变质量密度简支梁横向振动的模态局部化

Mode Localization in Transverse Vibration of Variable-density Simply Supported Beam

结构动力学领域的模态局部化现象最早由Hodges发现,主要发生在周期结构中,例如螺旋桨之类的循环对称结构、连续梁以及通信天线等大型空间桁架结构。发生模态局部化现象时,振动能量集中于结构局部,容易造成结构破坏。实际上,模态局部化现象已经造成了一些损害,特别是在航空航天领域。目前研究模态局部化现象时主要采用简单模型,例如周期分布的弹簧质量系统、均匀连续梁。O.O.Bendiksen研究了密度周期分布杆的纵向和扭转振动,求解了该问题的控制方程Mauthieu方程。本文基于Floquet解用Fourier级数法求解了变质量密度简支梁的横向振动问题,得到了固有频率和模态,并观察到了固有频率分组现象和模态局部化现象。求解特征值问题的过程中应用了连分式技术。有效的提高了计算精度。

Hodges firstly discovered the phenomenon of mode localization in structural dynamics. This phenomenon mainly takes place in periodic structures such as cyclic symmetrical structures, continuous beams and large space structures. When happening, this phenomenon converges most vibrating energy into the local of structures, then leads to the failure of structures. In fact, this phenomenon has brought some detriments into the reality, especially in aeronautics and astronautics. The research objects in study are currently limited in simply structures like spring-mass systems and continuous beams. The axial and torsional vibration of variable-density bars have been studied by O. O. Bendiksen. Based on the Floquet solution, here, vibration of variable-density simply supported beam is solved by Fourier Series Method. Nature frequencies and modes are obtained. The characteristics of nature frequencies group and mode localization are observed. The application of continue fraction improves calculation presision.