Stability of a Rotor Whose Cavity Has an Arbitrary Meridian and Is Partially Filled With Fluid

1985 ◽  
Vol 107 (4) ◽  
pp. 440-445 ◽  
Author(s):  
R. Wohlbru¨ck
Keyword(s):  
Boundary Conditions ◽  
Close Correlation ◽  
Experimental Investigations ◽  
The Finite Element Method ◽  
Field Equations ◽  
Equilibrium Conditions ◽  
Conical Cavity ◽  
Rotor Spinning ◽  
The Stability ◽  
Elastically Supported

The stability of an elastically supported rotor spinning with constant angular velocity is studied. The rotor has a cavity of arbitrary meridian and is partially filled with an ideal fluid. The motions of the system are governed by linearized equilibrium conditions for the rotor and field equations as well as boundary conditions for the fluid. Due to the arbitrary shape of the meridian, it is not possible to solve the boundary value problem in closed form. Therefore a variational expression is developed which satisfies the boundary conditions naturally. The variational problem is solved approximately by the finite element method. The results, incorporated in the equilibrium conditions for the rotor, lead to stability statements. For numerical and experimental investigations, two rotors, one with an elliptical and one with a conical cavity are used. The fill medium is water. There is a close correlation between numerical and experimental results.

scholarly journals STABILITY OF ROTATING CIRCULAR CYLINDRICAL SHELL SUBJECT TO AN AXIAL AND ROTATIONAL FLUID FLOW

2017 ◽  
Vol 18 (9) ◽  
pp. 84-97
Author(s):  
S.A. Bochkarev ◽  
V.P. Matveenko
Keyword(s):  
Finite Element Method ◽  
Fluid Flow ◽  
Boundary Conditions ◽  
Circular Cylindrical Shell ◽  
Loss Of Stability ◽  
Rotating Fluid ◽  
The Finite Element Method ◽  
The Stability ◽  
Rotating Shell ◽  
Simultaneous Rotation

This paper is concerned with the stability analysis of rotating cylindrical shells conveying a co-rotating fluid. The problem is solved by the finite element method for shells subjected to different boundary conditions. It has been found that the loss of stability for a rotating shell under the action of the fluid having both axial and circumferential velocity components depends on the type of boundary conditions imposed on the shell ends. The results of numerical calculations have shown that for different variants of boundary conditions a simultaneous rotation of shell and the fluid causes an increase or decrease in the critical velocity of axial fluid flow.

The Dynamics of MEMS Arches of Non-Ideal Boundary Conditions

2011 ◽  
Author(s):  
Sami A. Alkharabsheh ◽  
Mohammad I. Younis
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Electrostatic Force ◽  
Dynamical Behavior ◽  
Experimental Investigations ◽  
Harmonic Load ◽  
Ideal Boundary ◽  
Shooting Technique ◽  
Softening Behavior ◽  
The Stability

We present an investigation into the dynamics of MEMS arches when actuated electrically including the effect of their flexible (non-ideal) supports. First, the eigenvalue problem of a nonlinear Euler-Bernoulli shallow arch with torsional and transversal springs at the boundaries is solved analytically. Several results are shown to demonstrate the possibility of tuning the theoretically obtained natural frequencies of an arch to match the experimentally measured. Then, simulation results are shown for the forced vibration response of an arch when excited by a DC electrostatic force superimposed to an AC harmonic load. Shooting technique is utilized to find periodic motions. The stability of the captured periodic motion is examined using the Floquet theory. The results show several jumps in the response during snap-through motion and pull-in. Theoretical and experimental investigations are conducted on a microfabricated curved beam actuated electrically. Results show softening behavior and superharmonic resonances. It is demonstrated that non-ideal boundary conditions can have significant effect on the qualitative dynamical behavior of the MEMS arch, including its natural frequencies, snap-through behavior, and dynamic pull-in.

Vibration and Frequency Analysis of Non-Uniform Timoshenko Beams Subjected to Axial Forces

2003 ◽  
Author(s):  
A. R. Ohadi ◽  
H. Mehdigholi ◽  
E. Esmailzadeh
Keyword(s):  
Stability Analysis ◽  
Boundary Conditions ◽  
Axial Force ◽  
Frequency Equation ◽  
Axial Loads ◽  
Various Boundary Conditions ◽  
Axial Forces ◽  
First Case ◽  
The Stability ◽  
Elastically Supported

Dynamic and stability analysis of non-uniform Timoshenko beam under axial loads is carried out. In the first case of study, the axial force is assumed to be perpendicular to the shear force, while for the second case the axial force is tangent to the axis of the beam column. For each case, a pair of differential equations coupled in terms of the flexural displacement and the angle of rotation due to bending was obtained. The parameters of the frequency equation were determined for various boundary conditions. Several illustrative examples of uniform and non-uniform beams with different boundary conditions such as clamped supported, elastically supported, and free end mass have been presented. The stability analysis, for the variation of the natural frequencies of the uniform and non-uniform beams with the axial force, has also been investigated.

Vibrations of an Elastically Mounted Spinning Rotor Partially Filled With Liquid

1982 ◽  
Vol 104 (2) ◽  
pp. 389-396 ◽  
Author(s):  
G. Lichtenberg
Keyword(s):  
Equations Of Motion ◽  
Constant Angular Velocity ◽  
Two Dimensional ◽  
Field Equations ◽  
Stability Boundaries ◽  
Inviscid Incompressible Fluid ◽  
Wide Range ◽  
The Stability ◽  
Elastically Supported ◽  
Dimensional Surface

The stability of a rotor with a cylindrical cavity, spinning with constant angular velocity and partially filled with an inviscid, incompressible fluid is studied. The rotor is elastically supported on a vertically mounted massless shaft in overhung position. A set of coupled linearized spatial equations of motion of the rotor and field equations, as well as boundary conditions of the liquid, is established and solved, leading to a characteristic equation. First numerical results predict a wide range of rotor speeds, where the system performs unstable motions caused by a two-dimensional surface wave of the liquid. The stability boundaries are calculated for a flat rotor in dependence on the mass of the contained liquid and agree extremely well with experimental data.

Stability Threshold of Flexibly Supported Hybrid Gas Journal Bearings

1979 ◽  
Vol 101 (4) ◽  
pp. 451-457 ◽  
Author(s):  
Zbyszko Kazimierski ◽  
Krzysztof Jarzecki
Keyword(s):  
Dynamic Properties ◽  
Theoretical Calculations ◽  
Journal Bearings ◽  
Experimental Investigations ◽  
Elastic Supports ◽  
Stability Threshold ◽  
The Stability ◽  
Elastically Supported ◽  
Gas Bearing ◽  
Bearing System

Results of experimental investigations of the dynamic properties of elastic supports for gas bearings having the form of rubber O-rings are presented. Theoretical calculations of the stability threshold of an externally pressurized gas bearing system elastically supported by means of O-rings were performed. An experimental investigation of the stability threshold of this gas bearing system was made. Comparisons between theoretical and experimental results verify the theoretical model and illustrate the possibility of its application to design purposes.

Buckling of Circular Cylindrical Shells Using a Displacement Function

1974 ◽  
Vol 96 (4) ◽  
pp. 1322-1327
Author(s):  
Shun Cheng ◽  
C. K. Chang
Keyword(s):  
Boundary Conditions ◽  
Axial Compression ◽  
Cylindrical Shells ◽  
External Pressure ◽  
Displacement Function ◽  
Combined Loading ◽  
Trial Solution ◽  
Governing Differential Equation ◽  
Circular Cylindrical Shells ◽  
The Stability

The buckling problem of circular cylindrical shells under axial compression, external pressure, and torsion is investigated using a displacement function φ. A governing differential equation for the stability of thin cylindrical shells under combined loading of axial compression, external pressure, and torsion is derived. A method for the solutions of this equation is also presented. The advantage in using the present equation over the customary three differential equations for displacements is that only one trial solution is needed in solving the buckling problems as shown in the paper. Four possible combinations of boundary conditions for a simply supported edge are treated. The case of a cylinder under axial compression is carried out in detail. For two types of simple supported boundary conditions, SS1 and SS2, the minimum critical axial buckling stress is found to be 43.5 percent of the well-known classical value Eh/R3(1−ν2) against the 50 percent of the classical value presently known.

scholarly journals Dirichlet absorbing boundary conditions for classical and peridynamic diffusion-type models

2020 ◽  
Vol 66 (4) ◽  
pp. 773-793 ◽  
Author(s):  
Arman Shojaei ◽  
Alexander Hermann ◽  
Pablo Seleson ◽  
Christian J. Cyron
Keyword(s):  
Boundary Conditions ◽  
Materials Science ◽  
Unbounded Domains ◽  
Diffusion Models ◽  
Absorbing Boundary Conditions ◽  
The Finite Element Method ◽  
Absorbing Boundary ◽  
Diffusion Type ◽  
2D And 3D ◽  
Accuracy And Stability

Abstract Diffusion-type problems in (nearly) unbounded domains play important roles in various fields of fluid dynamics, biology, and materials science. The aim of this paper is to construct accurate absorbing boundary conditions (ABCs) suitable for classical (local) as well as nonlocal peridynamic (PD) diffusion models. The main focus of the present study is on the PD diffusion formulation. The majority of the PD diffusion models proposed so far are applied to bounded domains only. In this study, we propose an effective way to handle unbounded domains both with PD and classical diffusion models. For the former, we employ a meshfree discretization, whereas for the latter the finite element method (FEM) is employed. The proposed ABCs are time-dependent and Dirichlet-type, making the approach easy to implement in the available models. The performance of the approach, in terms of accuracy and stability, is illustrated by numerical examples in 1D, 2D, and 3D.

scholarly journals Determination of Transverse Shear Stiffness of Sandwich Panels with a Corrugated Core by Numerical Homogenization

Materials ◽  
2021 ◽  
Vol 14 (8) ◽  
pp. 1976
Author(s):  
Tomasz Garbowski ◽  
Tomasz Gajewski
Keyword(s):  
Transverse Shear ◽  
Shear Stiffness ◽  
The Finite Element Method ◽  
Homogenization Procedure ◽  
Plate Structures ◽  
Corrugated Board ◽  
Energy Equivalence ◽  
The Stability ◽  
Global Stiffness Matrix

Knowing the material properties of individual layers of the corrugated plate structures and the geometry of its cross-section, the effective material parameters of the equivalent plate can be calculated. This can be problematic, especially if the transverse shear stiffness is also necessary for the correct description of the equivalent plate performance. In this work, the method proposed by Biancolini is extended to include the possibility of determining, apart from the tensile and flexural stiffnesses, also the transverse shear stiffness of the homogenized corrugated board. The method is based on the strain energy equivalence between the full numerical 3D model of the corrugated board and its Reissner-Mindlin flat plate representation. Shell finite elements were used in this study to accurately reflect the geometry of the corrugated board. In the method presented here, the finite element method is only used to compose the initial global stiffness matrix, which is then condensed and directly used in the homogenization procedure. The stability of the proposed method was tested for different variants of the selected representative volume elements. The obtained results are consistent with other technique already presented in the literature.

scholarly journals Stability charts based on the finite element method for underground cavities in soft carbonate rocks: validation through case-study applications

2019 ◽  
Vol 19 (10) ◽  
pp. 2079-2095 ◽  
Author(s):  
Michele Perrotti ◽  
Piernicola Lollino ◽  
Nunzio Luciano Fazio ◽  
Mario Parise
Keyword(s):  
Finite Element ◽  
Safety Margin ◽  
Structural Elements ◽  
The Finite Element Method ◽  
Geometrical Features ◽  
Overall Stability ◽  
Underground Cavities ◽  
The Stability ◽  
Rock Mechanical Properties

Abstract. The stability of man-made underground cavities in soft rocks interacting with overlying structures and infrastructures represents a challenging problem to be faced. Based upon the results of a large number of parametric two-dimensional (2-D) finite-element analyses of ideal cases of underground cavities, accounting for the variability both cave geometrical features and rock mechanical properties, specific charts have been recently proposed in the literature to assess at a preliminary stage the stability of the cavities. The purpose of the present paper is to validate the efficacy of the stability charts through the application to several case studies of underground cavities, considering both quarries collapsed in the past and quarries still stable. The stability graphs proposed by Perrotti et al. (2018) can be useful to evaluate, in a preliminary way, a safety margin for cavities that have not reached failure and to detect indications of predisposition to local or general instability phenomena. Alternatively, for sinkholes that already occurred, the graphs may be useful in identifying the conditions that led to the collapse, highlighting the importance of some structural elements (as pillars and internal walls) on the overall stability of the quarry system.

Low Buckling Loads of Axially Compressed Conical Shells

1970 ◽  
Vol 37 (2) ◽  
pp. 384-392 ◽  
Author(s):  
M. Baruch ◽  
O. Harari ◽  
J. Singer
Keyword(s):  
Boundary Conditions ◽  
Axial Compression ◽  
Plane Boundary ◽  
Essential Difference ◽  
Physical Reason ◽  
Buckling Loads ◽  
Conical Shells ◽  
Simple Supports ◽  
Interaction Curves ◽  
The Stability

The stability of simply supported conical shells under axial compression is investigated for 4 different sets of in-plane boundary conditions with a linear Donnell-type theory. The first two stability equations are solved by the assumed displacement, while the third is solved by a Galerkin procedure. The boundary conditions are satisfied with 4 unknown coefficients in the expression for u and v. Both circumferential and axial restraints are found to be of primary importance. Buckling loads about half the “classical” ones are obtained for all but the stiffest simple supports SS4 (v = u = 0). Except for short shells, the effects do not depend on the length of the shell. The physical reason for the low buckling loads in the SS3 case is explained and the essential difference between cylinder and cone in this case is discussed. Buckling under combined axial compression and external or internal pressure is studied and interaction curves have been calculated for the 4 sets of in-plane boundary conditions.

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