The Global Interpolating EFG Method Applied For Dynamics Of Flexible Structures Undergoing Large Overall Motion

In the engineering fields characterized by the lightweight,high speed and high precision,researches on flexible multibody system dynamics in which the elastic deformation and the large overall motion are coupled,are of very important theoretical significance and application value.What's more,the discretization scheme of the flexible body is one of the most essential and basic issues.However,in the traditional numerical methods such as the finite element method(FEM),there are some limitations due to its dependence on elements connected by the pre-arranged nodes: 1)the mesh generation is necessary and costly especially for the complicated three-dimensional structures;2)with the theory of displacement gradient,the computational accuracy of the dynamic stress is low as the difficulty in constructing highly continuous shape functions;3)the static shape function of FEM is difficult to approximate the true deformation field of the elastic body with the rigid-flexible coupling effect and thus the modeling accuracy cannot be guaranteed.In recent decades,the element-free(or meshless)method has become an attractive alternative for scholars in all kinds of fields as its unique advantage.The approximate solution of this method is obtained based on a set of scattered nodes in the whole domain instead of elements and the construction of high-order gradient fields is very easy.The element-free Galerkin(EFG)method formulated by the moving least squares(MLS)is one of the most popular element-free methods due to the high accuracy and convergence rate.However,there are still two mian problems when applying the traditional EFG method to the flexible dynamics: one is the difficulty for imposition of boundary conditions,especially for the derivative variables;the other is the inconvenience in solving the system equations where both actual variables of the rigid motion and nodal parameters of flexible body coexist.To overcome the aforementioned defects,a kind of global interpolating EFG method is proposed by combing the generalized moving least squares(GMLS)and the complete transformation scheme which is proposed for imposition of boundary conditions in the traditional EFG method.There are some advantages of the improved EFG method: 1)the mesh partition is needless in the pretreatment and the imposition of both Dirichlet and Neumann boundary conditions can be achieved as easily and directly as the FEM;2)highly continuous shape functions can be obtained easily and the method has better computational accuracy than MLS;3)convenient operation in establishing the whole spatial discretized equations because of the global interpolating property,and what's more the actual nodal values instead of nodal parameters can be directly obtained from the discretized dynamic equations which improves the computational efficiency and makes the convenience in dealing with the rigid-flexible coupling terms as well as combining with other numerical algorithms.Therefore,the global interpolating EFG method is introduced into the flexible multibody dynamics.The research is prepared for the establishment of a new theoretical framework and numerical computational format in flexible multibody systems and then provides a reliable foundation for strength and fatigue design of mechanical parts with high speed.The main research work is summarized as follows:1)From the governing equations of the two-dimensional elastic mechanics to the Galerkin weak form,the principle of MLS,boundary conditions,discretization of dynamic equations,integration scheme and numerical programming,the basic theories of the traditional EFG method are described in detail.Comparison between GMLS shape function and Hermite element of Euler beam is conducted to validate the high computational accuracy of the dynamic stress of the EFG method in theory.Numerical results show that the EFG method is more preferable in solving dynamic stress than FEM,which provides a new idea in structural strength and fatigue design.2)Due to the limitations of the traditional EFG method directly applied for rigid-flexible coupling dynamics,a kind of global interpolating GMLS(IGMLS)shape function is constructed by combining the GMLS with the transformation scheme.Based on the IGMLS,the dynamic discretized equations of the Kirchhoff plate are formulated by the global interpolating EFG method.Given the convenience to obtain analytical solutions,the Euler beam and Kirchhoff plate are selected for numerical examples.Numerical results of modal and forced vibration analyses with different boundary conditions are given to demonstrate the advantages and good numerical performance of the proposed method in the traditional structural dynamics.The research lays the necessary foundation of the application of mesh-free method for rigid-flexible coupling dynamics.3)A global interpolating EFG method is developed for the nonlinear rigid-flexible coupling dynamics of structures undergoing large overall motion by using the IMLS and IGMLS shape functions.According to the Hamilton principle and the first order approximate coupling theory in which the geometric nonlinearity and longitudinal shrinking of the beam induced by the transverse displacement are considered,the element-free spatial discretized expressions of the rotating and axially moving hub-beam systems are formulated.Using the global interpolating EFG method,structural dynamics in a non-inertial frame and the rigid-flexible coupling dynamics are analyzed for these two flexible beam models.On the one hand,numerical results demonstrate the good convergence,high computational accuracy and acceptable computational efficiency of the proposed EFG method,which provides the theoretical basis for introduction of the mesh-free method into the flexible-coupling dynamics of complicated flexible multibody systems.On the other hand,the numerical results of the EFG method prove the rationality of the first-order approximate coupling model and the effectiveness of the first-order simplified model in the transverse vibration where the longitudinal deformation is neglected,which offers a new numerical scheme for research on the rigid-flexible coupling mechanism and modeling.4)A global interpolating EFG formulation is established for the transverse vibration of a kind of flexible structure which can axially move relative to the fixed base.Based on the theory of Euler's field,the dynamic governing equations of two kinds of axially moving continuum(or gyroscope continuum)are derived: the axially moving simply supported beam and the telescopic cantilever beam.By discretizing the transverse deformation variables using the standard GMLS shape function,the traditional EFG discretized equations with constant coefficient matrices for the axially moving simply supported beam as well as time-varying matrices for the telescopic cantilever beam are derived respectively according to the Hamilton principle.Based on this,the global interpolating EFG equations are obtained by the transformation relationship of coefficient matrices in the dynamic equations.Numerical analyses about the transverse vibration of both kinds of models show that the results obtained from the EFG method have highly agreement with the conclusions from the existing literatures,which supply a new way to study the nonlinear dynamic behavior of more complicated axially moving continuum.