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.

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

2007 ◽  
Vol 130 (1) ◽  
Author(s):  
Ioannis K. Chatjigeorgiou
Keyword(s):  
Boundary Layer ◽  
Mathematical Formulation ◽  
Constant Factor ◽  
Mode Shapes ◽  
Perturbation Approach ◽  
Problem Solution ◽  
Bending Vibration ◽  
Boundary Layer Problem ◽  
Boundary Layer Problems ◽  
Slender Structures

The present work treats the bending vibration problem for vertical slender structures assuming clamped connections at the ends. The final goal is the solution of the associated eigenvalue problem for calculating the natural frequencies and the corresponding mode shapes. The mathematical formulation accounts for all physical properties that influence the bending vibration of the structure including the variation in tension. The resulting model incorporates all principal characteristics, such as the bending stiffness, the submerged weight, and the function of the static tension. The governing equation is treated using a perturbation approach. The application of this method results in 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. Here, the term “boundary layer” is used to indicate the narrow region across which the dependent variable undergoes very rapid changes. The boundary layers adjacent to the clamped ends are associated with the fact that the stiffness (which is small) is the constant factor multiplied by the highest derivative in the governing differential equation.

Numerical Study of Boundary Layers With Reverse Wedge Flows Over a Semi-Infinite Flat Plate

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
R. Ahmad ◽  
K. Naeem ◽  
Waqar Ahmed Khan
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Numerical Study ◽  
Boundary Layer Equation ◽  
Wedge Angle ◽  
Adverse Pressure Gradient ◽  
Problem Solution ◽  
Weighted Residual Method ◽  
Boundary Layer Problem ◽  
Layer Problem

This paper presents the classical approximation scheme to investigate the velocity profile associated with the Falkner–Skan boundary-layer problem. Solution of the boundary-layer equation is obtained for a model problem in which the flow field contains a substantial region of strongly reversed flow. The problem investigates the flow of a viscous liquid past a semi-infinite flat plate against an adverse pressure gradient. Optimized results for the dimensionless velocity profiles of reverse wedge flow are presented graphically for different values of wedge angle parameter β taken from 0≤β≤2.5. Weighted residual method (WRM) is used for determining the solution of nonlinear boundary-layer problem. Finally, for β=0 the results of WRM are compared with the results of homotopy perturbation method.

In-Plane Vibration Modes of Arbitrarily Thick Disks

1997 ◽  
Author(s):  
Kevin I. Tzou ◽  
Jonathan A. Wickert ◽  
Adnan Akay
Keyword(s):  
Natural Frequencies ◽  
Arbitrary Dimension ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Three Dimensional Model ◽  
Bending Vibration ◽  
Annular Disk ◽  
Out Of Plane ◽  
Plane Bending

Abstract The three-dimensional vibration of an arbitrarily thick annular disk is investigated for two classes of boundary conditions: all surfaces traction-free, and all free except for the clamped inner radius. These two models represent limiting cases of such common engineering components as automotive and aircraft disk brakes, for which existing models focus on out-of-plane bending vibration. For a disk of significant thickness, vibration modes in which motion occurs within the disk’s equilibrium plane can play a substantial role in setting its dynamic response. Laboratory experiments demonstrate that in-plane modes exist at frequencies comparable to those of out-of-plane bending even for thickness-to-diameter ratios as small as 10−1. The equations for three-dimensional motion are discretized through the Ritz technique, yielding natural frequencies and mode shapes for coupled axial, radial, and circumferential deformations. This treatment is applicable to “disks” of arbitrary dimension, and encompasses classical models for plates, bars, cylinders, rings, and shells. The solutions so obtained converge in the limiting cases to the values expected from the classical theories, and to ones that account for shear deformation and rotary inertia. The three-dimensional model demonstrates that for geometries within the technologically-important range, the natural frequencies of certain in- and out-of-plane modes can be close to one another, or even identically repeated.

Comparison of Free Flexural Vibrations of Composite Plates or Panels Stiffened by a Central and a Non-Central Plate Strip

1999 ◽  
Author(s):  
U. Yuceoglu ◽  
V. Özerciyes
Keyword(s):  
Natural Frequencies ◽  
Plate Theory ◽  
Composite Plates ◽  
Mode Shapes ◽  
Mindlin Plate ◽  
Solution Technique ◽  
Orthotropic Plates ◽  
Bending Vibration ◽  
Central Plate ◽  
Plate Strip

Abstract The natural frequencies and the corresponding mode shapes of two classes of composite base plate or panel stiffened by a central or a non-central plate strip are analyzed and compared with each other. In each case, the base plates and the single, stiffening plate strips are assumed to be dissimilar orthotropic plates connected by a very thin, yet deformable adhesive layer. The free bending vibration problems for the two cases are formulated in terms of the Mindlin Plate Theory for orthotropic plates. The governing equations are reduced to a system of first order equations. The solution technique is the “Modified Version of the Transfer Matrix Method”. The effects of the bonded central and non-central stiffening strip on the mode shapes and the natural frequencies of the composite plate or panel system are investigated. Some important conclusions are drawn from the numerical and parametric studies presented.

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.

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.

In-Plane Vibration Modes of Arbitrarily Thick Disks

1998 ◽  
Vol 120 (2) ◽  
pp. 384-391 ◽  
Author(s):  
K. I. Tzou ◽  
J. A. Wickert ◽  
A. Akay
Keyword(s):  
Natural Frequencies ◽  
Arbitrary Dimension ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Three Dimensional Model ◽  
Bending Vibration ◽  
Annular Disk ◽  
Out Of Plane ◽  
Plane Bending

The three-dimensional vibration of an arbitrarily thick annular disk is investigated for two classes of boundary conditions: all surfaces traction-free, and all free except for the clamped inner radius. These two models represent limiting cases of such common engineering components as automotive and aircraft disk brakes, for which existing models focus on out-of-plane bending vibration. For a disk of significant thickness, vibration modes in which motion occurs within the disk’s equilibrium plane can play a substantial role in-setting its dynamic response. Laboratory experiments demonstrate that in-plane modes exist at frequencies comparable to those of out-of-plane bending even for thickness-to-diameter ratios as small as 10−1. The equations for three-dimensional motion are discretized through the Ritz technique, yielding natural frequencies and mode shapes for coupled axial, radial, and circumferential deformations. This treatment is applicable to “disks” of arbitrary dimension, and encompasses classical models for plates, bars, cylinders, rings, and shells. The solutions so obtained converge in the limiting cases to the values expected from the classical theories, and to ones that account for shear deformation and rotary inertia. The three-dimensional model demonstrates that for geometries within the technologically-important range, the natural frequencies of certain in- and out-of-plane modes can be close to one another, or even identically repeated.

scholarly journals Carbon Nanotube Structure Vibration Based on Non-local Elasticity

2016 ◽  
Vol 3 (1) ◽  
pp. 9-13 ◽  
Author(s):  
Abdelkadir Belhadj ◽  
Abdelkrim Boukhalfa ◽  
Sid Ahmed Belalia
Keyword(s):  
Carbon Nanotube ◽  
Analytical Procedure ◽  
Natural Frequencies ◽  
Differential Quadrature Method ◽  
Mode Shapes ◽  
Quadrature Method ◽  
Bending Vibration ◽  
Single Walled Carbon Nanotube ◽  
Non Local ◽  
Structure Vibration

This manuscript investigates the bending vibration dynamic of a single walled carbon nanotube (SWCNT) based on the theory of non-local elasticity. Fundamental natural frequencies and mode shapes of the SWCNT are computed by using a semi-analytical procedure called differential quadrature method (DQM), which gives accurate results in reference with the exact solution.

On the Three-Dimensional Turbulent Boundary Layer Generated by Secondary Flow

1960 ◽  
Vol 82 (1) ◽  
pp. 233-246 ◽  
Author(s):  
J. P. Johnston
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Secondary Flow ◽  
Three Dimensional ◽  
Cross Flow ◽  
Problem Solution ◽  
Boundary Layer Problem ◽  
Flow Components ◽  
Momentum Integral ◽  
The Relationship

A study of the secondary flow type of three-dimensional turbulent boundary layer is presented. Two objectives are achieved: (a) A mathematical model of the relationship between the cross-flow and main-flow components of the velocity vectors of the layer is established. (b) By utilization of the model some of the relationships required to carry out a boundary-layer problem solution by the use of the momentum-integral equations are developed.

scholarly journals Structural Dynamics of Real and Modeled Solanum Stamens: Implications for Pollen Ejection by Buzzing Bees

2021 ◽  
Author(s):  
Mark Jankauski ◽  
Riggs Ferguson ◽  
Avery L Russell ◽  
Stephen Buchmann
Keyword(s):  
Material Properties ◽  
Natural Frequencies ◽  
Vibration Mode ◽  
Element Model ◽  
Flowering Plant ◽  
Mode Shapes ◽  
Bending Vibration ◽  
Solanum Elaeagnifolium ◽  
Flowering Plant Species ◽  
Poricidal Anthers

An estimated 10% of flowering plant species conceal their pollen within tube-like anthers that dehisce through small apical pores (poricidal anthers). Bees extract pollen from poricidal anthers through a complex motor routine called floral buzzing, whereby the bee applies large vibratory forces to the flower stamen by rapidly contracting its flight muscles. The resulting deformation and pollen expulsion depend critically on the stamen's natural frequencies and vibration mode shapes, yet these properties remain unknown. We performed experimental modal analysis on Solanum elaeagnifolium stamens to quantify their natural frequencies and vibration modes. Based on morphometric and dynamic measurements, we developed a finite element model of the stamen to identify how variable material properties, geometry and bee weight could affect its dynamics. In general, stamen natural frequencies fell outside the reported floral buzzing range, and variations in stamen geometry and material properties were unlikely to bring natural frequencies within this range. However, inclusion of bee mass reduced natural frequencies to within the floral buzzing frequency range and gave rise to an axial-bending vibration mode. We hypothesize that floral buzzing bees exploit the large vibration amplification factor of this mode to increase anther deformation, which may facilitate pollen ejection.

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