Simulation of Planar Dynamic Mechanical Systems With Changing Topologies—Part 1: Characterization and Prediction of the Kinematic Constraint Changes

1991 ◽  
Vol 113 (1) ◽  
pp. 70-76 ◽  
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

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 1. 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 unforessen changes in topology. Part 2 presents the implementation of the characterizations into a simulation strategy and presents examples.

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

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.

Simulation of Planar Dynamic Mechanical Systems With Changing Topologies: Part II — Implementation Strategy and Simulation Results for Example Dynamic Systems

1987 ◽  
Author(s):  
B. J. Gilmore ◽  
R. J. Cipra
Keyword(s):  
Incidence Matrix ◽  
Impact Analysis ◽  
Equations Of Motion ◽  
Constraint Equation ◽  
Rigid Bodies ◽  
General Point ◽  
Kinematic Constraint ◽  
Simulation Strategy ◽  
Simulation Results ◽  
System Topology

Abstract Part II presents a technique which uses the general point to line kinematic constraint equation presented in Part I coupled with an incidence matrix to automatically formulate the kinematic constraint equations between contacting rigid bodies and solve the resulting dynamic equations of motion. The automatic determination of system connectivity coupled with impulse-momentum theory enables the strategy to handle impact between rigid bodies in line contact or between rigid bodies in topologically complex systems. The characterizations of the changes in the kinematic constraints, as described by Part I, are combined with the incidence matrix and impact analysis to form an automatic dynamic simulation strategy. The strategy automatically predicts, detects, and determines the changes in the system topology and then reformulates the equations of motion without the user specifying the kinematic constraint changes and resulting system topology. The simulation results of two example systems are presented. The examples illustrate the effect the boundaries have on a system’s motion and show that the only connectivity data required by the strategy is the initial system topology.

scholarly journals A Discussion of Low-Order Numerical Integration Formulas for Rigid and Flexible Multibody Dynamics

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Dan Negrut ◽  
Laurent O. Jay ◽  
Naresh Khude
Keyword(s):  
Numerical Experiments ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Real Life ◽  
Flexible Multibody Dynamics ◽  
Kinematic Constraint ◽  
Kinematic Constraints ◽  
Integration Formulas ◽  
Impact Phenomena ◽  
Low Order

The premise of this work is that the presence of high stiffness and/or frictional contact/impact phenomena limits the effective use of high order integration formulas when numerically investigating the time evolution of real-life mechanical systems. Producing a numerical solution relies most often on low-order integration formulas of which the paper investigates three alternatives: Newmark, HHT, and order 2 BDFs. Using these methods, a first set of three algorithms is obtained as the outcome of a direct index-3 discretization approach that considers the equations of motion of a multibody system along with the position kinematic constraints. The second batch of three algorithms draws on the HHT and BDF integration formulas and considers, in addition to the equations of motion, both the position and velocity kinematic constraint equations. Numerical experiments are carried out to compare the algorithms in terms of several metrics: (a) order of convergence, (b) energy preservation, (c) velocity kinematic constraint drift, and (d) efficiency. The numerical experiments draw on a set of three mechanical systems: a rigid slider-crank, a slider-crank with a flexible body, and a seven body mechanism. The algorithms investigated show good performance in relation to the asymptotic behavior of the integration error and, with one exception, result in comparable CPU simulation times with a small premium being paid for enforcing the velocity kinematic constraints.

Simulation of Planar Dynamic Mechanical Systems With Changing Topologies—Part 2: Implementation Strategy and Simulation Results for Example Dynamic Systems

1991 ◽  
Vol 113 (1) ◽  
pp. 77-83 ◽  
Author(s):  
B. J. Gilmore ◽  
R. J. Cipra
Keyword(s):  
Incidence Matrix ◽  
Impact Analysis ◽  
Equations Of Motion ◽  
Constraint Equation ◽  
Rigid Bodies ◽  
General Point ◽  
Kinematic Constraint ◽  
Simulation Strategy ◽  
Simulation Results ◽  
System Topology

Part 2 presents a technique which uses the general point to line kinematic constraint equation presented in part 1 coupled with an incidence matrix to automatically reformulate the kinematic constraint equations between contacting or separating rigid bodies and solve the resulting dynamic equations of motion. The automatic determination of system connectivity coupled with impulse-momentum theory enables the strategy to handle impact between rigid bodies in line contact or between rigid bodies in topologically complex systems. The characterizations of the changes in the kinematic constraints, as described by Part 1, are combined with the incidence matrix and impact analysis to form an automatic dynamic simulation strategy. The strategy automatically predicts, detects, and determines the changes in the system topology and then reformulates the equations of motion without the user specifying the kinematic constraint changes and resulting system topology. The simulation results of two example systems are presented. The examples illustrate the effect the boundaries have on a system’s motion and show that the only connectivity data required by the strategy is the initial system topology.

Cayley kinematics and the Cayley form of dynamic equations

2005 ◽  
Vol 461 (2055) ◽  
pp. 761-781 ◽  
Author(s):  
Andrew J. Sinclair ◽  
John E. Hurtado
Keyword(s):  
Euler Equations ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
Matrix Form ◽  
Dynamic Equations ◽  
Cayley Transform ◽  
Higher Dimensional ◽  
The Matrix ◽  
General Motion

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.

A Unified Framework for Dynamics and Lyapunov Stability of Holonomically Constrained Rigid Bodies

2005 ◽  
Vol 9 (4) ◽  
pp. 387-394 ◽  
Author(s):  
Khoder Melhem ◽  
◽  
Zhaoheng Liu ◽  
Antonio Loría ◽  
◽  
...  
Keyword(s):  
System Dynamics ◽  
Lyapunov Stability ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Minimal Realization ◽  
State Variables ◽  
Unified Framework ◽  
Constrained System ◽  
And Control

A new dynamic model for interconnected rigid bodies is proposed here. The model formulation makes it possible to treat any physical system with finite number of degrees of freedom in a unified framework. This new model is a nonminimal realization of the system dynamics since it contains more state variables than is needed. A useful discussion shows how the dimension of the state of this model can be reduced by eliminating the redundancy in the equations of motion, thus obtaining the minimal realization of the system dynamics. With this formulation, we can for the first time explicitly determine the equations of the constraints between the elements of the mechanical system corresponding to the interconnected rigid bodies in question. One of the advantages coming with this model is that we can use it to demonstrate that Lyapunov stability and control structure for the constrained system can be deducted by projection in the submanifold of movement from appropriate Lyapunov stability and stabilizing control of the corresponding unconstrained system. This procedure is tested by some simulations using the model of two-link planar robot.

Application of the Vector-Network Method to Constrained Mechanical Systems

1986 ◽  
Vol 108 (4) ◽  
pp. 471-480 ◽  
Author(s):  
Tai-Wai Li ◽  
Gordon C. Andrews
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Computer Applications ◽  
Kinematic Chains ◽  
Constrained Mechanical Systems ◽  
3 Dimensional ◽  
Methodical Approach ◽  
Dynamic Formulation ◽  
Reaction Forces ◽  
New Formulation

The vector-network technique is a methodical approach to formulating equations of motion for unconstrained dynamic systems, utilizing concepts from graph theory and vectorial mechanics; it is ideally suited to computer applications. In this paper, the vector-network theory is significantly improved and extended to include constrained mechanical systems with both open and closed kinematic chains. A new formulation procedure is developed in which new kinematic constraint elements are incorporated. The formulation is based on a modified tree/cotree classification, which deviates significantly from previous work, and reduces the number of equations of motions to be solved. The dynamic equations of motion are derived, with generalized accelerations and a subset of the reaction forces as solution variables, and a general kinematic analysis procedure is also developed, similar to that of the dynamic formulation. Although this paper restricts most discussions to two-dimensional (planar) systems, the new method is equally applicable to 3-dimensional systems.

Dynamics of Multirigid-Body Systems

1978 ◽  
Vol 45 (4) ◽  
pp. 889-894 ◽  
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.

Dynamic Analysis of Multi-Rigid-Body Systems

1974 ◽  
Vol 96 (3) ◽  
pp. 886-892 ◽  
Author(s):  
V. K. Gupta
Keyword(s):  
Digital Simulation ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Mechanism Analysis ◽  
Spatial Mechanism ◽  
Higher Derivatives ◽  
Reaction Forces ◽  
Representative Problem ◽  
Classical Vector ◽  
Vector Approach

A method is presented for formulating and solving the Newton-Euler equations of motion of a system of interconnected rigid bodies. The digital simulation may involve numerical integration of the kinematic equations as well as the dynamic equations. The reaction forces and torques resulting from rigid constraints imposed at the connecting joints are also determined. The derivation of kinematic expressions for first and higher derivatives is demonstrated based on direct differentiation of the rotation matrix in the spirit of the classical vector approach. A representative problem in spatial mechanism analysis is solved and illustrated with numerical results.

The Incorporation of Arc Boundaries and Stick/Slip Friction in a Rule-Based Simulation Algorithm for Dynamic Mechanical Systems with Changing Topologies

1993 ◽  
Vol 115 (3) ◽  
pp. 423-434 ◽  
Author(s):  
Inhwan Han ◽  
B. J. Gilmore ◽  
M. M. Ogot
Keyword(s):  
High Speed ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Coefficient Matrix ◽  
Design Tool ◽  
Simulation Algorithm ◽  
Kinematic Constraint ◽  
Stick Slip ◽  
Rule Based ◽  
Dynamic Mechanical

Many dynamic mechanical systems, such as parts-feeders and percussive power tools, are described by equations of motion which are discontinuous. The discontinuities result from kinematic constraint changes which are difficult to foresee, especially in presence of impact and friction. A simulation algorithm for these types of systems must be able to algorithmically predict and detect the kinematic constraint changes without any prior knowledge of the system’s motion. This paper presents a rule-based approach to the prediction and detection of kinematic constraint changes between bodies with arc and line boundaries. A new type of constraint change, constraint exchange, is characterized. When arc contact exists, stick/slip friction is the difference between pure rolling and rolling with slip. Therefore, stick/slip friction is included in the algorithm. A force constraint is applied to the equations of motion when additional kinematic constraints due to friction would render the coefficient matrix singular. The efficacy of the rule-based simulation algorithm as a design tool is demonstrated through the design and experimental validation of a parts-feeder. The parts-feeder design is validated through two means: (1) a frame-by frame comparison of simulation results with the part motion recorded by high speed video and (2) actual testing.

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