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Wave propagation control in periodic track structure through local resonance mechanism 被引量:10
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作者 WANG Ping YI Qiang +2 位作者 ZHAO Cai-you XING Meng-ting LU J 《Journal of Central South University》 SCIE EI CAS CSCD 2018年第12期3062-3074,共13页
Excessive vibration and noise radiation of the track structure can be caused by the operation of high speed trains.Though the track structure is characterized by obvious periodic properties and band gaps,the bandwidth... Excessive vibration and noise radiation of the track structure can be caused by the operation of high speed trains.Though the track structure is characterized by obvious periodic properties and band gaps,the bandwidth is narrow and the elastic wave attenuation capability within the band gap is weak.In order to effectively control the vibration and noise of track structure,the local resonance mechanism is introduced to broaden the band gap and realize wave propagation control.The locally resonant units are attached periodically on the rail,forming a new locally resonant phononic crystal structure.Then the tuning of the elastic wave band gaps of track structure is discussed,and the formation mechanism of the band gap is explicated.The research results show that a new wide and adjustable locally resonant band gap is formed after the resonant units are introduced.The phenomenon of coupling and transition can be observed between the new locally resonant band gap and the original band gap of the periodic track structure with the band gap width reaching the maximum at the coupling position.The broader band gap can be applied for vibration and noise reduction in high speed railway track structure. 展开更多
关键词 wave propagation control periodic track structure band gap local resonance mechanism transfer matrix
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Transformation method and wave control 被引量:3
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作者 Zheng Chang Jin Hu Geng-Kai Hu 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2010年第6期889-898,共10页
Transformation method provides an efficient way to control wave propagation by materials.The transformed relations for field and material during a transformation are essential to fulfill this method.We propose a syste... Transformation method provides an efficient way to control wave propagation by materials.The transformed relations for field and material during a transformation are essential to fulfill this method.We propose a systematic method to derive the transformed relations for a general physic process,the constraint conditions are obtained by considering geometrical and physical constraint during a mapping. The proposed method is applied to Navier's equation for elastodynamics,Helmholtz's equation for acoustic wave and Maxwell's equation for electromagnetic wave,the corresponding transformed relations are derived,which can be used in the framework of transformation method for wave control.We show that contrary to electromagnetic wave,the transformed relations are not uniquely determined for elastic wave and acoustic wave,so we have a freedom to choose them differently.Using the obtained transformed relations,we also provide some examples for device design,a concentrator for elastic wave,devices for illusion acoustic and illusion optics are conceived and validated by numerical simulations. 展开更多
关键词 wave control Elastic wave Acoustic wave Electromagnetic wave Local affine transformation
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A wide-angle wave control method of reducing torque ripple for brushless DC motor 被引量:3
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作者 吴春华 陈国呈 孙承波 《Journal of Shanghai University(English Edition)》 CAS 2007年第3期300-303,共4页
In order to deal with torque pulsation problem caused by traditional control method for brushless DC (BLDC) motor and to achieve high precision and good stability, a novel control strategy is proposed. Compared with... In order to deal with torque pulsation problem caused by traditional control method for brushless DC (BLDC) motor and to achieve high precision and good stability, a novel control strategy is proposed. Compared with the traditional control scheme, by using phase voltage as a control objective and making waveform of phase current approximately quasi-sinusoidal, torque ripple of BLDC motor is reduced from the original 14% to 3.4%, while toque is increased by 3.8%. Furthermore, by detecting zero-crossings of back electromotive force (BEMF) with non-conducting phases, sensorless control is realized. The new control strategy is simple. It can minimize torque ripple, increase torque, and realize sensorless control for BLDC motor. Simulation and experiments show good performance of BLDC motor by using the new control method. 展开更多
关键词 wide-angle wave control BLDC motor torque ripple SENSORLESS
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Simple Push-Type Wave Generating Method Using Digital Rotary Valve Control 被引量:6
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作者 Yi Liu Jiafei Zheng +3 位作者 Ruiyin Song Qiaoning Xu Junhua Chen Fangping Huang 《Chinese Journal of Mechanical Engineering》 SCIE EI CAS CSCD 2020年第1期221-231,共11页
The important parameters to describe waves are their amplitude and length.In order to make it easier to improve wave amplitude and facilitate wave experiment,a simple push-type wave generating method using digital rot... The important parameters to describe waves are their amplitude and length.In order to make it easier to improve wave amplitude and facilitate wave experiment,a simple push-type wave generating method using digital rotary valve control was proposed and different wave amplitudes were generated by the new method.After the mathematical model of the new method had been established,numerical analysis based on the linear wave theory was carried out by means of Matlab/Simulink software tools,and experiments were conducted on the push-type wave maker to ascertain the validity of the established model and the numerical simulation results.It shows that both experimental and theoretical results agree relatively well,and the plate motion frequency and amplitude of the push-type wave maker can be continuously adjusted and the various required regular waves can be obtained.Although the wave amplitude and length descends with the increasing of working frequency,the wave amplitude can be improved conveniently by setting the axial opening width of the valve and the oil supply pressure of system.The wave length remains unchanged with the axial opening width and the oil supply pressure change.The research indicates that different regular waves can be easily generated by the new method and the wave amplitude can be further improved in a certain plate motion frequency range. 展开更多
关键词 Push-type wave MAKER DIGITAL rotary valve control Mathematical modeling wave amplitude wave frequency
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Dynamic and inner-dressing control of four-wave mixing in periodically-driven atomic system 被引量:2
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作者 Yuan-Yuan Li Li Li +1 位作者 Yun-Zhe Zhang Lei Zhang 《Chinese Physics B》 SCIE EI CAS CSCD 2019年第10期327-331,共5页
Four-wave-mixing(FWM) process is examined by using density matrix formalism in a periodically-driven atomic medium. Numerical result shows that FWM signals can be controlled by selecting different dynamic parameters o... Four-wave-mixing(FWM) process is examined by using density matrix formalism in a periodically-driven atomic medium. Numerical result shows that FWM signals can be controlled by selecting different dynamic parameters of the probe field and strengths of the inner-dressing fields. It is also shown that the controllable FWM process is dominantly influenced by the evolution of atomic population difference and two-photon coherence. This dynamic and inner-dressing control of FWM is probably used for optimizing the optical nonlinear process and information processing. 展开更多
关键词 FOUR-wave-MIXING DYNAMIC control inner-dressing
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Optimal Active Control of Wave-Induced Vibration for Offshore Platforms 被引量:10
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作者 李华军 Sau-LonJamesHU TomotsukaTAKAYAMA 《China Ocean Engineering》 SCIE EI 2001年第1期1-14,共14页
An obvious motivation of this paper is to examine the effectiveness of the lateral vibration control of a jacket type offshore platform with an AMD control device, in conjunction with H-2 control algorithm, which is a... An obvious motivation of this paper is to examine the effectiveness of the lateral vibration control of a jacket type offshore platform with an AMD control device, in conjunction with H-2 control algorithm, which is an optimal frequency domain control method based on minimization of H-2 norm of the system transfer function In this study, the offshore platform is modeled numerically by use of the finite element method, instead of a lumped mass model This structural model is later simplified to be single-degree-of-freedom (SDOF) system by extracting the first vibration mode of the structure. The corresponding 'generalized' wave force is determined based on an analytical approximation of the first mode shape function, the physical wave loading being calculated from the linearized Morison equation. This approach facilitates the filter design for the generalized force. Furthermore, the present paper also intends to make numerical comparison between H-2 active control and the corresponding passive control using a TMD with the same device parameters. 展开更多
关键词 active control offshore platform VIBRATION wave loading
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Numerical evaluation of passive control of shock wave/boundary layer interaction on NACA0012 airfoil using jagged wall 被引量:3
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作者 Mojtaba Dehghan Manshadi Ramin Rabani 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2016年第5期792-804,共13页
Shock formation due to flow compressibility and its interaction with boundary layers has adverse effects on aerodynamic characteristics, such as drag increase and flow separation. The objective of this paper is to app... Shock formation due to flow compressibility and its interaction with boundary layers has adverse effects on aerodynamic characteristics, such as drag increase and flow separation. The objective of this paper is to appraise the practicability of weakening shock waves and, hence, reducing the wave drag in transonic flight regime using a two-dimensional jagged wall and thereby to gain an appropriate jagged wall shape for future empirical study. Different shapes of the jagged wall, including rectangular, circular, and triangular shapes, were employed. The numerical method was validated by experimental and numerical studies involving transonic flow over the NACA0012 airfoil, and the results presented here closely match previous experimental and numerical results. The impact of parameters, including shape and the length-to-spacing ratio of a jagged wall, was studied on aerodynamic forces and flow field. The results revealed that applying a jagged wall method on the upper surface of an airfoil changes the shock structure significantly and disintegrates it, which in turn leads to a decrease in wave drag. It was also found that the maximum drag coefficient decrease of around 17 % occurs with a triangular shape, while the maximum increase in aerodynamic efficiency(lift-to-drag ratio)of around 10 % happens with a rectangular shape at an angle of attack of 2.26?. 展开更多
关键词 Jagged wall Passive flow control Shock wave/boundary layer interaction Aerodynamic efficiency
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Robust Control Based on Feedback Linearization for Roll Stabilizing of Autonomous Underwater Vehicle Under Wave Disturbances 被引量:4
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作者 潘立鑫 金鸿章 王琳琳 《China Ocean Engineering》 SCIE EI 2011年第2期251-263,共13页
In the case of Autonomous Underwater Vehicle (AUV) navigating with low speed near water surface, a new method for design of roll motion controller is proposed in order to restrain wave disturbance effectively and im... In the case of Autonomous Underwater Vehicle (AUV) navigating with low speed near water surface, a new method for design of roll motion controller is proposed in order to restrain wave disturbance effectively and improve roll stabilizing performance. Robust control is applied, which is based on uncertain nonlinear horizontal motion model of AUV and the principle of zero speed fin stabilizer. Feedback linearization approach is used to transform the complex nonlinear system into a comparatively simple linear system. For parameter uncertainty of motion model, the controller is designed with mixed-sensitivity method based on H-infinity robust control theory. Simulation results show better robustness improved by this control method for roll stabilizing of AUV navigating near water surface. 展开更多
关键词 wave disturbances roll stabilization robust control zero-speed fin stabilizer
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Pentamode metamaterials for acoustic wave control 被引量:1
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作者 Zhaohong Wang Chengxin Cai +1 位作者 Yangyang Chu Guangshun Liu 《光电工程》 CAS CSCD 北大核心 2017年第1期122-122,共1页
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Research advances in enhanced coal seam gas extraction by controllable shock wave fracturing
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作者 Chaojun Fan Hao Sun +6 位作者 Sheng Li Lei Yang Bin Xiao Zhenhua Yang Mingkun Luo Xiaofeng Jiang Lijun Zhou 《International Journal of Coal Science & Technology》 EI CAS CSCD 2024年第3期1-31,共31页
With the continuous increase of mining in depth,the gas extraction faces the challenges of low permeability,great ground stress,high temperature and large gas pressure in coal seam.The controllable shock wave(CSW),as ... With the continuous increase of mining in depth,the gas extraction faces the challenges of low permeability,great ground stress,high temperature and large gas pressure in coal seam.The controllable shock wave(CSW),as a new method for enhancing permeability of coal seam to improve gas extraction,features in the advantages of high efficiency,eco-friendly,and low cost.In order to better utilize the CSW into gas extraction in coal mine,the mechanism and feasibility of CSW enhanced extraction need to be studied.In this paper,the basic principles,the experimental tests,the mathematical models,and the on-site tests of CSW fracturing coal seams are reviewed,thereby its future research directions are provided.Based on the different media between electrodes,the CSW can be divided into three categories:hydraulic effect,wire explosion and excitation of energetic materials by detonating wire.During the process of propagation and attenuation of the high-energy shock wave in coal,the shock wave and bubble pulsation work together to produce an enhanced permeability effect on the coal seam.The stronger the strength of the CSW is,the more cracks created in the coal is,and the greater the length,width and area of the cracks being.The repeated shock on the coal seam is conducive to the formation of complex network fracture system as well as the reduction of coal seam strength,but excessive shock frequency will also damage the coal structure,resulting in the limited effect of the enhanced gas extraction.Under the influence of ground stress,the crack propagation in coal seam will be restrained.The difference of horizontal principal stress has a significant impact on the shape,propagation direction and connectivity of the CSW induced cracks.The permeability enhancement effect of CSW is affected by the breakage degree of coal seam.The shock wave is absorbed by the broken coal,which may hinder the propagation of CSW,resulting in a poor effect of permeability enhancement.When arranging two adjacent boreholes for CSW permeability enhancement test,the spacing of boreholes should not be too close,which may lead to negative pressure mutual pulling in the early stage of drainage.At present,the accurate method for effectively predicting the CSW permeability enhanced range should be further investigated. 展开更多
关键词 controllable shock wave Permeability enhancement Gas extraction Basic principle Experimental test Mathematical models On-site test
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Approximate Optimal Tracking Control for Near-Surface AUVs with Wave Disturbances 被引量:3
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作者 YANG Qing SU Hao TANG Gongyou 《Journal of Ocean University of China》 SCIE CAS 2016年第5期789-798,共10页
This paper considers the optimal trajectory tracking control problem for near-surface autonomous underwater vehicles(AUVs) in the presence of wave disturbances. An approximate optimal tracking control(AOTC) approach i... This paper considers the optimal trajectory tracking control problem for near-surface autonomous underwater vehicles(AUVs) in the presence of wave disturbances. An approximate optimal tracking control(AOTC) approach is proposed. Firstly, a six-degrees-of-freedom(six-DOF) AUV model with its body-fixed coordinate system is decoupled and simplified and then a nonlinear control model of AUVs in the vertical plane is given. Also, an exosystem model of wave disturbances is constructed based on Hirom approximation formula. Secondly, the time-parameterized desired trajectory which is tracked by the AUV's system is represented by the exosystem. Then, the coupled two-point boundary value(TPBV) problem of optimal tracking control for AUVs is derived from the theory of quadratic optimal control. By using a recently developed successive approximation approach to construct sequences, the coupled TPBV problem is transformed into a problem of solving two decoupled linear differential sequences of state vectors and adjoint vectors. By iteratively solving the two equation sequences, the AOTC law is obtained, which consists of a nonlinear optimal feedback item, an expected output tracking item, a feedforward disturbances rejection item, and a nonlinear compensatory term. Furthermore, a wave disturbances observer model is designed in order to solve the physically realizable problem. Simulation is carried out by using the Remote Environmental Unit(REMUS) AUV model to demonstrate the effectiveness of the proposed algorithm. 展开更多
关键词 trajectory tracked simplified Tracking observer desired transformed successive quadratic coordinate
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Computer simulation of short-circuiting transfer welding under waveform control on inverter power source
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作者 杨立军 冯胜强 +3 位作者 董天顺School of Materials Science and Technology Hebei University of Technology Tianjin 李桓 《China Welding》 EI CAS 2007年第1期9-13,共5页
A simulation model is introduced about the non-linearity process of short-circuiting transfer in CO2 arc welding for displaying the interaction between the inverter power source and welding arc under waveform control.... A simulation model is introduced about the non-linearity process of short-circuiting transfer in CO2 arc welding for displaying the interaction between the inverter power source and welding arc under waveform control. In the simulation model, the feedback signals of current and voltage are taken respectively at the different phase in a short circuit periodic time and applied to the PWM (pulse width modulation) module in a model of inverter power source to control the output of power source. The simulation operation about the dynamic process of CO2 short-circuiting transfer welding is implemented on the founded simulation model with a peak arc current of 400 A and a peak voltage of 35 V, producing the dynamic arc waveforms which can embody the effect of inverter harmonic wave. The simulating waveforms are close to that of welding experiments. 展开更多
关键词 INVERTER SIMULATION short-circuiting transfer harmonic wave waveform control
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Analysis on influence of seismic travelling wave effect on semi-active control for long-span rigid-continuous bridge 被引量:1
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作者 亓兴军 李小军 周国良 《Acta Seismologica Sinica(English Edition)》 EI CSCD 2006年第2期199-206,共8页
The analysis approach of semi-active control for long-span rigid-continuous bridge under seismic travelling wave input is established. Magnetorheological dampers are set on the positions of the bridge beatings. The se... The analysis approach of semi-active control for long-span rigid-continuous bridge under seismic travelling wave input is established. Magnetorheological dampers are set on the positions of the bridge beatings. The semi-active control calculation and analysis are performed for a five-span rigid-continuous bridge under seismic travelling waves with different apparent surface velocities. The results indicate that travelling wave effect remarkably influences the uncontrolled seismic responses, the semi-active control seismic responses and vibration control effects for the long-span rigid-continuous bridge. It is disadvantageous to the responses of the beams and the piers under the travelling wave input with lower apparent surface velocity, and travelling wave effect can decrease the vibration control effects evidently. Therefore, the travelling wave effect should be considered for the selection of the parameter values of semi-active control system in order to get the designing control effect. 展开更多
关键词 travelling wave effect semi-active control long-span rigid-continuous bridge magnetorheoiogicai damper
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Feedback control of wave segments in an excitable medium
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作者 吴宁杰 高洪俊 应和平 《Chinese Physics B》 SCIE EI CAS CSCD 2013年第5期187-190,共4页
Depending on the excitability of the medium, a propagating wave segment will either contract or expand to fill the medium with spiral waves. This paper aims to introduce a simple mechanism of feedback control to stabi... Depending on the excitability of the medium, a propagating wave segment will either contract or expand to fill the medium with spiral waves. This paper aims to introduce a simple mechanism of feedback control to stabilize such an expansion or contraction. To do this, we lay out a feedback control system in a block diagram and reduce it into a bare, universal formula. Analytical and experimental findings are compared through a series of numerical simulations of the Barkley model. 展开更多
关键词 feedback control wave segments excitable medium
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Sea-Control Algorithm in Lake Wave Case
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作者 Yoshiki Uemura Kenji Kita Kazuyuki Matsumoto 《Applied Mathematics》 2024年第12期829-833,共5页
In previous studies, an isosceles triangular-type possibility distribution was employed to represent the analog waves of a Gaussian Process. The model was then projected onto actual waves using Zadeh’s extension prin... In previous studies, an isosceles triangular-type possibility distribution was employed to represent the analog waves of a Gaussian Process. The model was then projected onto actual waves using Zadeh’s extension principle of mapping (Hori et al., 2019). Furthermore, by applying Vague Set and Systems theory, it was shown that the actual waves followed a Gaussian process, and that the system could be efficiently controlled via Monte Carlo simulation. However, due to the use of fuzzy OR logic in the extension principle of mapping and wave synthesis, the resulting ambiguity increased significantly. To address this issue, a Possibility Markov Chain was proposed, incorporating possibility theory to mitigate the explosion of ambiguity. In this study, we propose a novel modeling approach that utilizes a possibility transition matrix without relying on fuzzy OR logic. Additionally, we introduce the Sea-Control Algorithm, which artificially introduces system error into the system function, thereby enabling modification of the possibility transition matrix through the deliberate manipulation of possibility information within the fuzzy system. 展开更多
关键词 Lake wave Modeling Lake wave control Possibility Markov Chain Analog Gaussian Process Vague Event
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Portable Dynamic Positioning Control System on A Barge in Short-Crested Waves Using the Neural Network Algorithm 被引量:3
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作者 FANG Ming-chung LEE Zi-yi 《China Ocean Engineering》 SCIE EI CSCD 2013年第4期469-480,共12页
This paper develops a nonlinear mathematical model to simulate the dynamic motion behavior of the barge equipped with the portable outboard Dynamic Positioning (DP) system in short-crested waves. The self-tuning Pro... This paper develops a nonlinear mathematical model to simulate the dynamic motion behavior of the barge equipped with the portable outboard Dynamic Positioning (DP) system in short-crested waves. The self-tuning Proportional- Derivative (PD) controller based on the neural network algorithm is applied to control the thrusters for optimal adjustment of the barge position in waves. In addition to the wave, the current, the wind and the nonlinear drift force are also considered in the calculations. The time domain simulations for the six-degree-of-freedom motions of the barge with the DP system are solved by the 4th order Runge-Kutta method which can compromise the efficiency and the accuracy of the simulations. The technique of the portable alternative DP system developed here can serve as a practical tool to assist those ships without being equipped with the DP facility while the dynamic positioning missions are needed. 展开更多
关键词 neural network PD controller dynamic positioning short-crested wave
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Optimal control of oscillation wave energy system using velocity premonition 被引量:2
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作者 CUI Xiao 《Instrumentation》 2015年第2期28-32,共5页
Recently ocean wave energy draw much more attention for its widespread,abundant and highly energy flux density properties.Extracting energy from incident wave however,is limited for the random and unstable power input... Recently ocean wave energy draw much more attention for its widespread,abundant and highly energy flux density properties.Extracting energy from incident wave however,is limited for the random and unstable power input.Motion control for WEC is a promising method to improve the energy absorption and some practical applications are also verified such as latching control.In this paper,an active control strategy is proposed to achieve maximum energy capture.The mathematical description shows that the active control has the characteristic of anti-causal and the wave premonition is necessary for controller design.But the fact of premonition time horizon is still unclear.In this paper,the premonition nature is described mathematically based on hydrodynamic theory.Furthermore,a simulation is also performed to study the impacting of premonition time horizon on WEC's properties and give a more insightful understanding of WEC active control. 展开更多
关键词 wave energy converter(WEC) active control PREMONITION time
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Semi-Active Control of Wave-Induced Vibration for Offshore Platforms by Use of MR Damper 被引量:8
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作者 李华军 土树青 秘春艳 《China Ocean Engineering》 SCIE EI 2002年第1期33-40,共8页
The objective of the present research is to examine the effectiveness of the lateral vibration control of wave-excited response of offshore platforms with magneto-rheological (MR) damper. In this study, the offshore p... The objective of the present research is to examine the effectiveness of the lateral vibration control of wave-excited response of offshore platforms with magneto-rheological (MR) damper. In this study, the offshore platform is simplified to be a singled degree-of-freedom (SDOF) system by extracting the first vibration mode of the structure. The external 'generalized' wave force is determined with a white noise via a designed filter. A semi-active control method based on optimal control theory is proposed considering that the yield stress of the MR damper can he varied continuously within a certain range. The dynamics of SDOF structure coupled with the MR damper is investigated. Numerical simulation demonstrates that the MR damper with this control strategy can significantly reduce the maximum responses and the root-mean-square (RMS) values. 展开更多
关键词 semi-active control offshore platform vibration reduction wave loading magneto-rheological damper
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Distributed predictive control of spiral wave in cardiac excitable media
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作者 甘正宁 成新明 《Chinese Physics B》 SCIE EI CAS CSCD 2010年第5期160-167,共8页
In this paper, we propose the distributed predictive control strategies of spiral wave in cardiac excitable media. The modified FitzHugh-Nagumo model was used to express the cardiac excitable media approximately. Base... In this paper, we propose the distributed predictive control strategies of spiral wave in cardiac excitable media. The modified FitzHugh-Nagumo model was used to express the cardiac excitable media approximately. Based on the control-Lyapunov theory, we obtained the distributed control equation, which consists of a positive eontrol-Lyapunov function and a positive cost function. Using the equation, we investigate two kinds of robust control strategies: the time-dependent distributed control strategy and the space-time dependent distributed control strategy. The feasibility of the strategies was demonstrated via an illustrative example, in which the spiral wave was prevented to occur, and the possibility for inducing ventricular fibrillation was eliminated. The strategies are helpful in designing various cardiac devices. Since the second strategy is more efficient and robust than the first one, and the response time in the second strategy is far less than that in the first one, the former is suitable for the quick-response control systems. In addition, our spatiotemporal control strategies, especially the second strategy, can be applied to other cardiac models, even to other reaction-diffusion systems. 展开更多
关键词 spiral wave FitzHugh-Nagumo model control-Lyapunov function
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Boundary Control for an Unstable Wave Equation in Annular with Symmetrical Initial Data
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作者 齐洁 张晶 《Journal of Donghua University(English Edition)》 EI CAS 2016年第3期467-472,共6页
A problem of boundary stabilization of a wave equation with anti-damping term in annular is considered.This term puts some eigenvalues of the open-loop system in the right half of the complex plane.Suppose the initial... A problem of boundary stabilization of a wave equation with anti-damping term in annular is considered.This term puts some eigenvalues of the open-loop system in the right half of the complex plane.Suppose the initial and boundary conditions are rotationally symmetric,the equation in two-dimensional(2-D)annular is transformed to an equivalent one-dimensional(1-D)equation in polar coordinates.A feedback law based on the backstepping method is designed.By a successive approximation,it's proved that there exists a unique solution of the integral kernel which weights the state feedback on boundary.It's also proved that the energy function of the closed-loop system decays exponentially,implying the exponential stability of the closed-loop system.The effectiveness of the control is illustrated with numerical simulations. 展开更多
关键词 boundary control BACKSTEPPING wave equation Lyapunov function ANNULAR
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