Elastic Accelerations Research Articles

Dynamics of Flexible Body Negotiating a Curve

When a rigid body negotiates a curve, the centrifugal force takes a simple form which is function of the body mass, forward velocity, and the radius of curvature of the curve. In this simple case of rigid body dynamics, curve negotiation does not lead to Coriolis forces. In the case of a flexible body negotiating a curve, on the other hand, the inertia of the body becomes function of the deformation, curve negotiations lead to Coriolis forces, and the expression for the deformation-dependent centrifugal forces becomes more complex. In this paper, the nonlinear constrained dynamic equations of motion of a flexible body negotiating a circular curve are used to develop a systematic procedure for the calculation of the centrifugal forces during curve negotiations. The floating frame of reference (FFR) formulation is used to describe the body deformation and define the nonlinear centrifugal and Coriolis forces. The algebraic constraint equations which define the motion trajectory along the curve are formulated in terms of the body reference and elastic coordinates. It is shown in this paper how these algebraic motion trajectory constraint equations can be used to define the constraint forces that lead to a systematic definition of the resultant centrifugal force in the case of curve negotiations. The embedding technique is used to eliminate the dependent variables and define the equations of motion in terms of the system degrees of freedom. As demonstrated in this paper, the motion trajectory constraints lead to constant generalized forces associated with the elastic coordinates, and as a consequence, the elastic velocities and accelerations approach zero in the steady state. It is also shown that if the motion trajectory constraints are imposed on the coordinates of a flexible body reference that satisfies the mean-axis conditions, the centrifugal forces take the same form as in the case of rigid body dynamics. The resulting flexible body dynamic equations can be solved numerically in order to obtain the body coordinates and evaluate numerically the constraint and centrifugal forces. The results obtained for a flexible body negotiating a circular curve are compared with the results obtained for the rigid body in order to have a better understanding of the effect of the deformation on the centrifugal forces and the overall dynamics of the body.

Dynamic modeling of parallel robots with flexible platforms

This paper presents a method for calculating the direct and inverse dynamic models of a parallel robot with a flexible platform. The system considered in this study is a Gough–Stewart 6-DOF parallel robot however the method is general and can be used for other structures. The platform of the parallel manipulator is considered as a flexible body and modeled using distributed flexibility while the links of the legs are considered as rigid. The direct dynamic model gives the elastic and Cartesian accelerations in terms of the input torques and the current state of the system i.e. the position and velocities of both the rigid and elastic variables. The inverse dynamic model calculates the elastic accelerations and the actuator torques from the current state variables and the desired acceleration of the platform.

Solid Deformable Multibody Dynamic Problems in Precision Manufacturing Systems

The recursive projection schemes used in most existing recursive methods for solid deformable structure dynamic problems in precision manufacturing systems 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 in engineering materials increases, the size of the coefficient matrix in the acceleration equations becomes large and consequently the use of these recursive methods for solving 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 multibody system. An amalgamated formulation that can be used to decouple the elastic and joint accelerations for different multibody manufacturing systems is then proposed. The use of the nonlinear recursive method developed in this paper is demonstrated using the open-loop and closedloop chains in precision manufacturing systems.

Analysis of mechanical vibrations and forces using amalgamated decoupling method in multibody mechanical systems

AbstractThe 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.

Decoupling joint and elastic accelerations in deformable mechanical systems using non‐linear recursive formulations

AbstractThe aim of this paper is to develop non‐linear recursive formulations for decoupling joint and elastic accelerations, while maintaining the non‐linear inertia coupling between rigid body motion and elastic deformation in deformable mechanical systems. The inertia projection schemes used in most existing recursive formulations for the dynamic analysis of deformable mechanisms lead to dense coefficient matrices in the equations of motion. 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. Consequently, the use of these recursive formulations for solving the joint and elastic accelerations becomes less efficient. In this paper, the non‐linear recursive formulations have been used to decouple the elastic and joint accelerations in deformable mechanical systems. The relationships between the absolute, elastic and joint variables and generalized Newton–Euler equations are used to develop systems of loosely coupled equations that have sparse matrix structure. By using the inertia matrix structure of deformable mechanical systems and the fact that joint reaction forces associated with elastic coordinates do represent independent variables, a reduced system of equations whose dimension is dependent of 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 use of the approaches developed in this investigation is illustrated using deformable open‐loop serial robot and closed‐loop four‐bar mechanical systems. Copyright © 2007 John Wiley & Sons, Ltd.

Dynamic recursive decoupling method for closed-loop flexible mechanical systems

In this paper, a new method for the dynamic analysis of a closed-loop flexible kinematic mechanical system is presented. The kinematic and force models are developed using absolute reference, joint relative, and elastic coordinates as well as joint reaction forces. This recursive formulation leads to a system of loosely coupled equations of motion. In a closed-loop kinematic chain, cuts are made at selected auxiliary joints in order to form spanning tree structures. Compatibility conditions and reaction force relationships at the auxiliary joints are adjoined to the equations of open-loop mechanical systems in order to form closed-loop dynamic equations. Using the sparse matrix structure of these equations and the fact that the joint reaction forces associated with elastic degrees of freedom do not represent independent variables, a method for decoupling the joint and elastic accelerations is developed. Unlike existing recursive formulations, this method does not require inverse or factorization of large non-linear matrices. It leads to small systems of equations whose dimensions are independent of the number of elastic degrees of freedom. The application of dynamic decoupling method in dynamic analysis of closed-loop deformable multibody systems is also discussed in this paper. The use of the numerical algorithm developed in this investigation is illustrated by a closed-loop flexible four-bar mechanism.

Nonlinear recursive method to solid deformable structure dynamic problems

AbstractThe 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.

Kinematic and Dynamic Analysis for Closed-Loop Flexible Manufacturing Systems Using Nonlinear Recursive Method

In this paper, a nonlinear recursive method for the dynamic and kinematic analysis of a closed-loop flexible manufacturing system is presented. The kinematic and dynamic models are developed using absolute reference, joint relative, and elastic coordinates as well as joint reaction forces. This recursive method leads to a system of loosely coupled equations of motion. In a closed-loop manufacturing system, cuts are made at selected secondary joints in order to form spanning tree structures. Compatibility conditions and reaction force relationships at the secondary joints are adjoined to the equations of open-loop manufacturing systems in order to form closed-loop kinematic and dynamic equations. Using the sparse matrix structure of these equations and the fact that the joint reaction forces associated with elastic degrees of freedom do not represent independent variables, a method for decoupling the joint and elastic accelerations is developed. Unlike existing recursive formulations, this method does not require inverse or factorization of large nonlinear matrices. The application of nonlinear recursive method in kinematic and dynamic analysis of closed-loop manufacturing systems is also discussed in this paper. The use of the numerical algorithm developed in this investigation is illustrated by a closed-loop flexible four-bar mechanism.

A new approach for dynamic analysis of flexible manipulator systems

In this paper, a new approach for dynamic analysis of the flexible multibody manipulator systems is described. The organization of the computer implementations which are used to automatically construct and numerically solve the system of loosely coupled dynamic equations expressed in terms of the absolute, joint and elastic coordinates is discussed. The main processor source code 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 coordinates. At the same time, the force module is used to numerically evaluate the generalized external and elastic forces associated with the absolute, joint and elastic coordinates. Computational efficiency is achieved by taking advantage of the structure of the resulting system of loosely coupled equations. The absolute, joint and elastic accelerations are integrated forward in time using direct numerical integration methods. The absolute positions and velocities can then be determined using the kinematic relationships. The flexible 2-DOF double-pendulum and spatial manipulator systems are used as illustrated examples to demonstrate and verify the application of the computational procedures discussed in this paper.

Dynamic Coupling Between the Joint and Elastic Coordinates in Flexible Mechanism Systems

The aim of this investigation is to develop an efficient proce dure for decoupling the joint and elastic accelerations while maintaining the nonlinear inertia coupling between the rigid body motion and the elastic deformation. The inertia projec tion schemes used in most existing recursive methods for the dynamic analysis of flexible robotics and mechanism systems 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 investigation 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 formulation that can be used to decouple the elastic and joint accelerations is then used to obtain a reduced system of equations whose dimension is in dependent of the number of elastic degrees of freedom of the system. This system can be solved for the joint accelerations as well as the joint reaction forces. The use of the procedure de veloped in this article is demonstrated using a four-bar flexible mechanism.

Projection methods in flexible multibody dynamics. Part I: Kinematics

In this paper a recursive projection method for the dynamic analysis of open-loop mechanical systems that consist of a set of interconnected deformable bodies is presented. The configuration of each body in the system is identified using a coupled set of reference and elastic co-ordinates. The absolute velocities and accelerations of leaf or child bodies in the open-loop system are expressed in terms of the absolute velocities and accelerations of the parent bodies and the time derivatives of the relative co-ordinates of the joints between the bodies. The dynamic differential equations of motion are developed for each link using the generalized Newton-Euler equations. The relationship between the actual joint reactions and the generalized forces combined with the kinematic relationships and the generalized Newton-Euler equations are used to develop a system of loosely coupled equations which has a sparse matrix structure. Using matrix partitioning and recursive projection techniques based on optimal block factorization an efficient solution for the system accelerations and joint reaction forces is obtained. This solution technique yields a much smaller operations count and can more effectively exploit vectorization and parallel processing. It also allows a systematic procedure for decoupling the joint and elastic accelerations.

Projection methods in flexible multibody dynamics. Part II: Dynamics and recursive projection methods

AbstractIn Part I of this paper the kinematic relationships between the absolute, elastic and joint accelerations are developed. In this paper, these kinematic equations are used with the generalized Newton‐Euler equations and the relationship between the actual and generalized reaction forces to develop a recursive projection algorithm for the dynamic analysis of open‐loop mechanical systems consisting of a set of interconnected rigid and deformable bodies. Optimal matrix permutation, partitioning and projection methods are used to eliminate the elastic accelerations while maintaining the inertia coupling between the rigid body motion and the elastic deformation. Recursive projection methods are then applied in order to project the inertia of the leaf bodies onto their parent bodies. This leads to an optimal symbolic factorization which recursively yields the absolute and joint accelerations, and the joint reaction forces. The method presented in this paper avoids the use of Newton‐Raphson algorithms in the numerical solution of the constrained dynamic equations of open‐loop kinematic chains since the joint accelerations are readily available from the solution of the resulting reduced system of equations. Furthermore, the method requires only the inversion or decomposition of relatively small matrices and the numerical integration of a minimum number of co‐ordinates. Open‐loop multibody robotic manipulator systems are used to compare the results and efficiency of the recursive methods with that of the augmented formulations that employ Newton‐Raphson algorithms.

Dynamics of flexible multibody space cranes using recursive projection methods

In this paper the computer implementation of the recursive projection methods in flexible multibody dynamics is described. The organization of the computer program which is used to automatically construct and numerically solve the system of loosely coupled dynamic equations expressed in terms of the absolute and joint coordinates is discussed. The main processor computer program consists of three main modules: constraint module, mass module and force module. The constraint module is used for the numerical evaluation of the relationship between the absolute and joint accelerations. The mass module is used for the numerical evaluation of the system inertia matrix as well as the nonlinear Coriolis and centrifugal forces associated with the absolute and elastic coordinates. The force module is used to evaluate the generalized external and elastic forces associated with the absolute, joint and elastic coordinates. Computational efficiency is achieved by taking advantage of the structure of the resulting system of loosely coupled equations. Elimination of the elastic coordinates and projection of the inertia of the child (leaf) bodies into the space of coordinates of the parent (root) bodies lead to small number of equations which can be solved for the joint accelerations. The joint and elastic accelerations are integrated forward in time using direct numerical integration methods. The absolute coordinates and velocities can then be determined using the kinematic relationships. The multibody space crane is used as an example to demonstrate the use of the computational procedure discussed in this paper.

Harmonic response of cylindrical and toroidal shells to an internal acoustic field. Part I: Theory

The coupled elastic and acoustic response of a system of cylindrical and toroidal shells enclosing an acoustic medium is presented in two parts. The first part develops the theory to model the elastodynamics of cylindrical and toroidal segments of shell. The elastic simulation is based on transfer matrices while the acoustic simulation adapts a Green’s function and curved surface elements. Equilibrium of the acoustic pressure and internal elastic reactions of the shell and compatibility between the acoustic and elastic accelerations at the shell–fluid interface determine the coupled response. The second part discusses numerical results based on the theory for both shell configurations with a plane of symmetry as well as asymmetric ones.

Kineto-elastodynamic analysis of mechanisms by finite element method

In the present paper a systematic finite element method for kineto-elastodynamic analysis of high speed mechanisms has been presented. Effects of number of divisions have been investigated and it is found that a certain minimum number of divisions of links is necessary to yield accurate results. A new approach for eliminating the singularity in the stiffness matrices of mechanisms has been developed and it has been shown in this paper that this new approach yields improved results with negligible additional effort. Finite element expressions for the coriolis, tangential and normal components of elastic accelerations have been derived for a moving link and a new geometric stiffness matrix has been developed to include the effects of the rigid body pinforces and the distributed axial rigid body inertia forces on the transverse vibrations of the links. The method has been further extended to take into account the nonlinear elastic axial forces. Several examples have been solved.