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.

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.

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.

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.

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.

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.

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.

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.

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.

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