Vibration of Plates Subject to Arbitrary In-Plane Loads—A Perturbation Approach

1973 ◽  
Vol 40 (4) ◽  
pp. 1023-1028 ◽  
Author(s):  
S. F. Bassily ◽  
S. M. Dickinson
Keyword(s):  
Perturbation Theory ◽  
Natural Frequencies ◽  
Perturbation Technique ◽  
Mode Shapes ◽  
Perturbation Approach ◽  
Rectangular Plates ◽  
Stress Distributions ◽  
Lateral Vibration ◽  
Approximate Frequency ◽  
Loading Parameter

Perturbation theory is used to obtain the natural frequencies and mode shapes of lateral vibration of rectangular plates under arbitrary in-plane loading in terms of a power series in a loading parameter, commencing from known approximate or “exact” solutions of the same plate under no in-plane loading. The range of loading over which the solution is applicable is shown to be extendible to practically any limit by the introduction of a stepwise perturbation technique. Numerical results demonstrating the applicability and accuracy of the analysis are presented for plates subject to uniform, linear, and parabolic in-plane stress distributions. Simple approximate frequency expressions are also given.

Hydroelastic Vibration of Rectangular Plates

1996 ◽  
Vol 63 (1) ◽  
pp. 110-115 ◽  
Author(s):  
Moon K. Kwak
Keyword(s):  
Boundary Conditions ◽  
Approximate Formula ◽  
Natural Frequencies ◽  
Mass Matrix ◽  
Ritz Method ◽  
Plate Boundary ◽  
Rigid Wall ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Virtual Mass

This paper is concerned with the virtual mass effect on the natural frequencies and mode shapes of rectangular plates due to the presence of the water on one side of the plate. The approximate formula, which mainly depends on the so-called nondimensionalized added virtual mass incremental factor, can be used to estimate natural frequencies in water from natural frequencies in vacuo. However, the approximate formula is valid only when the wet mode shapes are almost the same as the one in vacuo. Moreover, the nondimensionalized added virtual mass incremental factor is in general a function of geometry, material properties of the plate and mostly boundary conditions of the plate and water domain. In this paper, the added virtual mass incremental factors for rectangular plates are obtained using the Rayleigh-Ritz method combined with the Green function method. Two cases of interfacing boundary conditions, which are free-surface and rigid-wall conditions, and two cases of plate boundary conditions, simply supported and clamped cases, are considered in this paper. It is found that the theoretical results match the experimental results. To investigate the validity of the approximate formula, the exact natural frequencies and mode shapes in water are calculated by means of the virtual added mass matrix. It is found that the approximate formula predicts lower natural frequencies in water with a very good accuracy.

scholarly journals NASTRAN Analysis of a Turbine Blade and Comparison With Test and Field Data

1975 ◽  
Author(s):  
J. M. Allen ◽  
L. B. Erickson
Keyword(s):  
Turbine Blade ◽  
Natural Frequencies ◽  
Beam Theory ◽  
Mode Shapes ◽  
Stress Distributions ◽  
Free Standing ◽  
Element Analysis ◽  
Metallographic Examination ◽  
Stiffening Effect ◽  
Generalized Beam Theory

A NASTRAN finite element analysis of a free standing gas turbine blade is presented. The analysis entails calculation of the first four natural frequencies, mode shapes, and relative vibratory stresses, as well as deflections and stresses due to centrifugal loading. The stiffening effect of the centrifugal force field was accounted for by using NASTRAN’s differential stiffness option. Natural frequencies measured in a rotating test correlated well with computed results. Areas of maximum vibratory stress (fundamental mode) coincided with the three zones of crack initiation observed in a metallographic examination of a fatigue failure. Airfoil stress distributions were found to be significantly different from that predicted by generalized beam theory, especially near the airfoil-platform junction.

Combinations for the Free Vibration Behaviors of Anisotropic Rectangular Plates Under General Edge Conditions

1999 ◽  
Author(s):  
Yoshihiro Narita
Keyword(s):  
Free Vibration ◽  
Structural Design ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Technical Information ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Plate Models ◽  
Edge Conditions ◽  
Anisotropic Rectangular Plates

Abstract The free vibration behavior of rectangular plates provides important technical information in structural design, and the natural frequencies are primarily affected by the boundary conditions as well as aspect and thickness ratios. One of the three classical edge conditions, i.e., free, simple supported and clamped edges, may be used to model the constraint along an edge of the rectangle. Along the entire boundary with four edges, there exist a wide variety of combinations in the edge conditions, each yielding different natural frequencies and mode shapes. For counting the total number of possible combinations, the present paper introduces the Polya counting theory in combinatorial mathematics, and formulas are derived for counting the exact numbers. A modified Ritz method is then developed to calculate natural frequencies of anisotropic rectangular plates under any combination of the three edge conditions and is used to numerically verify the numbers. In numerical experiments, the number of combinations in the free vibration behaviors is determined for some plate models by using the derived formulas, and are corroborated by counting the numbers of different sets of the natural frequencies that are obtained from the Ritz method.

Non-linear free transverse vibration of thin rotating discs

2007 ◽  
Vol 221 (3) ◽  
pp. 467-473 ◽  
Author(s):  
H R Hamidzadeh
Keyword(s):  
Steady State ◽  
Free Vibration ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Stress Distributions ◽  
Rotating Speed ◽  
Rotating Discs ◽  
Non Linear ◽  
Modes Of Vibration ◽  
Free Transverse Vibration

An analytical method is adopted to determine modal characteristics of non-linear spinning discs. The disc is assumed to be isotropic and rotating under steady-state conditions. The effects of amplitude and rotating speed on natural frequencies are determined. The developed procedure is also capable of analysing natural frequencies of linear free vibration, which is independent of amplitude. Attention is confined to determine natural frequencies, mode shapes, stress distributions, and critical speeds for different numbers of nodal diameters. The developed procedure does not consider modes of vibration corresponding to nodal circles. Validity of this procedure is verified by comparing some of the computed results with those established for certain cases.

Natural Frequencies of Plate Systems using the Edge Effect Method

1967 ◽  
Vol 9 (4) ◽  
pp. 318-324 ◽  
Author(s):  
S. M. Dickinson ◽  
G. B. Warburton
Keyword(s):  
Natural Frequency ◽  
Edge Effect ◽  
Series Solution ◽  
Natural Frequencies ◽  
Rectangular Plates ◽  
Form Part ◽  
Approximate Frequency ◽  
Flexural Vibrations ◽  
Effect Method ◽  
Partial Success

In this paper the Bolotin edge effect method is used to consider the free flexural vibrations of systems built up from rectangular plates. The constituent plates of the systems are considered either to lie in the same plane and form part of a plate continuous over line supports or to lie in planes at right angles to each other, as in box constructions. The application of the edge effect method to single-and multi-plate systems is described and the approximate frequency equations for two two-plate systems are given. The first 10 natural frequency parameters for these two systems for particular side ratios are compared with those obtained using a series solution and agreement is shown to be good. A few frequency parameters for a closed box computed using the edge effect method and the series solution are also compared. The range of plate systems to which the edge effect method may be applied with complete success and the range to which it may be applied with only partial success are indicated. The sources of errors in the edge effect solutions are indicated.

Solution of the Boundary Layer Problems for Calculating the Natural Modes of Riser-Type Slender Structures

2006 ◽  
Author(s):  
Ioannis K. Chatjigeorgiou
Keyword(s):  
Boundary Layer ◽  
Natural Frequencies ◽  
Mathematical Formulation ◽  
Mode Shapes ◽  
Perturbation Approach ◽  
Problem Solution ◽  
Bending Vibration ◽  
Boundary Layer Problem ◽  
Boundary Layer Problems ◽  
Slender Structures

The present work treats the problem of the calculation of the natural frequencies and the corresponding bending vibration modes of vertical slender structures. The originality of the study lies on fact that for the derivation of natural frequencies and the corresponding mode shapes, all physical properties that influence the bending vibration of the structure were considered including the aspect of the variation of tension. The resulting mathematical formulation incorporates all principal contributions such as the bending stiffness, the weight and the tension variation. The governing equation is treated using a perturbation approach. The application of this method results to the development of two boundary layer problems at the ends of the structure. These problems are treated properly using a boundary layer problem solution methodology in order to obtain asymptotic approximations to the shape of the vibrating riser-type structure. It should be noted that in this work the term ‘boundary layer’ is not connected with fluid flows but it is used to indicate the narrow region across which the dependent variable undergoes very rapid changes. Frequently these narrow regions adjoin the boundaries of the domain of intersect, especially when the small parameter multiplies the highest derivative.

Analysis Behavior of a Rig Shafting Vibration Set Changes Bearing Parameters

2013 ◽  
Vol 437 ◽  
pp. 98-101 ◽  
Author(s):  
Van Thanh Ngo ◽  
Dan Mei Xie
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Natural Frequencies ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Lateral Vibration ◽  
Eigenvalues And Eigenvectors ◽  
Fem Model ◽  
Critical Speeds ◽  
Bearing Stiffness

Frequently, in the design of machines, some of parameters that directly affect the rotordynamics of the machines are not accurately known. In particular, bearing stiffness support is one such parameter. Taking a rig shafting as an example, this paper studies the lateral vibration of the rig shafting with multi-degree-of-freedom by using finite element method (FEM). The FEM model is created and the eigenvalues and eigenvectors are calculated and analyzed to find natural frequencies, critical speeds, mode shapes. Then critical speeds and mode shapes are analyzed by set bearing stiffness changes. The model permitted to identify the critical speeds and bearings that have an important influence on the vibration behavior.

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.

Free Vibration Characteristics of SPR Joints

2006 ◽  
Author(s):  
Xiaocong He ◽  
Ian Pearson ◽  
Ken Young
Keyword(s):  
Aluminum Alloy ◽  
Free Vibration ◽  
Torsional Vibration ◽  
Natural Frequencies ◽  
Vibration Characteristics ◽  
Advanced Materials ◽  
Mode Shapes ◽  
Stress Distributions ◽  
Numerical Examples ◽  
Different Characteristics

Self-piercing riveting (SPR) has drawn more attention in recent years because it can join some advanced materials that are hard to weld, such as aluminum alloy sheets. In this paper, the free torsional vibration characteristics of single lap-jointed encastre SPR beam are investigated in detail. The focus of the analysis is to reveal the influence on the torsional natural frequencies and mode shapes of the single lap-jointed encastre SPR beam of different characteristics of sheets to be jointed. Numerical examples show that the torsional natural frequencies increase significantly as the Young’s modulus of the sheets increase, but almost no change corresponding to the change in Poisson’s ratio of the sheets to be joint. The mode shapes show that there are different deformations in the jointed section of SPR beam compared with the reference encastre beam without joint. These different deformations may cause different natural frequency values and different stress distributions.

A Perturbation Solution of the Thickness Natural Vibrations of a Piezoelectric Plate

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Piotr Cupiał
Keyword(s):  
Boundary Conditions ◽  
Electrostatic Potential ◽  
Natural Vibration ◽  
Natural Frequencies ◽  
Piezoelectric Plate ◽  
Perturbation Solution ◽  
Mode Shapes ◽  
Perturbation Approach ◽  
Linear Piezoelectricity ◽  
Coupled Theory

This paper discusses a perturbation approach to the calculation of the natural frequencies and mode shapes for both the displacement and the electrostatic potential through-thickness vibration of an infinite piezoelectric plate. The problem is formulated within the coupled theory of linear piezoelectricity. It is described by a set of two coupled differential equations with unknown thickness displacement, the electrostatic potential and a general form of boundary conditions. A consistent perturbation solution to the natural vibration problem is described. An important element not present in the classical eigenvalue perturbation solution is that the small parameter appears in the boundary conditions; a way to handle this complication has been discussed. The natural frequencies and mode shapes obtained using the perturbation approach are compared to exact solutions, demonstrating the effectiveness of the proposed method. The advantage of the perturbation method derives from the fact that coupled piezoelectric results can be obtained from the elastic solution during the postprocessing stage.

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