The Flexible Rotating Beam Dynamic Analysis Based On Absolute Nodal Coordinate Method

With the development of science and technology and the actual demand, many light, flexible and high-speed multi-body systems, such as long mechanical arm of spacecraft, crankshaft system of space engine, etc. are widely applied to various fields. The large rigid motion of the systems will be strongly coupled with the deformation motion of flexible body during their operating periods. Previous research indicates that the traditional modeling based on the small deformation and small rotation assumptions cannot generate the accurate solutions for these flexible multi-body systems. The absolute nodal coordinate method (ANCM) is proposed and can be used to solve these problems with good performance. This method has increasingly become one of the very active fields of flexible multi-body dynamics in recent years.Based on the ANCM, the one-dimensional Euler beam and plane Rayleigh beam element models are established in this paper. The coordinates of element nodal of the two models are defined in the global coordinate system, and the motion of beam element is described using global absolute slope vector in place of rotation coordinate vector of traditional finite element method. The dynamic equations of motion for large deformation and large rotating flexible Euler beam and Rayleigh beam are derived using principle of virtual work and Lagrange equations based on geometrically nonlinear theory. The dynamic differential algebra equations have excellent characteristics. The mass matrix is constant, and Coriolis forces and centrifugal forces are both equal to zero. It is shown that the ANCM can be used to model accurately, and greatly reduce the nonlinearity of the dynamic equations even in large rotation and large deformation case.The Runge-Kutta method with variable step is used to solve the dynamic equations of motion of Euler beam model in this paper. Dynamic characteristics of this flexible beam model under large scale rotation are studied. The flexible Euler beam model is correctly verified by conservation of energy. Then the configuration diagram and dynamic characteristics of Euler beam model with different elastic modulus and element numbers are analyzed and compared respectively. After that, the displacement and speed in absolute coordinate system (ACS) are transformed into deformation and deformation speed in body coordinate system (BCS) using transform matrix. The dynamic characteristics of Euler beam under various conditions in BCS are studied and analyzed. The phase plane diagrams in ACS of corresponding flexible Euler beam endpoint and intermediate node are provided. For comparison, the analytical solution for dynamics of a rigid free falling pendulum with distributed mass only under its own gravity motion is derived. The phase plane diagrams in ACS of this pendulum free end are given in this paper, and compared with the results of the flexible Euler beam model. At last, the simulation and calculation of Rayleigh beam model with shear effect are achieved and the corresponding dynamic analysis is compared with those of Euler beam.