用于柔性转子主动控制的等几何Timoshenko 梁模型及其数值验证
Isogeometric Timoshenko Beam Model and Numerical Verification for Active Vibration Control of Flexible Rotor

为了克服采用标准有限元方法建立的转子模型自由度多不便直接用于控制器设计的问题,结合等几何分析方法的精度高、自由度少的特点,提出了把低阶等几何Timoshenko梁模型运用到柔性转子的主动控制中,并进行了数值验证。首先,给出了半离散等几何模型及在主动控制中作为转子控制输入信号的各类边界条件;其次,分别对比了高、低阶等几何模型的奇异值响应以及简支条件下的高、低阶模型的数值解与理论解;最后,在采用磁悬浮轴承支撑和给定分布不平衡力扰动的条件下,对转子进行了分散比例微分(PD)仿真控制。结果表明:所采用的低阶模型的奇异值响应在前6阶临界转速范围内与高阶模型基本一致;高阶模型的前10阶模态频率很好地吻合了理论解,低阶模型前4阶模态频率误差在0.2％以内;高、低阶等几何梁模型下的转子不平衡振动位移稳态响应的差别很小,该误差可看成工作转速下的同频小扰动。低阶等几何梁模型在低频范围的高精度验证了该方法所得低阶模型直接用于控制器设计的可行性。
A Timoshenko beam model with fewer degrees of freedom (DOFs) by isogeometric analysis method is proposed to be applied to active control of the flexible rotor and is numerically verified. This strategy solves the difficulties in the model from the standard FEM analysis, which cannot be directly employed for controller design due to the more DOFs. The isogeometric beam model with the semi？？discrete form and the various boundaries are considered as the input signals in the active control for the rotor, then the singular value responses of the isogeometric beam models with more and fewer DOFs are compared, and the numerical solutions of isogeometric beam models with more and fewer DOFs and theoretic solutions both with simple supports are also compared. The rotor supported by the magnetic bearing and under the disturbances by the given distributing unbalance forces is simulated by the decentralized PD method. The results show that the singular responses of the model with fewer DOFs almost agree with those with more DOFs within the range of the sixth critical speed; the numerical solutions with more DOFs agree well the theoretic solutions within the first ten modal frequencies and the relative error of the first four modal frequencies from the model with fewer DOFs is less than 0.2％; the deviation of the steady displacement responses from the unbalance vibration between the isogeometric models with more and fewer DOFs is small and their steady response deviation can be regarded as the small disturbances with the same frequency at the working speed. The feasibility of applying directly for controller design is verified by the higher accuracy of the fewer DOFs isogeometric beam model within the low frequency range