超音速气流影响下壁板的非线性动力学分析
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摘要
壁板是现代飞行器上一种很重要的结构单元。在高速气流作用下,壁板不仅会发生静态屈曲变形,而且常常会出现颤振现象而导致壁板结构的疲劳寿命降低,甚至产生破坏对航空结构的可靠性造成影响。另外,当高速飞行器机体发生振动时,由于高速气流和壁板支承运动的联合作用,壁板系统会发生包括内共振在内的更为复杂和严重的强迫振动,对结构安全性的危害极大。目前,超音速飞行器的结构设计正向柔性轻型方向发展,这使得壁板的稳定性问题越来越突出。因此,研究各种条件下壁板的稳定性和动力学行为对飞行器设计有很现实的指导意义。基于此,本文将对超音速气流作用下壁板的非线性动力学问题进行研究,具体内容如下:
(1)对超音速气流作用下的壁板系统进行力学建模。利用一阶活塞气动理论简化壁板外部受到的气动力,根据Von Karman大变形理论和Hamilton原理建立了超音速气流作用下壁板的横向运动偏微分控制方程。对方程无量纲化和离散化后,得到了常微分方程组。
(2)对壁板系统的稳定性和分岔问题进行了研究。利用罗斯-霍维兹判据得到了系统的稳定性区域,发现在某些参数区域内有多个平衡点共存的情况,并通过计算各平衡点处Jacobi矩阵的特征值确定了相应平衡点的稳定性,数值算例验证了理论分析的正确性。
(3)利用多元L-P法(MDLP)和增量谐波平衡法(IHB)对超音速气流和支承运动联合作用下壁板系统具有内共振的横向强迫振动进行研究,并对两种方法的结果进行比较分析。讨论了前两阶主共振和组合共振响应情况及内共振与外激励幅值的关系,并分析了各模态的振动情况。研究发现,在系统第2阶固有频率约为第1阶固有频率3倍的情况下,当外激励频率接近前两阶固有频率或其和的一半时,系统将发生内部共振,两个模态相互激励。但某些内共振的发生取决于外激励幅值的大小。
The panel is an important structure in modern aerospace vehicles. In the circumstance of high-speed airflow, a static deformation as buckling of the panel can occur, and the flutter may be encountered, which always leads to the fatigue failure of the panel and has harmful influence on the structure reliability. In addition, when the high-speed aircraft vibrates in flight, a complicated and intensive forced-oscillation, including internal resonance, would arise by the combination of high-speed airflow and the support motion, which greatly endangers the flight safety. Now, the structure design of the supersonic aircraft has a flexible and light trend, which makes the panel stability more and more significant. So, the study of the stability and dynamic behavior of the panel under various conditions has practical meanings for the aircraft design. In this dissertation, nonlinear dynamics of the panel in supersonic airflow is investigated. The main contents are as follows:
(1) The mechanical system of panels in supersonic airflow is modeled. The aerodynamic load is simplified In terms of the first-order piston theory. Appling the Von Karman deformation strain-displacement relation and Hamilton principle, the partial differential equations of transverse nonlinear vibration for the panels are derived. After nondimensionalization and discretization, a set of ordinary differential equations is obtained.
(2) The stability and bifurcations of the panel system are researched. According to the Routh-Hurwitz criterion, the stability regions of the system are obtained. It is found that in some parameter regions, several equilibrium points exist in one region, whose stabilities are determined by calculating the eigenvalues of the corresponding Jacobi matrices. Numerical examples are presented to support the conclusions.
(3) Forced vibrations with internal resonance of the panel system are studied through the multiple dimensions Lindstedt-Poincaré(MDLP) method, and the results obtained are compared with those of the incremental harmonic balance (IHB) method. The resonance responses of the first two modes and combination resonance are investigated and the relationship between the internal resonance and excitation amplitude is discussed. The motions of all modes are analyzed. The results show that the internal resonances occur as the excitation frequency is near the first, second natural frequency or half the sum of them under the condition that the second natural frequency is three times the first one, when the first two modes are excited by each other. However some of the internal resonances are decided by the excitation amplitude.
(1)对超音速气流作用下的壁板系统进行力学建模。利用一阶活塞气动理论简化壁板外部受到的气动力,根据Von Karman大变形理论和Hamilton原理建立了超音速气流作用下壁板的横向运动偏微分控制方程。对方程无量纲化和离散化后,得到了常微分方程组。
(2)对壁板系统的稳定性和分岔问题进行了研究。利用罗斯-霍维兹判据得到了系统的稳定性区域,发现在某些参数区域内有多个平衡点共存的情况,并通过计算各平衡点处Jacobi矩阵的特征值确定了相应平衡点的稳定性,数值算例验证了理论分析的正确性。
(3)利用多元L-P法(MDLP)和增量谐波平衡法(IHB)对超音速气流和支承运动联合作用下壁板系统具有内共振的横向强迫振动进行研究,并对两种方法的结果进行比较分析。讨论了前两阶主共振和组合共振响应情况及内共振与外激励幅值的关系,并分析了各模态的振动情况。研究发现,在系统第2阶固有频率约为第1阶固有频率3倍的情况下,当外激励频率接近前两阶固有频率或其和的一半时,系统将发生内部共振,两个模态相互激励。但某些内共振的发生取决于外激励幅值的大小。
The panel is an important structure in modern aerospace vehicles. In the circumstance of high-speed airflow, a static deformation as buckling of the panel can occur, and the flutter may be encountered, which always leads to the fatigue failure of the panel and has harmful influence on the structure reliability. In addition, when the high-speed aircraft vibrates in flight, a complicated and intensive forced-oscillation, including internal resonance, would arise by the combination of high-speed airflow and the support motion, which greatly endangers the flight safety. Now, the structure design of the supersonic aircraft has a flexible and light trend, which makes the panel stability more and more significant. So, the study of the stability and dynamic behavior of the panel under various conditions has practical meanings for the aircraft design. In this dissertation, nonlinear dynamics of the panel in supersonic airflow is investigated. The main contents are as follows:
(1) The mechanical system of panels in supersonic airflow is modeled. The aerodynamic load is simplified In terms of the first-order piston theory. Appling the Von Karman deformation strain-displacement relation and Hamilton principle, the partial differential equations of transverse nonlinear vibration for the panels are derived. After nondimensionalization and discretization, a set of ordinary differential equations is obtained.
(2) The stability and bifurcations of the panel system are researched. According to the Routh-Hurwitz criterion, the stability regions of the system are obtained. It is found that in some parameter regions, several equilibrium points exist in one region, whose stabilities are determined by calculating the eigenvalues of the corresponding Jacobi matrices. Numerical examples are presented to support the conclusions.
(3) Forced vibrations with internal resonance of the panel system are studied through the multiple dimensions Lindstedt-Poincaré(MDLP) method, and the results obtained are compared with those of the incremental harmonic balance (IHB) method. The resonance responses of the first two modes and combination resonance are investigated and the relationship between the internal resonance and excitation amplitude is discussed. The motions of all modes are analyzed. The results show that the internal resonances occur as the excitation frequency is near the first, second natural frequency or half the sum of them under the condition that the second natural frequency is three times the first one, when the first two modes are excited by each other. However some of the internal resonances are decided by the excitation amplitude.
引文
[1] Zhou R C, Xue D Y, Mei C. Finite element time domain-modal formulation for nonlinear of composit panels [J] . AIIA Journal, Vol. 32, NO. 10.1994: 2044-2052
[2]孟凡颖,钟腾育.用壁板颤振理论解决某系列飞机的方向舵蒙皮裂纹故障[J].飞机设计, 2000, (4): l-6
[3] Bolotin V V, Grishko A A, Kounadis A N, et al. Non-linear panel flutter in remote pos-critical domains[J]. International Journal of Non-Linear Mechanics,1988, 33(5): 753-764
[4] Bismarck-Nasr M N, Bones C A. Damping effects in nonlinear panel flutter[J]. AIAA Journal, 2000, 38(4):711-713
[5] Beldica C E, Hilton H H, Kubair D. Viscoelastic panel flutter-stablility,probabilities of failure and survival times[C]. Collection of Technical Papers-AIAA/ASME/ASCE/AHS/ASC Structural Dynamics and Materials Conference, 2001, 5: 3423-3433
[6] Epureanu B I, Tang L S, Paidoussis M P. Coherent structures and their influence on the dynamics of aeroelastic panels[J]. International Journal of Non-linear Mechanics, 2004, 39(6): 977-991
[7]杨智春,夏巍,孙浩.高速飞行器壁板颤振的分析模型和分析方法[J].应用力学学报, 2006, (4): 537-543
[8]张云峰,刘占生.粘弹性壁板颤振的非线性动力特性[J].推进技术, 2007, 28(1): 103-107
[9]张云峰,刘占生.粘弹壁板非线性颤振滞后特性分析[J].振动与冲击, 2007, 26(7): 101-105
[10] Dowell E H, Voss H M. Theoretical and experimental panel flutter studies in the mach number range of 1.0 to 5.0[J]. AIAA Journal, 1970, 8(9): 2022-2304
[11] Gray Jr C E, Mei C. Large amplitude finite element flutter analysis of composite panels in hypersonic flow[J]. AIAA Journal, 1993, 31(6): 1090-1099
[12] Mei C, Abdel-Motagaly K, Chen R. Review of nonlinear panel flutter at supersonic and hypersonic speeds[J]. Applied Mechanics Review, 1999, 52(10): 321-332
[13] Cheng G F, Mei C. Finite element modal formulation for hypersonic panel flutter analysis with thermal effects [R]. AIAA, 2003: 154-162
[14] Dowell E H. Aeroelasticity of Plates and Shells[M]. Leyden: Noordhoof International Publishing, 1975
[15]陈树辉.强非线性振动系统的定量分析方法[M].北京:科学出版社, 2007
[16] Lau S L, Cheung Y K. Amplitude incremental variational principle for non-linear vibration of elastic systems[J]. ASME Journal of Applied Mechanics, 1981, 48(4): 959-964
[17] Cheung Y K, Chen S H, Lau S L. Application of the incremental harmonic balance method to cubic non-linearity systems[J]. Journal of Sound and Vibration, 1990, 140(2): 273-286
[18] Zhang W Y, Huseyin K. Complex formulation of the IHB technique and its comparison with other methods[J]. Applied Mathematical Modelling, 2002, 26(1): 53-75
[19] Sze K Y, Chen S H, Huang J L. The incremental harmonic balance method for nonlinear vibration of axially moving beams[J]. Journal of Sound and Vibration, 2005, 281(3): 611-626
[20] Yeh Z F, Shih Y S. Critical load, dynamic characteristics and parametric instability of electrorheological material-based adaptive beams[J]. Computers & Structures, 2005, 83(25/26):2162-2174
[21] Wu G Y, Shih Y S. Analysis of dynamic instability for arbitrarily laminated skew plates[J]. Journal of Sound and Vibration, 2006, 292(1/2): 315-340
[22] Wu G Y. The analysis of dynamic instability on the large amplitude vibrations of a beam with transverse magnetic fields and thermal loads[J]. Journal of Sound and Vibration, 2007, 302(1/2): 167-177
[23]陈树辉,黄建亮,佘锦炎.轴向运动梁横向非线性振动研究[J].动力学与控制学报, 2004, 2(1): 40-45
[24]蔡铭,刘济科,李军.多自由度强非线性颤振分析的增量谐波平衡法[J].应用数学和力学, 2006, 27(7): 833-838
[25] Ashley H, Zartarian G. Piston Theory-A New aerodynamics tool for the aeroelastician [J]. Journal of Aeronautical Science, 1956, 23(10): 1109-1108
[26] Dowell E H. Nonlinear oscillations of a fluttering plate[J]. AIAA Journal, 1966, 4(7): 1267-1275
[27] Azzouz M S, Prezekop A, Guo X Y, et al. Nonlinear flutter of shallow shell under yawed supersonic flow using FEM[R]. AIAA Paper, 2003: 1514
[28] Abdel-Motagaly K, Duan B, Mei C. Active control of nonlinear panel flutter under yawed supersonic flow[R]. AIAA Paper 2003: 1368
[29] Moon S H, Hwang J S. Panel flutter suppression with an optimal controller based on the nonlinear model using piezoelectric materials[J]. Composite Structures, 2005, 68(3): 371-379
[30] Singha M K, Ganapathi M. A parametric study on supersonic flutter behavior of laminated composite skew flat panels[J]. Composite Structures, 2005, 69(1): 55-63
[31] Koo K N, Hwang W S. Effects of hysteretic and aerodynami damping on supersonic panel flutter of composite plates[J]. Journal of Sound and Vibration, 2004, 273: 569-583
[32] Bisplinghoof R L, Ashley H. Principles of Aeroelasticity[M]. New York: John Wiley and Sons, Inc, 1962
[2]孟凡颖,钟腾育.用壁板颤振理论解决某系列飞机的方向舵蒙皮裂纹故障[J].飞机设计, 2000, (4): l-6
[3] Bolotin V V, Grishko A A, Kounadis A N, et al. Non-linear panel flutter in remote pos-critical domains[J]. International Journal of Non-Linear Mechanics,1988, 33(5): 753-764
[4] Bismarck-Nasr M N, Bones C A. Damping effects in nonlinear panel flutter[J]. AIAA Journal, 2000, 38(4):711-713
[5] Beldica C E, Hilton H H, Kubair D. Viscoelastic panel flutter-stablility,probabilities of failure and survival times[C]. Collection of Technical Papers-AIAA/ASME/ASCE/AHS/ASC Structural Dynamics and Materials Conference, 2001, 5: 3423-3433
[6] Epureanu B I, Tang L S, Paidoussis M P. Coherent structures and their influence on the dynamics of aeroelastic panels[J]. International Journal of Non-linear Mechanics, 2004, 39(6): 977-991
[7]杨智春,夏巍,孙浩.高速飞行器壁板颤振的分析模型和分析方法[J].应用力学学报, 2006, (4): 537-543
[8]张云峰,刘占生.粘弹性壁板颤振的非线性动力特性[J].推进技术, 2007, 28(1): 103-107
[9]张云峰,刘占生.粘弹壁板非线性颤振滞后特性分析[J].振动与冲击, 2007, 26(7): 101-105
[10] Dowell E H, Voss H M. Theoretical and experimental panel flutter studies in the mach number range of 1.0 to 5.0[J]. AIAA Journal, 1970, 8(9): 2022-2304
[11] Gray Jr C E, Mei C. Large amplitude finite element flutter analysis of composite panels in hypersonic flow[J]. AIAA Journal, 1993, 31(6): 1090-1099
[12] Mei C, Abdel-Motagaly K, Chen R. Review of nonlinear panel flutter at supersonic and hypersonic speeds[J]. Applied Mechanics Review, 1999, 52(10): 321-332
[13] Cheng G F, Mei C. Finite element modal formulation for hypersonic panel flutter analysis with thermal effects [R]. AIAA, 2003: 154-162
[14] Dowell E H. Aeroelasticity of Plates and Shells[M]. Leyden: Noordhoof International Publishing, 1975
[15]陈树辉.强非线性振动系统的定量分析方法[M].北京:科学出版社, 2007
[16] Lau S L, Cheung Y K. Amplitude incremental variational principle for non-linear vibration of elastic systems[J]. ASME Journal of Applied Mechanics, 1981, 48(4): 959-964
[17] Cheung Y K, Chen S H, Lau S L. Application of the incremental harmonic balance method to cubic non-linearity systems[J]. Journal of Sound and Vibration, 1990, 140(2): 273-286
[18] Zhang W Y, Huseyin K. Complex formulation of the IHB technique and its comparison with other methods[J]. Applied Mathematical Modelling, 2002, 26(1): 53-75
[19] Sze K Y, Chen S H, Huang J L. The incremental harmonic balance method for nonlinear vibration of axially moving beams[J]. Journal of Sound and Vibration, 2005, 281(3): 611-626
[20] Yeh Z F, Shih Y S. Critical load, dynamic characteristics and parametric instability of electrorheological material-based adaptive beams[J]. Computers & Structures, 2005, 83(25/26):2162-2174
[21] Wu G Y, Shih Y S. Analysis of dynamic instability for arbitrarily laminated skew plates[J]. Journal of Sound and Vibration, 2006, 292(1/2): 315-340
[22] Wu G Y. The analysis of dynamic instability on the large amplitude vibrations of a beam with transverse magnetic fields and thermal loads[J]. Journal of Sound and Vibration, 2007, 302(1/2): 167-177
[23]陈树辉,黄建亮,佘锦炎.轴向运动梁横向非线性振动研究[J].动力学与控制学报, 2004, 2(1): 40-45
[24]蔡铭,刘济科,李军.多自由度强非线性颤振分析的增量谐波平衡法[J].应用数学和力学, 2006, 27(7): 833-838
[25] Ashley H, Zartarian G. Piston Theory-A New aerodynamics tool for the aeroelastician [J]. Journal of Aeronautical Science, 1956, 23(10): 1109-1108
[26] Dowell E H. Nonlinear oscillations of a fluttering plate[J]. AIAA Journal, 1966, 4(7): 1267-1275
[27] Azzouz M S, Prezekop A, Guo X Y, et al. Nonlinear flutter of shallow shell under yawed supersonic flow using FEM[R]. AIAA Paper, 2003: 1514
[28] Abdel-Motagaly K, Duan B, Mei C. Active control of nonlinear panel flutter under yawed supersonic flow[R]. AIAA Paper 2003: 1368
[29] Moon S H, Hwang J S. Panel flutter suppression with an optimal controller based on the nonlinear model using piezoelectric materials[J]. Composite Structures, 2005, 68(3): 371-379
[30] Singha M K, Ganapathi M. A parametric study on supersonic flutter behavior of laminated composite skew flat panels[J]. Composite Structures, 2005, 69(1): 55-63
[31] Koo K N, Hwang W S. Effects of hysteretic and aerodynami damping on supersonic panel flutter of composite plates[J]. Journal of Sound and Vibration, 2004, 273: 569-583
[32] Bisplinghoof R L, Ashley H. Principles of Aeroelasticity[M]. New York: John Wiley and Sons, Inc, 1962