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.

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

2007 ◽  
Author(s):  
Naresh Khude ◽  
Laurent O. Jay ◽  
Andrei Schaffer ◽  
Dan Negrut
Keyword(s):  
Numerical Integration ◽  
Multibody Dynamics ◽  
Numerical Experiments ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Dynamics Simulation ◽  
Stability And Convergence ◽  
Integration Formulas ◽  
Theoretical Results ◽  
Low Order

The premise of this work is that real-life mechanical systems limit the use of high order integration formulas due to the presence in the associated models of friction and contact/impact elements. In such cases producing a numerical solution necessarily relies on low order integration formulas. The resulting algorithms are generally robust and expeditious; their major drawback remains that they typically require small integration step-sizes in order to meet a user prescribed accuracy. This paper looks at three low order numerical integration formulas: Newmark, HHT, and BDF of order two. These formulas are used in two contexts. A first set of three methods is obtained by considering a direct index-3 discretization approach that solves for the equations of motion and imposes the position kinematic constraints. The second batch of three additional methods 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. The first objective of this paper is to review the theoretical results available in the literature regarding the stability and convergence properties of these low order methods when applied in the context of multibody dynamics simulation. When no theoretical results are available, numerical experiments are carried out to gauge order behavior. The second objective is to perform a set of numerical experiments to compare these six methods in terms of several metrics: (a) efficiency, (b) velocity constraint drift, and (c) energy preservation. A set of simple mechanical systems is used for this purpose: a double pendulum, a slider crank with rigid bodies, and a slider crank with a flexible body represented in the floating frame of reference formulation.

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 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.

Implicit Integration of the Equations of Multibody Dynamics in Descriptor Form

1997 ◽  
Author(s):  
Edward J. Haug ◽  
Mirela Iancu ◽  
Dan Negrut
Keyword(s):  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Kinetic Equations ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Integration Formulas ◽  
Multibody Dynamic ◽  
Constrained Equations

Abstract An implicit numerical integration approach, based on generalized coordinate partitioning of the descriptor form of the differential-algebraic equations of motion of multibody dynamics, is presented. This approach is illustrated for simulation of stiff mechanical systems using the well known Newmark integration method from structural dynamics. Second order Newmark integration formulas are used to define independent generalized coordinates and their first time derivative as functions of independent accelerations. The latter are determined as the solution of discretized equations obtained using the descriptor form of the equations of motion. Dependent variables in the formulation, including Lagrange multipliers, are determined to satisfy all the kinematic and kinetic equations of multibody dynamics. The approach is illustrated by solving the constrained equations of motion for mechanical systems that exhibit stiff behavior. Results show that the approach is robust and has the capability to integrate differential-algebraic equations of motion for stiff multibody dynamic systems.

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.

scholarly journals Dynamic Modeling and Stability Analysis of Mechanical Systems with Time-Varying Topologies

1993 ◽  
Vol 115 (4) ◽  
pp. 808-816 ◽  
Author(s):  
Yu Wang
Keyword(s):  
Stability Analysis ◽  
Dynamic Modeling ◽  
Local Stability ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Time Varying ◽  
Kinematic Constraints ◽  
Discrete Equations ◽  
Local Stability Analysis ◽  
Multiple Collisions

A model is developed for analyzing mechanical systems with a pair of bodies with topological changes in their kinematic constraints. It is built upon the concept of a Poincare´ map rather than following the traditional methods of differential equations. The model provides a set of well-defined and naturally-discrete equations of motion and is capable of giving physical insights of dynamic characteristics of deadbeat convergence of multiple collisions and periodic or chaotic responses. The development of a dynamic model and a local stability analysis are presented.

Control of Uncertain Nonlinear Multibody Mechanical Systems

2013 ◽  
Vol 81 (4) ◽  
Author(s):  
Firdaus E. Udwadia ◽  
Thanapat Wanichanon
Keyword(s):  
Closed Form ◽  
Error Bounds ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Real Life ◽  
Reference Trajectory ◽  
Control Force ◽  
Control Laws ◽  
Nominal System ◽  
The Face

Descriptions of real-life complex multibody mechanical systems are usually uncertain. Two sources of uncertainty are considered in this paper: uncertainties in the knowledge of the physical system and uncertainties in the “given” forces applied to the system. Both types of uncertainty are assumed to be time varying and unknown, yet bounded. In the face of such uncertainties, what is available in hand is therefore just the so-called “nominal system,” which is our best assessment and description of the actual real-life situation. A closed-form equation of motion for a general dynamical system that contains a control force is developed. When applied to a real-life uncertain multibody system, it causes the system to track a desired reference trajectory that is prespecified for the nominal system to follow. Thus, the real-life system's motion is required to coincide within prespecified error bounds and mimic the motion desired of the nominal system. Uncertainty is handled by a controller based on a generalization of the concept of a sliding surface, which permits the use of a large class of control laws that can be adapted to specific real-life practical limitations on the control force. A set of closed-form equations of motion is obtained for nonlinear, nonautonomous, uncertain, multibody systems that can track a desired reference trajectory that the nominal system is required to follow within prespecified error bounds and thereby satisfy the constraints placed on the nominal system. An example of a simple mechanical system demonstrates the efficacy and ease of implementation of the control methodology.

A unified tensor approach to the analysis of mechanical systems

1997 ◽  
Vol 211 (4) ◽  
pp. 313-322 ◽  
Author(s):  
A A Fogarasy ◽  
M R Smith
Keyword(s):  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Equation Of Motion ◽  
Newtonian Mechanics ◽  
Kinematic Constraint ◽  
Lagrangian Equation ◽  
Generalized Coordinates ◽  
Lagrangian Equations

It is shown in this paper that all methods of dynamic analysis of mechanisms used in practice can be derived from an invariant formed from the Lagrangian equation of motion. For the dynamic analysis of mechanisms subjected to kinematic constraint conditions, the Lagrangian equations of motion are far more suitable than the Newtonian approach. Since the Lagrangian equations are tensor equations, they are valid irrespective of what kind of generalized coordinates are used. This is not so, however, when the Newtonian approach is used. It is demonstrated by a simple example that a careless use of Newtonian mechanics can lead to erroneous results.

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

1991 ◽  
Author(s):  
Inhwan Han ◽  
Brian J. Gilmore ◽  
Mandara 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

Abstract 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 exist, 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.

Effect of Material and Geometric Parameters on the Steady-State Belt Stresses and Belt Slip for Flat Belt-Drives

2015 ◽  
Author(s):  
Cagkan Yildiz ◽  
Tamer M. Wasfy ◽  
Hatem M. Wasfy ◽  
Jeanne M. Peters
Keyword(s):  
Steady State ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Flexible Multibody Dynamics ◽  
Friction Model ◽  
Rigid Bodies ◽  
Geometric Parameters ◽  
Rubber Matrix ◽  
Axial Stiffness ◽  
Belt Drive

In order to accurately predict the fatigue life and wear life of a belt, the various stresses that the belt is subjected to and the belt slip over the pulleys must be accurately calculated. In this paper, the effect of material and geometric parameters on the steady-state stresses (including normal, tangential and axial stresses), average belt slip for a flat belt, and belt-drive energy efficiency is studied using a high-fidelity flexible multibody dynamics model of the belt-drive. The belt’s rubber matrix is modeled using three-dimensional brick elements and the belt’s reinforcements are modeled using one dimensional truss elements. Friction between the belt and the pulleys is modeled using an asperity-based Coulomb friction model. The pulleys are modeled as cylindrical rigid bodies. The equations of motion are integrated using a time-accurate explicit solution procedure. The material parameters studied are the belt-pulley friction coefficient and the belt axial stiffness and damping. The geometric parameters studied are the belt thickness and the pulleys’ centers distance.

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