结构极点配置中输出反馈增益矩阵的模态算法研究
|
摘要
随着科技的发展,结构的振动控制已经成为了亟待解决的课题。振动主动控制也因其优越的性能而成为国内外研究的热点,有关振动系统中极点及振型分配仍然是一个没有完全解决的问题,没有一个充要条件回答对任何振动系统那些极点及振型可分配那些极点及振型不可分配。极点分配是振动系统控制的主要目标之一,现在应用反馈控制方法控制振动系统的极点也是极点分配的主要方法之一,本文就是以离散特征值结构的极点配置在输出反馈控制下的模态控制算法作为研究内容,紧紧围绕这一问题,首先,在状态空间中通过离散结构在原始坐标和模态坐标下的状态空间方程来研究系统的振动与控制状况及控制系统的可控性、可观性问题;主要是研究极点配置的两种反馈控制方法:状态反馈控制和输出反馈控制。在两种反馈控制方法下振动系统可控、可观性与极点配置的关系。其次,由于单输入系统状态反馈孤立特征值系统完全可控性与所有极点可分配是等价的,但在自由度较高的振动系统中,由于测量技术和数值算法的问题,应用状态反馈控制进行全极点配置就很难实现了,因此本文重点研究输出反馈控制下局部振动控制的部分极点配置理论及输出反馈增益矩阵的模态算法。将此算法应用到“欧拉梁”振动系统中,通过计算分析结果,检验此算法的可行性。最后,展望了模态振动控制中极点配置方法研究的发展方向,指出以后有待进一步研究的理论及应用问题。
With the development of the science and technology, structure vibration control has been an important research field due to its urgent application need in actual engineering structures. Active vibration control has received considerable attention in recent years at homes and overseas because of its excellent advantages.
This dissertation deals with the modal control method of the pole assignment of the discrete eigenvalue,the control for local vibration method and related issues in a more in-depth theory and research, mainly composed of two parts :
1. The two pole assignment control method: state feedback control and output feedback control. In the structure of the state space, under two control methods, studying the relationship between the pole assignment and the controllability and observability of the vibration control system, i.e. the relationship between the complete the controllability and observability of the vibration control system and all the pole with corresponding eigenvector dispensability or some pole with corresponding eigenvector dispensability.
2. Between the vibration control system and the assignment control of a part of eigenvalue and their eigenvector, study pole assignment in output feedback control. Through this, the method is got, which is simple, enforceable output feedback control gain matrix account method. Make the part vibration of the structure controlled that not influence vibration action of the other part, i.e. output feedback control gain matrix modal arithmetic. This is the main part of this research.
The research work of this thesis consists of seven chapters:
1) On the basis of reading extensive literatures, summarize the development of the modal control and the method application in some areas.
2) In the state space, under original coordinate and modal coordinate; establish state space equation, base on space equation, and study the state space of the structure vibration analysis and control; the controllability and observability analysis of vibration system and the controllability and observability theory of the feedback control system.
3) Researches on the state feedback and pole assignment in the state space of the vibration control system; relations between the pole assignment under the control of the state feedback and the controllability and observability of the vibration control system and the computational method of the output feedback control gain matrix which need to be researched.
4) Researches on the output feedback and pole assignment in the state space of the vibration control system; relations between the pole assignment under the control of the output feedback and the controllability and observability of the vibration control system and the computational method of the output feedback control gain matrix which need to be researched.
5) In the state space of the control system, the structural state equation under the modal coordinates is set up according to the structural finite element equation; choosing methods according to the eigenvalue and eigenvector which is controlled; researches on the method of assigning the poles under the output feedback control, using the open-loop system vector projecting method and the least energy distribution method; in the modal coordinates, researches on the modal computing methods of the output feedback gain matrix under the open-loop system eigenvector projecting method and the least energy distribution method.
6) Take cantilever beam for example, based on Euler beam’s finite element model and equation of the cantilever beam for applying Euler subjunctive to analyze and operate. In the state space of structural vibration control system, building the state space equation, analysis controllability and observability of control system and accounting modal’s all eigenvalues and all eigenvectors, using method of output feedback control in pole assignment of vibration system. The deformation of a structure is caused mainly by the response of the lower modes, and the amplitude responses of higher modes are very small in contrast to the lower modes. So we mainly study the control of the lower modes. Firstly build finite element equation of structure and respective state space equation, compute some lower order modes. The goal is to control some lower modes and almost not to disturb the other modes. Secondly select control force distribution matrix B and sensor distribution matrix C and guarantee the system is controllable and observable. In order to keep the structure stable and protect from spill over, which means actuators and sensors are collocated, and request those actuators only generate translational force, which means only adding translational controls. Thirdly build state space equation under part of modal coordinates, Utilizing two important methods of modal computing method of output feedback control gain of projection method of open-loop system vectors of reassigned poles and minimum energy assignment to compute output feedback gains, and then apply the feedback gains to original state space equation of structure. From the control effects we obtain the results which we want. Under two control methods, the output feedback gains are computed by using original structural state space equation, four-order, six-order and eight-order state space equation of modal coordinates for first two eigenvalues assignment of original structure, respectively, and then apply those gains obtained under modal coordinates to original structural state space equation. Comparing the control effect of modal gains with one of original gain, it is verified that modal computing method is available.
7) The research work of this thesis is summarized, and future work on passive/active vibration control with modal control is proposed.
With the development of the science and technology, structure vibration control has been an important research field due to its urgent application need in actual engineering structures. Active vibration control has received considerable attention in recent years at homes and overseas because of its excellent advantages.
This dissertation deals with the modal control method of the pole assignment of the discrete eigenvalue,the control for local vibration method and related issues in a more in-depth theory and research, mainly composed of two parts :
1. The two pole assignment control method: state feedback control and output feedback control. In the structure of the state space, under two control methods, studying the relationship between the pole assignment and the controllability and observability of the vibration control system, i.e. the relationship between the complete the controllability and observability of the vibration control system and all the pole with corresponding eigenvector dispensability or some pole with corresponding eigenvector dispensability.
2. Between the vibration control system and the assignment control of a part of eigenvalue and their eigenvector, study pole assignment in output feedback control. Through this, the method is got, which is simple, enforceable output feedback control gain matrix account method. Make the part vibration of the structure controlled that not influence vibration action of the other part, i.e. output feedback control gain matrix modal arithmetic. This is the main part of this research.
The research work of this thesis consists of seven chapters:
1) On the basis of reading extensive literatures, summarize the development of the modal control and the method application in some areas.
2) In the state space, under original coordinate and modal coordinate; establish state space equation, base on space equation, and study the state space of the structure vibration analysis and control; the controllability and observability analysis of vibration system and the controllability and observability theory of the feedback control system.
3) Researches on the state feedback and pole assignment in the state space of the vibration control system; relations between the pole assignment under the control of the state feedback and the controllability and observability of the vibration control system and the computational method of the output feedback control gain matrix which need to be researched.
4) Researches on the output feedback and pole assignment in the state space of the vibration control system; relations between the pole assignment under the control of the output feedback and the controllability and observability of the vibration control system and the computational method of the output feedback control gain matrix which need to be researched.
5) In the state space of the control system, the structural state equation under the modal coordinates is set up according to the structural finite element equation; choosing methods according to the eigenvalue and eigenvector which is controlled; researches on the method of assigning the poles under the output feedback control, using the open-loop system vector projecting method and the least energy distribution method; in the modal coordinates, researches on the modal computing methods of the output feedback gain matrix under the open-loop system eigenvector projecting method and the least energy distribution method.
6) Take cantilever beam for example, based on Euler beam’s finite element model and equation of the cantilever beam for applying Euler subjunctive to analyze and operate. In the state space of structural vibration control system, building the state space equation, analysis controllability and observability of control system and accounting modal’s all eigenvalues and all eigenvectors, using method of output feedback control in pole assignment of vibration system. The deformation of a structure is caused mainly by the response of the lower modes, and the amplitude responses of higher modes are very small in contrast to the lower modes. So we mainly study the control of the lower modes. Firstly build finite element equation of structure and respective state space equation, compute some lower order modes. The goal is to control some lower modes and almost not to disturb the other modes. Secondly select control force distribution matrix B and sensor distribution matrix C and guarantee the system is controllable and observable. In order to keep the structure stable and protect from spill over, which means actuators and sensors are collocated, and request those actuators only generate translational force, which means only adding translational controls. Thirdly build state space equation under part of modal coordinates, Utilizing two important methods of modal computing method of output feedback control gain of projection method of open-loop system vectors of reassigned poles and minimum energy assignment to compute output feedback gains, and then apply the feedback gains to original state space equation of structure. From the control effects we obtain the results which we want. Under two control methods, the output feedback gains are computed by using original structural state space equation, four-order, six-order and eight-order state space equation of modal coordinates for first two eigenvalues assignment of original structure, respectively, and then apply those gains obtained under modal coordinates to original structural state space equation. Comparing the control effect of modal gains with one of original gain, it is verified that modal computing method is available.
7) The research work of this thesis is summarized, and future work on passive/active vibration control with modal control is proposed.
引文
[1] 黄建平,“梁结构振动复合主动控制方法研究”,南京航空航天大学硕士学位论文,2005.02
[2] 宋虎,“基于 FPGA 的振动主动控制技术研究”,南京理工大学硕士学位论文,2005.5
[3] 刘坤,刘翠响,李妍,“MATLAB 自动控制原理习题精解”,国防工业出版社,2004.6 第一版
[4] 吴麒,王诗宓,自动控制原理(第 2 版)(下册),清华大学出版社,2006.10
[5] 张希周,“自动控制原理”,重庆大学出版社,1996.2 第一版
[6] 张晋格,“控制系统 CAD-基于 MATLAB 语言”,机械工业出版社,2004.6
[7] J. F. Magni, ‘Robust modal control with a Toolbox for use with Matlab,’ Kluwer Academic/Plenum Publishers, New York 233 Spring Street, 2002.
[8] A. Preumont, ‘Vibration control of active structures: An introduction-2nd edition,’ Kluwer academic publishers, 2002
[9] 王晋莹,陈科进,“转动惯量和剪切变形对轴向受压梁振动特性的影响”,西安公路交通大学学报,Vol-17 No. l, Mar. 1997
[10] 王勖成,邵敏,“有限单元法基本原理和数值方法”,清华大学出版社,2002.10 第二版
[11] 陈侃松,何翔,“运用 MATLAB 语言进行控制系统极点配置的方法”,中南民族学院学报,Vol. 20 No. 2,Jun. 2001
[12] Guo-Feng Yao etc., “Modal computing method of output feedback control gain matrix in pole assignment of vibration system”, Mechanical system and signal processing (2007)(accepted by Mechanical system and signal processing).
[13] Guo-Feng Yao etc., “Controllability of Repeated-Eigenvalue Systems and Defective Systems”(2007)(submitted to Mechanical system and signal processing)
[14] J. F. Magni, ‘Exact pole assignment by output feedback part 3’, International journal of control, Vol. 45, No. 6, pp. 2021-2033, 1987.
[15] [美]约翰·F·加德纳,周进雄,张陵 译,“机构动态仿真—使用 MATLAB和 SIMULINK”,西安交通大学出版社,2002.9 第一版
[16] W. M. Wonham, ‘On pole assignment in multi-input controllable linear system,’IEEE Transactions on Automatic Control, Vol. Ac-12, pp. 660-665, Dec. 1967.
[17] E. J. Davison, ‘On pole assignment in linear system with incomplete state feedback,’ IEEE Transactions on Automatic Control, Vol. Ac-15, pp. 348-351, June. 1970.
[18] H. Kimura, ‘Pole assignment by gain output feedback,’ IEEE Transactions on Automatic Control, Vol. Ac-20, pp. 509-516, Aug. 1975.
[19] V. L. Syrmos, C. T. Abdallah, P. Dorato, et al, ‘Static output feedback—A survey,’ Automatica, Vol. 33, No. 2, pp. 125-137, Feb. 1997.
[20] B. Moore, ‘On the flexibility offered by state feedback in multivariable control,’ IEEE Transactions on Automatic Control, Vol. Ac-21, pp. 689-692, Oct.. 1976.
[21] 徐铭陶,肖明葵,“工程动力学振动与控制”,机械工业出版社
[22] 姚起杭,姚军,李岳锋,郝联盈,“飞机结构振动主动控制应用技术[1]”,应用力学学报,Vol 18, SI, 2001.9
[23] J. E. Mottershead and Y. M. Ram, ‘Inverse eigenvalue problems in vibration absorption: passive modification and active control’, Mechanical systems and signal processing, Vol. 20, pp. 5-44, 2006.
[24] Guo-Feng Yao,S.H.Chen,’Modal control algorithm on optimal control of intelligent structure’, Structural Engineering and Mechanics, Vol.15, No.4, 451-462, 2003.
[25] Guo-Feng Yao etc., “Model Control in the optimal shape control and the shape control research of a paraboloid antenna under different stimulations”, Smart Materials and Structures, 14(2005)191-196.
[26] 牟全臣. 黄文虎. 郑钢铁. 王心清. 张景绘,“航天结构主、被动一体化振动控制技术的研究现状和进展”,应用力学学报,2001 年 03 期。
[27] Y. M. Ram, “Dynamic structural modification”, The Shock and Vibration Digest 32(1)(2000)11-17.
[28] D. J. Ewins, “Modal Testing: Theory, Practice and Application”, second ed., Research Studies Press, Baldock, Hertfordshire, UK, 2000.
[29] M. T. Chu, Y. C. Kuo, W. W. Lin, “On inverse quadratic eigenvalue problems with partially prescribed eigenstructure”, SIAM Journal on Matrix Analysis and Applications 25(2003)995-1020.
[30] 顾仲权,马扣根,陈卫东,“振动主动控制”,国防工业出版社,1997
[31] 张义民,“机械振动力学”,吉林科学技术出版社,2000.9 第一版
[32] 马扣根,顾仲权,“振动结构的多输入一多输出实时辨识控制与试验研究”,航空学报,2000; 21(1): 60-63
[33] 闻邦椿,赵春雨,苏东海,熊万里,“机械系统的振动同步与控制同步”,科学出版社
[34] J. Kautsky, N. K. Nichols and P. Van Dooren, ‘Robust pole assignment in linear state feedback,’ International Journal of Control, Vol. 41, No.5, pp. 1129-1155, May 1985.
[35] Gene F. Franklin, J. David Powell, Abbas Emami-Naeini, ‘Feedback control of dynamic systems,’ Upper Saddle River, NJ: Pearson Prentice Hall, 2005(5th,edition)
[36] 姚军,李岳锋,“用压电陶瓷进行梁的主动振动控制研究”,应用力学学报,Vol. 14, SI, 1997.10
[37] 张义民,刘巧伶,闻邦椿,“随机结构系统的一般实矩阵特征值问题的概率分析”,力学季刊,Vol. 24 No .4 Dec. 2003
[38]Guo-Feng Yao, Su-Huan Chen, "A Block Iterative Algorithm of the Continuous Riccati Equation in the Optimal Shape Control", Computer Methods in Applied Mechanics and Engineering, Vol.187(2000):173-180.(SCI, EI)
[39] 周星德,汪凤泉,韩晓林,“框架结构智能主动控制的研究”,工程力学,Vol. 19 No.2 April 2002
[40] 陈翰馥,曹希仁,“系数未知的随机系统的极点配置”,中国科学,2000.6
[41]Su-Huan Chen, Guo-Feng Yao and Cheng Huang, "A New Intelligent Thin-Shell Element", Smart Materials and Structures, Vol.9(2000):10-18(.SCI)
[42] 张令弥,何柏庆,袁向荣,“设计灵敏度分析的迭代模态法”,“南京航天大学学报 ”Vol.26 No.3,June 1994
[43] 曹承志,屈红梅,路战红,杨晓波,“MATLAB 软件包中 SIMULINK 环境下直接转矩控制系统的仿真”,电机与控制学报,Vol. 5 No.2 June,2001
[44] 崔平,翁正新,“基于状态空间极点配置的倒立摆平衡控制”,实验室研究与探索,Vol. 22No.2,A pr. 2003
[45] Guo-Feng Yao, Su-Huan Chen, "The Precise Integration of Riccati Equation and Its Application in the Optimal Shape Control", Computer Methods in Applied Mechanics and Engineering, Vol.189(2000):141-148.(SCI, EI)
[46] 孙兆林,“MATLAB.X 图像处理”,北京:清华大学出版社,2002.5
[47] 王小华,陈庆伟,胡维礼,“悬臂梁的振动抑制研究”,南京理工大学学报,Vol .26 Supp,Dec .2002
[48] 李山虎,杨靖波,黄清华,陈德成,“轴向运动悬臂梁的独立模态振动控制—I 近似理论分析”,应用力学学报,Vol .19 No. l Mar.2002
[49] Su-Huan Chen, Guo-Feng Yao and Zong-Jie Cao, "A new Intelligent 3-D Thin Shell Element and the Optimal Dynamic Shape Control of a Paraboloid Antenna", Journal of Vibration and Control, Vol.6(2000):463-484.(SCI)
[50] 龚善初,“均质悬臂梁的频率与振型”,衡阳师专学报,Vol.19 No.6 Dec .1998
[51] 朱桂东,邵成勋,王本利,“伸展悬臂梁动态特性分析”,宇航学报,Vol. 18 No. 2 Apr. 1997
[52] S. H. Chen, G. F. Yao and H. D. Lian, A new piezoelectric shell element and its application in static shape control, Structural Engineering and Mechanics, Vol. 12, No 5, 2001, 491-506.(SCI, EI)
[53] 彭福军,马扣根,顾仲权,“结构响应主动控制的时域与频域实现”,南京航空航天大学学报,1995,27(2):249- 257
[2] 宋虎,“基于 FPGA 的振动主动控制技术研究”,南京理工大学硕士学位论文,2005.5
[3] 刘坤,刘翠响,李妍,“MATLAB 自动控制原理习题精解”,国防工业出版社,2004.6 第一版
[4] 吴麒,王诗宓,自动控制原理(第 2 版)(下册),清华大学出版社,2006.10
[5] 张希周,“自动控制原理”,重庆大学出版社,1996.2 第一版
[6] 张晋格,“控制系统 CAD-基于 MATLAB 语言”,机械工业出版社,2004.6
[7] J. F. Magni, ‘Robust modal control with a Toolbox for use with Matlab,’ Kluwer Academic/Plenum Publishers, New York 233 Spring Street, 2002.
[8] A. Preumont, ‘Vibration control of active structures: An introduction-2nd edition,’ Kluwer academic publishers, 2002
[9] 王晋莹,陈科进,“转动惯量和剪切变形对轴向受压梁振动特性的影响”,西安公路交通大学学报,Vol-17 No. l, Mar. 1997
[10] 王勖成,邵敏,“有限单元法基本原理和数值方法”,清华大学出版社,2002.10 第二版
[11] 陈侃松,何翔,“运用 MATLAB 语言进行控制系统极点配置的方法”,中南民族学院学报,Vol. 20 No. 2,Jun. 2001
[12] Guo-Feng Yao etc., “Modal computing method of output feedback control gain matrix in pole assignment of vibration system”, Mechanical system and signal processing (2007)(accepted by Mechanical system and signal processing).
[13] Guo-Feng Yao etc., “Controllability of Repeated-Eigenvalue Systems and Defective Systems”(2007)(submitted to Mechanical system and signal processing)
[14] J. F. Magni, ‘Exact pole assignment by output feedback part 3’, International journal of control, Vol. 45, No. 6, pp. 2021-2033, 1987.
[15] [美]约翰·F·加德纳,周进雄,张陵 译,“机构动态仿真—使用 MATLAB和 SIMULINK”,西安交通大学出版社,2002.9 第一版
[16] W. M. Wonham, ‘On pole assignment in multi-input controllable linear system,’IEEE Transactions on Automatic Control, Vol. Ac-12, pp. 660-665, Dec. 1967.
[17] E. J. Davison, ‘On pole assignment in linear system with incomplete state feedback,’ IEEE Transactions on Automatic Control, Vol. Ac-15, pp. 348-351, June. 1970.
[18] H. Kimura, ‘Pole assignment by gain output feedback,’ IEEE Transactions on Automatic Control, Vol. Ac-20, pp. 509-516, Aug. 1975.
[19] V. L. Syrmos, C. T. Abdallah, P. Dorato, et al, ‘Static output feedback—A survey,’ Automatica, Vol. 33, No. 2, pp. 125-137, Feb. 1997.
[20] B. Moore, ‘On the flexibility offered by state feedback in multivariable control,’ IEEE Transactions on Automatic Control, Vol. Ac-21, pp. 689-692, Oct.. 1976.
[21] 徐铭陶,肖明葵,“工程动力学振动与控制”,机械工业出版社
[22] 姚起杭,姚军,李岳锋,郝联盈,“飞机结构振动主动控制应用技术[1]”,应用力学学报,Vol 18, SI, 2001.9
[23] J. E. Mottershead and Y. M. Ram, ‘Inverse eigenvalue problems in vibration absorption: passive modification and active control’, Mechanical systems and signal processing, Vol. 20, pp. 5-44, 2006.
[24] Guo-Feng Yao,S.H.Chen,’Modal control algorithm on optimal control of intelligent structure’, Structural Engineering and Mechanics, Vol.15, No.4, 451-462, 2003.
[25] Guo-Feng Yao etc., “Model Control in the optimal shape control and the shape control research of a paraboloid antenna under different stimulations”, Smart Materials and Structures, 14(2005)191-196.
[26] 牟全臣. 黄文虎. 郑钢铁. 王心清. 张景绘,“航天结构主、被动一体化振动控制技术的研究现状和进展”,应用力学学报,2001 年 03 期。
[27] Y. M. Ram, “Dynamic structural modification”, The Shock and Vibration Digest 32(1)(2000)11-17.
[28] D. J. Ewins, “Modal Testing: Theory, Practice and Application”, second ed., Research Studies Press, Baldock, Hertfordshire, UK, 2000.
[29] M. T. Chu, Y. C. Kuo, W. W. Lin, “On inverse quadratic eigenvalue problems with partially prescribed eigenstructure”, SIAM Journal on Matrix Analysis and Applications 25(2003)995-1020.
[30] 顾仲权,马扣根,陈卫东,“振动主动控制”,国防工业出版社,1997
[31] 张义民,“机械振动力学”,吉林科学技术出版社,2000.9 第一版
[32] 马扣根,顾仲权,“振动结构的多输入一多输出实时辨识控制与试验研究”,航空学报,2000; 21(1): 60-63
[33] 闻邦椿,赵春雨,苏东海,熊万里,“机械系统的振动同步与控制同步”,科学出版社
[34] J. Kautsky, N. K. Nichols and P. Van Dooren, ‘Robust pole assignment in linear state feedback,’ International Journal of Control, Vol. 41, No.5, pp. 1129-1155, May 1985.
[35] Gene F. Franklin, J. David Powell, Abbas Emami-Naeini, ‘Feedback control of dynamic systems,’ Upper Saddle River, NJ: Pearson Prentice Hall, 2005(5th,edition)
[36] 姚军,李岳锋,“用压电陶瓷进行梁的主动振动控制研究”,应用力学学报,Vol. 14, SI, 1997.10
[37] 张义民,刘巧伶,闻邦椿,“随机结构系统的一般实矩阵特征值问题的概率分析”,力学季刊,Vol. 24 No .4 Dec. 2003
[38]Guo-Feng Yao, Su-Huan Chen, "A Block Iterative Algorithm of the Continuous Riccati Equation in the Optimal Shape Control", Computer Methods in Applied Mechanics and Engineering, Vol.187(2000):173-180.(SCI, EI)
[39] 周星德,汪凤泉,韩晓林,“框架结构智能主动控制的研究”,工程力学,Vol. 19 No.2 April 2002
[40] 陈翰馥,曹希仁,“系数未知的随机系统的极点配置”,中国科学,2000.6
[41]Su-Huan Chen, Guo-Feng Yao and Cheng Huang, "A New Intelligent Thin-Shell Element", Smart Materials and Structures, Vol.9(2000):10-18(.SCI)
[42] 张令弥,何柏庆,袁向荣,“设计灵敏度分析的迭代模态法”,“南京航天大学学报 ”Vol.26 No.3,June 1994
[43] 曹承志,屈红梅,路战红,杨晓波,“MATLAB 软件包中 SIMULINK 环境下直接转矩控制系统的仿真”,电机与控制学报,Vol. 5 No.2 June,2001
[44] 崔平,翁正新,“基于状态空间极点配置的倒立摆平衡控制”,实验室研究与探索,Vol. 22No.2,A pr. 2003
[45] Guo-Feng Yao, Su-Huan Chen, "The Precise Integration of Riccati Equation and Its Application in the Optimal Shape Control", Computer Methods in Applied Mechanics and Engineering, Vol.189(2000):141-148.(SCI, EI)
[46] 孙兆林,“MATLAB.X 图像处理”,北京:清华大学出版社,2002.5
[47] 王小华,陈庆伟,胡维礼,“悬臂梁的振动抑制研究”,南京理工大学学报,Vol .26 Supp,Dec .2002
[48] 李山虎,杨靖波,黄清华,陈德成,“轴向运动悬臂梁的独立模态振动控制—I 近似理论分析”,应用力学学报,Vol .19 No. l Mar.2002
[49] Su-Huan Chen, Guo-Feng Yao and Zong-Jie Cao, "A new Intelligent 3-D Thin Shell Element and the Optimal Dynamic Shape Control of a Paraboloid Antenna", Journal of Vibration and Control, Vol.6(2000):463-484.(SCI)
[50] 龚善初,“均质悬臂梁的频率与振型”,衡阳师专学报,Vol.19 No.6 Dec .1998
[51] 朱桂东,邵成勋,王本利,“伸展悬臂梁动态特性分析”,宇航学报,Vol. 18 No. 2 Apr. 1997
[52] S. H. Chen, G. F. Yao and H. D. Lian, A new piezoelectric shell element and its application in static shape control, Structural Engineering and Mechanics, Vol. 12, No 5, 2001, 491-506.(SCI, EI)
[53] 彭福军,马扣根,顾仲权,“结构响应主动控制的时域与频域实现”,南京航空航天大学学报,1995,27(2):249- 257