n Explicit Formulation of Singularity-Free Dynamic Equations of Mechanical Systems in Lagrangian Form---Part one: Single Rigid Bodies

The Cayley transform and the Cayley–transform kinematic relationships are an important and fascinating set of results that have relevance in N –dimensional orientations and rotations. In this paper these results are used in two significant ways. First, they are used in a new derivation of the matrix form of the generalized Euler equations of motion for N –dimensional rigid bodies. Second, they are used to intimately relate the motion of general mechanical systems to the motion of higher–dimensional rigid bodies. This approach can be used to describe an enormous variety of systems, one example being the representation of general motion of an N –dimensional body as pure rotations of an ( N + 1)–dimensional body.

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Dynamics of Multirigid-Body Systems

Journal of Applied Mechanics ◽

10.1115/1.3424437 ◽

1978 ◽

Vol 45 (4) ◽

pp. 889-894 ◽

Cited By ~ 109

Author(s):

R. L. Huston ◽

C. E. Passerello ◽

M. W. Harlow

Keyword(s):

Equations Of Motion ◽

Mechanical Systems ◽

Rigid Bodies ◽

Computer Algorithms ◽

Euler Parameters ◽

Closed Loops ◽

Explicit Formulation ◽

Relative Coordinates ◽

D'alembert's Principle

New and recently developed concepts and ideas useful in obtaining efficient computer algorithms for solving the equations of motion of multibody mechanical systems are presented and discussed. These ideas include the use of Euler parameters, Lagrange’s form of d’Alembert’s principle, quasi-coordinates, relative coordinates, and body connection arrays. The mechanical systems considered are linked rigid bodies with adjoining bodies sharing at least one point, and with no “closed loops” permitted. An explicit formulation of the equations of motion is presented.

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Modeling of Mechanical Systems Using Rigid Bodies and Transmission Line Joints

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.2802523 ◽

1999 ◽

Vol 121 (4) ◽

pp. 606-611 ◽

Cited By ~ 5

Author(s):

Petter Krus

Keyword(s):

Transmission Line ◽

Hydraulic System ◽

Large Scale ◽

Mechanical Systems ◽

Rigid Bodies ◽

Large Scale Systems ◽

Distributed Parameters ◽

Transmission Line Modeling ◽

Integration Algorithms ◽

Transmission Line Models

Dynamic simulation of systems, where the differential equations of the system are solved numerically, is a very important tool for analysis of the detailed behavior of a system. The main problem when dealing with large complex systems is that most simulation packages rely on centralized integration algorithms. For large scale systems, however, it is an advantage if the system can be partitioned in such a way that the parts can be evaluated with only a minimum of interaction. Using transmission line models, with distributed parameters, physically motivated pure time delays are introduced in the communication between components. These models can be used to represent both lines in a hydraulic system and springs in mechanical systems. As a result, components and subsystems can be simulated more independently of each other. In this paper it is shown how flexible joints based on transmission line modeling (TLM) with distributed parameters can be used to simplify modeling of large mechanical link systems interconnected with other physical domains. Furthermore, it provides a straightforward formulation for parallel processing.

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Simulation of Planar Dynamic Mechanical Systems With Changing Topologies: Part I — Characterization and Prediction of the Kinematic Constraint Changes

13th Design Automation Conference: Volume 2 — Robotics, Mechanisms, and Machine Systems ◽

10.1115/detc1987-0102 ◽

1987 ◽

Author(s):

B. J. Gilmore ◽

R. J. Cipra

Keyword(s):

Equations Of Motion ◽

Mechanical Systems ◽

Rigid Bodies ◽

State Variables ◽

Kinematic Constraint ◽

Kinematic Constraints ◽

Force Closure ◽

Simulation Strategy ◽

Reaction Forces ◽

Discontinuous Equations

Abstract Due to changes in the kinematic constraints, many mechanical systems are described by discontinuous equations of motion. This paper addresses those changes in the kinematic constraints which are caused by planar bodies contacting and separating. A strategy to automatically predict and detect the kinematic constraint changes, which are functions of the system dynamics, is presented in Part I. The strategy employs the concepts of point to line contact kinematic constraints, force closure, and ray firing together with the information provided by the rigid bodies’ boundary descriptions, state variables, and reaction forces to characterize the kinematic constraint changes. Since the strategy automatically predicts and detects constraint changes, it is capable of simulating mechanical systems with unpredictable or unforeseen changes in topology. Part II presents the implementation of the characterizations into a simulation strategy and presents examples.

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Dynamics of Mechanical Systems and the Generalized Free-Body Diagram—Part I: General Formulation

Journal of Applied Mechanics ◽

10.1115/1.2965372 ◽

2008 ◽

Vol 75 (6) ◽

Cited By ~ 13

Author(s):

József Kövecses

Keyword(s):

Configuration Space ◽

Dynamic Equilibrium ◽

Mechanical Systems ◽

Constrained Systems ◽

Dynamic Equations ◽

Analytical Mechanics ◽

Constrained Motion ◽

Equilibrium Equations ◽

Free Body Diagram ◽

Free Body

In this paper, we generalize the idea of the free-body diagram for analytical mechanics for representations of mechanical systems in configuration space. The configuration space is characterized locally by an Euclidean tangent space. A key element in this work relies on the relaxation of constraint conditions. A new set of steps is proposed to treat constrained systems. According to this, the analysis should be broken down to two levels: (1) the specification of a transformation via the relaxation of the constraints; this defines a subspace, the space of constrained motion; and (2) specification of conditions on the motion in the space of constrained motion. The formulation and analysis associated with the first step can be seen as the generalization of the idea of the free-body diagram. This formulation is worked out in detail in this paper. The complement of the space of constrained motion is the space of admissible motion. The parametrization of this second subspace is generally the task of the analyst. If the two subspaces are orthogonal then useful decoupling can be achieved in the dynamics formulation. Conditions are developed for this orthogonality. Based on this, the dynamic equations are developed for constrained and admissible motions. These are the dynamic equilibrium equations associated with the generalized free-body diagram. They are valid for a broad range of constrained systems, which can include, for example, bilaterally constrained systems, redundantly constrained systems, unilaterally constrained systems, and nonideal constraint realization.

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The Geometry of Virtual Work Dynamics in Screw Space

Journal of Applied Mechanics ◽

10.1115/1.2899535 ◽

1992 ◽

Vol 59 (2) ◽

pp. 411-417 ◽

Cited By ~ 2

Author(s):

Steven Peterson

Keyword(s):

Tangent Space ◽

Screw Theory ◽

Virtual Work ◽

Rigid Bodies ◽

Dynamic Equations ◽

Coordinate Space ◽

Dynamical Equations ◽

Least Action ◽

System Of Rigid Bodies ◽

Configuration Manifold

In this paper, screw theory is employed to develop a method for generating the dynamic equations of a system of rigid bodies. Exterior algebra is used to derive the structure of screw space from projective three space (homogeneous coordinate space). The dynamic equation formulation method is derived from the parametric form of the principle of least action, and it is shown that a set of screws exist which serves as a basis for the tangent space of the configuration manifold. Equations generated using this technique are analogs of Hamilton’s dynamical equations. The freedom screws defining the manifold’s tangent space are determined from the contact geometry of the joint using the virtual coefficient, which is developed from the principle of virtual work. This results in a method that eliminates all differentiation operations required by other virtual work techniques, producing a formulation method based solely on the geometry of the system of rigid bodies. The procedure is applied to the derivation of the dynamic equations for the first three links of the Stanford manipulator.

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Dynamics of Serial Multibody Systems Using the Decoupled Natural Orthogonal Complement Matrices

Journal of Applied Mechanics ◽

10.1115/1.2791809 ◽

1999 ◽

Vol 66 (4) ◽

pp. 986-996 ◽

Cited By ~ 62

Author(s):

S. K. Saha

Keyword(s):

Multibody Systems ◽

Degrees Of Freedom ◽

Inverse Dynamics ◽

Equations Of Motion ◽

Kinematic Chain ◽

Rigid Bodies ◽

Dynamic Equations ◽

Orthogonal Complement ◽

Forward Dynamics ◽

Velocity Constraints

Constrained dynamic equations of motion of serial multibody systems consisting of rigid bodies in a serial kinematic chain are derived in this paper. First, the Newton-Euler equations of motion of the decoupled rigid bodies of the system at hand are written. Then, with the aid of the decoupled natural orthogonal complement (DeNOC) matrices associated with the velocity constraints of the connecting bodies, the Euler-Lagrange independent equations of motion are derived. The De NOC is essentially the decoupled form of the natural orthogonal complement (NOC) matrix, introduced elsewhere. Whereas the use of the latter provides recursive order n—n being the degrees-of-freedom of the system at hand—inverse dynamics and order n3 forward dynamics algorithms, respectively, the former leads to recursive order n algorithms for both the cases. The order n algorithms are desirable not only for their computational efficiency but also for their numerical stability, particularly, in forward dynamics and simulation, where the system’s accelerations are solved from the dynamic equations of motion and subsequently integrated numerically. The algorithms are illustrated with a three-link three-degrees-of-freedom planar manipulator and a six-degrees-of-freedom Stanford arm.

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Discrete Adjoint Method for the Sensitivity Analysis of Flexible Multibody Systems

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4041237 ◽

2019 ◽

Vol 14 (2) ◽

Cited By ~ 2

Author(s):

Alfonso Callejo ◽

Valentin Sonneville ◽

Olivier A. Bauchau

Keyword(s):

Sensitivity Analysis ◽

Multibody Systems ◽

Adjoint Method ◽

Mechanical Systems ◽

Rigid Bodies ◽

Flexible Multibody Systems ◽

Design Parameters ◽

Integration Scheme ◽

Discrete Version ◽

Flexible Multibody

The gradient-based design optimization of mechanical systems requires robust and efficient sensitivity analysis tools. The adjoint method is regarded as the most efficient semi-analytical method to evaluate sensitivity derivatives for problems involving numerous design parameters and relatively few objective functions. This paper presents a discrete version of the adjoint method based on the generalized-alpha time integration scheme, which is applied to the dynamic simulation of flexible multibody systems. Rather than using an ad hoc backward integration solver, the proposed approach leads to a straightforward algebraic procedure that provides design sensitivities evaluated to machine accuracy. The approach is based on an intrinsic representation of motion that does not require a global parameterization of rotation. Design parameters associated with rigid bodies, kinematic joints, and beam sectional properties are considered. Rigid and flexible mechanical systems are investigated to validate the proposed approach and demonstrate its accuracy, efficiency, and robustness.

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Force Equilibrium Approach for Linearization of Constrained Mechanical System Dynamics

Journal of Mechanical Design ◽

10.1115/1.1541631 ◽

2003 ◽

Vol 125 (1) ◽

pp. 143-149 ◽

Cited By ~ 15

Author(s):

Ju Seok Kang ◽

Sangwoo Bae ◽

Jang Moo Lee ◽

Tae Oh Tak

Keyword(s):

Mechanical Systems ◽

Nonlinear Dynamic Analysis ◽

Differential Algebraic Equations ◽

Dynamic Equations ◽

Small Displacement ◽

Algebraic Equations ◽

Small Motion ◽

Constrained Mechanical Systems ◽

External Forces ◽

Force Equilibrium

The purpose of this study is to derive a linearized form of dynamic equations for constrained mechanical systems. The governing equations for constrained mechanical systems are generally expressed in terms of Differential-Algebraic Equations (DAEs). Conventional methods of linearization are based on the perturbation of the nonlinear DAE, where small amounts of perturbations are taken to guarantee linear characteristics of the equations. On the other hand, the proposed linearized dynamic equations are derived directly from a force equilibrium condition, not from the DAEs, with small motion assumption. This approach is straightforward and simple compared to conventional perturbation methods, and can be applicable to any constrained mechanical systems that undergo small displacement under external forces. The modeling procedure and formulation of linearized dynamic equations are demonstrated by the example of a vehicle suspension system, a typical constrained multibody system. The solution is validated by comparison with conventional nonlinear dynamic analysis and modal test results.

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