ME451
Kinematics and Dynamics
of Machine Systems
Dynamics of Planar Systems
December 9, 2010
Solving Index 3 DAEs using Newmark Method
© Dan Negrut, 2010
ME451, UW-Madison
Before we get started…
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Last Time:


Started discussion on the numerical solution of DAEs of Multibody
Dynamics using Newmark’s method
Today

Finish the discussion started last time





Marks the end of the Dynamics Analysis of Mechanical Systems chapter
Discuss the final exam
Class evaluation forms need to be filled out
Finish a bit early today, have to introduce Hammad in the graduate
student seminar at 12 noon.
Next time: last class

Equilibrium Analysis and Inverse Dynamics
2
Solution Strategy: Important Slide

This slide explains the strategy through which the numerical solution; i.e., an
approximation of the actual solution of the dynamics problem, is produced

Step 1: two integration formulas (Newmark in our case) are used to express
the positions and velocities as functions of accelerations


These are also called “discretization formulas”
Step 2: everywhere in the constrained equations of motion, the positions and
velocities are replaced using the discretization formulas and expressed in
terms of the acceleration

This is the most important step, since through this “discretization” the differential
problem is transformed into an algebraic problem


The algebraic problem, which effectively amounts to solving a nonlinear system, is
approached using Newton’s method (so again, we need to produce a Jacobian)
Step 3: solve a nonlinear system to find the acceleration and the Lagrange
multipliers
3
Stage 3: The Discretization
of the Constrained Equations of Motion

The Discretized Equations Solved at each Time-Step tn+1:
8
Än + 1 + © Tq (q n + 1 )¸ n + 1 ¡ Q A (t n + 1 ; q n + 1 ; q_n + 1 ) = 0
< Mq
:

4
1
¯h2
© (q n + 1 ; t n + 1 ) = 0
Above you see
and
functions of the accelerations:
, but remember that they are
Again, these are Newmark’s formulas
that express the generalized position
and velocity as functions of
generalized accelerations
Solving the Nonlinear System =0
~ Newton Method ~
5

Based on Newmark Integration formulas, the Jacobian is calculated as:

Corrections Computed as :
Note: subscripts “n+1”
dropped to keep
presentation simple
Simple Pendulum Example:
Looking at Stage 3
6
First.
At each time step (solving for the state of the system at tn+1…)
Increment tn+1 = tn + h
Define initial guess for accelerations
q_dotdot and lambdas (take values
from previous time step tn)
Find initial conditions for
general coordinate positions
and velocities that satisfy
position and velocity
constraint equations
Iterate to solve nonlinear
system obtained after
Newmark discretization of
DAEs.
Use Newton’s equations of
motion along with the
acceleration constraint
equations to find general
coordinate accelerations and
lambdas at time = 0.
Update tn+1 position and
velocity using Newmark’s
formulas and the most recent
values for acceleration and
lambdas.
8
>
<
2
>
: q
_n+ 1
h
q n + h q_n + h2 ( 1 ¡ 2¯) Ä
q n + 2¯Ä
q n+ 1
h
i
q_n + h ( 1 ¡ ° ) Ä
qn + ° Ä
q n+ 1
q n+ 1 =
=
i
Get Jacobian of discretized equation. Use
most recent q, q_dot, q_dotdot, and lambda
at n+1
Use most recent values at n+1 to compute
what each equation equals. Each equation
will probably not equal zero. These values
are called the residuals.
Yes. Refine
guess.
2
Compute the
correction
vector.
"
#
6
6
±Ä
q
= 66
±¸
4
Ã
M + ¯h 2
Correct the most recent
acceleration and lambda to
get better approximations for
accelerations and lambdas.
8
>
>
<
>
>
:
MÄ
q + © Tq ( q ) ¸ ¡ Q A ( q_; q ; t ) = 0
1 © q; t =
(
)
¯h 2
@( © T
QA
q¸ )
¡ @@
@q
q
0
3¡ 1
!
A
Q
¡ ° h @@
q_
© Tq 77
7
7
5
©q
"
q
Ä
¸
#( new)
0
"
=
q
Ä
¸
#( old)
"
¡
±q
Ä
±¸
#
Compute the largest value present in the correction factor matrix
and compare to a specified limit. This is the convergence check
(largest correction is easily computed as infinity norm in MATLAB)
Is value greater than
limit?
No.
7
¢ª ( Ä
q ( ol d) ; ¸ ol d)
Store acceleration and lambda for time tn+1. Use this accelerations to compute the
position and velocity at tn+1 and store these quantities as well.
residual
Newton-Type Methods:
[Geometric Interpretation]
8
Solving the Dynamics Problem using
Quasi-Newton Method
Än + 1 and ¸ n + 1 = 0 t he following nonlinear syst em:
² What are we aft er? We want t o solve for q
2
Än + 1 ; ¸ n + 1 ) ´ 4
ª (q
Än + 1 + © Tq (q n + 1 )¸ n + 1 ¡ Q A (t n + 1 ; q n + 1 ; q_n + 1 )
Mq
1
¯h2
3
5= 0
© (q n + 1 ; t n + 1 )
² Expression of J, t he Jacobian of t he discret izat ion nonlinear syst em:
2
µ
¶
@( © Tq ¸ )
@Q A
QA
2
M + ¯h
¡ @q
¡ ° h @@_
h
i
@q
q
6
@ª
@ª
6
J´
=
@q
Ä
@¸
4
©q
3
©Tq
7
7
5
0
² Here's what you are missing in simEngine2D:
{ Sensit ivity of t he react ion forces wit h respect t o t he generalized posit ions q:
@(©Tq ¸ )
@q
{ Sensit ivity of t he generalized applied forces wit h respect t o t he generalized posit ions q:
@Q A
@q
_
{ Sensit ivity of t he generalized applied forces wit h respect t o t he generalized velocit ies q:
@Q A
@q_
Solving the Dynamics Problem using
Quasi-Newton Method
Än + 1 ; ¸ n + 1 ) = 0,
² Conclusion from previous slide: when solving t he discret izat ion nonlinear syst em ª ( q
~
we are not going t o work wit h t he exact Jacobian J, but wit h an approximat ion of it called J.
² We will t hus rely on a Quasi-Newt on met hod since we dont have all t he sensit ivit ies available in
simEngine2D
² Expression of J, t he act ual Jacobian:
2
µ
@( © Tq ¸ )
2
¡
h
i
@q
6 M + ¯h
@ª
@ª
6
J´
= 4
@q
Ä
@¸
¶
A
@Q
@q
©q
~ t he subst it ut e Jacobian:
² Expression of J,
2
M
J~ ´ 4
©q
©Tq
0
3
5
3
A
Q
¡ ° h @@_
q
©Tq 7
7
5
0
Solving the Dynamics Problem using
Quasi-Newton Method
² Quant it ies dropped from t he expression of t he act ual Jacobian:
@(©Tq ¸ )
¯h
@q
2
¯h
Q
2@
A
@q
@Q A
°h
@q_
² Import ant quest ion: When can we get away wit h neglect ing t hese t erms?
{ T his should be ok if t he st ep-size h is very small: h ¼ 0:001. Not e: t his is a rule of
t humb, not always small enough...
² All quant it ies t hat we decided t o neglect are fairly st raight forward t o comput e, but we are not
going t o pursue t hem in ME451
² All t hese quant it ies are evaluat ed fully in ADAMS, which works wit h an [approximat ely]
accurat e Jacobian
First.
At each time step (solving for the state of the system at tn+1…)
Increment tn+1 = tn + h
Define initial guess for accelerations
q_dotdot and lambdas (take values
from previous time step tn)
Find initial conditions for
general coordinate positions
and velocities that satisfy
position and velocity
constraint equations
Iterate to solve nonlinear
system obtained after
Newmark discretization of
DAEs.
Use Newton’s equations of
motion along with the
acceleration constraint
equations to find general
coordinate accelerations and
lambdas at time = 0.
Update tn+1 position and
velocity using Newmark’s
formulas and the most recent
values for acceleration and
lambdas.
8
>
<
2
>
: q
_n+ 1
h
q n + h q_n + h2 ( 1 ¡ 2¯) Ä
q n + 2¯Ä
q n+ 1
h
i
q_n + h ( 1 ¡ ° ) Ä
qn + ° Ä
q n+ 1
q n+ 1 =
=
Get Jacobian of discretized equation. Use
most recent q, q_dot, q_dotdot, and lambda
at n+1
Use most recent values at n+1 to compute
what each equation equals. Each equation
will probably not equal zero. These values
are called the residuals.
Yes. Refine
guess.
8
>
>
<
>
>
:
MÄ
q + © Tq ( q ) ¸ ¡ Q A ( q_; q ; t ) = 0
1 © q; t =
(
)
¯h 2
0
Compute the
correction
vector.
Correct the most recent
acceleration and lambda to
get better approximations for
accelerations and lambdas.
"
q
Ä
¸
#( new)
"
=
q
Ä
¸
#( old)
"
¡
±q
Ä
±¸
#
Compute the largest value present in the correction factor matrix
and compare to a specified limit. This is the convergence check
(largest correction is easily computed as infinity norm in MATLAB)
Is value greater than
limit?
No.
12
Store acceleration and lambda for time tn+1. Use this accelerations to compute the
position and velocity at tn+1 and store these quantities as well.
residual
i
Final Exam Info

Tuesday, December 21, 5:05 PM. Room: ME1245
For simEngine2D, please make sure you support all the modeling elements that you
were assigned to handle in the MATLAB assignment 3:
http://sbel.wisc.edu/Courses/ME451/2010/Documents/MATLAB/matlabAssignment03.pdf

In terms of forces/torques, please make sure you support the ones you were
assigned to handle in the MATLAB assignment 5:
http://sbel.wisc.edu/Courses/ME451/2010/Documents/MATLAB/matlabAssignment05.pdf


13
Note: only the last 15 points (out of 100) of the final exam are tied to simEngine2D.
In other words, if you don’t even have a simEngine2D and answer all the other
questions right you’ll end up with a score of 85.
Final Exam, Rules of Engagement
² De¯ne an acf and adm pair of ¯les associat ed wit h t he type of analysis t hat you are supposed
t o carry out and t he model t hat you were given.
² Run simulat ions (K inemat ics, Dynamics, and Equilibrium) using t he simulat ion engine.
² Generat e a set of plot s t hat show t he t ime evolut ion of an at t ribut e of t he model (t he mot ion
of a point , t he value of a react ion force as a funct ion of t ime, et c.)
² Answer quest ions t hat are more t heoret ical in nat ure. For inst ance, \ How has been t he
Lagrange Mult iplier T heorem used in deriving t he equat ions of mot ion?" , \ Why are init ial
condit ions import ant in t he cont ext of Dynamics analysis?" , et c.
² Email t he TA and class inst ruct or a zipped direct ory t hat cont ains your code, adm/ acf ¯les,
png plot s of your result s, and t he answer t o t he t heoret ical quest ions. T he naming convent ion for t his direct ory should be \ Last NameME451.zip" . For inst ance, \ Negrut ME451.zip" .
T he answers t o t he t heoret ical quest ions should be typed in MS-Word in a ¯le called FinalExam.doc[x].
Final Exam, Comments
² If your simEngine2D does not use MAT LAB, it will be your responsibility t o have t he compiler
support allowing you t o run t he software during t he Final Exam.
² Running your simEngine2D code should also report t he amount of t ime it t ook for complet ing
a simulat ion. T his informat ion should also be included in t he email t o t he TA & inst ruct or.
² I will not insist on having simEngine2D t hat you use during t he exam be implement ed exclusively by you. However, in good fait h, you will have t o indicat e in t he email t hat you will be
sending t o t he TA & t he inst ruct or t he percent age of your cont ribut ion t o t he simEngine2D
code t hat you are using for t he exam. I will t hen underst and t hat t he remaining percent came
from code writ t en by ot her ME451 colleague[s]. T his is ¯ne, but should be acknowledged.
² If you cont ribut ed more t han 66% t o your simEngine2D, you qualify for ent ering t he race for
t he fast est solver. Winning t hat race t ranslat es int o an aut omat ic A-grade in t he course.
² One ot her aut omat ic A grade might be assigned for t he most general, ° exible, and neat ly
organized simEngine2D code.
Course Evaluation

Please provide comments on the back of the form

16
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
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
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
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