单向偏心粘弹性梁弯扭耦合振动复模态分析

Complex Modal Analysis of Bending-torsional Coupled Vibration of Viscoelastic Beam with Single Cross-sectional Symmetry

对单向偏心等截面粘弹性梁,考虑偏心引起的弯扭耦合作用。将运动方程写成状态方程形式,利用复模态正交性将其解耦成为若干个广义复振子的求解和叠加问题;使用跟踪结构边界条件矩阵行列式零点的方法求得复频率和复模态,进而可以求得粘弹性偏心梁在任意初始条件和外部激励下的动力响应。通过算例,从结构复频率、复模态幅值和幅角、在不同频率简谐集中力作用下结构动力响应等方面综合分析了粘弹性阻尼和弯扭耦合的影响。计算结果表明,在粘弹性阻尼作用下,衰减系数随振型阶数而增大,振动频率随之不断减小;单纯弯曲和扭转振动的固有频率分布影响各阶复模态中弯扭耦合作用的强弱。通过与有限元法计算结果比较,验证了本文方法的合理性。

The bending-torsional coupled vibration of viscoelastic beam with uniform cross section due to the offset between the mass center and shear center was dealt.The motion equations of the beam were rewritten in the form of state equations and then transformed into the combination of a set of complex oscillators based on the complex modal superposition method.The complex frequencies and modes were obtained by tracing zero points of the determinant of the structural boundary condition matrix;and then,the dynamic responses of the beam under arbitrary initial conditions and external excitations were obtained.Through the numerical examples,the effects of visco-elastic damping and bending-torsional coupling were analyzed by means of a comprehensive study of the complex frequencies,amplitude and phase angle of complex modes,and the structural dynamic responses to concentrated harmonic excitations with different frequencies.The results show that,due to the effect of visco-elastic damping,the attenuation coefficient increases with the increasing of modal order,while the vibrational frequencies decrease with it;the effects of bending-torsional coupling were determined by the distribution of natural frequencies of pure bending and pure torsional vibrations.The proposed method is validated by the comparison with finite element method.