On an Implementation of the Hilber-Hughes-Taylor Method in the Context of Index 3 Differential-Algebraic Equations of Multibody Dynamics (DETC2005-85096)

2006 ◽  
Vol 2 (1) ◽  
pp. 73-85 ◽  
Author(s):  
Dan Negrut ◽  
Rajiv Rampalli ◽  
Gisli Ottarsson ◽  
Anthony Sajdak
Keyword(s):  
Size Control ◽  
Differential Algebraic Equations ◽  
Integration Step ◽  
Algebraic Equations ◽  
Step Size ◽  
Stopping Criteria ◽  
Flexible Bodies ◽  
Engine Model ◽  
Numerical Integrator ◽  
Speedup Factor

The paper presents theoretical and implementation aspects related to a numerical integrator used for the simulation of large mechanical systems with flexible bodies and contact/impact. The proposed algorithm is based on the Hilber-Hughes-Taylor (HHT) implicit method and is tailored to answer the challenges posed by the numerical solution of index 3 differential-algebraic equations that govern the time evolution of a multibody system. One of the salient attributes of the algorithm is the good conditioning of the Jacobian matrix associated with the implicit integrator. Error estimation, integration step-size control, and nonlinear system stopping criteria are discussed in detail. Simulations using the proposed algorithm of an engine model, a model with contacts, and a model with flexible bodies indicate a 2 to 3 speedup factor when compared against benchmark MSC.ADAMS runs. The proposed HHT-based algorithm has been released in the 2005 version of the MSC.ADAMS/Solver.

The Newmark Integration Method for Simulation of Multibody Systems: Analytical Considerations

2005 ◽  
Author(s):  
B. Gavrea ◽  
D. Negrut ◽  
F. A. Potra
Keyword(s):  
Convergence Analysis ◽  
Size Control ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Step Size ◽  
Generalized Coordinates ◽  
Numerical Integrator ◽  
Space Reduction ◽  
System Solution ◽  
Computationally Intensive

When simulating the behavior of a mechanical system, the time evolution of the generalized coordinates used to represent the configuration of the model is computed as the solution of a combined set of ordinary differential and algebraic equations (DAEs). There are several ways in which the numerical solution of the resulting index 3 DAE problem can be approached. The most well-known and time-honored algorithms are the direct discretization approach, and the state-space reduction approach, respectively. In the latter, the problem is reduced to a minimal set of potentially new generalized coordinates in which the problem assumes the form of a pure second order set of Ordinary Differential Equations (ODE). This approach is very accurate, but computationally intensive, especially when dealing with large mechanical systems that contain flexible parts, stiff components, and contact/impact. The direct discretization approach is less but nevertheless sufficiently accurate yet significantly faster, and it is the approach that is considered in this paper. In the context of direct discretization methods, approaches based on the Backward Differentiation Formulas (BDF) have been the traditional choice for more than 20 years. This paper proposes a new approach in which BDF methods are replaced by the Newmark formulas. Local convergence analysis is carried out for the proposed method, and step-size control, error estimation, and nonlinear system solution related issues are discussed in detail. A series of two simple models are used to validate the method. The global convergence analysis and a computational-efficiency comparison with the most widely used numerical integrator available in the MSC.ADAMS commercial simulation package are forthcoming. The new method has been implemented successfully for industrial strength Dynamic Analysis simulations in the 2005 version of the MSC.ADAMS software and used very effectively for the simulation of systems with more than 15,000 differential-algebraic equations.

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.

Application of Scaling to Multibody Dynamics Simulations

2015 ◽  
Author(s):  
William Prescott
Keyword(s):  
Equations Of Motion ◽  
Industrial Applications ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Step Size ◽  
Virtual Lab ◽  
Long Run ◽  
Multibody Dynamic ◽  
Dynamics Simulations ◽  
Stability Issues

This paper will examine the importance of applying scaling to the equations of motion for multibody dynamic systems when applied to industrial applications. If a Cartesian formulation is used to formulate the equations of motion of a multibody dynamic system the resulting equations are a set of differential algebraic equations (DAEs). The algebraic components of the DAEs arise from appending the joint equations used to model revolute, cylindrical, translational and other joints to the Newton-Euler dynamic equations of motion. Stability issues can arise in an ill-conditioned Jacobian matrix of the integration method this will result in poor convergence of the implicit integrator’s Newton method. The repeated failures of the Newton’s method will require a small step size and therefore simulations that require long run times to complete. Recent advances in rescaling the equations of motion have been proposed to address this problem. This paper will see if these methods or a variant addresses not only stability concerns, but also efficiency. The scaling techniques are applied to the Gear-Gupta-Leimkuhler (GGL) formulation for multibody problems by embedding them into the commercial multibody code (MBS) Virtual. Lab Motion and then use them to solve an industrial sized automotive example to see if performance is improved.

scholarly journals Analysis of Orbital Dynamic Equation in Navigation for a Mars Gravity-Assist Mission

2012 ◽  
Vol 65 (3) ◽  
pp. 531-548 ◽  
Author(s):  
Xin Ma ◽  
Xiaolin Ning ◽  
Jiancheng Fang
Keyword(s):  
Navigation System ◽  
Dynamic Equation ◽  
Autonomous Navigation ◽  
Impact Factors ◽  
Integration Step ◽  
Step Size ◽  
Gravity Assist ◽  
Efficient System ◽  
Numerical Integrator ◽  
The Impact

Gravity Assist (GA) is a kind of transfer orbit technology widely used in interplanetary missions, which highly depends on navigation performance to succeed. The Orbital Dynamic Equation is an essential component in the navigation system, affected by factors including the numerical integrator, perturbing planets, integration step size, gravitational constant and planet ephemerides. To analyse the impact factors mentioned above and investigate an efficient system model, the propagation and navigation results are carried out in a Mars-assist explorer scenario; a specific case study is also provided in this paper. The results indicate that the planetary ephemeris uncertainty and integration size are the dominant error sources, and the integration step size is the dominant impact factor on the real-time performance. In this specific case, the ‘Orbital Dynamic Equation’ considering Sun and Mars perturbation is suggested for integration by RK4 with 60 s integration step size. The conclusions drawn by this study are particularly useful in the design, construction, and analysis of an autonomous navigation system for a GA explorer.

On the Use of the HHT Method in the Context of Index 3 Differential Algebraic Equations of Multibody Dynamics

2005 ◽  
Author(s):  
Dan Negrut ◽  
Rajiv Rampalli ◽  
Gisli Ottarsson ◽  
Anthony Sajdak
Keyword(s):  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Body System ◽  
Vehicle Model ◽  
Good Efficiency ◽  
Numerical Integrator ◽  
Starting Point ◽  
Implementation Aspects ◽  
Multi Body ◽  
Simulation Package

The paper presents theoretical and implementation aspects related to a new numerical integrator available in the 2005 version of the MSC.ADAMS/Solver C++. The starting point for the new integrator is the Hilber-Hughes-Taylor method (HHT, also known as α-method) that has been widely used in the finite element community for more than two decades. The method implemented is tailored to answer the challenges posed by the numerical solution of index 3 Differential Algebraic Equations that govern the time evolution of a multi-body system. The proposed integrator was tested with more than 1,600 models prior to its release in the 2005 version of the simulation package MSC.ADAMS. In this paper an all-terrain-vehicle model with flexible chassis is used to prove the good efficiency and accuracy of the method.

A First Principle Engine Model for Up-Front Design: Bearing Models and Example Results

2001 ◽  
Author(s):  
Zheng-Dong Ma ◽  
N. C. Perkins
Keyword(s):  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Hydrodynamic Bearing ◽  
Engine Model ◽  
Trade Offs ◽  
Engine Design ◽  
Design Cycle ◽  
Relative Coordinates ◽  
Noise Vibration

Abstract The design of new engine concepts requires an engineering tool that can quickly estimate noise, vibration and durability metrics at the very onset of the engine design cycle. In (Ma et al., 2000), we presented an engine modeling template (EMT) to support up-front engine design. The engine models generated from the EMT use the minimum set of generalized coordinates to represent engine dynamics. This is achieved by employing a pre-selected set of relative coordinates. The resulting engine model is cast as a (minimum) set of ordinary differential equations in lieu of the differential-algebraic equations that result from using commercial multibody dynamics codes. The resulting models then enjoy greater computational efficiency. In (Ma et al., 2000), we formulate the equations of motion for the engine and its major components. The objective of this paper is to review the numerical results obtained from sample engine designs and to discuss several tradeoffs between model accuracy and efficiency. Attention focuses on the trade-offs resulting from several bearing models, including linear and nonlinear spring-damper bearing models, and hydrodynamic bearing models based on the Reynolds equation. Results computed using these bearing models are critically compared.

Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Olivier A. Bauchau ◽  
Alexander Epple ◽  
Carlo L. Bottasso
Keyword(s):  
Physical Properties ◽  
Augmented Lagrangian ◽  
Equations Of Motion ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Integration Schemes

This paper addresses practical issues associated with the numerical enforcement of constraints in flexible multibody systems, which are characterized by index-3 differential algebraic equations (DAEs). The need to scale the equations of motion is emphasized; in the proposed approach, they are scaled based on simple physical arguments, and an augmented Lagrangian term is added to the formulation. Time discretization followed by a linearization of the resulting equations leads to a Jacobian matrix that is independent of the time step size, h; hence, the condition number of the Jacobian and error propagation are both O(h0): the numerical solution of index-3 DAEs behaves as in the case of regular ordinary differential equations (ODEs). Since the scaling factor depends on the physical properties of the system, the proposed scaling decreases the dependency of this Jacobian on physical properties, further improving the numerical conditioning of the resulting linearized equations. Because the scaling of the equations is performed before the time and space discretizations, its benefits are reaped for all time integration schemes. The augmented Lagrangian term is shown to be indispensable if the solution of the linearized system of equations is to be performed without pivoting, a requirement for the efficient solution of the sparse system of linear equations. Finally, a number of numerical examples demonstrate the efficiency of the proposed approach to scaling.

scholarly journals Earthquake dynamic response of large flexible multibody systems

2013 ◽  
Vol 4 (1) ◽  
pp. 131-137 ◽  
Author(s):  
E. V. Zahariev
Keyword(s):  
Flexible Structures ◽  
High Amplitude ◽  
Differential Algebraic Equations ◽  
Dynamic Equations ◽  
Motion Simulation ◽  
Algebraic Equations ◽  
Power Generator ◽  
Flexible Bodies ◽  
Wind Power Generator ◽  
Generalized Newton

Abstract. In the paper dynamics of large flexible structures imposed on earthquakes and high amplitude vibrations is regarded. Precise dynamic equations of flexible systems are the basis for reliable motion simulation and analysis of loading of the design scheme elements. Generalized Newton–Euler dynamic equations for rigid and flexible bodies are applied. The basement compulsory motion realized because of earthquake or wave propagation is presented in the dynamic equations as reonomic constraints. The dynamic equations, algebraic equations and reonomic constraints compile a system of differential algebraic equations which are transformed to a system of ordinary differential equations with respect to the generalized coordinates and the reactions due to the reonomic constraints. Examples of large flexible structures and wind power generator dynamic analysis are presented.

scholarly journals DYNAMIC SIMULATION OF EVAPORATOR IN ETHANOL BIOREFINERY

2019 ◽  
Vol 49 (1) ◽  
pp. 65-70
Author(s):  
J. FERREIRA ◽  
B. L. NOGUEIRA ◽  
A. R. SECCHI
Keyword(s):  
First Generation ◽  
Differential Algebraic Equations ◽  
Generation Process ◽  
Algebraic Equations ◽  
Transient Model ◽  
Tube Heat Exchanger ◽  
2Nd Generation ◽  
Numerical Integrator ◽  
Concentration Step ◽  
Second Generation Ethanol

Biorefineries producing ethanol from sugarcane are, mostly, plants that obtain ethanol from first-generation process. Second-generation ethanol is currently being studied and evaluated technologically and economically. One of the stages of large energy consumption, common to 1st and 2nd generation processes, is the concentration step by evaporator systems. In the present work a phenomenological transient model of Robert type evaporator is implemented using EMSO simulator. The physicochemical properties were calculated by the thermodynamic package VRTherm. The calandria section of the evaporator was modeled as a shell-and-tube heat exchanger, followed by a flash vessel for phase separation using level and pressure controllers. The finitedifferences method was used to discretize the spatial variable in the obtained partial differential equations, and the numerical integrator DASSLC was the code used to solve the resulting system of differential-algebraic equations. The obtained compositions of the liquid and vapor phases and temperatures are in agreement with data obtained from literature. In order to compare the results between a simpler and a more detailed models, flash and Robert type evaporator models were evaluated. 

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