Numerical methods for Engineers and Scientists

For a free vibration problem of a thermoelastic hollow sphere into the context of the generalized thermoelasticity theory with one relaxation time, exact analytic solutions are obtained with the use of eigenvalue approach. Both the inner and outer curved surfaces of the sphere are considered stress-free and isothermal surfaces. The dispersion relations for the existence of various types of possible modes of vibrations in the considered hollow sphere are derived. The numerical results have been presented graphically in respect of natural frequencies, thermoelastic damping, and frequency shift.

Analytical Solution for a Free Vibration of a Thermoelastic Hollow Sphere

We come now to the connecting link between linear transformations and matrices. Our approach will be to consider first the case of a linear transformation 

$$ {\text{T}}:{R^3} \to {R^3} $$

in some detail and then abstract the salient features to the general case. Let us therefore suppose given a fixed linear transformation $ {\text{T}}:{R^3} \to {R^3} $. As usual we will denote by E1, E2, E3 the standard basis vectors (1,0,0), (0, 1, 0), (0, 0, 1) in ℝ3. Let

$${\text{T}}({{\text{E}}_1}){\text{ = }}({a_{11}},{a_{21}},{a_{31}}){\text{T}}({{\text{E}}_2}){\text{ = }}({a_{12}},{a_{22}},{a_{32}}){\text{T}}({{\text{E}}_3}){\text{ = }}({a_{13}},{a_{23}},{a_{33}})$$

.

Matrices and linear transformations

The free vibrations of an axisymmetric functionally graded, transversely isotropic, thermoelastic hollow cylinder have been modeled and analyzed. The material has been assumed to be graded according to a simple power law in the radial coordinate. The Laplace transform method has been used to solve the problem. The complex Laplace transform parameter has directly been used to find the natural frequencies of free vibrations without performing inversion of the transforms. The frequency equations of free vibrations in a hollow cylinder have been solved by using the software Maple. The natural frequencies of the first ten modes for different values of the grading index have been computed numerically for zinc material. The numerical results for radial stress, circumferential stress, temperature change, frequency shift, and thermoelastic damping (inverse quality factor) have been presented graphically. The closed form solutions obtained here are interesting and allow further parametric studies of functionally graded structures. The inhomogeneity parameter is useful in design and it can be tailored for specific applications as a controller.

Analysis of free vibrations in axisymmetric functionally graded thermoelastic cylinders

The vibration of a functionally graded axisymmetric nonlocal thermoelastic hollow sphere with dual-phase-lag effect is addressed in this paper. Surfaces of the sphere are assumed to be thermally insulated or isothermal and stress free. According to a simple power law, the material is assumed to be graded in the radial direction. The linear theory of modified thermoelasticity with a dual phase lag based on Eringen’s nonlocal elasticity is employed to model this problem. The Matrix Frobenius method of continued power series is introduced to derive the analytical solutions. The phase velocity relations for the existence of various modes of vibrations in the designed hollow sphere are derived in compact forms. In order to explore the attributes of vibrations, the fixed-point numerical iteration technique is used to solve the secular equations. The numerical computations for the material crust in respect of the natural frequencies, thermoelastic damping and the frequency shifting are presented graphically using MATLAB software tools.

Vibration analysis of functionally graded thermoelastic nonlocal sphere with dual-phase-lag effect

Purpose – The purpose of this paper is to analyze the free vibrations in a stress free and thermally insulated (or isothermal), homogeneous, transversely isotropic, solid cylinder based on three‐dimensional coupled thermoelasticity, which is initially undeformed and kept at uniform temperature.Design/methodology/approach – The displacement potential functions have been introduced in the equations of motion and heat conduction in order to decouple the purely shear and longitudinal motions. The system of governing partial differential equations is reduced to four second‐order coupled ordinary differential equations in radial coordinate by using the method of separation of variables. The matrix Frobenius method of extended power series is employed to obtain the solution of coupled ordinary differential equations along the radial coordinate. The convergence analysis of matrix Frobenius method has been successfully carried out.Findings – The purely transverse mode is found to be independent of the rest of the ...

Modeling of thermoelastic damping and frequency shift of vibrations in a transversely isotropic solid cylinder

The aim of the paper is to study three-dimensional free vibrations in a homogeneous transversely isotropic, thermoelastic hollow cylinder, which is initially undeformed and at uniform temperature. The surfaces of the cylinder are subjected to stress free and thermally insulated or isothermal boundary conditions. The displacement potential functions have been introduced in the equations of motion and heat conduction to decouple purely shear and longitudinal motions. The purely transverse wave is not affected by thermal field. By using the method of separation of variables, the system of governing partial differential equations is reduced to four second-order coupled ordinary differential equations in radial coordinate. The Matrix Frobenius method of the extended power series is employed to obtain the solution in radial direction. To illustrate the analytic results, the numerical solution of various relations and equations has been carried out to compute the lowest frequency, frequency shift and damping fac...

Vibration Analysis of a Transversely Isotropic Hollow Cylinder by Using the Matrix Frobenius Method

The three-dimensional free vibrations in a simply supported homogeneous isotropic, thermally conducting, circular cylinder of finite length with an eccentrically located inner circular cavity have been studied. The surfaces of the cylinder are subjected to stress free and thermally insulated or isothermal boundary conditions. The three-dimensional linear theory of coupled thermoelasticity has been employed to model the problem. The displacement potential functions have been introduced to decouple purely shear and longitudinal motions. The purely transverse wave has been found to remain unaffected due to thermal field. The translation addition theorem for cylindrical wave functions along with orthogonal series expansions has been used to develop the exact solution. To illustrate the analytical results, the numerical solution of some relations and equations have been obtained to compute lowest frequency and dissipation factor versus eccentricities, for selected length to radius ratio and radius ratio of hol...

Free Vibration Analysis of an Eccentric Thermoelastic Hollow Cylinder

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

H. Singh

Author Tools

Chat about Author

Papers

Modeling of thermoelastic damping and frequency shift of vibrations in a transversely isotropic solid cylinder

Vibration Analysis of a Transversely Isotropic Hollow Cylinder by Using the Matrix Frobenius Method

Free Vibration Analysis of an Eccentric Thermoelastic Hollow Cylinder