Методическое пособие по решению задач геометрического моделирования в системе MathCAD. Найханов В.В. - 27 стр.

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27
ORIGIN 0:= M50:=
a0 v() 1 3v
2
− 2v
3
+:= a1 v() 3v
2
2v
3
−:= b0 v() v 2v
2
− v
3
+:= b1 v() v
2
− v
3
+:=
i0M..:= T
i
i
2π
M
⋅:= X
i
sin 2 T
i
⋅
()
:= Y
i
sin 3 T
i
⋅
()
:=
i0M1−..:= h
i
T
i1+
T
i
−:=
i1M1−..:= μ
i
h
i
h
i
h
i1−
+
:= λ
i
1 μ
i
−:=
c
M
3
X
1
X
0
−
h
0
()
2
X
M
X
M1−
−
h
M1−
()
2
+
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
⋅:=A
MM,
1
h
M1−
:=A
MM1−,
2
h
M1−
:=A
M1,
1
h
0
:=A
M0,
2
h
0
:=
c
0
0:=A
0M,
1−:=A
00,
1:=
c
i
3 μ
i
X
i1+
X
i
−
h
i
⋅λ
i
X
i
X
i1−
−
h
i1−
⋅+
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
⋅:=A
ii 1+,
μ
i
:=A
ii,
2:=A
ii1−,
λ
i
:=i1M1−..:=
A
ij,
0:=j0M..:=i0M..:=
i0M..:= j0M..:= B
ij,
0:=
i1M1−..:= B
ii1−,
λ
i
:= B
ii,
2:= B
ii 1+,
μ
i
:= d
i
3 μ
i
Y
i1+
Y
i
−
h
i
⋅λ
i
Y
i
Y
i1−
−
h
i1−
⋅+
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
⋅:=
B
00,
1:= B
0M,
1−:= d
0
0:=
B
M0,
2
h
0
:= B
M1,
1
h
0
:= B
MM1−,
2
h
M1−
:= B
MM,
1
h
M1−
:= d
M
3
Y
1
Y
0
−
h
0
()
2
Y
M
Y
M1−
−
h
M1−
()
2
+
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
⋅:=
pX A
1−
c⋅:= pY B
1−
d⋅:=
xt() E 0←
z
tT
i
−
h
i
←
EX
i
a0 z()⋅ X
i1+
a1 z()⋅+ pX
i
b0 z()⋅ h
i
⋅+ pX
i1+
b1 z()⋅ h
i
⋅+←
T
i
t≤ T
i1+
<if
i0M1−..∈for
E
:=
yt() E 0←
z
tT
i
−
h
i
←
EY
i
a0 z()⋅ Y
i1+
a1 z()⋅+ pY
i
b0 z()⋅ h
i
⋅+ pY
i1+
b1 z()⋅ h
i
⋅+←
T
i
t≤ T
i1+
<if
i0M1−..∈for
E
:=
                                                                                                                     27

ORIGIN:= 0                         M := 50
                               2            3                                        2             3                                            2          3                                   2       3
a0( v ) := 1 − 3v + 2v                                        a1( v ) := 3v − 2v                            b0( v ) := v − 2v + v                                      b1( v ) := −v + v

                                                                                               (        )                                   (              )
                                                       2π
i := 0 .. M                        T := i ⋅                                  X := sin 2 ⋅ T                                Y := sin 3 ⋅ T
                                    i                  M                      i                         i                    i                         i

i := 0 .. M − 1                    h := T                     −T
                                    i             i+ 1                   i

                                                          h
                                                              i
i := 1 .. M − 1                    μ i :=                                         λi := 1 − μ i
                                              h +h
                                                  i               i− 1

i := 0 .. M           j := 0 .. M                      A             := 0
                                                              i, j
                                                                                                                                                           ⎛                       −X                      X −X
                                                                                                                                                                                                                           i− 1 ⎞
                                                                                                                                                                        X
                                                                                                                                        c := 3 ⋅ ⎜ μ i ⋅
                                                                                                                                                                          i+ 1             i                i                   ⎟
i := 1 .. M − 1                     A                 := λi                   A           := 2              A              := μ i                                                              + λi ⋅
                                        i , i− 1                                   i, i                         i , i+ 1                 i                 ⎜                  h                                 h              ⎟
                                                                                                                                                           ⎝                      i                                 i− 1       ⎠
A          := 1 A                    := −1                     c := 0
    0, 0                   0, M                                   0
                  2                               1                                                2                                    1                                         ⎡ X1 − X0                 X
                                                                                                                                                                                                                M
                                                                                                                                                                                                                        −X
                                                                                                                                                                                                                               M −1 ⎤
A          :=              A             :=                          A                    :=                A                :=                                c       := 3 ⋅ ⎢                        +                                ⎥
    M,0                        M,1                                       M , M −1                                   M,M                                            M
                  h
                      0
                                              h
                                                   0
                                                                                               h
                                                                                                   M −1
                                                                                                                                    h
                                                                                                                                        M −1                                      ⎢
                                                                                                                                                                                  ⎣       ( 0)
                                                                                                                                                                                          h
                                                                                                                                                                                                   2
                                                                                                                                                                                                                (   h
                                                                                                                                                                                                                        M −1   )   2    ⎥
                                                                                                                                                                                                                                        ⎦

i := 0 .. M               j := 0 .. M                     B          := 0
                                                              i, j
                                                                                                                                                               ⎛        Y             −Y                   Y −Y                 ⎞
                                                                                                                                         d := 3 ⋅ ⎜ μ i ⋅
                                                                                                                                                                            i+ 1           i                i              i− 1 ⎟
i := 1 .. M − 1                     B                  := λi                   B          := 2              B              := μ i                                                              + λi ⋅
                                        i , i− 1                                   i, i                         i , i+ 1                     i                 ⎜              h                                 h              ⎟
                                                                                                                                                               ⎝                      i                             i− 1       ⎠
B          := 1           B             := −1                     d := 0
    0, 0                   0, M                                      0

                  2                               1                                                 2                                    1                                         ⎡ Y1 − Y0                Y
                                                                                                                                                                                                                M
                                                                                                                                                                                                                        −Y
                                                                                                                                                                                                                               M −1 ⎤
B           :=             B             :=                          B                    :=                    B             :=                               d       := 3 ⋅ ⎢                         +                               ⎥
    M,0                        M,1                                       M , M −1                                   M,M                                            M
                  h
                      0
                                                  h
                                                      0
                                                                                               h
                                                                                                   M −1
                                                                                                                                    h
                                                                                                                                        M −1                                       ⎢
                                                                                                                                                                                   ⎣      ( h 0)   2
                                                                                                                                                                                                                ( h M − 1)          2   ⎥
                                                                                                                                                                                                                                        ⎦
             −1                                                          −1
pX := A               ⋅c                          pY := B                     ⋅d

 x( t ) :=        E← 0
                  for i ∈ 0 .. M − 1
                      if T ≤ t < T
                               i                   i+ 1
                                          t−T
                                                          i
                               z←
                                              h
                                                  i
                               E ← X ⋅ a0( z) + X                                    ⋅ a1( z) + pX ⋅ b0( z) ⋅ h + pX                                       ⋅ b1( z) ⋅ h
                                              i                               i+ 1                              i                   i               i+ 1                      i
                  E

 y ( t) :=        E← 0
                  for i ∈ 0 .. M − 1
                          if T ≤ t < T
                               i                      i+ 1
                                          t−T
                                                          i
                               z←
                                              h
                                                   i
                               E ← Y ⋅ a0( z) + Y                                    ⋅ a1( z) + pY ⋅ b0( z) ⋅ h + pY                                       ⋅ b1( z) ⋅ h
                                              i                               i+ 1                              i                   i               i+ 1                      i
                  E

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