Backward Differentiation Formula and Newmark-Type Index-2 and Index-1 Integration Schemes for Constrained Mechanical Systems

2019 ◽  
Vol 15 (2) ◽  
Author(s):  
T. Meyer ◽  
P. Li ◽  
B. Schweizer
Keyword(s):  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
Differentiation Formula ◽  
Constrained Mechanical Systems ◽  
Time Points ◽  
Backward Differentiation Formula ◽  
Integration Schemes ◽  
Intermediate Time ◽  
Dae Systems ◽  
Index 2

Abstract Various methods for solving systems of differential-algebraic equations (DAE systems) are known from literature. Here, an alternative approach is suggested, which is based on a collocated constraints approach (CCA). The basic idea of the method is to introduce intermediate time points. The approach is rather general and may basically be applied for solving arbitrary DAE systems. Here, the approach is discussed for constrained mechanical systems of index-3. Application of the presented formulations for nonmechanical higher index DAE systems is also possible. We discuss index-2 formulations with one intermediate time point and index-1 implementations with two intermediate time points. The presented technique is principally independent of the time discretization method and may be applied in connection with different time integration schemes. Here, implementations are investigated for backward differentiation formula (BDF) and Newmark-type integrator schemes. A direct application of the presented approach yields a system of discretized equations with larger dimensions. The increased dimension of the discretized system of equations may be considered as the main drawback of the presented technique. The main advantage is that the approach may be used in a very straightforward manner for solving rather arbitrary multiphysical DAE systems with arbitrary index. Hence, the method might, for instance, be attractive for general purpose DAE integrators, since the approach is not tailored for special DAE systems (e.g., constrained mechanical systems). Numerical examples will demonstrate the straightforward application of the approach.

Solving Differential-Algebraic Equation Systems: Alternative Index-2 and Index-1 Approaches for Constrained Mechanical Systems

2015 ◽  
Vol 11 (4) ◽  
Author(s):  
Bernhard Schweizer ◽  
Pu Li
Keyword(s):  
Algebraic Equation ◽  
Mechanical Systems ◽  
Order Reduction ◽  
Integration Time ◽  
Differential Algebraic Equation ◽  
Constrained Mechanical Systems ◽  
Dae Systems ◽  
Index 2 ◽  
Runge Kutta ◽  
Hidden Constraints

Regarding constrained mechanical systems, we are faced with index-3 differential-algebraic equation (DAE) systems. Direct discretization of the index-3 DAE systems only enforces the position constraints to be fulfilled at the integration-time points, but not the hidden constraints. In addition, order reduction effects are observed in the velocity variables and the Lagrange multipliers. In literature, different numerical techniques have been suggested to reduce the index of the system and to handle the numerical integration of constrained mechanical systems. This paper deals with an alternative concept, called collocated constraints approach. We present index-2 and index-1 formulations in combination with implicit Runge–Kutta methods. Compared with the direct discretization of the index-3 DAE system, the proposed method enforces also the constraints on velocity and—in case of the index-1 formulation—the constraints on acceleration level. The proposed method may very easily be implemented in standard Runge–Kutta solvers. Here, we only discuss mechanical systems. The presented approach can, however, also be applied for solving nonmechanical higher-index DAE systems.

scholarly journals Numerical solution for consistent initial conditions of constrained mechanical systems

2003 ◽  
Vol 25 (3) ◽  
pp. 170-185
Author(s):  
Dinh Van Phong
Keyword(s):  
Numerical Solution ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Flexible Approach ◽  
Constrained Mechanical Systems ◽  
Consistent Initial Values

The article deals with the problem of consistent initial values of the system of equations of motion which has the form of the system of differential-algebraic equations. Direct treating the equations of mechanical systems with particular properties enables to study the system of DAE in a more flexible approach. Algorithms and examples are shown in order to illustrate the considered technique.

On the Geometric Stabilization for Discrete Hamiltonian Systems With Holonomic Constraints

2009 ◽  
Author(s):  
Hiroaki Yoshimura ◽  
Kenji Soya
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Geometric Constraint ◽  
Differentiation Formula ◽  
Holonomic Constraints ◽  
Constrained Mechanical Systems ◽  
Constraint Stabilization ◽  
Backward Differentiation Formula ◽  
External Forces ◽  
Baumgarte Stabilization

This paper develops a discrete Hamiltonian system with holonomic constraints with Geometric Constraint Stabilization. It is first shown that constrained mechanical systems with nonconservative external forces can be formulated by using canonical symplectic structures in the context of Hamiltonian systems. Second, it is shown that discrete holonomic Hamiltonian systems can be developed via the discretization based on the Backward Differentiation Formula and also that geometric constraint stabilization can be incorporated into the discrete Hamiltonian systems. It is demonstrated that the proposed method enables one to stabilize constraint violations effectively in comparison with conventional methods such as Baumgarte Stabilization and Gear–Gupta–Leimkuhler Stabilization, together with an illustrative example of linkage mechanisms.

Simulation of Constrained Mechanical Systems—Part II: Explicit Numerical Integration

2012 ◽  
Vol 79 (4) ◽  
Author(s):  
David J. Braun ◽  
Michael Goldfarb
Keyword(s):  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Direct Consequence ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Constrained Motion ◽  
Constrained Mechanical Systems ◽  
Holonomic And Nonholonomic Constraints ◽  
Long Time ◽  
Constraint Correction

This paper presents an explicit to integrate differential algebraic equations (DAEs) method for simulations of constrained mechanical systems modeled with holonomic and nonholonomic constraints. The proposed DAE integrator is based on the equation of constrained motion developed in Part I of this work, which is discretized here using explicit ordinary differential equation schemes and applied to solve two nontrivial examples. The obtained results show that this integrator allows one to precisely solve constrained mechanical systems through long time periods. Unlike many other implicit DAE solvers which utilize iterative constraint correction, the presented DAE integrator is explicit, and it does not use any iteration. As a direct consequence, the present formulation is simple to implement, and is also well suited for real-time applications.

Force Equilibrium Approach for Linearization of Constrained Mechanical System Dynamics

2003 ◽  
Vol 125 (1) ◽  
pp. 143-149 ◽  
Author(s):  
Ju Seok Kang ◽  
Sangwoo Bae ◽  
Jang Moo Lee ◽  
Tae Oh Tak
Keyword(s):  
Mechanical Systems ◽  
Nonlinear Dynamic Analysis ◽  
Differential Algebraic Equations ◽  
Dynamic Equations ◽  
Small Displacement ◽  
Algebraic Equations ◽  
Small Motion ◽  
Constrained Mechanical Systems ◽  
External Forces ◽  
Force Equilibrium

The purpose of this study is to derive a linearized form of dynamic equations for constrained mechanical systems. The governing equations for constrained mechanical systems are generally expressed in terms of Differential-Algebraic Equations (DAEs). Conventional methods of linearization are based on the perturbation of the nonlinear DAE, where small amounts of perturbations are taken to guarantee linear characteristics of the equations. On the other hand, the proposed linearized dynamic equations are derived directly from a force equilibrium condition, not from the DAEs, with small motion assumption. This approach is straightforward and simple compared to conventional perturbation methods, and can be applicable to any constrained mechanical systems that undergo small displacement under external forces. The modeling procedure and formulation of linearized dynamic equations are demonstrated by the example of a vehicle suspension system, a typical constrained multibody system. The solution is validated by comparison with conventional nonlinear dynamic analysis and modal test results.

scholarly journals Variable step 2-point block backward differentiation formula for index-1 differential algebraic equations

ScienceAsia ◽  
2014 ◽  
Vol 40 (5) ◽  
pp. 375
Author(s):  
Naghmeh Abasi ◽  
Mohamed Bin Suleiman ◽  
Zarina Bibi Ibrahim ◽  
Hamisu Musa ◽  
Faranak Rabiei
Keyword(s):  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Differentiation Formula ◽  
Backward Differentiation Formula ◽  
Variable Step

Subset Selection Methods for Design Sensitivity Analysis of Constrained Dynamic Systems

1990 ◽  
Author(s):  
Jun-Tien Twu ◽  
Prakash Krishnaswami ◽  
Rajiv Rampalli
Keyword(s):  
Dynamic Systems ◽  
Correct Choice ◽  
Design Sensitivity ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Selection Methods ◽  
Constrained Motion ◽  
Differentiation Formula ◽  
Backward Differentiation Formula ◽  
Solution Methods

Abstract In the Lagrangian formulation of the constrained motion of mechanical systems, a system of Differential-Algebraic Equations is generally encountered. The popular Backward Differentiation Formula for the numerical solution of such problems leads to an over-determined system of equations. The correct choice of a proper exactly determined subset can greatly enhance the performance of a solution algorithm. In this paper, we discuss four solution methods with different choices of subsets. Three numerical examples are solved to compare the accuracy and efficiency of these methods.

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