1:1内共振环形桁架天线的稳定性分析
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摘要
复杂的太空环境易导致环形桁架天线产生大幅非线性振动,严重影响天线的稳定性和结构性能.在大幅的非线性振动中不同模态之间的能量可以相互传递和转换,将引起天线产生内振动.因此内共振在天线的大幅振动中起着重要的作用.本文所研究的环形桁架天线被简化成等效圆柱壳模型,建立其非线性动力学方程.根据有限元模态分析,得出环形桁架天线第四阶模态和第五阶模态的振动频率接近1:1.所以本文利用理论分析和数值计算研究1:1内共振情形下环形桁架天线的局部动力学性质.即系统受到小扰动后平衡点的稳定性.详细研究了两种不同形式下的平衡点的稳定性,即系统特征根为双零和两个负数时的稳定性以及系统特征根为双零和一对纯虚复数时的稳定性.利用中心流形理论、非线性变换、Routh-Hurwitz判据得到平衡点的稳定区域、不稳定区域和临界分岔曲线.最后通过数值模拟验证理论分析.
The antenna is subjected to the complicated space environment,which results in nonlinear large amplitude vibration. These nonlinear vibrations severely affect the stability and performance of the structure. Due to the energy exchange between different modes,the circular mesh antenna is easier to produce complex internal resonance. The internal resonance plays a dominant role in large amplitude nonlinear vibrations of the circular mesh antenna. The circular mesh antenna is simplified as the equivalent circular cylindrical shell in this paper.According to the finite element analysis,it is possible that the 1:1 internal resonance between the fourth order mode and the fifth order mode of the circular mesh antenna occurs. Therefore,the local dynamics of the structure with 1:1 internal resonances are investigated by using theoretical analysis and numerical simulations. In this paper,the stability of the equilibrium point under small perturbations is studied. Two types of critical equilibrium points are considered,such as a double zero and two negative eigenvalues,as well as a double zero and a pair of purely imaginary eigenvalues. The stable regions and the unstable regions of the initial equilibrium point and the critical bifurcation curves are obtained by the center manifold theorem,nonlinear-transform and Routh-Hurwitz criterion. Base on the averaged equation of the circular mesh antenna,the trajectories of the initial equilibrium point are examined in order to verify theoretical results.
The antenna is subjected to the complicated space environment,which results in nonlinear large amplitude vibration. These nonlinear vibrations severely affect the stability and performance of the structure. Due to the energy exchange between different modes,the circular mesh antenna is easier to produce complex internal resonance. The internal resonance plays a dominant role in large amplitude nonlinear vibrations of the circular mesh antenna. The circular mesh antenna is simplified as the equivalent circular cylindrical shell in this paper.According to the finite element analysis,it is possible that the 1:1 internal resonance between the fourth order mode and the fifth order mode of the circular mesh antenna occurs. Therefore,the local dynamics of the structure with 1:1 internal resonances are investigated by using theoretical analysis and numerical simulations. In this paper,the stability of the equilibrium point under small perturbations is studied. Two types of critical equilibrium points are considered,such as a double zero and two negative eigenvalues,as well as a double zero and a pair of purely imaginary eigenvalues. The stable regions and the unstable regions of the initial equilibrium point and the critical bifurcation curves are obtained by the center manifold theorem,nonlinear-transform and Routh-Hurwitz criterion. Base on the averaged equation of the circular mesh antenna,the trajectories of the initial equilibrium point are examined in order to verify theoretical results.
引文
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13 Wei Z C,Zhang W.Hidden hyperchaotic attractors in a modified Lorenz-Stenflo system with only one stable equilibrium.International Journal of Bifurcation and Chaos,2014,24(10):1450127
14 Xu X,Hu H Y,Wang H L.Stability,bifurcation and chaos of a delayed oscillator with negative damping and delayed feedback control.Nonlinear Dynamics,2007,49(49):117~129
2 Zhang W,Chen J,Sun Y.Nonlinear breathing vibrations and chaos of a circular truss antenna with 1∶2 internal resonance.International Journal of Bifurcation and Chaos,2016,26(5):1650077
3 Sun Y,Zhang W,Yao M H.Multi-pulse chaotic dynamics of circular mesh antenna with 1∶2 internal resonance.International Journal of Applied Mechanics,2017,9(4):1750060
4胡海岩,田强,张伟等.大型网架式可展开空间结构的非线性动力学与控制.力学进展,2013,43(4):390~414(Hu H Y,Tian Q,Zhang W,et al.Nonlinear dynamics and control of large deployable space structures composed of trusses and meshes.Advances in Mechanics,2013,43(4):390~414(in Chinese))
5 Gao X M,Jin D P,Hu H Y.Internal resonances and their bifurcations of a rigid-flexible space antenna.International Journal of Non-Linear Mechanics,2017,94:160~173
6范震,郭翔鹰,张伟.大型空间环型天线结构动力学分析.动力学与控制学报,2016,14(1):41~47(Fan Z,Guo X Y,Zhang W.Dynamics analysis of large deployable mesh antenna structure.Journal of Dynamics and Control,2016,14(1):41~47(in Chinese))
7 Yu P,Bi Q.Analysis of non-linear dynamics and bifurcations of a double pebdulum.Journal of Sound and Vibration,1998,217(4):691~736
8 Zhang D M,Chen F Q.Stability and bifurcation of a cantilever functionally graded material plate subjected to transversal excitation.Meccanica,2015,50(6):1403~1418
9 Yu W Q,Chen F Q,Li N,et al.Stability and bifurcation dynamics for a nonlinear controlled system subjected to parametric excitation.Archive Applied Mechanics,2016,87(3):479~487
10 Zhang L,Chen F C.Stability and bifurcation for limit cycle oscillations of an airfoil with external store.Nonlinear Dynamics,2016,10(1):1403~1418
11 Wei Z C,Pham V T,Kapitaniak T,et al.Bifurcation analysis and circuit realization for multiple-delayed WangChen system with hidden chaotic attractors.Nonlinear Dynamics,2016,85(3):1635~1650
12 Wei Z C,Yang Q G.Dynamical analysis of a new autonomous 3-D chaotic system only with stable equilibria.Nonlinear Analysis:Real World Applications,2011,12(1):106~118
13 Wei Z C,Zhang W.Hidden hyperchaotic attractors in a modified Lorenz-Stenflo system with only one stable equilibrium.International Journal of Bifurcation and Chaos,2014,24(10):1450127
14 Xu X,Hu H Y,Wang H L.Stability,bifurcation and chaos of a delayed oscillator with negative damping and delayed feedback control.Nonlinear Dynamics,2007,49(49):117~129