An Algebraic Method for Analyzing Dynamic Systems

1
An Algebraic Method for Analyzing Dynamic Systems

• Wenqin Zhou, G. J. Reid and D. J. Jeffrey
• University of Western Ontario, Canada
• MMRC 2004

2
Outline

A
C
B
DynaFlex
RifSimp Implicit RifSimp
Numerical solver Implicit numerical solver
3
Traditional Subsystem Models
Parameters
Subsystem
Inputs
Outputs
Subsystem

• Inputs, outputs, and parameters are numeric
• Subsystems combined at simulation time

4
Symbolic Subsystem Models
Parameters
Relationships defined for Internal Variables
Output Expressions
Input Expressions

• Inputs, parameters and outputs are symbolic
  expressions
• Subsystems combined at formulation time

5
Introduction on DynaFlex

• Symbolic manipulation applied to mechanical
  systems to get the governing dynamical equations.
  
• Using graph theory to describe the mechanical
  system as input file, then automatically generate
  the dynamical equations.4
• Easy to get physical insight into the system
• Good for communication
• Facilitates subsequent real-time simulations.

6
DynaFlex Output

• The general forms for multibody dynamical system
  is
• Here is a DAE system of second order but usually
  of high differential index.

7
3D Spinning Top
The model equations for a top from DynaFlex
are without algebraic
constraints. They are
Fig1 The three-dimensional top. The centre of
mass is at and , Gravity
acts in the direction.
8
Two-dimensional slider crank

Fig2 The two-dimensional slider crank. The arm
of length and mass rotates and causes
the mass attached to the end of the arm of
length to move left and right. Each arm has
mass and moment of inertia , for
.
9
2D Slider Crank Equations
The model from the DynaFlex is
with the algebraic constraints
. where

10
The Spinning Disk
11
The Rigid Spatial Slider-Crank
12
How to solve these symbolic models?

• Three ways for solving these symbolic models
• A?B Just symbolic solve these equations. Too
  complicated for Maple to get analytical
  solutions.
• A?C Directly numeric solve these models. Hard to
  get the consistent initialization of numerical
  solution procedures.
• A?B?C Symbolic simplify then numeric solve. Such
  as using differentiation and elimination methods
  to get a simplified canonical form and including
  all constraints for the system. Then the initial
  value problem of the original DAE system has a
  unique solution. Like using Maple package RifSimp
  and diffalg.

13
Advantages and Difficulties

• The advantages for using \Dyn\ are
• Easy to get physical insight into the system
• Good for communication
• Facilitates subsequent real-time simulations.

• The difficulties with \Dyn\ are
• The generation of Large Expressions
• It is hard to analyse the equations when they are
  very big.

14
How does RifSimp work?

• For example, if the input system is ODE system,
• Step 1 define a ranking. like the default
  ranking
• Step 2 classify the whole system into two parts
  
• leading linear and leading
  non-linear
• Step3 solves for their highest
  derivatives, until it can no longer find any such
  equations. Case split based on the coefficients
  of the leading derivatives yields a binary tree
  of cases
• Step4 differentiated , then reduced
  with respect to the current set of and
  then with respect to the . 2,3

15
Application to 3D Tops

• First need to convert trig functions to
  polynomials with and
• get rational
  polynomial differential equations.
• Getting 24 cases and 9 cases after recording
  physical facts, like and . (Maple
  worksheet).
• Pay price the total degree increased and the
  complexity is higher.

16
Some Difficulties

• Matrix inverse for solving . For
  example if we try to invert a 6 by 6 matrix, the
  output from Maple is huge. So it is hard to get
  symbolic canonical forms.
• Membership test. If the leading linear part is
  too lengthy, it is harder to do reductions
  symbolically with respect to such lengthy
  equations.

17
New idea Implicit Rif-form

• Definition Implicit Rif form Let be a
  ranking. A system is said to be
  in implicit reduced involutive form if there
  exist derivatives such that is
  leading linear in with respect to
  (i.e. ) and
• is in reduced involutive form. 1

18
Application to multibody systems

• Theorem To the general multibody dynamical
  models, with the ranking defined by
  
• , the system
• are in implicit
  rif-form with
• in definition given by
• and
  where
• Please refer 1 for the proof.

19
Applications

• If without constraints, like 3d top, we can get
  implicit rif-form faster than doing matrix
  inversion but not all the cases from Rifsimp case
  split option.
• If with constraints, like 2d slider crank, we
  only need to do differentiation twice, then we
  get implicit rif-form, i.e. generic case, saving
  ourselves from membership testing.
• If we want to do case splitting, then we need to
  case splitting on .
• To higher index problem and PDAE problem we also
  can use the similar idea, as RifSimp works with
  PDE with constraints.

20
Reference

• 1 W. Zhou, D.J. Jeffrey, G.J. Reid, C. Schmitke
  and J. McPhee. Implicit Reduced Involutive Forms
  and Their Application to Engineering Multibody
  Systems. Submitted to Springer-Lecture notes in
  Computer Science (2004).
• 2 A.D. Wittkopf, G.J. Reid. The Reduced
  Involutive Form Package. Maple Software Package.
  First distributed as part of Maple 7 (2001).
• 3 A.D. Wittkopf. Algorthims and Implementation
  for Differential Elimination. Ph.D. Thesis. Simon
  Fraser University, Burnaby (2004).
• 4 P. Shi, J. McPhee. DynaFlex Users Guide,
  Systems Design Engineering, University of
  Waterloo (2002).