Modelovanje i simulacija dinamickog sistema

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  • 8/10/2019 Modelovanje i simulacija dinamickog sistema

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    : :

    : .

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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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    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

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    .

    f(x,y) (14)

    .

    f(x,y)

    y(x).

    ode45

    ,

    .

    . .

    .

    .

    26

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