Numerical Investigation on Dynamics of the Tendon System of A TLP by Applying Absolute Nodal Coordinate Formulation

In the present study,the dynamics of the tendon system of a tension-leg platform(TLP)is investigated through the absolute nodal coordinate formulation(ANCF).Based on the energy conversion principle,the stiffness,generalized elastic force,external load and mass matrices of the element are deduced to perform the element assembling by using the finite element method.Then the motion equation of the tendon/riser is established.In this study,the TLP in the International Ship Structures Committee(ISSC)model under the first and second wave forces is considered as the case study.The simulation is performed in the MATLAB environment.Moreover,the accuracy and reliability of the programs are verified for cases of beam model with theoretical solutions.It is found that the motion response of tendons is affected by the TLP movement and environmental load,simultaneously.Then,the motion response is calculated using the SESAM software and exported as the boundary of ANCF tendons.Finally,the static and dynamic characteristics of the four tendons of ISSC TLP are analyzed systematically by the ANCF method.Performed analysis proves the effectiveness and feasibility of the ANCF method.It is concluded that the proposed method is a powerful scheme for calculating the dynamics of tendon/riser in the field of ocean engineering.

Contact dynamics of elasto-plastic thin beams simulated via absolute nodal coordinate formulation

Under the frame of multibody dynamics, the contact dynamics of elasto-plastic spatial thin beams is numerically studied by using the spatial thin beam elements of absolute nodal coordinate formulation（ANCF）. The internal force of the elasto-plastic spatial thin beam element is derived under the assumption that the plastic strain of the beam element depends only on its longitudinal deformation.A new body-fixed local coordinate system is introduced into the spatial thin beam element of ANCF for efficient contact detection in the contact dynamics simulation. The linear isotropic hardening constitutive law is used to describe the elasto-plastic deformation of beam material, and the classical return mapping algorithm is adopted to evaluate the plastic strains. A multi-zone contact approach of thin beams previously proposed by the authors is also introduced to detect the multiple contact zones of beams accurately, and the penalty method is used to compute the normal contact force of thin beams in contact. Four numerical examples are given to demonstrate the applicability and effectiveness of the proposed elasto-plastic spatial thin beam element of ANCF for flexible multibody system dynamics.

Planar Weak Form Quadrature Beam Elements Based on Absolute Nodal Coordinate Formulation: A Structural Mechanics Approach

A planar nonlinear weak form quadrature beam element of arbitrary number of axial nodes is proposed on the basis of the absolute nodal coordinate formulation (ANCF). Elastic forces of the element are established through geometrically exact beam theory, resulting in good consistency with classical beam theory. Two examples with strong geometrical nonlinearity are presented to verify the effec-tiveness of the formulation.

Dynamic Analysis and Optimal Parameter Design of FlexibleComposite Structures via Absolute Nodal Coordinate Formulation

The composite structure with the dielectric elastomer and soft materials is the main form of theactuators in soft robots. However, the theoretical model is hard to obtain due to the nonlinear large deformationof materials. In this paper, a new composite element model is established based on the absolute nodal coordinateformulation. The consistent deformation conditions at the contact interface between two thin plates are deduced.The hyperelastic constitutive model and the dielectric elastomer constitutive model are introduced for the twothin plates. Then the dynamic model is established to study the dynamic behaviors of the composite flexiblestructure with various parameters. The results show that the nonlinear deformation appears obviously whenthe flexible composite plate structure is driven by various voltages, and the warping deformation becomes moreobvious with the increase of the voltage. The width and thickness of the driven thin plate influence the stabilityof the whole structure. With the decrease of the width or thickness, the deformation of the structure is moreconsistent with obvious periodicity, and the control performance is improved. Finally, the structural parametersof the composite structures are optimized to improve the control performance based on the dynamic performance.Additionally, smaller width and thickness parameters are preferred to obtain better performance in the design offlexible actuator of soft robot.

A new absolute nodal coordinate formulation beam element with multilayer circular cross section

A systematic numerical integration method is applied to the absolute nodal coordinate formulation(ANCF)fully parameterized beam element with smooth varying and continuous cross section.Moreover,the formulation for the integration points and weight coefficients are given in the method which is used to model the multilayer beam with a circular cross section.To negate the effect of the bending stiffness for the element used to model the high-voltage electrical wire,the general continuum mechanical approach is adjusted.Additionally,the insulation cover for some particular types of the wire is described by the nearly incompressible Mooney-Rivlin material model.Finally,a static problem is presented to prove the accuracy and convergence properties of the element,and a dynamic problem of a flexible pendulum is simulated whereby the balance of the energy can be ensured.An experiment is carried out in which a wire is released as a pendulum and falls on a steel rod.The configurations of the wire are captured by a high-speed camera and compared with the simulation results.The feasibility of the wire model can therefore be demonstrated.

Modeling Method and Application of Rational Finite Element Based on Absolute Nodal Coordinate Formulation

In multibody system dynamics, the absolute nodal coordinate formulation （ANCF） uses power functions as interpolating polynomials to describe the displacement field. It can get accurate results for flexible bodies that undergo large deformation and large rotation. However, the power functions are irrational representation which cannot describe the complex shapes precisely, especially for circular and conic sections. Different from the ANCF representation, the rational absolute nodal coordinate formulation （RANCF） utilizes rational basis functions to describe geometric shapes, which allows the accurate representation of complicated displacement and deformation in dynamics modeling. In this paper, the relationships between the rational surface and volume and the RANCF finite element are provided, and the generalized transformation matrices are established correspondingly. Using these transformation matrices, a new four-node three-dimensional RANCF plate element and a new eight-node three-dimensional RANCF solid element are proposed based on the RANCF. Numerical examples are given to demonstrate the applicability of the proposed elements. It is shown that the proposed elements can depict the geometric characteristics and structure configurations precisely, and lead to better convergence in comparison with the ANCF finite elements for the dynamic analysis of flexible bodies.

Coupled attitude-vibration analysis of an E-sail using absolute nodal coordinate formulation

In this study,the effects of solar wind on an electric sail(E-sail)are modeled and analyzed using an absolute nodal coordinate formulation(ANCF).First,the thrust of the charged metal tether that makes up the E-sail was analyzed and a model was established.Numerical simulations of a single metal tether were performed.Then,an overall E-sail model was established using the connection matrix,and E-sails subjected to different angular velocities were compared.Simulation results of the ANCF model and a dumbbell model were compared at different angular velocities.The results confirm that with a relatively high angular velocity,the flexible metal chain can be approximately regarded as a rigid body.However,with a small angular velocity,the flexibility of the metal chain cannot be ignored.

Deployment dynamics of a simplified spinning IKAROS solar sail via absolute coordinate based method

The spinning solar sail of large scale has been well developed in recent years. Such a solar sail can be considered as a rigid-flexible multibody system mainly composed of a spinning central rigid hub, a number of flexible thin tethers, sail membranes, and tip masses. A simplified interplanetary kite-craft accelerated by radiation of the Sun （IKAROS） model is established in this study by using the absolute-coordinate-based （ACB） method that combines the natural coordinate formulation （NCF） describing the central rigid hub and the absolute nodal coordinate formulation （ANCF） describing flexible parts. The initial configuration of the system in the second-stage deployment is determined through both dynamic and static analyses. The huge set of stiff equations of system dynamics is solved by using the generalized-alpha method, and thus the deployment dynamics of the system can be well understood.

Nonlinear formulation for flexible multibody system with large deformation

In this paper, nonlinear modeling for flexible multibody system with large deformation is investigated. Absolute nodal coordinates are employed to describe the displacement, and variational motion equations of a flexible body are derived on the basis of the geometric nonlinear theory, in which both the shear strain and the transverse normal strain are taken into account. By separating the inner and the boundary nodal coordinates, the motion equations of a flexible multibody system are assembled. The advantage of such formulation is that the constraint equations and the forward recursive equations become linear because the absolute nodal coordinates are used. A spatial double pendulum connected to the ground with a spherical joint is simulated to investigate the dynamic performance of flexible beams with large deformation. Finally, the resultant constant total energy validates the present formulation.

INTEGRATION OF NON-UNIFORM RATIONAL B-SPLINES GEOMETRY AND RATIONAL ABSOLUTE NODAL COORDINATES FORMULATION FINITE ELEMENT ANALYSIS

This investigation is intended to develop a computer procedure for the integration of NURBS geometry and the rational absolute nodal coordinate formulation （RANCF） finite element analysis. A linear transformation is given that can be used to convert the NURBS curve to RANCF cable element mesh retaining the same geometry and the same degree of continuity, including the discussion of continuity control and mesh refinement. The green strain tensor is used to establish the nonlinear dynamic equations with numerical examples to demonstrate the use of the procedure in the dynamic analysis of flexible bodies.

Computational dynamics of soft machines

Soft machine refers to a kind of mechanical system made of soft materials to complete sophisticated missions, such as handling a fragile object and crawling along a narrow tunnel corner, under low cost control and actuation. Hence, soft machines have raised great challenges to computational dynamics. In this review article, recent studies of the authors on the dynamic modeling, numerical simulation, and experimental validation of soft machines are summarized in the framework of multibody system dynamics. The dynamic modeling approaches are presented first for the geometric nonlinearities of coupled overall motions and large deformations of a soft component, the physical nonlinearities of a soft component made of hyperelastic or elastoplastic materials, and the frictional contacts/impacts of soft components, respectively. Then the computation approach is outlined for the dynamic simulation of soft machines governed by a set of differential-algebraic equations of very high dimensions, with an emphasis on the efficient computations of the nonlinear elastic force vector of finite elements. The validations of the proposed approaches are given via three case studies, including the locomotion of a soft quadrupedal robot, the spinning deployment of a solar sail of a spacecraft, and the deployment of a mesh reflector of a satellite antenna, as well as the corresponding experimental studies. Finally, some remarks are made for future studies.

An improved dynamic model for a silicone material beam with large deformation

Dynamic modeling for incompressible hyperelastic materials with large deformation is an important issue in biomimetic applications. The previously proposed lower-order fully parameterized absolute nodal coordinate formulation（ANCF） beam element employs cubic interpolation in the longitudinal direction and linear interpolation in the transverse direction, whereas it cannot accurately describe the large bending deformation. On this account, a novel modeling method for studying the dynamic behavior of nonlinear materials is proposed in this paper. In this formulation, a higher-order beam element characterized by quadratic interpolation in the transverse directions is used in this investigation. Based on the Yeoh model and volumetric energy penalty function, the nonlinear elastic force matrices are derived within the ANCF framework. The feasibility and availability of the Yeoh model are verified through static experiment of nonlinear incompressible materials. Furthermore,dynamic simulation of a silicone cantilever beam under the gravity force is implemented to validate the superiority of the higher-order beam element. The simulation results obtained based on the Yeoh model by employing three different ANCF beam elements are compared with the result achieved from a commercial finite element package as the reference result. It is found that the results acquired utilizing a higher-order beam element are in good agreement with the reference results,while the results obtained using a lower-order beam element are different from the reference results. In addition, the stiffening problem caused by volumetric locking can be resolved effectively by applying a higher-order beam element. It is concluded that the proposed higher-order beam element formulation has satisfying accuracy in simulating dynamic motion process of the silicone beam.

Adaptive ANCF method and its application in planar flexible cables

In the conventional absolute nodal coordinate formulation(ANCF), the model is pre-meshed, the number,distribution and type of elements are unchangeable during the simulation. In addition, the deformations of a flexible body are space-varying and time-varying, one cannot predict when, where, and how the deformations will occur. Therefore, in order to obtain a satisfactory accuracy during the whole simulation, the model is usually densely meshed, but it will result in a loss of computational efficiency. In this study,an adaptive absolute nodal coordinate formulation(AANCF)is proposed to optimize the accuracy and efficiency of flexible dynamics. The movement features of flexible bodies are analyzed, and the conventional and adaptive ANCF methods are compared. Then the adaptive computation strategy is presented. The discretization errors come from the inability of interpolation functions of individual elements to capture the complexity of the exact solution, so the mesh can be adaptively optimized by changing the element sizes or the orders of interpolation functions during dynamic computation. Important issues of AANCF, including error estimation,mesh updating, and performance of the AANCF model, are analyzed and discussed in detail. A theoretical model of a planar AANCF cable is presented, where the strategies of dividing and merging elements are discussed. Moreover, the continuity of dynamic variables is deduced, and the mean factors that affect the continuity are obtained, which is very important for the subsequent continuity optimization. Thesimulation results indicate that the distribution of elements varies with time and space, and the elements are denser in large-deformed domains. The AANCF model improved the computational accuracy and efficiency, but the system energy is discontinuous when the elements are merged. Therefore,a continuity-optimized AANCF model is given based on the previous continuity analysis, the results show that the accuracy and continuity of energy are further improved by the continuity-optimized AANCF model.

基于绝对节点坐标法的聚酯系泊缆粘弹性研究

The taut mooring system using synthetic fiber ropes has overcome the shortcomings such as the large self-weight of the mooring lines and provides better mooring performance for the floating structures.The polyester rope has attracted much attention among numerous synthetic fiber rope materials due to its lightweight,low price,corrosion resistance,and high strength.Thus,the mooring characteristics of it are worth studying.Polyester mooring lines are flexible in deep water,when a marine structure is moored by them,the geometric nonlinearity such as large displacement,large stretch,and large bending deformation,and the material nonlinearity like viscoelastic of the polyester ropes become complex integrated problems to be studied.Considering the nonlinear phenomenon,the simulation and calculation of a polyester line were carried out by the absolute nodal coordinate formulation(ANCF)in this paper since the ANCF method has advantages in dealing with the significant deformation problems of the flexible structures.In addition,a chain mooring line was also simulated for comparison,and the results show that the polyester ropes reduce the self-weight of the mooring lines and provide sufficient mooring strength at the same time,and the nonlinear phenomenon of the polyester ropes is different from that of the chain mooring lines.

New insight into the stability and dynamics of fluid-conveying supported pipes with small geometric imperfections

In several previous studies,it was reported that a supported pipe with small geometric imperfections would lose stability when the internal flow velocity became sufficiently high.Recently,however,it has become clear that this conclusion may be at best incomplete.A reevaluation of the problem is undertaken here by essentially considering the flow-induced static deformation of a pipe.With the aid of the absolute nodal coordinate formulation(ANCF)and the extended Lagrange equations for dynamical systems containing non-material volumes,the nonlinear governing equations of a pipe with three different geometric imperfections are introduced and formulated.Based on extensive numerical calculations,the static equilibrium configuration,the stability,and the nonlinear dynamics of the considered pipe system are determined and analyzed.The results show that for a supported pipe with the geometric imperfection of a half sinusoidal wave,the dynamical system could not lose stability even if the flow velocity reaches an extremely high value of 40.However,for a supported pipe with the geometric imperfection of one or one and a half sinusoidal waves,the first-mode buckling instability would take place at high flow velocity.Moreover,based on a further parametric analysis,the effects of the amplitude of the geometric imperfection and the aspect ratio of the pipe on the static deformation,the critical flow velocity for buckling instability,and the nonlinear responses of the supported pipes with geometric imperfections are analyzed.

TLP平台张力筋与顶张紧立管动态响应的数值和实验研究

In this study,the dynamics of the tendon/top tension riser(TTR)system of a tension-leg platform(TLP)are investigated through an experiment and by using absolute nodal coordinate formulation(ANCF).First,the model test of the TLP system is conducted in the water tank of Harbin Engineering University to examine the motion response of the TLP and the dynamic response characteristics of the tendon and TTR.The test scale ratio is set to 1:66.3.Then,on the basis of the ANCF,the stiffness,external load,and mass matrices of the element are deduced to establish the motion equation of the tendon/riser.Finally,the static and dynamic characteristics of the tendon/TTR system of TLP are analyzed systematically by using the ANCF method.The results are compared with commercial software and test results.The motion response of tendon/TTR is affected by the TLP movement and environmental load simultaneously.The analysis proves the effectiveness and accuracy of the ANCF method despite the low number of riser units,suggesting the superiority of the ANCF method for calculating the dynamics of tendon/riser in the field of ocean engineering.

Utilization of nonlinear vibrations of soft pipe conveying fluid for driving underwater bio-inspired robot

Creatures with longer bodies in nature like snakes and eels moving in water commonly generate a large swaying of their bodies or tails,with the purpose of producing significant frictions and collisions between body and fluid to provide the power of consecutive forward force.This swaying can be idealized by considering oscillations of a soft beam immersed in water when waves of vibration travel down at a constant speed.The present study employs a kind of large deformations induced by nonlinear vibrations of a soft pipe conveying fluid to design an underwater bio-inspired snake robot that consists of a rigid head and a soft tail.When the head is fixed,experiments show that a second mode vibration of the tail in water occurs as the internal flow velocity is beyond a critical value.Then the corresponding theoretical model based on the absolute nodal coordinate formulation(ANCF)is established to describe nonlinear vibrations of the tail.As the head is free,the theoretical modeling is combined with the computational fluid dynamics(CFD)analysis to construct a fluid-structure interaction(FSI)simulation model.The swimming speed and swaying shape of the snake robot are obtained through the FSI simulation model.They are in good agreement with experimental results.Most importantly,it is demonstrated that the propulsion speed can be improved by 21%for the robot with vibrations of the tail compared with that without oscillations in the pure jet mode.This research provides a new thought to design driving devices by using nonlinear flow-induced vibrations.

Rotation-based finite elements:reference-configuration geometry and motion description

Infinitesimal-rotation finite elements allow creating a linear problem that can be exploited to systematically reduce the number of coordinates and obtain efficient solutions for a wide range of applications,including those governed by nonlinear equations.This paper discusses the limitations of conventional infinitesimal-rotation finite elements(FE)in capturing correctly the initial stress-free reference-configuration geometry,and explains the effect of these limitations on the definition of the inertia used in the motion description.An alternative to conventional infinitesimal-rotation finite elements is a new class of elements that allow developing inertia expressions written explicitly in terms of constant coefficients that define accurately the reference-configuration geometry.It is shown that using a geometrically inconsistent(GI)approach that introduces the infinitesimal-rotation coordinates from the outset to replace the interpolation-polynomial coefficients is the main source of the failure to capture correctly the reference-configuration geometry.On the other hand,by using a geometrically consistent(GC)approach that employs the position gradients of the absolute nodal coordinate formulation(ANCF)to define the infinitesimal-rotation coordinates,the reference-configuration geometry can be preserved.Two simple examples of straight and tapered beams are used to demonstrate the basic differences between the two fundamentally different approaches used to introduce the infinitesimal-rotation coordinates.The analysis presented in this study sheds light on the differences between the incremental co-rotational solution procedure,widely used in computational structural mechanics,and the non-incremental floating frame of reference formulation(FFR),widely used in multibody system(MBS)dynamics.

New ANCF solid-beam element:relationship with Bézier volume and application on leaf spring modeling

An eight-node solid-beam element based on absolute nodal coordinate formulation(ANCF)which uses cubic interpolation at the longitudinal direction and linear at the transverse direction is proposed.The element can accurately discretize the geometry represented by the Bézier volume with the same basis function order.The continuity property of the proposed beam element is verified.Advantages of the proposed element can be found in the application of leaf spring modeling.The parabolically varying thickness of the leaf can be accurately described.Components in leaf spring such as the spring eye and the rubber bushing can be assembled efficiently since the proposed element can be connected at any direction.The ANCF reference node is used to represent the rigid components and the revolute joints in the vehicle suspension system.Static I-shaped cross-section cantilever beam and flexible pendulum model are used to test the accuracy and dynamic performance of the proposed solid-beam element.The leaf spring model shows the flexibility of the element in modeling the complex mechanical system and the balance between the accuracy and the efficiency.

Dynamic modeling and analysis of the looped space tether transportation system based on ANCF

A high‐fidelity multibody‐system dynamic model of the looped tether transportation system(L‐TTS)is proposed in this study to study its large deformation as well as large overall motion.The absolute nodal coordinate formulation(ANCF)‐based gradient‐deficient beam element is employed to establish the accurate model of the two flexible tethers subject to large deformations.The relative movement of climbers along tethers is described by using the sliding joint model based on ANCF.To reduce the collision risks between tethers and climbers,two libration suppression strategies,namely,the decelerated motion of climbers relative to tethers and multiple climbers per tether are investigated in this study.Several numerical simulations not only validate the effectiveness of the two strategies in reducing the collision risks between climbers and tethers,the overall librations of L‐TTS,and the magnitudes of the longitudinal elastic force of tethers,but also verify the good performance of the high‐fidelity model proposed in this study for dynamic simulation of the L‐TTS in microgravity conditions.