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8/10/2019 Modelovanje i simulacija dinamickog sistema
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: :
: .
8/10/2019 Modelovanje i simulacija dinamickog sistema
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8/10/2019 Modelovanje i simulacija dinamickog sistema
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, ,
, . ,
MATLAB.
,
SIMULINK a. (m,k,c),
(F) (x(0), x'(0)).
:
m1=20 kg;
m2=35 kg;
k1=200 N/cm;
k2=100 N/cm;
k3=75 N/cm;
c1=10 Ns/cm;
c2=20 Ns/cm;
c3=30 Ns/cm;
:
-
,
- CDMATLAB.
3
8/10/2019 Modelovanje i simulacija dinamickog sistema
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.
.
.
,
, ,
. ,
() .
, ,
. ,
.
(
) .
, , m( 1) F(t)
x(t).
:
).t(xm)t(F..
= (1)
1.
4
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2, c. F(t)
x(t), :
).t(cx)t(Fo = (2)
2.
3.
:
),t(xb)t(F.
t = (3)
Ft , b () .
,
, ,
.
3.
5
8/10/2019 Modelovanje i simulacija dinamickog sistema
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4,
, .
4.
F(t) x(t),
),t(F)t(cx)t(xb)t(xm...
=++ (4)
,
5().
5(),
F(t) M(t), x1(t) x2(t)
1(t) i 2(t), m .
5.
, ,
.
, , , .
.
6
8/10/2019 Modelovanje i simulacija dinamickog sistema
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6.
. 1 .
k2 1 2, .
:
( ) ( ) 1..
111
.
2
.
21221
.
111 xmxFxxcxxkxcxk =+
++ (5)
( ) ( ) ( ) 1122211
2
.
2211
.
1
..
1
m:/xFxkkkxxcccxxm =++++ (6)
( ) ( )( )xF
m
1x
m
k
m
kkxx
m
c
m
ccxx 1
1
2
1
2
1
2112
.
1
2
1
211
.
1
..
=+
++
+ (7)
( ) ( ) ( )2
1
2
1
21
12
.
1
2
1
211
.
1
11
..
xm
k
m
kkxx
m
c
m
ccx
m
xFx +
++
+= (8)
7
8/10/2019 Modelovanje i simulacija dinamickog sistema
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2 .
:
( ) ( ) 2..
221
.
2
.
21222
.
323 xmxFxxcxxkxcxk =+
+ (9)
( ) ( ) ( ) 22123221.
2322
.
2
..
2 m:/xFxkkkxxcccxxm =++++ (10)
( ) ( )( )xF
m
1x
m
k
m
kkxx
m
c
m
ccxx 2
2
1
2
2
2
32
21
.
2
2
2
322
.
2
..
=+
++
+ (11)
( ) ( ) ( )1
2
2
2
32
21
.
2
2
2
322
.
2
22
..
xm
k
m
kkxx
m
c
m
ccx
m
xFx +
++
+= (12)
,
:
=
+
++
+
++
)t(F
)t(F
x
x
)kk(k
k)kk(
x
x
)cc(c
c)cc(
x
x
m0
0m
2
1
2
1
322
221
.
2
.
1
322
221
..
2
..
1
2
1
(13)
8
8/10/2019 Modelovanje i simulacija dinamickog sistema
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.
.
.
,
.
. ,
.
.
() F(x,y,y') = 0
y' = f(x,y).
:
y = f(x,y), y(x0) = y0, (14)
x0 y0
a < x
8/10/2019 Modelovanje i simulacija dinamickog sistema
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h . 0,2 0,1.
.
y(x) x+h:
( )( ) ( ),!n
xynhxy
0n
nn
=
=+ (16)
( ) ( ) ( ) ( ) ( ) ++++=+ x'''y!3
hx''y
!2
hx'yhxyhxy
32
(17)
.
.
. (x, x+h)
. (x, y), (, ) ( 7).
w1 w
2 y
:
( ) ( ) ( )
++=+
~~
21 y,xfhwy,xfhwxyhxy (18)
7.
10
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(, ) R:[x, x+h] x [y, y+k]
( )y,xfhk = ,
:
)y,x(fhyhyy
hxx
~
~
+=+=
+=
(19)
w1 w2 ''
(16) .
y x:
( ) ( ) ( ) ( ) ( ),hOx''y!2hx'yhxyhxy 3
2
++++=+ (20)
y y:
( ) ( ) ( ),hOh!2
fffhfxyhxy 32
yx++
+++=+
(21)
f (x, y).
, . h , w1=1 w2=0. . ,
h2.
.
f(, ) . :
( )
( )
( ) ( )( ) ( )32yyxy2xx
yx
~~
hOfhffhhf2hf
!2
1hfhfy,xf
fhy,hxfy,xf
+++
+++=
=++=
(22)
11
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f(, ) (18) :
( ) ( )
( ) ( ) ( ) +++++=+++++=+
2
yx221
y
2
2x
2
221
hfffwfhwwxy
ffhwfhwfhwfhwxyhxy
(23)
(21) (23) h2
:
w1+w2 = 1, w2= 1/2, w2= 1/2, (24)
, , ,
. = =1, w1 = w2 = 1/2.
= =1 (, )
R:[x, x+h] x [y, y+k],
, (x, x+h)
(x, y) (x+h, y+hf).
, .
= =1, w1 = w2 = 1/2 (18) (19)
:
( ) ( ) ( )fhy,hxfh2
1fh
2
1fhwxyhxy 1 +++++=+ (25)
:y = y(x ) f = f(x,y). F1=hf(x, y) F2=hf(x+h, y+F1)
(25) :
( ) ( ) ( )21 FF
2
1xyhxy ++=+ (26)
(25) (26) (26)
. , F1
F2,
y(x, h).
12
8/10/2019 Modelovanje i simulacija dinamickog sistema
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, , w1, w2 (24) .
.
.
:
( ) ( ) ( )4321 FF2F2F6
1xyhxy ++++=+ (27)
:
( )
( )34
23
12
1
Fy,hxfhF
F2
1y,
2
hxfhF
F2
1
y,2
h
xfhF
y,xfhF
++=
++=
++=
=
(28)
,
.
:
( )( )z,y,xg'z
z,y,xf'y
=
= (29)
:
( )00
00
z)x(zyxy
=
= (30)
( 27)
4 , F1, F2, F3,
F4 ( )xyy= G1, G2, G3,G4 ( )xzz=
13
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( ) ( ) ( )
( ) ( ) ( )4321
4321
GG2G2G6
1xzhxz
FF2F2F6
1xyhxy
++++=+
++++=+
(31)
, (27),
:
( )
( )z,y,xghG
z,y,xfhF
1
1
=
=
(32)
( )
( )334
334
223
223
112
112
Gz,Fy,hxfhG
Gz,Fy,,hxfhF
_______________________________
G2
1z,F
2
1y,
2
hxfhG
G2
1z,F
2
1y,
2
hxfhF
______________________________
G2
1
z,F2
1
y,2
h
xfhG
G2
1z,F
2
1y,
2
hxfhF
+++=
+++=
+++=
+++=
+++=
+++=
(33)
,
.
, :
( )Y,xF'Y = (34)
: ( ) 00 YxY = .
Y , Y' ,
F . x
x ,
14
8/10/2019 Modelovanje i simulacija dinamickog sistema
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. , . y1 = x
y1 = 1 y1(x0)=x0.
:
( )( ) 00 YxY
Y,xF'Y
==
(35)
n :
( )
( )
( ) )y,,y,y(fx'y
)y,,y,y(fx'y
)y,,y,y(fx'y
n21nn
n2122
n2111
=
=
=
( )
( )
( ) 0,n0n
0,202
0,101
yxy
yxy
yxy
=
=
=
:
( )( ) 00 YxY
YF'Y
=
=, :
[ ]
[ ]
[ ]
( ) ( ) ( )[ ]T0n02010
T
n21
T
n21
T
n21
xy,,xy,xyY
f,,f,fF
'y,,'y,'y'Y
y,,y,yY
=
=
=
=
:
( ) ( ) ( )4321 FF2F2F6
1xYhxY ++++=+ , :
( )
( )34
23
12
1
FYFhF
F2
1YFhF
F2
1YFhF
y,xFhF
+=
+=
+=
=
15
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ATLABA
ATLAB , ode23 ode45,
. (ode23)
.
ode45
.
n n
.
.
function xprim = ime(t, x), t , x .
.
ATLABU
[t, x] = ode23(@ime, [tp, tk], x0) [t, x] = ode45(@ime, [tp, tk], x0),
: @ime , tp
, tk , x0 .
: t ,
x . x0
tp,
tk. ODE () ,
.
16
8/10/2019 Modelovanje i simulacija dinamickog sistema
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(ode45)
( )
6.
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )1
2
2
2
32
21
.
2
2
2
322
.
2
22
..
2
1
2
1
2112
.
1
2
1
211
.
1
11
..
2123221
.
2322
.
2
..
2
1222112
.
2211
...
11
xm
k
m
kkxx
m
c
m
ccx
m
xFx
xm
k
m
kkxx
m
c
m
ccx
m
xFx
____________________________________________________
xFxkkkxxcccxxm
xFxkkkxxcccxxm
++
++
=
++
++
=
=++++
=++++
(36)
:.
2
.
24
.
1
.
132211 xxx;xxx;xx;xx ====== , 4321 x;x;x;x -
:
( ) ( ) ( )
( ) ( ) ( )
2
24
2
323
2
22
2
321
2
2.
4
1
14
1
23
1
212
1
21
1
21.
3
42
.
31
.
m
xFx
m
ccx
m
cx
m
kkx
m
kx
m
xFx
m
cx
m
ccx
m
kx
m
kkx
xx
xx
++
++
=
+++
++
=
=
=
(37)
ATLAB
MKC.m :
f uncti on xpr =MKC( t , x) ;
F1=300; %Si l a u [ N]
F2=200; %Si l a u [ N] c1=10; %Ul j ni pr i gusi vac 1 u [ Ns/ cm] c2=20; %Ul j ni pr i gusi vac 2 u [ Ns/ cm] c3=30; %Ul j ni pr i gusi vac 3 u [ Ns/ cm] m1=20; %masa t er et a 1 u [ kg] m2=30; %masa t er et a 2 u [ kg] k1=200; %Krutost opr uge 1 u [ N/ cm] k2=100; %Krutost opr uge 2 u [N/ cm]
k3=75; %Kr utost opruge 2 u [ N/ cm]
17
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8/10/2019 Modelovanje i simulacija dinamickog sistema
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7.
8.
( 7)
. .
19
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MATLAB
, .
.
,
''click and drag'' .
,
, .
, (
MATLAB ).
,
.
, .
MATLAB Workspace ''.mat'' ,
.
simulink MATLAB . SimulinkLibrary Browser .
Simulink Library Browser FileNewModel.
Simulink Library Browser
.
.
(sample time )
(+ )
( )
20
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(
, . -1 ).
display
( )
.
Format task bar u
. . Format Flip Block 180,
Format Flip Block 90 .
:
Sources
Continous ili Math
Sinks
21
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, .
( defaultu 10s). : Simulation Simulation Parameters Solver,
.
Simulation Start( , Ctrl+T).
displey ,
.
(. ) ,
. Workspace MATLAB .
Workspace Simulation Simulation Parameters
Workspace I/O Save to Workspace, Workspace
( default, tout
yout ). .
, MATLABA a plot(tout, yout).
6 :
(38)
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )1
2
2
2
3221
.
2
2
2
322
.
2
22
..
2
1
2
1
2112
.
1
2
1
211
.
1
11
..
2123221
.
2322
.
2
..
2
1222112
.
2211
...
11
xm
k
m
kkxx
m
c
m
ccx
m
xFx
xm
k
m
kkxx
m
c
m
ccx
m
xFx
____________________________________________________
xFxkkkxxcccxxm
xFxkkkxxcccxxm
++
++
=
++
++
=
=++++
=++++
22
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Podaci_Seminarski.m
.
Podaci_Seminarski.m
cl ear ; cl c; %=======================================================================% PODACI O SI STEMU %
%=======================================================================
F1=100; %Si l a u [ N] T_F1=0; %Vr i j eme poj avl j i vanj a si l e F1 u [s] F2=100; %Si l a u [ N]
T_F2=0; %Vr i j eme poj avl j i vanj a si l e F2 u [s] %- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - c1=10; %Ul j ni pr i gusi vac 1 u [ Ns/ cm]
c2=20; %Ul j ni pr i gusi vac 2 u [ Ns/ cm] c3=30; %Ul j ni pr i gusi vac 3 u [ Ns/ cm] %- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - m1=20; %masa t er et a 1 u [ kg] m2=30; %masa t er et a 2 u [ kg] %- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - k1=200; %Krutost opr uge 1 u [ N/ cm] k2=100; %Krutost opr uge 2 u [ N/ cm] k3=75; %Krutost opr uge 3 u [ N/ cm] %- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Crtanje_Seminarski.m
%======================================================================% CRTANJ E GRAFI KA%======================================================================f i gur e( 1) pl ot ( t , v1, ' b' , t , v2, ' r ' ) gr i d ont i t l e( ' Vr emenski odzi v br zi na mehani ckog t r ansl atornog si st ema' ) xl abel ( ' Vr i j eme [s] ' ) l egend( ' Br zi na - v1' , ' Br zi na - v2' ) yl abel ( ' Br zi na [cm/ s] ' ) gr i d of f
f i gur e( 2) pl ot ( t , x1, ' b' , t , x2, ' r ' ) gr i d ont i t l e( ' Vr emenski odzi v pomj eraj a mehani ckog t r ansl atornog si st ema' ) xl abel ( ' Vr i j eme [s] ' ) l egend( ' Pomj er aj - x1' , ' Pomj er aj - x2' ) yl abel ( ' Pomj eraj [ cm] ' ) gr i d of f
23
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38. 9.
9.
, ( x v)
Workspace MATLAB .
Crtanje_Seminarski.m ( 10) ( 11)
.
24
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10.
11.
25
8/10/2019 Modelovanje i simulacija dinamickog sistema
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.
f(x,y) (14)
.
f(x,y)
y(x).
ode45
,
.
. .
.
.
26
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