Systematic Construction of the Equations of Motion for Multibody Systems Containing Closed Kinematic Loops

1989 ◽  
Author(s):  
P. E. Nikravesh ◽  
G. Gim
Keyword(s):  
Equations Of Motion ◽  
Transformation Matrix ◽  
Velocity Transformation ◽  
Constraint Equations ◽  
Advantages And Disadvantages ◽  
Lagrange Multiplier Technique ◽  
Joint Coordinates ◽  
Minimum Number ◽  
Kinematic Loops ◽  
Equations Of Motions

Abstract This paper presents a systematic method for deriving the minimum number of equations of motion for multibody system containing closed kinematic loops. A set of joint or natural coordinates is used to describe the configuration of the system. The constraint equations associated with the closed kinematic loops are found systematically in terms of the joint coordinates. These constraints and their corresponding elements are constructed from known block matrices representing different kinematic joints. The Jacobian matrix associated with these constraints is further used to find a velocity transformation matrix. The equations of motions are initially written in terms of the dependent joint coordinates using the Lagrange multiplier technique. Then the velocity transformation matrix is used to derive a minimum number of equations of motion in terms of a set of independent joint coordinates. An illustrative example and numerical results are presented, and the advantages and disadvantages of the method are discussed.

Systematic Construction of the Equations of Motion for Multibody Systems Containing Closed Kinematic Loops

1993 ◽  
Vol 115 (1) ◽  
pp. 143-149 ◽  
Author(s):  
P. E. Nikravesh ◽  
Gwanghun Gim
Keyword(s):  
Equations Of Motion ◽  
Transformation Matrix ◽  
Velocity Transformation ◽  
Constraint Equations ◽  
Advantages And Disadvantages ◽  
Lagrange Multiplier Technique ◽  
Joint Coordinates ◽  
Minimum Number ◽  
Kinematic Loops ◽  
Equations Of Motions

This papers presents a systematic method for deriving the minimum number of equations of motion for multibody system containing closed kinematic loops. A set of joint or natural coordinates is used to describe the configuration of the system. The constraint equations associated with the closed kinematic loops are found systematically in terms of the joint coordinates. These constraints and their corresponding elements are constructed from known block matrices representing different kinematic joints. The Jacobian matrix associated with these constraints is further used to find a velocity transformation matrix. The equations of motions are initially written in terms of the dependent joint coordinates using the Lagrange multiplier technique. Then the velocity transformation matrix is used to derive a minimum number of equations of motion in terms of a set of independent joint coordinates. An illustrative example and numerical results are presented, and the advantages and disadvantages of the method are discussed.

A General and Efficient Method for Dynamic Analysis of Mechanical Systems Using Velocity Transformations

1986 ◽  
Vol 108 (2) ◽  
pp. 176-182 ◽  
Author(s):  
S. S. Kim ◽  
M. J. Vanderploeg
Keyword(s):  
Equations Of Motion ◽  
General Purpose ◽  
Rigid Bodies ◽  
Transformation Matrix ◽  
Cartesian Coordinates ◽  
Velocity Transformation ◽  
Joint Coordinates ◽  
Relative Coordinates ◽  
General Purpose Computer ◽  
New Formulation

This paper presents a new formulation for the equations of motion of interconnected rigid bodies. This formulation initially uses Cartesian coordinates to define the position of the system, the kinematic joints between bodies, and forcing functions on and between bodies. This makes initial system definition straightforward. The equations of motion are then derived in terms of relative joint coordinates through the use of a velocity transformation matrix. The velocity transformation matrix relates relative coordinates to Cartesian coordinates. It is derived using kinematic relationships for each joint type and graph theory for identifying the system topology. By using relative coordinates, the equations of motion are efficiently integrated. Use of both Cartesian and relative coordinates produces an efficient set of equations without loss of generality. The algorithm just described is implemented in a general purpose computer program. Examples are used to demonstrate the generality and efficiency of the algorithms.

A Computational Method for the Dynamic Analysis of Planar Linkages With Applications

2001 ◽  
Author(s):  
Hazem A. Attia
Keyword(s):  
Closed Loop ◽  
Equations Of Motion ◽  
Open Loop ◽  
Rigid Bodies ◽  
Transformation Matrix ◽  
Computational Method ◽  
Constraint Forces ◽  
Constrained System ◽  
Velocity Transformation ◽  
Planar Linkages

Abstract This paper presents a computational method for generating the equations of motion of planar linkages consisting of interconnected rigid bodies. The formulation uses initially the rectangular Cartesian coordinates of an equivalent constrained system of particles to define the configuration of the mechanism. This results in a simple and straightforward procedure for generating the equations of motion. The equations of motion are then derived in terms of relative joint variables through the use of a velocity transformation matrix. The velocity transformation matrix relates the relative joint velocities to the Cartesian velocities. For the open loop case, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For the closed loop case, suitable joints should be cut and few cut-joints constraint equations should be included for each closed loop. Examples are used to demonstrate the generality and efficiency of the proposed method.

Multibody Techniques in Offshore Dynamics With an Application to a Crane Ship

2004 ◽  
Author(s):  
Joost den Haan ◽  
Hermione van Zutphen
Keyword(s):  
Efficient Method ◽  
Time Domain ◽  
Equations Of Motion ◽  
Response Prediction ◽  
Transformation Method ◽  
Computationally Efficient ◽  
Velocity Transformation ◽  
Offshore Engineering ◽  
Initial Results ◽  
Equations Of Motions

In this paper multibody dynamics techniques are reviewed for their applicability to offshore engineering, for example to cases like installation, decommissioning, salvage, offloading, pipe laying and dredging. The basis of every simulation is formed by the equations of motion. Multibody techniques have the advantage over analytical methods that they are generic: the equations of motions are derived by a computer algorithm rather than by hand. This allows for a large variety of systems to be analyzed with a single program. The review shows that the topology dependent semi-recursive velocity transformation method (VTM) provides a computationally efficient method for spatial time domain simulations. As an application a crane vessel payload-pendulation problem is studied. The initial results of this study suggest that multibody techniques in offshore dynamics provide better insight and possibly higher accuracy in response prediction than other methods.

A Recursive Householder Transformation for Complex Dynamical Systems With Constraints

1988 ◽  
Vol 55 (3) ◽  
pp. 729-734 ◽  
Author(s):  
F. M. L. Amirouche ◽  
Tongyi Jia ◽  
Sitki K. Ider
Keyword(s):  
Dynamical Systems ◽  
General Solution ◽  
Equations Of Motion ◽  
New Method ◽  
Systems Dynamics ◽  
Complex Dynamical Systems ◽  
Governing Equations ◽  
Householder Transformation ◽  
Constraint Equations ◽  
Computer Automation

A new method is presented by which equations of motion of complex dynamical systems are reduced when subjected to some constraints. The method developed is used when the governing equations are derived using Kane’s equations with undetermined multipliers. The solution vectors of the constraint equations are determined utilizing the recursive Householder transformation to obtain a Pseudo-Uptriangular matrix. The most general solution in terms of new independent coordinates is then formulated. Methods previously used for handling such systems are discussed and the new method advantages are illustrated. The procedures developed are suitable for computer automation and especially in developing generic programs to study a large class of systems dynamics such as robotics, biosystems, and complex mechanisms.

Analysis of the Nonlinear Dynamics of a Railway Vehicle Wheelset

1973 ◽  
Vol 95 (1) ◽  
pp. 28-35 ◽  
Author(s):  
E. Harry Law ◽  
R. S. Brand
Keyword(s):  
Lateral Displacement ◽  
Equations Of Motion ◽  
Vertical Displacement ◽  
Linear Analysis ◽  
Design Parameters ◽  
Railway Vehicle ◽  
Nonlinear Variation ◽  
Constraint Equations ◽  
The Stability ◽  
Nonlinear Equations Of Motion

The nonlinear equations of motion for a railway vehicle wheelset having curved wheel profiles and wheel-flange/rail contact are presented. The dependence of axle roll and vertical displacement on lateral displacement and yaw is formulated by two holonomic constraint equations. The method of Krylov-Bogoliubov is used to derive expressions for the amplitudes of stationary oscillations. A perturbation analysis is then used to derive conditions for the stability characteristics of the stationary oscillations. The expressions for the amplitude and the stability conditions are shown to have a simple geometrical interpretation which facilitates the evaluation of the effects of design parameters on the motion. It is shown that flange clearance and the nonlinear variation of axle roll with lateral displacement significantly influence the motion of the wheelset. Stationary oscillations may occur at forward speeds both below and above the critical speed at which a linear analysis predicts the onset of instability.

Global dynamical time for binary point-like particle systems

2020 ◽  
Vol 17 (09) ◽  
pp. 2050131
Author(s):  
Osvaldo M. Moreschi
Keyword(s):  
Equations Of Motion ◽  
Minkowski Spacetime ◽  
Particle Systems ◽  
Lorentzian Geometry ◽  
Field Equations ◽  
Retarded Time ◽  
Dynamical Time ◽  
The Difference ◽  
Multiple Particle ◽  
Equations Of Motions

A geometrical construction for a global dynamical time for binary point-like particle systems, modeled by relativistic equations of motions, is presented. Thus, we provide a convenient tool for the calculation of the dynamics of recent models for the dynamics of black holes that use individual proper times. The construction is naturally based on the local Lorentzian geometry of the spacetime considered. Although in this presentation we are dealing with the Minkowskian spacetime, the construction is useful for gravitational models that have as a seed Minkowski spacetime. We present the discussion for a binary system, but the construction is obviously generalizable to multiple particle systems. The calculations are organized in terms of the order of the corresponding relativistic forces. In particular, we improve on the Darwin and Landau–Lifshitz approaches, for the case of electromagnetic systems. We discuss the possibility of a Lagrangian treatment of the retarded effects, depending on the nature of the relativistic forces. The higher-order contractions are based on a Runge–Kutta type procedure, which is used to calculate the quantities at the required retarded time, by increasing evaluations of the forces at intermediate times. We emphasize the difference between approximation orders in field equations and approximation orders in retarded effects in the equations of motion. We show this difference by applying our construction to the binary electromagnetic case.

Invariant Modeling of Mechanical Systems in Terms of Quasi-Velocities and Quasi-Forces

2009 ◽  
Author(s):  
Farhad Aghili
Keyword(s):  
Motion Control ◽  
Equations Of Motion ◽  
Control Action ◽  
Constraint Equations ◽  
Weighting Matrix ◽  
Rotational Components ◽  
Tracking Force ◽  
Invariant Formulation ◽  
Invariance Problem ◽  
Multi Body

A gauge-invariant formulation for deriving the dynamic equations of constrained multi-body systems (MBS) in terms of (reduced) quasi–velocities is presented. This formulation does not require any weighting matrix to deal with the gauge-invariance problem when both translational and rotational components are involved in the generalized coordinates or in the constraint equations. Moreover, in this formulation the equations of motion are decoupled from those of constrained force and each system has its own independent input. This allows the possibility to develop a simple force control action that is totally independent from the motion control action facilitating a hybrid force/motion control. Tracking force/motion control of constrained multi-body systems based on a combination of feedbacks on the vectors of the quasi–velocities and the configuration variables are presented.

Dynamics of Constrained Multibody Systems

1984 ◽  
Vol 51 (4) ◽  
pp. 899-903 ◽  
Author(s):  
J. W. Kamman ◽  
R. L. Huston
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Governing Equations ◽  
Dynamical Equations ◽  
Closed Loops ◽  
Constrained Multibody Systems ◽  
Constraint Equations ◽  
Cable Dynamics ◽  
Hanging Chain ◽  
A Chain

A new automated procedure for obtaining and solving the governing equations of motion of constrained multibody systems is presented. The procedure is applicable when the constraints are either (a) geometrical (for example, “closed-loops”) or (b) kinematical (for example, specified motion). The procedure is based on a “zero eigenvalues theorem,” which provides an “orthogonal complement” array which in turn is used to contract the dynamical equations. This contraction, together with the constraint equations, forms a consistent set of governing equations. An advantage of this formulation is that constraining forces are automatically eliminated from the analysis. The method is applied with Kane’s equations—an especially convenient set of dynamical equations for multibody systems. Examples of a constrained hanging chain and a chain whose end has a prescribed motion are presented. Applications in robotics, cable dynamics, and biomechanics are suggested.

A Recursive Formulation for Real-Time Dynamic Simulation of Mechanical Systems

1991 ◽  
Vol 113 (2) ◽  
pp. 158-166 ◽  
Author(s):  
Dae-Sung Bae ◽  
Ruoh-Shih Hwang ◽  
Edward J. Haug
Keyword(s):  
Real Time ◽  
Dynamic Simulation ◽  
Recursive Algorithm ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Time Dynamic ◽  
Time Simulation ◽  
Kinematic Loops ◽  
Multi Body ◽  
Recursive Equations

A new recursive algorithm for real-time dynamic simulation of mechanical systems with closed kinematic loops is presented. State vector kinematic relations that represent translational and rotational motion are defined to simplify the formulation and to relieve computational burden. Recursive equations of motion are first derived for a single loop multi-body system. Faster than real-time performance is demonstrated for a closed loop manipulator, using an Alliant FX/8 multiprocessor. The algorithm is extended to multi-loop, multi-body systems for parallel processing real-time simulation in companion papers [1, 2] where performance of the algorithm on a shared memory multi-processor is compared with that achieved with other dynamic simulation algorithms.

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