利用波动法研究曲梁结构中的波形转换和能量传递

基于Flugge理论，建立了薄壁均质常曲率曲梁面内运动的6阶微分控制方程，得到了曲梁的频散特性曲线和6种波的轴向位移和径向位移的比值，推导了位移和内力响应的表达式以及物理域和波数域的变换矩阵。利用波的传递和反射矩阵对曲梁和半无限长直梁耦合时的能量传递系数和反射系数进行了求解分析。对于半无限长直梁中给定的拉伸波或弯曲波入射，得到了和频率，曲率半径和伸展角度相关的各种波传递和反射的能量系数表达式。数值结果表明，纵波和弯曲波在经过曲梁结构之后发生了波形转换，并研究了能量传递和反射系数随频率，伸展角度，曲梁曲率半径和截面尺寸比的变化。结果表明，无限长直梁和曲梁耦合系统中，低频时，经过曲梁反射和传递后的弯曲波和纵波会相互转化；高频时弯曲波和纵波都能够没有散射地通过曲梁而进行传播。为改善高频时曲梁中的能量衰减效果，研究了在曲梁结构中插入单个、多个中间支撑或阻振质量块时的能量传递和反射系数。结果表明，阻振质量块能够很好地阻止高频时曲梁中能量的传递，对于周期分布的多个阻振质量块，能量传递系数随频率的变化存在周期结构的阻带特征。这些研究结果为曲梁结构的设计提供定性的理论基础。

Application of wave approach in wave mode conversion and energy transmission of curved beams

The six-order coupled differential equations were derived to describe in-plane motion of a thin,uniform,curved beam with constant curvature based on Flugge’s theory.The dispersion relationships and the ratios of tangential displacement to radial displacement for six wave components were obtained.Expressions for displacement,internal force and transfer matrices between physical domain and wave domain were derived.Energy transmission and reflection coefficients for the coupling between a curved beam and a semi-infinite straight beam were derived based on the wave propagation,reflection and transmission matrices.For a given incident tensional or flexural wave in a semi-infinite straight beam,the related energy coefficients expressions of the reflected and transmitted waves were determined as functions of frequency,curvature and bend angle.Numerical examples for energy transmission and reflection coefficients with different frequencies,bend angles,curvatures and cross-section sizes were presented.Numerical results showed that wave transmission and reflection in a curved beam introduces wave mode conversion,an incident wave of one type can induce reflected and transmitted waves of the other type;in lower frequency range,the wave mode conversion takes place for the reflected and transmitted wave;while,in higher frequency range,the incident tensional or flexural wave can transmit without attenuation and conversion.In order to enhance the energy attenuation ability in higher frequency range,a curved beam with inserted single/multiple supports or vibration isolation masses were investigated.The simulation results showed that the vibration isolation masses can impede wave propagation in higher frequency range and the behavior of multiple vibration isolation masses with uniform distribution has band pass/stop characteristics usually demonstrated by periodic structures.This study provided a qualitative guide for design of curved beam structures.