Three-Dimensional Vibrations of Twisted Cantilevered Parallelepipeds

1986 ◽  
Vol 53 (3) ◽  
pp. 614-618 ◽  
Author(s):  
A. Leissa ◽  
K. I. Jacob
Keyword(s):  
Finite Element ◽  
Twist Angle ◽  
Ritz Method ◽  
Three Dimensional ◽  
Algebraic Polynomials ◽  
Thick Plates ◽  
Cantilevered Beams ◽  
Element Theory

A large number of references dealing with the vibrations of twisted, cantilevered beams and plates exist in the literature. These works show considerable disagreement concerning the effect of twist angle upon frequencies. The present work is the first three-dimensional study of the problem. Displacement components are assumed in the form of algebraic polynomials which satisfy the fixed face conditions exactly, and which are mathematically complete. The Ritz method is then applied. Accurate frequencies are calculated for twisted thick plates and are compared with ones obtained recently by others using beam, shell, and finite element theory.

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.

Three-Dimensional Vibration Analysis of Thick, Complete Conical Shells

2004 ◽  
Vol 71 (4) ◽  
pp. 502-507 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W. Leissa
Keyword(s):  
Present Method ◽  
Shell Theory ◽  
Ritz Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Algebraic Polynomials ◽  
Shells Of Revolution ◽  
Conical Shells ◽  
Thin Shell Theory ◽  
Time Periodic

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution. Unlike conventional shell theories, which are mathematically two-dimensional (2D), the present method is based upon the 3D dynamic equations of elasticity. Displacement components ur,uz, and uθ in the radial, axial, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z-directions. Potential (strain) and kinetic energies of the conical shells are formulated, 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 of the conical shells. Novel numerical results are presented for thick, complete conical shells of revolution based upon the 3D theory. Comparisons are also made between the frequencies from the present 3D Ritz method and a 2D thin shell theory.

Three-dimensional vibration of rotating functionally graded beams

2017 ◽  
Vol 24 (15) ◽  
pp. 3292-3306 ◽  
Author(s):  
Jianshi Fang ◽  
Ding Zhou ◽  
Yun Dong
Keyword(s):  
Chebyshev Polynomials ◽  
Ritz Method ◽  
Three Dimensional ◽  
Functionally Graded ◽  
Gradient Index ◽  
Coupling Dynamic ◽  
Cantilevered Beams ◽  
Stiffening Effect ◽  
Large Deformation Problems ◽  
Space Method

The three-dimensional free vibration and time response of rotating functionally graded (FG) cantilevered beams are studied. Material properties of functionally graded beams are assumed to change gradually through both the width and the thickness in power-law form. The second-kind Lagrange’s equations are used in conjunction with the Ritz method to derive the comprehensive coupling dynamic equations for the axial, chordwise, and flapwise motions. The trial functions of deformations are taken as the products of the Chebyshev polynomials and the corresponding boundary functions. Nonlinear coupling deformations are considered to capture the dynamic stiffening effect due to the rotating motion. The influences of the material gradient index and rotational speed on modal characteristics are investigated by the state space method. The eigenvalue loci veering phenomena with modal conversions are exhibited. The time responses indicate that the deformations of rotating functionally graded beams are greatly affected by the material gradient index. It is shown that for large deformation problems, using Chebyshev polynomials is more efficient in computing precision and robustness than using other polynomials.

3D Vibration Analysis of Combined Shells of Revolution

2019 ◽  
Vol 19 (02) ◽  
pp. 1950005 ◽  
Author(s):  
Jae-Hoon Kang
Keyword(s):  
Legendre Polynomials ◽  
Thin Shell ◽  
Ritz Method ◽  
Three Dimensional ◽  
Surface Strain ◽  
Mode Shapes ◽  
Algebraic Polynomials ◽  
Shells Of Revolution ◽  
Thick Shells ◽  
Admissible Functions

A three-dimensional (3D) method of analysis is presented for determining the natural frequencies and the mode shapes of combined hemispherical–cylindrical shells of revolution with and without a top opening by the Ritz method. Instead of mathematically two-dimensional (2D) conventional thin shell theories or higher-order thick shell theories, the present method is based upon the 3D dynamic equations of elasticity. Mathematically, minimal or orthonormal Legendre polynomials are used as admissible functions in place of ordinary simple algebraic polynomials which are usually applied in the Ritz method. The analysis is based upon the circular cylindrical coordinates instead of the shell coordinates which are normal and tangent to the shell mid-surface. Strain and kinetic energies of the combined shell of revolution with and without a top opening 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 Legendre polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Numerical results are presented for the combined shells of revolution with or without a top opening, which are completely free and fixed at the bottom of the combined shells. The frequencies from the present 3D Ritz method are compared with those from 2D thin shell theories by previous researchers. The present analysis is applicable to very thick shells as well as very thin shells.

FREE VIBRATIONS OF COMBINED HEMISPHERICAL–CYLINDRICAL SHELLS OF REVOLUTION WITH A TOP OPENING

2013 ◽  
Vol 14 (01) ◽  
pp. 1350023 ◽  
Author(s):  
JAE-HOON KANG
Keyword(s):  
Present Method ◽  
Thin Shell ◽  
Natural Frequencies ◽  
Cylindrical Shells ◽  
Ritz Method ◽  
Three Dimensional ◽  
Free Vibrations ◽  
Two Dimensional ◽  
Algebraic Polynomials ◽  
Shells Of Revolution

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies of joined hemispherical–cylindrical shells of revolution with a top opening. Unlike conventional shell theories, which are mathematically two-dimensional (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 algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the joined shells 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 2D thin shell theories.

scholarly journals Three-Dimensional Elasticity-Based Solutions for Natural Vibrations of Low Aspect Ratio Compressor Blades

1993 ◽  
Author(s):  
Oliver G. McGee
Keyword(s):  
Aspect Ratio ◽  
Twist Angle ◽  
Ritz Method ◽  
Three Dimensional ◽  
Natural Vibrations ◽  
Compressor Blades ◽  
Low Aspect Ratio ◽  
Plate And Shell ◽  
Convergence Study ◽  
Three Dimensional Elasticity

This paper offers three-dimensional (3-D) vibration frequency solutions for low aspect ratio compressor blades. The Ritz method is used to minimize the 3-D elasticity-based dynamical energies with displacements approximated by mathematically complete polynomials satisfying the clamped boundary conditions exactly. The accuracy of the method is established by a convergence study explicitly showing the influence of solution determinant size. Several tables are presented which show the variation of natural frequencies with twist angle in the presence of skewness of low aspect ratio compressor blades. Results obtained using the present Ritz method are used to elucidate those frequency solutions which are inaccessible using beam, plate and shell theories, since kinematic constraints associated with these theories are eliminated in the present 3-D approach.

Mechanical properties of the three-dimensional compression-twist cellular structure

2019 ◽  
Vol 39 (7-8) ◽  
pp. 260-277
Author(s):  
Wei Zhang ◽  
Xiaoyu Bai ◽  
Bowen Hou ◽  
Yadong Sun ◽  
Xiao Han
Keyword(s):  
Mechanical Properties ◽  
Finite Element Analysis ◽  
Finite Element ◽  
Cellular Structure ◽  
Cell Structure ◽  
Twist Angle ◽  
Three Dimensional ◽  
Thickness Ratio ◽  
Cell Length ◽  
Element Analysis

The cellular structure can exhibit many special mechanical behaviors due to its variable cell shape. A three-dimensional compression-twist cell structure based on the rotation mechanism of two-dimensional chiral cell structure is developed, which has twist deformation under axial compression. The shape of three-dimensional compression-twist cell structure is determined through cell angle, cell length, and thickness ratio. Analytical expressions of effective Young’s modulus, Poisson’s ratio, and twist angle are derived by using beam theory, which have a good agreement with the finite element calculations and the deformation process of the cell is discussed. To work on the effect of geometric parameters of cell on the mechanical properties, a finite element analysis model of compression-twist cell structure is carried out, which shows the process of elastic and plastic deformation under compression. Effects of cell angle, cell length, and thickness ratio are fully discussed, which indicate that cell angle has obvious nonlinear effect on relative twist angle and could stiffen it. Finally, a compression-twist cell structure sample is made through three-dimensional printing, and an in-plane compressive experiment is carried out to prove analytical and finite element analysis results.

Vibration Characteristics of Low Aspect Ratio Compressor Blades

1971 ◽  
Vol 93 (1) ◽  
pp. 103-112 ◽  
Author(s):  
Ralph Petricone ◽  
Fernando Sisto
Keyword(s):  
Aspect Ratio ◽  
Contact Point ◽  
Twist Angle ◽  
Ritz Method ◽  
Three Dimensional ◽  
Beam Theory ◽  
Vibration Characteristics ◽  
Mode Shapes ◽  
Compressor Blades ◽  
Low Aspect Ratio

This paper presents the results of a study of the vibration characteristics of low aspect ratio compressor blades. The treatment is based on thin shell theory and the Rayleigh-Ritz method is used to obtain the eigenvectors and eigenvalues. The object is to elucidate those characteristics which are inaccessible using beam theory. Results are presented which show the variation of the natural frequencies and mode shapes with angle of twist, aspect ratio, and angle of inclination of the base of the blade. A three-dimensional plot of the bending mode frequencies versus aspect ratio and twist angle is presented. Although the surfaces describing the variation of frequencies for specific modes do not intersect, there is a point of contact. This contact point is significant in the transition of mode shapes along the frequency surfaces. It is demonstrated that the “stiff-direction” or “in-plane” vibration of the untwisted plate evolves into coupled bending modes as the twist angle increases from zero and that the character of these modes changes in the vicinity of the contact point.

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.

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