Geometric Nonlinear Dynamic Analysis Of Flexible Beams Using Variable-length Corotational Beam Elements

Flexible structures with variable length are widely used in engineering applications,such as spacecraft antennas,space elevators,robot arms and rope-pulley systems.These dynamic systems mainly have two characteristics:one is the geometrical nonlinear deformation of the structure in the form of large displacement,large rotation and small strain;the other is that the length of the system changes with time.These dynamic systems can be modelled by a sliding beam or a flexible beam with moving boundaries and/or moving loads.For the nonlinear dynamic analysis of these systems,the traditional finite element method(FEM)in the Lagrange description(element length is fixed)is most widely applicable.However,to accurately describe the position of boundary and load,we need to divide the structrue into very small elements.This manner will undoubtedly reduce the computational efficiency of numerical solution and increase the consumption of computer memory.An ingenious method is to discretize the variable length flexible structure by using variable-length elements.We can keep the element node consistent with the moving boundary or the position of the load.This manner will make up the defect of the traditional Lagrange element.To consider the geometrical nonlinearity of the flexible beam,corotational method(CRM)is a good choice.This method has a very high computational accuracy and is versatile.Under the element independent framework,various assumptions can be introduced to describe the local displacement field of the element.However,the traditional CRM is based on the Lagrange description.In this dissertation,CRM is further extended and different forms of variable-domain corotational beam elements are proposed to investigate the dynamic problems of 2D sliding beams,3D sliding beams,viscoelastic curved beams with moving boundaries and loads,flexible beams with multiple non-material supports and arresting gears,respectively.The details can be summarized as:1)A 2D variable-length element is developed based on CRM for the geometric nonlinear dynamic analysis of 2D sliding beams.To establish an element-independent framework,the same cubic shape functions are used to derive the elastic force vector and the inertia force vector to ensure the consistency of the element,and the shape functions are used to describe the local displacements.All kinds of standard elements can be embedded within this framework.Therefore,the presented method is more versatile than previous approaches.To consider the shear deformation,the sliding beam(a system of changing mass)is discretized using a fixed number of variable-domain interdependent interpolation elements(IIE).In addition,the nonlinear axial strain and the rotary inertia are also considered in this dissertation.The nonlinear motion equations are derived by using the extended Hamilton's principle and solved by combining the Newton-Raphson method and the Hilber-Hughes-Taylor(HHT)method.Furthermore,the closed-form expressions of the iterative tangent matrix and the residual force vector are obtained.2)The 2D variable-length corotational element is further extended to the 3D case to investigate the 3D sliding beam problem which consider a flexible beam slide through a revolute-prismatic joint.The beam manipulated by the revolute-prismatic joint can undergo large overall motion and slide through the joint.There are two main difficulties in the dynamic analysis of the model:1.Considering the geometric nonlinearity of the flexible beam,the configuration space of this beam is a nonlinear differentiable manifold(R3 × SO(3));2.Large 3D spatial rotation and translation of the beam will occur under the control of the re volute-prismatic joint.CRM is a good choice to investigate this kind of geometric nonlinear problem.In this dissertation,rotation variable is introduced to parameterize the rotation matrix to accurately describe the spatial configuration of beams.By introducing a rotational frame,the rigid body motions generated by not only the deformation of the beam but also the revolute-prismatic joint can be removed conveniently.Then,the pure deformation of the element can be easily measured in the rotational frame(the local system).In the corotational frame,various assumptions can be introduced to describe the local displacement field of the element.To consider the shear deformation and rotary inertia,the ?E is introduced.3)To consider a more general model,an arbitrary Lagrangian-Eulerian(ALE)formulation based on the consistent corotational method is presented for the geometric nonlinear dynamic analysis of 2D curved viscoelastic beams.In the ALE description,mesh nodes can be moved in some arbitrarily specified way,which is convenient for investigating problems with moving boundaries and loads.By introducing a corotational frame,the rigid-body motion of an element can be removed.Then,the pure deformation and the deformation rate of the element can be measured in the local frame.This method can avoid rigid-body motion damping.In addition,the elastic force vector,the inertia force vector and the internal damping force vector are derived with the same shape functions to ensure the consistency and independency of the element.In this dissertation,the interdependent interpolation element(IIE)and the Kelvin-Voigt model are introduced in the local frame to consider the shear deformation,rotary inertia and viscoelasticity.Moreover,the presented method can consider the arbitrary curved initial geometry of a beam.4)A 2D non-material variable-domain corotational element(NVCE),which the element length depends on the deformation of the structure and is not a a prescribed function of time as the previous sections,is developed to perform a nonlinear dynamic analysis of a flexible beam with multiple non-material supports and arresting gears.On the arresting gear,the impact of the carrier aircraft will result in the arresting cable sliding out of the deck sheaves and undergoing large deformation.The constraints of the deck sheaves on the cable depends on the current configuration of the cable.By using NVCEs presented in this paper,these constraints can be accurately described.For the dynamic analysis of the arresting cable,a cable element can be embedded into the corotational frame to improve the computational efficiency.In the numerical examples,the propagation mechanism of longitudinal waves and kink waves and the dynamic characteristics of the arresting system are investigated.