scholarly journals Co-Simulation of Algebraically Coupled Dynamic Subsystems

2001 ◽  
Author(s):  
Bei Gu ◽  
H. Harry Asada
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Discrete Time ◽  
Sliding Mode ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Step Size ◽  
Sliding Manifold ◽  
Sliding Manifolds ◽  
Algebraic Constraints

Abstract This paper analyzes the problem of Co-Simulation. The term Co-Simulation is used to describe a large dynamic system that is simulated by running a group of independently coded subsystem simulators. Very commonly, the Co-Simulation of subsystems faces incompatible boundary conditions, i.e., causal conflicts. These causal conflicts cannot be directly resolved, due to the nonlinearity and/or difficulties in modification of coded subsystem simulators. Causal conflicts result in algebraic constraints. Boundary Condition Coordinators (BCCs) are designed to calculate boundary conditions based on subsystem models and their algebraic constraints. The Co-Simulation, which is modeled as Differential Algebraic Equations, then relies on BCC to provide compatible boundary conditions for subsystem simulators. The high index constraint is reduced to index one by defining a sliding manifold. Different ways of enforcing the sliding manifold are discussed: A new Discrete-Time Sliding Mode (DTSM) controller is devised to serve as a BCC, enforcing sliding manifolds and providing boundary conditions. The multi-rate scheme can guarantee Co-Simulation stability at any given step size of all subsystem simulators, provided the subsystem simulators are tested stable at that step size. An example is given to demonstrate the DTSM method. Advantages and possible future improvements are discussed.

Discrete-Time Inverse Dynamics Control of Flexible Joint Robots

1992 ◽  
Vol 114 (2) ◽  
pp. 229-233 ◽  
Author(s):  
K. P. Jankowski ◽  
H. Van Brussel
Keyword(s):  
Discrete Time ◽  
Control Algorithm ◽  
Inverse Dynamics ◽  
Differential Algebraic Equations ◽  
Control Methods ◽  
Algebraic Equations ◽  
Computationally Efficient ◽  
Electromechanical Actuators ◽  
Flexible Joint Robots ◽  
Inverse Dynamics Control

This paper focuses on the problem of the application of inverse dynamics control methods to robots with flexible joints and electromechanical actuators. Due to drawbacks of the continuous-time inverse dynamics, discussed in the paper, a new control strategy in discrete-time is presented. The proposed control algorithm is based on numerical methods conceived for the solution of index-three systems of differential-algebraic equations. The method is computationally efficient and admits low sampling frequencies. The results of numerical experiments confirm the advantages of the designed control algorithm.

scholarly journals A distributional solution framework for linear hyperbolic PDEs coupled to switched DAEs

2020 ◽  
Vol 32 (4) ◽  
pp. 455-487
Author(s):  
R. Borsche ◽  
D. Kocoglu ◽  
S. Trenn
Keyword(s):  
Partial Differential Equations ◽  
Boundary Conditions ◽  
Electric Power ◽  
Unique Solvability ◽  
Differential Algebraic Equations ◽  
Hyperbolic Partial Differential Equations ◽  
Power Lines ◽  
Algebraic Equations ◽  
Distributional Solution ◽  
Telegraph Equations

AbstractA distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations and switched differential-algebraic equations (DAEs) which are coupled via boundary conditions. The unique solvability is then characterize in terms of a switched delay DAE. The theory is illustrated with an example of electric power lines modelled by the telegraph equations which are coupled via a switching transformer where simulations confirm the predicted impulsive solutions.

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.

Projection Method With Minimal Correction Procedure for Numerical Simulation of Constrained Dynamics

2017 ◽  
Author(s):  
Patrick S. Heaney ◽  
Gene Hou
Keyword(s):  
Projection Method ◽  
Numerical Technique ◽  
Correction Method ◽  
Differential Algebraic Equations ◽  
Constrained Systems ◽  
Correction Procedure ◽  
Algebraic Equations ◽  
Differential Algebraic Equation ◽  
Index Reduction ◽  
Algebraic Constraints

This paper describes a numerical technique for simulating the dynamics of constrained systems, which are described generally by differential-algebraic equations. The Projection Method for index reduction of a differential-algebraic equation and a minimal correction procedure are described. This procedure ensures algebraic constraints are satisfied during the numerical integration of the reduced index system of differential equations. Two examples illustrate how the method can be utilized to solve constrained multibody and rotational dynamics problems. The efficiency and accuracy of the proposed index-reduction and minimal correction method are then evaluated.

scholarly journals A Numerical Method for a System of Fractional Differential-Algebraic Equations Based on Sliding Mode Control

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1134
Author(s):  
Yongpeng Tai ◽  
Ning Chen ◽  
Lijin Wang ◽  
Zaiyong Feng ◽  
Jun Xu
Keyword(s):  
Fractional Calculus ◽  
Sliding Mode Control ◽  
Numerical Method ◽  
Present Method ◽  
Sliding Mode ◽  
Constitutive Relations ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Mode Control ◽  
Fractional Differential

Fractional calculus is widely used in engineering fields. In complex mechanical systems, multi-body dynamics can be modelled by fractional differential-algebraic equations when considering the fractional constitutive relations of some materials. In recent years, there have been a few works about the numerical method of the fractional differential-algebraic equations. However, most of the methods cannot be directly applied in the equations of dynamic systems. This paper presents a numerical algorithm of fractional differential-algebraic equations based on the theory of sliding mode control and the fractional calculus definition of Grünwald–Letnikov. The algebraic equation is considered as the sliding mode surface. The validity of the present method is verified by comparing with an example with exact solutions. The accuracy and efficiency of the present method are studied. It is found that the present method has very high accuracy and low time consumption. The effect of violation corrections on the accuracy is investigated for different time steps.

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.

Multi-Splitting Waveform Relaxation Methods for Solving the Initial Value Problem of Linear Integral-Differential-Algebraic Equations

2011 ◽  
Vol 403-408 ◽  
pp. 1763-1766
Author(s):  
Xiao Lin Lin ◽  
Yuan Sang ◽  
Hong Wei ◽  
Li Ming Liu ◽  
Yu Mei Wang ◽  
...  
Keyword(s):  
Spectral Radius ◽  
Discrete Time ◽  
Initial Value Problem ◽  
Differential Algebraic Equations ◽  
Waveform Relaxation ◽  
Algebraic Equations ◽  
Initial Value ◽  
Relaxation Methods ◽  
Discrete Time Case ◽  
Linear Integral

We present the multi-splitting waveform relaxation (MSWR) methods for solving the initial value problem of linear integral-differential-algebraic equations. Based on the spectral radius of the derived operator by decoupled process, a convergent condition is proposed for the MSWR method. Finally we discussed the convergent condition of discrete-time case of MSWR.

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