CHAPTER 11. Numerical Method in Dynamics : Solution Method for DAE

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CHAPTER 11. Numerical Method in Dynamics
Solution Method for DAE
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Kinematic analysis

• Kinematic analysis

Kinematics is the study of the position,
velocity, and acceleration of a system of
interconnected bodies that make up a multibody
systems, independent of the forces that produce
the motion. Thus, to perform kinematic analysis,
number of driver must be given as the same number
of the system D.O.F gtKinematically driven
system
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Kinematic analysis

• Assembly of a system
• Using unconstrained minimization method, minimize
  following cost
• function to find out assembled configuration.
• Position analysis
• Newton-Raphson method for n non-linear equations
  in n unknowns

• Taylor series expansion with first order yields
  following liberalized system

and setting

• When to stop iteration?

Velocity analysis
Acceleration analysis
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Kinematic analysis

• Computational flow chart

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Dynamic analysis

• Dynamic analysis
• Solving differential algebraic equations of motion

• With initial conditions

• To make use of mathematical library for solving
  differential equations,
• linear equation solver is necessary to explicitly
  compute acceleration
• and Lagrange multiplier in the form of

• Forming second order differential equation into
  two sets of first order
• differential equations as

where
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Dynamic analysis

• However, integrated value a, 4 must be satisfied
  following constraint equations.

• Reducing DAE system into ODE system
• Theoretically it is possible to reduce DAE system
  into ODE system by
• choosing independent generalized coordinates, to
  make use of
• mathematical foundation of ODE theory such as
  existence and
• uniqueness.
• Partitioning generalized coordinate into a set of
  independent and a set
• of dependent coordinate, using implicit function
  theorem as,

• as long as dependent coordinate u can be
  expressed by
• Independent coordinate v as

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Dynamic analysis

• Partitioning DAE system

• using dependent and independent coordinates
  yields following
• equations

(1)

• using velocity relationship

• explicit form of dependent velocity is

• explicit form of dependent acceleration is

(2)
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Dynamic analysis

• From the first equation of eq(1)

(3)

• Substituting eqs (2) and (3) into the 2nd
  equation of eq(1) yields

where

• However, this ODE form of differential equation
  is only valid when
• implicit function theorem is valid.
• To solve DAE system, there are three method with
  explicit numerical
• integrator
• GCP (generalized coordinate partitioning method)
• Constraint stabilization method
• Hybrid method