THREE-DIMENSIONAL VIBRATION ANALYSIS OF SOLID CYLINDERS OF POLYGONAL CROSS-SECTION USING THEp-RITZ METHOD

1997 ◽  
Vol 200 (4) ◽  
pp. 505-518 ◽  
Author(s):  
K.M. Liew ◽  
K.C. Hung ◽  
M.K. Lim
Keyword(s):  
Cross Section ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Three Dimensional

Three-Dimensional Vibration Analysis of Twisted Cylinder With Sectorial Cross Section

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
D. Zhou ◽  
S. H. Lo
Keyword(s):  
Cross Section ◽  
Chebyshev Polynomial ◽  
Numerical Solutions ◽  
Twist Angle ◽  
Ritz Method ◽  
Three Dimensional ◽  
Cylindrical Domain ◽  
Linear Elasticity Theory ◽  
Boundary Functions ◽  
Polynomial Series

The three-dimensional (3D) free vibration of twisted cylinders with sectorial cross section or a radial crack through the height of the cylinder is studied by means of the Chebyshev–Ritz method. The analysis is based on the three-dimensional small strain linear elasticity theory. A simple coordinate transformation is applied to map the twisted cylindrical domain into a normal cylindrical domain. The product of a triplicate Chebyshev polynomial series along with properly defined boundary functions is selected as the admissible functions. An eigenvalue matrix equation can be conveniently derived through a minimization process by the Rayleigh–Ritz method. The boundary functions are devised in such a way that the geometric boundary conditions of the cylinder are automatically satisfied. The excellent property of Chebyshev polynomial series ensures robustness and rapid convergence of the numerical computations. The present study provides a full vibration spectrum for thick twisted cylinders with sectorial cross section, which could not be determined by 1D or 2D models. Highly accurate results presented for the first time are systematically produced, which can serve as a benchmark to calibrate other numerical solutions for twisted cylinders with sectorial cross section. The effects of height-to-radius ratio and twist angle on frequency parameters of cylinders with different subtended angles in the sectorial cross section are discussed in detail.

scholarly journals Higher-Order Hierarchical Models for the Free Vibration Analysis of Thin-Walled Beams

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
G. Giunta ◽  
S. Belouettar
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Vibration Analysis ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Approximation Order ◽  
Higher Order ◽  
Geometrical Parameters ◽  
Thin Walled ◽  
Three Dimensional Finite Element

This paper addresses a free vibration analysis of thin-walled isotropic beams via higher-order refined theories. The unknown kinematic variables are approximated along the beam cross section as aN-order polynomial expansion, whereNis a free parameter of the formulation. The governing equations are derived via the dynamic version of the Principle of Virtual Displacements and are written in a unified form in terms of a “fundamental nucleus.” This latter does not depend upon order of expansion of the theory over the cross section. Analyses are carried out through a closed form, Navier-type solution. Simply supported, slender, and short beams are investigated. Besides “classical” modes (such as bending and torsion), several higher modes are investigated. Results are assessed toward three-dimensional finite element solutions. The numerical investigation shows that the proposed Unified Formulation yields accurate results as long as the appropriate approximation order is considered. The accuracy of the solution depends upon the geometrical parameters of the beam.

3D Vibration Analysis of Hermetic Capsules by Using Ritz Method

2017 ◽  
Vol 17 (03) ◽  
pp. 1750040
Author(s):  
Jae-Hoon Kang
Keyword(s):  
Eigenvalue Problem ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Legendre Polynomials ◽  
Present Method ◽  
Ritz Method ◽  
Three Dimensional ◽  
Dynamic Equations ◽  
Vibration Frequencies ◽  
Good Agreement

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies of a hermetic capsule comprising a cylinder closed with hemi-ellipsoidal caps at both ends. Unlike conventional shell theories, which are mathematically 2D, the present method is based upon the 3D dynamic equations of elasticity. Displacement components [Formula: see text], [Formula: see text], and [Formula: see text] in the radial, circumferential, and axial directions, respectively, are taken to be periodic in [Formula: see text] and in time, and the Legendre polynomials in the r and z directions instead of ordinary ones. Potential (strain) and kinetic energies of the hermetic capsule are formulated, and the Ritz method is used to solve the eigenvalue problem, thereby yielding upper bound values of the frequencies. As the degree of the Legendre polynomials is increased, frequencies converge to the exact values. Typical convergence studies are carried out for the first five frequencies. The frequencies from the present 3D method are in good agreement with those obtained from other 3D approach and 2D shell theories proposed by previous researchers.

3-D Free Vibration Analysis of Functionally Graded Thick Circular and Annular Plates on Elastic Foundation

2010 ◽  
Author(s):  
Vahid Tajeddini ◽  
Abdolreza Ohadi ◽  
Mojtaba Sadighi
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Small Strain ◽  
Functionally Graded ◽  
Different Boundary Conditions ◽  
Fg Plates

This paper describes a study of three-dimensional free vibration analysis of thick circular and annular functionally graded (FG) plates resting on Pasternak foundation. The formulation is based on the linear, small strain and exact elasticity theory. Plates with different boundary conditions are considered and the material properties of the FG plate are assumed to vary continuously through the thickness according to power law. The kinematic and the potential energy of the plate-foundation system are formulated and the polynomial-Ritz method is used to solve the eigenvalue problem. Convergence and comparison studies are done to demonstrate the correctness and accuracy of the present method. With respect to geometric parameters, elastic coefficients of foundation and different boundary conditions some new results are reported which maybe used as a benchmark solution for future researches.

scholarly journals Three-dimensional vibration analysis of prisms with isosceles triangular cross-section

2009 ◽  
Vol 80 (6) ◽  
pp. 699-710 ◽  
Author(s):  
Ding Zhou ◽  
Y. K. Cheung ◽  
S. H. Lo ◽  
F. T. K. Au
Keyword(s):  
Cross Section ◽  
Vibration Analysis ◽  
Three Dimensional ◽  
Triangular Cross Section

scholarly journals Three-dimensional vibration analysis of a torus with circular cross section

2002 ◽  
Vol 112 (6) ◽  
pp. 2831-2839 ◽  
Author(s):  
D. Zhou ◽  
F. T. K. Au ◽  
S. H. Lo ◽  
Y. K. Cheung
Keyword(s):  
Cross Section ◽  
Vibration Analysis ◽  
Three Dimensional ◽  
Circular Cross Section
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