Effects of Forward/Backward Whirl Mechanism on Nonlinear Normal Modes of a Rotor/Stator Rubbing System

2015 ◽  
Vol 137 (5) ◽  
Author(s):  
Yanhua Chen ◽  
Jun Jiang
Keyword(s):  
Response Spectrum ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Frequency Component ◽  
Harmonic Response ◽  
Backward Whirl ◽  
Coupling Stiffness ◽  
Nonlinear Normal ◽  
The Stability ◽  
The Relationship

In this paper, the effects of forward and backward whirl mechanism on the existence and the stability of multiple nonlinear normal modes (NNMs) in a four degree-of-freedom (DOF) rotor/stator rubbing system with cross-coupling stiffness and dry friction are investigated analytically. The NNMs may possess either positive or negative modal frequencies, corresponding, respectively, to the inherent motions of forward or backward whirl, and can be either stable or unstable. The relationship between the NNMs, regarding to their stability, and the forced system responses of the system is of great interest. It is found that a stable NNM corresponds to a forced harmonic response with the same whirl direction and frequency as the NNM, and an unstable NNM may still influence some forced system responses by contributing a frequency component equal to the modal frequency to the response spectrum.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

Localized and Non-Localized Normal Modes in a Strongly Nonlinear Discrete System

1991 ◽  
Author(s):  
Alexander F. Vakakis
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Linear Vibration ◽  
Mode Localization ◽  
Linear Mode ◽  
Strongly Nonlinear ◽  
System A ◽  
Coupling Stiffness ◽  
Nonlinear Normal ◽  
Nonlinear Discrete System

Abstract The free oscillations of a strongly nonlinear, discrete oscillator are examined by computing its “nonsimilar nonlinear normal modes.” These are motions represented by curves in the configuration space of the system, and they are not encountered in classical, linear vibration theory or in existing nonlinear perturbation techniques. For an oscillator with weak coupling stiffness and “mistiming,” both localized and nonlocalized modes are detected, occurring in small neighborhoods of “degenerate” and “global” similar modes of the “tuned” system. When strong coupling is considered, only nonlocalized modes are found to exist. An interesting result of this work is the detection of mode localization in the “tuned” periodic system, a result with no counterpart in existing theories on linear mode localization.

An Energy-Based Formulation for Computing Nonlinear Normal Modes in Undamped Continuous Systems

1994 ◽  
Vol 116 (3) ◽  
pp. 332-340 ◽  
Author(s):  
M. E. King ◽  
A. F. Vakakis
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Conservation Of Energy ◽  
Nonlinear Modes ◽  
One Dimensional ◽  
Linear Differential ◽  
Nonlinear Normal ◽  
The Stability ◽  
Nonlinear Elastic

The nonlinear normal modes of a class of one-dimensional, conservative, continuous systems are examined. These are free, periodic motions during which all particles of the system reach their extremum amplitudes at the same instant of time. During a nonlinear normal mode, the motion of an arbitrary particle of the system is expressed in terms of the motion of a certain reference point by means of a modal function. Conservation of energy is imposed to construct a partial differential equation satisfied by the modal function, which is asymptotically solved using a perturbation methodology. The stability of the detected nonlinear modes is then investigated by expanding the corresponding variational equations in bases of orthogonal polynomials and analyzing the resulting set of linear differential equations with periodic coefficients by Floquet analysis. Applications of the general theory are given by computing the nonlinear normal modes of a simply-supported beam lying on a nonlinear elastic foundation, and of a cantilever beam possessing geometric nonlinearities.

DETERMINING THE CHARACTERISTICS OF A SELF-EXCITED OSCILLATION IN ROTOR/STATOR SYSTEMS FROM THE INTERACTION OF LINEAR AND NONLINEAR NORMAL MODES

2010 ◽  
Vol 20 (12) ◽  
pp. 4137-4150 ◽  
Author(s):  
JUN JIANG ◽  
ZHIQIANG WU
Keyword(s):  
Dry Friction ◽  
Dominant Role ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Nonlinear Dynamical ◽  
Backward Whirl ◽  
Excited Oscillation ◽  
Linear And Nonlinear ◽  
Contact Surfaces ◽  
Nonlinear Normal

In this paper, the linear and nonlinear modes of the unforced coupled rotor/stator system from a general rotor/stator model, which accounts for both the dynamics of the rotor and the stator as well as the friction and the deformation at the contact surfaces, are derived. The bifurcations of the nonlinear normal modes are analyzed based on the constrained bifurcation theory with the linear normal modes as the constraints. Then, the existence boundaries and the backward whirl frequencies of dry friction backward whirl — a hazardous self-excited oscillation in rotor/stator systems — of this model are derived. It is found by analysis that many inherent characteristics of the dry friction backward whirl can be derived from the information of the interaction of the linear and the nonlinear normal modes of the coupled rotor/stator system, such as the number of existence regions and their position relationship, the minimal friction on the contact surfaces that may induce the self-excited oscillation, the upper limits of the backward whirl frequencies of the response, and more. This study has well demonstrated the dominant role of the interaction of the linear and the nonlinear normal modes in deciding the characteristics of some nonlinear dynamical behaviors.

Nonlinear Normal Modes in the Presence of Internal Resonances: An Energy-Based Approach

1995 ◽  
Author(s):  
Melvin E. King ◽  
Alexander F. Vakakis
Keyword(s):  
Normal Mode ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Weakly Nonlinear ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

Abstract In this work, modifications to existing energy-based nonlinear normal mode (NNM) methodologies are developed in order to investigate internal resonances. A formulation for computing resonant NNMs is developed for discrete, or discretized for continuous systems, sets of weakly nonlinear equations with uncoupled linear terms (i.e systems in modal, or canonical, form). By considering a canonical framework, internal resonance conditions are immediately recognized by identifying commensurable linearized natural frequencies. Additionally, the canonical formulation allows for a single (linearized modal) coordinate to parameterize all other (modal) coordinates during a resonant modal response. Energy-based NNM methodologies are then applied to the canonical equations and asymptotic solutions are sought. In order to account for the resonant modal interactions, it will be shown that high-order terms in the O(1) solutions must be considered. Two applications (‘3:1’ resonances in a two-degree-of-freedom system and ‘3:1’ resonance in a hinged-clamped beam) are then considered by which to demonstrate the application of the resonant NNM methodology. Resonant normal mode solutions are obtained and the stability characteristics of these computed modes are considered. It is shown that for some responses, nonlinear modal relations do not exist in the context of physical coordinates and thus the transformation to canonical coordinates is necessary in order to define appropriate NNM relations.

Nonlinear Modal Analysis and Energy Localization in a Bladed Disk Assembly

2008 ◽  
Author(s):  
F. Georgiades ◽  
M. Peeters ◽  
G. Kerschen ◽  
J. C. Golinval ◽  
M. Ruzzene
Keyword(s):  
Modal Analysis ◽  
Nonlinear Oscillations ◽  
Periodic Structures ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Energy Localization ◽  
Bladed Disk ◽  
Bladed Disk Assembly ◽  
Nonlinear Modal Analysis ◽  
Nonlinear Normal

The objective of this study is to carry out modal analysis of nonlinear periodic structures using nonlinear normal modes (NNMs). The NNMs are computed numerically with a method developed in [18] that is using a combination of two techniques: a shooting procedure and a method for the continuation of periodic motion. The proposed methodology is applied to a simplified model of a perfectly cyclic bladed disk assembly with 30 sectors. The analysis shows that the considered model structure features NNMs characterized by strong energy localization in a few sectors. This feature has no linear counterpart, and its occurrence is associated with the frequency-energy dependence of nonlinear oscillations.

scholarly journals Comparison of Galerkin, POD and Nonlinear-Normal-Modes Models for Nonlinear Vibrations of Circular Cylindrical Shells

2006 ◽  
Author(s):  
M. Amabili ◽  
C. Touze´ ◽  
O. Thomas
Keyword(s):  
Nonlinear Vibrations ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Harmonic Excitation ◽  
Nonlinear Responses ◽  
Nonlinear Normal ◽  
The Galerkin Method ◽  
Proper Orthogonal

The aim of the present paper is to compare two different methods available to reduce the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell. The two methods are: the proper orthogonal decomposition (POD) and an asymptotic approximation of the Nonlinear Normal Modes (NNMs) of the system. The structure used to perform comparisons is a water-filled, simply supported circular cylindrical shell subjected to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency. A reference solution is obtained by discretizing the Partial Differential Equations (PDEs) of motion with a Galerkin expansion containing 16 eigenmodes. The POD model is built by using responses computed with the Galerkin model; the NNM model is built by using the discretized equations of motion obtained with the Galerkin method, and taking into account also the transformation of damping terms. Both the POD and NNMs allow to reduce significantly the dimension of the original Galerkin model. The computed nonlinear responses are compared in order to verify the accuracy and the limits of these two methods. For vibration amplitudes equal to 1.5 times the shell thickness, the two methods give very close results to the original Galerkin model. By increasing the excitation and vibration amplitude, significant differences are observed and discussed.

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