State-Space Based Implicit Integration of the Differential-Algebraic Equations of Multibody Dynamics

1999 ◽  
Author(s):  
Dan Negrut ◽  
Edward J. Haug
Keyword(s):  
State Space ◽  
Mechanical System ◽  
Multibody Dynamics ◽  
Reference Frames ◽  
Differential Algebraic Equations ◽  
The State ◽  
Algebraic Equations ◽  
Implicit Integration ◽  
Euler Parameters ◽  
Generalized Coordinates

Abstract Three methods for the state-space based implicit integration of differential-algebraic equations of multibody dynamics are summarized and numerically compared. In the state-space approach, the time evolution of a mechanical system is characterized using a number of generalized coordinates equal with the number of degrees of freedom of the system. In this paper these independent generalized coordinates are a subset of the Cartesian position coordinates and orientation Euler parameters of body centroidal reference frames. Depending on the method, the independent generalized coordinates are implicitly integrated and dependent quantities (including Lagrange multipliers) are determined to satisfy constraint equations at position, velocity, and acceleration levels. Five computational algorithms based on the proposed methods are used to simulate the motion of a stiff 14-body vehicle model. Results show that the proposed methods deal effectively with challenges posed by stiff mechanical system simulation. A comparison with a state-space based explicit algorithm for the simulation of the same model indicates a speed-up of approximately two orders of magnitude.

Dwelling on the Connection Between SO(3) and Rotation Matrices in Rigid Multibody Dynamics – Part 2: Comparison Against Formulations Using Euler Parameters or Euler Angles

2021 ◽  
Author(s):  
Jay Taves ◽  
Alexandra Kissel ◽  
Dan Negrut
Keyword(s):  
Multibody Dynamics ◽  
Group Structure ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Implicit Time ◽  
Euler Angles ◽  
Algebraic Equations ◽  
Euler Parameters ◽  
Generalized Coordinates ◽  
Orientation Matrix

Abstract We compare three solution approaches that use the index 3 set of differential algebraic equations (DAEs) to solve the constrained multibody dynamics problem through straight discretization via an implicit time integrator. The first approach is described in a companion paper and dwells on the connection between the orientation matrix and the SO(3) group. Its salient point is that the orientation matrix A is a problem unknown, directly computed without resorting to the use of other position-level generalized coordinates such as Euler angles or Euler parameters. The second approach employs Euler angles as part of the position-level generalized coordinates, and uses them to subsequently evaluate the orientation matrix A. The third approach replaces the Euler angles with Euler parameters (quaternions). The numerical integration method of choice in this contribution is first order implicit Euler. We report a similar number of iterations for convergence for all solution implementations (called rA, rε, and rp); we also observed an approximately twofold speedup of rA over rp and rε. The tests were carried out in conjunction with three models: simple pendulum, slider crank, and four-link mechanism. These simulation results were obtained using two Python simulation engines that were developed independently as part of this formulation comparison undertaking. The codes are available in a GitHub public repository and were developed to provide two different perspectives on the formulation performance issue. The improvements in simulation speed are traced back to a simpler form of the equations of motion and more concise Jacobians that enter the numerical solution. It remains to investigate whether these speed gains carry to higher order integration formulas, where the underlying Lie-group structure of SO(3) brings additional complexity in the rA solution.

scholarly journals Assessment of Linearization Approaches for Multibody Dynamics Formulations

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Francisco González ◽  
Pierangelo Masarati ◽  
Javier Cuadrado ◽  
Miguel A. Naya
Keyword(s):  
Mechanical System ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Practical Applications ◽  
Linearization Methods ◽  
Highly Nonlinear ◽  
Wide Range ◽  
The Way

Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear differential-algebraic equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analyses are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics. A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.

Globally Independent Coordinates for Real-Time Vehicle Simulation

1998 ◽  
Vol 122 (4) ◽  
pp. 575-582 ◽  
Author(s):  
Radu Serban ◽  
Edward J. Haug
Keyword(s):  
State Space ◽  
Real Time ◽  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Vehicle Simulation ◽  
Generalized Coordinates ◽  
Dynamics Of Multibody Systems ◽  
Improved Performance ◽  
Control Forces

Models of the dynamics of multibody systems generally result in a set of differential-algebraic equations (DAE). State-space methods for solving the DAE of motion are based on reduction of the DAE to ordinary differential equations (ODE), by means of local parameterizations of the constraint manifold that must be often modified during a simulation. In this paper it is shown that, for vehicle multibody systems, generalized coordinates that are dual to suspension and/or control forces in the model are independent for the entire range of motion of the system. Therefore, these additional coordinates, together with Cartesian coordinates describing the position and orientation of the chassis, form a set of globally independent coordinates. In addition to the immediate advantage of avoiding the computationally expensive redefinition of local parameterization in a state-space formulation, the existence of globally independent coordinates leads to efficient algorithms for recovery of dependent generalized coordinates. A topology based approach to identify efficient computational sequences is presented. Numerical examples with realistic vehicle handling models demonstrate the improved performance of the proposed approach, relative to the conventional Cartesian coordinate formulation, yielding real-time for vehicle simulation. [S1050-0472(00)00404-9]

Rosenbrock-Nystrom Integrator for SSODE of Mechanical Systems

2003 ◽  
Author(s):  
Adrian Sandu ◽  
Dan Negrut ◽  
Corina Sandu ◽  
Edward J. Haug ◽  
Florian A. Potra
Keyword(s):  
Mechanical System ◽  
Time Evolution ◽  
Degrees Of Freedom ◽  
State Of The Art ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Generalized Coordinates ◽  
System A ◽  
Analytical Derivatives

When performing dynamic analysis of a constrained mechanical system, a set of index 3 Differential-Algebraic Equations (DAE) describes the time evolution of the system. The paper presents a state-space based method for the numerical solution of the resulting DAE. A subset of so called independent generalized coordinates, equal in number to the number of degrees of freedom of the mechanical system, is used to express the time evolution of the mechanical system. The second order statespace ordinary differential equations (SSODE) that describe the time variation of independent coordinates are numerically integrated using a Rosenbrock type formula. For stiff mechanical systems, the proposed algorithm is shown to significantly reduce simulation times when compared to state of the art existent algorithms. The better efficiency is due to the use of an L-stable integrator and a rigorous and general approach to providing analytical derivatives required by it.

scholarly journals Differential Algebraic Equations of MOS Circuits and Jump Behavior

2012 ◽  
Vol 10 ◽  
pp. 327-332
Author(s):  
P. Sarangapani ◽  
T. Thiessen ◽  
W. Mathis
Keyword(s):  
State Space ◽  
Current Model ◽  
Equivalent Circuit Model ◽  
Geometric Approach ◽  
Differential Algebraic Equations ◽  
The State ◽  
Switching Behavior ◽  
Algebraic Equations ◽  
Flip Flop ◽  
Mos Circuits

Abstract. Many nonlinear electronic circuits showing fast switching behavior exhibit jump effects which occurs when the state space of the electronic system contains a fold. This leads to difficulties during the simulation of these systems with standard circuit simulators. A method to overcome these problems is by regularization, where parasitic inductors and capacitors are added at the suitable locations. However, the transient solution will not be reliable if this regularization is not done in accordance with Tikhonov's Theorem. A geometric approach is taken to overcome these problems by explicitly computing the state space and jump points of the circuit. Until now, work has been done in analyzing example circuits exhibiting this behavior for BJT transistors. In this work we apply these methods to MOS circuits (Schmitt trigger, flip flop and multivibrator) and present the numerical results. To analyze the circuits we use the EKV drain current model as equivalent circuit model for the MOS transistors.

An Accurate Spatial Discretization and Substructural Method With Application to Moving Elevator Cable-Car Systems: Part I—Methodology

2011 ◽  
Author(s):  
H. Ren ◽  
W. D. Zhu
Keyword(s):  
Differential Algebraic Equations ◽  
Distributed Parameter ◽  
Algebraic Equations ◽  
Spatial Discretization ◽  
Matching Conditions ◽  
Generalized Coordinates ◽  
Axial Forces ◽  
Linear Algebraic Equations ◽  
Dependent Variables ◽  
Elevator Cable

A spatial discretization and substructure method is developed to calculate the dynamic responses of one-dimensional systems, which consist of length-variant distributed-parameter components such as strings, rods, and beams, and lumped-parameter components such as point masses and rigid bodies. The dependent variable, such as the displacement, of a distributed-parameter component is decomposed into boundary-induced terms and internal terms. The boundary-induced terms are interpolated from the boundary motions, and the internal terms are approximated by an expansion of trial functions that satisfy the corresponding homogeneous boundary conditions. All the matching conditions at the interfaces of the components are satisfied, and the expansions of the dependent variables of the distributed-parameter components absolutely and uniformly converge. The spatial derivatives of the dependent variables, which are related to the internal forces/moments, such as the axial forces, bending moments, and shear forces, can be accurately calculated. Assembling the component equations and the geometric matching conditions that arise from the continuity relations leads to a system of differential algebraic equations (DAEs). When some matching conditions are linear algebraic equations, some generalized coordinates can be represented by others so that the number of the generalized coordinates can be reduced. The methodology is applied to moving elevator cable-car systems in Part II of this work.

Well-Posed Formulations for Nonholonomic Mechanical System Dynamics

2020 ◽  
Vol 15 (10) ◽  
Author(s):  
Edward J. Haug
Keyword(s):  
System Dynamics ◽  
Mechanical System ◽  
Lagrange Multiplier ◽  
Variational Formulation ◽  
Direct Consequence ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
First Order ◽  
Well Posed ◽  
Problem Data

Abstract Four formulations of nonholonomic mechanical system dynamics, with both holonomic and differential constraints, are presented and shown to be well posed; i.e., solutions exist, are unique, and depend continuously on problem data. They are (1) the d'Alembert variational formulation, (2) a broadly applicable manifold theoretic extension of Maggi's equations that is a system of first-order ordinary differential equations (ODE), (3) Lagrange multiplier-based index 3 differential-algebraic equations (index 3 DAE), and (4) Lagrange multiplier-based index 0 differential-algebraic equations (index 0 DAE). The ODE formulation is shown to be well posed, as a direct consequence of the theory of ODE. The variational formulation is shown to be equivalent to the ODE formulation, hence also well posed. Finally, the index 3 DAE and index 0 DAE formulations are shown to be equivalent to the variational and ODE formulations, hence also well posed. These results fill a void in the literature and provide a theoretical foundation for nonholonomic mechanical system dynamics that is comparable to the theory of ODE.

scholarly journals A Generalized State-Space Aeroservoelastic Model Based on Tangential Interpolation

Aerospace ◽  
2019 ◽  
Vol 6 (1) ◽  
pp. 9 ◽  
Author(s):  
David Quero ◽  
Pierre Vuillemin ◽  
Charles Poussot-Vassal
Keyword(s):  
State Space ◽  
Reference Model ◽  
Differential Algebraic Equations ◽  
Summation Method ◽  
Algebraic Equations ◽  
Minimal Order ◽  
New Approach ◽  
Model Based ◽  
Generalized State Space ◽  
Generalized State

In this work, a new approach for the generation of a generalized state-space aeroservoelastic model based on tangential interpolation is presented. The resulting system of differential algebraic equations (DAE) is reduced to a set of ordinary differential equations (ODE) by residualization of the non-proper part of the transfer function matrix. The generalized state-space is of minimal order and allows for the application of the force summation method (FSM) for the aircraft loads recovery. Compared to the classical rational function approximation (RFA) approach, the presented method provides a minimal order realization with exact interpolation of the unsteady aerodynamic forces in tangential directions, avoiding any selection of poles (lag states). The new approach is applied first for the generation of an aerodynamic model for the bidimensional unsteady incompressible flow in the time domain. Next, an application on the generation of an aeroservoelastic model for loads evaluation of the flutter reduced order assessment (FERMAT) model under atmospheric disturbances is done, showing an excellent agreement with the reference model in the frequency domain. The proposed aeroservoelastic model of minimal order is suited for loads analysis and multivariable control design, and an application to a gust loads alleviation (GLA) strategy is shown.

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