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.

Vibration Analysis of Complete Conical Shells with Variable Thickness

2014 ◽  
Vol 14 (04) ◽  
pp. 1450001 ◽  
Author(s):  
Jae-Hoon Kang
Keyword(s):  
Vibration Analysis ◽  
Present Method ◽  
Shell Theory ◽  
Ritz Method ◽  
Three Dimensional ◽  
Dynamic Equations ◽  
Algebraic Polynomials ◽  
Bottom Edge ◽  
Conical Shells ◽  
Varying Thickness

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies of complete (not truncated) conical shells with linearly varying thickness. The complete conical shells free or clamped at the bottom edge with a free vertex are investigated. Unlike conventional shell theories, which are mathematically 2D, the present method is based upon the 3D dynamic equations of elasticity. Displacement components ur, uθ and uz in the radial, circumferential and axial directions, respectively, are taken to be periodic in θ and in time, and expressed by algebraic polynomials in the r- and z-directions. Potential (strain) and kinetic energies of the complete conical shell are formulated. The Ritz method is used to solve the eigenvalue problem, yielding the upper bound values of the frequencies by minimization. As the degree of the polynomials is increased, frequencies converge to the exact values, with four-digit exactitude demonstrated for the first five frequencies. The frequencies from the present 3D method are compared with those from other 3D approaches and 2D shell theory 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.

Free Vibration of Curvilinearly Stiffened Shallow Shells

2015 ◽  
Vol 137 (3) ◽  
Author(s):  
Peng Shi ◽  
Rakesh K. Kapania ◽  
C. Y. Dong
Keyword(s):  
Free Vibration ◽  
Shell Theory ◽  
Ritz Method ◽  
Three Dimensional ◽  
Beam Theory ◽  
Shell Element ◽  
Shallow Shells ◽  
First Order ◽  
Stiffened Shells ◽  
Good Agreement

The free vibration of curvilinearly stiffened shallow shells is investigated by the Ritz method. Based on the first-order shear deformation shell theory and three-dimensional (3D) curved beam theory, the strain and kinetic energies of the stiffened shells are introduced. The stiffener can be placed anywhere within the shell, without the need for having the stiffener and shell element nodes coincide. Numerical results with different geometrical shells and boundary conditions and different stiffener locations and curvatures are analyzed to verify the feasibility of the presented Ritz method for solving the problems. The results show good agreement with those using other methods, e.g., using a converged set of results obtained by Nastran.

scholarly journals An Improved Fourier Series Method for the Free Vibration Analysis of the Three-Dimensional Coupled Beams

2016 ◽  
Vol 2016 ◽  
pp. 1-18 ◽  
Author(s):  
Runze Zhang ◽  
Yipeng Cao ◽  
Wenping Zhang ◽  
Hongbo Li ◽  
Xiangmei Li
Keyword(s):  
Fourier Series ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Present Method ◽  
Vibration Suppression ◽  
Three Dimensional ◽  
Fourier Method ◽  
Free Vibration Analysis ◽  
Fourier Cosine ◽  
Coupled Beams

This paper presents a free vibration analysis of three-dimensional coupled beams with arbitrary coupling angle using an improved Fourier method. The displacement and rotation of the coupled beams are represented by the improved Fourier series which consisted of Fourier cosine series and closed-form auxiliary functions. The coupling and boundary conditions are accomplished by setting coupling and boundary springs and assigning corresponding stiffness values to the springs. Modal parameters are determined through the application of Rayleigh-Ritz procedure to the system energy formulation. The accuracy and convergence of the present method are demonstrated by finite element method (FEM) result. Investigation on vibration of the propulsion shafting structure shows the extensive applicability of present method. The studies on the vibration suppression devices are also reported.

Three-Dimensional Vibration Analysis of Solid and Hollow Hemispheres Having Varying Thicknesses With and Without Axial Conical Holes

2004 ◽  
Vol 10 (2) ◽  
pp. 199-214 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W Leissa
Keyword(s):  
Vibration Analysis ◽  
Present Method ◽  
Ritz Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Algebraic Polynomials ◽  
Shells Of Revolution ◽  
Time Periodic ◽  
Hemispherical Shells

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of solid and hollow hemispherical shells of revolution of arbitrary wall thickness having arbitrary constraints on their boundaries. Unlike conventional shell theories, which are mathematically two-dimensional, the present method is based upon the 3D dynamic equations of elasticity. Displacement components u \#966;, u z, and u \#952; in the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in \#952;, and algebraic polynomials in the \#966;-direction and zdirection. Potential (strain) and kinetic energies of the hemispherical shells are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Novel numerical results are presented for solid and hollow hemispheres with linear thickness variation. The effect on frequencies of a small axial conical hole is also discussed. Comparisons are made for the frequencies of completely free, thick hemispherical shells with uniform thickness from the present 3D Ritz solutions and other 3D finite element ones.

scholarly journals Three-Dimensional Free Vibration Analysis of Gears with Variable Thickness Using the Chebyshev–Ritz Method

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Yi Li ◽  
Ming Lv ◽  
Shi-ying Wang ◽  
Hui-bin Qin ◽  
Jun-fan Fu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Free Vibration ◽  
Variable Thickness ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Admissible Function ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Element Method

To reflect vibration more comprehensively and to satisfy the machining demand for high-order frequencies, we presented a three-dimensional free vibration analysis of gears with variable thickness using the Chebyshev–Ritz method based on three-dimensional elasticity theory. We derived the eigenvalue equations. We divided the gear model into three annular parts along the locations of the step variations, and the admissible function was a Ritz series that consisted of a Chebyshev polynomial multiplying boundary function. The convergence study demonstrated the high accuracy of the present method. We used a hammering method for a modal experiment to test two annular plates and one gear’s eigenfrequencies in a completely free condition. We also applied the finite element method to solve the eigenfrequencies. Through a comparative analysis of the frequencies obtained by these three methods, we found that the results achieved by the Chebyshev–Ritz method were close to those obtained from the experiment and finite element method. The relative errors of four sets of data were greater than 4%, and the errors of the other 48 sets were less than 4%. Thus, it was feasible to use the Chebyshev–Ritz method to solve the eigenfrequencies of gears with variable thickness.

Vibration Analysis of Shallow Spherical Domes with Non-Uniform Thickness

2017 ◽  
Vol 17 (02) ◽  
pp. 1750016 ◽  
Author(s):  
Jae-Hoon Kang
Keyword(s):  
Vibration Analysis ◽  
Present Method ◽  
Shell Theory ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Three Dimensional ◽  
Algebraic Polynomials ◽  
Uniform Thickness ◽  
3D Finite Element ◽  
3D Finite Element Method

A three-dimensional (3D) method of analysis is presented for determining the natural frequencies of shallow spherical domes with non-uniform thickness. Unlike conventional shell theories, which are mathematically two dimensional (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 meridional, circumferential, and normal directions, respectively, are taken to be periodic in [Formula: see text] and in time, and algebraic polynomials in the [Formula: see text] and z directions. Potential (strain) and kinetic energies of the shallow spherical domes with non-uniform thickness are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Natural frequencies are presented for different boundary conditions. The frequencies from the present 3D method are compared with those from a 2D exact method, a 2D thick shell theory, and a 3D finite element method by previous researchers.

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