Derivation of Governing Equations For Self-Acting Foil Bearings

1967 ◽  
Vol 89 (3) ◽  
pp. 334-339 ◽  
Author(s):  
E. J. Barlow
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Bending Stiffness ◽  
The Self ◽  
Governing Equations ◽  
Foil Bearings ◽  
Foil Bearing

Contained is the derivation of the equations for the self-acting foil bearing. These equations include the effects of bending stiffness of the tape and of compressibility of the lubricant. They are nonlinear, and the boundary conditions are divided equally between the two ends of the tape. These complications even make obtaining numerical solutions difficult. Linearized solutions are derived for large wrap angles neglecting the bending stiffness of the tape.

scholarly journals Numerical Boundary Conditions for Solar Magnetic Hydrodynamic Flows Using the Method of Characteristics

1996 ◽  
Vol 154 ◽  
pp. 149-153
Author(s):  
S. T. Wu ◽  
A. H. Wang ◽  
W. P. Guo
Keyword(s):  
Boundary Conditions ◽  
Method Of Characteristics ◽  
Numerical Solutions ◽  
The Self ◽  
Time Dependent ◽  
Solar Plasma ◽  
The Method Of Characteristics ◽  
Mhd Simulations ◽  
Self Consistent ◽  
Dynamic Structures

AbstractWe discuss the self-consistent time-dependent numerical boundary conditions on the basis of theory of characteristics for magnetohydrodynamics (MHD) simulations of solar plasma flows. The importance of using self-consistent boundary conditions is demonstrated by using an example of modeling coronal dynamic structures. This example demonstrates that the self-consistent boundary conditions assure the correctness of the numerical solutions. Otherwise, erroneous numerical solutions will appear.

scholarly journals On the consistency of two-phase local/nonlocal piezoelectric integral model

2021 ◽  
Vol 42 (11) ◽  
pp. 1581-1598
Author(s):  
Yanming Ren ◽  
Hai Qing
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Integral Model ◽  
Governing Equations ◽  
Two Phase ◽  
Differential Model ◽  
Piezoelectric Beam ◽  
General Strain

AbstractIn this paper, we propose general strain- and stress-driven two-phase local/nonlocal piezoelectric integral models, which can distinguish the difference of nonlocal effects on the elastic and piezoelectric behaviors of nanostructures. The nonlocal piezoelectric model is transformed from integral to an equivalent differential form with four constitutive boundary conditions due to the difficulty in solving intergro-differential equations directly. The nonlocal piezoelectric integral models are used to model the static bending of the Euler-Bernoulli piezoelectric beam on the assumption that the nonlocal elastic and piezoelectric parameters are coincident with each other. The governing differential equations as well as constitutive and standard boundary conditions are deduced. It is found that purely strain- and stress-driven nonlocal piezoelectric integral models are ill-posed, because the total number of differential orders for governing equations is less than that of boundary conditions. Meanwhile, the traditional nonlocal piezoelectric differential model would lead to inconsistent bending response for Euler-Bernoulli piezoelectric beam under different boundary and loading conditions. Several nominal variables are introduced to normalize the governing equations and boundary conditions, and the general differential quadrature method (GDQM) is used to obtain the numerical solutions. The results from current models are validated against results in the literature. It is clearly established that a consistent softening and toughening effects can be obtained for static bending of the Euler-Bernoulli beam based on the general strain- and stress-driven local/nonlocal piezoelectric integral models, respectively.

An Experimental Study of High-Speed Rotors Supported by Air-Lubricated Foil Bearings—Part 1: Rotation in Pressurized and Self-Acting Foil Bearings

1969 ◽  
Vol 91 (3) ◽  
pp. 477-493 ◽  
Author(s):  
L. Licht
Keyword(s):  
Experimental Study ◽  
Temperature Rise ◽  
High Speed ◽  
Air Gap ◽  
The Self ◽  
Wear Characteristics ◽  
Thrust Bearings ◽  
Foil Bearings ◽  
Foil Bearing ◽  
Mode Of Operation

A high-speed rotor, supported by an air-lubricated foil bearing, is rotated in both the vertical and horizontal attitudes at speeds in excess of 60,000 rpm. The rotor is stable and free from “half-frequency” or “fractional-frequency” whirl instability encountered in conventional gas bearings. External pressurization is applied to separate the foil surfaces from the journal during the initial and final stages of rotation, with adequate self-acting support and foil separation established at relatively low transition speeds. In the pressurized mode of operation, the system is characterized by a series of ultra-harmonic resonances, of sharply defined frequencies, related by fractions to speeds of synchronous resonance. In the self-acting mode of operation, the response of the system to residual imbalance is influenced by both the foil bearing and by the pressurized thrust bearings. The magnitude of the air gap (clearance) is determined at various rotational speeds and compared with theoretically predicted results. The temperature rise of the foil with speed is measured at various locations in order to assess its contribution to clearance growth. The journal and foil surfaces are examined and it is found that the foil bearing is endowed with excellent wipe-wear characteristics.

Dynamic Responses of a Self-Moving Precision Positioning Stage Impacted by a Spring-Mounted Piezoelectric Actuator

2003 ◽  
Vol 125 (4) ◽  
pp. 658-661 ◽  
Author(s):  
Rong-Fong Fung, ◽  
Yung-Tien Liu, ◽  
Tai-Kun Huang, ◽  
Toshiro Higuchi,
Keyword(s):  
Piezoelectric Actuator ◽  
Numerical Solutions ◽  
High Accuracy ◽  
The Self ◽  
Dynamic Responses ◽  
Nanometer Scale ◽  
The Finite Element Method ◽  
Governing Equations ◽  
Precision Positioning ◽  
Positioning Stage

The piezoelectric actuator (PA) has been used for precision positioning from micrometer down to nanometer scale. In this paper, a spring-mounted PA is designed to achieve a high accuracy and self-moving ability in precision positioning motion. The contact force between the hammer and the self-moving stage, and the friction force of Leuven’s model caused between the grinded groove and the self-moving stage are considered. The governing equations of the system are formulated by using the finite-element method (FEM). The numerical solutions are provided to compare with the experimental results, and demonstrate the well agreement of the present theoretical formulations.

Vibration Considerations in Foil-Bearing Design

1999 ◽  
Vol 66 (2) ◽  
pp. 432-438 ◽  
Author(s):  
A. A. Renshaw
Keyword(s):  
Slip Flow ◽  
Bending Stiffness ◽  
The Other ◽  
Bearing Surface ◽  
Web Handling ◽  
Foil Bearings ◽  
Viscous Forces ◽  
Foil Bearing ◽  
Bearing Design ◽  
Two Parameters

The semianalytic foil-bearing solution algorithm of Eshel and Elrod (1965) is extended to the solution of the linearized, free vibration problem for one-dimensional self-pressurized foil bearings. The results demonstrate that unwanted variations in the spacing between the moving foil and the stationary bearing surface can be eliminated through proper design. The penetration depth through which vibration of the free span penetrates into the foil bearing is determined by two exponential exponents, one describing inlet penetration, the other describing outlet penetration. When the inlet exponent is large and negative and the outlet exponent is large and positive, there is negligible coupling between the vibration of the free spans and the vibration of the spacing between the foil and the stationary bearing surface. This decoupling is desirable in magnetic recording and web handling applications and can be achieved by properly selecting two dimensionless parameters, one describing the ratio of the viscous forces to the tape tension, the other describing the ratio of the tape transport speed to the wave speed in the tape. The values of these two parameters in current designs of both magnetic tape recording and web-handling devices are consistent with the design goal of minimizing foil vibration over the bearing. The inlet and outlet exponents are the roots of a fourth-order polynomial, and, in most cases, good estimates for these roots can be found without explicitly solving the foil-bearing problem. The effects of the air compressibility, tape bending stiffness, and slip flow are also investigated. Tape bending stiffness is found to play a significant role in vibration coupling. These results provide new insight into the influence of vibration on foil-bearing design.

A Theoretical Study of the Dynamic Behavior of Foil Bearings

1971 ◽  
Vol 93 (1) ◽  
pp. 133-142 ◽  
Author(s):  
T. B. Barnum ◽  
H. G. Elrod
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Dynamic Behavior ◽  
Theoretical Study ◽  
Compressibility Factor ◽  
Frequency Parameter ◽  
Numerical Results ◽  
Foil Bearings ◽  
Foil Bearing

The differential equations and boundary conditions for the problem of a foil bearing subjected to small variations in tape tension are developed. The numerical results show that the fluctuations of the foil are a function or the frequency parameter ω* = 2r0ωε1/3H0*/U and of the compressibility factor θ = T0/par0.

Core Variation in an Externally Pressurized Converging Thrust Bearing with Bingham Lubricant

2016 ◽  
Vol 852 ◽  
pp. 428-434 ◽  
Author(s):  
I. Jayakaran Amalraj ◽  
G. Alexander Raymand
Keyword(s):  
Film Thickness ◽  
Boundary Conditions ◽  
Yield Surface ◽  
Constitutive Equations ◽  
Thrust Bearing ◽  
Numerical Solutions ◽  
Flow Region ◽  
Core Region ◽  
Governing Equations ◽  
The Core

The effects of angle of convergence on the shape and thickness of the core are analyzed theoretically by considering variable film thickness in an externally pressurized circular thrust bearing. Using the assumptions of the lubrication theory, modified Reynold’s equation and the governing equations are obtained. Using the boundary conditions of the problem in the constitutive equations we get the velocity of the core region as well as flow region. By considering the equilibrium of an element in the yield surface, an algebraic equation to determine the thickness of the yield surface is derived. Numerical solutions are obtained for the thickness of yield surface and velocities for various values of Bingham Numbers and the angle of convergence.

scholarly journals Vibration of rotating circular cylindrical shells with distributed springs

2021 ◽  
Vol 37 ◽  
pp. 346-358
Author(s):  
Fuchun Yang ◽  
Xiaofeng Jiang ◽  
Fuxin Du
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Shell Theory ◽  
Rotation Speed ◽  
Cylindrical Shells ◽  
Free Vibrations ◽  
Governing Equations ◽  
Natural Response ◽  
Circular Cylindrical Shells ◽  
The Galerkin Method

Abstract Free vibrations of rotating cylindrical shells with distributed springs were studied. Based on the Flügge shell theory, the governing equations of rotating cylindrical shells with distributed springs were derived under typical boundary conditions. Multicomponent modal functions were used to satisfy the distributed springs around the circumference. The natural responses were analyzed using the Galerkin method. The effects of parameters, rotation speed, stiffness, and ratios of thickness/radius and length/radius, on natural response were also examined.

Thermal Post-Buckling Analysis of Functionally Graded Composite Plates with Nonlinear Aerodynamic Forces

2010 ◽  
Vol 123-125 ◽  
pp. 280-283
Author(s):  
Chang Yull Lee ◽  
Ji Hwan Kim
Keyword(s):  
Material Properties ◽  
Numerical Solutions ◽  
Virtual Work ◽  
Shear Deformation Theory ◽  
Composite Plates ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Governing Equations ◽  
Functionally Graded Composite ◽  
Post Buckling

The post-buckling of the functionally graded composite plate under thermal environment with aerodynamic loading is studied. The structural model has three layers with ceramic, FGM and metal, respectively. The outer layers of the sandwich plate are different homogeneous and isotropic material properties for ceramic and metal. Whereas the core is FGM layer, material properties vary continuously from one interface to the other in the thickness direction according to a simple power law distribution in terms of the volume fractions. Governing equations are derived by using the principle of virtual work and numerical solutions are solved through a finite element method. The first-order shear deformation theory and von-Karman strain-displacement relations are based to derive governing equations of the plate. Aerodynamic effects are dealt by adopting nonlinear third-order piston theory for structural and aerodynamic nonlinearity. The Newton-Raphson iterative method applied for solving the nonlinear equations of the thermal post-buckling analysis

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