Nonlinear Numerical Prediction of Gas Foil Bearing Stability and Unbalanced Response

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
Sébastien Le Lez ◽  
Mihaï Arghir ◽  
Jean Frêne
Keyword(s):  
Dry Friction ◽  
Stability Limit ◽  
Time Step ◽  
Structural Dynamic ◽  
Friction Forces ◽  
Foil Bearings ◽  
Mass Unbalance ◽  
Foil Bearing ◽  
Highly Nonlinear ◽  
The Stability

One of the main interests of gas foil bearings lies in their superior rotordynamic characteristics compared with conventional bearings. A numerical investigation on the stability limit and on the unbalanced response of foil bearings is presented in this paper. The main difficulty in modeling the dynamic behavior of such bearings comes from the dry friction that occurs within the foil structure. Indeed, dry friction is highly nonlinear and is strongly influenced by the dynamic amplitude of the pressure field. To deal with these nonlinearities, a structural dynamic model has been developed in a previous work. This model considers the entire corrugated foil and the interactions between the bumps by describing the foil bearing structure as a multiple degrees of freedom system. It allows the determination of the dynamic friction forces at the top and at the bottom of the bumps by simple integration of ordinary differential equations. The dynamic displacements of the entire corrugated sheet are then easily obtained at each time step. The coupling between this structural model and a gas bearing prediction code is presented in this paper and allows performing full nonlinear analyses of a complete foil bearing. The bearing stability is the first investigated problem. The results show that the structural deflection enhances the stability of compliant surface bearings compared with rigid ones. Moreover, when friction is introduced, a new level of stability is reached, revealing the importance of this dissipation mechanism. The second investigated problem is the unbalanced response of foil bearings. The shaft trajectories depict a nonlinear jump in the response of both rigid and foil bearings when the value of the unbalance increases. Again, it is evidenced that the foil bearing can support higher mass unbalance before this undesirable step occurs.

Nonlinear Numerical Prediction of Gas Foil Bearings Stability and Unbalanced Response

2008 ◽  
Author(s):  
Se´bastien Le Lez ◽  
Mihai¨ Arghir ◽  
Jean Frene
Keyword(s):  
Dry Friction ◽  
Stability Limit ◽  
Time Step ◽  
Structural Dynamic ◽  
Friction Forces ◽  
Foil Bearings ◽  
Mass Unbalance ◽  
Foil Bearing ◽  
Non Linear ◽  
The Stability

One of the main interests of gas foil bearings lies in their superior rotordynamic characteristics compared to conventional bearings. A numerical investigation on the stability limit and on the unbalanced response of foil bearings is presented in this paper. The main difficulty in modeling the dynamic behavior of such bearings comes from the dry friction that occurs within the foil structure. Indeed, dry friction is highly non linear and is strongly influenced by the dynamic amplitude of the pressure field. To deal with these non linearities, a structural dynamic model has been developed in a previous work. This model considers the entire corrugated foil and the interactions between the bumps by describing the foil bearing structure as a multiple degrees of freedom system. It allows the determination of the dynamic friction forces at the top and at the bottom of the bumps by simple integration of ordinary differential equations. The dynamic displacements of the entire corrugated sheet are then easily obtained at each time step. The coupling between this structural model and a gas bearing prediction code is presented in this paper and allows performing full non linear analyses of a complete foil bearing. The bearing stability is the first investigated problem. The results show that the structural deflection enhances the stability of compliant surface bearings compared to rigid ones. Moreover, when friction is introduced, a new level of stability is reached revealing the importance of this dissipation mechanism. The second investigated problem is the unbalanced response of foil bearings. The shaft trajectories depict a non linear jump in the response of both rigid and foil bearings when the value of the unbalance increases. Again, it is evidenced that the foil bearing can support higher mass unbalance before this undesirable step occurs.

Stability analysis of rigid rotors supported by gas foil bearings coupled with electromagnetic actuators

2019 ◽  
Vol 234 (2) ◽  
pp. 427-443 ◽  
Author(s):  
Kamal Kumar Basumatary ◽  
Gaurav Kumar ◽  
Karuna Kalita ◽  
Sashindra K Kakoty
Keyword(s):  
Stability Analysis ◽  
Equations Of Motion ◽  
Electromagnetic Actuator ◽  
Electromagnetic Actuators ◽  
Foil Bearings ◽  
Damping Characteristics ◽  
Foil Bearing ◽  
Model Combining ◽  
The Stability ◽  
Rigid Rotors

Rotors supported on gas foil bearings have low damping characteristics, which limits its application. A possible solution could be an integration of a gas foil bearing with an electromagnetic actuator. This paper discusses the effect of electromagnetic actuators on the stability of a rotor supported on gas foil bearings. A coupled dynamic model combining the dynamics of gas foil bearing and electromagnetic actuator has been developed. The fluid film forces from the gas foil bearings and the electromagnetic forces from the electromagnetic actuators are integrated into the equations of motion of the rotor. The sub-synchronous vibration present in case of conventional gas foil bearings is reduced and the stability band of the rotor is increased due to the implementation of electromagnetic actuator.

Numerical Investigation on the Vibration of Steam Turbine Inlet Valves and the Feedback to the Dynamic Flow Field

2015 ◽  
Author(s):  
Clemens Bernhard Domnick ◽  
Friedrich-Karl Benra ◽  
Dieter Brillert ◽  
Hans Josef Dohmen ◽  
Christian Musch
Keyword(s):  
Steam Turbine ◽  
Structural Dynamics ◽  
Fluid Dynamic ◽  
Dynamic Flow ◽  
Piston Rings ◽  
Time Step ◽  
Structural Dynamic ◽  
Turbine Inlet ◽  
Valve Stem ◽  
Highly Nonlinear

The power output of steam turbines is controlled by steam turbine inlet valves. These valves have a large flow capacity and dissipate in throttled operation a huge amount of energy. Due to that, high dynamic forces occur in the valve which can cause undesired valve vibrations. In this paper, the structural dynamics of a valve are analysed. The dynamic steam forces obtained by previous computational fluid dynamic (CFD) calculations at different operating points are impressed on the structural dynamic finite element model (FEM) of the valve. Due to frictional forces at the piston rings and contact effects at the bushings of the valve plug and the valve stem the structural dynamic FEM is highly nonlinear and has to be solved in the time domain. Prior to the actual investigation grid and time step studies are carried out. Also the effect of the temperature distribution within the valve stem is discussed and the influence of the valve actuator on the vibrations is analysed. In the first step, the vibrations generated by the fluid forces are investigated. The effects of the piston rings on the structural dynamics are discussed. It is found, that the piston rings are able to reduce the vibration significantly by frictional damping. In the second step, the effect of the moving valve plug on the dynamic flow in the valve is analysed. The time dependent displacement of the valve is transferred to CFD calculations using deformable meshes. With this one way coupling method the response of the flow to the vibrations is analysed.

scholarly journals The Dynamic Characteristics of a Turborotor Simulator Supported on Gas-Lubricated Foil Bearings—Part 2: Operation With Heating and Thermal Gradients

1970 ◽  
Vol 92 (4) ◽  
pp. 650-659 ◽  
Author(s):  
L. Licht
Keyword(s):  
Dynamic Characteristics ◽  
High Speed ◽  
Elevated Temperatures ◽  
Load Capacity ◽  
Temperature Gradients ◽  
Thermal Gradients ◽  
Foil Bearings ◽  
Foil Bearing ◽  
Geometrical Imperfections ◽  
The Stability

A high-speed rotor, supported by gas-lubricated foil bearings, is free from self-excited whirl and displays no loss of load capacity when vibrated at frequency equal half the rotational speed [1]. It is demonstrated here that in addition to tolerance of geometrical imperfections, misalignment, and foreign particles [3, 4], the foil bearing performs well at elevated temperatures and accommodates appreciable temperature gradients. The foil bearing is endowed with superior wipe-wear characteristics, and the flexibility of the foil accounts not only for the stability of the foil bearing but also for its forgiveness with respect to distortion, contamination, and contact.

Link-Spring Model of Bump-Type Foil Bearings

2009 ◽  
Author(s):  
Kai Feng ◽  
Shigehiko Kaneko
Keyword(s):  
Experimental Data ◽  
Film Thickness ◽  
Wide Spectrum ◽  
Load Capacity ◽  
Ambient Air ◽  
Radial Clearance ◽  
Interaction Forces ◽  
Friction Forces ◽  
Foil Bearings ◽  
Foil Bearing

The field experiences of gas foil bearings (GFBs) from the 1960s prove that GFBs offer several advantages over traditional oil bearings and rolling element bearings. They have the potential to be applied in a wide spectrum of turbomachinery. Bump-type foil bearings, which are considered as the best structure for GFBs, can be simply described as a hydrodynamic bearing utilizing the ambient air as the lubricant and a smooth shell supported by a corrugated bump foil as the bearing surface. However, the performance predictions of bump-type foil bearings are difficult due to mechanical complexity of the support elastic structure, especially for the effects of four factors, elasticity of bump foil, interaction forces between bumps, friction forces at contact surfaces, and local deflection of top foil. In this investigation, an analytical model of bump-type foil bearings considering the effects of all above factors is presented. In this model, each bump of the bump strip is simplified to two rigid links and a horizontally spaced spring, whose stiffness is determined from Castigliano’ theorem. Then, interaction forces and friction forces can be coupled with the bump flexibility though the horizontal elementary spring. The local deflection of top foil is described using a Finite Element model and added to the film thickness for the pressure prediction with the Reynolds’ equation. The bump deflections of a strip with ten bumps under different load distributions are calculated with the presented model and the predictions show consistency with published results. Moreover, the predicted bearing load and film thickness of a full bump-type foil bearing using this model are very close to the experimental data. Also, radial clearance and friction force variations in the foil bearing are noted to change the stiffness of bump significantly. And the predictions from the calculation with a proper selection of radial clearance and friction coefficients show extremely good agreement with the experimental data. The assumption of minimum reachable film thickness is based on experimental data to determine the load capacity of bearing. The results demonstrate that the radial clearance of foil bearing has an optimum value for the maximum load capacity.

scholarly journals Micro-Slip as a Loss of Determinacy in Dry-Friction Oscillators

2019 ◽  
Vol 29 (06) ◽  
pp. 1930015 ◽  
Author(s):  
S. Webber ◽  
M. R. Jeffrey
Keyword(s):  
Differential Equations ◽  
Dry Friction ◽  
Dynamical Theory ◽  
Stick Slip ◽  
Friction Forces ◽  
Mechanical Oscillators ◽  
The Stability ◽  
Stick Slip Motion ◽  
Friction Oscillators ◽  
Slip Motion

Dry-friction contacts in mechanical oscillators can be modeled using nonsmooth differential equations, and recent advances in dynamical theory are providing new insights into the stability and uniqueness of such oscillators. A classic model is that of spring-coupled masses undergoing stick-slip motion on a rough surface. Here, we present a phenomenon in which multiple masses transition from stick to slip almost simultaneously, but suffer a brief loss of determinacy in the process. The system evolution becomes many-valued, but quickly collapses back down to an infinitesimal set of outcomes, a sort of “micro-indeterminacy”. Though fleeting, the loss of determinacy means masses may each undergo different microscopic sequences of slipping events, before all masses ultimately slip. The microscopic loss of determinacy is visible in local changes in friction forces, and in creating a bistability of global stick-slip oscillations. If friction forces are coupled between the oscillators then the effect is more severe, as solutions are compressed instead onto two (or more) macroscopically different outcomes.

Compliant Foil Bearing Structural Stiffness Analysis: Part I—Theoretical Model Including Strip and Variable Bump Foil Geometry

1992 ◽  
Vol 114 (2) ◽  
pp. 394-400 ◽  
Author(s):  
C.-P. Roger Ku ◽  
H. Heshmat
Keyword(s):  
Theoretical Model ◽  
Variable Load ◽  
Distribution Profile ◽  
Friction Forces ◽  
Stiffness Analysis ◽  
Foil Bearings ◽  
Foil Bearing ◽  
Load Distributions ◽  
Fixed End

This paper presents a theoretical model of corrugated foil strip (bump foil) deformation in compliant foil bearings and dampers. The friction forces between bump foils and the housing or the top foil, local interaction forces, variable load distributions, and bump geometries are taken into consideration. Following the trend of earlier published experimental data, the bumps near the fixed end have a much higher predicted stiffness (lower deflection) than those near the free end. Higher friction coefficients tend to increase stiffness and may pin down bumps near the fixed end. An increase in the friction coefficient between the top foil and the bump is a more effective method of achieving both Coulomb damping and higher stiffness. In addition to bump geometry, the load distribution profile greatly influences bump stiffness. A follow-up paper will present the experimental verification and discuss the comparison between theoretical and experimental results.

A Fully Coupled 3D Finite Element Analysis for a Bump-Type Compliant Foil Bearing

2011 ◽  
Author(s):  
Serdar Aksoy ◽  
M. Bulut Coskun ◽  
Mahmut F. Aksit
Keyword(s):  
Finite Element ◽  
Film Formation ◽  
Load Capacity ◽  
Foil Surface ◽  
Element Analysis ◽  
Foil Bearings ◽  
Foil Bearing ◽  
Element Approach ◽  
The Stability ◽  
Fully Coupled

A bump-type foil bearing consists of a compliant corrugated sheet metal supporting structure that is covered by a thin top foil surface. Bumps serve as stiffener and damping elements to increase the stability of the system while top foil creates necessary wedge surface for aerodynamic film formation. Foil bearings are hard to analyze as flexible foil surface deforms and changes shape while aerodynamic film pressure forms. Fully coupled finite element approach is presented. To increase the accuracy, the geometry is discretized with quadratic hexagonal elements. The gas film is modeled by compressible Reynolds equation. The effect of velocity on load capacity will be investigated to reveal the performance of the code.

Removing the stability limit of the explicit finite-difference scheme with eigenvalue perturbation

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. A93-A98 ◽  
Author(s):  
Yingjie Gao ◽  
Jinhai Zhang ◽  
Zhenxing Yao
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Unit Circle ◽  
Finite Difference Scheme ◽  
Stability Limit ◽  
Time Step ◽  
Dispersion Error ◽  
Time Dispersion ◽  
Explicit Finite Difference ◽  
The Stability

The explicit finite-difference scheme is popular for solving the wave equation in the field of seismic exploration due to its simplicity in numerical implementation. However, its maximum time step is strictly restricted by the Courant-Friedrichs-Lewy (CFL) stability limit, which leads to a heavy computational burden in the presence of small-scale structures and high-velocity targets. We remove the CFL stability limit of the explicit finite-difference scheme using the eigenvalue perturbation, which allows us to use a much larger time step beyond the CFL stability limit. For a given time step that is within the CFL stability limit, the eigenvalues of the update matrix would be distributed along the unit circle; otherwise, some eigenvalues would be distributed outside of the unit circle, which introduces unstable phenomena. The eigenvalue perturbation can normalize the unstable eigenvalues and guarantee the stability of the update matrix by using an arbitrary time step. The update matrix can be preprocessed before the numerical simulation, thus retaining the computational efficiency well. We further incorporate the forward time-dispersion transform (FTDT) and the inverse time-dispersion transform (ITDT) to reduce the time-dispersion error caused by using an unusually large time step. Our numerical experiments indicate that the combination of the eigenvalue perturbation, the FTDT method, and the ITDT method can simulate highly accurate waveforms when applying a time step beyond the CFL stability limit. The time step can be extended even toward the Nyquist limit. This means that we could save many iteration steps without suffering from time-dispersion error and stability problems.

An Improved Structure-Dependent Explicit Method for Structural Dynamics

2008 ◽  
Author(s):  
Shuenn-Yih Chang ◽  
Chiu-Li Huang
Keyword(s):  
Low Frequency ◽  
Conditional Stability ◽  
Explicit Method ◽  
Time Step ◽  
Structural Dynamic ◽  
Linear Elastic ◽  
Acceleration Method ◽  
The Stability ◽  
Stiffness Softening ◽  
Hardening System

An explicit method is presented herein whose coefficients are determined from the initial structural properties of the analyzed system. Thus, it is structure-dependent. This method has a great stability property when compared to the previously published method [6], which is unconditionally stable for linear elastic and any instantaneous stiffness softening systems while it only has conditional stability for an instantaneous stiffness hardening system. The most important improvement of this method is that it has unconditional stability for general instantaneous stiffness hardening systems in addition to linear elastic and instantaneous stiffness softening systems. This implies that a time step may be selected base on accuracy consideration only and the stability problem might be neglected. Hence, many computational efforts can be saved in the step-by-step solution of a general structural dynamic problem, where the response is dominated by the low frequency modes and the high frequency responses are of no great interest, when compared to an explicit method, such as the Newmark explicit method, and an implicit method, such as the constant average acceleration method.

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