Projective geometric algebra for flexible multibody dynamics: implementation and validation through benchmark problems

In this paper, issues related to the dynamic modelling of constrained deformable bodies that undergo large rigid body displacements are discussed. Particular attention is focused on finite element formulations. It is shown that the use of nodal co‐ordinates and shape functions to describe the finite rotation of some of the commonly used finite elements leads to a linearization of the kinematics and dynamic relationships. The structure of the non‐linear dynamic equations that govern the motion of deformable bodies that undergo large displacements is examined. Comments on the finite element formulation of the invariants of motion, the definition of the generalized forces and moments in flexible body dynamics and the computational strategy used for the automatic generation of the equations of motion are made. The computer formulation of the joint constraints between deformable bodies as well as the numerical algorithms currently used in many of the general purpose computer programs that are based on the augmented formulation are discussed. A decoupled joint‐elastic acceleration recursive formulation is also presented. This formulation leads to a small system of acceleration equations whose dimensions are independent of the number of the elastic degrees of freedom of the system. In this paper, the coupling between the displacements of the deformable bodies is classified as kinematic, inertia and elastic. In view of this classification, comments on the validity of using the updated finite element Lagrangian formulation and the 4 × 4 transformation matrix in the dynamic analysis of flexible multibody systems are made. The coupling between the finite rotation and the wave motion in constrained deformable bodies is also discussed.

<div class="htmlview paragraph">In this paper methods for the nonlinear dynamic analysis of multibody vehicle systems are presented. A total Lagrangian finite element formulation for deformable bodies of the vehicle systems that undergo large translational and rotational displacements is developed. A set of nonlinear dynamic differential equations are developed in a closed form for each finite element in terms of the reference as well as the nodal elstic coordinates and their time derivatives. These equations are expressed in terms of a set of element invariants that depend on the assumed displacement field. The invariants of the deformable body that undergoes large reference translational and rotational displacements can be obtained by assembling the invariants of its elements using a standard finite element procedure. The nonlinear dynamic equations of the deformable body can then be obtained by assembling the nonlinear element equations. Both the consistent and lumped mass finite element formulations are discussed. The dynamic formulations presented in this paper are exemplified using multibody vehicle systems that consist of interconnected rigid and deformable bodies. Numerical results are obtained using the general purpose computer program <b>DAMS</b> (Dynamic Analysis of Multibody Systems).</div>

This paper presents a finite element implementation of plates and shells for the analysis of flexible multibody systems. The developments are set within the framework of the motion formalism that (1) uses configuration and motion to describe the kinematics of flexible multibody systems, (2) couples their displacement and rotation components by recognizing that configuration and motion are members of the Special Euclidean group, and (3) resolves all tensors components in local frames. The formulation based on the motion formalism (1) provides a theoretical framework that streamlines the formulation of shell elements, (2) leads to governing equations of motion that are objective, intrinsic, and present a reduced order of nonlinearity, (3) improves the efficiency of the solution process, (4) circumvents the shear locking phenomenon that plagues shell formulations based on classical kinematic descriptions, and (5) prevents the occurrence of singularities in the treatment of finite rotation. Numerical examples are presented to illustrate the advantageous features of the proposed formulation.

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.

Computational structural analysis of spatial multibody systems based on mobility criteria

Clearance is inevitable for manufacture and assembly in the revolute joints of multibody systems. Excessive value of joint clearance will lead to the poor dynamic performance of mechanism, namely noise, vibration and fatigue failure. This paper presents the development of a dynamic model for planar multibody systems with clearance joints. Based on the continuous contact law, the contact force model proposed by Lankarani and Nikravesh is employed to describe the contact–impact phenomenon between pin and hole. And, the incorporation of the friction influence on revolute joint clearance is conducted by a modified friction force model. According to the geometric relationship between contact elements, the kinematics of the multibody systems is mapped into the global coordinate systems. Additionally, an experimental test platform is set up whereby a slider–crank mechanism with two clearance joints is used as an example to demonstrate the correctness and effectiveness of the proposed approach. Finally, the effects of joint clearance on the dynamic characteristics of planar multibody systems are investigated.

Methods and identities for computation of kinematic and kinetic derivatives required for a broad spectrum of multibody system analyses are presented. Analyses such as implicit numerical integration of the differential–algebraic equations of multibody dynamics, dynamic sensitivity analysis, and workspace analysis are shown to require computation of three derivatives of algebraic constraint functions and first derivatives of inertia and force expressions. Computationally efficient derivative calculation methods and associated identities are presented for Cartesian generalized coordinates, with Euler parameters for orientation. Results presented enable practical and efficient computation of all derivatives required in multibody mechanical system analysis.

This paper presents a generalization of the divide and conquer algorithm for sensitivity analysis of dynamic multibody systems based on direct differentiation. While similar sensitivity analysis approach has been demonstrated for multi-rigid and multi-flexible systems in tree topologies and a limited set of kinematically closed loop topologies, this paper presents the generalization of these approaches to systems in generalized topologies including many coupled kinematically closed loops. This generalization retains the efficient complexity of the underlying formulations i.e. linear and logarithmic complexity in serial and parallel implementation. Other than the computational efficiency, the advantages of this method include concurrent sensitivity analysis with forward dynamics, no numerical artifacts arising from parametric perturbation and significantly reduced data storage compared to traditional methods. An interesting application of this work in control of multibody systems is discussed.

In this work, a methodology for dynamic analysis of rigid-flexible multibody systems with elastohydrodynamic (EHD) lubricated joints is presented. The EHD lubricated cylindrical joint is formulated by the Natural Coordinate Formulation (NCF) and the twenty-node hexahedral element of Absolute Nodal Coordinate Formulation (ANCF), being the lubricant pressure determined through the resolution of the Reynolds’ equation employing the finite difference method. The outcomes are validated with those obtained by using the commercial software ADINA. It is demonstrated that the bearing flexibility plays a significant role in the system responses, extends the lubricant distribution space and reduces the lubricant pressure.

Dynamic recursive decoupling method for closed-loop flexible mechanical systems

In mechanical systems, the effect of friction on the dynamic response of the mechanism is unavoidable. In order to describe the friction behavior at the revolute joint of mechanical systems more accurately, the friction coefficient is modeled as a nonlinear function with respect to pressure and rotational speed based on material-level friction tests. Combined with the Ambrósio friction model, we propose a friction model with nonlinear friction coefficient, and the dynamics of a crank-slider mechanism containing a revolute clearance joint is analyzed based on the proposed model. The results show that the friction model with nonlinear friction coefficient can improve the accuracy of numerical simulation of the system and promote the capability of prediction of dynamic response of complex mechanical systems with joint clearance.

On rigid components and joint constraints in nonlinear dynamics of flexible multibody systems employing 3D geometrically exact beam model

Analytical formulas for kinematic and kinetic derivatives needed in multibody system analyses are derived. A broad spectrum of problems, including implicit numerical integration, dynamic sensitivity analysis, and kinematic workspace analysis, require evaluation of first derivatives of generalized inertia and force expressions and at least three derivatives of algebraic constraint functions. In the setting of a formulation based on Cartesian generalized coordinates with Euler parameters for orientation, basic identities are developed that enable practical and efficient computation of all derivatives required for a large number of multibody mechanical system analyses. The formulation is verified through application to a spatial slider crank mechanism and a 14 body vehicle model. Efficiency of computation using the expressions derived is compared with results obtained employing finite differences, showing significant computational advantage using the analytically derived expressions.

The kinematic and dynamic analysis of the high-degree of freedom (dof) hybrid systems that using serial and parallel robots together on fixed or mobile platform such as legged robots and service robots, is a rather difficult and complex, time consuming problem and a topic of great interest for researchers. There are many educational and commercial toolboxes which analyze serial and parallel robots separately. With the toolbox developed here, high-dof serial, parallel and hybrid robotics systems used on both fixed and mobile platforms can be designed and studied easily. Forward and inverse kinematics, Jacobian matrix, singularities, trajectory planning, torques and forces in joint and operational space can be examined with different robot designs. Using the toolbox that proposed in this paper, robot designer, researchers, students and educators can easily develop their own robotics systems and visualize them in the virtual reality environment provided by MATLAB. Also, it helps understanding fundamentals of high-dof multi-body hybrid robotics systems through user friendly interactive simulation. The most of existing toolboxes in the literature allow only user to analyze the predefined robots in their library with limited interactive capabilities. In some of them, the user can define a new robot with limited dof. But, the user must already have a prior knowledge about designing process of robots. In the proposed toolbox, designing and defining any new robotics systems become very easy. What makes this toolbox unique compared with the existing toolboxes is that any type of robot structures with high dof can be developed, analyzed and studied in detail.

Optimal design of multibody systems is of primary importance to researchers working in various fields. The goal of the paper is to present a novel Hamiltonian based approach for finding design sensitivities of multibody systems (MBS) through the use of direct differentiation method. The total derivatives of Hamiltonian equations of motion (EOM) based on the augmented Lagrangian in terms of the design are determined in the text. An algorithm for the evaluation of a gradient of an objective function is also delivered. Sample test case is presented that demonstrates the validity of the proposed approach.