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.

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.

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.

scholarly journals The Motion of Halley's Comet from 837 to 1910

1972 ◽  
Vol 45 ◽  
pp. 155-155
Author(s):  
J. L. Brady
Keyword(s):  
Numerical Integration ◽  
Numerical Experiments ◽  
Equations Of Motion ◽  
Secular Term ◽  
Run Off ◽  
Halley's Comet ◽  
Perihelion Passage ◽  
Made In

Numerical experiments have been made in an attempt to remove the residuals of P/Halley and link the seven apparitions from 1456 to 1910. All efforts to link more than two apparitions using Newtonian equations have invariably failed. However, by the addition of a secular term to the equations of motion, the four apparitions from 1910 back to 1682 can be linked by a numerical integration which represents the observations to contemporary accuracy. When this integration is continued, the apparitions of 1607, 1531, and 1456 show residuals of less than one day in the time of perihelion passage. Prior to 1456 the residuals begin to run off but, with the exception of 1222 and 1066, the apparitions back to 837 show residuals no greater than four days in the time of perihelion passage. The residuals of 20 days in 1222 and 7 days in 1066 appear anomalous but can be made reasonable if the Chinese records are adopted in preference to the European records.

Stabilization Method for Numerical Integration of Multibody Mechanical Systems

1998 ◽  
Vol 120 (4) ◽  
pp. 565-572 ◽  
Author(s):  
Shih-Tin Lin ◽  
Ming-Chong Hong
Keyword(s):  
Numerical Integration ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Constraint Equation ◽  
Differential Algebraic Equations ◽  
Integration Step ◽  
Algebraic Equations ◽  
Stabilization Method ◽  
Step Size ◽  
Integration Methods

The object of this study is to solve the stability problem for the numerical integration of constrained multibody mechanical systems. The dynamic equations of motion of the constrained multibody mechanical system are mixed differential-algebraic equations (DAE). In applying numerical integration methods to this equation, constrained equations and their first and second derivatives must be satisfied simultaneously. That is, the generalized coordinates and their derivatives are dependent. Direct integration methods do not consider this dependency and constraint violation occurs. To solve this problem, Baumgarte proposed a constraint stabilization method in which a position and velocity terms were added in the second derivative of the constraint equation. The disadvantage of this method is that there is no reliable method for selecting the coefficients of the position and velocity terms. Improper selection of these coefficients can lead to erroneous results. In this study, stability analysis methods in digital control theory are used to solve this problem. Correct choice of the coefficients for the Adams method are found for both fixed and variable integration step size.

scholarly journals An Intuitive Introduction to Fractional and Rough Volatilities

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 994
Author(s):  
Elisa Alòs ◽  
Jorge A. León
Keyword(s):  
Malliavin Calculus ◽  
Numerical Experiments ◽  
Implied Volatility ◽  
Natural Approach ◽  
Volatility Models ◽  
Implied Volatility Surface ◽  
Short Memory ◽  
White Type ◽  
Volatility Surface ◽  
Theoretical Results

Here, we review some results of fractional volatility models, where the volatility is driven by fractional Brownian motion (fBm). In these models, the future average volatility is not a process adapted to the underlying filtration, and fBm is not a semimartingale in general. So, we cannot use the classical Itô’s calculus to explain how the memory properties of fBm allow us to describe some empirical findings of the implied volatility surface through Hull and White type formulas. Thus, Malliavin calculus provides a natural approach to deal with the implied volatility without assuming any particular structure of the volatility. The aim of this paper is to provides the basic tools of Malliavin calculus for the study of fractional volatility models. That is, we explain how the long and short memory of fBm improves the description of the implied volatility. In particular, we consider in detail a model that combines the long and short memory properties of fBm as an example of the approach introduced in this paper. The theoretical results are tested with numerical experiments.

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