Exact solutions of variable-arc-length elasticas under moment gradient

1997 ◽  
Vol 5 (5) ◽  
pp. 529-539 ◽  
Author(s):  
Somchai Chucheepsakul ◽  
Geeraphong Thepphitak ◽  
Chien Ming Wang
Keyword(s):  
Exact Solutions ◽  
Arc Length ◽  
Moment Gradient

Exact Solutions for Out-of-Plane Vibration of Curved Nonuniform Beams

2000 ◽  
Vol 68 (2) ◽  
pp. 186-191 ◽  
Author(s):  
S. Y. Lee ◽  
J. C. Chao
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Natural Frequencies ◽  
Variable Coefficients ◽  
Physical Parameters ◽  
Arc Length ◽  
Order Ordinary Differential Equation ◽  
Constant Radius ◽  
Out Of Plane ◽  
Center Angle

The governing differential equations for the out-of-plane vibrations of curved nonuniform beams of constant radius are derived. Two physical parameters are introduced to simplify the analysis, and the explicit relations between the torsional displacement, its derivative and the flexural displacement are derived. With these explicit relations, the two coupled governing characteristic differential equations can be decoupled and reduced to one sixth-order ordinary differential equation with variable coefficients in the out-of-plane flexural displacement. It is shown that if the material and geometric properties of the beam are in arbitrary polynomial forms, then the exact solutions for the out-of-plane vibrations of the beam can be obtained. The derived explicit relations can also be used to reduce the difficulty in experimental measurement. Finally, two limiting cases are considered and the influence of taper ratio, center angle, and arc length on the first two natural frequencies of the beams are illustrated.

The visual perception of arc length on a real surface

1997 ◽  
Author(s):  
J. Farley Norman ◽  
Joseph S. Lappin ◽  
Hideko F. Norman
Keyword(s):  
Visual Perception ◽  
Real Surface ◽  
Arc Length

Dimensions of plane and solid angles and their units in the International System of Units (SI)

2020 ◽  
pp. 26-32
Author(s):  
M. I. Kalinin ◽  
L. K. Isaev ◽  
F. V. Bulygin
Keyword(s):  
Solid Angle ◽  
International System ◽  
International Committee ◽  
Plane Angle ◽  
Arc Length ◽  
Weights And Measures ◽  
International System Of Units ◽  
System Of Units ◽  
In Series ◽  
Solid Angles

The situation that has developed in the International System of Units (SI) as a result of adopting the recommendation of the International Committee of Weights and Measures (CIPM) in 1980, which proposed to consider plane and solid angles as dimensionless derived quantities, is analyzed. It is shown that the basis for such a solution was a misunderstanding of the mathematical formula relating the arc length of a circle with its radius and corresponding central angle, as well as of the expansions of trigonometric functions in series. From the analysis presented in the article, it follows that a plane angle does not depend on any of the SI quantities and should be assigned to the base quantities, and its unit, the radian, should be added to the base SI units. A solid angle, in this case, turns out to be a derived quantity of a plane angle. Its unit, the steradian, is a coherent derived unit equal to the square radian.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

Conserved Quantities for the General Linear Heat Equation

2016 ◽  
pp. 4437-4439
Author(s):  
Adil Jhangeer ◽  
Fahad Al-Mufadi
Keyword(s):  
Evolution Equation ◽  
Heat Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Approach ◽  
General Linear ◽  
Conserved Quantities ◽  
Linear Heat ◽  
The Family ◽  
Partial Lagrangian

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

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