载流导线激励弹性约束轴向变速运动梁主-参联合共振

PRINCIPAL⁃PARAMETRIC JOINT RESONANCE OF AN AXIALLY VARIABLE⁃SPEED MOVING BEAM WITH ELASTIC RESTRAINT EXCITED BY CURRENT⁃CARRYING WIRES

研究了两端弹性约束轴向变速运动梁的主-参联合共振问题。根据哈密顿原理,推导导电梁在平行载流导线激发磁场中的磁弹性非线性振动方程。由两端弹性约束边界条件得到振型函数,再对方程进行伽辽金积分法离散,然后利用多尺度法得到稳态运动的近似解析解,并对稳态解的稳定性进行了分析。通过算例,用数值方法对解析解结果进行验证,并分析不同参数对共振振幅及频率的影响。结果表明,系统发生主-参联合共振时,弹簧刚度和外激励力主要影响系统振幅,而轴向速度、电流、轴向力对振幅和共振频率都有影响;系统的稳定区域随弹簧刚度的增大而增加。

The principal⁃parametric joint resonance of an axially variable⁃speed moving beam with elastic restraint at both ends was studied.Based on Hamilton principle,the magneto⁃elastic nonlinear vibration equation of the beam was derived,in which magnetic field generated by parallel current⁃carrying wires was introduced in modelling.The mode shape function was obtained by means of elastic boundary conditions,and then the Galerkin method was employed to separate time⁃items and space⁃items from each other.The approximate analytical solution and the stability of steady⁃state motion were achieved by multi⁃scale method.In analyzing the computational results,the reliability of the analytical solutions were verified by numerical calculation results corresponding to parameters.In addition,the effects of different parameters on the resonance amplitude and frequency were analyzed.The results show that the stiffnesses of springs and the external excitation force may only affect the amplitude of the system,and the axial velocity,the current and the axial force may lead to the variation of the amplitude and frequency of the resonance,where the stability region increases with the increase of the stiffness of springs.