A Simplified and Consistent Nonlinear Transient Analysis Method for Gas Bearing: Extension to Flexible Rotors

2015 ◽  
Vol 137 (9) ◽  
Author(s):  
Mohamed Amine Hassini ◽  
Mihai Arghir
Keyword(s):  
Transient Analysis ◽  
Rational Functions ◽  
Air Bearings ◽  
Analysis Method ◽  
New Approach ◽  
Gas Bearings ◽  
Fluid Forces ◽  
Nonlinear Transient ◽  
Nonlinear Transient Analysis ◽  
The Stability

A simplified, new method for evaluating the nonlinear fluid forces in air bearings was recently proposed (Hassini, M. A., and Arghir, M., 2012, “Simplified Nonlinear Transient Analysis Method for Gas Bearings,” ASME J. Tribol., 134(1), p. 011704). The method is based on approximating the frequency dependent linearized dynamic coefficients at several eccentricities, by second-order rational functions. A set of ordinary differential equations is then obtained using the inverse of Laplace transform linking the fluid forces components to the rotor displacements. Coupling these equations with the equations of motion of the rotor leads to a system of ordinary differential equations where displacements and velocities of the rotor and the fluid forces come as unknowns. The numerical results stemming from the proposed approach showed good agreement with the results obtained by solving the full nonlinear transient Reynolds equation coupled to the equation of motion of a point mass rotor. However, the method (Hassini, M. A., and Arghir, M., 2012, “Simplified Nonlinear Transient Analysis Method for Gas Bearings,” ASME J. Tribol., 134(1), p. 011704) requires a special treatment to ensure continuity of the values of the fluid forces and their first derivatives. More recently, the same authors (Hassini, M. A., and Arghir, M., 2013, “A New Approach for the Stability Analysis of Rotors Supported by Gas Bearings,” ASME Paper No. GT2013-94802) showed the benefits of imposing the same set of stable poles to the rational functions approximating the impedances. These constrains simplified the expressions of the fluid forces and avoided the introduction of false poles. The method in (Hassini, M. A., and Arghir, M., 2013, “A New Approach for the Stability Analysis of Rotors Supported by Gas Bearings,” ASME Paper No. GT2013-94802) was applied in the frame of the small perturbation analysis for calculating Campbell and stability diagrams. This approach also enhances the consistency of the fluid forces approximated with the same set of poles because they become naturally continuous over the whole bearing clearance while their increments were not. The present paper shows how easily the new formulation may be applied to compute the nonlinear response of systems with multiple degrees of freedom such as a flexible rotor supported by two air bearings.

A New Approach for the Stability Analysis of Rotors Supported by Gas Bearings

2013 ◽  
Vol 136 (2) ◽  
Author(s):  
Mohamed Amine Hassini ◽  
Mihai Arghir
Keyword(s):  
Rational Function ◽  
Dynamic Characteristics ◽  
Transient Analysis ◽  
Transfer Functions ◽  
Analysis Method ◽  
Gas Bearings ◽  
Rational Transfer ◽  
Nonlinear Transient ◽  
Nonlinear Transient Analysis ◽  
The Stability

A simplified nonlinear transient analysis method for gas bearings was recently published by the authors (Hassini, M. A., and Arghir, M., 2012, “Simplified Nonlinear Transient Analysis Method for Gas Bearings,” J. Tribol., 134(1), 011704). The method uses the fact that linearized dynamic characteristics of gas bearings, namely the impedances, can be approximated by rational transfer functions. The method gave good results if the rational transfer function approach approximated the linearized dynamic characteristics well. Indeed, each of the four complex impedances Zαβ,α,β={x,y} had one or two poles depending on the order of the rational function that were used. These poles appear as supplementary eigenvalues of the extended matrix of the homogeneous system of first order differential equations describing the model of the rotor. They govern the stability of the dynamic model in the same way as the original eigenvalues do and therefore they impose non-negligible constraints on the rational function approximation of the impedances of gas bearings. The present improvement of the method overrides this problem. The basic idea is to impose the same set of poles for Zxx, Zxy, Zyx, and Zyy. By imposing this constraint, the poles are stable and the introduction of artificial instability or erratic eigenvalues is avoided. Campbell and stability diagrams naturally taking into account the variation of the dynamic coefficients with the excitation frequency can now be easily plotted. For example, the method is used for analyzing the stability of rigid and flexible rotors supported by two identical gas bearings modeled with second order rational transfer functions. The method can be applied to any bearing or seal whose impedance is approximated by rational transfer functions.

Comparison Between Linear and Nonlinear Transient Analysis Techniques to Find the Stability of a Rigid Rotor

1999 ◽  
Vol 121 (1) ◽  
pp. 198-201 ◽  
Author(s):  
Ram Turaga ◽  
A. S. Sekhar ◽  
B. C. Majumdar
Keyword(s):  
Transient Analysis ◽  
Stability Limit ◽  
Journal Bearings ◽  
Quantitative Comparison ◽  
Rigid Rotor ◽  
Analysis Techniques ◽  
Analysis Technique ◽  
Nonlinear Transient ◽  
Nonlinear Transient Analysis ◽  
The Stability

The subsynchronous whirl stability limit of a rigid rotor supported on two symmetrical finite journal bearings has been studied using the linearised perturbation method and the nonlinear transient analysis technique. A quantitative comparison for journal bearings with different l/d ratios has been provided.

A Simplified and Consistent Nonlinear Transient Analysis Method for Gas Bearing: Extension to Flexible Rotors

2014 ◽  
Author(s):  
Amine Hassini ◽  
Mihai Arghir
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Rational Functions ◽  
Special Treatment ◽  
Air Bearings ◽  
Fluid Forces ◽  
Stability Diagrams ◽  
Nonlinear Transient

A simplified, new method for evaluating the nonlinear fluid forces in air bearings was recently proposed in [1]. The method is based on approximating the frequency dependent linearized dynamic coefficients at several eccentricities, by second order rational functions. A set of ordinary differential equations is then obtained using the inverse of Laplace Transform linking the fluid forces components to the rotor displacements. Coupling these equations with the equations of motion of the rotor lead to a system of ordinary differential equations where displacements and velocities of the rotor and the fluid forces come as unknowns. The numerical results stemming from the proposed approach showed good agreement with the results obtained by solving the full nonlinear transient Reynolds equation coupled to the equation of motion of a point mass rotor. However the method [1] requires a special treatment to ensure continuity of the values of the fluid forces and their first derivatives. More recently, the same authors [2] showed the benefits of imposing the same set of stable poles to the rational functions approximating the impedances. These constrains simplified the expressions of the fluid forces and avoided the introduction of false poles. The method in [2] was applied in the frame of the small perturbation analysis for calculating Campbell and stability diagrams. This approach enhances also the consistency of the fluid forces approximated with the same set of poles because they become naturally continuous over the whole bearing clearance while their increments were not. The present paper shows how easily the new formulation may be applied to compute the nonlinear response of systems with multiple degrees of freedom such as a flexible rotor supported by two air bearings.

Nonlinear transient analysis of moderately thick rectangular composite plates

2010 ◽  
Vol 114 (1157) ◽  
pp. 437-444
Author(s):  
H. Tanriöver ◽  
E. Şenoca
Keyword(s):  
Transient Analysis ◽  
Solution Process ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Composite Plates ◽  
Laminated Composite Plates ◽  
Element Analysis ◽  
Geometrically Nonlinear Analysis ◽  
Nonlinear Transient ◽  
Nonlinear Transient Analysis

Abstract This paper presents an analytical-numerical methodology for the geometrically nonlinear analysis of laminated composite plates under dynamic loading. The methodology employs Galerkin technique, in which suitable polynomials are chosen as trial functions. In the solution process, Newmark’s scheme for time integration, and modified Newton-Raphson method for the solution of resulting nonlinear equations are used. In the formulation, first order shear deformation theory based on Mindlin’s hypothesis and von Kármán type geometric nonlinearity are considered. The results are compared to that of finite strips, and Chebyshev series published elsewhere. The method is found to determine closely both the displacements and the stresses. A finite element analysis has also been carried out for the validation of the results. The present method can be efficiently and easily applied for the nonlinear transient analysis of laminated composite plates with various boundary conditions.

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