Absolute Nodal Coordinate Formulation Research Articles

Constrained Large-Displacement Thermal Analysis

Abstract Two different cases are encountered in the thermal analysis of solids. In the first case, continua are not subject to boundary and motion constraints and all material points experience same displacement-gradient changes as the result of application of thermal loads. In this case, referred to as unconstrained thermal expansion, the thermal load produces uniform stress-free motion within the continuum. In the second case, point displacements due to boundary and motion constraints are restricted, and therefore, continuum points do not move freely when thermal loads are applied. This second case, referred to as constrained thermal expansion, leads to thermal stresses and its study requires proper identification of the independent coordinates which represent expansion degrees-of-freedom. To have objective evaluation and comparison between the two cases of constrained and unconstrained thermal expansion, the reference-configuration geometry is accurately described using the absolute nodal coordinate formulation (ANCF) finite elements. ANCF position-gradient vectors have unique geometric meanings as tangent to coordinate lines, allowing systematic description of the two different cases of unconstrained and constrained thermal expansions using multiplicative decomposition of the matrix of position-gradient vectors. Furthermore, generality of the approach for large-displacement thermal analysis requires using the Lagrange–D'Alembert principle for proper treatment of algebraic constraint equations. Numerical results are presented to compare two different expansion cases, demonstrate use of the new approach, and verify its results by comparing with conventional finite element (FE) approaches.

Implementation implications on the performance of ANCF simulations

Implementation choices when writing multibody-dynamics software can be critically important to the overall performance. This is no different when applied to the Absolute Nodal Coordinate Formulation to include non-linear flexibility in the simulation. While reducing the number of floating point operations is important when developing efficient implementations, the architecture and constraints of the target hardware also need to be carefully considered. A modified calculation method of the internal forces with a new derivation and a new calculation method for the Jacobian of the internal forces is presented with both mathematical and software implementation details that attempt to include these hardware aspects into the implementation. Several different software implementations are compared running on physical hardware to demonstrate the cost of not optimizing the implementation and to show significant performance gains are possible when better aligning the implementation to the hardware architecture even while maintaining the same number of floating point operations.

Vortex-induced vibrations of a top tensioned riser subjected to flows with spanwise varying directions

Vortex-induced vibrations (VIVs) of a top tensioned riser (TTR) subjected to flows with spanwise varying directions have been simulated and studied in this paper. The riser is simplified as an Euler–Bernoulli beam and its dynamics is described using the finite element method with the Absolute Nodal Coordinate Formulation (ANCF). The hydrodynamic forces due to vortex shedding are modelled with wake oscillators attached to nodes along the riser. The model has been firstly used to simulate an experiment and acceptable model results have been obtained in comparison with the measurements. Then, the VIVs of the TTR under two flow conditions, one with uniform and the other with linearly sheared velocity, have been simulated. In both cases, the direction of the flow varies linearly along the riser. Simulation results have shown different response characteristics at small and high flow velocities with respect to the frequency, amplitude and vibrating direction of the riser. In general, at a small flow velocity where the response of the riser is dominated by the standing wave, the variation in the flow direction has a major influence on the VIV of the riser. However, when the travelling wave is predominant at a high flow velocity, the effect of the spanwise varying flow direction is insignificant. The energy transfer between the structure and flow has been studied to reveal the possible mechanism that leads to these differences in the responses of the riser at small and high flow velocities. The results of the present paper suggest that the engineering simplification of ignoring the influence of flow direction variation on VIV may only be applicable at high flow velocities when the response of the riser is dominated by the travelling wave.

Spatial weak form quadrature beam elements based on absolute nodal coordinate formulation

A new spatial nonlinear beam element allowing for arbitrary number of nodes is proposed. The element which embodies the idea of weak form quadrature is established on the basis of the absolute nodal coordinate formulation (ANCF). The order of Taylor expansion over the cross section can be chosen as desired, enabling the element to cope with complex cross-sectional behavior such as high-order shear and warping deformation. Four typical examples are presented and satisfactorily results are obtained, demonstrating the effectiveness of the formulation for the analysis of spatial beams with strong geometrical nonlinearity.

Numerical and Experimental Study on the Dynamics of the Tendon/Top Tension Riser System of a Tension-Leg Platform

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.

Kinematic Modeling and Characterization of Soft Parallel Robots

Parallel robots with rigid transmission mechanisms have been widely developed to improve the speed, precision, and load capability. However, it is still challenging in promoting soft parallel robots due to the difficulty in accurate kinematic modeling of soft continuum links. In this article, we present a general framework on the design, kinematic modeling, model-based characterization, and control for a class of soft parallel robots. The designed soft parallel robot consists of three fiber-reinforced soft pneumatic actuators, a base stage, and an output stage. With the introduction of the mathematical toolkit of the absolute nodal coordinate formulation, we develop a continuum-based model to describe and parameterize both the global complex configuration and the local large deformation of the soft parallel robot. In this sense, the mappings among the defined kinematic spaces of the robot can be characterized through force analysis. Based on the developed model, we next analyze the robot’s workspace and stiffness with different design parameters, which are also verified by a set of experiments. Finally, we establish a model-based trajectory tracking controller for the soft parallel robot. The experimental results demonstrate that with the feedforward controller, the end effector of the soft parallel robot can well follow the desired trajectories under different output velocities, where the average positioning error is about 2.6–3.9% of the maximum length of the workspace.

Large deformations of hyperelastic curved beams based on the absolute nodal coordinate formulation

Compared with traditional linear elastic materials, the soft structure composed of incompressible hyperelastic materials has not only geometrical nonlinearity but also material nonlinearity during deformation. In this paper, the absolute nodal coordinate formulation (ANCF) is used to study the large deformations and large overall motions of incompressible hyperelastic curved beams. A novel large deformation dynamic modeling method for curved beams made of hyperelastic materials is proposed, in which a simplified Neo-Hookean model is combined with the one-dimensional ANCF beam element. The elastic force vector is calculated according to the exact expression of curvature. The dynamic equations are derived by using the virtual work principle. The dynamic responses of a cantilever silica gel beam under gravity are calculated based on the present method and compared with those of the improved low-order beam element (ILOBE), high-order beam element (HOBE), and commercial finite element analysis software (ANSYS). Simulation results show that the proposed method can accurately describe the large deformation and large overall motion of the beam, and has better computational efficiency. Research in this paper provides an efficient dynamic model for the dynamics analysis of soft robot arms.

An adaptive triangular element of absolute nodal coordinate formulation for thin plates and membranes

An adaptive triangular element based on the absolute nodal coordinate formulation is developed for thin plates and membranes in this work. An elegant theory for thin plate and membrane is first generated under the Kirchhoff–Love assumptions, where the Green strain tensor is decomposed into membrane strain and bending strain tensors. The Hsieh–Clough–Tocher triangular element is extended into the frame of the absolute nodal coordinate formulation for the first time and corresponding shape functions are then derived. The elastic force and Jacobian matrix of the membrane element are explicitly derived by the multiplication of constant tensors and generalized coordinates, resulting in higher efficiency than the plate element. The same group of generalized coordinates is used in membrane and plate elements. Based on this characteristic, a novel adaptive algorithm is proposed based on the stress states to determine at which Gauss points only the membrane strain needs to be considered. The accuracy and adaptability are validated using several benchmark problems. The present adaptive element can improve efficiency without loss of accuracy in cases where most of the membrane is fully tensioned and a part of the local region is slack or wrinkled.

An integrated framework of surface accuracy prediction for clearance-affected extendible support structures with dimensional deviations and elastic deformations

The surface accuracy of the extendible support structure (ESS) greatly influences the imaging resolution of spaceborne antennas. Therefore, considering dimensional deviations, joint clearances and elastic deformations comprehensively, this paper proposes an enhanced deterministic prediction framework to analyze the surface accuracy of the spatial overconstrained ESS. On the one hand, unlike the previous studies which neglect the elastic deformations and connection constraints of antenna panels, these factors and the link flexibility are fully incorporated into the constructed deformation analysis model via the absolute nodal coordinate formulation. Meanwhile, the elastic forces triggered by dimensional deviations and structural deformations are also determined with the skills of invariant matrices for the flexible elements. On the other hand, accounting for axial and radial clearances in spatial revolute joints, the contact conditions and contact forces are described comprehensively, and the contact detection strategy is further put forward for all potential contact scenarios. Afterwards, an integrated accuracy prediction framework is developed to quantify the interaction among dimensional deviations, joint clearances and elastic deformations. Finally, the dynamic segmentation modeling is proposed to determine the surface accuracy of the ESS, and the experiments verify that the developed method is of good prediction accuracy.

The usability and limitations of the various absolute nodal coordinate beam elements subjected to torsional and bi-moment loading

This paper explores the usability of several beam elements based on the Absolute Nodal Coordinate Formulation (ANCF) subjected to torsional and bi-moment loading scenarios. ANCF beam element solutions are compared against an analytical solution and beam and solid finite element solutions. When subjected to bi-moment loading, most of studied higher-order ANCF beam elements suffer from strong “non-smooth” displacement continuity between elements, and therefore they should be used with caution. The distribution of warping displacement was shown to be unacceptable for four-node higher-order ANCF beam elements. Numerical results revealed that the higher-order elements failed to characterize local loading scenarios such as a bi-moment. This problem can be alleviated using very fine beam discretization, which would be more computationally intensive.

Simulation of membrane deployment accounting for the nonlinear crease effect based on absolute nodal coordinate formulation

With the advantages of lightweight and high stowing efficiency, the origami membrane has a broad application prospect in the space structure. As an improvement of the existing deployment simulations, the behavior of crease is considered and both static and dynamic simulation are conducted in this paper. The origami membrane is modeled as a flexible multibody system and its deployment is simulated based on the absolute nodal coordinate formulation. The nonlinear effect of the crease to resist deployment is integrated into the multibody system via the virtual work principle. The facet is modeled by the thin shell element and the crease is constructed as a nonlinear torsional spring with specific position and gradient constraints. The behavior of the crease is parameterized based on the experimental data. The modeling approach is verified by the numerical/experimental reference of the Z-folding structure in the existing literature. For the multi-crease origami membrane model considering the crease effect, a form-finding method to obtain the initial configuration of the flexible origami is proposed based on the principle of minimum potential energy. The deployment of the classical Miura-ori unit membrane structure is analyzed. The magnitude of driving force is estimated based on the force analysis of rigid deployment. Based on the simulation result of flexible deployment, the driving force is planned to achieve a stable deployment with a constant increase speed in the deployment ratio. This work is expected to provide a reference for the design of space membrane structure, the estimation of driving force, and optimizing the deployment strategy of an origami membrane.

Tribo-dynamic performances and vibration transmission of lubricated translational joints in marine engines

• A new tribo-dynamic model for lubricated translational joints in two-stroke marine engines is established to study its dynamic and lubrication performances. • The fit clearance and profile are investigated so as to reveal their influences on the tribo-dynamic performances. • The use of a quadratic profile is an effective way to form a stable wedge-shaped oil film and avoid unnecessary collisions. • The vibration energy of lubricated translational joints through an optimal quadratic profile is greatly reduced in each frequency band. In this paper, a new tribo-dynamic coupling model for lubricated translational joints in two-stroke marine engines is established to study its dynamic and lubrication performances. Dynamic responses of spatial rigid-flexible multi-body systems are solved by a unified absolute nodal coordinate formulation (ANCF). Mixed lubrication performances are described by the modified Reynolds equation and solved by the finite element method. According to the coupling model, the fit clearance and profile are investigated to uncover their influences on the tribo-dynamic performances. The results show that the use of a quadratic profile is an effective way to form a stable wedge-shaped oil film and avoid unnecessary collisions. An optimal quadratic profile of crosshead is suggested in the light of the above analysis, and its influence on the transmission characteristics of vibration signals in the joint is analyzed. It can be concluded that the vibration energy through the optimal profile is greatly reduced in each frequency band.

Dynamic modeling of a fish tail actuated by IPMC actuator based on the absolute nodal coordinate formulation

Ionic polymer metal composite (IPMC) is a smart material with low driving voltage, high energy conversion, light weight and simple structure. It is suitable for driving the tail fin of small bionic fish. Actuators made of this material are quite flexible, and they are laborious to accurately describe the complex deformation of structures. Absolute nodal coordinate formulation (ANCF) has great merits in describing large deformation and large displacement problems. In this paper, the feasibility of ANCF in dynamic modeling of a fish tail actuated by IPMC actuator will be explored for the first time. The tail fin is simplified into two forms: a pure flexible IPMC beam and a hybrid beam that consists of an IPMC beam and a fixed rigid beam. The ANCF one-dimensional two-node beam element is adopted. Considering the influence of deformation curvature, the exact expression of generalized elastic force is obtained. The proposed dynamic model can describe the large deformation problem of the flexible beam. The dynamic responses of the pure IPMC beam tail fin and the hybrid tail fin driven by square wave voltage are calculated. The displacement changes of end nodes are compared, and the swing law under different driving voltage amplitude, frequency, and fluid resistance is analyzed. It is found that the driving efficiency of IPMC can be improved by increasing the driving voltage amplitude, reducing the voltage frequency, and reducing the fluid resistance in a certain range. This research would be beneficial to the study of the large swing motion mechanism of bionic robotic fish.

Novel algorithm for flexible multibody systems with hybrid uncertainties

In this paper, a sparse polynomial chaos-Legendre metamodel based on compressed sensing is proposed to quantify the hybrid uncertainties in flexible multibody systems. Based on the fully parameterized beam elements of absolute nodal coordinate formulation, the differential-algebraic equations of the multibody system with hybrid uncertain parameters are established firstly. Secondly, the dynamical response of the multibody system can be expanded by using the polynomial chaos-Legendre metamodel, and the matrix form of dynamical response can be transformed into vector form by Kronecker product. Then, the coefficients of the polynomial are sparsely reconstructed by using the subspace pursuit algorithm. Finally, the effectiveness of sparse polynomial chaos-Legendre metamodel based on compressed sensing is verified through numerical examples. Especially, compared with the traditional methods, the sampling size of the proposed method is less than the number of degrees of freedom of unknown coefficients in the polynomial surrogate model. Therefore, the proposed sparse polynomial chaos-Legendre metamodel based on compressed sensing can effectively alleviate the “curse of dimensionality”.

Bifurcation and stability analysis of static equilibrium configuration of curved pipe conveying fluid

We present bifurcation and stability analysis of static equilibrium configurations of a clamped-clamped curved pipe conveying fluid in this paper. To capture large deformations of the pipe, this pipe is modeled with an absolute nodal coordinate formulation (ANCF). A technique of parameter continuation is used to solve the equilibria of the governing equation of this pipe system. Effects of external force, flow velocity and arc angle on the nonlinear responses of the curved pipe conveying fluid are discussed in detail. In the case of the curved pipe without any external loadings, the system admits two stable equilibria and an unstable equilibrium when the flow velocity exceeds a critical value on which a saddle-node bifurcation occurs. This critical flow velocity that characterizes the onset of the multistability is increased as the arc angle of the pipe increases. In addition, a pitchfork bifurcation occurs along the branch of the unstable equilibria. When the curved pipe is subjected to a concentrated force at its midpoint, the pipe deforms only in symmetrical mode shapes and undergoes a snap-through buckling under variations in the concentrated force. In contrast, when the curved pipe is subject to a gravity and the arc angle of the pipe exceeds a critical value, a pitchfork bifurcation buckling is observed under variations in the gravity. Along the secondary branch of this pitchfork bifurcation, equilibria in asymmetric mode shapes are found. Finally, effects of system parameters on these critical buckling loads are carefully explored.

Consistent Strain-Based Multifidelity Modeling for Geometrically Nonlinear Beam Structures

Abstract Conventional multifidelity modeling for slender structures such as folding-wing aircraft and offshore wind turbines does not allow the generation of multiple fidelity models that consistently use the same external force model, which complicates simulation program and design process. Although consistent absolute nodal coordinate formulation (ANCF)-based multifidelity modeling was recently proposed to address this inconsistency, it still has the following four problems: (1) a large number of generalized coordinates, (2) a large number of Lagrange multipliers, (3) difficulty in constraining high-frequency axial deformation, and (4) a lack of lower-fidelity models. The lower-fidelity models that have not yet been developed are torsion-only beam, extension-only truss, and bending-only beam models. The objective of this study was to develop a novel consistent strain-based multifidelity modeling framework that addresses these problems by leveraging new vector–strain transformations from ANCF to the strain-based beam formulation. We employed a hydrodynamic force model based on Morison's equation as an example to demonstrate all fidelity models obtained from the proposed strain-based framework consistently use the same external force model. We conducted five simulations to verify the proposed models. The consistent external force model for the hydrodynamic force was then validated by comparison with experimental data. The simulation results concurred with those of conventional models and experiments. Low-fidelity models exhibited over 98% reduction in calculation time compared to high-fidelity models, which helps in conceptual and initial designs that require a large number of parametric simulations.

Dynamic analysis and regulation of the flexible pipe conveying fluid with a hard-magnetic soft segment

The recently developed hard-magnetic soft (HMS) materials can play a significant role in the actuation and control of medical devices, soft robots, flexible electronics, etc. To regulate the mechanical behaviors of the cantilevered pipe conveying fluid, the present work introduces a segment made of the HMS material located somewhere along the pipe length. Based on the absolute node coordinate formulation (ANCF), the governing equations of the pipe conveying fluid with an HMS segment are derived by the generalized Lagrange equation. By solving the derived equations with numerical methods, the static deformation, linear vibration characteristic, and nonlinear dynamic response of the pipe are analyzed. The result of the static deformation of the pipe shows that when the HMS segment is located in the middle of the pipe, the downstream portion of the pipe centerline will keep a straight shape, providing that the pipe is stable with a relatively low flow velocity. Therefore, it is possible to precisely regulate the ejection direction of the fluid flow by changing the magnetic and fluid parameters. It is also found that the intensity and direction of the external magnetic field greatly affect the stability and dynamic response of the pipe with an HMS segment. In most cases, the magnetic actuation increases the critical flow velocity for the flutter instability of the pipe system and suppresses the vibration amplitude of the pipe.

Dynamic and sliding mode control of space netted pocket system capturing and attitude maneuver non-cooperative target

Abstract. Similar to a space flying net, the capture field of the space netted pocket system is large and it can be applied to capture space non-cooperative targets flexibly. To maintain the stability of the space netted pocket system, eight inflatable rods are used as the supporting structure of the net surface. In this paper, a space netted pocket system is designed and modeled. Based on ANCF (absolute nodal coordinate formulation), a dynamic model of the complex space rope net system is established, and then an accurate model of closing rope considering the variable length is derived by introducing mass flow element. A double closed-loop sliding control method is designed to maintain the stable attitude of the service spacecraft. An extended observer is applied to estimate and compensate for the disturbances due to the uncertainty of the contact and flexibility in the system. Finally, the dynamic model and control method is verified through the simulation of the virtual prototype. Results show that the service spacecraft can maintain the attitude stability during target captured process and can track the desired angle during attitude maneuver. The flexible deformation and collision cause great disturbance to the service spacecraft, and the extended observer can improve the control accuracy from 10−3 to 10−4.

Approximation of pre-twisted Achilles sub-tendons with continuum-based beam elements

Achilles sub-tendons are materially and geometrically challenging structures that can nearly undergo around 15% elongation from their pre-twisted initial states during physical activities. Sub-tendons’ cross-sectional shapes are subject-specific, varying from simple to complicated. Therefore, the Achilles sub-tendons are often described by three-dimensional elements that lead to a remarkable number of degrees of freedom. On the other hand, the continuum-based beam elements in the framework of the absolute nodal coordinate formulation have already been shown to be a reliable and efficient replacement for the three-dimensional continuum elements in some special problems. So far, that element type has been applied only to structures with a simple cross-section geometry. To computationally efficiently describe a pre-twisted Achilles sub-tendon with a complicated cross-section shape, this study will develop a continuum-based beam element based on the absolute nodal coordinate formulation with an arbitrary cross-section description. To demonstrate the applicability of the developed beam element to the Achilles sub-tendons, 16 numerical examples are considered. During these numerical tests, the implemented cross-section descriptions agreed well with the reference solutions and led to faster convergence rates in comparison with the solutions provided by commercial finite element codes. Furthermore, it is demonstrated that in the cases of very complicated cross-sectional forms, the commercial software ANSYS provides inflated values for the elongation deformation in comparison with ABAQUS (about 6.2%) and ANCF (about 9.4%). Additionally, the numerical results reveal a possibility to model the whole sub-tendons via coarse discretization with high accuracy under uniaxial loading. This demonstrates the huge potential for use in biomechanics and also in multibody applications, where the arbitrary cross-section of beam-like structures needs to be taken into account.

Dynamic modeling and simulation of rigid-flexible coupling cable system by absolute nodal coordinate formulation

The rigid-flexible coupling cable system under large deformation is studied, and the beam element from absolute node coordinate formulation is used to establish flexible cable body of the system. Different numerical integral algorithms are discussed for solving the rigid-flexible cable system and an integration strategy which combines Implicit Euler with Minimum Residual Method (MINRES) is proposed. The influence of the position and number of rigid components and different the lengths of the flexible elements on the system dynamics are analyzed. With constant total mass of the system, higher number of rigid components and their uniform distribution contribute to stabilization of the swing of the flexible cable body. When the total length of the cable is constant, increasing the number of beam elements enhances the nonlinear characteristics of the swing motion and damages the stability. The influence of different factors on the movement of large deformation flexible cable body is obtained through modeling and simulation of the rigid-flexible coupling cable system.