An Analytical Solution Procedure to Analyze Material Nonlinear Behavior of Laminated Beams

2007 ◽  
Vol 26 (4) ◽  
pp. 391-404
Author(s):  
Cesim Ataş ◽  
Onur Sayman ◽  
Hasan Çallioğlu
Keyword(s):  
Analytical Solution ◽  
Nonlinear Behavior ◽  
Solution Procedure ◽  
Laminated Beams ◽  
Material Nonlinear

scholarly journals Vibration analysis of hard-coating laminated plate considering strain-dependent characteristic

2018 ◽  
Vol 37 (4) ◽  
pp. 789-800 ◽  
Author(s):  
Wei Sun ◽  
Xiaozhou Liu ◽  
Jixiang Jiang
Keyword(s):  
Composite Plate ◽  
Nonlinear Behavior ◽  
Coating Material ◽  
Hard Coating ◽  
Nonlinear Phenomenon ◽  
Laminated Plate ◽  
Lagrange’S Equation ◽  
Calculation Results ◽  
Material Nonlinear ◽  
Strain Dependent

An analytical modeling method of hard-coating laminated plate under base excitation was studied considering strain-dependent characteristic of coating material (i.e. a kind of material nonlinear behavior). For convenience, the strain-dependent characteristic of hard-coating material was characterized by polynomial, and the material parameters were divided into two parts: linearity and nonlinearity. Hard coating was regarded as a special layer in the analysis and Lagrange’s equation was used to acquire nonlinear equation of motion of the hard-coating laminated plate. Based on Newton–Raphson method, the procedure of solving resonant response and resonant frequency of composite plate was presented. Finally, a T300/QY89l1 laminated plate with NiCoCrAlY + YSZ hard coating was chosen to demonstrate the proposed method, the linear and nonlinear vibrations of the composite plate were solved, and only the linear results were validated by ANSYS software. The results reveal that there is a big difference between the calculation results considering the nonlinearity of coating material and the linear results, which means the laminated plate displays soft nonlinear phenomenon because of depositing coating.

scholarly journals A Full-Fledged Analytical Solution to the Quantum Harmonic Oscillator for Undergraduate Students of Science and Engineering

Physics ◽  
2020 ◽  
Vol 2 (4) ◽  
pp. 541-570
Author(s):  
Arturo Rodríguez-Gómez ◽  
Ana Laura Pérez-Martínez
Keyword(s):  
Analytical Solution ◽  
Harmonic Oscillator ◽  
Undergraduate Students ◽  
Solution Procedure ◽  
Quantum Harmonic Oscillator ◽  
Web Based ◽  
Chemistry Students ◽  
Physics Students ◽  
Step Procedure ◽  
And Mathematics

The quantum harmonic oscillator is a fundamental piece of physics. In this paper, we present a self-contained full-fledged analytical solution to the quantum harmonic oscillator. To this end, we use an eight-step procedure that only uses standard mathematical tools available in natural science, technology, engineering and mathematics disciplines. This solution is accessible not only for physics students but also for undergraduate engineering and chemistry students. We provide interactive web-based graphs for the reader to observe the shape of the wave functions for an electron and a proton when both are subject to the same potential. Each of the eight steps in our solution procedure is treated as a separate problem in order to allow the reader to quickly consult any step without the need to review the entire article.

On the dynamics of bistable micro/nano resonators: Analytical solution and nonlinear behavior

2015 ◽  
Vol 20 (3) ◽  
pp. 1078-1089 ◽  
Author(s):  
Farid Tajaddodianfar ◽  
Hossein Nejat Pishkenari ◽  
Mohammad Reza Hairi Yazdi ◽  
Ehsan Maani Miandoab
Keyword(s):  
Analytical Solution ◽  
Nonlinear Behavior

An Analytical Solution of the Generalized Phase-Lagging Equation for Ultra-Fast Heat Transfer in One-Dimensional Semi-Infinite Domain

2012 ◽  
Author(s):  
Vladimir Kulish ◽  
Kirill V. Poletkin
Keyword(s):  
Analytical Solution ◽  
Domain Boundary ◽  
Integral Form ◽  
Solution Procedure ◽  
Generalized Derivatives ◽  
Infinite Domain ◽  
One Dimensional ◽  
Confluent Hypergeometric Functions ◽  
The Laplace Transform ◽  
Derivatives Of

The paper presents an integral solution of the generalized one-dimensional phase-lagging heat equation with the convective term. The solution of the problem has been achieved by the use of a novel technique that involves generalized derivatives (in particular, derivatives of non-integer orders). Confluent hypergeometric functions, known as Whittaker’s functions, appear in the course of the solution procedure, upon applying the Laplace transform to the original transport equation. The analytical solution of the problem is written in the integral form and provides a relationship between the local values of the temperature and heat flux. The solution is valid everywhere within the domain, including the domain boundary.

Analytical Solution for Three-Dimensional, Unsteady Heat Conduction in a Multilayer Sphere

2016 ◽  
Vol 138 (10) ◽  
Author(s):  
Suneet Singh ◽  
Prashant K. Jain ◽  
Rizwan-uddin
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Coordinate System ◽  
Three Dimensional ◽  
Solution Procedure ◽  
Polar Coordinate System ◽  
Legendre Functions ◽  
Spherical Coordinates ◽  
Azimuthal Direction ◽  
Spherical Coordinate

An analytical solution has been obtained for the transient problem of three-dimensional multilayer heat conduction in a sphere with layers in the radial direction. The solution procedure can be applied to a hollow sphere or a solid sphere composed of several layers of various materials. In general, the separation of variables applied to 3D spherical coordinates has unique characteristics due to the presence of associated Legendre functions as the eigenfunctions. Moreover, an eigenvalue problem in the azimuthal direction also requires solution; again, its properties are unique owing to periodicity in the azimuthal direction. Therefore, extending existing solutions in 2D spherical coordinates to 3D spherical coordinates is not straightforward. In a spherical coordinate system, one can solve a 3D transient multilayer heat conduction problem without the presence of imaginary eigenvalues. A 2D cylindrical polar coordinate system is the only other case in which such multidimensional problems can be solved without the use of imaginary eigenvalues. The absence of imaginary eigenvalues renders the solution methodology significantly more useful for practical applications. The methodology described can be used for all the three types of boundary conditions in the outer and inner surfaces of the sphere. The solution procedure is demonstrated on an illustrative problem for which results are obtained.

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