横向磁场中导电壳体的亚谐波共振与稳定性

Subharmonic Resonances and Stability of Conductive Thin Shell in Transverse Magnetic Field

研究了磁场环境中受机械载荷作用圆柱壳体的三阶亚谐波共振问题。在给出横向磁场和机械动载共同作用下导电圆柱薄壳的振动方程基础上,应用伽辽金积分法,并进行无量纲化处理,导出了相应的非线性振动微分方程。应用多尺度法对三阶亚谐波共振问题进行求解,得到了稳态运动下的幅频响应方程及共振解的存在域和稳定性条件。通过算例,给出了共振非平凡解的存在域以及反映振幅与各参数关系的曲线图,讨论了电磁与机械参量对壳体振动的影响。

The nonlinear 3-step subharmonic resonances of a thin cylindrical shell in a magnetic field subjected to mechanical loadings are evaluated. The vibration equations of the thin conductive cylindrical shell under a uniform transverse magnetic and mechanical loadings are obtained. Adopting the Galerkin method, the differential equation of the nonlinear vibration is derived consequently. The 3-step subharmonic resonances are analyzed by the method of multiple scales. The expression of the existing region of nontrivial solutions, the amplitude-frequency resonance equations and the stabilities of the solutions in steady state are derived respectively. By the numerical calculation, the existing region of nontrivial solutions of subharnomic resonances and the amplitude-frequency curves are plotted. The effects of magnetic inductions, the thickness of the shell and the amplitudes of excitations on the vibration of the system are discussed.