Application of the method of direct separation of motions to the parametric stabilization of an elastic wire
Application of the method of direct separation of motions to the parametric stabilization of an elastic wire
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DOI：
10.1007/s11071-008-9331-9
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发表时间：
2008-02
影响因子：
5.6
通讯作者：
E. Shishkina;I. Blekhman;M. Cartmell;S. Gavrilov
中科院分区：
文献类型：
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作者：
E. Shishkina;I. Blekhman;M. Cartmell;S. Gavrilov
本文研究了运动直接分离方法在分布式系统研究中的应用。提出了一种方法，可以直接将该方法应用于初始运动方程，并满足运动的慢速和快速分量产生的所有边界条件。该方法是通过一个关于所谓的印度魔法绳魔术的经典问题来证明的(Blekhman等人在振动力学选择主题中，第11卷，第139-149页，[2004]；Champneys和Fraser在Proc. R. Soc中。Lond。[2000]；在SIAM J. apple。数学，65(1):267-298,[2004]；弗雷泽和钱普尼斯案。Lond。[2002]；Galan et al. in J. Sound Vib. 280:359-377,[2005])，其中上部垂直位置不稳定的金属丝由于其底部支撑点的垂直振动而稳定。钢丝被建模为具有垂直振动下端的重伯努利-欧拉梁。在此基础上，得到了非激励系统临界弯曲刚度的振动修正的显式公式。
The paper considers the application of the method of direct separation of motions to the investigation of distributed systems. An approach is proposed which allows one to apply the method directly to the initial equation of motion and to satisfy all boundary conditions, arising for both slow and fast components of motion. The methodology is demonstrated by means of a classical problem concerning the so-called Indian magic rope trick (Blekhman et al. in Selected topics in vibrational mechanics, vol. 11, pp. 139–149, [2004]; Champneys and Fraser in Proc. R. Soc. Lond. A 456:553–570, [2000]; in SIAM J. Appl. Math. 65(1):267–298, [2004]; Fraser and Champneys in Proc. R. Soc. Lond. A 458:1353–1373, [2002]; Galan et al. in J. Sound Vib. 280:359–377, [2005]), in which a wire with an unstable upper vertical position is stabilized due to vertical vibration of its bottom support point. The wire is modeled as a heavy Bernoulli–Euler beam with a vertically vibrating lower end. As a result of the treatment, an explicit formula is obtained for the vibrational correction to the critical flexural stiffness of the nonexcited system.