Study on Nonlinear Dynamic Response of an Unbalanced Rotor Supported on Ball Bearing

2009 ◽  
Vol 131 (6) ◽  
Author(s):  
G. Chen
Keyword(s):  
Nonlinear Dynamic ◽  
Ball Bearing ◽  
Period Doubling ◽  
Dynamic Responses ◽  
Hertzian Contact ◽  
Frequency Spectra ◽  
Rotating Speed ◽  
Unbalanced Rotor ◽  
Motion Period ◽  
Contact Position

An unbalanced rotor dynamic model supported on ball bearings is established. In the model, three nonlinear factors of ball bearing are considered, namely, the clearance of bearing, nonlinear Hertzian contact force between balls and races, and the varying compliance vibrations because of periodical change in contact position between balls and races. The numerical integration method is used to obtain the nonlinear dynamic responses; the effects of the rotating speed and the bearing clearance on dynamic responses are analyzed; and the bifurcation plots, the phase plane plots, the frequency spectra, and the Poincaré maps are used to carry out the analyses of bifurcation and chaotic motion. Period doubling, quasiperiod loop breaking, and mechanism of intermittency are observed as the routes to chaos.

scholarly journals Nonlinear Dynamics of the High-Speed Rotating Plate

2018 ◽  
Vol 2018 ◽  
pp. 1-23 ◽  
Author(s):  
Minghui Yao ◽  
Li Ma ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamic ◽  
Composite Plate ◽  
High Speed ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Dynamic Responses ◽  
Phase Portraits ◽  
Partial Differential ◽  
Rotating Speed ◽  
Rotating Blades

High speed rotating blades are crucial components of modern large aircraft engines. The rotating blades under working condition frequently suffer from the aerodynamic, elastic and inertia loads, which may lead to large amplitude nonlinear oscillations. This paper investigates nonlinear dynamic responses of the blade with varying rotating speed in supersonic airflow. The blade is simplified as a pre-twist and presetting cantilever composite plate. Warping effect of the rectangular cross-section of the plate is considered. Based on the first-order shear deformation theory and von-Karman nonlinear geometric relationship, nonlinear partial differential dynamic equations of motion for the plate are derived by using Hamilton’s principle. Galerkin approach is applied to discretize the partial differential governing equations of motion to ordinary differential equations. Asymptotic perturbation method is exploited to derive four-degree-of-freedom averaged equation for the case of 1 : 3 internal resonance-1/2 sub-harmonic resonance. Based on the averaged equation, numerical simulation is used to analyze the influence of the perturbation rotating speed on nonlinear dynamic responses of the blade. Bifurcation diagram, phase portraits, waveforms and power spectrum prove that periodic motion and chaotic motion exist in nonlinear vibration of the rotating cantilever composite plate.

scholarly journals Nonlinear Dynamic Characteristics of Marine Rotor-Bearing System under Heaving Motion

2019 ◽  
Vol 2019 ◽  
pp. 1-16 ◽  
Author(s):  
Yongchao Han ◽  
Ming Li
Keyword(s):  
Dynamic Characteristics ◽  
Nonlinear Dynamic ◽  
Dynamic Performance ◽  
Chaotic Oscillation ◽  
Mathematic Model ◽  
Rotating Speed ◽  
Period 2 ◽  
Rotor Bearing ◽  
Motion Period ◽  
Bearing System

In this paper, the influence of the heaving motion on the nonlinear dynamic behavior of the rotor-bearing system is considered. First, a mathematic model of the marine rotor-bearing system is developed on the short bearing theory in the noninertial reference system, in which the heaving motion is taken into account. Then its dynamic characteristics are analyzed based on the numerical integration method, such as the bifurcation diagram, the largest Lyapunov exponents (LLE), the steady-state response, and the rotor orbit and its Poincaré map. The results indicate that heaving motion has a great effect on the dynamics of the rotor system, which exhibits a period 1 motion at low rotating speed, with the increase of the rotating speed, the phenomena of the quasiperiodic, period 2, and double Hopf bifurcations appear. Its dynamic performance presents a period 1 motion, period 2, quasiperiodic, and chaotic oscillation.

Dynamic analysis of rotor-bearing system by considering the transverse crack on rotor

2019 ◽  
Author(s):  
N. Upadhyay ◽  
P. K Kankar
Keyword(s):  
Deformation Theory ◽  
Transverse Crack ◽  
Dynamic Behaviour ◽  
Ball Bearing ◽  
Equation Of Motion ◽  
Hertzian Contact ◽  
Rotating Speed ◽  
Deep Groove ◽  
Deep Groove Ball Bearing ◽  
Unbalance Force

In this study, a new improved theoretical model of rotorbearing system has been presented to analyse the behaviour of the system due to the transverse crack on the rotor. Firstly, a mathematical model of the system with a transverse crack on rotor has been developed. In the modelling, the rotor is taken as Timoshenko beam and the unbalance force also included, which vary with rotating speed. The rotor is supported by two healthy deep groove ball bearing at both ends. The contact between balls and races of the bearings is considered as nonlinear spring, whose stiffness is obtained by Hertzian contact deformation theory. After the modelling of the rotor, the equation of motion has been derived which represents the dynamic behaviour of the system. Bifurcation diagrams are used to investigate the influence of depth and size of the crack on the dynamic behaviour of rotor ball-bearings system. Results indicate that if the depth and size of the crack increase the system becomes highly chaotic and unstable.

Nonlinear Dynamic Response Analysis of Rotor-Ball Bearing-Stator Coupling System Including Unbalance-Rubbing Coupling Faults

2007 ◽  
Author(s):  
Guo Chen ◽  
Rong Tao Hou
Keyword(s):  
Ball Bearing ◽  
Dynamic Responses ◽  
Response Analysis ◽  
Nonlinear Dynamic Response ◽  
Rotating Speed ◽  
Coupling Faults ◽  
Mass Unbalance ◽  
Coupling System ◽  
Chaos Motion ◽  
Rotor Mass

In this paper, a new rotor-ball bearing-stator coupling system dynamic model is established. In the model, the rotor mass unbalance and rubbing faults are included, and the nonlinear factors of ball bearing such as the clearance of bearing, nonlinear Hertzian contract force between balls and races, and the varying compliance vibration coming from the periodical variety of contact positions between balls and races are modeled. The numerical integral method is employed to obtain system’s responses, and the vibration amplitude-rotating speed curve, bifurcation plot, phase plane plot, shaft centre orbits, frequency spectrum and Poincare´ map are used to carry out the analysis of bifurcation and chaos motion, and the effects of rotational speed, rubbing stiffness, rotor eccentricity, bearing house-stator stiffness, and stator-foundation stiffness on dynamic responses are analyzed, and the non-linear dynamic characteristics of rotor-ball bearing-stator system under unbalance and rubbing coupling faults are discovered.

scholarly journals Nonlinear Torsional Dynamics of Star Gearing Transmission System of GTF Gearbox

2020 ◽  
Vol 2020 ◽  
pp. 1-15 ◽  
Author(s):  
Siyu Wang ◽  
Rupeng Zhu
Keyword(s):  
Nonlinear Dynamic ◽  
Damping Ratio ◽  
Global Bifurcation ◽  
Period Doubling ◽  
Transmission System ◽  
Dynamic Responses ◽  
Nonlinear Dynamic Model ◽  
Chaotic Motions ◽  
Meshing Stiffness ◽  
And Control

Considering time-varying meshing stiffness, comprehensive errors, and piecewise backlash nonlinearities of gear and spline, a torsional nonlinear dynamic model of star gear-rotor coupling transmission system of (Geared Turbofan Engine) GTF aeroengine is established. By using the Runge–Kutta numerical integration method, the dynamic responses are solved, analyzed, and illustrated with the bifurcation parameters including input rotational speed, gear backlash, damping ratio, and comprehensive meshing errors. The motions of the star gearing system and diverse nonlinear dynamic characteristics are identified through global bifurcation, FFT spectra, Poincaré map, and the phase diagram. The results reveal that the star gear-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, and mutation are demonstrated. These results provide an understanding of undesirable torsional dynamic motion for the GTF transmission system and provide a reference for the design and control of gear system.

scholarly journals Understanding nonlinear vibration behaviours in high-power ultrasonic surgical devices

2015 ◽  
Vol 471 (2176) ◽  
pp. 20140906 ◽  
Author(s):  
Andrew Mathieson ◽  
Andrea Cardoni ◽  
Niccolò Cerisola ◽  
Margaret Lucas
Keyword(s):  
Nonlinear Dynamic ◽  
Maxillofacial Surgery ◽  
Period Doubling ◽  
Dynamic Responses ◽  
Hysteresis Curve ◽  
Minimal Risk ◽  
Half Wavelength ◽  
Surgical Devices ◽  
Power Ultrasonic ◽  
New Generation

Ultrasonic surgical devices are increasingly used in oral, craniofacial and maxillofacial surgery to cut mineralized tissue, offering the surgeon high accuracy with minimal risk to nerve and vessel tissue. Power ultrasonic devices operate in resonance, requiring their length to be a half-wavelength or multiple-half-wavelength. For bone surgery, devices based on a half-wavelength have seen considerable success, but longer multiple-half-wavelength endoscopic devices have recently been proposed to widen the range of surgeries. To provide context for these developments, some examples of surgical procedures and the associated designs of ultrasonic cutting tips are presented. However, multiple-half-wavelength components, typical of endoscopic devices, have greater potential to exhibit nonlinear dynamic behaviours that have a highly detrimental effect on device performance. Through experimental characterization of the dynamic behaviour of endoscopic devices, it is demonstrated how geometrical features influence nonlinear dynamic responses. Period doubling, a known route to chaotic behaviour, is shown to be significantly influenced by the cutting tip shape, whereas the cutting tip has only a limited effect on Duffing-like responses, particularly the shape of the hysteresis curve, which is important for device stability. These findings underpin design, aiming to pave the way for a new generation of ultrasonic endoscopic surgical devices.

Experimental and Numerical Studies on Nonlinear Dynamic Behavior of Rotor System Supported by Ball Bearings

2010 ◽  
Vol 132 (8) ◽  
Author(s):  
Changqing Bai ◽  
Hongyan Zhang ◽  
Qingyu Xu
Keyword(s):  
Finite Element ◽  
Dynamic Behavior ◽  
Nonlinear Dynamic ◽  
High Speed ◽  
Ball Bearing ◽  
Rotor System ◽  
Hertzian Contact ◽  
Ball Bearings ◽  
Flexible Rotor ◽  
Nonlinear Dynamic Behavior

Ball bearings are important mechanical components in high-speed turbomachinery that is liable for severe vibration and noise due to the inherent nonlinearity of ball bearings. Using experiments and the numerical approach, the nonlinear dynamic behavior of a flexible rotor supported by ball bearings is investigated in this paper. An experimental ball bearing-rotor test rig is presented in order to investigate the nonlinear dynamic performance of the rotor systems, as the speed is beyond the first synchroresonance frequency. The finite element method and two-degree-of-freedom dynamic model of a ball bearing are employed for modeling the flexible rotor system. The discrete model of a shaft is built with the aid of the finite element technique, and the ball bearing model includes the nonlinear effects of the Hertzian contact force, bearing internal clearance, and so on. The nonlinear unbalance response is observed by experimental and numerical analysis. All of the predicted results are in good agreement with experimental data, thus validating the proposed model. Numerical and experimental results show that the resonance frequency is provoked when the speed is about twice the synchroresonance frequency, while the subharmonic resonance occurs due to the nonlinearity of ball bearings and causes severe vibration and strong noise. The results show that the effect of a ball bearing on the dynamic behavior is noticeable in optimum design and failure diagnosis of high-speed turbomachinery.

Numerical modeling of dynamic response of miniature multi-impact electromagnetic device for low and wide range frequencies energy harvesting

2018 ◽  
Vol 233 (7) ◽  
pp. 2400-2409 ◽  
Author(s):  
Jianxiong Zhu ◽  
Nuh S Yuksek ◽  
M Almasri ◽  
Zaichun Feng
Keyword(s):  
Numerical Modeling ◽  
Nonlinear Dynamic ◽  
Low Frequency ◽  
Period Doubling ◽  
Nonlinear Oscillators ◽  
Dynamic Responses ◽  
Phase Portraits ◽  
Vibration Source ◽  
External Vibration ◽  
Wide Range

In this study, numerical modeling of nonlinear dynamic responses of miniature electromagnetic energy harvesters is reported for multiple impacts using limited amplitude and low-frequency excitations (0.5–3 g, 10–40 Hz). When an external vibration source frequency approaches oscillators’ resonate frequencies (15 Hz and 30 Hz), these oscillators strongly impact onto a stiffer cantilever resulting in a much higher frequency vibration (1 kHz) in accordance with a large frequency up-conversion factor ∼33.3–66.6. The Lorentz force and the nonlinear oscillators together resulted in complicated nonlinear dynamic responses of the cantilever, such as period doubling, superharmonic, or chaotic. Furthermore, the instantaneous generated power of miniature electromagnetic harvester was dramatically enhanced with 3 μW, and the enhancement came from the more the number of oscillators, the lesser the air damping, and appropriate frequencies from external vibration sources. Moreover, the free tip of the cantilever in the system with both of the cube nonlinear oscillators and the linear oscillators were carefully analyzed by the phase portraits to demonstrate its dynamic responses behavior.

scholarly journals Mechanism and Characteristics of Global Varying Compliance Parametric Resonances in a Ball Bearing

2020 ◽  
Vol 10 (21) ◽  
pp. 7849
Author(s):  
Zhiyong Zhang ◽  
Thomas Sattel ◽  
Yujie Zhu ◽  
Xuan Li ◽  
Yawei Dong ◽  
...  
Keyword(s):  
Ball Bearing ◽  
Primary Resonance ◽  
Period Doubling ◽  
Hertzian Contact ◽  
Rolling Bearings ◽  
Bearing Clearance ◽  
Parametric Resonances ◽  
Rolling Elements ◽  
The Stability ◽  
Varying Compliance

Varying compliance (VC) is an unavoidable form of parametric excitation in rolling bearings and can affect the stability and safety of the bearing and its supporting rotor system. To date, we have investigated VC primary resonance in ball bearings, and in this paper other parametric VC resonance types are addressed. For a classical ball bearing model with Hertzian contact and clearance nonlinearities between the rolling elements and raceway, the harmonic balance and alternating frequency/time domain (HB–AFT) method and Floquet theory are adopted to analyze the VC parametric resonances and their stabilities. It is found that the 1/2-order subharmonic resonances, 2-order superharmonic resonances, and various VC combination resonances, such as the 1-order and 2-order summed types, can be excited, thus resulting in period-1, period-2, period-4, period-8, period-35, quasi-period, and even chaotic VC motions in the system. Furthermore, the bifurcation and hysteresis characteristics of complex VC resonant responses are discussed, in which cyclic fold, period doubling, and the second Hopf bifurcation can occur. Finally, the global involution of VC resonances around bearing clearance-free operations (i.e., adjusting the bearing clearance to zero or one with low interference) are provided. The overall results extend the investigation of VC parametric resonance cases in rolling bearings.

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