A Probabilistic Analysis for Vibration of Rotating Beams With Uncertain Parameters

2005 ◽  
Author(s):  
Z. Q. Wang ◽  
L. Q. An ◽  
Z. Z. Peng
Keyword(s):  
Present Method ◽  
Natural Frequencies ◽  
Perturbation Technique ◽  
Ritz Method ◽  
Joint Stiffness ◽  
Random Variable ◽  
Uncertain Parameters ◽  
Vibration Modes ◽  
Matrix Perturbation ◽  
Rotating Speed

In this paper, the material properties, sectional properties, rotating speed and the boundary condition of beam are treated as random variable. A Ritz method is used to derive the eigenvalues equation of the rotating beam with uncertain parameters. A matrix perturbation technique is employed to obtain probabilistic characters of the natural frequencies and vibration modes for free vibration. The sensitivity analysis is performed to study the influence of parametric changes on the variation of nature frequencies. The effects of variation in material propertied, rotating speed and the joint stiffness on the deterministic part of natural frequencies and vibration modes, the covariance and the coefficient of variance (COV) for the nature frequencies are investigated. A numerical example is studied by the present method and the Monte Carlo simulation to establish the validity of the present method.

A Probabilistic Analysis for Static and Dynamic Frequency of Blade With Uncertain Boundary Conditions

2005 ◽  
Author(s):  
Z. Q. Wang ◽  
L. Q. An ◽  
Z. Z. Peng
Keyword(s):  
Boundary Condition ◽  
Probabilistic Analysis ◽  
Natural Frequencies ◽  
Perturbation Technique ◽  
Ritz Method ◽  
Joint Stiffness ◽  
Matrix Perturbation ◽  
Sensitivity Matrix ◽  
The Matrix ◽  
Uncertain Boundary

A probabilistic analysis method is developed for frequencies analysis of turbine blade with uncertain boundary condition at the root of blade. The Ritz method is used to derive the eigenvalues equation of the rotating blade with uncertain root boundary condition. The matrix perturbation technique is employed for the probabilistic analysis to obtain the deterministic part of natural frequencies and vibration modes, the sensitivity matrix, the covariance matrix and the coefficient of variance (COV) for the natural frequencies. The effects of variations in the expectation and the variance of joint stiffness on the expectation and the variance of the natural frequencies are investigated.

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 Internal Characteristics Research on the Turbocharger Rotor

2012 ◽  
Vol 516-517 ◽  
pp. 709-713
Author(s):  
Jun Sheng Zhao ◽  
Liao Ping Hu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Natural Frequencies ◽  
Vibration Modes ◽  
The Finite Element Method ◽  
Rotating Speed ◽  
Bearing Stiffness ◽  
Film Stiffness ◽  
Test Result ◽  
Element Method

In order to obtain the vibration characteristics of the turbocharger rotor system, its natural frequencies and vibration modes are approached by the experimental method and the finite element method. The modal parameters are obtained by modal tests, and are calculated by the finite element method with the software ANSYS8.0 at the meantime. The influence is approached mainly for the oil film stiffness of the floating-ring bearings and the rotor’s rotating speed. The FEM result is certificate compared with the modal test result. The influence of the rotating speed is greater on the rotor natural frequency than that of the bearing stiffness. It provides the theoretical basis for turbocharger’s vibration control and structure improve.

Measurement of Young’s Modulus and Shear Modulus of Some Structures Welded Using Flux Cored Wire by Vibration Tests

2014 ◽  
Vol 216 ◽  
pp. 151-156 ◽  
Author(s):  
Liviu Bereteu ◽  
Mircea Vodǎ ◽  
Gheorghe Drăgănescu
Keyword(s):  
Shear Modulus ◽  
Arc Welding ◽  
Natural Frequencies ◽  
Free Vibrations ◽  
Vibration Modes ◽  
Cored Wire ◽  
Flux Cored Arc Welding ◽  
Longitudinal Elastic Modulus ◽  
Vibration Tests ◽  
Non Destructive

The aim of this work was to determine by vibration tests the longitudinal elastic modulus and shear modulus of welded joints by flux cored arc welding. These two material properties are characteristic elastic constants of tensile stress respectively torsion stress and can be determined by several non-destructive methods. One of the latest non-destructive experimental techniques in this field is based on the analysis of the vibratory signal response from the welded sample. An algorithm based on Pronys series method is used for processing the acquired signal due to sample response of free vibrations. By the means of Finite Element Method (FEM), the natural frequencies and modes shapes of the same specimen of carbon steel were determined. These results help to interpret experimental measurements and the vibration modes identification, and Youngs modulus and shear modulus determination.

scholarly journals Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffnesses

2020 ◽  
Vol 11 (1) ◽  
pp. 127
Author(s):  
Fuchun Yang ◽  
Dianrui Wang
Keyword(s):  
High Speed ◽  
Differential Operators ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Distribution Area ◽  
Vibration Modes ◽  
Governing Equations ◽  
Vibration Properties ◽  
Wide Range ◽  
Forced Responses

Vibration properties of high-speed rotating and revolving planet rings with discrete and partially distributed stiffnesses were studied. The governing equations were obtained by Hamilton’s principle based on a rotating frame on the ring. The governing equations were cast in matrix differential operators and discretized, using Galerkin’s method. The eigenvalue problem was dealt with state space matrix, and the natural frequencies and vibration modes were computed in a wide range of rotation speed. The properties of natural frequencies and vibration modes with rotation speed were studied for free planet rings and planet rings with discrete and partially distributed stiffnesses. The influences of several parameters on the vibration properties of planet rings were also investigated. Finally, the forced responses of planet rings resulted from the excitation of rotating and revolving movement were studied. The results show that the revolving movement not only affects the free vibration of planet rings but results in excitation to the rings. Partially distributed stiffness changes the vibration modes heavily compared to the free planet ring. Each vibration mode comprises several nodal diameter components instead of a single component for a free planet ring. The distribution area and the number of partially distributed stiffnesses mainly affect the high-order frequencies. The forced responses caused by revolving movement are nonlinear and vary with a quasi-period of rotating speed, and the responses in the regions supported by partially distributed stiffnesses are suppressed.

Vibration-Based Crack Detection in L-Frames

2014 ◽  
Vol 658 ◽  
pp. 261-268
Author(s):  
Jean Louis Ntakpe ◽  
Gilbert Rainer Gillich ◽  
Florian Muntean ◽  
Zeno Iosif Praisach ◽  
Peter Lorenz
Keyword(s):  
Strain Energy ◽  
Crack Detection ◽  
Natural Frequencies ◽  
Vibration Modes ◽  
Structural Elements ◽  
Element Analysis ◽  
Accurate Localization ◽  
Mathematical Expressions ◽  
Measurement Results ◽  
Non Destructive

This paper presents a novel non-destructive method to locate and size damages in frame structures, performed by examining and interpreting changes in measured vibration response. The method bases on a relation, prior contrived by the authors, between the strain energy distribution in the structure for the transversal vibration modes and the modal changes (in terms of natural frequencies) due to damage. Using this relation a damage location indicator DLI was derived, which permits to locate cracks in spatial structures. In this paper an L-frame is considered for proving the applicability of this method. First the mathematical expressions for the modes shapes and their derivatives were determined and simulation result compared with that obtained by finite element analysis. Afterwards patterns characterizing damage locations were derived and compared with measurement results on the real structure; the DLI permitted accurate localization of any crack placed in the two structural elements.

Natural Frequencies and Mode Shapes of Elliptic Plates With Boundary Characteristic Orthogonal Polynomials As Assumed Shape Functions

1991 ◽  
Author(s):  
C. Rajalingham ◽  
R. B. Bhat ◽  
G. D. Xistris
Keyword(s):  
Coordinate System ◽  
Orthogonal Polynomials ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Mode Shapes ◽  
Polar Coordinate System ◽  
Free Boundaries ◽  
Natural Modes ◽  
Polar Coordinate ◽  
Boundary Characteristic

Abstract The natural frequencies and natural modes of vibration of uniform elliptic plates with clamped, simply supported and free boundaries are investigated using Rayleigh-Ritz method. A modified polar coordinate system is used to investigate the problem. Energy expressions in Cartesian coordinate system are transformed into the modified polar coordinate system. Boundary characteristic orthogonal polynomials in the radial direction, and trigonometric functions in the angular direction are used to express the deflection of the plate. These deflection shapes are classified into four basic categories, depending on its symmetrical or antisymmetrical property about the major and minor axes of the ellipse. The first six natural modes in each of the above categories are presented in the form of contour plots.

Natural Frequencies of Parallelogrammic Plates Using Characteristic Orthogonal Polynomials in Rayleigh-Ritz Method

1992 ◽  
Author(s):  
L. T. Lee ◽  
W. F. Pon
Keyword(s):  
Boundary Conditions ◽  
Coordinate System ◽  
Orthogonal Polynomials ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Energy Functional ◽  
Comprehensive Treatment ◽  
Energy Functionals ◽  
Simply Supported ◽  
Cartesian Coordinate

Abstract Natural frequencies of parallelogrammic plates are obtained by employing a set of beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. The orthogonal polynomials are generalted by using a Gram-Schmidt process, after the first member is constructed so as to satisfy all the boundary conditions of the corresponding beam problems accompanying the plate problems. The strain energy functional and kinetic energy functionals are transformed from Cartesian coordinate system to a skew coordinate system. The natural frequencies obtained by using the orthogonal polynomial functions are compared with those obtained by other methods with all four edges clamped boundary conditions and greet agreements are found between them. The natural frequencies for parallelogrammic plates with other boundary conditions, such as four edges simply supported, clamped-free and simply supported-free, are also obtained. This method is considered as a better and accurate comprehensive treatment for this type of problems.

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