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.

Deep Learning of (Periodic) Minimal Coordinates for Multibody Simulations

2020 ◽  
Author(s):  
Andrea Angeli ◽  
Frank Naets ◽  
Wim Desmet
Keyword(s):  
Neural Network ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Multibody Model ◽  
Network Training ◽  
The Neural Network ◽  
Continuous Description ◽  
Minimal Coordinates ◽  
Reduced Space ◽  
Multi Body

Abstract Mechanical systems are typically described through multi-body models with redundant coordinates, related by imposed constraints, where the dynamics is expressed with Differential Algebraic Equations. Alternatively, for rigid models, it may be preferable to employ minimal coordinates that do not require additional constraints, thus leading to Ordinary Differential Equations. However, to reduce a general multibody model to minimal coordinates and perform the simulation in the reduced space, the mapping between the minimal coordinates and the full coordinates is required. In this work, it is proposed to approximate such mapping using a neural network. In order to avoid overfitting and guarantee a continuous description of the solution manifold, the multibody dynamics information are included in the neural network training. The particular case where periodic minimal coordinates are required is treated and validated. In general, the methodology can be used when the mapping is unknown such as for spatial mechanisms with closed loops.

Variational Analysis of a Two Link Slider-Crank Mechanism Using Polynomial Chaos Theory

2017 ◽  
Author(s):  
Paul S. Ryan ◽  
Sarah C. Baxter ◽  
Philip A. Voglewede
Keyword(s):  
Chaos Theory ◽  
Closed Loop ◽  
Polynomial Chaos ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Advantages And Disadvantages ◽  
Multi Body ◽  
Polynomial Chaos Theory ◽  
Body Dynamic ◽  
Crank Mechanism

Variation occurs in many closed loop multi-body dynamic (MBD) systems in the geometry, mass, or forces. Understanding how MBD systems respond to variation is imperative for the design of a robust system. However, simulation of how variation propagates into the solution is complicated as most MBD systems cannot be simplified into to a system of ordinary differential equations (ODE). This paper investigates polynomial chaos theory (PCT) as a means of quantifying the effects of uncertainty in a closed loop MBD system governed by differential algebraic equations (DAE). To demonstrate how PCT could be used, the motion of a two link slider-crank mechanism is simulated with variation in the link lengths. To validate and show the advantages and disadvantages of PCT in closed loop MBD systems, the PCT approach is compared to Monte Carlo simulations.

scholarly journals Solution of Time-Varying Index-2 Linear Differential Algebraic Control Systems Via A Variational Formulation Technique

2021 ◽  
pp. 3656-3671
Author(s):  
Ghazwa F. Abd ◽  
Radhi A. Zaboon
Keyword(s):  
Approximate Solution ◽  
Variational Formulation ◽  
Direct Method ◽  
Differential Algebraic Equations ◽  
Time Varying ◽  
Algebraic Equations ◽  
Good Efficiency ◽  
Physical Problems ◽  
Index 2 ◽  
Linear Differential

    This paper deals with finding an approximate solution to the index-2 time-varying linear differential algebraic control system based on the theory of variational formulation. The solution of index-2 time-varying differential algebraic equations (DAEs) is the critical point of the equivalent variational formulation. In addition, the variational problem is transformed from the indirect into direct method by using a generalized Ritz bases approach. The approximate solution is found by solving an explicit linear algebraic equation, which makes the proposed technique reliable and efficient for many physical problems. From the numerical results, it can be implied that very good efficiency, accuracy, and simplicity of the present approach are obtained.

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.

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. 

Planar Multi-body Dynamics of a Tracked Vehicle using Imaginary Wheel Model for Tracks

2017 ◽  
Vol 67 (4) ◽  
pp. 460
Author(s):  
Ilango Mahalingam ◽  
Chandramouli Padmanabhan
Keyword(s):  
Interaction Model ◽  
Differential Algebraic Equations ◽  
Rigid Bodies ◽  
Contact Forces ◽  
Algebraic Equations ◽  
Tracked Vehicle ◽  
The Road ◽  
Wheel Model ◽  
Longitudinal Slip ◽  
Multi Body

<p class="p1">Off-road vehicles achieve their mobility with the help of a track system. A track has large number of rigid bodies with pin joints leading to computational complexity in modelling the dynamic behaviour of the system. In this paper, a new idea is proposed, where the tracks are replaced by a set of imaginary wheels connected to the road wheels using mechanical links. A non-linear wheel terrain interaction model considering longitudinal slip is used to find out the normal and tangential contact forces. A linear trailing arm suspension, where a road arm connecting the road wheel and chassis with a rotational spring and damper system is considered. The differential algebraic equations (DAEs) from the multi-body model are derived in Cartesian coordinates and formulated using augmented formulation. The augmented equations are solved numerically using appropriate stabilisation techniques. The novel proposition is validated using experimental measurements done on a tracked vehicle.</p>

Study on the Multi-Body Dynamic Characteristics of FPSO Soft Yoke Mooring System Based on Symplectic Algorithm

2019 ◽  
Author(s):  
Wenhua Wu ◽  
Baicheng Lyu ◽  
Ji Yao ◽  
Qianjin Yue ◽  
Zhang Yantao ◽  
...  
Keyword(s):  
Stress State ◽  
Dynamic Characteristics ◽  
Degrees Of Freedom ◽  
Single Point ◽  
Differential Algebraic Equations ◽  
Mooring System ◽  
Algebraic Equations ◽  
Restoring Force ◽  
Hamilton System ◽  
Multi Body

Abstract The soft yoke single-point mooring (SYMS) system is the main mooring approach for the floating production storage and offloading (FPSO) unit. As a typical multi-rigid-body system, a SYMS consists of the single-point turret, yoke, mooring legs, and mooring support. It releases the rotational degrees of freedom of an FPSO through the combined effects of multiple joint structures, so as to deliver the weather-vane effect of the FPSO. In this paper, a multi-body dynamics model of the soft yoke mooring system was established. To deal with the difficult integration in the process of solving differential-algebraic equations, a symplectic numerical integration method was proposed on the basis of the Zu Chongzhi method. The proposed solution format had simple symplectic property automatically satisfying the Hamilton system, as well as a high accuracy in solving nonlinear systems. The measured data of the FPSO’s six degrees of freedom (6DoF) under two different sea conditions were selected, and the mooring restoring force of the SYMS was calculated. The calculated results showed that the symplectic solution method could the actual stress state of the structures with more obvious dynamic characteristics. Furthermore, the displacement and stress state of the single-body structures, such as the mooring legs and yoke, and the analysis result could comprehensively evaluate the overall working state of the SYMS.

From Off-Line to Real Time Simulations by Model Reduction and Modular Vehicle Modelling

2003 ◽  
Author(s):  
E. Pankiewicz ◽  
W. Rulka
Keyword(s):  
Real Time ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Vehicle Suspensions ◽  
Environmental Models ◽  
Control Units ◽  
Applied Forces ◽  
Multi Body ◽  
Real Time Simulations

Over the last couple of years there has been a growing tendency in the automotive industry to use more active control systems with mechatronic components, realized by electronic control units (ECU). As the ECU is a highly complex system and also for reasons of safety and costs the ECU-test is not carried out in road tests, but at a much earlier time, i.e. in the laboratory, using Hardware-in-the-Loop (HiL) simulation. This requires real-time environmental models. This paper shows how the transition from offline multi-body-system (MBS) models to real-time simulation can be achieved automatically by using a component oriented reduction procedure for vehicle suspensions, which keeps the full component parameterization. This necessitates a transformation of the equations of motion from differential algebraic equations (DAE) into ordinary differential equations (ODE). For use in HiL-simulators the equations of motion with open interfaces of the applied forces and torques are provided to be integrated in a modular dynamic ride model.

Utilization of Mathematical Software Packages in Chemical Engineering Research

2005 ◽  
Vol 5 (2) ◽  
pp. 125
Author(s):  
Ang Wee Lee ◽  
Nayef Mohamed Ghasem ◽  
Mohamed Azlan Hussain
Keyword(s):  
Differential Equations ◽  
Computer Algebra ◽  
Large Scale ◽  
Chemical Engineering ◽  
Nonlinear Differential Equations ◽  
Differential Algebraic Equations ◽  
Good Choice ◽  
Computer Algebra Systems ◽  
Algebraic Equations ◽  
Starting Point

Using Fortran taken as the starting point, we are now on the sixth decade of high-level programming applications. Among the programming languages available, computer algebra systems (CAS) appear to be a good choice in chemical engineering can be applied easily. Until the emergence of CAS, the assistance from a specialized group for large-scale programming is justified. Nowadays, it is more effective for the modern chemical engineer to rely on his/her own programming ability for problem solving. In the present paper, the abilities of Polymath, Maple, Matlab, Mathcad, and Mathematica in handling differential equations are illustrated for differential-algebraic equations, large system of nonlinear differential equations, and partial differential equations. The programming of solutions with these CAS are presented, contrasted, and discussed in relation to chemical engineering problems. Keywords: Computer algebra systems (CAS),computer simulation,Mathcad, Mathematica,Mathlab and numerical methods.

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