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The aim of this paper is to develop an efficient method for decoupling joint and elastic accelerations, while maintaining the nonlinear inertia coupling between rigid body motion and elastic body deformation. Almost all of the existing recursive methods for analysis of flexible open-loop and closed-loop multibody mechanical systems lead to dense coefficient matrices in the equations of motion, and consequently there are strong dynamic couplings between the joint and elastic coordinates. When the number of elastic degrees of freedom increases, the size of the coefficient matrix in the equations of motion becomes large and unfortunately the use of these methods for solving for the joint and elastic accelerations becomes less efficient. This paper discusses the problems associated with the inertia projection schemes used in the existing recursive methods, and it is shown that decoupling the joint and elastic accelerations using these methods requires the factorization of nonlinear matrices whose dimensions depend on the number of elastic degrees of freedom of the system. An amalgamated method that can be used to decouple the elastic and joint accelerations is then proposed. In this amalgamated decoupling formulation, the relationships between the absolute, elastic, and joint variables and the generalized Newton–Euler equations are used to develop systems of loosely coupled equations that have sparse matrix structure. Utilizing the inertia matrix structure of flexible manufacturing systems and the fact that the joint reaction forces associated with the elastic coordinates do represent independent variables, a reduced system of equations whose dimension is dependent on the number of elastic degrees of freedom is obtained. This system can be solved for the joint accelerations as well as for the joint reaction forces. The application of the procedure developed in this paper is illustrated using flexible open-loop robotic manipulators and closed-loop crank-slider mechanical systems. Copyright © 2007 John Wiley & Sons, Ltd.

In this paper, a non-linear recursive formulation is developed for kinematic and dynamic analysis of open-loop mechanical systems. The non-linear equations of motion are developed for deformable links that undergo large translational and rotational displacements. These equations are formulated in terms of a set of time invariant scalars and matrices that depend on the spatial co-ordinates as well as the assumed displacement field, and these time invariant quantities represent the dynamic coupling between the rigid-body modes and elastic deformations. A new recursive formulation is presented for solving equations of motion for open-loop chains consisting of interconnected rigid and deformable open-loop mechanical systems. This formulation is expressed by the recursive relationships and the generalized non-linear equations for deformable mechanical systems to obtain a large system of loosely coupled equations of motion. The main processor program consists of three main modules: constraint module, mass module and force module. The constraint module is used to numerically evaluate the relationship between the absolute and joint accelerations. The mass module is used to numerically evaluate the system mass matrix as well as the non-linear Coriolis and centrifugal forces associated with the absolute, joint and elastic co-ordinates. Simultaneously, the force module is used to numerically evaluate the generalized external and elastic forces associated with the absolute, joint and elastic co-ordinates. Computational efficiency is achieved by taking advantage of the structure of the resulting system of loosely coupled equations. The solution techniques used in this investigation yield a much smaller operations count and can more efficiently implement in any computer. The algorithms and solutions presented in this paper are illustrated by using an industrial robotic manipulator system. The numerical results using this formulation are also presented and discussed in this paper. Copyright © 2006 John Wiley & Sons, Ltd.

The recursive projection schemes used in most existing recursive methods for solid deformable structure dynamic problems lead to dense coefficient matrices in the acceleration equations and consequently there is a strong dynamic coupling between the joint and elastic coordinates. When the number of elastic degrees of freedom increases, the size of the coefficient matrix in the acceleration equations becomes large and consequently the use of these recursive methods for solving for the joint and elastic accelerations becomes less efficient. This paper discusses the problems associated with the recursive projection schemes used in the existing recursive methods, and it is shown that decoupling the joint and elastic accelerations using the nonlinear recursive method requires the factorization of nonlinear matrices whose dimensions are independent of the number of elastic degrees of freedom of the open-loop and closed-loop system. An amalgamated formulation that can be used to decouple the elastic and joint accelerations is then proposed. The use of the nonlinear recursive method developed in this paper is demonstrated using an open-loop free-falling deformable double pendulum system and a closed-loop four-bar mechanism. Copyright © 2006 John Wiley & Sons, Ltd.