scholarly journals Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic Excitation

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Xindong Ma ◽  
Shuqian Cao
Keyword(s):  
Natural Frequency ◽  
Coupling Effect ◽  
Excitation Frequency ◽  
Time Interval ◽  
Periodic Excitation ◽  
Bursting Oscillations ◽  
Fold Bifurcation ◽  
Exciting Frequency ◽  
Bursting Oscillation ◽  
Parameter Condition

The coupling effect of two different frequency scales between the exciting frequency and the natural frequency of the Shimizu-Morioka system with slow-varying periodic excitation is investigated. First, based on the analysis of the equilibrium states, homoclinic bifurcation, fold bifurcation, and supercritical Hopf bifurcation are observed in the system under a certain parameter condition. When the exciting frequency is much smaller than the natural frequency, we can regard the periodic excitation as a slow-varying parameter. Second, complicated dynamic behaviors are analyzed when the slow-varying parameter passes through different bifurcation points, of which the mechanisms of four different bursting patterns, namely, symmetric “homoclinic/homoclinic” bursting oscillation, symmetric “fold/Hopf” bursting oscillation, symmetric “fold/fold” bursting oscillation, and symmetric “Hopf/Hopf” bursting oscillation via “fold/fold” hysteresis loop, are revealed with different values of the parameterbby means of the transformed phase portrait. Finally, we can find that the time interval between two symmetric adjacent spikes of bursting oscillations exhibits dependency on the periodic excitation frequency.

Bursting Oscillations and the Mechanism with Sliding Bifurcations in a Filippov Dynamical System

2018 ◽  
Vol 28 (12) ◽  
pp. 1850146 ◽  
Author(s):  
Rui Qu ◽  
Yu Wang ◽  
Guoqing Wu ◽  
Zhengdi Zhang ◽  
Qinsheng Bi
Keyword(s):  
Dynamical System ◽  
Natural Frequency ◽  
Stable Equilibrium ◽  
Multiple Scales ◽  
Hopf Bifurcations ◽  
Periodic Excitation ◽  
Nonsmooth Boundary ◽  
Bursting Oscillations ◽  
First Case ◽  
Exciting Frequency

The main purpose of the paper is to investigate the effect of multiple scales in frequency domain on the complicated oscillations of Filippov system with discontinuous right-hand side. A relatively simple model based on the Chua’s circuit with periodic excitation is introduced as an example. When the exciting frequency is far less than the natural frequency, implying that an order gap between the exciting frequency and the natural frequency exists, the whole exciting term can be considered as a slow-varying parameter, based on which the bifurcations of the two subsystems in different regions divided by the nonsmooth boundary are presented. Two typical cases are considered, which correspond to different distributions of equilibrium branches as well as the related bifurcations. In the first case, periodic symmetric Hopf/Hopf-fold-sliding bursting oscillations can be obtained, in which Hopf bifurcations may cause the alternations between the quiescent states and the spiking states, while fold bifurcations connect the two quiescent states moving along the stable equilibrium branches and sliding along the nonsmooth boundary, respectively. While the second case is the periodic symmetric fold/fold-fold-sliding bursting, where the fold bifurcations not only lead to the alternations between the quiescent states and the spiking states, but also connect the two quiescent states moving along the stable equilibrium branches and sliding along the nonsmooth boundary, respectively. It is pointed out that, different from the bursting oscillations in smooth dynamical systems in which the bifurcations may cause the alternations between quiescent states and spiking states, in the nonsmooth system, bifurcations may not only lead to the alternations, but also connect different forms of quiescent states. Furthermore, in the Filippov system, sliding movement along the nonsmooth boundary can be observed, the mechanism of which is presented based on the analysis of the two subsystems in different regions.

Vibration Diagnosis Featuring Blade-Shaft Coupling Effect of Turbine Rotor Models

2010 ◽  
Vol 133 (2) ◽  
Author(s):  
Norihisa Anegawa ◽  
Hiroyuki Fujiwara ◽  
Osami Matsushita
Keyword(s):  
Natural Frequency ◽  
Rotational Speed ◽  
Coupling Effect ◽  
Rotating Shaft ◽  
Cross Sectional ◽  
Additional Resonance ◽  
Exciting Frequency ◽  
Excited Vibration ◽  
Out Of Plane ◽  
Blade System

As is well known, zero and one nodal diameter (k=0 and k=1) modes of a blade system interact with the shaft system. The former couples with torsional and/or axial shaft vibrations, and the latter with bending shaft vibrations. This paper addresses the latter. With respect to k=1 modes, we discuss, from experimental and theoretical viewpoints, in-plane blades and out-of-plane blades attached radially to a rotating shaft. We found that when we excited the shaft at the rotational speed of Ω=|ωb−ωs| (where ωb is the blade natural frequency, ωs the shaft natural frequency, and Ω is the rotational speed), the exciting frequency ν=ωs induced shaft-blade coupling resonance. In addition, in the case of the in-plane blade system, we encountered an additional resonance attributed to deformation caused by gravity. In the case of the out-of-plane blade system, we experienced two types of abnormal vibrations. One is the additional resonance generated at Ω=ωb/2 due to the unbalanced shaft and the anisotropy of bearing stiffness. The other is a flow-induced, self-excited vibration caused by galloping due to the cross-sectional shape of the blade tip because this instability disappeared in the rotation test inside a vacuum chamber. The two types of abnormal vibrations occurred at the same time, and both led to the entrainment phenomenon, as identified by our own frequency analysis technique.

Fast–Slow Dynamics and Bifurcation Mechanism in a Novel Chaotic System

2019 ◽  
Vol 29 (10) ◽  
pp. 1930028
Author(s):  
Lan Huang ◽  
Guoqing Wu ◽  
Zhengdi Zhang ◽  
Qinsheng Bi
Keyword(s):  
Natural Frequency ◽  
Chaotic System ◽  
Three Dimensional ◽  
Harmonic Excitation ◽  
Phase Portraits ◽  
Fast Subsystem ◽  
Bursting Oscillations ◽  
Exciting Frequency ◽  
Different Types ◽  
Multiple Coexisting Attractors

This paper proposes a novel three-dimensional chaotic system with multiple coexisting attractors, where different values of a constant control parameter may drive the chaotic behaviors to evolve from single-scroll to double-scroll attractors. When the controlling term is replaced by a periodic harmonic excitation where the exciting frequency is far less than the natural frequency, chaotic movement may disappear, while periodic bursting oscillations will take place. Based on the fact that during a period defined by the natural frequency, the exciting term keeps almost a constant, the whole exciting term can be regarded as a slow-varying parameter resulting in a generalized autonomous system, its equilibrium branches as well as the related bifurcations occurring with the variation of the slow-varying parameter are derived. With the increase of the exciting amplitude, asymmetric and symmetric bursting attractors can be observed, for which the mechanism can be analyzed by the overlap of the equilibrium branches and the transformed phase portraits. With different values of the exciting amplitude corresponding to the change region of the slow-varying parameter, different bifurcations such as fold and Hopf bifurcations may involve the bursting structures, leading to different types of bursting oscillations. Furthermore, the phase space can be divided into two regions by a line boundary because of the symmetry of the vector field. When the trajectory from one region returning to the region arrives at the boundary, two asymmetric bursting attractors located in different regions coexist, which are symmetric to each other. However, when the trajectory passes across the boundary, an enlarged symmetric bursting attractor can be observed, whose trajectory connects the two original asymmetric attractors. Furthermore, it is found that when the trajectory runs along a stable equilibrium branch to the bifurcation point, it may move almost strictly along an unstable equilibrium branch of the fast subsystem because of the delay influence of the bifurcation.

Dynamic Modeling and Analysis of a Rotating Piezoelectric Smart Beam

2018 ◽  
Vol 18 (01) ◽  
pp. 1850003 ◽  
Author(s):  
En Lu ◽  
Wei Li ◽  
Xuefeng Yang ◽  
Yuqiao Wang ◽  
Yufei Liu
Keyword(s):  
Natural Frequency ◽  
Coupling Effect ◽  
Structural Parameters ◽  
Active Vibration Control ◽  
Excitation Frequency ◽  
Piezoelectric Actuators ◽  
Transverse Displacement ◽  
Assumed Mode Method ◽  
Smart Beam ◽  
Piezoelectric Actuators And Sensors

In active vibration control study, piezoelectric actuators and sensors are bonded on the surface of a beam. They can change the frequency and modal characteristics of the system. This paper presents an analysis of the frequency response to a rotating piezoelectric smart beam. Hamilton’s principle along with the assumed mode method are employed to derive the governing equations of the first-order approximate coupling model for the piezoelectric smart beam. The coupling is taken into account as the second-order coupling effect of the axial elongation caused by the transverse displacement of the beam. Then, the equations are transformed into a dimensionless form after identifying the necessary parameters. The dimensionless natural frequencies of the piezoelectric smart beam corresponding to the bending and stretching vibrations are obtained through a numerical simulation, with comparison made of those of the beam with no actuator or sensor. Furthermore, the implication is investigated of the structural parameters and bond location on the piezoelectric actuators and sensors. Besides, the common case of a smart beam bonded with multiple pairs of piezoelectric actuators and sensors is studied, and the effects of the first natural frequency and tip deformation are analyzed. The research provides a theoretical reference for the optimization of structural parameters and location of piezoelectric actuators and sensors, thereby preventing the resonance when the excitation frequency is approximately equal to the natural frequency of the beam.

Vibration Diagnosis Featuring Blade-Shaft Coupling Effect of Turbine Model Rotors

2010 ◽  
Author(s):  
Norihisa Anegawa ◽  
Hiroyuki Fujiwara ◽  
Osami Matsushita
Keyword(s):  
Natural Frequency ◽  
Rotational Speed ◽  
Coupling Effect ◽  
Rotating Shaft ◽  
Analysis Technique ◽  
Additional Resonance ◽  
Exciting Frequency ◽  
Out Of Plane ◽  
Blade System ◽  
Bearing Stiffness

It is well known that zero and one nodal diameter (k = 0 and k = 1) modes of a blade system interact with the shaft system. The former is coupling with torsional and/or axial shaft vibrations, and the latter with bending shaft vibrations. This paper deals with the latter. With respect to k = 1 modes, we discuss experimentally and theoretically in-plane blades and out-of-plane blades attached radially to a rotating shaft. We found that when we excited the shaft at the rotational speed of Ω = |ωb – ωs| (where ωb = blade natural frequency, ωs = shaft natural frequency and Ω = rotational speed), the exciting frequency ν = ωs induced shaft-blade coupling resonance. In addition, in the case of the in-plane blade system, we encountered an additional resonance attributed to deformation caused by gravity. In the case of the out-of-plane blade system, we experienced two types of abnormal vibrations. One is the additional resonance generated at Ω = ωb / 2 due to the unbalanced shaft and the anisotropy of bearing stiffness. The other is a flow-induced self-exited vibration caused by galloping due to the cross-section shape of the blade tip, because this instability disappeared at the rotation test inside a vacuum chamber. Both occurred at the same time, and both led to the entrainment phenomenon, which was identified by our own frequency analysis technique.

Frequency Tunable Isolator Based on Shape Memory Alloy for Effective Shock and Vibration Suppression

2013 ◽  
Author(s):  
Ho-Kyeong Jeong ◽  
Juho Lee ◽  
Jae-Hung Han
Keyword(s):  
Natural Frequency ◽  
Structural Integrity ◽  
Frequency Tuning ◽  
Dynamic Pressure ◽  
Excitation Frequency ◽  
Low Frequency ◽  
Design Procedure ◽  
Shock Attenuation ◽  
Exciting Frequency ◽  
Frequency Tunable

A Shock and vibration isolator is widely used due to its simplicity and effectiveness. It attenuates vibration energy when the external excitation frequency is more than about 2 times its natural frequency, while the vibration around its natural frequency is generally amplified. However, an exciting frequency often varies so that it is difficult to avoid the vibration amplification. In particular, when these amplification phenomena occur in the low frequency domain, induced large vibration displacements degrade the structural integrity. This paper introduces a novel frequency tunable isolator proposed by the present authors. The isolator uses SMA wires as actuator as well as the isolation materials. The isolator material is a compressed mesh washer isolator using the pseudoelasticity of SMA. Frequency tune of the isolator can be easily achieved through a simple electric circuit. Thus, this isolator can be widely applied to various vibration and shock environments such as in aircrafts and motor vehicles. Particularly, the detail design procedure is presented here for the adaptive shock isolator for launch vehicle in order to achieve both shock attenuation performance and avoidance of the vibration amplification. Launch vehicles experience severe dynamic environment during the flight phase. Specially, pyroshock generated from the several separation events could result in malfunctions of electric components and low frequency vibration below 100 Hz at the maximum dynamic pressure phase could reduce the structural integrity of payload. The resonant frequency of the isolator is selectively controlled in two modes by using an adaptive mechanical system with compressing the isolation materials. The isolator was successfully designed and various test results with frequency tuning are presented, in this paper.

scholarly journals Bursting Oscillation and Its Mechanism of a Generalized Duffing–Van der Pol System with Periodic Excitation

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Youhua Qian ◽  
Danjin Zhang ◽  
Bingwen Lin
Keyword(s):  
Numerical Simulations ◽  
Phase Portrait ◽  
Time Scales ◽  
Bifurcation Analysis ◽  
Hot Spot ◽  
Periodic Excitation ◽  
Fast Subsystem ◽  
Van Der Pol ◽  
Bursting Oscillations ◽  
Bursting Oscillation

The complex bursting oscillation and bifurcation mechanisms in coupling systems of different scales have been a hot spot domestically and overseas. In this paper, we analyze the bursting oscillation of a generalized Duffing–Van der Pol system with periodic excitation. Regarding this periodic excitation as a slow-varying parameter, the system can possess two time scales and the equilibrium curves and bifurcation analysis of the fast subsystem with slow-varying parameters are given. Through numerical simulations, we obtain four kinds of typical bursting oscillations, namely, symmetric fold/fold bursting, symmetric fold/supHopf bursting, symmetric subHopf/fold cycle bursting, and symmetric subHopf/subHopf bursting. It is found that these four kinds of bursting oscillations are symmetric. Combining the transformed phase portrait with bifurcation analysis, we can observe bursting oscillations obviously and further reveal bifurcation mechanisms of these four kinds of bursting oscillations.

Vibration of Bending-Torsion Coupled Resonance in a Rotor

2013 ◽  
Author(s):  
Hiroyuki Fujiwara ◽  
Tadashi Tsuji ◽  
Osami Matsushita
Keyword(s):  
Numerical Simulations ◽  
Natural Frequency ◽  
Torsional Vibration ◽  
Rotational Speed ◽  
Coupling Effect ◽  
The Other ◽  
Resonance Phenomenon ◽  
Horizontal Direction ◽  
Bending Vibration ◽  
Rotor Systems

In certain rotor systems, bending-torsion coupled resonance occurs when the rotational speed Ω (= 2π Ωrps) is equal to the sum/difference of the bending natural frequency ωb (= 2π fb) and torsional natural frequency ωθ(= 2πfθ). This coupling effect is due to an unbalance in the rotor. In order to clarify this phenomenon, an equation was derived for the motion of the bending-torsion coupled 2 DOF system, and this coupled resonance was verified by numerical simulations. In stability analyses of an undamped model, unstable rotational speed ranges were found to exist at about Ωrps = fb + fθ. The conditions for stability were also derived from an analysis of a damped model. In rotational simulations, bending-torsion coupled resonance vibration was found to occur at Ωrps = fb − fθ and fb + fθ. In addition, confirmation of this resonance phenomenon was shown by an experiment. When the rotor was excited in the horizontal direction at bending natural frequency, large torsional vibration appeared. On the other hand, when the rotor was excited by torsion at torsional natural frequency, large bending vibration appeared. Therefore, bending-torsion coupled resonance was confirmed.

Nonlinear Vibrations in an Elastic Structure Subjected to Vertical Excitation and Coupled With Liquid Sloshing in a Rectangular Tank

1997 ◽  
Author(s):  
Takashi Ikeda
Keyword(s):  
Theoretical Analysis ◽  
Natural Frequency ◽  
Liquid Surface ◽  
Excitation Frequency ◽  
Harmonic Balance Method ◽  
Liquid Sloshing ◽  
Elastic Structure ◽  
Vertical Excitation ◽  
Rectangular Tank ◽  
Resonance Curves

Abstract The nonlinear coupled vibrations of an elastic structure and liquid sloshing in a rectangular tank, partially filled with liquid, are investigated. The structure containing the tank is vertically subjected to a sinusoidal excitation. In the theoretical analysis, the resonance curves for the responses of the structure and liquid surface are presented by the harmonic balance method, when the natural frequency of the structure is equal to twice the natural frequency of one of the sloshing modes. From the theoretical analysis, the following predictions have been obtained: (a) Due to the nonlinearity of the fluid force, harmonic oscillations appear in the structure, while subharmonic oscillations occur on the liquid surface, (b) the shapes of the resonance curves markedly change depending on the liquid depth, and (c) when the detuning condition is slightly deviated, almost periodic oscillations and chaotic oscillations appear at certain intervals of the excitation frequency. These were qualitatively in good agreement with the experimental results.

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