压电材料双曲壳热弹耦合作用下的混沌运动
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摘要
运用弹性力学有限变形基本理论推导出了压电材料双曲壳在外激力和温度场作用下的非线性振动方程和协调方程.通过Bubnov-Galerkin原理,得到该结构的非线性动力学方程.利用Melnikov方法,得到系统产生Smale马蹄变换意义下混沌的条件,用四阶Runge-Kutta法编写程序对系统进行数值求解,并绘制出相应的分岔图、Lyapunov指数图、相轨迹图以及Poincaré截面图,分析了温度场对压电材料双曲壳系统的非线性特性的影响.仿真结果表明,随着温度的升高,系统的混沌与周期区交替出现,温度场的改变可影响和控制系统的振动特性.
Piezoelectric material, which exhibits excellent electro-mechanical conversion properties, is widely used in smart sensors and structures for sonar systems, weather detection and remote sensing. Hyperbolic shell structure made of piezoelectric material is liable to break down when it is used in high temperature environment, which is caused by the unexpected chaotic dynamic motion under the coupling effect of thermal filed and force field. Therefore, the chaotic nonlinear dynamic vibration of simply-supported piezoelectric material hyperbolic shell is studied under the combined action of temperature field and simple harmonic excitation. Based on the theory of finite deformation, the non-linear vibration equation and coordination equation of the hyperbolic shell are established. The non-linear dynamic equation of the structure is obtained by the Bubnov-Galerkin principle. The corresponding undisturbed Hamilton system has a homoclinic orbit. Using Melnikov function, the chaotic motion condition of the dynamic system under the criterion of Smale-horseshoe transformation is obtained. Furthermore, the mathematical model is established by Simulink software and the numerical simulations are performed by the fourth-order Runge-Kutta method. The simulation results accord well with those from the Melnikov method. The bifurcation diagram, the Lyapunov exponent diagram, the phase diagram and Poincaré section diagram are acquired to analyze the influence of temperature field on the non-linear characteristic of piezoelectric material hyperbolic shell system. When the temperature is close to 32?C and 41?C, the Lyapunov index is less than 0 and the corresponding movement of the system is in the periodic zone, which is the same as that for a temperature range from 36?C to 37?C. When the Lyapunov index is greater than 0, the corresponding movement of the system is in chaos zone. Therefore, the change of temperature has an additional effect on the stiffness of the system which affects the vibration of the system. The chaos and periodic zones of the system alternate with the increase of temperature and the vibration characteristics of the system can be controlled by changing the temperature field.Therefore, adjusting the temperature field can control the motion state of the system, which helps to improve reliability of the structure.
Piezoelectric material, which exhibits excellent electro-mechanical conversion properties, is widely used in smart sensors and structures for sonar systems, weather detection and remote sensing. Hyperbolic shell structure made of piezoelectric material is liable to break down when it is used in high temperature environment, which is caused by the unexpected chaotic dynamic motion under the coupling effect of thermal filed and force field. Therefore, the chaotic nonlinear dynamic vibration of simply-supported piezoelectric material hyperbolic shell is studied under the combined action of temperature field and simple harmonic excitation. Based on the theory of finite deformation, the non-linear vibration equation and coordination equation of the hyperbolic shell are established. The non-linear dynamic equation of the structure is obtained by the Bubnov-Galerkin principle. The corresponding undisturbed Hamilton system has a homoclinic orbit. Using Melnikov function, the chaotic motion condition of the dynamic system under the criterion of Smale-horseshoe transformation is obtained. Furthermore, the mathematical model is established by Simulink software and the numerical simulations are performed by the fourth-order Runge-Kutta method. The simulation results accord well with those from the Melnikov method. The bifurcation diagram, the Lyapunov exponent diagram, the phase diagram and Poincaré section diagram are acquired to analyze the influence of temperature field on the non-linear characteristic of piezoelectric material hyperbolic shell system. When the temperature is close to 32?C and 41?C, the Lyapunov index is less than 0 and the corresponding movement of the system is in the periodic zone, which is the same as that for a temperature range from 36?C to 37?C. When the Lyapunov index is greater than 0, the corresponding movement of the system is in chaos zone. Therefore, the change of temperature has an additional effect on the stiffness of the system which affects the vibration of the system. The chaos and periodic zones of the system alternate with the increase of temperature and the vibration characteristics of the system can be controlled by changing the temperature field.Therefore, adjusting the temperature field can control the motion state of the system, which helps to improve reliability of the structure.
引文
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[13]Cao Z,Hao Y X,Gu X J,Li Z N 2018 Acta Mech.Solida39 284(in Chinese)[曹洲,郝育新,顾晓军,李珍妮2018固体力学学报39 284]
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[15]Zhang Q C,Tian R L,Wang W 2008 Acta Phys.Sin.57 2799(in Chinese)[张琪昌,田瑞兰,王炜2008物理学报57 2799]
[16]Ji Y,Bi Q S 2009 Acta Phys.Sin.58 4431(in Chinese)[季颖,毕勤胜2009物理学报58 4431]
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[18]Liu R H 2011 Nonlinear Bending,Stability and Vibration of Reticulated Shell Structure(Beijing:Science Press)p9(in Chinese)[刘人怀2011网壳结构的非线性弯曲、稳定和振动(北京:科学出版社)第9页
[19]Zhang G Q 2004 Ph.D.Dissertation(Tianjin:Tianjin University)(in Chinese)[张国清2004博士学位论文(天津:天津大学)]
[20]Ding H L 1996 Ph.D.Dissertation(Beijing:Tsinghua University)(in Chinese)[丁红丽1996博士学位论文(北京:清华大学)]
[21]Liu S N 1963 Acta Mech.Sin.6 61(in Chinese)[刘世宁1963力学学报6 61]
[22]Wang J F,Su W B,Wang C M 2011 Theory and Application of Piezoelectric Vibration(Beijing:Science Press)pp231-243(in Chinese)[王矜奉,苏文斌,王春明2011压电振动理论及应用(北京:科学出版社)]第231-243页
[2]Yeh Y L,Chen C K,Lai H Y 2002 Chaos,Solitons and Fractals 13 1493
[3]Wang P,Chen S M 2013 J.Mech.Engin.49 82(in Chinese)[王平,陈蜀梅2013机械工程学报49 82]
[4]Wang P,Chen S M,Wang Z R 2013 J.Vib.Shock 32129(in Chinese)[王平,陈蜀梅,王知人2013振动与冲击32 129]
[5]Zhu W G 2011 Ph.D.Dissertation(Hebei:Yanshan University)(in Chinese)[朱为国2011博士学位论文(河北:燕山大学)]
[6]Xue C X,Zhang S Y,Shu X F 2010 Acta Phys.Sin.596599(in Chinese)[薛春霞,张善元,树学锋2010物理学报59 6599]
[7]Chen Z J,Zhang S Y,Zheng K 2010 Acta Phys.Sin.594017(in Chinese)[陈赵江,张淑仪,郑凯2010物理学报59 4017]
[8]Xue C X,Ren X J 2014 J.Measur.Sci.Instrum.5 93
[9]Pan E 2001 J.Appl.Mech.68 608
[10]Hao Y X,Chen L H,Zhang W 2008 J.Sound Vib.312862
[11]Li L 2011 Ph.D.Dissertation(Shanxi:Xi’an University of Architecture and Technology)(in Chinese)[栗蕾2011博士学位论文(陕西:西安建筑科技大学)]
[12]Zhou Y Z 2008 M.S.Thesis(Zhejiang:Zhejiang University)(in Chinese)[周运朱2008硕士学位论文(浙江:浙江大学)]
[13]Cao Z,Hao Y X,Gu X J,Li Z N 2018 Acta Mech.Solida39 284(in Chinese)[曹洲,郝育新,顾晓军,李珍妮2018固体力学学报39 284]
[14]Zhang Q C,Wang W,He X J 2008 Acta Phys.Sin.575384(in Chinese)[张琪昌,王炜,何学军2008物理学报57 5384]
[15]Zhang Q C,Tian R L,Wang W 2008 Acta Phys.Sin.57 2799(in Chinese)[张琪昌,田瑞兰,王炜2008物理学报57 2799]
[16]Ji Y,Bi Q S 2009 Acta Phys.Sin.58 4431(in Chinese)[季颖,毕勤胜2009物理学报58 4431]
[17]Gu H Z,Aditi C,Li J M 1999 Int.J.Solids Struc.376479
[18]Liu R H 2011 Nonlinear Bending,Stability and Vibration of Reticulated Shell Structure(Beijing:Science Press)p9(in Chinese)[刘人怀2011网壳结构的非线性弯曲、稳定和振动(北京:科学出版社)第9页
[19]Zhang G Q 2004 Ph.D.Dissertation(Tianjin:Tianjin University)(in Chinese)[张国清2004博士学位论文(天津:天津大学)]
[20]Ding H L 1996 Ph.D.Dissertation(Beijing:Tsinghua University)(in Chinese)[丁红丽1996博士学位论文(北京:清华大学)]
[21]Liu S N 1963 Acta Mech.Sin.6 61(in Chinese)[刘世宁1963力学学报6 61]
[22]Wang J F,Su W B,Wang C M 2011 Theory and Application of Piezoelectric Vibration(Beijing:Science Press)pp231-243(in Chinese)[王矜奉,苏文斌,王春明2011压电振动理论及应用(北京:科学出版社)]第231-243页