory of Bending

1953 ◽  
Vol 20 (4) ◽  
pp. 511-514
Author(s):  
Julius Miklowitz
Keyword(s):  
Boundary Conditions ◽  
Present Method ◽  
Shear Force ◽  
Beam Element ◽  
Transverse Load ◽  
Flexural Wave ◽  
Infinite Beam ◽  
Coupled Equations ◽  
Wave Solutions ◽  
The Moment

Abstract Presented here is a new method for deriving flexural wave solutions for the Timoshenko bending theory. The method is based on a breakdown of the total deflection into its bending and shear components. Instead of treating the full Timoshenko equation (1) an equivalent set of coupled equations, representing the rotational and translatory motions of the beam element, is solved. The advantages of this method stem from (a) the simplicity of the associated expressions for the moment and shear force, which are the elementary bending theory relations, and (b) the well-defined nature of the related boundary conditions. The latter is particularly important since it is difficult to define the proper boundary conditions associated with the full Timoshenko equation. This is evidenced in the works of Uflyand (2) and Dengler and Goland (3), both of which are concerned with wave solutions for the infinite beam under the action of a concentrated transverse load. The quoted work (3) points out the erroneous boundary conditions used in the Uflyand work (2). The present method is applied to the same case treated in the works (2, 3). Agreement is shown with the Dengler and Goland solution. The Uflyand solution is shown to have meaning when interpreted properly. The derivation of transforms for other beam cases, both finite and infinite, by the present method has also been included in this work.

Solitary Waves and Chaotic Behavior in Large-Deflection Beam

2010 ◽  
Vol 29-32 ◽  
pp. 28-34
Author(s):  
Yi Qing Zhou ◽  
Zhi Fang Liu ◽  
Shan Yuan Zhang
Keyword(s):  
Elliptic Function ◽  
Wave Solution ◽  
Large Deflection ◽  
Chaotic Behavior ◽  
Solitary Wave Solution ◽  
Heteroclinic Orbit ◽  
Flexural Wave ◽  
Threshold Condition ◽  
Jacobi Elliptic Function ◽  
Homoclinic Point

The motion equation of nonlinear flexural wave in large-deflection beam is derived from Hamilton's variational principle using the coupling of flexural deformation and midplane stretching as key source of nonlinearity and taking into account transverse, axial and rotary inertia effects. The system has homoclinic or heteroclinic orbit under certain conditions, the exact periodic solutions of nonlinear wave equation are obtained by means of Jacobi elliptic function expansion. The solitary wave solution and shock wave solution is given when the modulus of Jacobi elliptic function in the degenerate case. It is easily thought that the introduction of damping and external load can result in break of homoclinic (or heteroclinic) orbit and appearance of transverse homoclinic point. The threshold condition of the existence of transverse homoclinic point is given by help of Melnikov function. It shows that the system has chaos property under Smale horseshoe meaning.

Towards quantitative simulations of inelastic electron diffraction patterns and images

1992 ◽  
Vol 50 (2) ◽  
pp. 1170-1171
Author(s):  
Z. L. Wang
Keyword(s):  
Phonon Scattering ◽  
Incident Beam ◽  
Dynamical Theory ◽  
Inelastic Electron ◽  
Additional Advantage ◽  
Coupled Equations ◽  
Atomic Vibrations ◽  
Diffraction Patterns ◽  
Primary Advantage ◽  
The One

A new dynamical theory has been developed based on Yoshioka's coupled equations for describing inelastic electron scattering in thin crystals. Compared to existing theories, the primary advantage of this theory is that the incoherent summation of the diffracted intensities contributed by electrons after exciting vast numbers of different excited states has been evaluated before any numerical calculation. An additional advantage is that the phase correlations of atomic vibrations are considered, so that full lattice dynamics can be combined in the phonon scattering calculation. The new theory has been proven to be equivalent to the inelastic multislice theory, and has been applied to calculate energy-filtered diffraction patterns and images formed by phonon, single electron and valence scattered electrons.A calculated diffraction pattern of elastic and phonon scattered electrons for a parallel incident beam case is in agreement with the one observed (Fig. 1), showing thermal diffuse scattering (TDS) streaks and Kikuchi pattern.

Problems on the Gravitational Wave and the Repulsive Gravitation

2016 ◽  
pp. 4422-4429
Author(s):  
C. Y. Lo
Keyword(s):  
Black Holes ◽  
Gravitational Wave ◽  
Einstein Equation ◽  
Wave Solution ◽  
Dynamic Solution ◽  
Pure Mathematics ◽  
Non Linear

It is exciting that the gravitational wave has been confirmed, according to the announcement of LIGO. This would be the time to fix the Einstein equation for the gravitational wave and the nonexistence of the dynamic solution. As a first step, theorists should improve their pure mathematics on non-linear mathematics and related physical considerations beyond Einstein. Then, it is time to rectify the Einstein equation that has no gravitational wave solution which Einstein has recognized, and no dynamic solution that Einstein failed to see. A problem is that physicists in LIGO did not know their shortcomings. Also, in view of the far distance of the sources, it is very questionable that the physicists can determine they are from black holes. Moreover, since the repulsive gravitation can also generate a gravitational wave, the problem of gravitational wave is actually far more complicated than we have known. A useful feature of the gravitational wave based on repulsive gravitation is that it can be easily generated on earth. Thus this can be a new tool for communication because it can penetrate any medium.

Flexural Wave Band Gaps in Phononic Crystal Thick Plates with Defects

2019 ◽  
Author(s):  
Edson Jansen Pedrosa de Miranda Junior ◽  
Jose Maria Campos dos Santos
Keyword(s):  
Phononic Crystal ◽  
Band Gaps ◽  
Flexural Wave ◽  
Wave Band ◽  
Thick Plates

Ciliary Flow of Casson Nanofluid with the Influence of MHD having Carbon Nanotubes

2020 ◽  
Vol 16 ◽  
Author(s):  
Adel Alblawi ◽  
Saba Keyani ◽  
S. Nadeem ◽  
Alibek Issakhov ◽  
Ibrahim M. Alarifi
Keyword(s):  
Carbon Nanotubes ◽  
Velocity Profile ◽  
Pressure Gradient ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Pressure Rise ◽  
Shear Stresses ◽  
Left Wall ◽  
Coupled Equations ◽  
The Right

Objective: In this paper, we consider a model that describes the ciliary beating in the form of metachronal waves along with the effects of Magnetohydrodynamic fluid over a curved channel with slip effects. This work aims at evaluating the effect of Magnetohydrodynamic (MHD) on the steady two dimensional (2-D) mixed convection flow induced in carbon nanotubes. The work is done for both the single wall nanotube and multiple wall nanotube. The right wall and the left wall possess a metachronal wave that is travelling along the outer boundary of the channel. Methods: The wavelength is considered as very large for cilia induced MHD flow. The governing linear coupled equations are simplified by considering the approximations of long wavelength and small Reynolds number. Exact solutions are obtained for temperature and velocity profile. The analytical expressions for the pressure gradient and wall shear stresses are obtained. Term for pressure rise is obtained by applying Numerical integration method. Results: Numerical results of velocity profile are mentioned in a table form, for various values of solid volume fraction, curvature, Hartmann number [M] and Casson fluid parameter [ζ]. Final section of this paper is devoted to discussing the graphical results of temperature, pressure gradient, pressure rise, shear stresses and stream functions. Conclusion: Velocity profile near the right wall of the channel decreases when we add nanoparticles into our base fluid, whereas an opposite behaviour is depicted near the left wall due to ciliated tips whereas the temperature is an increasing function of B and ߛ and decreasing function of ߶.

Dynamics of Electrodiffusion Friction Probes. I. Shape-Dependent Potentiostatic Transient

1997 ◽  
Vol 62 (3) ◽  
pp. 397-419 ◽  
Author(s):  
Ondřej Wein ◽  
Václav Sobolík
Keyword(s):  
Simple Formula ◽  
Flow Direction ◽  
Voltage Step ◽  
Convex Shape ◽  
Exact Theory ◽  
Working Electrode ◽  
Arbitrary Convex

An exact theory is given of the voltage-step transient under limiting diffusion conditions for an electrodiffusion friction probe of arbitrary convex shape. The actual transient courses are given for the strip, circular, elliptic, triangular, and rectangular probes of any orientation with respect to the flow direction. A simple formula for any probe with a single working electrode of convex shape is suggested to facilitate the calibration of electrodiffusion probes based on the voltage-step transient.

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