Periodic Solutions in Rotor Dynamic Systems With Nonlinear Supports: A General Approach

1989 ◽  
Vol 111 (2) ◽  
pp. 187-193 ◽  
Author(s):  
C. Nataraj ◽  
H. D. Nelson
Keyword(s):  
Dynamic Systems ◽  
Original Problem ◽  
Squeeze Film ◽  
Algebraic Equations ◽  
Trigonometric Collocation ◽  
Nonlinear Algebraic Equations ◽  
The Stability ◽  
Future Work ◽  
Rotor Dynamic ◽  
Large Rotor

A new quantitative method of estimating steady state periodic behavior in nonlinear systems, based on the trigonometric collocation method, is outlined. A procedure is developed to analyze large rotor dynamic systems with nonlinear supports by the use of the above method in conjunction with Component Mode Synthesis. The algorithm discussed is seen to reduce the original problem to solving nonlinear algebraic equations in terms of only the coordinates associated with the nonlinear supports and is a big improvement over commonly used integration methods. The feasibility and advantages of the procedure so developed are illustrated with the help of an example of a typical rotor dynamic system with an uncentered squeeze film damper. Future work on the investigation of the stability of the periodic response so obtained is outlined.

Prediction of Periodic Response of Rotor Dynamic Systems With Nonlinear Supports

1995 ◽  
Author(s):  
Yu Wang
Keyword(s):  
Dynamic Systems ◽  
Nonlinear Differential Equations ◽  
Squeeze Film ◽  
Element Formulation ◽  
Algebraic Equations ◽  
Periodic Response ◽  
Nonlinear Algebraic Equations ◽  
Nodal Displacements ◽  
The Time Domain ◽  
Rotor Dynamic

Abstract A numerical-analytical method for estimating steady-state periodic behavior of nonlinear rotordynamic systems is presented. Based on a finite element formulation in the time domain, this method transforms the nonlinear differential equations governing the motion of large rotor dynamic systems with nonlinear supports into a set of nonlinear algebraic equations with unknown temporal nodal displacements. A procedure is proposed to reduce the resulting problem to solving nonlinear algebraic equations in terms of the coordinates associated with the nonlinear supports only. The result is a simple and efficient approach for predicting all possible fundamental and sub-harmonic responses. Stability of the periodic response is readily determined by a direct use of Floquet’s theory. The feasibility and advantages of the proposed method are illustrated with two examples of rotor-bearing systems of deadband supports and squeeze film dampers, respectively.

Prediction of Periodic Response of Rotor Dynamic Systems With Nonlinear Supports

1997 ◽  
Vol 119 (3) ◽  
pp. 346-353 ◽  
Author(s):  
Yu Wang
Keyword(s):  
Dynamic Systems ◽  
Nonlinear Differential Equations ◽  
Squeeze Film ◽  
Element Formulation ◽  
Algebraic Equations ◽  
Periodic Response ◽  
Nonlinear Algebraic Equations ◽  
Nodal Displacements ◽  
The Time Domain ◽  
Rotor Dynamic

A numerical-analytical method for estimating steady-state periodic behavior of nonlinear rotordynamic systems is presented. Based on a finite element formulation in the time domain, this method transforms the nonlinear differential equations governing the motion of large rotor dynamic systems with nonlinear supports into a set of nonlinear algebraic equations with unknown temporal nodal displacements. A procedure is proposed to reduce the resulting problem to solving nonlinear algebraic equations in terms of the coordinates associated with the nonlinear supports only. The result is a simple and efficient approach for predicting all possible fundamental and sub-harmonic responses. Stability of the periodic response is readily determined by a direct use of Floquet’s theory. The feasibility and advantages of the proposed method are illustrated with two examples of rotor-bearing systems of deadband supports and squeeze film dampers, respectively.

Stability and Bifurcation of Unbalanced Response of a Squeeze Film Damped Flexible Rotor

1994 ◽  
Vol 116 (2) ◽  
pp. 361-368 ◽  
Author(s):  
J. Y. Zhao ◽  
I. W. Linnett ◽  
L. J. McLean
Keyword(s):  
Power Spectra ◽  
Squeeze Film ◽  
Integration Scheme ◽  
Flexible Rotor ◽  
Jump Phenomenon ◽  
Stability And Bifurcation ◽  
Trigonometric Collocation ◽  
Numerical Integration Scheme ◽  
The Stability ◽  
Nonsynchronous Vibrations

The stability and bifurcation of the unbalance response of a squeeze film damper-mounted flexible rotor are investigated based on the assumption of an incompressible lubricant together with the short bearing approximation and the “π” film cavitation model. The unbalanced rotor response is determined by the trigonometric collocation method and the stability of these solutions is then investigated using the Floquet transition matrix method. Numerical examples are given for both concentric and eccentric damper operations. Jump phenomenon, subharmonic, and quasi-periodic vibrations are predicted for a range of bearing and unbalance parameters. The predicted jump phenomenon, subharmonic and quasi-periodic vibrations are further examined by using a numerical integration scheme to predict damper trajectories, calculate Poincare´ maps and power spectra. It is concluded that the introduction of unpressurized squeeze film dampers may promote undesirable nonsynchronous vibrations.

A GLOBAL PERIOD-1 MOTION OF A PERIODICALLY EXCITED, PIECEWISE-LINEAR SYSTEM

2005 ◽  
Vol 15 (06) ◽  
pp. 1945-1957 ◽  
Author(s):  
SANTHOSH MENON ◽  
ALBERT C. J. LUO
Keyword(s):  
Linear System ◽  
Numerical Simulations ◽  
Piecewise Linear ◽  
Dynamical Behavior ◽  
Analytical Prediction ◽  
Algebraic Equations ◽  
Piecewise Linear System ◽  
Nonlinear Algebraic Equations ◽  
The Stability ◽  
Nonsmooth Dynamical Systems

The period-1 motion of a piecewise-linear system under a periodic excitation is predicted analytically through the Poincaré mapping and the corresponding mapping sections formed by the switch planes pertaining to the two constraints. The mapping relationship generates a set of nonlinear algebraic equations from which the period-1 motion is determined analytically. The stability and bifurcation of the period-1 motion are determined, and numerical simulations are carried out for confirmation of the analytical prediction of period-1 motion. An unsymmetrical stable period-1 motion is observed. This investigation helps us understand the dynamical behavior of period-1 motion in the piecewise-linear system and more efficiently obtain other periodic motions and chaos through numerical simulations. The similar methodology presented in this paper can be used for other nonsmooth dynamical systems.

scholarly journals Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order integro-differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
H. Jafari ◽  
S. Nemati ◽  
R. M. Ganji
Keyword(s):  
Differential Equations ◽  
Chebyshev Polynomials ◽  
Original Problem ◽  
High Accuracy ◽  
Operational Matrices ◽  
Variable Order ◽  
Algebraic Equations ◽  
Numerical Tests ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations

AbstractIn this research, we study a general class of variable order integro-differential equations (VO-IDEs). We propose a numerical scheme based on the shifted fifth-kind Chebyshev polynomials (SFKCPs). First, in this scheme, we expand the unknown function and its derivatives in terms of the SFKCPs. To carry out the proposed scheme, we calculate the operational matrices depending on the SFKCPs to find an approximate solution of the original problem. These matrices, together with the collocation points, are used to transform the original problem to form a system of linear or nonlinear algebraic equations. We discuss the convergence of the method and then give an estimation of the error. We end by solving numerical tests, which show the high accuracy of our results.

scholarly journals Using the Modal Method in Rotor Dynamic Systems

2003 ◽  
Vol 9 (3) ◽  
pp. 181-196
Author(s):  
Eduard Malenovský
Keyword(s):  
Finite Element Method ◽  
Collocation Method ◽  
Dynamic Systems ◽  
Theoretical Basis ◽  
The Finite Element Method ◽  
Transient Responses ◽  
Technical Application ◽  
Trigonometric Collocation ◽  
Rotor Dynamic ◽  
Modal Method

This article deals with computational modeling of nonlinear rotor dynamic systems. The theoretical basis of the method of dynamic compliances and the modal method, supplemented by the method of trigonometric collocation, are presented. The main analysis is focused on the solutions of the eigenvalue problem and steady-state and transient responses. The algorithms for solving this range of problems are presented. The finite element method, the method of dynamic compliances, and the modal method are supplemented by the trigonometric collocation method. The theoretical analysis is supplemented by the solution of a model task, which is focused on the application of the trigonometric collocation method. The solution of a technical application, which is a pump, is presented in this article.

scholarly journals Heuristic Determination of Resolving Controls for Exact and Approximate Controllability of Nonlinear Dynamic Systems

2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
Asatur Zh. Khurshudyan
Keyword(s):  
Dynamic Systems ◽  
Approximate Controllability ◽  
State Constraints ◽  
Nonlinear Dynamic Systems ◽  
Algebraic Equations ◽  
Control Functions ◽  
Nonlinear Algebraic Equations ◽  
Free Parameters ◽  
Nonlinear State

Dealing with practical control systems, it is equally important to establish the controllability of the system under study and to find corresponding control functions explicitly. The most challenging problem in this path is the rigorous analysis of the state constraints, which can be especially sophisticated in the case of nonlinear systems. However, some heuristic considerations related to physical, mechanical, or other aspects of the problem may allow coming up with specific hierarchic controls containing a set of free parameters. Such an approach allows reducing the computational complexity of the problem by reducing the nonlinear state constraints to nonlinear algebraic equations with respect to the free parameters. This paper is devoted to heuristic determination of control functions providing exact and approximate controllability of dynamic systems with nonlinear state constraints. Using the recently developed approach based on Green’s function method, the controllability analysis of nonlinear dynamic systems, in general, is reduced to nonlinear integral constraints with respect to the control function. We construct parametric families of control functions having certain physical meanings, which reduce the nonlinear integral constraints to a system of nonlinear algebraic equations. Regimes such as time-harmonic, switching, impulsive, and optimal stopping ones are considered. Two concrete examples arising from engineering help to reveal advantages and drawbacks of the technique.

Using the Modal and Trigonometric Collocation Methods in Rotor Dynamic Systems

2004 ◽  
Vol 126 (2) ◽  
pp. 229-234 ◽  
Author(s):  
Eduard Malenovsky´
Keyword(s):  
Dynamic Systems ◽  
Theoretical Basis ◽  
Collocation Methods ◽  
The Finite Element Method ◽  
Trigonometric Collocation ◽  
Steady State Responses ◽  
State Responses ◽  
Rotor Dynamic ◽  
Modal Method ◽  
Transient And Steady State

This article deals with the computational modeling of nonlinear rotor dynamic systems. The theoretical basis of the modal method, and combination with the method of dynamic compliances supplemented by the method of trigonometric collocation, is presented. The main analysis is focused on the solutions of transient and steady state responses. The algorithms for solving this range of problems are presented. The finite element method is the basis for both methods. The theoretical analysis is supplemented with a solution of an example model.

scholarly journals Measurements of Squeeze Film Bearing Forces to Demonstrate the Effect of Fluid Inertia

1984 ◽  
Author(s):  
John A. Tichy
Keyword(s):  
Dynamic Systems ◽  
High Speed ◽  
Hydrodynamic Lubrication ◽  
Reynolds Numbers ◽  
Lubrication Theory ◽  
Squeeze Film ◽  
Fluid Inertia ◽  
Viscous Forces ◽  
Rotor Dynamic ◽  
Inertia Forces

Squeeze film dampers are commonly applied to high speed rotating machinery, such as aircraft engines, to reduce vibration problems. The theory of hydrodynamic lubrication has been used for the design and modeling of dampers in rotor dynamic systems despite typical modified Reynolds numbers in applications between ten and fifty. Lubrication theory is strictly valid for Reynolds numbers much less than one, which means that fluid viscous forces are much greater than inertia forces. Theoretical papers which account for fluid inertia in squeeze films have predicted large discrepancies from lubrication theory, but these results have not found wide acceptance by workers in the gas turbine industry. Recently, experimental results on the behavior of rotor dynamic systems have been reported which strongly support the existence of large fluid inertia forces. In the present paper direct measurements of damper forces are presented for the first time. Reynolds numbers up to ten are obtained at eccentricity ratios 0.2 and 0.5. Lubrication theory underpredicts the measured forces by up to a factor of two (100% error). Qualitative agreement is found with predictions of earlier improved theories which include fluid inertia forces.

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