Stability of a Stationary Flexible Cantilevered Plate in an Axial Flow

2014 ◽  
Author(s):  
Katsuhisa Fujita ◽  
Keiji Matsumoto
Keyword(s):  
Stability Analysis ◽  
Velocity Potential ◽  
Fluid Velocity ◽  
Fluid Pressure ◽  
Axial Flow ◽  
Complex Eigenvalue ◽  
Flexible Plate ◽  
Moving Plate ◽  
Cantilevered Plate ◽  
The Stability

As the flexible plates, the papers in printing machines, the thin plastic and metal films, and the fluttering flag are enumerated. In this paper, the flexible plate is assumed to be stationary in an axial flow although both the stationary plate and the axially moving plate can be thought. The fluid is assumed to be treated as an ideal fluid in a subsonic domain, and the fluid pressure is calculated using the velocity potential theory. The coupled equation of motion of a flexible cantilevered plate is derived in consideration of the added mass, added damping and added stiffness, respectively. The velocity potential is obtained by assuming the unsteady axial fluid velocity to be zero at the trailing edge of a flexible cantilevered plate, neglecting the effect of a circulation. The complex eigenvalue analysis is performed for the stability analysis. In order to investigate the validity of the proposed analysis, another stability analysis is also performed by using the non-circulatory aerodynamic theory. The comparison between both solutions is investigated and discussed. Changing the velocities of a fluid and the specifications of a plate as parametric studies, the effects of these parameters on the stability of a flexible cantilevered plate are investigated.

Stability of an Axially Moving Flexible Cantilevered Plate in an Axial Flow

2015 ◽  
Author(s):  
Katsuhisa Fujita ◽  
Keiji Matsumoto
Keyword(s):  
Stability Analysis ◽  
Dynamic Instability ◽  
Axial Flow ◽  
Flexible Plate ◽  
Moving Plate ◽  
Axially Moving Plate ◽  
Cantilevered Plate ◽  
Axially Moving ◽  
Band Saw ◽  
Surrounding Fluid

When a flexible plate of machines and structures is in axial flow, increasing the flow velocity induces the dynamic instability. Also, the similar instability occurs for a thin film wound to a coil form and a band saw driven with high transport speed even if we consider there is no effect of fluid. However, if a flexible plate is very light, and the rigidity is weak, the dynamic instability of a plate is thought to be affected both by the transport speed of plate and by the surrounding fluid. In this paper, the stability analysis is performed using the velocity potential based on the boundary conditions between the surrounding fluid and the plate. The numerical studies are performed as for a cantilevered plate. The case of an axially moving plate in a still fluid, the case of a stationary plate subjected to an axial flow, and the case of an axially moving plate in neglecting the effect of surrounding fluid are examined. And, the effect of the transport speed of cantilevered plate on the dynamic stability is discussed. Furthermore, another stability analysis is also performed by using the non-circulatory aerodynamic theory, and the numerical analysis results are examined and discussed.

Fluid Pressures on Unanchored Rigid Flat-Bottom Cylindrical Tanks Under Action of Uplifting Acceleration

2009 ◽  
Vol 132 (1) ◽  
Author(s):  
Tomoyo Taniguchi ◽  
Yoshinori Ando
Keyword(s):  
Velocity Potential ◽  
Fluid Velocity ◽  
Fluid Pressure ◽  
Angular Acceleration ◽  
Bottom Plate ◽  
Center Line ◽  
Cylindrical Tank ◽  
Cylindrical Coordinates ◽  
Flat Bottom ◽  
Cylindrical Tanks

To protect flat-bottom cylindrical tanks against severe damage from uplift motion, accurate evaluation of accompanying fluid pressures is indispensable. This paper presents a mathematical solution for evaluating the fluid pressure on a rigid flat-bottom cylindrical tank in the same manner as the procedure outlined and discussed previously by the authors (Taniguchi, T., and Ando, Y., 2010, “Fluid Pressures on Unanchored Rigid Rectangular Tanks Under Action of Uplifting Acceleration,” ASME J. Pressure Vessel Technol., 132(1), p. 011801). With perfect fluid and velocity potential assumed, the Laplace equation in cylindrical coordinates gives a continuity equation, while fluid velocity imparted by the displacement (and its time derivatives) of the shell and bottom plate of the tank defines boundary conditions. The velocity potential is solved with the Fourier–Bessel expansion, and its derivative, with respect to time, gives the fluid pressure at an arbitrary point inside the tank. In practice, designers have to calculate the fluid pressure on the tank whose perimeter of the bottom plate lifts off the ground like a crescent in plan view. However, the asymmetric boundary condition given by the fluid velocity imparted by the deformation of the crescent-like uplift region at the bottom cannot be expressed properly in cylindrical coordinates. This paper examines applicability of a slice model, which is a rigid rectangular tank with a unit depth vertically sliced out of a rigid flat-bottom cylindrical tank with a certain deviation from (in parallel to) the center line of the tank. A mathematical solution for evaluating the fluid pressure on a rigid flat-bottom cylindrical tank accompanying the angular acceleration acting on the pivoting bottom edge of the tank is given by an explicit function of a dimensional variable of the tank, but with Fourier series. It well converges with a few first terms of the Fourier series and accurately calculates the values of the fluid pressure on the tank. In addition, the slice model approximates well the values of the fluid pressure on the shell of a rigid flat-bottom cylindrical tank for any points deviated from the center line. For the designers’ convenience, diagrams that depict the fluid pressures normalized by the maximum tangential acceleration given by the product of the angular acceleration and diagonals of the tank are also presented. The proposed mathematical and graphical methods are cost effective and aid in the design of the flat-bottom cylindrical tanks that allow the uplifting of the bottom plate.

Stability of a String in Axial Flow

1985 ◽  
Vol 107 (4) ◽  
pp. 421-425 ◽  
Author(s):  
G. S. Triantafyllou ◽  
C. Chryssostomidis
Keyword(s):  
Stability Analysis ◽  
Frictional Coefficient ◽  
Axial Flow ◽  
Added Mass ◽  
Equation Of Motion ◽  
Bending Stiffness ◽  
Diameter Ratio ◽  
Constant Speed ◽  
Hamilton’S Principle ◽  
The Stability

The equation of motion of a long slender beam submerged in an infinite fluid moving with constant speed is derived using Hamilton’s principle. The upstream end of the beam is pinned and the downstream end is free to move. The resulting equation of motion is then used to perform the stability analysis of a string, i.e., a beam with negligible bending stiffness. It is found that the string is stable if (a) the external tension at the free end exceeds the value of a U2, where a is the “added mass” of the string and U the fluid speed; or (b) the length-over-diameter ratio exceeds the value 2Cf/π, where Cf is the frictional coefficient of the string.

scholarly journals Flutter of Long Flexible Cylinders in Axial Flow

2006 ◽  
Author(s):  
E. de Langre ◽  
M. P. Paidoussis ◽  
Y. Modarres-Sadeghi ◽  
O. Doare´
Keyword(s):  
Stability Analysis ◽  
Axial Flow ◽  
Equation Of Motion ◽  
External Flow ◽  
Transverse Motion ◽  
Control Parameters ◽  
Limit Case ◽  
Finite Region ◽  
Flexible Cylinder ◽  
The Stability

We consider the stability of a thin flexible cylinder considered as a beam, when subjected to axial flow and fixed at the up-stream end only. A linear stability analysis of transverse motion aims at determining the risk of flutter as a function of the governing control parameters such as the flow velocity or the length of the cylinder. Stability is analysed applying a finite difference scheme in space to the equation of motion expressed in the frequency domain. It is found that, contrary to previous predictions based on simplified theories, flutter may exist for very long cylinders, provided that the free downstream end of the cylinder is well-streamlined. More generally, a limit regime is found where the length of the cylinder does not affect the characteristics of the instability, and the deformation is confined to a finite region close to the downstream end. These results are found complementary to solutions derived for shorter cylinders and are confirmed by linear computations using a Galerkin method. A link is established to similar results on long hanging cantilevered systems with internal or external flow. The limit case of vanishing bending stiffness, where the cylinder is modelled as a string, is analysed and related to previous results. A simple model for the behaviour of long cylinders is proposed.

Fluid Pressure on Rectangular Tank Consisting of Rigid Side Walls and Rectilinearly Deforming Bottom Plate Due to Uplift Motion

2008 ◽  
Author(s):  
Tomoyo Taniguchi ◽  
Toru Segawa
Keyword(s):  
Velocity Potential ◽  
Fluid Velocity ◽  
Fluid Pressure ◽  
Angular Acceleration ◽  
Parabolic Partial Differential Equation ◽  
Bottom Plate ◽  
Flat Bottom ◽  
Fourier Cosine ◽  
Rectangular Tank ◽  
Rigid Walls

Although the uplift motion of flat bottom cylindrical shell tanks has been considered to contribute toward various damage to tanks, the mechanics were not fully understood. As well as computing uplift displacement of the tanks, accurate prediction of fluid pressure accompanied with uplift motion is indispensable to prevent the tanks from severe damage. This paper mathematically derives the fluid pressure on a rectangular tank with unit depth consisting of rigid walls and rectilinearly deforming bottom plate accompanied with the uplift motion. Assuming perfect fluid and velocity potential, the continuity equation is given by Laplace equation. The fluid velocity imparted by motion of rigid walls, immobile bottom plate and rectilinearly deformed bottom plate of tank accompanied with uplift of the tank constitutes boundary conditions. Since this problem is set as the parabolic partial differential equation of Neumann problem, the velocity potential is solved with Fourier-cosine expansion. Derivatives of the velocity potential with respect to time give the fluid pressure at arbitrary point inside the tank. The proposed mathematical solution well converges with a first few terms of Fourier series. Diagrams that depict the fluid pressure normalized by product of angular acceleration and diagonals of the tank are presented.

scholarly journals Stability analysis and breakup length calculations for steady planar liquid jets

2010 ◽  
Vol 668 ◽  
pp. 384-411 ◽  
Author(s):  
M. R. TURNER ◽  
J. J. HEALEY ◽  
S. S. SAZHIN ◽  
R. PIAZZESI
Keyword(s):  
Stability Analysis ◽  
Velocity Profile ◽  
Shear Layer ◽  
Fluid Velocity ◽  
Finite Thickness ◽  
Liquid Jet ◽  
Absolute Instability ◽  
Breakup Length ◽  
The Stability ◽  
Thickness Shear

This study uses spatio-temporal stability analysis to investigate the convective and absolute instability properties of a steady unconfined planar liquid jet. The approach uses a piecewise linear velocity profile with a finite-thickness shear layer at the edge of the jet. This study investigates how properties such as the thickness of the shear layer and the value of the fluid velocity at the interface within the shear layer affect the stability properties of the jet. It is found that the presence of a finite-thickness shear layer can lead to an absolute instability for a range of density ratios, not seen when a simpler plug flow velocity profile is considered. It is also found that the inclusion of surface tension has a stabilizing effect on the convective instability but a destabilizing effect on the absolute instability. The stability results are used to obtain estimates for the breakup length of a planar liquid jet as the jet velocity varies. It is found that reducing the shear layer thickness within the jet causes the breakup length to decrease, while increasing the fluid velocity at the fluid interface within the shear layer causes the breakup length to increase. Combining these two effects into a profile, which evolves realistically with velocity, gives results in which the breakup length increases for small velocities and decreases for larger velocities. This behaviour agrees qualitatively with existing experiments on the breakup length of axisymmetric jets.

Fluid Pressure on Rectangular Rigid Tanks Due to Uplift Motion

2007 ◽  
Author(s):  
Tomoyo Taniguchi ◽  
Yoshinori Ando
Keyword(s):  
Cylindrical Shell ◽  
Velocity Potential ◽  
Fluid Velocity ◽  
Fluid Pressure ◽  
Design Procedure ◽  
Parabolic Partial Differential Equation ◽  
Bottom Plate ◽  
Flat Bottom ◽  
Side Walls ◽  
Unit Depth

This paper mathematically derives fluid pressure on a rectangular rigid tank with unit depth, which models a section of a center part of a flat-bottom cylindrical shell tank, accompanied with uplift motion. Employing boundary conditions consisting of fluid velocity imparted by motion of the side walls and bottom plate of the tank along with uplifting, the equation of continuity of fluid given by the Laplace equation is solved as the parabolic partial differential equation of Neumann problem. The fluid pressure is given by a function of the velocity potential. Comparison of mathematical results with numerical ones based on explicit FE analysis corroborates its accuracy and applicability on design procedure of flat-bottom cylindrical shell tanks.

scholarly journals Stability Analysis in Determining Safety Drilling Fluid Pressure Windows in Ice Drilling Boreholes

Energies ◽  
2018 ◽  
Vol 11 (12) ◽  
pp. 3378
Author(s):  
Han Zhang ◽  
Dongbin Pan ◽  
Lianghao Zhai ◽  
Ying Zhang ◽  
Chen Chen
Keyword(s):  
Stability Analysis ◽  
Oil And Gas ◽  
Drilling Fluid ◽  
Fluid Pressure ◽  
Failure Criteria ◽  
Borehole Wall ◽  
Polar Regions ◽  
Borehole Stability ◽  
Oil And Gas Exploration ◽  
The Stability

Borehole stability analysis has been well studied in oil and gas exploration when drilling through rock formations. However, a related analysis of ice borehole stability has never been conducted. This paper proposes an innovative method for estimating the drilling fluid pressure window for safe and sustainable ice drilling, which has never been put forward before. First, stress concentration on a vertical ice borehole wall was calculated, based on the common elastic theory. Then, three failure criteria, the Mogi–Coulomb, teardrop, and Derradji-Aouat criteria, were used to predict the stability of the ice borehole for an unbroken borehole wall. At the same time, fracture mechanics were used to analyze the stable critical pressure for a fissured wall. Combining with examples, our discussion shows how factors like temperature, strain rate, ice fracture toughness, ice friction coefficient, and fracture/crack length affect the stability of the borehole wall. The results indicate that the three failure criteria have similar critical pressures for unbroken borehole stability and that a fissured borehole could significantly decrease the safety drilling fluid pressure window and reduce the stability of the borehole. The proposed method enriches the theory of borehole stability and allows drillers to adjust the drilling fluid density validly in ice drilling engineering, for potential energy exploration in polar regions.

Flutter of a Rectangular Cantilevered Plate

2006 ◽  
Author(s):  
Christophe Eloy ◽  
Claire Souilliez ◽  
Lionel Schouveiler
Keyword(s):  
Fourier Transforms ◽  
Flexural Rigidity ◽  
Fluid Pressure ◽  
Axial Flow ◽  
Three Dimensions ◽  
Fluid Viscosity ◽  
Mass Ratio ◽  
Critical Flow Velocity ◽  
Plate Aspect Ratio ◽  
The Stability

We address theoretically the stability of a cantilevered rectangular plate in an uniform and incompressible axial flow. We assume that the fluid viscosity and the plate viscoelastic damping are negligible. In this limit, a flutter instability arises from a competition between the destabilising fluid pressure and the stabilising flexural rigidity of the plate. The flutter modes are assumed to be two-dimensional but the potential flow is calculated in three dimensions in the asymptotic limit of large plate span. Using a Galerkin method and Fourier transforms, we are able to predict the flutter modes, their frequencies and growth rates. The critical flow velocity is calculated as a function of the mass ratio and the plate aspect ratio. We demonstrate a new result: a plate of finite span is more stable than a plate of infinite span.

Spatial stability and the onset of absolute instability of Batchelor's vortex for high swirl numbers

2007 ◽  
Vol 583 ◽  
pp. 27-43 ◽  
Author(s):  
L. PARRAS ◽  
R. FERNANDEZ-FERIA
Keyword(s):  
Stability Analysis ◽  
Temporal Stability ◽  
Axial Flow ◽  
Reynolds Numbers ◽  
Absolute Instability ◽  
Spatial Stability ◽  
The Past ◽  
Trailing Vortices ◽  
The Stability ◽  
Large Aircraft

Batchelor's vortex has been commonly used in the past as a model for aircraft trailing vortices. Using a temporal stability analysis, new viscous unstable modes have been found for the high swirl numbers of interest in actual large-aircraft vortices. We look here for these unstable viscous modes occurring at large swirl numbers (q > 1.5), and large Reynolds numbers (Re >103), using a spatial stability analysis, thus characterizing the frequencies at which these modes become convectively unstable for different values of q, Re, and for different intensities of the uniform axial flow. We consider both jet-like and wake-like Batchelor's vortices, and are able to analyse the stability for Re as high as 108. We also characterize the frequencies and the swirl numbers for the onset of absolute instabilities of these unstable viscous modes for large q.

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