Nonlinear Dynamic Analysis on Planetary Gears-Rotor System in Geared Turbofan Engines

2019 ◽  
Vol 29 (06) ◽  
pp. 1950076 ◽  
Author(s):  
Lanlan Hou ◽  
Shuqian Cao
Keyword(s):  
Nonlinear Dynamic Analysis ◽  
Period Doubling ◽  
Nonlinear Behavior ◽  
Rotor System ◽  
Chaotic Motions ◽  
Planetary Gears ◽  
Gear Noise ◽  
Turbofan Engines ◽  
Vibration Responses ◽  
Nonlinear Analytical Model

Rotor fatigue and gear noise triggered by nonlinear vibration are the key concerns in Geared Turbofan (GTF) engine which features a new configuration by introducing planetary gears into low-pressure compressor. A nonlinear analytical model of the GTF planetary gears-rotor system is developed, where the torsional effect of rotor and pivotal parameters from gears are incorporated. The nonlinear behavior of the model can be obtained by focusing on the relative torsional vibration responses between gear and rotor. The torsional nonlinear responses are illustrated with bifurcation diagrams, the largest Lyapunov exponents (LLE), Poincaré maps, phase diagrams and spectrum waterfall. Numerical results reveal that the gears-rotor system exhibits abundant torsional nonlinear behaviors, including multiperiodic, quasi-periodic, and chaotic motions. Furthermore, the roads to chaos via quasi-periodicity, period-doubling scenario, mutation and intermittence are demonstrated. The ring gear stiffness at a low value can propel the system into chaos. The damping may complicate the motion, i.e. the system may enter chaos with increasing damping. These results provide an understanding of undesirable torsional dynamic motion for the GTF engine rotor system and therefore serve as a useful reference for engineers in designing and controlling such system.

Shooting With Deflation Algorithm-Based Nonlinear Response and Neimark-Sacker Bifurcation and Chaos in Floating Ring Bearing Systems

2016 ◽  
Vol 12 (3) ◽  
Author(s):  
Sitae Kim ◽  
Alan B. Palazzolo
Keyword(s):  
Nonlinear Response ◽  
Nonlinear Behavior ◽  
Rotor System ◽  
Operating Conditions ◽  
Fluid Film ◽  
Stable Limit ◽  
Bifurcation And Chaos ◽  
Chaotic Motions ◽  
Response Characteristics ◽  
Computation Efficiency

The double-sided fluid film force on the inner and outer ring surfaces of a floating ring bearing (FRB) creates strong nonlinear response characteristics such as coexistence of multiple orbits, Hopf bifurcation, Neimark-Sacker (N-S) bifurcation, and chaos in operations. An improved autonomous shooting with deflation algorithm is applied to a rigid rotor supported by FRBs for numerically analyzing its nonlinear behavior. The method enhances computation efficiency by avoiding previously found solutions in the numerical-based search. The solution manifold for phase state and period is obtained using arc-length continuation. It was determined that the FRB-rotor system has multiple response states near Hopf and N-S bifurcation points, and the bifurcation scenario depends on the ratio of floating ring length and diameter (L/D). Since multiple responses coexist under the same operating conditions, simulation of jumps between two stable limit cycles from potential disturbance such as sudden base excitation is demonstrated. In addition, this paper investigates chaotic motions in the FRB-rotor system, utilizing four different approaches, strange attractor, Lyapunov exponent, frequency spectrum, and bifurcation diagram. A numerical case study for quenching the large amplitude motion by adding unbalance force is provided and the result shows synchronization, i.e., subsynchronous frequency components are suppressed. In this research, the fluid film forces on the FRB are determined by applying the finite element method while prior work has utilized a short bearing approximation. Simulation response comparisons between the short bearing and finite bearing models are discussed.

scholarly journals Nonlinear Dynamic Analysis of Rotor Rub-Impact System

2019 ◽  
Vol 2019 ◽  
pp. 1-20
Author(s):  
Youfeng Zhu ◽  
Zibo Wang ◽  
Qiang Wang ◽  
Xinhua Liu ◽  
Hongyu Zang ◽  
...  
Keyword(s):  
Periodic Motion ◽  
Chaotic Motion ◽  
Nonlinear Dynamic Analysis ◽  
Period Doubling ◽  
Time History ◽  
Rotor System ◽  
Quasiperiodic Motion ◽  
Rotor Bearing ◽  
Motion Period ◽  
Bearing System

A dynamic model of a double-disk rub-impact rotor-bearing system with rubbing fault is established. The dynamic differential equation of the system is solved by combining the numerical integration method with MATLAB. And the influence of rotor speed, disc eccentricity, and stator stiffness on the response of the rotor-bearing system is analyzed. In the rotor system, the time history diagram, the axis locus diagram, the phase diagram, and the Poincaré section diagram in different rotational speeds are drawn. The characteristics of the periodic motion, quasiperiodic motion, and chaotic motion of the system in a given speed range are described in detail. The ways of the system entering and leaving chaos are revealed. The transformation and evolution process of the periodic motion, quasiperiodic motion, and chaotic motion are also analyzed. It shows that the rotor system enters chaos by the way of the period-doubling bifurcation. With the increase of the eccentricity, the quasi-periodicity evolution is chaotic. The quasiperiodic motion evolves into the periodic three motion phenomenon. And the increase of the stator stiffness will reduce the chaotic motion period.

scholarly journals Periodic, Quasi-Periodic, and Chaotic Motions to Diagnose a Crack on a Horizontally Supported Nonlinear Rotor System

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2059
Author(s):  
Nasser A. Saeed ◽  
Mohamed S. Mohamed ◽  
Sayed K. Elagan
Keyword(s):  
Period Doubling ◽  
Qualitative Change ◽  
Crack Size ◽  
Rotor System ◽  
Numerical Results ◽  
Chaotic Motions ◽  
Simple Method ◽  
Superharmonic Resonance ◽  
Whirling Motion ◽  
Period 2

This work aims to diagnose the crack size of a nonlinear rotating shaft system based on the qualitative change of the system oscillatory characteristics. The considered system is modeled as a two-degree-of-freedom horizontally supported nonlinear Jeffcott rotor system. The influence of the crack size on the system whirling motion for the primary, superharmonic, and subharmonic resonance cases are investigated utilizing the bifurcation diagram, Poincaré map, frequency spectrum, and whirling orbit. The obtained numerical results revealed that the cracked system whirling motion is subjected to a continuous qualitative change as the crack size increases for the superharmonic resonance case, where the system can exhibit period-1, period-2, quasi-periodic, period-3, period-doubling, chaotic, and period-2 motions, sequentially. In addition, an asymmetry is observed in the system whirling orbit due to both the shaft weight and shaft crack. Moreover, it is found that the disk eccentricity does not affect the nature of these motions. Accordingly, we illustrated a simple method to diagnose the existence of such a crack and to quantify its size via monitoring the system lateral vibrations at the superharmonic resonance. Finally, all the obtained numerical results are concluded and a comparison with already published work is included.

scholarly journals Dynamic response of a cracked rotor-bearing-seal system

2019 ◽  
Vol 288 ◽  
pp. 01006
Author(s):  
Huang Zhiwei
Keyword(s):  
Structural Parameters ◽  
Crack Depth ◽  
Period Doubling ◽  
Coupled System ◽  
Rotor System ◽  
Nonlinear Phenomena ◽  
Cracked Rotor ◽  
Chaotic Motions ◽  
Rotor Bearing ◽  
Opening And Closing

With the structural parameters of the rotation machinery increased, the seal fluid induced force imposed on the rotor will significantly increase. Taking a cracked rotor-bearing-seal system as object the non-linear dynamic behaviours of the coupled system are investigated using numerical integration method. For the crack in shaft, the segmented switch function is applied to express the process of the crack’s opening and closing in rotation, and the cross stiffness originated from crack is considered. Various nonlinear phenomena compressing periodic, quasi-periodic and chaotic motions in the rotor system are analysed. The research results show that the unstable form of the rotor system is Hopf bifurcation when the crack depth is smaller. The influence to the response of the system increased along with the crack depth, the unstable form of the coupled system is period-doubling bifurcation. The seal clearance has an important impact on stability of the cracked rotor system. In other words, the fluid induced force can reduce the periodic responses of the rotor. It is indicated that this study can contribution to a further understanding of the nonlinear of such a rotor-bearing system with crack and seal.

Dynamic Analysis of Planetary Gears With Bearing Clearance

2012 ◽  
Vol 7 (4) ◽  
Author(s):  
Yi Guo ◽  
Robert G. Parker
Keyword(s):  
Harmonic Balance Method ◽  
Period Doubling ◽  
Nonlinear Behavior ◽  
Nonlinear Effects ◽  
Lumped Parameter Model ◽  
Solution Stability ◽  
Planetary Gears ◽  
Gear Mesh ◽  
Bearing Clearance ◽  
Lumped Parameter

This study investigates the dynamics of planetary gears where nonlinearity is induced by bearing clearance. Lumped-parameter and finite element models with bearing clearance, tooth separation, and gear mesh stiffness variation are developed. The harmonic balance method with arc length continuation is applied to the lumped-parameter model to obtain the dynamic response. Solution stability is analyzed using Floquet theory. Rich nonlinear behavior is exhibited, consisting of nonlinear jumps, a hardening effect induced by the transition from no bearing contact to contact, and softening induced by tooth separation. Bearings of the central members (sun, carrier, and ring) impact against the bearing races near resonances, which leads to coexisting solutions in wide speed ranges, grazing bifurcation, and chaos. Secondary Hopf and period-doubling bifurcations are the routes to chaos. Input torque can suppress some of the nonlinear effects caused by bearing clearance.

scholarly journals Quasi-Static Characteristics and Vibration Responses Analysis of Helical Geared Rotor System with Random Cumulative Pitch Deviations

2020 ◽  
Vol 10 (12) ◽  
pp. 4403
Author(s):  
Bing Yuan ◽  
Geng Liu ◽  
Lan Liu
Keyword(s):  
Degrees Of Freedom ◽  
Transmission Error ◽  
Rotor System ◽  
Dynamic Responses ◽  
Contact Ratio ◽  
Tooth Contact Analysis ◽  
Long Period ◽  
Light Load ◽  
Vibration Responses ◽  
Loaded Tooth Contact Analysis

As one of the long period gear errors, the effects of random cumulative pitch deviations on mesh excitations and vibration responses of a helical geared rotor system (HGRS) are investigated. The long-period mesh stiffness (LPMS), static transmission error (STE), as well as composite mesh error (CMS), and load distributions of helical gears are calculated using an enhanced loaded tooth contact analysis (LTCA) model. A dynamic model with multi degrees of freedom (DOF) is employed to predict the vibration responses of HGRS. Mesh excitations and vibration responses analysis of unmodified HGRS are conducted in consideration of random cumulative pitch deviations. The results indicate that random cumulative pitch deviations have significant effects on mesh excitations and vibration responses of HGRS. The curve shapes of STE and CMS become irregular when the random characteristic of cumulative pitch deviations is considered, and the appearance of partial contact loss in some mesh cycles leads to decreased LPMS when load torque is relatively low. Vibration modulation phenomenon can be observed in dynamic responses of HGRS. In relatively light load conditions, the amplitudes of sideband frequencies become larger than that of mesh frequency and its harmonics (MFIHs) because of relatively high contact ratio. The influences of random cumulative pitch deviations on the vibration responses of modified HGRS are also discussed.

Bifurcations and Chaos in Forced, Coupled Pendula

2001 ◽  
Author(s):  
Kazuyuki Yagasaki
Keyword(s):  
Averaging Method ◽  
Nonlinear Behavior ◽  
Melnikov Method ◽  
Original System ◽  
Chaotic Motions ◽  
Small Perturbations ◽  
Small Oscillations ◽  
Codimension One ◽  
Averaged Systems ◽  
Direct Numerical Integration

Abstract We consider forced, coupled pendula and show that they exhibit very complicated dynamics using the averaging method and Melnikov-type techniques. First, the averaged system for small oscillations of the pendula near the hanging state is analyzed. Codimension-one and -two local bifurcations at which several non-synchronized periodic orbits and quasiperiodic orbits are born in the original system are detected. The validity of the theoretical results is demonstrated by comparison with direct numerical integration results. Moreover, chaotic motions, which result from the Shilnikov type phenomena in the averaged systems, are observed in numerical simulations. Second, the second-order averaging method is applied to small perturbations of rotary orbits with no damping and external forcing. Analyzing the averaged system, we can describe nonlinear behavior in the original system. Finally, using a generalization of Melnikov method, we prove the occurrence of many other homo-clinic phenomena, which also yield chaotic dynamics.

Nonlinear Dynamic Analysis of Atomic Force Microscopy

2006 ◽  
Author(s):  
Hossein Nejat Pishkenari ◽  
Mehdi Behzad ◽  
Ali Meghdari
Keyword(s):  
Atomic Force Microscopy ◽  
Equations Of Motion ◽  
Basin Of Attraction ◽  
Nonlinear Dynamic Analysis ◽  
Nonlinear Behavior ◽  
Random Excitation ◽  
System Response ◽  
Force Microscopy ◽  
Atomic Force ◽  
Exciting Frequency

This paper is devoted to the analysis of nonlinear behavior of amplitude modulation (AM) and frequency modulation (FM) modes of atomic force microscopy. For this, the microcantilever (which forms the basis for the operation of AFM) is modeled as a single mode approximation and the interaction between the sample and cantilever is derived from a van der Waals potential. Using perturbation methods such as Averaging, and Fourier transform nonlinear equations of motion are analytically solved and the advantageous results are extracted from this nonlinear analysis. The results of the proposed techniques for AM-AFM, clearly depict the existence of two stable and one unstable (saddle) solutions for some of exciting parameters under deterministic vibration. The basin of attraction of two stable solutions is different and dependent on the exciting frequency. From this analysis the range of the frequency which will result in a unique periodic response can be obtained and used in practical experiments. Furthermore the analytical responses determined by perturbation techniques can be used to detect the parameter region where the chaotic motion is avoided. On the other hand for FM-AFM, the relation between frequency shift and the system parameters can be extracted and used for investigation of the system nonlinear behavior. The nonlinear behavior of the oscillating tip can easily explain the observed shift of frequency as a function of tip sample distance. Also in this paper we have investigated the AM-AFM system response under a random excitation. Using two different methods we have obtained the statistical properties of the tip motion. The results show that we can use the mean square value of tip motion to image the sample when the excitation signal is random.

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