一类具有C-F边界的旋转薄壁梁振动方程整体解的存在性

Existence of global solutions for vibration equations of a rotating thin-walled beam with C-F boundary

航空发动机作为大型飞机的心脏,为大型飞机提供动力,是航空工业发展的重要支柱.同时,航空发动机的制造水平也是每个国家国力、科技和工业基础的重要体现.航空发动机零部件中最多也最重要的当属发动机叶片,其重要性主要体现在高速旋转时转子叶片发生的振动故障.为了研究高速旋转时转子叶片发生的振动,在C-F边界条件下考虑了一类圆截面旋转薄壁梁振动方程,其中C-F边界为常见的叶片左端固定(clamped boundary)、右端自由振动边界(free boundary).首先,将振动方程组简化为向量方程,并给出向量方程弱解的定义;其次,借助Faedo-Galerkin方法构造了振动方程的近似解,并利用常微分方程解的存在性理论证明了近似解存在的唯一性;最后,利用能量方法和Gronwall不等式得到了近似解的一致估计,进而得到了方程整体弱解存在的唯一性.利用Gagliardo-Nirenberg不等式、Young不等式和Holder不等式得到了整体弱解的高阶估计,进而证明了整体强解的存在性.

As the heart of large aircraft,aero-engines provide power for large aircraft and are an important pillar of aviation development.At the same time,the construction level of aero-engines is also an important manifestation of the national strength,technology and industrial.The most important component of aero-engine is engine blade,which is mainly reflected in the vibration fault of rotor blade when rotating at high speed.In order to research the vibration of the rotor blade during high-speed rotation,a vibration equation of a circular-section rotating thin-walled beam is considered with C-F boundary conditions,where the C-F boundary is the clamped boundary at the left and the free boundary at the right.Firstly,the vibration equations were reduced to vector equations,and the definition of the weak solution was given.Secondly,the approximate solution of vibration equation was constructed by Faedo-Galerkin method and the existence and uniqueness of the approximate solution was proved by using the existence theory of the ordinary differential equation.Finally,by using the energy method and the Gronwall inequality,the consistent estimation of approximate solutions was obtained,and the existence and uniqueness of the global weak solution was proved.By applying Gagliardo-Nirenberg inequality,Young inequality and Holder inequality,the higher order estimation of global weak solutions was deduced,and the existence of global strong solutions was proved.