solving engineering system dynamics problems with matlab
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Copyright © 2007, New Age International (P) Ltd., PublishersPublished by New Age International (P) Ltd., Publishers
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74 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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86 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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MATLAB TUTORIAL 157
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158 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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172 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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MATLAB TUTORIAL 173
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174 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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MATLAB TUTORIAL 175
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178 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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180 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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182 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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Bode Diagram
Mag
nitu
de (
dB)
Pha
se (
deg)
Frequency (rad/sec)
40
20
0
– 20
– 40
– 60– 45
– 90
– 135
– 18010
– 110
010
110
2
9+
<4 4!
18
)18
6 56
;6 56- +#4 %6!F+ #6-
$641 + %"44- ? "$$$
7+ =4 4
184 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
1.4
1.2
0.8
0.6
0.4
0.2
0
1
0 0.5 1.51 2 2.5 3 3.5 4 4.5 5Time (sec)
Step Response
Am
plitu
de
:? 0
>?(<<5)(<!<"<%)!#
>>
+; 0 ( 7+ =4 4
H C C+&
!#AA5
A!A"A%
&
!#E!A!A5#
E"A"E4A#$E!A$"
Bode Diagram
Mag
nitu
de (
dB)
Pha
se (
deg)
Frequency (rad/sec)
40
20
0
– 20
– 40
– 60– 45
– 90
– 135
– 18010
– 110
010
110
2
MATLAB TUTORIAL 185
9+
<5
;@
);@
$ ##
;$ ##- +$4 #F+ $5-
# %$6$61 + 6#65#- ? $566"
1.4
1.2
0.8
0.6
0.4
0.2
0
1
0 0.5 1.51 2 2.5 3 3.5 4 4.5 5Time (sec)
Step Response
Am
plitu
de
%7))
39
( " (
" 9!*
& . *.'/0'1 / :
7 &
98:8;-+
= 3
+'
+(<#<!)'
,3.
3+ +
(+! +!) ##'
,3!.
3!+! +!
,333!.
186 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
33*3!
= 9
9, +9.'
/ 0?9*3'
/
(,1 9.!9)
I+&
3
&
E4A5E!A$
3!
&
<E#A$E"<$%E4A! $%%"E!<! "6#A6 $55#
E#A$E"A$%E4A! $%%"E!A! "6#A6 $55#
333!
&
<E#A$E"<$%E4A! $%%"E!<! "6#A6 $55#
E%A55E5A!5$E$A# 6""E#A5 665#E"A$ $64$E4
A4 655E!A# %$5
>9"
+7+ =4 "0
1.2
1
0.8
0.6
0.4
0.2
0
– 0.20 1 2 3 4 5 6 7
Am
plitu
de
Time (sec)
Step Response
%
MATLAB TUTORIAL 187
&8 *.'/0'11
3"
( + , ( "
3+
! +
, ( " ( , ( $
( $ ( ( , (
98:8;-+
; +
#'
()(4)( 5#)'
0
G & ; 7+ =4 #
20
– 20
– 40– 60– 80
– 100– 90
– 135
– 180
– 225
– 27010
– 110
010
110
2
Frequency (rad/sec)
Bode Diagram
Mag
nitu
de (
dB)
Pha
se (
deg)
0
&
98:8;-+
;
(5#5%)'
(%!5)'
0
G & ; 7+ =4 #
188 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
10
0
– 10
– 20
– 30
– 45
– 90
– 135
0
10– 1
100
101
102
Frequency (rad/sec)
Bode Diagram
Mag
nitu
de (
dB)
Pha
se (
deg)
! !
" # $
# % # &! # ! # %&
'()
*''+,')-
. /
!
"#
$"%%&%%%%
'(')(''
0 *#"
+((,-
./10
01'2"
"1'%.3
42.
+"567
+"87
MATLAB TUTORIAL 189
+"97"
+":7:
+"474
'0,'2:
+";
.+!
<++(
=++"(
/
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
?&/?/@?/'@/1'
A+2,2"(
/
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
?//?//
=-B*#","-6"C")D)E)
5
4
3
2
1
0
– 1
– 2
– 3
– 4
– 5– 3 – 2 – 1 0 1 2 3 4
x 10– 3
Real Axis
x 10– 3 Nyquist Diagram
Imag
inar
y A
xis
"
2 # % # 3!4
190 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
,'"-*'','
"-*','"-
.5
,*'%%
')'(')(''
7D+4
4"=*"4
.40
:F .40
7<,,"+"
+((,-F
"F3%1'
+"5"474
+"C+#*1'7
+":"7:
+":"G7'0,':
+"F-7F
'0,'2:"
+""+"G7
.+!
7<,,-+"
+((,-:
":3
+"5"474
+"C+#*'G7
+":"7:
+":"G7'0,':
+"F-7F
F1'/F
+"F-+"G7F
.+!
MATLAB TUTORIAL 191
H,+++":H=IHG+""(
,+
,
7G,""
7D+
,*'%%
')'(')(''
7D+4
4"=*"4
.40
FJJ +"40
G
'0,'
<++(
=++"(
>>>>>>>>>>>>>>>>>>>>>>>>>>
?/?/E
=*"4&'
=-G,"-6"C")D))
50
0
– 50
– 100– 90
– 135
– 180
– 225
– 27010
– 210
– 110
010
1
Frequency (rad/sec)
Bode Diagmram
Mag
nitu
de (
dB)
Pha
se (
deg)
""&'C+#*1'E)':"')'1:"G%)EE
F-%1')'E"+"G)EE
192 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
""&'C+#*'G):"')@&1:"G%')1&
F-%)1@F-+"GE&)&'
H,+++":H=IHG+""(
=++"(
>>>>>>>>>>>>>>>>>>>>>>>>>
?/?/E
=*""&'
)'''
F
E)1@
J
E
J
)'&
G
)E'E
=-G,"-6"C")D)
50
0
– 50
– 100– 90
– 135
– 180
– 225
– 27010
– 210
– 110
010
1
Frequency (rad/sec)
Bode Diagmram
Mag
nitu
de
(dB
)P
hase
(d
eg)
'
,'"" 67-
+=
!
" 8
9 # % # 3&
MATLAB TUTORIAL 193
*''')
'0"-
*'0)0
''),'")
. /
7D+
'
7D+4
4"=*"4
=
=.!40
.=
",<,>,+#*+
:F .=
: $:
:
:G'0,':
+((,-:
":3')'
+"G6"-7
.+!
<++(
=++"(
/
>>>>>>>>>>>>>>>>>>>>>>>>
?/?/
=*"4&'
194 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
=++"(
&'/''
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
?/?//''
:
)E
:G
')@@
)@
G6"-')1'E
=-G,"-6"C")D)1)
40
20
– 20
– 40
– 60100
135
90
45
0
0
– 4510
– 110
010
110
2
Frequency (rad/sec)
Bode Diagram
Mag
nitu
de (
dB)
Pha
se (
deg)
*''
'-
!
8 3
9 % # ! # 3
. /
7K">++
''
'(')('
,"
,%
+"
MATLAB TUTORIAL 195
",K">++
$,.,L
*,.,K">+"
$)'&)'K">+"
$)')'M
10
8
6
4
2
00 2 4 6 8 10
t Sec
Unit-ramp responseU
nit-
ram
p in
put a
nd o
utpu
t
Unit-ramp input
Output
!
6 % !
$ # 3: # 378
9 # & # 33 # 3: # 37
',
00+,0' -
.;,
''E'
E'
'(')'('
+
,
+"
",K">+
$,.,L
*,.,M
196 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
1.6
1.4
1.2
0.8
0.6
0.4
0.2
0
1
0 5 10 15 20t Sec
Unit-step response
Out
put y
(t)
78+"+'7@'7"",,
6-",3')/
6-",3')@/
+"N"%0')'
+"N"
')&''
*$ $
!N"%0')'
!N"
)''
$N+-*$%
$N+-
')@
''6-",')@1;3)'%
,"N"%0')'
,"N"
E)'''
<'''''
'',
8
9 % !
= # 3!
# 3!
# & #
*',
>7-$
MATLAB TUTORIAL 197
7K">++>,"
''
1
'(')('
,"
,%
+"
",K">++
$,.,L
*,.,M
$)'E)K">+"
$E)&)M
10
8
6
4
2
00 2 4 6 8 10
t Sec
Out
put
Unit-ramp input
Output
Unit-ramp response
!"
79$%')
''
1
'(')(
$%')0
,"
,%
+"
",8"$%')
$,.,L
198 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
1.2
1
0.8
0.6
0.4
0.2
0
– 0.20 2 4 6 8 10 12
t Sec
Response to input r1 = exp(– 0.7t)In
put a
nd O
utpu
t
Input r1 = exp (– 0.7 )t
Output
+ +
!" #!$
%&
!"#$
"#$
%%$
&$
'(! !$
)*(
(+''( &
,-
( &
MATLAB TUTORIAL 199
14
12
10
8
6
4
2
00 2 4 6 8 10
t Sec
Response to input ( ) = 3 + r t t
Out
put
c(
) a
nd in
put r
()
= 3
+
tt
t
!
& '
( ) *
( ) + +
, $ !
$ ! $ - $ + $ .
()
.
,
++ + + +
+
+ + + +.
/
,
/%&
!"#$
" /#$
,%% $
, % % $
, %%$
,.%%$
,%%/$
,",, ,,.,#$
200 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
*- ' !,$
*"//#$('*
)*(
(+- '0()
+('
!)(*('
8
6
4
2
0
– 2
– 4
– 6
– 8– 10 – 5 0 5
Real axis
Root-locus plot of G(s) H(s)
Imag
inar
y ax
is
"#
$0 ' '
( +
,
$ 1 $ 2
"* 3 !
%&
"#$
*'
'
/.&
/.
.
%&
!"#$
MATLAB TUTORIAL 201
"#$
*- ' !$
(''1 *
)*(
(+2- '0(
5
0
– 5– 6 – 4 – 2 0
Root Locus
Real Axis
Imag
Axi
s
$"%
& 0
*
( #! +
4
$ 5! $ 1 $ 4
'63 (
3
"#$
*'
'
.&
.
/
363
!"#$
"#$
1 (' !
*" #$('*$(''1 *
)*(
(41 ('0(
202 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
2
1
0
1.5
0.5
– 0.5
– 1
– 1.5
– 2– 2– 3 – 1 0 1 2 3
Real Axis
Nyquist Diagram
Ima
gin
ary
Axi
s
&'()
56**72'(6*()660-(*-!06-*((-
(6-'2''!(' '
* 0
*
( #, $ !
$ 4 $ .
& 63 ( ,#4"45"455
(
,
++ +
/
,
++ +
%&
!"#$
"/#$
'%%$
"*(!'#1 (' !'$
"* (! '#1 ('8 !'$
"*(!'#1 ('8 !'$
*(!* (! *(!
*"#$('*
)*(
(41 ('()*!'
+('
!)(*('
9
9
9
MATLAB TUTORIAL 203
3
2
1
0
– 1
– 2
– 3– 3 – 2 – 1 0 1 2 3
Real Axis
Nyquist diagram
Imag
inar
y ax
is
*'()+
*0*
( #1
$ 4 $ !
& 63
(
( *0
(+ +
. + +
%63 ( 7(
!"#$
".#$
! "#$
".#$
1 (' !
6
: **6
1 (' !
*"#$('*
)*(
/(
(
204 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
5
4
3
2
1
0
– 1
– 2
– 3
– 4
– 5– 5 0 5
Real Axis
Nyquist Diagram
Imag
inar
y A
xis
'()#
,8 '89!+:"
#7+;< ! & *
)1s5.0(s
7
+Gc(s)+
–
,-(
;*!;()< /0*6-'2('- &< 6
' !06)-*( ('062' (')(= >
? ?;*6-'2 & < 7?62
*'0*0 -(6*7*'6)0(-(-06)(=2*'0*0 -(
6'(*-'2 &< (')(=
/? ???
56-!'*! '-*( ?-!'*56'(!'0*!0
-!'*('
(,
++
MATLAB TUTORIAL 205
s = –2 + j2 3
jωj2
j1.5
j1
j0.5
–j0.510–1–2–3–4
90º 120º
σ
s = –2 + j 3
jω
–j0.510–1–2–3–4
90º
σ
j2
j1.5
j1
j0.530º
. " ## # - "/
#" 0 1 # 2
,#
07'-6@*06-!'* 6606-!'*
! '-.(**6=6')0?';()< /
A- (',
.
$
$
56)(,('(0*!6-((
. <
,
= − +
++ +
* ,
.
. <
= − +
+
* (
.
++
562*'0*0 -(06-!'''!(')(=
(
+
.
++
.
+
. +
56-'2*'0*0 -(06*()(''!('
.
. + +
56-!'''!'-'2*'0*0 -(('
. + +
206 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
BCB5DBE*)*!6 (2'*''- *='06*()(-!2
'''!'(')(=756 (2'*''- *='*'67(;()< /
%&
!" .#$
" .#$
!-" #$
-" . #$
%%$
' !$
' !--$
,-
'
*()(''!
.:!'''!
)*(
(F(2,+'''0*()(,'!:!','!
1.5
0.5
1
00 10.5 1.5 2 2.5 3 3.5 4 4.5 5
Original system
Compensated system
Out
puts
t Sec
Unit-Step Responses of Original System and Compensated System
,
38 '89!+2
#74$<4=
)s5.0(
12Gc(s)
+
– Lead compensator
3
MATLAB TUTORIAL 207
;*'(*-'2&<6)-*( (067
2'6*()((')(=?? ?56*0*6)0(-(-
(')(=
/????
A-6-!'*! '-*( ?
G'--!'*060*!
(,
++
-6'6@*06-!'***(66')
0?606-!'*! '-';()<
jωj2
j1
–j1
10–1–2
90º
σ–3
34 "/ # 0
351 #
56*0* (,
++
76*9! '(0*!6!)( -(('
<
,
= − +
++
* 9
<
= − +
++
56*0*6-!'*-!'
(
++
5600*7**'0*0 -(('
(
.
++
208 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
!
"
"
++ + +
!#$%$!#$
&
'(')'*
!+, , -.
!+$ , -.
!+, , " -.
!+$ " -.
!,/,,/$,.
$! 00.
! 00.
0$01200012
134 5 3 6 42
7 142
18 2
70,99016 2
7$0$"9013 2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
00 1 2 3 4 5 6 7 8 9 10
Uncompensated system
Compensated systemOut
puts
t Sec
Unit-Step Responses of Uncompensated and Compensated Systems
MATLAB TUTORIAL 209
'(')'*
!+, , , $-.
!+$ , $ ,-.
!+, , , " -.
!+$ " ,-.
!,/,,/$:.
$! 00.
! 00.
00$0120001#2
135 5 3 6 42
7 142
1; 8 2
1201 201#201 2
15
10
5
00 5 10 15
Inpu
t and
Out
puts
Uncompensated SystemCompensated System
t Sec
Unit-Ramp Responses of Uncompensated and Compensated Systems
!"#$#
%#&'(&()&## * !#&#"*
" #*$# *#+(#+#
#&, #*# * # # #
&-# *# #*# **#
# *&
210 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
8s4.3s
12 ++s
)1bs)(1as(K +++
–
++
D(s)
! $ $.
+ +
!
" 9 $ $
. + + + + +
! " 9
. . . .+ + + + + +
;<
=0ξ!,:ω!">
!#$,
<
?$,?@,:@"?"!?$,?"?$A
!?$"?:A?$A,
< <$
?"?.?9?.?.?.!,
"?.!$"
9?.?.!:A
.!$A,
!,,AA:0?!,
B;C
!
+ $-.
+ + +!
+ +$A,,,AAA , $
!+ +$,A ":9 $:,
MATLAB TUTORIAL 211
DB;C0
! $" :A $A,
+ + +! $, " $A
+ + +
0 E0
→ ∞
!,
→
!,
→
$, " $A
+ + +$
!,
(')'*
,
7F <G=0
*# #*# *
!+, , $ ,-.
!+$ $" :A $A,-.
!,/,,$/:.
+$0/$0-! 00.
0$
15 34 C ; 2
7 142
18 C ; 2
14
12
10
8
6
4
2
0
– 2
– 40 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t Sec
Response to Unit-Step Disturbance Inputx 10– 3
Out
put t
o D
istu
rban
ce In
put
212 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
0
!
$,A ":9 $:,
$" :A $A,
+ ++ + +
!
$,A "9 $A,
$" :A $A,
+ ++ + +
(')'*
, 7
FH$F0<
*# # *
!+, $,A"9 $A,-.
!+$ $" :A $A,-.
!,/,,$/:.
+0/0-! 00.
0
15 34 5; 2
7 142
18 5; 2
1.4
1.2
1
0.8
0.6
0.4
0.2
00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Response to Unit-Step Reference Input
Out
put t
o R
efer
ence
Inpu
t
t Sec
% "$#%#&'(&(01#
#.##" *#"&
MATLAB TUTORIAL 213
)1s5.1s)(5s)(3s(s
)5s2s(K2
2
++++++
–
+R(s) C(s)
I
!
:
: $: $
.
+ ++ + + +
!
: "
:
: 9 , $:
.
+ ++ + + +
'(')'*
$
H(')'*
!+, , , $ :-.
!+$ : 9 , $: ,-.
0
2!+#9 #: :-.72.71<2
15)B2
5
4
3
2
1
0
– 1
– 2
– 3
– 4
– 5– 8 – 6 – 4 – 2 0 2
Real Axis
Root Locus
Ima
g A
xis
$0 '
ω7
214 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
ω70 !ω<
:?:"?9?,?$:?.??:!,
ω:?:ω"?9ω?,?.ω?$:?.ω?:.!,
+:ω"#,?.ω?:.-?+ω:#9ω?$:?.ω-!,< <E0 0
:ω"#,?Iω?:.!, $
ω:#9ω?$:?.ω !,
<
ω!, ω"#9ω?$:?.!,
.!" 9 $:
− ω + ω −"
4 <"<$0
:ω"#+,?J#ω"?9ω#$:-ω#:ω"?F,ω#F:!,
,:ωA#ω"?:F:ω#F:!,
< (')'*
H(')'*
!+,: , # , :F: , #F:-.
-* *#3
!
#"F9A?$$:F#
#"F9A#$$:F#
"F9A?$$:F#
"F9A#$$:F#
,9$::
#,9$::
ω7ω!,9$::
.!" ,9$:: 9 ,9$:: $:
− + × −!$:9"ω!,9$::
.0 .
$:9"KIK,
%"$#%#&'(&(+3
#"
##.##"&
MATLAB TUTORIAL 215
K5s
3s
++
)3s(s
32 +
R(s) C(s)+
–
!I
:
++
+! "
9 $:
.
++ +
'(')'*
H
!+, , , $ -.
!+$ 9 $: , ,-.
0
2!+#A " #: :-.72.71<2
15)B2
5
4
3
2
1
0
– 1
– 2
– 3
– 4
– 5– 6 – 4 – 2 0 2 4
Real Axis
Root Locus
Imag
Axi
s
0I
7
0<
<
"?9?$:?.?.!,
216 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
4 !ω<0ω"?9ω?$:ω?.ω?.!,
ω"#$:ω?.?ω#9ω?.!,
< <E0 0
ω"#$:ω?.!, $
ω#9ω?.!,
<
ω!, #9ω?.!,
4 .<$0
ω"#$:ω?@9
ω!,
ω"?ω!,
ω!,ω!%4ω!<ω70 ω!<
.
.!9
ω!
9
@!"
0 .
"K.K,
%"$#%#&'(&((3
#"
#2*.* ##ζ#
#)&4
#
*# *2*#&
)5s)(3s(s
K
+++
–
(')'*
!+, , , $-.
!+$ 9 $: ,-.
0
2!+#A " #: :-.72.71<2
MATLAB TUTORIAL 217
15)B2
5
4
3
2
1
0
– 1
– 2
– 3
– 4
– 5– 6 – 4 – 2 0 2 4
Real Axis
Root Locus
Imag
Axi
s
Dζ ,LζL$θω7
jω
θ
ωn
–ζωnσ
Constant ζ line
0
θ!
ζωω
!ζ
'ζ!,A !#,F:?
,LL∞ . ζ ,A!#,F:?
<
!#,F:? $
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218 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
!" #$ ! %$ &" ' ($)'#%
* *
*)
*
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*
,"& *
&
+*&*
-)"' # $'& ! '&*#)," !
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,")#.#$$#) %"&##!/0,102$!
3 45
++
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3 45
3 45
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!
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;# #7-):
MATLAB TUTORIAL 219
%# #7< $ :
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
00 5 10 15 20 25 30 35 40
Time (sec)
Step Response
Am
plit
ude
t sec
!"##$"
%&
'(
) * +,)-. * (,
/#$%%
01(!0-2!(-
1)2)(0.6ss(s
1
++R(s) C(s)+
–
=) # !)$ &"&
'β
+β +
+
+β
β>
?&! β
''
+ +
@ "! " )'#)%)
220 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
6) %
# →
''
# →
+β +
'
# →
'
# →
+ +
="")$ % &" %. $& ) .
=) $#"2 ! &
'ω
ω ω + ω +
," ! ) ' $" . !#) '&'ω "!*"$#"$" !& A")""" "% #
40
20
0
– 20
– 40– 90
– 135
– 180
– 225
– 27010
– 110
010
1
Frequency (rad/sec)
Bode Diagram
Mag
nitu
de (
dB)
Pha
se (
deg)
!" "
!
," & # !)$ &"$" ) '&"2 ! "& ##AA"$)&$" !)$ &"&) .&"$" ) '-)"& )%)$! $" !&A B"! )'& )%&")$ % #) "' # =)"")& )%ωB-)")& )%& #"! )'& )%"&) "$" ) ' % ##0# A"!'$" ! ## ) )) &"# ! !# )%" # ! )$ ," $" ! A ," $" !# & " )$ $.#$ && )A ω B6))""! )'& )% BC!" ! ) '2"# !)$ !'") % ")"") 2
MATLAB TUTORIAL 221
,"& #!
β
β,"")& )%ωβ,")$"$#&"# !)$ .
β B
6)" && )&"# !)$ !'%
'
++
+
+
-)"! ) #) # β " '
β
,"&")$ '
'
++
,"$.#$ && )&")$ %"&
''
++ + +
," ! $" . !#) '&'ω'ω "!*&
40
20
0
– 20
– 40– 90
– 135
– 180
– 225
– 27010
– 110
010
1
Frequency (rad/sec)
Bode Diagram
Mag
nitu
de(d
B)
Pha
se(d
eg)
100
50
0
– 50
– 100
222 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
,"$" !&")$ % A" ' # ,"! ! 2 ))$ #," )'#)%) ," ")$. % &" "" % "# ' #%
=" .$$ . $$&")$ % "! # )$ %,")#.#$ && )&").$ )$ % !'%
3
4
*
++ + + +
3
4
+ + +
$)'#%
0/0,102$! " .$ . $$&")$. )$ %!'#," #! .$$) ' . $$) ' "!* *$)'#%
D 8.$$
3 45
3 45
)3 45
)3 45
EE*5
354$ 5
354$ ))5
$#7:7:
!
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3 45
3 45
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EE5
374$ 5
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MATLAB TUTORIAL 223
1.4
1.2
0.8
0.6
0.4
0.2
0
1
0 5 10 15 20 25 30 35
Out
puts
Unit-Step Responses of Compensated and Uncompensated Systems
Compensated System
Uncompensated System
t sec
!
#78.9 $9$&F$ 8)$ -%:
;# #7-):
%# #7< $ :
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;*78)$ -%:
20
18
16
14
12
10
8
6
4
2
00 2 4 6 8 10 12 14 16 18 20
Uncompensated System
Compensated System
t Sec
Out
put
s
Unit-Ramp Responses of Compensated and Uncompensated Systems
224 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
#"##$#$"#&8%&
'
) * ,) * 0,
/'),$%(-1(!
0-2!(-
=) # !.# )$ &"&
'
+ +
β β+ +
," $.#$ & & ) & " )$ % ''
# →
'" )'#)%) " '
%
# →
''
# →
'
+ +
,"&
0/0,102$! $#"2 ! !'# " ! "!
3 45
3 45
40
20
0
– 20
– 40
– 90
– 135
– 180
– 225
– 27010
– 1 100
Frequency (rad/sec)
Bode Diagram
Mag
nitu
de (
dB)
Pha
se (
deg)
– 60
– 80
101
102
##$%&''#(
MATLAB TUTORIAL 225
!"$" !&" )$ %A")""" "% #,! # !.# )$ )" ! )'& )%"$" . !#) '&'ω"$" )'& )%ω B=) #)"! )'& )% B )"" "$" .# !# ω B A0!## !.# )$ ) $'" &$" .# !#
=) &")& )&"$" .# !$&"# !.# )$ F"!")& )%ωB)$!"(&"$" .# !$&")$ ) #"! )'& )% ω B ")& )%ωBβ"' # &β,"' # &β) &") &"# $&")$
," ; $" .# !#φ!'%+αBβ" '!'
φ
β −β +
β)$φ*+A= A$" !) #)β6)
β,"")& )%ωBβ ω," && )&"$" .# !$&"# !.# )$
++
+ +
,"! )'& )%ω B G'G& 2,".&&"# !.# )$ ) 2 ω B""! )'& )% " $# !"#&#$2B) $ !" !""$2 B,")&"# ".2# 2#!'")& ),")& )&"# $ ω* B ω* B6)" && )&"# $&"# !.# )$ !'%
*
*
++
+ +
2%)!" && )&"# ! # $&")$ ) &" && )'&"# !.# )$ !
'*
*
++
++
+ ++ +
,"2 ! &"# !.# )$ ' %"&##!/0,102$! ," #!$#"!&
D
3 45
3 * 45
226 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
0
– 5
– 10
– 15
– 2090
45
0
– 45
– 9010
– 310
– 210
– 110
010
110
2
Frequency (rad/sec)
Bode Diagram
Mag
nitu
de (
dB)
Pha
se (
deg)
#)
,"$.#$ && )&")$ %
'' *
*
+ ++ +
+ +
*
+
** +*
+ ++ + + +
," ! $" . !# ) ' & " ! $.#$ & & )'' " "2 ! &!," ! !&##!/0,102$! H" " $#% # !")') &##E
3 * 45
&3 45
)'&
**+*
3 + 45
3 **+*45
MATLAB TUTORIAL 227
100
50
0
– 50
– 100– 45
– 90
– 135
– 180
– 225
– 27010
– 310
– 210
– 110
010
110
2
Frequency (rad/deg)
Bode Diagram
Pha
se (
deg)
Mag
nitu
de (
dB)
#*
!" "$" !! ! )'#).%) ## &
+
H '' *
*
+ ++ +
+ +
3
4
' '
' ' +
*
+
** +* +
+ ++ + + + +
," .$$ %"&##!/0,102$! " .$$) '"!
228 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
!""#$ %&
1.2
1
0.8
0.6
0.4
0.2
00 5 10 20 25 30 35 40
Am
litud
e
1.4
15Time (sec)
Step Response
'()""#(%"*)*%(#""+,-'.-/'()""#012"3"(""#01'()"23"*)("+4566
"&
7!)!""#($ %&
8)*'2&
%)*-&
MATLAB TUTORIAL 229
20
18
16
14
12
10
8
6
4
2
00 2 4 6 8 10 12 14 16 18 20
Time (sec)
Unit-Ramp Response of the Designed System
Am
plitu
de
! " " #$
#%&
!$'
+ +
'()))#92% )0)())""# #"(
")"
.(#92%2"""()"#:6;<(),)"#*ω)(#"
∠(ω:6;
'(/"))("+4566=#",-'.-/")
230 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
20
0
– 20
– 40
– 90
– 135
– 180
– 225
– 27010
– 1
Frequency (rad/sec)
Bode Diagram
Mag
nitu
de (
dB)
Pha
se (
deg)
– 60
100
101
4"4566=(9())"#;)"22)(#92%ω=)0'()"#(ω)(#92%(:/'() ()#%
" /
" 6
) *)'+,',-.
!$ /#&0 10 /
23
'
K
s(0.1s 1)(s 1)+ +Gc(s)+
–
>"()2")"
α
+α +
+
+α
2 3):+()3
→
α
+α +
'
+ + α
MATLAB TUTORIAL 231
. )#?α ∧'(
∧
'(/"))"#
+ +
6
+ +
"*)*%(#""+,-'.-/")'(/"))("+4566
*"
50
0
– 50
– 100– 90
– 135
– 180
– 225
– 27010
– 1 100
Frequency (rad/sec)
Bode Diagram
Mag
nitu
de (
dB)
Pha
se (
deg)
101
102
4"4566(()))));)@/23%4")())"#;2
φ ;:;1;;'()8()); 2
φ
− α+ α
φ ;
α"*)).2(""α
'"(2"#92ω0)ω0α"#()2")"+"()()8()))φ "22)("2)"#(+"2"
232 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
#92"ω0 α '()""#("#2)"()23)
ω0 α "(2""#(10α1(3*%
(
( ω =α
+ ω+ ωα
α
2
α
@=@/
'()"#A(ωA:=@/+(2(2"""ω@)0'(#"+2()(+)2""3#92%ω
α ω B@@
α ω
α
@
=6
" @
=6
++
) ∧
α
C2
@
=6
++
=@
=6@
++
'(""")##2"
=@
=6@
++
+ +
6
6=
=6 =6
++ + +
'(2""")##2"
4
5
6
6=
=6 =6 =
++ + + +
'(#""+,-'.-/")"2("23)("+4566
6=
=6 =6 =
MATLAB TUTORIAL 233
7 !""#>") %&
8)* 2&
%)*D&
1.4
1.2
1
0.8
0.6
0.4
0.2
00 0.5 1 1.5 2 2.5 3 3.5
Am
plitu
de
Step Response
t sec Time (sec)
'(#""+,-'.-/")"2()"23)("+4566
6=
=6 =6 =
"
7!)!""#>") %&
8)* 2&
%)*7!)E) %D&
234 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
5
4
4.5
3.5
3
2.5
2
1.5
1
0.5
00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t Sec
Unit-Ramp Response of Compensated System
Uni
t-R
amp
Inpu
t and
Sys
tem
Out
put c
(t)
!6 * '7'
6
8
8
8
8
= @ @ 6@
− − − −
6
8
8
8
8
1
6
8
8
8
8
9.'(#""+,-'.-/")%(""#(3%'("23("+4566@
:=:@:@:6@
4
:
4:
MATLAB TUTORIAL 235
8)* 2&
%)*D&
'("("+4566@
350
300
250
200
150
100
50
00 10 20 30 40 50 60 70 80
Time (sec)
Step ResponseA
mp
litud
e
t sec
!
)%+(4:)+"*)(")("+4566@
20
15
10
5
0
– 50 10 20 30 40 50 60 70
Time (sec)
Impulse Response
Am
plitu
de
!
"6 * '
4
5
,
,# " -
" # " ,; " -
236 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
-,-'.-/")""*))))2)""#(%3*"+
FF6
FF6=
FF
')##2"
61
G61G16=1
FF0040:
::6=:
4
6
:
4"(,-'.-/"+"*)(#""+))29)"
6
8
8
8
6=
− − −
6
8
8
8
1
6
6
8
8
8
1
#$5 *)'+,'/%
'
)8s()6s()4s()1s(
)5s()3s(10
++++++R(s) C(s)+
–
#$
MATLAB TUTORIAL 237
H <
()3))*#"&
&
IJ:6::::=:@
'&
'#*)2J:
))=&
>""2)"2)#")"&
4444:4
#4
<()3))*#")"&
#
#4
4#44
,")#"&
&
IJ:6::::=:@
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)
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161
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238 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
!"
#$ %
#$
&'"'( )
#*$#*$
#*$#*$#*$#*$
MATLAB TUTORIAL 239
#$
&'"'( )
#*$#*$
#*$#*$#+**$
%
240 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
!
25ss8s10s7s
502345 +++++
R(s) C(s)
","( -)
" #$%
./"" 0
,
1 23
#,4$
14442 #,4$3
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"#$
",#$
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& '
,
5, )
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MATLAB TUTORIAL 241
(#$) *& + & & *&
,
8s9s3s7s
7s5234 ++++
+R(s) C(s)
#$
2345
234
s7s5s9s
8s7s12s3s
+++++++
R(s) C(s)
#$
#$,12
,
12
#,4$
5, )
*
+*+*+**
14442 #,4$3
",#$
"#$
242 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
",#$
"#$
,123
123
#,4$
5, )
+*+*+**
+*+*+*+
14442 #,4$3
",#$3
"#$
",#$
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+ & &#$
!
,
-
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. ! .
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. , 0
!
,
-
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!
/ .
&
MATLAB TUTORIAL 243
! . .
. . !
!
,
-
-
-
5, 6 ( !7
#$112
-
− − −
−
8 #$
− + − +
−
99"#$%
997
991323
99#13*3*23
9913323
99: ##$:
1'#+*:*$4 '#+*:*$2
1:'#+*:+*:*$*'#*$4:'#+*:+*:*$2
++ *& '
-
-
2
!
,
− − −
!
,
-
-
3!
&
24! ,5 !
,
-
-
+ & 67 &#$
− − −
1 2
5, #$7
#$#1$1 2
+ − +
244 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
1 2
1# $# $ 2 + + +
+ − +
1 2 + +1 2
+ +
+
+ +
1 2
1 2
99"#$%
991323
991323
99123
993
991,42 #444$
,
5, !
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-
. ; .
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− −
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.
/
;
,
21!0/2-
133323
133323
123
3
"#444$
----
-
-
-
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MATLAB TUTORIAL 245
&
-
-
-
-
----
+
&
,,
14442 #,4$3
2 6&)+7
5, )
+*+*+**
+*+*+*+
& + +
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0 = ,
− − − −
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24,<>!,5-?-6.72
.
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991332
991332
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246 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
99
991,(4(2 #))))$
,(
*:
(
992 6&)+7
5, )
+*
++*
99"#($
"
: ;&#$
#$
#$ +
)9s)(7s)(5s)(2s(
)6s)(4s(10
++++++
R(s) C(s)+
–
;#$%
#$. !0
.#$0
@A#124124$
.#$0
!<#44$
1,42#4.B0$3
. 0
MATLAB TUTORIAL 247
14442#,4$
",#$3
. !" 0
%B "#$
%B ",#$
%B "#$
.#$%0
.#$0
="<#124124$
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14442 #,4$
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#$ !
>
#$
&'"'( )
#*$#*$
#*$#*$#*$#*$
>
#$
&'"'( )
#*$#*$
#*$#*$#+**$
>
248 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
!""# $
% %&
% % % %
!""# $
%& %
%& % '%%&
() )
MATLAB TUTORIAL 249
&
&
&
&&&
! "# $
%
& ' (
( ) *
( ( +
− − −
%
, -( * *.%/ %"(# ,
(
(
(
250 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
**+,,-,
**+,,-,
**+-,
**,
**0...
%%%
%
%
%
%
%
%
%%%
(//01
**0
0.7
0.6
0.5
0.4
0.3
0.2
0.1
00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time (sec)
Step Response
Am
plitu
de
MATLAB TUTORIAL 251
*
'
%
%
( *
+( 1
− −
%
%
%* *
( *
*
'
*
'
* (
( *
%
%
- *! '.
$
2 311 # / 1/
** 2
**+,-,
**3+,-,
**(+,-,
**4+,-,
**1...
0
– 50
– 1000
– 45– 90
– 135
– 100
0
100
180
0
– 18010
010
110
210
310
010
110
210
3
From : U(1) From : U(2)
Bode Diagram
To :
Y(2
)To
: Y
(2)
To :
Y(1
)To
: Y
(1)
Ph
ase
(deg
) ; M
agn
itu
de
(dB
)
Frequency (rad/sec)
34
*
'
%
%
( *
+( 1
−
%
%
%(
+(
, -* (.
%
%
5 -(.
$
5)6 #/ 1 #//. 768/
) 16/#) 168/... 68/....
252 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
**9:2;3#
**+,-,
**3+,-,
**(+-,
**4+-,
**68/.3.(.4
**#1
**<768/=
0.8
0.6
0.4
0.2
0
– 0.2
– 0.4
– 0.6
– 0.8
– 1 – 0.5 0 0.5 1Real Axis
Nyquist Diagram
Imag
inar
y A
xis
!
*
'
%
%
* *
1 (
− −
%
%
%* *
* (
*
'
*
'
* (
( *
%
%
%( (
( (
$
6
*
*
"7 #
"7 #
ωω
! '
*
"7 #
"7 #
ωω
! *
'
"7 #
"7 #
ωω
! '
'
"7 #
"7 #
ωω
34 * ' 8
6 6$
2/11/ 16/#:2;3) 168/
...
**9:2;3#
**+,-,
MATLAB TUTORIAL 253
**+,-,
**+,-,
**+,-,
**68/...
0.5
– 0.5
0
2
0
– 2
– 1 0 1 2 – 1 0 1
From : U(1) From : U(2)Nyquist Diagram
Imag
inar
y A
xis
To :
Y(2
)To
: Y
(1)
Real Axis
!
"9 ! !
*
'
%
%
* *$&
' (
− −
%
%
5 *$&
(
, -* (.
%
%
$
**2:
**+,-,
**+,-,
**+-,
**+-,
**+.%.-... ,
**.
**#1
**<>0=
**? <5)=
**6 <@//=
254 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
0.6
0.5
0.4
0.3
0.2
0.1
0
– 0.1
– 0.2
– 0.30 2 4 6 8 10 12
t Sec
Unit-Step response
Out
put
"#$%&
**2:
**+,-,
**+,-,
**+-,
**+-,
**97A #1 1//8/
** +B. ,-,
**+,-,
**+-,
**+-,
**+8.%.-... ,
**%+-C%=,.%...<=
**#1
**<>0 =
**? <5)=
**6 <@// 1/0 /=
**?..<@//=
MATLAB TUTORIAL 255
20
18
16
14
12
10
8
6
4
2
00 5 10 15 20
t Sec
Unit-ramp response
Out
put
and
unit-
ram
p in
put
Output
"#$%&
**2:
**+,-,
**+,-,
**+-,
**+-,
**/...
1.5
0.5
1
0
– 0.5
– 10 2 4 6 8 10 12
Time (sec)
Impulse Response
Am
plitu
de
"#$%&
9 6 $
*
'
%
%
* *
1 (
− −
%
%
%* *
* (
*
'
256 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
*
'
* (
( *
%
%
%( (
( (
*
'
**2
**+.,-,
**+,-,
**+,-,
**+,-,
**...
From : U(1) From : U(2)
Bode Diagram
To :
Y(2
)To
: Y
(1)
Am
plit
ud
e
0.4
0.2
– 0.2
– 0.4
0
2
1
0
1.5
0.5
0 2 4 6 8 10 12 0 2 4 6 8 10 12
Time (sec)
9
$
*
'
+
%
%
%
,
& '& &
* ( (
( * (
− − −
*
'
+
%
%
%
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(
, -( '& &.
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+
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5 -(.
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**+,,-,
**+-,
**+-,
MATLAB TUTORIAL 257
**+.%.-... ,
**.
**#1
**<>0=
**? <5)=
**6 <@// =
1.4
1.2
0.8
0.6
0.4
0.2
0
1
0 1 2 3 4 5 6 7 8 9 10t Sec
Unit-response
Out
put
y(t
)
#$%&
: ;
& '& & (
* ( ( (
( * ( (
( '& & (
− − −
(
(
(
( '& & (
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*
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+( '& & (-+ (-
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**+,,-,
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258 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
!"#$ !
%#&'(
)#&*!+#!++
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
00 0.5 1 1.5 2 2.5 3 3.5 54.54
t Sec
Unit-ramp response
Inpu
t and
out
put
! "#$%&'
(#$%&
,-
&..
.
MATLAB TUTORIAL 259
#!/&&0.&
#!
,
#!/0.
#!
,
#!/&&0.&
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.
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260 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
!" , . !).
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MATLAB TUTORIAL 261
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262 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
.=7.7-7:>6!2$#%
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MATLAB TUTORIAL 263
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264 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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MATLAB TUTORIAL 265
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266 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
52++62 #$ 24!!1?K, #!
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MATLAB TUTORIAL 267
) )")" ! )! !""!8@5'=>'A !) !
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!
ζω ω
7φ 2
!
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268 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
!
52++62 #$ # 64
Response to initial excitations15
10
5
0
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–100 1 2 3 4 5 6
t(s)
x(t)
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Response to initial excitations4
3.5
3
2.5
2
1.5
1
0.5
00 0.5 1 1.5 2 2.5 3
t(s)
)
ζ=0.05
ζ=0.1
ζ=0.2
ζ=1.3
ζ=1.5
ζ=2.0
MATLAB TUTORIAL 269
) )" =>'#$%&"ω !D
!ζ'K! !@A7 @A0D'
Response to initial excitations4.5
4
3.5
3
2.5
2
1.5
1
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00 1 2 3 4 5 6
t(s)
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270 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
6 ξ...9.ωGω,! 6
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mk
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)
MATLAB TUTORIAL 271
$) " 0676OLD7µ'
!µ6''$ ! @A'7 @A' )@A#$%&'
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6 6
6
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272 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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MATLAB TUTORIAL 273
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274 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
m
k c
v h
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dcos1 2
h
d
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MATLAB TUTORIAL 275
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276 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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MATLAB TUTORIAL 277
I1I2
M1(t)M2(t)
θ1(t) θ2(t)
k2[θ1(t)– θ2(t)]k1θ1(t)
)&
52D+#! 6$!#3!&)
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J
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7 FL FL
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278 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
42ω 26D+!()6 (#!#!ΘΘΘ.5 #3(6(! #! 42#3
.
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1
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x
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MATLAB TUTORIAL 279
52D+#! 6$!#
I θ<MθI :
42 I*
MFL
J
.
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−
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.
θ θ
I
.
I
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280 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
η
′ ω
C !
!− τ ω τ τ∫
. .
++ α + ω
.I
*
7
( ! α α− ω − ω ω
η. . ..
−− α + ω
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7 . .
.
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MATLAB TUTORIAL 281
52 ! 3!&)
θηΘ<η.Θ..N ,-
42 η
C
6
6∑ 76
η. .
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5
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6 6 6 6
=
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282 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
θ. 9.,:9 ! :;,- . ! :;,-
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MATLAB TUTORIAL 283
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284 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
ω.:;FL
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Θ.
:;
.
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304 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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306 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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308 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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312 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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314 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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316 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB
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MATLAB TUTORIAL 317
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