Research On Three-dimensional Flexible Large-deformation Beam Formulations Based On Rotaional Coordianate Descriptions

With the rapid development of computer technology,multibody system dynamics,which is relying on large-scale numerical simulation technology,is more and more widely used in various engineering fields,such as aerospace,mechanical systems,wind turbines,robots and so on.With the wide use of lightweight materials,deformation of flexible bodies becomes larger and larger.Whether dynamic models of flexible bodies can effectively and efficiently deal with problems of large displacements,large deformations,large rotations and rigid-flexible coupling,has become key points and difficulty of flexible multibody dynamic theory.Developing efficient and accurate formulations of a beam,plate and shell for multibody dynamic analysis is the key to the study of flexible multibody dynamic theory,and it is also the theoretical basis of the development of general flexible multibody dynamic software.In the thesis,three-dimensional beam formulations based on rotational coordinate descriptions are developed for beams with large displacements,large deformations and large rotations.The formulations can greatly reduce numbers of elements and generalized coordinates.The current formulation is then used to calculate the equilibrium and dynamic responses of an elevator traveling cable with arbitrarily moving ends.Specific contents and some conclusions are shown as follows.(1)A singularity-free C0-continuous formulation of a three-dimensional inextensible Euler-Bernoulli beam with large deformations and large rotations is developed.The position of the centroid line of the beam is integrated from its slope,and position vectors of nodes of beam elements are no longer used as generalized coordinates.Euler parameters are used to characterize orientations of cross-sections of the beam,which can resolve the singularity problem caused by Euler angles.The hyper-spherical interpolation function is used to guarantee that the normalization constraint equation of Euler parameters is always satisfied.It can only guarantee that Euler parameters are continuous at nodes of beam elements and is C0 continuous.Several numerical examples are presented to demonstrate the performance of the current formulation.It is shown that the current formulation can achieve the same accuracy as the finite element method(FEM)and absolute node coordinate formulation(ANCF)with a fewer number of coordinates.(2)A singularity-free C1-continuous formulation of a three-dimensional inextensible Euler-Bernoulli beam with large deformations and large rotations is developed.The beam can be either straight or initially curved.A new C1-continuous interpolation function for Euler parameters is developed in the current formulation,which results in continuous first derivatives of Euler parameters and material curvatures at nodes of beam elements.Several numerical examples are presented to demonstrate the performance of the current formulation.The current formulation can be applied to static and dynamic problems and can achieve the same accuracy as the FEM,ANCF,and geometrically exact beam formulation with much fewer elements and generalized coordinates.(3)A singularity-free formulation of a three-dimensional extensible EulerBernoulli beam with large deformations and large rotations is developed.The position vector of the centroid line of the beam is integrated from its slope,which is expressed by Euler parameters and the normal strain.Governing equations of the beam and constraint equations are derived using Lagrange's equations for systems with constraints.After verification of the current formulation with a benchmark problem,it is used to calculate the equilibrium and dynamic responses of a round elevator traveling cable with arbitrarily moving ends.Equilibria of the traveling cable with different car positions are calculated;only trivial equilibria of the cable with no closed loops can be found from its three-dimensional model.Natural frequencies of the traveling cable and corresponding mode shapes are calculated.In-plane natural frequencies do not change much with the car position compared with out-of-plane ones.Free responses of the traveling cable due to vertical motion of the car and forced responses due to in-plane and out-of-plane building sways are studied.The vertical motion of the car can affect the lateral response of the traveling cable,but it has almost no effect on its out-of-plane response.While building sways can affect both lateral and out-of-plane responses of the traveling cable,they have little effect on its vertical responses.(4)An accurate singularity-free and locking-free formulation of a threedimensional shear-deformable beam with large deformations and large rotations is developed.The beam is extensible and shear-deformable,and it can be initially straight or curved.The position of the centroid line of the beam is integrated from its slope that is related to the rotation of a corresponding cross-section and stretch and shear strains.While stretch and shear strains are interpolated using polynomial functions,Euler parameters are interpolated using a C1-continous interpolation function to guarantee that curvatures of the centroid line of the beam are continuous at nodes of its elements.Numerical results show that the current formulation does not suffer from shear locking and Poisson locking problems and it can greatly reduce numbers of elements and generalized coordinates.(5)A new locking-free formulation of a three-dimensional shear-deformable beam with large deformations and large rotations is developed.The rotation is parametrized by a rotation vector,which has a clear and intuitive physical meaning.Since the rotation vector and stretch-shear strain vector are interpolated using polynomial functions,interpolation function is not unique and has much flexibility.It not only avoids the complex interpolation of Euler parameters,but also avoids normalization constraint equations,which greatly reduces the difficulty of numerical simulation.A rescaling strategy is adopted to resolve the singularity problem when there is only one singular point at a time instant,which is the case for most engineering applications.Results show that the current formulation do not suffer from shear and Poisson locking problems and it uses much fewer numbers of elements and generalized coordiantes to yield converged results.