研究温度场和磁场中功能梯度(FG)圆柱薄壳1:3阶超谐波共振问题。针对金属-陶瓷FG圆柱薄壳，采用热传导方程表征一维温度场沿壳体厚度分布特性；结合热弹性理论，并考虑几何非线性，确定了壳体物理中面下的本构关系；基于磁弹性理论，建立磁场环境下壳体所受洛伦兹力和磁化力模型。应用哈密顿变分原理推得磁热弹耦合非线性振动方程。利用伽辽金积分法得到两端简支FG圆柱薄壳的无量纲化非线性振动微分方程，确定了由静磁力和热内力所产生的静挠度特征方程。应用多尺度法对系统的1:3阶超谐波共振问题进行近似解析求解，得到系统稳态运动下的幅频响应方程，并应用李雅普诺夫稳定性理论对稳态解进行稳定性分析。通过数值算例，得到不同物理和几何参数影响下系统的共振幅值变化曲线图、动相轨迹图及多值解的区域划分图。结果表明：系统体现为硬弹簧特性；增大激励力幅值、陶瓷侧温度，临界多值解的分岔点右移，共振区域变大；增大磁场强度，稳定解振幅减小。
Abstract
In this paper, the 1:3 order superharmonic resonance of ferromagnetic functionally graded (FG) cylindrical shells is studied in temperature and magnetic fields. For the metal-ceramic FG cylindrical thin shell, the heat conduction equation is used to describe the distribution characteristics of one-dimensional temperature field along the thickness of shell. Based on the thermoselastic theory, and considering geometric nonlinearity, the constitutive relationships on physical neutral surface are determined. According to the magneto-elastic theory, the models of Lorentz force and magnetization force in magnetic field are established. The Hamilton variational principle is employed to derive the nonlinear magneto-thermoelastic coupling vibration equation. Applying the Galerkin method obtains the dimensionless vibration differential equation for the simply supported cylindrical thin shell. The nonlinear characteristic equation of static deflection generated by magnetization force and thermal stress is determined. The multi-scale method is applied to obtain the amplitude frequency equation of 1:3 order superharmonic resonance. Moreover, the stability of steady-state solutions is analyzed by Lyapunov theory. The amplitude curves, dynamic phase trajectories, and regional division of multi-valued solutions under the influence of the different physical and geometric parameters are plotted through numerical examples. The results indicate that the system exhibits hard spring characteristic; With the increase of excitation amplitude and ceramic side temperature, the bifurcation points corresponding to critical multi-valued solutions shift to right, meanwhile the resonance regions expand. In addition, the stable amplitude of superharmonic resonance decreases with the increase of magnetic field strength.
关键词
功能梯度圆柱壳 /
磁弹性 /
温度场 /
超谐波共振 /
多尺度法
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Key words
functionally graded cylindrical shells /
magneto-elasticity /
temperature field /
superharmonic resonance /
multi-scale method
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