Absolute Coordinate Formulation Research Articles

Absolute coordinate formulation and generalized component mode synthesis with rigid body coordinates

The dynamics of linearly-elastic multibody (MB) systems is conventionally modeled via the floating frame of reference formulation (FFRF); however, its equations of motion (EOMs) involve significantly more nonlinear terms and quantities than alternative formulations, such as the absolute coordinate formulation (ACF) and generalized component mode synthesis (GCMS). This large number of operations required makes computer implementations of the FFRF laborious as well as error-prone and introduces more complexity in general. These issues associated with the FFRF, and the fact that the formulations are mathematically equivalent as shown by the authors, render the ACF and its relatives appealing alternatives due to their simplistic equation structures. To make these alternatives even more appealing, this contribution proposes an improved ACF and GCMS, which (i) reduces the nonlinearity in the EOMs compared to their standard versions and (ii) eliminates the necessity to calculate the rigid body (RB) motion from the global nodal displacement field to obtain the flexible part of the degrees of freedom (DOFs) and the rotation matrix. The proposed EOMs feature a constant mass matrix, a corotated stiffness matrix in the flexible part, and a “small” nonlinear stiffness matrix in the RB rotation part. Moreover, attaching the moving reference frame to the center of mass of the underlying rigid body and employing linearized Tisserand and rotation matrix constraints eliminates coupling terms within the mass matrix and yields implementation-friendly EOMs to analyze the dynamics of linear-elastic flexible MB systems.

A comparative study of the principal methods for the analytical formulation and the numerical solution of the equations of motion of rigid multibody systems

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.

On the Computational Methods for Solving the Differential-Algebraic Equations of Motion of Multibody Systems

In this investigation, different computational methods for the analytical development and the computer implementation of the differential-algebraic dynamic equations of rigid multibody systems are examined. The analytical formulations considered in this paper are the Reference Point Coordinate Formulation based on Euler Parameters (RPCF-EP) and the Natural Absolute Coordinate Formulation (NACF). Moreover, the solution approaches of interest for this study are the Augmented Formulation (AF) and the Udwadia–Kalaba Equations (UKE). As shown in this paper, the combination of all the methodologies analyzed in this work leads to general, effective, and efficient multibody algorithms that can be readily implemented in a general-purpose computer code for analyzing the time evolution of mechanical systems constrained by kinematic joints. This study demonstrates that multibody algorithm based on the combination of the NACF with the UKE turned out to be the most effective and efficient computational method. The conclusions drawn in this paper are based on the numerical results obtained for a benchmark multibody system analyzed by means of dynamical simulations.

Dynamic Analysis of Planar Rigid Multibody Systems modeled using Natural Absolute Coordinates

This paper deals with the dynamic simulation of rigid multibody systems described with the use of two-dimensional natural absolute coordinates. The computational methodology discussed in this investigation is referred to as planar Natural Absolute Coordinate Formulation (NACF). The kinematic representation used in the planar NACF is based on a vector of generalized coordinates that includes two translational coordinates and four rotational parameters. In particular, the set of natural absolute coordinates is employed for describing the global location and the geometric orientation relative to the general configuration of a planar rigid body. The kinematic description utilized in the planar NACF is based on the separation of variable principle. Therefore, a constant symmetric positive-definite mass matrix and a zero inertia quadratic velocity vector associated with the centrifugal and Coriolis inertia effects enter in the formulation of the equations of motion. However, since a redundant set of rotational parameters is used in the kinematic description of the planar NACF for defining the geometric orientation of a rigid body, the introduction of a set of intrinsic normalization conditions is necessary for the mathematical formulation of the algebraic constraint equations. Thus, the intrinsic constraint equations associated with the natural absolute coordinates must be properly taken into account in addition to the extrinsic constraint equations that model the kinematic pairs which form the mechanical joints. This investigation discusses in details the mathematical derivation and the numerical implementation of the multibody system differential-algebraic equations of motion elaborated in the context of the planar NACF. For this purpose, simple geometric considerations are employed in the paper to develop the algebraic equations associated with the intrinsic and extrinsic constraints, whereas the fundamental principles of classical mechanics are utilized for the formal deduction of the dynamic equations. By using the augmented formulation, the index-three form of the differential-algebraic equations of motion is reduced to the corresponding index-one counterpart in order to be able to apply the Udwadia-Kalaba approach for the analytical calculation of the multibody system generalized acceleration vector. Furthermore, in the numerical implementation of the equations of motion based on the planar NACF, the direct correction method is utilized for stabilizing the algebraic constraint equations at both the position and velocity levels. The direct correction approach represents a new methodology recently developed in the field of multibody system dynamics for treating the algebraic constraint equations leading to physically correct and numerically stable dynamic simulations. A standard numerical integration algorithm is employed for obtaining an approximate solution of the nonlinear dynamic equations derived by using the planar NACF. The numerical implementation of a general-purpose multibody computer program based on the planar NACF is demonstrated in the paper considering four simple benchmark examples of rigid multibody systems.

On the Lagrange multipliers of the intrinsic constraint equations of rigid multibody mechanical systems

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.

A natural absolute coordinate formulation for the kinematic and dynamic analysis of rigid multibody systems

The purpose of this paper was to present the key features of a novel coordinate formulation for the analytical description of the motion of rigid multibody systems, namely the natural absolute coordinate formulation (NACF). As it is shown in this work, the kinematic and dynamic analysis of rigid multibody systems can be significantly enhanced employing the NACF. In particular, this formulation combines the main advantages of the natural coordinate formulation (NCF), such as the remarkable property of leading to a constant mass matrix and to zero centrifugal and Coriolis generalized inertia forces, with the generality and the effectiveness of the reference point coordinate formulation (RPCF), which is essentially represented by the possibility to develop and assemble the equations of motion of a multibody system together with the algebraic equations which model the joint constraints in a systematic manner. Moreover, a new computational method hereinafter referred to as the robust generalized coordinate partitioning algorithm is also introduced in this work. The robust generalized coordinate partitioning algorithm can be successfully utilized to numerically solve the index-one form of the multibody system equations of motion formulated by using the proposed NACF as well as the well-known RPCF. In particular, the computational procedure presented in this paper owes its robustness to the combination of the main ideas of the well-established generalized coordinate partitioning method, which is commonly employed to cope with the drift phenomenon of the constraint equations at the position and velocity levels when an index-one formulation of the equations of motion is considered, with the more general and advanced constraint enforcement technique at the acceleration level represented by the fundamental equations of constrained motion. In fact, the fundamental equations of constrained motion represent an effective and efficient method able to calculate analytically the generalized constraint forces relative to a multibody system subjected to a general set of redundant holonomic and/or nonholonomic constraint equations by using the Gauss principle of least constraint, thus avoiding the definition of the Lagrange multipliers. The fundamental equations of constrained motion are remarkably effective when used for modeling the dynamic behavior of rigid multibody systems mathematically represented employing the NACF, as it is shown in this paper. Four simple benchmark multibody systems are also examined in order to exemplify the application of the principal concepts developed in the paper.

Automated Kinematic Synthesis of Planar Mechanisms with Revolute Joints#

Kinematic synthesis of mechanisms at both the type and dimensional levels is a major challenge in the field of mechanism design. This paper presents a new approach for tackling this issue using the framework of genetic algorithms (GAs). The genome employed for a pin-connected planar mechanism includes two parts: a resizable joint-link incidence matrix representing the topology, and three variable-length real-number arrays storing geometric parameters, initial conditions, and task specific parameters. A multibody analysis engine based on an absolute coordinate formulation returns the kinematic simulation results for each genome. The GA can explore different types of mechanisms by applying genetic operators to selected genomes. The evolutionary process is not dependent on the results obtained from enumeration of mechanisms. Case studies on mechanism path generation problems show the applicability of this approach to the kinematic synthesis of pin-connected planar mechanisms.

Component mode synthesis with constant mass and stiffness matrices applied to flexible multibody systems

AbstractIn flexible multibody systems, component mode synthesis is used to reduce the size of system matrices of the single bodies from millions to a few hundred, while maintaining the full coupling between body deformation and overall rigid body motion. The combination of clamped eigenmodes and static deformation modes results in a dense and non‐diagonal structure of the system matrices which is computationally costly. In the present paper, the conventional concept of the floating frame of reference is transformed to a description based on absolute coordinates and a corotational linearization of the deformation. The constant mass matrix and the corotated stiffness matrix need to be factorized only once for the whole simulation. In a planar example, component mode synthesis is applied to this absolute coordinate formulation by using twice as many deformation modes as usual, while the solution of the overall system becomes less expensive due to efficient factorization. The size of the system matrices needs to be increased considerably for each axis of rotation. However, the computational complexity for the factorization reduces from cubic to quadratic. Numerical examples are presented in order to demonstrate the difference between standard procedures and the methodology now proposed. Copyright © 2007 John Wiley & Sons, Ltd.

Nonlinear constraints in the absolute coordinate formulation

The modeling of deformation in flexible multibody systems is still under intensive investigation. While the floating frame of reference formulation has become a standard for the modeling of deformable moving bodies, formulations based on absolute coordinates are comparatively new. The recently developed absolute nodal coordinate formulation uses solely nodal position and slopes as degrees of freedom for structural elements. The numerical treatment is similar to the absolute coordinate formulation, which is investigated in the following. An efficient formulation based on absolute coordinates with a reduced strain tensor (similar to corotational formulations) has been derived recently, and the analogy to the floating frame of reference formulation has been shown. The efficiency of this formulation is based on a co-rotated stiffness matrix which is factorized only once, however, only linear constraints have been treated up to now. The present paper treats the extension of the advantages of the co-rotated formulation with respect to nonlinear constraints, damping and contact. Constraints are discussed for the special cases of linear and nonlinear dependence on the deformation degrees of freedom. The (implicit) Newmark scheme is used to illustrate the time stepping procedure within the present method. In the case of a small number of nonlinear constraints, a single time step can be performed by solving only a small system of nonlinear equations by means of Newton's method. A numerical example shows a method to reduce the number of constraints and illustrates the computational advantages of the proposed method.

An absolute coordinate formulation for three-dimensional flexible pipe analysis

A rather general absolute coordinate formulation is presented for the analysis of flexible pipe structure. The present formulation uses a similar line of approach adopted by Paulling and Webster (1986), which avoids the need of having to use infinitestiminal or finite rotations as nodal variables at the local floating frames of reference by expressing the structural response characteristics in terms of globally based position coordinate and its derivative to allow an accurate non-linear description of the pipe kinematics to be made, and extend the idea to include bending–torsion coupling, axial–radial coupling and bend restriction effect. The present formulation is also capable of handling internal flow effect, pre-bent section and irregular seabed interaction with friction effect, thus allowing an alternative general tool to be developed for riser, pipeline and flexible hose analyses.

The Absolute Coordinate Formulation with Elasto-Plastic Deformations

The present work contributes to the field of multibody systems with respect to the absolute coordinate formulation with a reduced expression of the strain energy and a non-linear constitutive model. Standard methods for multibody systems lead to highly non-linear terms either in the mass matrix or in the stiffness matrix and the most expensive part in the solution of the equations of motion is the assembling of these matrices, the computation of the Jacobian of the non-linear system and the solution of a linear system with the system matrices. In the present work, a consistent simplification of the equations of motion with respect to small deformations but large rigid-body motions is performed. The absolute coordinate formulation is used, therefore the total displacements are the unknowns. This formulation leads to a constant mass matrix while the non-linear stiffness matrix is composed of the constant small strain stiffness matrix rotated by the underlying rigid body rotation. Plastic strains are introduced by an additive split of the strain into an elastic and a plastic part, a yield condition and an associative flow rule. The decomposition of strain has to be performed carefully in order to obey the principle of objectivity for plasticity under large rigid body rotations. As an example, a two-dimensional plate which is hinged at one side and driven by a harmonic force at the opposite side is considered. Plastic deformation is assumed to occur due to extreme environmental influences or due to failure of some attached parts like a defect bearing.