Vibration Analysis of Helicoidal Plates

1996 ◽  
Vol 210 (6) ◽  
pp. 501-517
Author(s):  
C L Ko
Keyword(s):  
Vibration Analysis ◽  
Natural Frequencies ◽  
Turbine Blades ◽  
Free Vibrations ◽  
Governing Equations ◽  
Higher Order Modes ◽  
Helical Angle ◽  
System Equations ◽  
The Finite Difference Method ◽  
Good Agreement

Governing equations for free vibrations of thin orthotropic or isotropic helicoidal plates are formulated by expressing displacements in terms of components in a helical orthogonal coordinate system. Equations derived for isotropic helicoidal plates are applied to the vibration problem of twisted rectangular turbine blades without camber. These twisted cantilever rectangular blades are simulated to be isotropic helicoidal plates with their angles of twist approximated to be their centre-line helical angles. Natural frequencies are calculated by solving the eigenvalue problem using the finite difference method and are compared to experimental measurements reported in the literature. Reasonably good agreement has been found for flexural modes; however, some discrepancies have also been observed for higher-order modes due to the inaccuracy of modelling the blade geometry as that of a helicoidal plate and due to the approximation of assuming the angle of twist to be the helical angle.

Free Vibrations of Beams on Viscoelastic Pasternak Foundations

2015 ◽  
Vol 744-746 ◽  
pp. 1624-1627
Author(s):  
Li Peng ◽  
Ying Wang
Keyword(s):  
Natural Frequencies ◽  
Free Vibrations ◽  
Mode Analysis ◽  
Analysis Method ◽  
Governing Equations ◽  
Complex Mode ◽  
Transverse Vibrations ◽  
Quadrature Methods ◽  
Foundation Parameters ◽  
Good Agreement

This paper investigates free transverse vibrations of finite Euler–Bernoulli beams resting on viscoelastic Pasternak foundations. The differential quadrature methods (DQ) are applied directly to the governing equations of the free vibrations. Under the simple supported boundary condition, the natural frequencies of the transverse vibrations are calculated, and compared with the results of the complex mode analysis method. The numerical results obtained by using the DQ and the complex mode methods are in good agreement for the first seven order natural frequencies, but with the growth of the orders, the small quantitative differences between them increase. The effects of the foundation parameters on the natural frequencies are also studied in numerical examples.

Non-local vibration of simply supported nano-beams: Higher-order modes

2017 ◽  
Vol 231 (2) ◽  
pp. 75-81
Author(s):  
Mehdi Masoumi ◽  
Masoud Masoumi
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Higher Order ◽  
Rotary Inertia ◽  
Governing Equations ◽  
Local Vibration ◽  
Simply Supported ◽  
Higher Order Modes ◽  
Non Local

In this article, the effects of some parameters, including rotary inertia, non-local parameter, and length-to-thickness ratio, on natural frequencies are studied for both classical and non-local theories. For Timoshenko beam, the equations of motion and the boundary conditions are derived from Hamilton’s principle and then non-local constitutive equations of Eringen are employed to altogether formulate the problem. Afterward, obtained governing equations are used to study the free vibrations of a Timoshenko’s simply supported nano-beam. And finally, the effects of above-mentioned parameters on estimated frequencies in classical and non-local elasticity theories are investigated. Results show that the discrepancy between the frequencies of higher-order vibration modes obtained from two theories increases and also significant reductions in natural frequencies occur when the rotary inertia is considered in the computations.

scholarly journals The Sensitivity Analysis for Life Estimation of Turbine Blades

1997 ◽  
Author(s):  
O. Repetski ◽  
K. Zainchkovski
Keyword(s):  
Vibration Analysis ◽  
Dynamic Stress ◽  
Natural Frequencies ◽  
Turbine Blades ◽  
Stress Sensitivity ◽  
Life Estimation ◽  
Sensitivity Coefficients ◽  
Sensitivity Distribution ◽  
Design Variables ◽  
Forced Vibration Analysis

The proposed algorithm permits one to determine the sensitivity coefficients of natural frequencies and dynamic displacements and stresses in the free- and forced vibration analysis. This algorithm is presented in a computer program with the help of finite element method (FEM). The design variables is the thickness of the blades. Usually a maximum resonance accounts more than half the damage and deterioration of machine components. The analysis of dynamic stress sensitivity distribution for this resonance permits us to control for both the endurance of machines and their components based on the thickness. In this study the sensitivity coefficients for both free vibration, dynamic resonances are investigated by acceleration and braking the regimes.

Forced Harmonic Response of Grouped Blade Systems: Part II—Application

1996 ◽  
Vol 118 (1) ◽  
pp. 137-145 ◽  
Author(s):  
L. F. Wagner ◽  
J. H. Griffin
Keyword(s):  
Numerical Model ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Turbine Blades ◽  
Modal Identification ◽  
Harmonic Response ◽  
Model Based ◽  
Grouped Blades ◽  
Flexible Disk ◽  
Symmetric Structures

The vibration of grouped blades on a flexible disk should, for purposes of economy and clarity of modal identification, be analyzed using procedures developed for cyclically symmetric structures. In this paper, a numerical model, based on the theory of cyclically symmetric structures, is applied to the vibration analysis, and in particular, the harmonic response, of a flexible disk supporting a number of groups, or packets, of turbine blades. Results are presented to show variations in the modal participation factors as a function of such parameters as disk flexibility, blade density, and the total number of assembled groups. It is also shown that many characteristics of the system spectra of natural frequencies are strongly dependent on the number of blade groups.

The Free Vibrations of Stiffened Drill Strings With Static Curvature

1967 ◽  
Vol 89 (1) ◽  
pp. 23-29 ◽  
Author(s):  
D. A. Frohrib ◽  
R. Plunkett
Keyword(s):  
Natural Frequencies ◽  
Free Vibrations ◽  
Drill String ◽  
Bending Stiffness ◽  
Mode Shapes ◽  
Static Tension ◽  
Lateral Vibration ◽  
Governing Equations ◽  
Deflection Curve ◽  
Wide Range

The natural frequencies of lateral vibration of a long drill string in static tension under its own weight are primarily the same as those of the equivalent catenary. These frequencies and the mode shapes are affected to a certain extent by the bending stiffness and to a greater extent by the static deflection curve due to lateral deflection of the bottom end. In this paper, the governing equations are derived and general solutions are given in an asymptotic expansion with the bending stiffness as the parameter. Specific numerical results are given in dimensionless form for the first three natural frequencies for a very wide range of horizontal tension and several appropriate values of bending stiffness for zero vertical static force at the bottom.

scholarly journals Vibration Analysis of Axially Functionally Graded Non-Prismatic Euler-Bernoulli Beams Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Vibration Analysis ◽  
Mass Density ◽  
Functionally Graded ◽  
Governing Equations ◽  
Mass System ◽  
The Finite Difference Method

This paper presents an approach to the vibration analysis of axially functionally graded (AFG) non-prismatic Euler-Bernoulli beams using the finite difference method (FDM). The characteristics (cross-sectional area, moment of inertia, elastic moduli, and mass density) of AFG beams vary along the longitudinal axis. The FDM is an approximate method for solving problems described with differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, differential equations were formulated with finite differences, and additional points were introduced at the beam’s ends and at positions of discontinuity (supports, hinges, springs, concentrated mass, spring-mass system, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. Vibration analysis of AFG non-prismatic Euler-Bernoulli beams was conducted with this model, and natural frequencies were determined. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of AFG non-prismatic Euler-Bernoulli beams, considering the damping. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was always increased through a grid refinement.

scholarly journals Kirchhoff-Love Plate Theory: First-Order Analysis, Second-Order Analysis, Plate Buckling Analysis, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference ◽  
Vibration Analysis ◽  
Plate Theory ◽  
Second Order ◽  
Governing Equations ◽  
Order Analysis ◽  
First Order ◽  
Order Polynomial ◽  
The Finite Difference Method ◽  
Point Stencil

This paper presents an approach to the Kirchhoff-Love plate theory (KLPT) using the finite difference method (FDM). The KLPT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. Generally in the case of KLPT, the finite difference approximations are derived based on the fourth-order polynomial hypothesis (FOPH) and second-order polynomial hypothesis (SOPH) for the deflection surface. The FOPH is made for the fourth and third derivative of the deflection surface while the SOPH is made for its second and first derivative; this leads to a 13-point stencil for the governing equation. In addition, the boundary conditions and not the governing equations are applied at the plate edges. In this paper, the FOPH was made for all of the derivatives of the deflection surface; this led to a 25-point stencil for the governing equation. Furthermore, additional nodes were introduced at plate edges and at positions of discontinuity (continuous supports/hinges, incorporated beams, stiffeners, brutal change of stiffness, etc.), the number of additional nodes corresponding to the number of boundary conditions at the node of interest. The introduction of additional nodes allowed us to apply the governing equations at the plate edges and to satisfy the boundary and continuity conditions. First-order analysis, second-order analysis, buckling analysis, and vibration analysis of plates were conducted with this model. Moreover, plates of varying thickness and plates with stiffeners were analyzed. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. In first-order, second-order, buckling, and vibration analyses of rectangular plates, the results obtained in this paper were in good agreement with those of well-established methods, and the accuracy was increased through a grid refinement.

scholarly journals Free Vibrations of a Reddy-Bickford Multi-Span Beam Carrying Multiple Spring-Mass Systems

2011 ◽  
Vol 18 (5) ◽  
pp. 709-726 ◽  
Author(s):  
Yusuf Yesilce
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Structural Elements ◽  
Mass System ◽  
Assembly Technique

The structural elements supporting motors or engines are frequently seen in technological applications. The operation of machine may introduce additional dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko single-span beams carrying a number of spring-mass system and multi-span beams carrying multiple spring-mass systems are plenty, but the free vibration analysis of Reddy-Bickford multi-span beams carrying multiple spring-mass systems has not been investigated by any of the studies in open literature so far. This paper aims at determining the exact solutions for the natural frequencies and mode shapes of Reddy-Bickford beams. The model allows analyzing the influence of the shear effect and spring-mass systems on the dynamic behavior of the beams by using Reddy-Bickford Beam Theory (RBT). The effects of attached spring-mass systems on the free vibration characteristics of the 1–4 span beams are studied. The natural frequencies of Reddy-Bickford single-span and multi-span beams calculated by using the numerical assembly technique and the secant method are compared with the natural frequencies of single-span and multi-span beams calculated by using Timoshenko Beam Theory (TBT); the mode shapes are presented in graphs.

Multistage Coupling of Eight Mistuned Bladed Disk on a Solid Shaft: Part 1—Free Vibration Analysis

2012 ◽  
Author(s):  
Romuald Rzadkowski ◽  
Artur Maurin
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Rotor Blades ◽  
Mode Shapes ◽  
Bladed Disk ◽  
Centrifugal Stiffening

Considered here was the effect of multistage coupling on the dynamics of a rotor consisting of eight mistuned bladed discs on a solid shaft. Each bladed disc had a different number of rotor blades. Free vibrations were examined using finite element representations of rotating single blades, bladed discs, and the entire rotor. In this study the global rotating mode shapes of eight flexible mistuned bladed discs on shaft assemblies were calculated, taking into account rotational effects such as centrifugal stiffening. The thus obtained natural frequencies of the blade, shaft, bladed disc and entire shaft with discs were carefully examined to discover resonance conditions and coupling effects. This study found that mistuned systems cause far more intensive multistage coupling than tuned ones. The greater the mistuning, the more intense the multistage coupling.

Image sequences and wavelets for vibration analysis: Part 1: Edge detection and extraction of natural frequencies

2002 ◽  
Vol 216 (9) ◽  
pp. 885-900 ◽  
Author(s):  
S Patsias ◽  
W J Staszewski ◽  
G R Tomlinson
Keyword(s):  
Edge Detection ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Detection Algorithm ◽  
Image Sequences ◽  
Image Features ◽  
Structural Vibration ◽  
Contact Method ◽  
Edge Detection Algorithm ◽  
Good Agreement

The paper presents the application of the wavelet edge detection algorithm for structural vibration analysis based on image sequences. A number of different edge detection operators are briefly discussed and illustrated using a simple image example. The wavelet-based edge detection algorithm is then used to identify natural frequencies of a simple cantilever beam. The paper discusses a number of important issues that need to be confronted when using image sequences for vibration analysis. These include correspondence of image features from image to image and image calibration. The paper compares the natural frequencies obtained using such an approach and also another non-contact method and finds them in good agreement.

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