Modeling, Analysis and Simulation of 3D Elastohydrodynamic Revolute Joints in Multibody Systems

The inverse dynamic model for a rigid-flexible 3-RRRU parallel robot is deduced by Lagrange's equations, wherein the rigid links and flexible links are described by natural coordinate formulation (NCF) and absolute nodal coordinate formulation (ANCF), respectively. To eliminate the Poisson locking effect, the elastic forces of the flexible links are calculated by a selective reduced integration scheme. The generalized-a method is employed to solve the dynamic equations under kinematic constraints. Because of the differences in convergence rates of the coordinates for the rigid links and flexible links, the rigid geometric constraints are subject to be violated during the iterative process. Therefore, a new method based on NCF, called Transient Rigid Kinematic Correction (TRKC), is proposed to preserve the constraints for the multibody system. The dynamic solution results are compared with rigid robotic system, which show that flexible links affect the angular displacement, velocity and torque of the actuated joints.

A comprehensive computational methodology to study the coupling dynamics of a geared multibody system supported by ElastoHydroDynamic (EHD) lubricated cylindrical joints is proposed throughout this work. The geared multibody system is described by using the Absolute-Coordinate-Based (ACB) method that combines the Natural Coordinate Formulation (NCF) describing rigid bodies and the Absolute Nodal Coordinate Formulation (ANCF) characterizing the flexible bodies. Based on the finite-short bearing approach, the EHD lubrication condition for the cylindrical joints supporting the geared system is considered here. The lubrication forces developed at the cylindrical joints are obtained by solving the Reynolds’ equation via the finite difference method. For the evaluation of the normal contact forces of gear pair along the Line Of Action (LOA), the time-varying mesh stiffness, mesh damping and Static Transmission Error (STE) are utilized. The time-varying mesh stiffness is calculated by using the Chaari’s methodology. The forces of sliding friction along the Off-Line-Of-Action (OLOA) are computed by using the Coulomb friction models with a time-varying coefficient of friction under the EHD lubrication condition of gear teeth. Finally, two numerical examples of application are presented to demonstrate and validate the proposed methodology.

Dynamics and control of a spatial rigid-flexible multibody system with multiple cylindrical clearance joints

Large deployable hoop-truss antennas are widely used in satellite communication systems for their high antenna gain and small packed volume. Knowledge of the deployment dynamics of the large deployable hoop-truss antenna and its coupling with spacecraft attitude motion are essential for the design of deployment mechanisms and attitude control systems. In this paper, a spacecraft with two flexible solar panels, which are reduced by Craig–Bampton component modal synthesis, is modeled by natural coordinate formulation. A large deployable hoop-truss antenna is modeled with the absolute nodal coordinate formulation. Both lead to a constant-mass matrix without the centrifugal and Coriolis forces. The whole dynamics model is used to solve the attitude and the reaction moment during the deployment process. Nonsynchronous deployment phenomena due to the structural flexibility and the decay of the driving force are investigated, and results show that these phenomena can increase the stresses suffered by the hoop truss, which should be considered in the design of a large deployable hoop-truss antenna.

The performance of the absolute nodal coordinate formulation (ANCF) tetrahedral element in the analysis of liquid sloshing is evaluated in this paper using a total Lagrangian nonincremental solution procedure. In this verification study, the results obtained using the ANCF tetrahedral element are compared with the results of the ANCF solid element which has been previously subjected to numerical verification and experimental validation. The tetrahedral-element model, which allows for arbitrarily large displacements including rotations, can be systematically integrated with computational multibody system (MBS) algorithms that allow for developing complex sloshing/vehicle models. The new fluid formulation allows for systematically increasing the degree of continuity in order to obtain higher degree of smoothness at the element interface, eliminate dependent variables, and reduce the model dimensionality. The effect of the fluid/container interaction is examined using a penalty contact approach. Simple benchmark problems and complex railroad vehicle sloshing scenarios are used to examine the performance of the ANCF tetrahedral element in solving liquid sloshing problems. The simulation results show that, unlike the ANCF solid element, the ANCF tetrahedral element model exhibits nonsmoothness of the free surface. This difference is attributed to the gradient discontinuity at the tetrahedral-element interface, use of different meshing rules for the solid- and tetrahedral-elements, and the interaction between elements. It is shown that applying curvature-continuity conditions leads, in general, to higher degree of smoothness. Nonetheless, a higher degree of continuity does not improve the solution accuracy when using the ANCF tetrahedral elements.

In this paper, a new continuum-based pantograph/catenary model based on the absolute nodal coordinate formulation (ANCF) is proposed and used to develop an effective method to control the contact force which arises from the pantograph/catenary interaction. In the proposed new model, only one ANCF gradient vector is used in the formulation of the pantograph/catenary contact conditions, thereby allowing for using the proposed approach for both fully parameterized and gradient deficient ANCF finite elements. A three-dimensional multibody system (MBS) model of a pantograph mounted on a train is developed using a nonlinear augmented MBS formulation. In order to take into account the catenary large deformation, ANCF finite elements are used. The contact between the pantograph and the catenary system is ensured using a sliding joint constraint whereas the contact between the rail vehicle wheels and the train track is modelled using an elastic contact formulation. In addition to the use of the new MBS approach to model the pantograph/catenary interaction, the contact force between the pantograph and the catenary is computed using a simpler lumped parameter model which describes the pan-head and the plunger subsystem dynamics. In order to reduce the standard deviation of the contact force without affecting its mean value, a control actuator is used between the pan-head and the plunger. To this end, three types of control laws for the control action are designed to improve the contact quality both in the transient phase and in the steady state phase of the pantograph/catenary interaction. The first control law proposed features a feedback structure whereas the second and the third control strategies employ a feedback plus feed-forward architecture. In order to demonstrate the effectiveness of the proposed method, the results of a set of numerical simulations with and without the controllers are presented.

It is an exceptional success when multibody dynamics researchers Multibody System Dynamics journal one of the most highly ranked journals in the last 10 years. In the inaugural issue, Professor Schiehlen wrote an interesting article explaining the roots and perspectives of multibody system dynamics. Professor Shabana also wrote an interesting article to review developments in flexible multibody dynamics. The application possibilities of multibody system dynamics have grown wider and deeper, with many application examples being introduced with multibody techniques in the past 10 years. In this paper, the development of multibody dynamics is briefly reviewed and several applications of multibody dynamics are described according to the author’s research results. Simulation examples are compared to physical experiments, which show reasonableness and accuracy of the multibody formulation applied to real problems. Computer simulations using the absolute nodal coordinate formulation (ANCF) were also compared to physical experiments; therefore, the validity of ANCF for large-displacement and large-deformation problems was shown. Physical experiments for large deformation problems include beam, plate, chain, and strip. Other research topics currently being carried out in the author’s laboratory are also briefly explained.

The main contribution of this paper is to demonstrate the feasibility of using one computational environment for developing accurate geometry as well as performing the analysis of detailed biomechanics models. To this end, the finite element (FE) absolute nodal coordinate formulation (ANCF) and multibody system (MBS) algorithms are used in modeling both the contact geometry and ligaments deformations in biomechanics applications. Two ANCF approaches can be used to model the rigid contact surface geometry. In the first approach, fully parameterized ANCF volume elements are converted to surface geometry using parametric relationship that reduces the number of independent coordinate lines. This parametric relationship can be defined analytically or using a spline function representation. In the second approach, an ANCF surface that defines a gradient deficient thin plate element is used. This second approach does not require the use of parametric relations or spline function representations. These two geometric approaches shed light on the generality of and the flexibility offered by the ANCF geometry as compared to computational geometry (CG) methods such as B-splines and NURBS (Non-Uniform Rational B-Splines). Furthermore, because B-spline and NURBS representations employ a rigid recurrence structure, they are not suited as general analysis tools that capture different types of joint discontinuities. ANCF finite elements, on the other hand, lend themselves easily to geometric description and can additionally be used effectively in the analysis of ligaments, muscles, and soft tissues (LMST), as demonstrated in this paper using the knee joint as an example. In this study, ANCF finite elements are used to define the femur/tibia rigid body contact surface geometry. The same ANCF finite elements are also used to model the MCL and LCL ligament deformations. Two different contact formulations are used in this investigation to predict the femur/tibia contact forces; the elastic contact formulation which allows for penetrations and separations at the contact points, and the constraint contact formulation in which the nonconformal contact conditions are imposed as constraint equations, and as a consequence, no separations or penetrations at the contact points are allowed. For both formulations, the contact surfaces are described in a parametric form using surface parameters that enter into the ANCF finite element geometric description. A set of nonlinear algebraic equations that depend on the surface parameters is developed and used to determine the location of the contact points. These two contact formulations are implemented in a general MBS algorithm that allows for modeling rigid and flexible body dynamics.

The goal of this investigation is to perform a comparative analysis of the principal methodologies employed for the analytical formulation and the numerical solution of the equations of motion of rigid multibody mechanical systems. In particular, three formulation approaches are considered in this work for the analytical formulation of the equations of motion. The multibody formulation strategies discussed in this paper are the Reference Point Coordinate Formulation with Euler Angles (RPCF-EA), the Reference Point Coordinate Formulation with Euler Parameters (RPCF-EP), and the Natural Absolute Coordinate Formulation (NACF). Moreover, five computational algorithms are considered in this investigation for the development of effective and efficient solution procedures suitable for the numerical solution of the equations of motion. The multibody computational algorithms discussed in this paper are the Augmented Formulation (AF), the Embedding Technique (ET), the Amalgamated Formulation (AMF), the Projection Method (PM), and the Udwadia-Kalaba Equations (UKE). The multibody formulation approaches and solution procedures analyzed in this work are compared in terms of generality, versatility, ease of implementation, accuracy, effectiveness, and efficiency. In order to perform a general comparative study, four benchmark multibody systems are considered as numerical examples. The comparative study carried out in this paper demonstrates that all the methodologies considered can handle general multibody problems, are computationally effective and efficient, and lead to consistent numerical solutions.

The floating frame of reference (FFR) formulation and the absolute nodal coordinate(ANC) formulation are often used for the modeling of multibody systems. In the presentwork, a reduced strain model is derived for the ANC formulation which is equivalent tothe (small strain) FFR formulation. The reduced strain model is based on a co-rotatedreference configuration and the deformation is assumed to be small with respect to thisconfiguration. This configuration is described by a translation vector and a rotation matrix which are both determined from the motion of the body with respect to its fixedreference. The ANC formulation with reduced strain leads to a constant mass matrix. The stiffness matrix consists of two parts: The most significant part depends on the small-strain stiffness matrix of the body in the fixed reference configuration which is rotated by the rotation matrix and the second part is small and nonlinearly depending on the strain tensor. Both formulations represent displacements and deformations differently but lead to exactly thesame results in the case of equivalent floating reference configurations. Different aspects of both formulations are shown in a 2D example problem of a rotating hinged plate. A detailed description of the modeling in both cases as well as numerical results are presented.

This paper examines the limitations of using B-spline representation as an analysis tool by comparing its geometry with the nonlinear finite element absolute nodal coordinate formulation (ANCF) geometry. It is shown that while both B-spline and ANCF geometries can be used to model non-structural discontinuities using linear connectivity conditions, there are fundamental differences between B-spline and ANCF geometries. First, while B-spline geometry can always be converted to ANCF geometry, the converse is not true; that is, ANCF geometry cannot always be converted to B-spline geometry. Second, because of the rigid structure of the B-spline recurrence formula, there are restrictions on the order of the parameters and basis functions used in the polynomial interpolation; this in turn can lead to models that have significantly larger number of degrees of freedom as compared to those obtained using ANCF geometry. Third, in addition to the known fact that B-spline does not allow for straight forward modeling of T-junctions, B-spline representation cannot be used in a straight forward manner to model structural discontinuities. It is shown in this investigation that ANCF geometric description can be used to develop new spatial chain models governed by linear connectivity conditions which can be applied at a preprocessing stage allowing for an efficient elimination of the dependent variables. The modes of the deformations at the definition points of the joints that allow for rigid body rotations between ANCF finite elements are discussed. The use of the linear connectivity conditions with ANCF spatial finite elements leads to a constant inertia matrix and zero Coriolis and centrifugal forces. The fully parameterized structural ANCF finite elements used in this study allow for the deformation of the cross section and capture the coupling between this deformation and the stretch and bending. A new chain model that employs different degrees of continuity for different coordinates at the joint definition points is developed in this investigation. In the case of cubic polynomial approximation, C1 continuity conditions are used for the coordinate line along the joint axis; while C0 continuity conditions are used for the other coordinate lines. This allows for having arbitrary large rigid body rotation about the axis of the joint that connects two flexible links. Numerical examples are presented in order to demonstrate the use of the formulations developed in this paper.

In the past decade, Absolute Nodal Coordinate Formulation (ANCF), which is kind of a finite element method, has been developed for flexible multibody systems with large deformation and large rotation. Almost all studies for ANCF are dedicated to the development of expression ability of flexible multibody system’s behaviors, for example, a study of expansion of ANCF to three dimensional beam, the improvement of computational performance for numerical analysis and so on. On the other hand, there are few studies which extract controllers from the mathematical expressions derived by ANCF. The main aim of this study is to propose a controller design procedure by the use of the mathematical expression which is derived by ANCF. A flexible beam is introduced as a controlled object and the control torque is applied to the one end of the beam. Control objective is to rotate the beam to the desired position and suppress the residual vibration of the beam. In order to derive the mathematical expression for controller design, a kind of ANCF which uses continuum mechanics approach is employed. It is shown that some assumptions and manipulations of the mathematical expression derived by that method result in the linear equation of motion with some uncertainties and the resultant equation has a form suitable for controller design based on μ synthesis framework which is one of the robust control design method. Using μ synthesis framework, controllers are derived for some design parameters and the derived controllers are applied to the controlled object. The validity of the procedure for controller design is shown by numerical simulations and the possibilities and future works of the proposed controller design procedure by the use of mathematical expression by ANCF is discussed.

This paper deals with the Lagrange multipliers corresponding to the intrinsic constraint equations of rigid multibody mechanical systems. The intrinsic constraint equations are algebraic equations that are associated with nonminimal sets of orientation parameters employed for the kinematic representation of large finite rotations. Two coordinate formulations are analyzed in this investigation, namely the reference point coordinate formulation (RPCF) with Euler parameters and the natural absolute coordinate formulation (NACF). While the RPCF with Euler parameters employs the four components of a unit quaternion as rotational coordinates, the NACF directly uses the orthonormal set of nine direction cosines for describing the orientation of a rigid body in the three-dimensional space. In the multibody approaches based on the RPCF with Euler parameters and on the NACF, the use of a nonminimal set of rotational coordinates facilitates a general and systematic formulation of the differential–algebraic equations of motion. Considering the basic equations of classical mechanics, the fundamental problem of constrained motion is formalized and solved in this paper by using a special form of the Udwadia–Kalaba method. By doing so, the Udwadia–Kalaba equations are employed for obtaining closed-form analytical solutions for the Lagrange multipliers associated with the intrinsic constraint equations that appear in the differential–algebraic dynamic equations developed by using the RPCF with Euler parameters and the NACF multibody approaches. Two simple numerical examples support the analytical results found in this paper.

Three formulations for a flexible spatial beam element for dynamic analysis are compared: a finite element method (FEM) formulation, an absolute nodal coordinate (ANC) formulation with a continuum mechanics approach and an ANC formulation with an elastic line concept where the shear locking of the asymmetric bending mode is suppressed by the application of the Hellinger–Reissner principle. The comparison is made by means of an eigenfrequency analysis on two stylized problems. It is shown that the ANC continuum approach yields too large torsional and flexural rigidity and that shear locking suppresses the asymmetric bending mode. The presented ANC formulation with the elastic line concept resolves most of these problems.

In this investigation, a curved beam element is developed for the analysis of large deformation of flexible multi-body systems using the absolute nodal coordinate formulation. In the existing beam element in this formulation, because the elastic forces are defined using the Green-Lagrange strain tensor as a volume element, locking phenomenon associated with the shear and cross-section deformation leads to erroneously stiffer bending characteristics. In order to circumvent this drawback, the Hellinger-Reissner variational principle is applied to modify the shear stress distribution, whereas the assumed strain method is employed to alleviate the locking associated with the cross-section deformation. The consistent tangent stiffness matrices are derived for the curved beam element and used with implicit integration methods. Numerical examples are presented in order to demonstrate the performance of the curved beam element developed in this investigation.