Rotating Blade Vibration Analysis Using Shells

1981 ◽  
Author(s):  
A. W. Leissa ◽  
J. K. Lee ◽  
A. J. Wang
Keyword(s):  
Shallow Shell ◽  
Shell Theory ◽  
Body Force ◽  
Ritz Method ◽  
Mode Shapes ◽  
Finite Element Analyses ◽  
Blade Vibration ◽  
Disk Radius ◽  
Rotating Blade ◽  
Turbomachinery Blades

Shallow shell theory and the Ritz method are employed to determine the frequencies and mode shapes of turbomachinery blades having both camber and twist, rotating with non-zero angles of attack. Frequencies obtained for different degrees of shallowness and thickness are compared with results available in the literature, obtained from finite element analyses of nonrotating blades. Frequencies are also determined for a rotating blade, showing the effects of changing the (1) angular velocity of rotation (2) disk radius and (3) angle of attack, as well as the significance of the most important body force terms.

Rotating Blade Vibration Analysis Using Shells

1982 ◽  
Vol 104 (2) ◽  
pp. 296-302 ◽  
Author(s):  
A. W. Leissa ◽  
J. K. Lee ◽  
A. J. Wang
Keyword(s):  
Shallow Shell ◽  
Shell Theory ◽  
Body Force ◽  
Ritz Method ◽  
Mode Shapes ◽  
Finite Element Analyses ◽  
Blade Vibration ◽  
Disk Radius ◽  
Rotating Blade ◽  
Turbomachinery Blades

Shallow shell theory and Ritz method are employed to determine the frequencies and mode shapes of turbomachinery blades having both camber and twist, rotating with non-zero angles of attack. Frequencies obtained for different degrees of shallowness and thickness are compared with results available in the literature, obtained from finite element analyses of nonrotating blades. Frequencies are also determined for a rotating blade, showing the effects of changing the (1) angular velocity of rotation (2) disk radius and (3) angle of attack, as well as the significance of the most important body force terms.

Vibrations of Blades With Variable Thickness and Curvature by Shell Theory

1984 ◽  
Vol 106 (1) ◽  
pp. 11-16 ◽  
Author(s):  
J. K. Lee ◽  
A. W. Leissa ◽  
A. J. Wang
Keyword(s):  
Experimental Data ◽  
Cylindrical Shell ◽  
Flat Plate ◽  
Variable Thickness ◽  
Shallow Shell ◽  
Shell Theory ◽  
Ritz Method ◽  
Mode Shapes ◽  
Uniform Thickness ◽  
Turbomachinery Blades

A procedure for analyzing the vibrations of rotating turbomachinery blades has been previously developed. This procedure is based upon shallow shell theory, and utilizes the Ritz method to determine frequencies and mode shapes. However, it has been limited heretofore to blades of uniform thickness, uniform curvature, and/or twist and rectangular planform. The present work shows how the procedure may be generalized to eliminate the aforementioned restrictions. Nonrectangular planforms are dealt with by a suitable coordinate transformation. This, as well as variable thickness, curvature and twist, require using numerical integration. The procedure is demonstrated on four examples of cantilevered blades for which theoretical and experimental data have been previously published: (1) flat plate with spanwise taper, (2) flat plate with chordwise taper, (3) twisted plate with chordwise taper, and (4) cylindrical shell with chordwise taper.

Vibrations of Twisted Rotating Blades

1984 ◽  
Vol 106 (2) ◽  
pp. 251-257 ◽  
Author(s):  
A. W. Leissa ◽  
J. K. Lee ◽  
A. J. Wang
Keyword(s):  
Shallow Shell ◽  
Shell Theory ◽  
Analytical Procedure ◽  
Ritz Method ◽  
Two Dimensional ◽  
Algebraic Polynomials ◽  
Shape Information ◽  
Rotating Blades ◽  
Turbomachinery Blades ◽  
Twisted Blade

The literature dealing with vibrations of turbomachinery blades is voluminous, but the vast majority of it treats the blades as beams. In a previous paper a two-dimensional analytical procedure was developed and demonstrated on simple models of blades having camber. The procedure utilizes shallow shell theory along with the classical Ritz method for solving the vibration problem. Displacement functions are taken as algebraic polynomials. In the present paper the method is demonstrated on blade models having camber. Comparisons are first made with results in the literature for nonrotating twisted plates and various disagreements between results are pointed out. A method for depicting mode shape information is demonstrated, permitting one to examine all three components of displacement. Finally, the analytical procedure is demonstrated on rotating twisted blade modes, both without and with camber.

Comparison of Beam and Shell Theories for the Vibrations of Thin Turbomachinery Blades

1983 ◽  
Vol 105 (2) ◽  
pp. 383-392 ◽  
Author(s):  
A. W. Leissa ◽  
M. S. Ewing
Keyword(s):  
Shallow Shell ◽  
Shell Theory ◽  
Free Vibrations ◽  
Two Dimensional ◽  
Analysis Method ◽  
One Dimensional ◽  
Low Aspect Ratio ◽  
Turbomachinery Blades ◽  
Dimensional Analysis Method ◽  
Beam Theories

A great deal of published literature exists which analyzes the free vibrations of turbomachinery blades by means of one-dimensional beam theories. Recently, a more accurate, two-dimensional analysis method has been developed based upon shallow shell theory. The present paper summarizes the two types of theories and makes quantitative comparisons of frequencies obtained by them. Numerical results are presented for cambered and/or twisted blades of uniform thickness. Significant differences between the theories are found to occur, especially for low aspect ratio blades. The causes of these differences are discussed.

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.

scholarly journals Applicability and Limitations of Simplified Elastic Shell Theories for Vibration Modelling of Double-Walled Carbon Nanotubes

2021 ◽  
Vol 7 (3) ◽  
pp. 61
Author(s):  
Matteo Strozzi ◽  
Oleg V. Gendelman ◽  
Isaac E. Elishakoff ◽  
Francesco Pellicano
Keyword(s):  
Carbon Nanotubes ◽  
Shell Theory ◽  
Cylindrical Shells ◽  
Ritz Method ◽  
Elastic Shell ◽  
Mode Shapes ◽  
Simplified Models ◽  
Circular Cylindrical Shells ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

The applicability and limitations of simplified models of thin elastic circular cylindrical shells for linear vibrations of double-walled carbon nanotubes (DWCNTs) are considered. The simplified models, which are based on the assumptions of membrane and moment approximate thin-shell theories, are compared with the extended Sanders–Koiter shell theory. Actual discrete DWCNTs are modelled by means of couples of concentric equivalent continuous thin, circular cylindrical shells. Van der Waals interaction forces between the layers are taken into account by adopting He’s model. Simply supported and free–free boundary conditions are applied. The Rayleigh–Ritz method is considered to obtain approximate natural frequencies and mode shapes. Different aspect and thickness ratios, and numbers of waves along longitudinal and circumferential directions, are analysed. In the cases of axisymmetric and beam-like modes, it is proven that membrane shell theory, differently from moment shell theory, provides results with excellent agreement with the extended Sanders–Koiter shell theory. On the other hand, in the case of shell-like modes, it is found that both membrane and moment shell theories provide results reporting acceptable agreement with the extended Sanders–Koiter shell theory only for very limited ranges of geometries and wavenumbers. Conversely, for shell-like modes it is found that a newly developed, simplified shell model, based on the combination of membrane and semi-moment theories, provides results in satisfactory agreement with the extended Sanders–Koiter shell theory in all ranges.

Comparisons of Paraboloidal Shells and Sinusoidal-Shaped Shells in Natural Frequencies

2019 ◽  
Vol 24 (3) ◽  
pp. 451-457
Author(s):  
Yeong-Bin Yang ◽  
Jae-Hoon Kang
Keyword(s):  
Thin Shell ◽  
Shell Theory ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Thin Shells ◽  
Thick Shell ◽  
Cylindrical Coordinates ◽  
Thin Shell Theory

Natural frequencies and mode shapes are obtained for a sinusoidal-shaped shell of revolution by using the Ritz method from a three-dimensional (3-D) analysis instead of a mathematically two-dimensional (2-D) thin shell theory or high order thick shell theory. The present analysis uses circular cylindrical coordinates instead of 3-D shell coordinates, which have been used in traditional shell analyses. Convergence studies can analyze the first five frequencies to four-digit exactitude. Results are given for a variety of shallow and deep sinusoidal-shaped shells with different boundary conditions. The sinusoidal-shaped shells are very similar to paraboloidal shells in shape. The frequencies of the sinusoidal-shaped shells from the present 3-D method are compared with those from 2-D thin shell theories for paraboloidal shells. The present 3-D method is applicable to very thick as well as thin shells.

Vibration and Buckling Analysis of a Rotating Annular Plate Subjected to a Compressive Centrifugal Body Force

2018 ◽  
Vol 18 (07) ◽  
pp. 1850097
Author(s):  
Y. B. Yang ◽  
J. H. Kang
Keyword(s):  
Body Force ◽  
Annular Plate ◽  
Ritz Method ◽  
Angular Speed ◽  
Plane Elasticity ◽  
Outer Edge ◽  
Outer Radius ◽  
Mode Shapes ◽  
Rotating Annular Plate ◽  
Significant Figures

The free vibration and buckling of a rotating annular plate with constant angular speed free at the inner edge and fixed at the outer edge subjected to a compressive centrifugal body force are analyzed using the Ritz method. Exact stress components and radial displacement of the rotating annular plate are obtained via the plane elasticity. Convergence studies in the frequencies and the critical buckling angular speed are made up to four significant figures. The natural frequencies and the corresponding mode shapes and the critical buckling angular speeds are presented for the rotating annular plates with various angular speeds and ratios of the inner radius to the outer radius.

Comparisons of Paraboloidal Shells and Sinusoidal-Shaped Shells in Natural Frequencies

2019 ◽  
Vol 24 (2) ◽  
pp. 451-457
Author(s):  
Yeong-Bin Yang ◽  
Jae-Hoon Kang
Keyword(s):  
Thin Shell ◽  
Shell Theory ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Thin Shells ◽  
Thick Shell ◽  
Cylindrical Coordinates ◽  
Thin Shell Theory

Natural frequencies and mode shapes are obtained for a sinusoidal-shaped shell of revolution by using the Ritz method from a three-dimensional (3-D) analysis instead of a mathematically two-dimensional (2-D) thin shell theory or high order thick shell theory. The present analysis uses circular cylindrical coordinates instead of 3-D shell coordinates, which have been used in traditional shell analyses. Convergence studies can analyze the first five frequencies to four-digit exactitude. Results are given for a variety of shallow and deep sinusoidal-shaped shells with different boundary conditions. The sinusoidal-shaped shells are very similar to paraboloidal shells in shape. The frequencies of the sinusoidal-shaped shells from the present 3-D method are compared with those from 2-D thin shell theories for paraboloidal shells. The present 3-D method is applicable to very thick as well as thin shells.

scholarly journals Comparison of Beam and Shell Theories for the Vibrations of Thin Turbomachinery Blades

1982 ◽  
Author(s):  
A. W. Leissa ◽  
M. S. Ewing
Keyword(s):  
Shallow Shell ◽  
Shell Theory ◽  
Free Vibrations ◽  
Two Dimensional ◽  
Analysis Method ◽  
One Dimensional ◽  
Low Aspect Ratio ◽  
Turbomachinery Blades ◽  
Dimensional Analysis Method ◽  
Beam Theories

A great deal of published literature exists which analyzes the free vibrations of turbomachinery blades by means of one-dimensional beam theories. Recently, a more accurate, two-dimensional analysis method has been developed based upon shallow shell theory. The present paper summarizes the two types of theories and makes quantitative comparisons of frequencies obtained by them. Numerical results are presented for cambered and/or twisted blades of uniform thickness. Significant differences between the theories are found to occur, especially for low aspect ratio blades. The causes of these differences are discussed.

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