轴向运动软夹层梁横向振动分析

Analysis of transverse vibration of axially moving soft sandwich beam

针对传统夹层梁沿厚度方向不可压缩缺点,以上下约束层与夹心层中面横向位移为独立变量,提出全新的夹层梁理论。将夹层内任意点横向位移假设沿厚度方向变化的二次待定多项式,利用界面位移协调条件,获得以夹心层中面、上下约束层中面横向位移表示的夹心层横向位移模式,由此获得厚度方向正应变及相应剪应变。基于Hamilton原理,建立轴向运动软夹层梁横向振动控制方程组,用Galerkin法求解控制方程。研究表明,软夹层梁一阶模态为上下约束层与夹层一起作横向运动,两层之间无相对变形,与传统夹层梁理论一致;软夹层梁二阶模态为上下约束层向两相反方向运动,软夹层中面相对上下约束层不动,夹层处于上下拉伸或压缩状态;软夹层梁三阶模态为上下约束层向同一方向运动,夹心层中面向相反方向运动,夹心层上下处于不同变形状态(拉或压)。通过对振型、模态函数、自由振动响应、轴向运动速度对频率影响等因素分析表明,传统夹层梁模型为软夹层梁模型的特殊形式。

In consideration of the shortcoming of traditional sandwich beam theory that the sandwich beam is assumed to be incompressible in thickness direction, a new sandwich beam theory was proposed by introducing independent variables in terms of the displacements of top face sheet,middle plane of soft core and bottom face sheet.The displacement of soft core was approximated by a second order polynomial in thickness direction.Using continuity conditions along the face sheets and soft core,the transverse displacement of the soft core was solved.The normal strain and shearing strain of the soft core in thickness direction were also obtained.Based on the Hamilton principle,the governing equation of the system was established.The Galerkin truncation method was used to solve the governing equation.It is found that：the first mode of soft sandwich beam displays that the face sheets and soft core move together in transverse direction and there is no relative deformation between the face sheets and soft core,this case is consistent with the traditional sandwich beam theory;the second mode of soft sandwich beam shows that the two face sheets move in opposite direction and the middle plane of soft core does not move,so the soft core is in the state of tension or in compression;the third mode of soft sandwich beam displays that the two face sheet move in the same direction and the soft core moves in opposite direction,so the upper part and lower part of soft core are in different deformation state （tension or compression）.By inspecting modal shapes,mode functions,responses of free vibration,the effect of axially moving velocity on frequencies and so on,it is concluded that the incompressible model of sandwich beam is only a special form of the soft sandwich beam model proposed in the paper.