8/13/2019 Ch 10-1.doc

1/30

Chapter 10

Dynamic response using numerical methods

10.1 Numerical methods: introduction

The Euler Method

Consider the model

constrrydt

dy== , (10.1-1)

From the definition of derivative,

ttty

dtdy

+= )(lim0

If the time increment t is chosen small enough, the derivative

can be replaced by the approimate epression

t

tytty

dt

dy

+

)()((10.1-!)

"ssume that the right-hand side of (10.1-1) remains constantover the time interval ),( ttt + , and replace (10.1-1) by the

follo#ing approimation$

)()()(

tryt

tytty=

+

orttrytytty +=+ )()()( (10.1-%)

&his techni'ue for replacing a differential e'uation #ith a

difference e'uation is the uler method.

'uation (10.1-%) can be #ritten in more convenient form as

follo#s. "t the initial time ot , (10.1-%) gives

ttrytytty ooo +=+ )()()(

et ttt o +=1 . &hen

161

8/13/2019 Ch 10-1.doc

2/30

ttrytytty +=+ )()()( 111

&hat is, (10.1-%) is applied successively at the instants kt , #herektt kk +=+1 , and #e can #rite in general

ttrytyty kkk +=+ )()()( 1 (10.1-*)

It is convenient to introduce a time inde k such that

)( kk tyy

=

&hus, (10.1-*) becomes

,...!,1,0,)1(1 =+=+=+ kytrtryyky kkk (10.1-+)In this form, it is easily seen that the continuous variable )(ty

has been represented by the discrete-time variable ky . 'uation

(10.1-+) is a recursion relation or difference e'uation. It can be

solved recursively (se'uentially) at the instants ,...!,1,0=k ,

starting #ith the initial value )( oo tyy = .

In cases #here the subscript notation is inconvenient, the

variable ky is #ritten as )(ky . Care must be taen not to

confuse )(ky #ith )( kty . For ,...!,1,0=k , )(ky represents )(ty at

the values of ,...,, !1 tttt o= In this notation, (10.1-+) becomes

,...!,1,0),()1()1( =+=+ kkytrky (10.1-)ith ra constant, #e can easily solve the original differential

e'uation (10.1-1), so #e really have no need for (10.1-).

/o#ever, this is not the case for a general e'uation of the form

),( vyfy =

(10.1-)

"ssume the right-hand side to be constant over the interval),( 1+kk tt and e'ual to the value )(),(2 kk tvtyf . ith the

approimation (10.1-!), #e have the uler method for (10.1-).

)(),(2)()( 1 kkkk tvtytftyty +=+ (10.1-3)

162

8/13/2019 Ch 10-1.doc

3/30

&his is easily programmed on a computer. &he values of,),(, ttyt

oo and any parameters in the functions f and v must

be read into the program along #ith a stopping criterion.

&he uler method is the simplest algorithm for numerical

solution of a differential e'uation, but it usually gives the least

accurate results.

Example 10.1

Compare the eact solution and the uler method using a step

si4e 1.0=t for

1)0(, ==

ytyy #here 10 t

&he uler algorithm (10.1-3) for this model is

)()1.01()( 1 kkk tytty +=+ (10.1-5)

For this case, the eact solution can be obtained by separating

variables and direct integration. First #e #rite

tdty

dy =

and integrate to obtain

=tty

y

tdty

dy

0

)(

)0(

163

8/13/2019 Ch 10-1.doc

4/30

or

!ln

!)(

)0(

ty

ty

y =

&his can be rearranged as

)!6ep()0()( !tyty =

&he follo#ing table compares the results.kt )( kty

act uler

0.1 1.00+ 1.0000.! 1.0!0 1.010

0.5 1.*55 1.*15

1.0 1.*5 1.+*

&he error in the uler method increases #ith time. It is in error

by 0.+7 in the first step8 by the tenth step, the error is .157.

&he accuracy can be improved by using a smaller value of t

.

164

8/13/2019 Ch 10-1.doc

5/30

Selecting a step size

&hus far, #e have not considered the effect of the 9run time9 of

a numerical method. &his is the time re'uired for the soft#are to

perform the calculations, and it is fre'uently an important factor

in selecting an algorithm. It is difficult to mae a prior

determination of the run time for a given algorithm.

&he process of selecting a proper si4e t begins by considering

the dynamics of the model and ho# fast the input changes #ith

time. " trial value of t is then selected that is small in relation

to these considerations. &he trial solution for the response is

obtained #ith this value. &he step si4e is then reduced

significantly, and the responses are compared. If they agree tothe number of significant figures specified by the analyst, the

correct ans#er is taen to be the last solution. If not, t is again

reduced by a significant factor and the process is repeated until

the solution converges.

In strictly analytical terms, as t approaches 4ero, the solution

of the difference e'uation approaches that of the corresponding

differential e'uation. &his is the reason for using a small t .

&he error produced #hen t is not small enough to representthe differential e'uation accurately is termed truncation error.

/o#ever, numerical considerations as #ell as computer time

re'uire not maing t too small. &he reason is that the effect of

round-off error is greater for smaller t due to the larger

number of iterations re'uired.

165

8/13/2019 Ch 10-1.doc

6/30

10.2 Advanced numerical methods

In light of the large variety of e'uation types that are nonlinear,

it is no #onder that many different numerical methods eist for

solving them. /ere, #e consider t#o methods that are generally

useful. &he first is a so-called predictor-corrector method based

on the uler method but #ith greater accuracy. &he second

method is the :unge-;utta family of algorithms.

Trapezoidal integration

8/13/2019 Ch 10-1.doc

7/30

&he difference bet#een the uler and trape4oidal integration

methods is sho#n in Figure 10.1. &he shaded area represents the

approimation to the integral

+1

)(k

k

t

t

dv

=oth methods use a rectangle of #idth t . &he uler method

taes the height of the rectangle to be )( ktv and approimates

the integral as$

+

=

1

)()(

k

k

t

t

kttvdv

>n the other hand, the trape4oidal method taes the rectangle?s

height as the average value @v of )( ktv and )( 1+ktv andapproimates the integral #ith (10.!-%)

167

8/13/2019 Ch 10-1.doc

8/30

Figure 10.1 Comparison of the uler and trape4oidal integration

approimations. &he shaded area represents the approimation

to the integral. (a) uler. (b) &rape4oidal.

redictor!corrector methods

"s can be seen from Figure 10.1, the uler method can haveserious deficiency in problems, #here the variables are rapidly

changing, because the method assumes the variables are

constant over the time interval t . "s a result, its truncation

error can be sho#n to be the order of !)( t .

>ne #ay of improving the method #ould be by using a better

approimation to the right-hand side of the model

168

8/13/2019 Ch 10-1.doc

9/30

),,( tvyfdt

dy= (10.!-+)

#here v is the input function. &he uler approimation is

[ ]kkkkk ttvtyfttyty ),(),()()( 1 +=+ (10.!-)

8/13/2019 Ch 10-1.doc

10/30

Euler predictor:

),,(1 kkkkk tvyftyx +=+ (10.!-5)

Trapezoidal corrector:

[ ]),,(),,(!

1111 ++++ +

+= kkkkkkkk tvxftvyft

yy (10.!-10)

&his algorithm is sometimes called the modified uler method.

/o#ever, note that any algorithm can be tried as a predictor or a

corrector. &hus many methods can be classified as predictor-corrector, but #e #ill limit our treatment to the modified uler

method. &he truncation error for this method is of the order of%)( t , a significant improvement over the !)( t error of the

basic uler method.

For purposes of comparison #ith the :unge-;utta methods to

follo#, #e can epress the modified uler method as$

),,(1 kkk tvyftg = (10.!-11)

),,( 11! ttvgyftg kkk ++= + (10.!-1!)

)(!

1!11

ggyykk ++=+ (10.!-1%)

170

8/13/2019 Ch 10-1.doc

11/30

Example 10.2

Ase the modified uler method #ith 1.0=t to solve the

e'uation 1)0(, == ytydt

dyup to time 0.1=t .

8/13/2019 Ch 10-1.doc

12/30

replace 1+kx on the right-hand side of that e'uation in order to

get an improved estimate of 1+ky . &his is repeated until the

desired accuracy is achieved8 then the net t interval is treated

the same #ay. /o#ever, this modification obviously increases

the compleity of the programming.

"unge!#utta methods

&he &aylor series representation forms the basis of several

methods for solving differential e'uations, including the :unge-

;utta methods. &he &aylor series may be used to represent the

solution )( tty + in terms of )(ty and its derivatives as follo#s.

...)()(-

1)()(

!

1)()()( %! ++++=+

tyttyttyttytty (10.!-1*)

&he re'uired derivatives are calculated from the differential

e'uation. For an e'uation of the form

),( ytfdtdy = (10.!-1+)

these derivatives are

),()( ytfty =

dt

dfty =

)(

etc.,)(!

!

dt

fdty =

(10.!-1)

#here (10.!-1+) is to be used to epress the derivatives in terms

of only )(ty . If these derivatives can be found, (10.!-1*) can be

used to march for#ard in time. In practice, the high order

derivatives can be difficult to calculate, and the series (10.!-1*)

is truncated at some term. &he number of terms ept in the

series thus determines its accuracy. If terms up to and including

172

8/13/2019 Ch 10-1.doc

13/30

nthderivative of y are retained, the truncation error at each step

is of the order of the first term dropped-namely,

1

11

)C1(

)(

+

++

+

n

nn

dt

yd

n

t

(10.!-1)

sometimes the closed-form epression for the series can be

recogni4ed. For eample, consider the e'uation

0,!

8/13/2019 Ch 10-1.doc

14/30

)(1

)()(

tyta

tytty

=+ (10.!-15)

8/13/2019 Ch 10-1.doc

15/30

8/13/2019 Ch 10-1.doc

16/30

1!1 =+ww (10.!-!+)

!

11 =w

(10.!-!)

!

1! =w (10.!-!)

&hus, the family of second-order :unge-;utta algorithms is

categori4ed by the parameters ),,,( !1 ww , one of #hich can be

chosen independently. &he choice%

!= minimi4es the

truncation error term. For 1= , the :unge-;utta algorithm(10.!-!1)-(10.!-!%) corresponds to the trape4oidal integration

rule if f is a function of only t , and is the same as the

predictor-corrector algorithm (10.!-5)-(10.!-10) for a general),( tyf .

%ourth!order algorithms

&he algorithm is

**%%!!111 gwgwgwgwyy kk ++++=+ (10.!-!3)

)(,2

)(,2

),(

),(

1%%%%%!%%*

1!!!!!%

111!

1

gggyhthfg

ggyhthfg

gyhthfg

ythfg

kk

kk

kk

kk

++++=

+++=

++=

=

(10.!-!5)

Comparison #ith the &aylor series yields eight e'uations for the

ten parameters. &hus, t#o parameters can be chosen in light of

other considerations. &hree common choices are as follo#s.

176

8/13/2019 Ch 10-1.doc

17/30

1.Gill's method. &his choice minimi4es the number of memory

locations re'uired to implement the algorithm and thus is #ell

suited for programmable calculators.

!611

!618!611

18!61

8%6)!611(8%6)!611(8-61

%

%!

%!1

%!*1

+=

==

===

+====

wwww

(10.!-%0)

2.Ralston's method. &his choice minimi4es a bound on the

truncation error.

0+05-+1-.%81+3+5-*.083%!3-*-.%

18*++%!+.08*.0

1113*3.08!0++%+-0.18++1*30--.081*-0!3.0

%!%

%!1

*%!1

===

===

====

wwww

(10.!-%1)

&.lassical method. &his method reduces to

8/13/2019 Ch 10-1.doc

18/30

Example 10.3

Ase the classical :unge-;utta parameter values (10.!-%!) #ith a

step si4e 1.0== th to solve

1.0toup1)0(, ===

tytyy

&he algorithm for the classical parameter value is

)101(0,2

)!

1

!

1(

!

1,

!

12

)!

1,

!

1(

),(

-

1

%

1

%

1

-

1

1%!*

1!%

1!

1

*%!11

gggyhthfg

ggyhthfg

gyhthfg

ythfg

ggggyy

kk

kk

kk

kk

kk

++++=

+++=

++=

=

++++=+

For the given problem, tyf = . For the first iteration,1,0,0 00 === ytk

and

0100+01!+.0)00+01!+.01)(1.00(1.0

00+01!+.0)00+.0!

11)(0+.00(1.0

00+.0)0!

1

1)(0+.00(1.0

0)0(1.0

*

%

!

1

=++=

=++=

=++=

==

g

g

g

g

&hus,

00+01!+!1.1)0100+01!+.0(-

1)00+01!+.0(

%

1)00+.0(

%

10

-

11)1.0(

1 =++++== yy

&his ans#er agrees #ith the eact solution )!6ep( !t . &he rest of

the solution is as follo#s.t 0.1 0.! 0.5 1.0

)(ky 1.00+01!+!1 1.0!0!01%* 1.*55%0!%! 1.*3!100

&he eact solution at 0.1=t , to ten figures, is 1.*3!1!1. &he

numerical solution is correct to seven figures.

178

8/13/2019 Ch 10-1.doc

19/30

10.& Conversion to state varia'le $orm

&here are several #ays of converting a model to state variable

form. If the variables representing the energy storage in the

system are chosen as the state variables, the model must then be

manipulated algebraically to produce the standard state variable

form. &he re'uired algebra is not al#ays obvious. /o#ever, #e

note here that the model

=

)(,,...,,1

1

tvdt

yd

dt

dyytf

dt

ydn

n

n

n

(10.%-1)

#ith the input )(tv can al#ays put into state variable form by

the follo#ing choice of state variables$

1

1

!

1

=

=

=

n

n

ndt

ydx

dt

dyx

yx

(10.%-!)

&he resulting state model is$

[ ])(,,...,,, !1

1

%!

!1

tvxxxtfx

xx

xx

xx

nn

nn

=

=

=

=

(10.%-%)

&he numbering scheme is not uni'ue8 in fact, #e often find thefollo#ing state variable choice$

179

8/13/2019 Ch 10-1.doc

20/30

yy

dt

dyy

dt

ydy

dt

ydy

n

n

n

n

n

n

=

=

=

=

1

!

!

!

1

1

1

(10.%-*)

#hich gives the model

[ ]

1

!%

1!

111 )(,,...,,,

=

=

=

=

nn

nn

yy

yy

yy

tvyyytfy

(10.%-+)

Coupled higher order models

'uation (10.%-1) is some#hat general in that it can be

nonlinear, but it does not cover all cases. For eample, considerthe coupled higher order model

),,,,(1

= zzyytfy (10.%-)

),,,,(!

= zzyytfz (10.%-)

Choose the state variables as$

=

=

=

=

zx

zx

yx

yx

*

%

!

1

(10.%-3)

&hen the state model is$

180

8/13/2019 Ch 10-1.doc

21/30

),,,,(

),,,,(

*%!1!*

*%

*%!11!

!1

xxxxtfx

xx

xxxxtfx

xx

=

=

=

=

(10.%-5)

State e(uations $rom trans$er $unctions

e no# illustrate a method for using the transfer function to

rearrange the diagram so that state variables can be identified.

&he order of the system, and therefore the number of state

variables re'uired, can be found by eamining the denominator

of the transfer functions. If the denominator polynomial is oforder n, then nstate variables are re'uired. Dhysical

considerations (integral causality) can be used to sho# that the

outputs of integration processes )61( s can be chosen as state

variables.

&o illustrate ho# a state model can be derived from a transfer

function, consider the follo#ing first-order model #ith

numerator dynamics$

1

10

)(

)()(

as

!s!

s"

s#sT

+

+== (10.%-10)

#here the input is u and the output is y . Cross-multiplying and

reverting to the time domain, #e obtain

u!u!yay 101 +=+

(10.%-11)

"n alternative #ay of obtaining the differential e'uation is by

dividing the numerator and denominator of (10.%-10) by s .

)(

)(

61

6)(

1

10

s"

s#

sa

s!!sT =

+

+= (10.%-1!)

181

8/13/2019 Ch 10-1.doc

22/30

&he essence of the techni'ue is to obtain a 919 in the

denominator, #hich is then used to isolate )(s# . Cross-

multiplying gives

[ ] )()()(1

)()()()( 0111

01 s"!s#as"!

ss"

s

!s"!s#

s

as# +=++=

&he term multiplying s61 is the input to an integrator8 the

integrator?s output can be selected as a state variable x . &hus,

[ ] [ ])()()(1)()(1)(

)()()(

011111

0

s"!as$as"!s

s#as"!s

s$

s"!s$s#

==

+=

or

u!a!xax )( 0111 +=

(10.%-1%)

u!xy0

+= (10.%-1*)

&he bloc diagram is sho#n in Figure 10.!. Compare (10.%-11)

#ith (10.%-1%), and note that the derivative of the input does not

appear in the latter form.

Figure 10.! &he bloc diagram for (10.%-1%) and (10.%-1*).

182

8/13/2019 Ch 10-1.doc

23/30

Eo#, consider the second-order model for the mechanical

vibration problem. Its transfer function is$

kcsmss%

s$sT

++==

!

1

)(

)()( (10.%-1+)

ivide by !ms to obtain a unity term.

!

!

661

61

)(

)()(

mskmsc

ms

s%

s$sT

++==

Cross-multiply and rearrange to identify the integration

operators.

+=

+=

)()(11

)(1

)(1

)()()(!!

s$m

ks%

mss$

m

c

s

s%ms

s$ms

ks$

ms

cs$

(10.%-1)

efining the state variables as the outputs of the integrators, #esee that

=

+==

)()(11

)(

)()(1

)()(

!

!1

s$m

ks%

mss$

s$s$m

c

ss$s$

or

=

+=

)()(11

)(

)()(1

)(

1!

!11

s$m

ks%

mss$

s$s$m

c

ss$

Figure 10.%a is the bloc diagram derived from these relations.

183

8/13/2019 Ch 10-1.doc

24/30

Figure 10.% &he bloc diagram for the vibration model (10.%-

1+). (a) iagram for the state model (10.%-1) and (10.%-13). (b)

iagram for the state model (10.%-15) and (10.%-!0).

&he corresponding state e'uations in standard form are$

!11 xxm

cx +=

(10.%-1)

fm

xm

kx

11!

+=

(10.%-13)

#here xx =1 .

184

8/13/2019 Ch 10-1.doc

25/30

'uation (10.%-1) can also be arranged as follo#s. et

)()(

)()(

1!

1

ss$s$

sm$s$

=

=

&hen

= )()()(

1)( 1!! s$

m

ks$

m

cs%

ss$

&hese relations give the diagram sho#n in Figure 10.%b. &he

resulting state model #ith mxx =1 is

!1 xx =

(10.%-15)

fxm

cx

m

kx +=

!1! (10.%-!0)

185

8/13/2019 Ch 10-1.doc

26/30

State varia'le models and numerator dynamics

&he general time-invariant model #ith numerator dynamics can

be #ritten as$

01

1

1

01

1

1

...

...

)(

)(

asasasa

!s!s!s!

s&

s#

n

n

n

n

n

n

n

n

++++

++++=

(10.%-!1)

ividing the numerator and denominator byn

nsa gives

nnn

nn

nn

sss

sss

s&

s#

++++

++++=

01

11

1

0

1

1

1

1

...1

...

)(

)(

(10.%-!!)

#here nii a! 6= and nii aa 6= . &his gives

{ }...)()(2)()()( !!1

11

1

00

!

!!

1

11

+++=

++

+

+=

s#s&ss#s&ss&

#'s()s&'s(*+,-......

#'s()s&'s( *+, -

#'s()s+&'s(,-&'s(-#'s(

nnnnn

*n

*

n*n*

nnn

e choose the output of each integrator )61( s to be a state

variable. &hus,

)()(2)(

.

.

.

)()()(2)(

)()()(2)(

)()()(

00

1

%!!

1

!

!11

1

1

1

s#s&ss$

s$s#s&ss$

s$s#s&ss$

s$s&s#

n

nn

nn

n

=

+=

+=

+=

(10.%-!%)

Ase the first e'uation in (10.%-!%) to eliminate G(s) in the

remaining e'uations. &hus,

)()()()(.

.

)()()()()(

)()()()()(

1000

%1!!!!

!11111

s$s&ss$

s$s$s&ss$

s$s$s&ss$

nn

nnnn

nnnn

=

+=

+=

186

8/13/2019 Ch 10-1.doc

27/30

&his gives the differential e'uation set

100

%1!!

!111

!

1

xvx

xxvx

xxvx

n

n

n

n

n

=

+=

+=

(10.%-!*)

#here

niii =

(10.%-!+)

1xvy

n += (10.%-!)

&he original initial conditions of the problem are given in terms

of y , as .),0(),0(),0( etcyyy

'uations (10.%-!*) and (10.%-!) can

be used to find the relation bet#een )0(ix and ),...0(),0(),(

yytv

From (10.%-!), at 0=t $

)0()0()0(1 vyx n= (10.%-!)

"lso, from the first e'uation in (10.%-!*), #e have

vxxx nn 1111!

+= (10.%-!3)

187

8/13/2019 Ch 10-1.doc

28/30

ifferentiating (10.%-!) and substituting gives

vvvyy

vvyvy

vxvyx

nnnn

nnnnn

nnn

111

111

111!

)(

++=

+=

+=

Henerali4ing this procedure and denoting ii dtyd 6 by )(iy gives

the result$

vvvv

yyyx

inin

i

n

i

n

in

i

n

i

i

11

)!(

1

)1(

1

)!(

1

)1(

...2

...

++

+

+++

+++=

(10.%-!5)

#here ni ,...,!,1=

&hus the initial values )0(ix can be found from (10.%-!5) given

the initial values of ),(),( tvty and their derivatives. Interpreting0=t as e'uivalent to = 0t , #e can tae 0...)0()0( ===

vv . In this

case (10.%-!5) gives

)0(...)0()0()0( 1)!(

1

)1(

yyyx ini

n

i

i +

+++= (10.%-%0)

&he transformation to state variable form is summari4ed in

&able 10.1

188

8/13/2019 Ch 10-1.doc

29/30

&able 10.1 " state variable form for numerator dynamics

For eample, consider

-+10

!%

)(

)(!

!

++

++=

ss

ss

s&

s#(10.%-%1)

#here 0for0)(and,)0(,*)0( ===

ttvyy . &he state model (10.%-

!*) and (10.%-!) is$

!11 +.0!+.0 xxvx +=

(10.%-%!)

1! -.01*.0 xvx =

(10.%-%%)

11.0 xvy += (10.%-%*)

&he initial conditions are, from (10.%-%0),

*)0()0(1 == yx (10.%-%+)

189

8/13/2019 Ch 10-1.doc

30/30

5!)0(+.0)0()0(! =+=+=

yyx (10.%-%)

'uations (10.%-%!)-(10.%-%*) are interpreted for 0>t . &hus,

from (10.%-%*)-(10.%-%+), #e see that*)0(1.0)0( ++=+ vy

. If)(tv

is a unit step, then 1)0( =+v and 1.*)0( =+y .