Nonlinear analysis and experiment for magnetoelastic buckling

首先，考虑电磁力和弹性力与挠度的三次非线性，以及电流的不对称分量，对梁在电磁力作用下的突变进行了理论分析。其次，从理论上提出了一种稳定磁弹性屈曲的方法，并通过数值计算验证了该方法的有效性。第三，利用一个简单的两杆系统在电磁力作用下的磁弹性屈曲，从理论和实验上研究了库仑阻尼对磁弹性屈曲的影响。主要研究结果如下：1.利用Liapunov-Schmidt方法和中心流形理论，得到了系统的分岔方程，并分析了系统屈曲模态的互补性。对于忽略补数的情况，对解的误差进行了估计，结果表明第三阶模态对后屈曲状态的振型影响大于第二阶模态.当一个不对称的部件 关于我们 当不存在电流的入口时，由于与梁在纵向方向上的拉伸相关联的弹性力的三次非线性分量，发生超临界干草叉分叉。因此，即使在后屈曲状态下，梁的挠度也保持有限值。在电流不对称分量存在的情况下，理论上证明了尖点突变的发生.我们提出了一种方法来稳定的磁弹性屈曲通过控制扰动的分歧，这种方法的适用条件是由梁的初始偏转和切换时间的控制之间的关系。通过数值仿真验证了所提出控制方法的有效性，并讨论了切换延时对控制效果的影响.在不存在库仑阻尼的情况下，在前屈曲的情况下通常存在一个稳态，但当存在库仑阻尼时，存在无限个稳态。随着磁弹性屈曲点的接近，稳态区域增大。另外，当分叉是超临界的情况下，库仑阻尼产生的非平凡的稳定的稳定状态，存在于库仑阻尼的情况下，在附近的稳态区域。另一方面，当分叉是亚临界的情况下，库仑阻尼产生的非平凡的不稳定的稳定状态，在附近存在的库仑阻尼的情况下，稳定状态区域。少

First, the catastrophe of the beam subjected to electromagnetic force is theoretically analyzed by taking into account the cubic nonlinearity of the electromagnetic and elastic forces with the deflection, and also of the unsymmetrical component of the current. Second, a method to stabilize the magnetoelastic buckling is theoretically proposed, and its validity is numerically confirmed. Third, the effect of Coulomb damping on the magnetoelastic buckling is theoretically and experimentally examined using a simple two rods system subjected to electromagnetic force. The main results are as follows :1. Using Liapunov-Schmidt method and center manifold theory, the bifurcation equation is obtained and the complement of the buckling mode is analyzed. For the case of neglecting the complement, the error of the solution is estimated, and it is shown that the third mode has a larger effect than the second mode on the mode shape of the postbuckling state.2. In the case when an unsymmetrical compon … More ent of the current is not present, a supercritical pitchfork bifurcation occurs due to the cubic nonlinear component of the elastic force associated with the stretching of the beam in the longitudinal direction. As a result, even in the postbuckling state, the deflection of the beam maintains a finite value. In the case when an unsymmetrical component of the current is present, it is shown theoretically that a cusp type catastrophe occurs.3. We present a method to stabilize the magnetoelastic buckling by controlling the perturbation of the bifurcation ; the condition of applicability of this method is governed by a relationship between the initial deflection of the beam and the switching time of control. The validity of the proposed control method is confirmed by numerical simulation, and also the effect of the delay of the switching is discussed.4. In the absence of Coulomb damping, there is generally one steady state in the case of prebuckling, but when Coulomb damping exists, there are infinite steady states. With approaching the magnetoelastic buckling point, the steady state region increases. Also for the case when the bifurcation is supercritical, Coulomb damping generates steady state region in the neighborhood of the nontrivial stable steady states which exist in the absence of Coulomb damping. On the other hand, for the case when the bifurcation is subcritical, the Coulomb damping generates steady state region in the neigh borhood of the nontrivial unstable steady states which exist in the absence of Coulomb damping. Less