scholarly journals Bifurcation Study of Thin Plate with an All-Over Breathing Crack

2016 ◽  
Vol 2016 ◽  
pp. 1-19 ◽  
Author(s):  
Lihua Chen ◽  
Jian Xue ◽  
Zhijie Zhang ◽  
Wei Zhang
Keyword(s):  
Boundary Conditions ◽  
Thin Plate ◽  
Nonlinear Dynamic ◽  
Crack Depth ◽  
Dynamic Responses ◽  
Phase Portraits ◽  
Open Crack ◽  
Breathing Crack ◽  
Dynamic Behaviors ◽  
Cracked Plate

An all-over breathing crack on the plate surface having arbitrary depth and location is assumed to be nonpropagating and parallel to one side of the plate. Based on a piecewise model, the nonlinear dynamic behaviors of thin plate with the all-over breathing crack are studied to analyze the effect of external excitation amplitudes and frequencies on cracked plate with different crack parameters (crack depth and crack location). Firstly, the mode shape functions of cracked thin plate are obtained by using the simply supported boundary conditions and the boundary conditions along the crack line. Then, natural frequencies and mode functions of the cracked plate are calculated, which are assessed with FEM results. The stress functions of thin plate with large deflection are obtained by the equations of compatibility in the status of opening and closing of crack, respectively. To compare with the effect of breathing crack on the plate, the nonlinear dynamic responses of open-crack plate and intact plate are analyzed too. Lastly, the waveforms, bifurcation diagrams, and phase portraits of the model are gained by the Runge-Kutta method. It is found that complex nonlinear dynamic behaviors, such as quasi-periodic motion, bifurcation, and chaotic motion, appear in the breathing crack plate.

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.

Chaos and Nonlinear Dynamic Analysis of Porous Air Bearing System

2015 ◽  
Vol 764-765 ◽  
pp. 204-207
Author(s):  
Cheng Chi Wang ◽  
Jui Pin Hung
Keyword(s):  
Dynamic Behavior ◽  
Nonlinear Dynamic ◽  
Effective Means ◽  
Nonlinear Dynamic Analysis ◽  
Transformation Method ◽  
Dynamic Responses ◽  
Air Bearing ◽  
Dynamic Behaviors ◽  
Rotor Mass ◽  
Bearing System

The chaos and nonlinear dynamic behaviors of porous air bearing system are studied by a hybrid numerical method combining the finite difference method (FDM) and differential transformation method (DTM). The numerical results are verified by two different schemes including hybrid method and FDM and the current analytical results are found to be in good agreement. Furthermore, the results reveal the changes which take place in the dynamic behavior of the bearing system as the rotor mass is increased. From the dynamic responses of the rotor center, they reveal complex dynamic behaviors including periodic, sub-harmonic motion and chaos. The results of this study provide an understanding of the nonlinear dynamic behavior of PAB systems characterized by different rotor masses. Specifically, the results have shown that system exists chaotic motion over the ranges of rotor mass 10.66≤Mr<13.7kg. The proposed method and results provide an effective means of gaining insights into the porous air bearing systems.

Dynamic Modelling and Nonlinear Characteristics of a Cracked Bolt-Disc Combined Rotor System

2021 ◽  
Author(s):  
Yi Liu ◽  
Heng Liu ◽  
Yuan Li ◽  
Nanshan Wang
Keyword(s):  
Stiffness Matrix ◽  
Crack Depth ◽  
Dynamic Modelling ◽  
Rotor System ◽  
Correction Coefficient ◽  
Local Defect ◽  
Open Crack ◽  
Rotating Shaft ◽  
Dynamic Behaviors ◽  
Crack Location

Abstract Different from the crack on the rotating shaft, the crack on the bolt which is a connecting part of the bolt-disc combined rotor is a kind of local defect. The local crack on the bolt under high pretension is always in open state, and it increases the overall vibration of the combined rotor significantly in practice. This paper studies the modelling of the crack on the bolt and nonlinear dynamic behaviors of the cracked bolt-disc rotor system. The circumferential bolts with a transverse open crack are treated as several bar elements under the assumption that each bolt has the same original tensile extension length. The cracked correction coefficient is introduced to describe the decreasing amount of bolt's tension due to crack. After this coefficient is obtained according to finite element method, the stiffness matrix of circumferential bolts with crack is built based on total potential energy. The dynamic model consists of a time-independent stiffness matrix for perfect bolts, a time-variant reductive stiffness and an additional moment. As a result, the crack in bolt reduces rotor's nonlinear stability and leads to greater vibration and fluctuation. In addition, crack depth has much larger influence than crack location on the dynamic behaviors.

scholarly journals Nonlinear Dynamic Behaviors of Rotated Blades with Small Breathing Cracks Based on Vibration Power Flow Analysis

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Hailong Xu ◽  
Zhongsheng Chen ◽  
Yeping Xiong ◽  
Yongmin Yang ◽  
Limin Tao
Keyword(s):  
Nonlinear Dynamic ◽  
Power Flow ◽  
Flow Analysis ◽  
Crack Model ◽  
Breathing Crack ◽  
Dynamic Behaviors ◽  
Power Flow Analysis ◽  
Local Flexibility ◽  
Vibration Power Flow ◽  
Nonlinear Behaviors

Rotated blades are key mechanical components in turbomachinery and high cycle fatigues often induce blade cracks. Accurate detection of small cracks in rotated blades is very significant for safety, reliability, and availability. In nature, a breathing crack model is fit for a small crack in a rotated blade rather than other models. However, traditional vibration displacements-based methods are less sensitive to nonlinear characteristics due to small breathing cracks. In order to solve this problem, vibration power flow analysis (VPFA) is proposed to analyze nonlinear dynamic behaviors of rotated blades with small breathing cracks in this paper. Firstly, local flexibility due to a crack is derived and then time-varying dynamic model of the rotated blade with a small breathing crack is built. Based on it, the corresponding vibration power flow model is presented. Finally, VPFA-based numerical simulations are done to validate nonlinear behaviors of the cracked blade. The results demonstrate that nonlinear behaviors of a crack can be enhanced by power flow analysis and VPFA is more sensitive to a small breathing crack than displacements-based vibration analysis. Bifurcations will occur due to breathing cracks and subharmonic resonance factors can be defined to identify breathing cracks. Thus the proposed method can provide a promising way for detecting and predicting small breathing cracks in rotated blades.

scholarly journals Monitoring breathing cracks of a beam-like bridge subjected to moving vehicle using wavelet spectrum

2013 ◽  
Vol 35 (2) ◽  
pp. 131-145
Author(s):  
Khoa Viet Nguyen
Keyword(s):  
Stiffness Matrix ◽  
Crack Depth ◽  
Wavelet Spectrum ◽  
Accurate Estimation ◽  
Open Crack ◽  
The Finite Element Method ◽  
Breathing Crack ◽  
Moving Vehicle ◽  
Beam Height ◽  
Crack Position

In this paper a wavelet spectrum technique for monitoring the breathing crack phenomenon of a beam-like bridge subjected to moving vehicle is presented. The stiffness of element with a breathing crack is modeled as a time dependent stiffness matrix using the finite element method. The stiffness matrix of the structure at each moment depends on the curvature of the structure at the crack position. The breathing crack phenomenon can be detected by analysing the instantaneous frequency (IF) of the system using the wavelet spectrum. When the crack depth is large, the crack area might be determined by the significant peak in the IF. The simulation results show that when the crack “breaths” the amplitude of the vibration obtained from the vehicle is smaller than in the case of an open crack. This is a warning when using the amplitude of the dynamic response to estimate the crack depth when there is a breathing crack in the structure. Therefore, it is important to distinguish the open crack and breathing crack to obtain a more accurate estimation of the crack depth. The results showed that crack with a depth as small as 10% of the beam height can be detected by the method. The proposed method can be applied with a noise level up to 10%.

scholarly journals Nonlinear Dynamic Responses of a Honeycomb Sandwich Plate Subject to Transverse Excitations

2018 ◽  
Vol 151 ◽  
pp. 01003
Author(s):  
Dongmei Wang ◽  
Wei Zhang ◽  
Minghui Yao ◽  
Yinli Liu
Keyword(s):  
Nonlinear Dynamic ◽  
Sandwich Plate ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Dynamic Responses ◽  
Phase Portraits ◽  
Period 2 ◽  
Transverse Excitation ◽  
Honeycomb Sandwich ◽  
Honeycomb Sandwich Plate

Nonlinear dynamic behaviors of a simply supported honeycomb sandwich plate subjected to the transverse excitations are investigated in this paper. Based on the classical thin plate theory and Von Karman large deformation theory, the governing equation of motion for the honeycomb sandwich plate is established by using the Hamilton principle. The nonlinear governing partial differential equation is discretized to the ordinary differential equations by differential quadrature method and then solved by Runge-Kutta-Fehlberg method. Based on the numerical simulations, combined with nonlinear dynamic theory, the influences of the frequency and amplitude of the transverse excitation are investigated respectively by using the bifurcation diagrams, Poincare maps and phase portraits. The results exhibit the existence of the period-1, period-2 and chaotic responses with the variation of the excitations, which demonstrate that those motions appear alternately.

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.

Parametric Influences on the Nonlinear Dynamic Responses of a Rotor-Bearing-Foundation-Labyrinth Seal System

2017 ◽  
Author(s):  
Enjie Zhang ◽  
Yinghou Jiao ◽  
Zhaobo Chen ◽  
Wenchao Mo ◽  
Shuai Wang
Keyword(s):  
Nonlinear Dynamic ◽  
Labyrinth Seal ◽  
Rotor System ◽  
Operating Conditions ◽  
Dynamic Responses ◽  
Lubricating Oil ◽  
Geometrical Parameters ◽  
Dynamic Behaviors ◽  
Highly Nonlinear ◽  
Rotor Bearing

The modern engineering industries rely heavily on the reliable operation of rotating machinery, e.g., steam turbine and gas turbine. These rotating machineries are inevitable to be excited by the unbalance mass forces, the oil film forces and seal forces. Moreover, the turbines installed in an aircraft as well as vessel are also excited by the base vibration. In order to retain the healthy operation and prolong the interval between overhauls, an enormous amount of experimental and theoretical investigations have been focused on the dynamic behaviors of the rotor system. The dynamic characteristics of the rotor system influenced by the single source of vibration, such as unbalance, flowing lubricating oil, sealing medium etc., and combined sources of vibration have also been thoroughly researched. However, the dynamic responses of the rotor-bearing-foundation system subjected to labyrinth seal forces have seldom been studied. Furthermore, the previous analyses of the rotor dynamics mostly were linear. In fact, the fluid film forces are strongly nonlinear functions of the displacement and velocity of the rotor. As a result, the rotordynamics of the turbine is highly nonlinear. It is not accurate enough to be considered from a linear point of view. Applying the energy method, this paper established a dynamic model of the rotor-bearing-foundation-labyrinth seal system. The influences of the geometrical parameters and operating conditions, such as mass eccentricities, inlet pressure and rotational speed etc., on the nonlinear dynamic behaviors of the rotor system are numerically studied. The responses of the same system excited by one side of and both sides of base movement are also comparatively analyzed by means of spectrum cascades, bifurcation diagrams and whirl orbits as well as Poincaré maps.

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