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.

Direct Differentiation Methods for the Design Sensitivity of Multibody Dynamic Systems

1996 ◽  
Author(s):  
Radu Serban ◽  
Jeffrey S. Freeman
Keyword(s):  
Sensitivity Analysis ◽  
Dynamic Systems ◽  
Design Sensitivity ◽  
Differential Algebraic Equations ◽  
Mechanism Analysis ◽  
Algebraic Equations ◽  
First Order ◽  
Direct Differentiation ◽  
Multibody Dynamic ◽  
Standard Techniques

Abstract Methods for formulating the first-order design sensitivity of multibody systems by direct differentiation are presented. These types of systems, when formulated by Euler-Lagrange techniques, are representable using differential-algebraic equations (DAE). The sensitivity analysis methods presented also result in systems of DAE’s which can be solved using standard techniques. Problems with previous direct differentiation sensitivity analysis derivations are highlighted, since they do not result in valid systems of DAE’s. This is shown using the simple pendulum example, which can be analyzed in both ODE and DAE form. Finally, a slider-crank example is used to show application of the method to mechanism analysis.

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

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.

Differential–Algebraic Equations and Dynamic Systems on Manifolds

2016 ◽  
Vol 52 (3) ◽  
pp. 408-418 ◽  
Author(s):  
Iu. G. Kryvonos ◽  
V. P. Kharchenko ◽  
N. M. Glazunov
Keyword(s):  
Dynamic Systems ◽  
Differential Algebraic Equations ◽  
Algebraic Equations

Component Based Modeling of Electromechanical Systems

2005 ◽  
Author(s):  
Shilpa A. Vaze ◽  
Prakash Krishnaswami ◽  
James DeVault
Keyword(s):  
Design Sensitivity ◽  
Differential Algebraic Equations ◽  
Single Step ◽  
System Response ◽  
Algebraic Equations ◽  
System Behavior ◽  
Governing Equations ◽  
Electromechanical Systems ◽  
System Equations ◽  
Saturation Effects

Simulation methods for electromechanical systems should accommodate their interdisciplinary nature and the fact that these systems often display qualitative changes in system behavior during operation, such as saturation effects and changes in kinematic structure. Current approaches are either based on deriving the system equations by applying a single formulation to all problem domains, or they are based on trying to integrate different software packages/modules to solve the interdisciplinary problem. In this paper, we present a component-based approach which allows the governing equations of each component to be defined in terms of its natural variables. The different component equations are then brought together to form a single system of differential-algebraic equations (DAE’s), which can be numerically solved to obtain the system response. The fact that we have an explicit, unified form of the system governing equations means that this formulation can be easily extended to design sensitivity analysis and optimization of electromechanical systems (EMS). The formulation includes monitor functions which can be used to detect when a qualitative system change has occurred, and to switch to a new set of governing equations to reflect this change. A single step integrator is used to make it easier to switch to a new system behavior, since this will always require a restart of the integrator. There is considerable flexibility in how the components can be defined, and connections between components are themselves modeled as special types of components. Examples of components from the mechanical and electrical side are presented, and two numerical examples are solved to illustrate the efficacy of the proposed method. One example is a link that is driven by a DC motor through a gearbox. The results of this example were verified against Simulink, and good agreement was observed. The second example is a motor driven slider-crank mechanism. The method can be extended to include components from any domain, such as hydraulics, thermal, controls, etc., as long as the governing equations can be written as DAE’s.

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.

A PID Type Constraint Stabilization Method for Numerical Integration of Multibody Systems

2011 ◽  
Vol 6 (4) ◽  
Author(s):  
Shih-Tin Lin ◽  
Ming-Wen Chen
Keyword(s):  
Numerical Integration ◽  
Pid Controller ◽  
Controller Design ◽  
Equations Of Motion ◽  
Constraint Equation ◽  
Correct Choice ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Stabilization Method ◽  
Constraint Stabilization

The dynamic equations of motion of the constrained multibody mechanical system are mixed differential-algebraic equations (DAEs). The numerical solution of the DAE systems solved using ordinary-differential equation (ODE) solvers may suffer from constraint drift phenomenon. 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. Baumgarte’s method is a proportional-derivative (PD) type controller design. In this paper, an Iintegrator controller is included to form a proportional-integral-derivative (PID) controller so that the steady state error of the numerical integration can be reduced. Stability analysis methods in the digital control theory are used to find out the correct choice of the coefficients for the PID controller.

STEADY-STATE METHODS OF DIFFERENTIAL-ALGEBRAIC EQUATIONS IN CIRCUIT SIMULATION

2005 ◽  
Vol 14 (02) ◽  
pp. 383-393
Author(s):  
YAO-LIN JIANG
Keyword(s):  
Steady State ◽  
Dynamic Systems ◽  
Krylov Subspace ◽  
Circuit Simulation ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Linear Dynamic Systems ◽  
Autonomous Case ◽  
Newton Iterations ◽  
Pseudo Inverse

In the paper, we study the steady-state methods of large dynamic systems. For a nonlinear system of differential-algebraic equations with a known period T, we decouple it, in function space, into linear subsystems by quasilinearization. The resulting linear dynamic systems can be solved by a waveform Krylov subspace method. For the autonomous case, that is, the period T is unknown, the well-known shooting process can be applied where Newton iterations are computed with pseudo-inverse.

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