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Russian Pages 103 Year 2005
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MathCAD , .
MathCAD,
1.1.
-
,
.
.
.
. .
,2
3
15.7 .
/
3
, /
3
,
/ 3.
: V := 2 ⋅cm
3
M := 15.7 ⋅gm : ρ :=
M V
γ := ρ ⋅g −3
ρ = 7.85 × 10
−3
gm⋅mm
−3
ρ = 7.85 gm⋅cm
ρ = 7.85 × 10 kgm 3
-3
γ = 7.698 × 10 m newton 4 -3
400
1.2. [9].
.
3
298
: kJ := 10 ⋅J 3
n := 3 ⋅mol
R := 8.3143 ⋅
T1 := 298 ⋅K
J mol⋅K
T2 := 400 ⋅K : W := n⋅R⋅( T2 − T1) W = 2.544 kJ
200°
1.3. 20°
. 220° ,
,
Q
1.5 0.54
/ · .
. : kJ := 10 ⋅J 3
M := 1.5kg
T1 := ( 20 + 273)K
T2 := ( 220 + 273)K C := 0.54
kJ
kg⋅K
:
Q := C ⋅M ⋅( T2 − T1)
Q = 162 kJ
t2 := T1 , T1 + 20 ⋅K .. T2 Q ( t2) := C ⋅( t2 − T1) ⋅M Q ( t2) =
0 16.2 32.4 48.6 64.8 81 97.2 113.4 129.6 145.8 162
200
kJ
180 160 140 120 Q( t2) kJ
100 80 60 40 20 300 325 350 375 400 425 450 475 500 t2 K
1.4.
α
.
, [6].
v0
: x0 := 0 ⋅m
α := 45 ⋅deg
y0 := 0 ⋅m a := −g
v0 := 100 ⋅
m sec
:
t := 0 ⋅sec , 0.1 ⋅sec .. 14.5 ⋅sec
vx0 := v0⋅cos (α )
vy0 := v0⋅sin(α )
vx( t) := vx0
vy( t) := vy0 + a ⋅t
x( t) := x0 + vx0⋅t
y( t) := y0 + vy0⋅t +
a 2
⋅( t)
2
200 y( t) m
0
500
1000
x( t ) m
⎛ vy( t) ⎞ ⎟ ⎝ vx( t) ⎠
α ( t) := atan⎜
-
50
α ( t) 0
deg
5
10
15
50 t sec
1.5.
.
,
σ.
. 1.1, [17].
. 1.1.
1.5
:
kN := 10 ⋅ N
MPa := 10 ⋅ Pa
3
ORIGIN ≡ 1
6
n := 2
i := 1 .. n
H1 := 10 ⋅ mm
1
B1 := 10 ⋅ mm
2 1
H2 := 20 ⋅ mm B2 := 10 ⋅ mm
2
L1 := 2 ⋅ m
1
L2 := 1 ⋅ m
2
E1 := 2 ⋅ 10 ⋅ MPa 5
1
E2 := 2 ⋅ 10 ⋅ MPa 5
2
F1 := 10 ⋅ kN
1
F2 := −40 ⋅ kN
2
:
→ ⎯ A := ( B ⋅ H)
⎛ 1⎞ 2 ⎟ cm ⎝ 2⎠
A= ⎜
⎛ 10 ⎞ ⎟ kN ⎝ −40 ⎠
F = ⎜
∑ i
σ i :=
i
=1 Ai
Fi
⎛ 100 ⎞ ⎟ MPa ⎝ −150 ⎠
σ = ⎜ δ i :=
∑
n + 1−i
i
⎛ σ n + 1−i ⋅ Ln + 1−i ⎞ ⎜ ⎟ En + 1−i ⎝ ⎠ =1
⎛ 0.25 ⎞ ⎟ mm ⎝ − 0.75 ⎠
δ = ⎜
σ 2 ⋅ L2 E2 σ 1 ⋅ L1 E1
1.6.
= − 0.75 mm
2
+
1
σ 2 ⋅ L2 E2
= 0.25 mm
.
, .
U( x , y) :=
q
(2
2 ⋅π ⋅λ ⋅ x + y
)
1
2 2
: q := 1000
,
λ := 0.4
,
U ( x , y) :=
q
(2
2 ⋅π ⋅λ ⋅ x + y
)
/(c * )
1
2 2
: 1.6.1. i := 0 .. 10
j := 0 .. 10
xi := −100 + 20.00000000000001 ⋅i
yj := −100 + 20.00000000000001 ⋅ j Mi, j := U ( xi , yj)
M
1.6.2.
CreateMesh
M := CreateMesh( U , −100 , 100 , −100 , 100 , 100 , 100)
M
1.7. s ( t)
. [12]. g⋅
v
,
a
s
t
t + v0⋅t + s0 2 . 2
1.7.1.
t : g⋅
s ( t)
2
t
2
+ v0⋅t + s0 :
" d s ( t) dt v( t) dd dt dt
/
"
g⋅t + v0
g⋅t + v0
s ( t)
a ( t)
1.7.2.
/
g
g
, , :
t := 0 ⋅sec , 0.1 ⋅sec .. 2 ⋅sec v0 := 1 ⋅
s0 := 0 ⋅m
m sec
s ( t) := g⋅
t + v0⋅t + s0 2 2
v( t) := g⋅t + v0
:
9.8066500000002 g 9.80665
d v( t) dt d2 dt
2
s ( t ) 9.8066499999998
9.8066499999996 0
0.5
1
1.5
2
t sec
1.8.
1 2 ⋅a ⋅π. 4
π.
.
(
(0, 0), (a, 0), (a, a), (0, a)) ,
a
[12]
,
S = ∫ f ( x )dx . a
0
4.
π.
1.8.1. : ⌠ ⎮ f ( x) dx ⌡0 a
S f ( x)
a a −x 2
f1( x)
2
:
⌠ 2 ⎮ a dx → a ⌡0 a
⌠ 1 ⎮ 2 ⎮ a2 − x2 dx → 1 ⋅ a2 ⋅ π ⌡0 4 a
(
)
π
⎤ ⎡ ⌠a 1 ⎢ ⎥ ⎮ ⎢⎮ 2 2 2 ⎥ ⎢ ⌡ ( a − x ) dx ⎥ ⎣ 0 ⎦ ⋅ 4 → π = 3.142 ⌠ ⎮ a dx ⌡0 a
1.8.2. [18] a := 5
N := 1000
: c
a
:
i := 0 .. N
Xi := 0 + i ⋅
a N
m := length( X) − 1 Xm − X0 m
Yi := a
−3
= 5 × 10
Y1i := a − ( Xi) 2
2
∑
m−1 ⎞ Xm − X0 ⎛⎜ X0 + Xm I := ⋅ + Yi⎟ ⎜ 2 ⎟ m i =0 ⎝ ⎠
∑
m−1 ⎞ Xm − X0 ⎛⎜ X0 + Xm I1 := ⋅ + Y1i⎟ ⎜ 2 ⎟ m i =0 ⎝ ⎠
⌠ ⎮ a dx → 25 = 25 ⌡0
I = 25.012
a
⌠ 1 ⎮ 2 ⎮ a2 − x2 dx → 25 ⋅ π = 19.635 ⌡0 4
(
)
PI :=
I1 ⋅4 I
a
π
π = 3.142 π − PI π
2. 2.1. 60 ?
I1 = 19.66
PI = 3.144
= −0.076 %
. .
. 40
/ ;
: V0 := 40 ⋅
h := 60 ⋅m
m sec
−g = −9.807 msec
-2
: 2.1.1. t := 0 ⋅sec , 2 ⋅sec .. 10 ⋅sec
F ( t) := V0⋅t −
g 2 ⋅t 2
t := 0 ⋅sec .. 10 ⋅sec
F ( t) =
0 m 35.097 60.387 75.87 81.547 77.417 63.48 39.737 6.187 -37.169 -90.332
100
50 F( t ) m h
0
m 50
100
0
5 t sec
"
/ / " : (2, 60), (6, 60)
10
2.1.2. C
t. "
/
/
a ⋅t + b ⋅t + c 2
"
0
⎛ 1⎞⎤ ⎡ ⎡ ⎜ 2⎟⎥ ⎢ 1 ⎢ ⎝ ⎠ 2 ⋅ ⎣ − b + ( b − 4 ⋅ a ⋅c ) ⎢ ⎦ ( 2 ⋅ a ) ⎢ ⎢ ⎛ 1⎞⎤ ⎡ ⎜ 2⎟⎥ ⎢ 1 ⎢ ⎝ ⎠ 2 ( ) − ⋅ − b b − 4 ⋅ a ⋅ c ⎣ ⎦ ⎢ ⎣ ( 2 ⋅a ) a :=
−g 2
b := V0
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
c := − h
⎛ 1⎞⎤ ⎡ ⎜ ⎟ ⎝ 2⎠⎥ 1 ⎢ 2 t1 := ⋅ ⎣ − b + b − 4 ⋅a ⋅ c ⎦ ( 2 ⋅a)
(
⎛ 1⎞⎤ ⎡ ⎜ ⎟ ⎝ 2⎠⎥ 1 ⎢ 2 ⋅⎣ − b − b − 4 ⋅ a ⋅ c t2 := ⎦ ( 2 ⋅a)
)
(
t1 = 1.981 sec 2.1.3.
)
t2 = 6.177 sec
"root"
TOL = 1 × 10
−3
f ( t) := − h + V0 ⋅t −
t := 0 ⋅sec
g 2 ⋅t 2
t_1 := root ( f ( t) , t) t_1 = 1.981 sec
t := 10 ⋅sec
t_2 := root ( f ( t) , t) t_2 = 6.177 sec
2.1.4.
"polyroot" − h + V0 ⋅t −
g 2 ⋅t 2
0
−h a 0 := m
a 1 :=
V0
a 0 ⋅ t + a 1 ⋅t + a 2 ⋅ t 0
1
⎛ −g ⎞ ⎜ ⎟ ⎝ 2 ⎠ a 2 := m
m sec
t0 = 1.981
t := polyroots( a)
2.1.5.
h := 60
TOL = 1 × 10
−3
sec
"Find"
g := 9.81
Given
0
−3
− h + V0 ⋅t −
f ( t) := Find( t)
g 2 ⋅t 2
f ( 0) = 1.981
f ( 10 ) = 6.173 2.1.6.
"Find"
Given 0
−h + V0 ⋅ t −
g 2 ⋅t 2
2
t1 = 6.177
V0 := 40
CTOL = 1 × 10
2
0 − 60 ⎞ ⎜⎛ ⎟ a = ⎜ 40 ⎟ ⎜ − 4.903 ⎟ ⎝⎝ .903⎠ ⎠
⎛ 1⎞⎤ ⎛ 1⎞⎤ ⎡ ⎡ ⎡ ⎜ 2⎟⎥ ⎜ 2⎟⎥ ⎢ 1 ⎢ ⎢ ⎝ ⎠ ⎝ ⎠ 1 2 2 Find( t) → ⎢ ⋅ ⎣ 2 ⋅ V0 + 2 ⋅ (V0 − 2 ⋅ g ⋅ h) ⋅ ⎣ 2 ⋅ V0 − 2 ⋅ ( V0 − 2 ⋅ g ⋅ h) ⎦ ⎦ ( 2 ⋅ g ) ( 2 ⋅ g ) ⎣
2.2.
.
40 .
/ .
,
,
.
V0 := 40 ⋅
h := 60 ⋅m
: m sec
−g = −9.807 msec
-2
: 2.2.1.
t := 0 ⋅sec .. 10 ⋅sec
F ( t) := V0 ⋅t −
g 2 ⋅t 2
100
F( t ) 0
m
100
0
5
10
t sec
" 2.2.2.
t := 0 ⋅sec Given F ( t)
/ / : (4, 81.547) "Minerr"
100 ⋅m
t_max := Minerr ( t) t_max = 4.079 sec
F ( t_max) = 81.577 m 2.2.3.
t= [0, 10] "Maximize"
t := 0 ⋅sec Given
0 ⋅sec ≤ t ≤ 10 ⋅sec
tmax := Maximize( F , t)
tmax = 4.079 sec F tmax = 81.577 m
(
)
⎤ ⎥ ⎥ ⎦
"
2.2.4.
F(t) >= 0 "Minimize"
t := 0 ⋅ sec Given
F ( t) ≥ 0
t_begin := Minimize ( F , t) t_begin = 0 sec
F ( t_begin ) = 0 m t := 10 ⋅ sec Given
F ( t) ≥ 0
t_end := Minimize ( F , t) t_end = 8.158 sec
F ( t_end ) = − 1.652 × 10
−5
t := 0 ⋅ sec .. 10 ⋅ sec
m
100 F ( t) m
50
F ( t_max ) m F ( t_begin ) m F ( t_end ) m
0
5
10
50
100 t t_max , , t_begin , t_end sec sec
1 2
2.3.
. ;
АВ В Q=1 α = 30°, h = 1 , L = 3 [2].
P=2 . 2.1).
(
А
, –
Y
B L C
A
α
Q h
X . 2.1.
2.3
.
. 2.2.
А
, ,
Y
B Q
N C
P
Ya Xa
A
X
. 2.2.
2.3
: kN := 10 ⋅N 3
P := 2 ⋅kN
N := Q
Q := 1 ⋅kN
L := 3 ⋅m
XA := P
YA := P
h := 1 ⋅m
: 2.3.1.
"Find"
Given XA − N ⋅cos ( 60 ⋅deg)
0
YA − P + N ⋅cos ( 30 ⋅deg) − Q
0
L N ⋅h −P ⋅ ⋅cos ( 30 ⋅deg) + − Q ⋅L⋅cos ( 30 ⋅deg) 2 sin( 30 ⋅deg) 1.299 ⎞ ⎜⎛ ⎟ Find( XA , YA , N) = ⎜ 0.75 ⎟ kN ⎜ 2.598 ⎟ ⎝ ⎠
0
2.3.2.
"Find"
Given XA − N ⋅cos ( 60 ⋅deg)
0
YA − P + N ⋅cos ( 30 ⋅deg) − Q
0
L N ⋅h −P ⋅ ⋅cos ( 30 ⋅deg) + − Q ⋅L⋅cos ( 30 ⋅deg) 2 sin( 30 ⋅deg)
0
Find( XA , YA , N) "
/
/
"
1 ( P + 2 ⋅Q) ⎡⎢ ⋅L⋅cos ( 30 ⋅deg) ⋅sin( 30 ⋅deg) ⋅ ⋅cos ( 60 ⋅deg) 2 h ⎢ ⎢ −1 (−2 ⋅P ⋅h + L⋅cos ( 30 ⋅deg) 2⋅sin( 30 ⋅deg) ⋅P + 2 ⋅L⋅cos ( 30 ⋅deg) 2⋅sin( 30 ⋅deg) ⋅Q − 2 ⋅Q ⋅h) ⎢ ⋅ h ⎢ 2 ⎢ ( P + 2 ⋅Q ) 1 ⋅L⋅cos ( 30 ⋅deg) ⋅sin( 30 ⋅deg) ⋅ ⎢ h 2 ⎣
⎥⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
2.3.3. :
P := 2
Q := 1
N := Q
XA := P
L := 3
h := 1
YA := P
:
XA − N ⋅ cos ( 60 ⋅ deg)
0
YA − P + N ⋅ cos ( 30 ⋅ deg) − Q −P ⋅
0
N⋅h L ⋅ cos ( 30 ⋅ deg) + − Q ⋅ L ⋅ cos ( 30 ⋅ deg) 2 sin( 30 ⋅ deg)
⎛ 1 0 − cos ( 60 ⋅ deg) ⎜ ⎜ 0 1 cos ( 30 ⋅ deg) A := ⎜ h ⎜ 0 0 sin ( 30 ⋅ deg) ⎝ X := A
−1
⎞ ⎟ ⎟ D := ⎟ ⎟ ⎠
0
0 ⎛ ⎜ Q+P ⎜ ⎜ L ⎜ P ⋅ ⋅ cos ( 30 ⋅ deg) + Q ⋅ L ⋅ cos ( 30 ⋅ deg) 2 ⎝
⋅D
1.299 ⎞ ⎜⎛ ⎟ X = ⎜ 0.75 ⎟ ⎜ 2.598 ⎟ ⎝ ⎠
X := lsolve( A , D)
⎛⎜ 1.299 ⎞⎟
X = ⎜ 0.75 ⎟
⎜ 2.598 ⎟ ⎝ ⎠
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
2.3.4. XA − N ⋅cos ( 60 ⋅deg)
0
YA − P + N ⋅cos ( 30 ⋅deg) − Q
0
L N ⋅h −P ⋅ ⋅cos ( 30 ⋅deg) + − Q ⋅L⋅cos ( 30 ⋅deg) 2 sin( 30 ⋅deg)
⎛ 1 0 −cos ( 60 ⋅deg) ⎜ ⎜ 0 1 cos ( 30 ⋅deg) ⎜ h ⎜ 0 0 sin ( 30 ⋅deg) ⎝
0
0 ⎞ ⎛ ⎟ ⎜ Q+P ⎟ ⋅⎜ ⎟ ⎜ L ⎟ ⎜ P ⋅ ⋅cos ( 30 ⋅deg) + Q ⋅L⋅cos ( 30 ⋅deg) ⎠ ⎝ 2
⎞ ⎟ ⎟→ ⎟ ⎟ ⎠
−1
cos ( 60 ⋅deg) ⎞ ⎛1 ⎡ ⋅sin( 30 ⋅deg) ⋅⎜ ⋅P ⋅L⋅cos ( 30 ⋅deg) + Q ⋅L⋅cos ( 30 ⋅deg) ⎟ ⎢ h ⎝2 ⎠ ⎢ ⎢ Q + P − cos ( 30 ⋅deg) ⋅sin( 30 ⋅deg) ⋅⎜⎛ 1 ⋅P ⋅L⋅cos ( 30 ⋅deg) + Q ⋅L⋅cos ( 30 ⋅deg) ⎟⎞ ⎢ h ⎝2 ⎠ ⎢ 1 ⎛1 ⎞ ⎢ ⋅sin( 30 ⋅deg) ⋅⎜ ⋅P ⋅L⋅cos ( 30 ⋅deg) + Q ⋅L⋅cos ( 30 ⋅deg) ⎟ h ⎣ ⎝2 ⎠
2.4.
.
(
[9]:
( . ). 0
=
log( P)
12.486 −
log( P)
7.884 −
3160 T
1860 T
0
: log( P)
12.486 −
log( P)
7.884 −
3160
1860 T
: P := 1
T := 1
Given log( P)
12.486 −
log( P)
7.884 −
3160 T
1860 T
⎛ P⎞ ⎜ ⎟ := Find( P , T) ⎝ T⎠
P = 19.934
P
= 20
.
. (2.666·103
).
T
T = 282.486
.
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .)
2.5.
.
, [9].
.
,
2%
: P1_0 − P1 P1_0
x2
n2 ( n2 + n1)
P1_0 := 100 P1 := 98 n1 :=
1000 18
-
n2
: n2 := 0
x2 := 0
Given P1_0 − P1 P1_0
x2
n2 ( n2 + n1)
⎛ x2 ⎞ ⎜ ⎟ := Find( x2 , n2) ⎝ n2 ⎠
2.6. ,
. – 50 %
n2 = 1.134
. , 10 %
70 % 40 %
30 %
.
,
– 80 % ,
15 %
20 % .
[10]? ш
.
u–
,v– .
,w– 15 %
,
,
0.3u + 0v + 0.1w = 0.15 . u+v+w 0.7u + 0.8v + 0.5w . u+v+w
. :
x=
(
u v w , y= , z= . u+v+w u+v+w u+v+w : 0.3x+0.1z–0.15=0 x+y+z–1=0, ), ц
0.7x+0.8y+0.5z. :
x, y, z ц .
.
, :
⎧0.7 x + 0.8 y + 0.5 → max ⎪ 0.3 x + 0.1z − 0.15 = 0 ⎪ ⎨ x+ y + z =1 ⎪ ⎪⎩ x ≥ 0, y ≥ 0, z ≥ 0
⎧0.7 x + 0.8 y + 0.5 → min ⎪ 0.3x + 0.1z − 0.15 = 0 ⎪ ⎨ x+ y+ z =1 ⎪ ⎪⎩ x ≥ 0, y ≥ 0, z ≥ 0
: F ( x , y , z) := 0.7 ⋅ x + 0.8 ⋅ y + 0.5 ⋅ z : x := 0
y := 0
z := 0
Given 0.3 ⋅ x + 0.1 ⋅ z − 0.15 x+ y+ z
0
1
x≥ 0
y≥ 0
z≥ 0
P := Minimize( F , x , y , z)
⎛⎜ 0.25 ⎞⎟ P = ⎜ 0 ⎟ ⎜ 0.75 ⎟ ⎝ ⎠ x := 0
y := 0
z := 0
Given 0.3 ⋅ x + 0.1 ⋅ z − 0.15 x+ y+ z
0
1
x≥ 0
y≥ 0
P := Maximize( F , x , y , z)
z≥ 0
⎛⎜ 0.5 ⎞⎟ P = ⎜ 0.5 ⎟ ⎜ 0 ⎟ ⎝ ⎠
3. 3.1. (
π [10]. (0, 0), (a, 0), (a, a), (0, a)).
. , ,
, «
». .
4.
a
1 2 ⋅a ⋅π. 4 π.
3.1.1.
(
.
1.8)
⌠ 2 ⎮ a dx → a ⌡0 a
⌠ 1 ⎮ 2 ⎮ a2 − x2 dx → 1 ⋅ a2 ⋅ π ⌡0 4 a
(
)
π
⎡ ⌠a ⎤ 1 ⎢ ⎥ ⎮ ⎢⎮ 2 2 2 ⎥ ⎢ ⌡ (a − x ) dx ⎥ ⎣ 0 ⎦ ⋅ 4 → π = 3.142 ⌠ ⎮ a dx ⌡0 a
3.1.2.
:
ORIGIN := 1 a := 5
c
N := 1000 :
i := 1 .. N
x := 0 , 0.01 .. a
y ( x) :=
a −x 2
2
X i := rnd ( a )
Y i := rnd ( a )
6
4
y ( x) Yi
2
0
n :=
∑
2
x, X i
4
6
"
⎡⎣⎡⎣ ( Xi) 2 + ( Y i) 2 ⎤⎦ ≤ a2⎤⎦
N
i
0
=1
π
PI :=
n ⋅4 N
PI = 3.14
"
n = 785
π = 3.142 -
π − PI π
N = 0.051 %
a
3.2.
.
)
24 . ~ 600
II (CuSO4),
(V = 20.0 . ,
250 25
5–6
.
,
, (
–
; ,
2Cu 2 + + 4 J − (
Na2S2O4 HCl H2 SO4 . : ) → Cu2 J 2 ↓ + J 2 ,
).
–
J 2 + 2 S 2O32 − → 2 J − + S 4O32 − .
Cu2+
N T ⋅ VT ⋅ ЭCu ⋅ 250 = 63,54 ⋅ 10 ⋅ N T ⋅ VT , 25 = 0.09132); VT – , 0.5 . 100 20 , . 3.1, mi –
qCu 2+ =
NT –
( NT 5 ,
,
. . , qi –
. 3.1
qi 600,0 601.5 602.5 603 604 604.5 605
mi 1 1 1 1 2 2 4
qi 605.5 606 606.5 607 607.5 608 608.5
mi 3 1 3 4 4 5 4
qi 609 609.5 610 610.5 611 611.5 612
mi 3 6 5 5 6 4 4
qi 612.5 613 613.5 614 614.5 615 615.5
mi 5 4 3 3 3 2 2
[19]. 3.2.1.
i := 0 .. 50
qi 616 616.5 617 617.5 618 618.5 621
mi 2 1 2 1 1 1 1
data1 i := 600 600.5 601 601.5 602 602.5 603 603.5 604 604.5 605 605.5 606 606.5 607 607.5 608 608.5 609 609.5 610 610.5 611 611.5 612 612.5 613 613.5 614 614.5 615 615.5 616 616.5 617 617.5 618 618.5 619 619.5 620 620.5 621
data2 i := 1 0 0 1 0 1 1 0 2 2 4 3 1 3 4 4 5 4 3 6 5 5 6 4 4 5 4 3 3 3 2 2 2 1 2 1 1 1 0 0 0 0 1
〈〉 data 0
:= data1
〈〉 data 1
3.2.2.
:= data2
(
)
〈〉 n := length data 0 n = 43 N :=
∑ (data 〈 〉 ) n −1
1
=0 N = 100
i
i
3.2.3.
(
) − min(data
〈〉 ω := max data 0
ω = 21 3.2.4.
)
〈0〉
∑ (data 〈 〉 ) ⋅(data 〈 〉 ) n −1
mx :=
=0
i
1
0
i
i
N
mx = 610.235 3.2.5. j := 0 .. n − 1 E j :=
∑ (data 〈 〉 ) j
j
=0
E20 = 50 med :=
1
(data
med = 610.25
〈0〉
j
E21 = 55
)20 + (data 2
〈0〉
)21
N = 100
3.2.6.
∑ ⎡⎣⎡⎣(data 〈 〉 ) − mx⎤⎦ ⎤⎦ ⋅(data 〈 〉 ) n −1
D :=
k
=0
1
k
k
N−1
D = 16.098 σ := ( D)
2
0
1 2
σ = 4.012 3.2.7.
h := 0.5 i := 0 ..
(max(data 〈0〉 ) − min(data 〈0〉 )) h
(
〈〉 xi := min data 0 3.2.8. P :=
) + (i⋅h)
〈1〉
data
N
3.2.9. 0.1
Pi
0.05
0 600
605
610
615
620
625
xi
3.2.10.
∑ ⎡⎣⎡⎣(data 〈 〉 ) − mx⎤⎦ ⎤⎦ ⋅(data 〈 〉 ) n −1
sk :=
k
3
0
=0
1
k
σ
sk = 0.03579
3
k
⋅
N
( N − 1 ) ⋅( N − 2 )
3.2.11.
∑ ⎡⎣⎡⎣(data 〈 〉 ) − mx⎤⎦ ⎤⎦ ⋅(data 〈 〉 ) n −1
ku :=
k
=0
4
0
1
k
σ
4
k
⋅
N ⋅( N + 1 )
( N − 1) ⋅( N − 2) ⋅( N − 3)
−
3 ⋅ ( N − 1)
2
( N − 2) ⋅( N − 3)
ku = −0.11443 3.2.12. X := min( x) , min( x) + h .. max( x)
mx = 610.235
⎡ −( X − mx) 2⎥⎤ exp⎢ ⎢ 2 ⋅σ 2 ⎥ ⎣ ⎦ G ( X) :=
σ = 4.012
σ ⋅ 2 ⋅π
med = 610.25
0.15
0.1
G( X) P h
0.05
0 600
X =
G ( X) =
600 600.5 601 601.5 602 602.5 603 603.5 604 604.5 605 605.5 606 606.5 607 607.5
3.841·10 -3 5.238·10 -3 7.033·10 -3 9.296·10 -3 0.012 0.016 0.02 0.024 0.03 0.036 0.042 0.05 0.057 0.064 0.072 0.079
605
610
0 1 2 3 4 5 6 x= 7 8 9 10 11 12 13 14 15
X, x
615
0 600 600.5 601 601.5 602 602.5 603 603.5 604 604.5 605 605.5 606 606.5 607 607.5
620
625
0 1 2 3 4 5 6 P = 7 h 8 9 10 11 12 13 14 15
0 0.02 0 0 0.02 0 0.02 0.02 0 0.04 0.04 0.08 0.06 0.02 0.06 0.08 0.08
3.2.13. 0,9
qchisq( 0.9 , n − 1 − 2) = 51.805
⎡⎢ ⎛ Pi ⎞ 2 ⎥⎤ ⎜ ⎟ ⎢⎝ h ⎠ ⎥ 3 ⎢ G ( x ) − N⎥ = −4.298 × 10 i ⎣ ⎦ =0
∑ n −1
i
⎡⎢ ⎛ Pi ⎞ 2 ⎥⎤ ⎜ ⎟ ⎢⎝ h ⎠ ⎥ ⎢ G ( x ) − N⎥ < qchisq( 0.9 , 43 − 3) = 1 i ⎣ ⎦ =0
∑ n −1
i
3.3.
.
. 3.2, .
READPRN
«tit.txt»,
. 3.2 600.0 601.5 602.5 603.0
607.0 607.0 607.0 607.5
609.5 609.5 609.5 609.5
611.0 611.5 611.5 611.5
613.5 614.0 614.0 614.0
604.0 604.0 604.5 604.5 605.0 605.0 605.0 605.0 605.5 605.5 605.5 606.0 606.5 606.5 606.5 607.0 3.3.1.
607.5 607.5 607.5 608.0 608.0 608.0 608.0 608.0 608.5 608.5 608.5 608.5 609.0 609.0 609.0 609.5
read := READPRN ( "tit.TXT")
(
data := stack read
3.3.2.
3.3.3.
〈0〉
, read
〈1〉
611.5 612.0 612.0 612.0 612.0 612.5 612.5 612.5 612.5 612.5 613.0 613.0 613.0 613.0 613.5 613.5
, read
〈2〉
614.5 614.5 614.5 615.0 615.0 615.5 615.5 616.0 616.0 616.5 617.0 617.0 617.5 618.0 618.5 621.0
, read
〈3〉
N := length( data ) N = 100
ω := max( data ) − min( data )
ω = 21 3.3.4.
609.5 610.0 610.0 610.0 610.0 610.0 610.5 610.5 610.5 610.5 610.5 611.0 611.0 611.0 611.0 611.0
mx := mean( data )
mx = 610.235
min( data ) = 600 max( data ) = 621
, read
〈4〉
)
3.3.5.
med := median( data) med = 610.25
3.3.6.
D := Var( data) D = 16.098
3.3.7.
Stdev( data) = 16.098 2
σ := Stdev( data)
σ = 4.012 3.3.8.
k := 5 ⋅ log( N)
max( data) − min( data)
h := 2
k
max( data) − min( data) h
n := 11
i := 0 .. n
xi := 600 + i ⋅ h
V := hist( x , data)
3.3.9. P :=
V N
= 2.1 = 10.5
k = 10
3.3.10. 0 600 602 604 606 608 610 612 614 616 618 620 622
0 1 2 3 4 x = 5 6 7 8 9 10 11
0 2 2 11 12 18 20 16 10 6 2 1
0 1 2 3 V = 4 5 6 7 8 9 10
0 1 2 3 P = 4 5 6 7 8 9 10
0 0.02 0.02 0.11 0.12 0.18 0.2 0.16 0.1 0.06 0.02 0.01
20
V
0 600
610
620
x
3.3.11.
3.3.12.
3.3.13.
skew ( data ) = 0.03579 kurt ( data ) = − 0.11443 G := dnorm (x , mx , σ )
mx = 610.235 σ = 4.012
med = 610.25
0.1
G P
0.05
h
0 600
610
620 x
0 1 2 3 4 x= 5 6 7 8 9 10 11
0 600 602 604 606 608 610 612 614 616 618 620 622
0 0 3.841·10 -3 1 0.012 2 0.03 3 0.057 4 0.085 G = 5 0.099 6 0.09 7 0.064 8 0.035 9 0.015 10 5.144·10 -3 11 1.35·10 -3
0 0 0.01 1 0.01 2 0.055 3 0.06 P 0.09 = 4 h 5 0.1 6 0.08 7 0.05 8 0.03 9 0.01 10 5·10 -3
3.3.14. 0,9
qchisq( 0.9 , n − 1 − 2) = 13.362 j := 0 .. n − 1
∑ n −1
j
=0
∑ n −1
j
3.4.
.
=0
8
⎛ Pj ⎞ ⎜ − G j+ 1⎟ h ⎝ ⎠ = 0.072 2
Pj h
⎛ Pj ⎞ ⎜ − G j+ 1⎟ h ⎝ ⎠ < qchisq( 0.9 , 10 − 3) = 1 2
Pj h
1.76 %.
0.08 %.
, [19].
S8 95-
: n := 8
mx := 1.76 ⋅ %
S_8 := 0.08 ⋅ %
γ := 0.95
: α := 1 − γ
α = 0.05
⎛ ⎝
tx := qt ⎜ 1 −
α ⎞ , n − 1⎟ 2 ⎠
(
)
tx = 2.365 δ := tx ⋅
S_8 n
δ = 0.067 % mx − δ = 1.693 % mx + δ = 1.827 %
4. 4.1.
y = φ( x )
. ,
. 4.1 [12]. 4.1
X Y
1 3
2 4
y = ax + b .
3 2.5
5 0.5 .
y = φ( x )
4.1.1. a
b.
∑
Given
⎤ 2 ⎡⎣Yi − ( a ⋅Xi + b)⎤⎦ ⎥ ⎥ ⎣ i =0 ⎦ ⎡ N−1
d ⎢ da ⎢
∑
⎡ N−1
d ⎢ db ⎢
⎣ i =0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Find( a , b) → ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎛ ⎜ X := ⎜ ⎜ ⎜ ⎝
⎤ 2 ⎡⎣Yi − ( a ⋅Xi + b)⎤⎦ ⎥ 0 ⎥ ⎦ N − 1 N−1 N−1 ⎛ ⎞ ⎜ Yi⋅Xi⋅N − Xi⋅ Yi⎟ ⎜ ⎟ i =0 i =0 ⎝ i =0 ⎠ 2 ⎡ N−1 ⎛ N−1 ⎞ ⎤ ⎢ ⎥ 2 ⎜ Xi⎟ ( Xi) ⋅N − ⎜ ⎢ ⎟ ⎥ ⎣ i =0 ⎝ i =0 ⎠ ⎦ N−1 N−1 N−1 ⎡ N−1 ⎤ 2 ⎢ Yi⋅Xi⎥ Xi) − Xi⋅ Yi⋅ ( ⎢ ⎥ i =0 i =0 i =0 ⎣ i =0 ⎦ 2 ⎡ N−1 ⎛ N−1 ⎞ ⎤ ⎢ ⎥ 2 ⎜ Xi⎟ ( Xi) ⋅N − ⎜ ⎢ ⎟ ⎥ ⎣ i =0 ⎝ i =0 ⎠ ⎦
∑
∑ ∑
∑
∑
∑ ∑
∑ ∑ ∑
⎛ ⎜ Y := ⎜ ⎜ ⎜ ⎝
1 ⎞
⎟ 2 ⎟ 3 ⎟ ⎟ 5 ⎠
N := length( X)
N = 4
0
∑
∑
∑
3 ⎞
⎟
4 ⎟ 2.5 ⎟
⎟
0.5 ⎠
∑
N− 1 N−1 ⎛ N− 1 ⎞ ⎜ Yi ⋅Xi ⋅N − Xi ⋅ Yi⎟ ⎜ ⎟ i =0 i =0 i =0 ⎝ ⎠ a := 2⎤ ⎡ N− 1 N − 1 ⎛ ⎞ ⎢ ⎥ 2 Xi ⎟ Xi) ⋅N − ⎜ ( ⎢ ⎜ ⎟ ⎥ ⎣ i =0 ⎝ i =0 ⎠ ⎦
∑
∑
∑
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
∑
∑
∑
N− 1 N− 1 N−1 ⎡ N− 1 ⎤ 2 ⎢ Xi) + Yi ⋅ Yi⋅Xi⎥ Xi ⋅ − − ( ⎢ ⎥ i =0 i =0 i =0 i =0 ⎣ ⎦ b := 2⎤ ⎡ N− 1 N − 1 ⎛ ⎞ ⎢ ⎥ 2 Xi⎟ Xi) ⋅N − ⎜ ( ⎢ ⎜ ⎟ ⎥ ⎣ i =0 ⎝ i =0 ⎠ ⎦
∑
∑
a = − 0.743
b = 4.543
x := 1 , 1.1 .. 5
y( x) := a ⋅x + b 4 y ( x) Y
2
0
0
2
x, X
4
approximation exptriment_point
6
4.1.2. a
a := 0
b := 0
b
Given
∑
N−1
=0
∑ i
N−1
i
=0
( Yi⋅Xi) − a ⋅ ∑ ( Xi) N−1
Yi − a ⋅
∑ N−1
i
=0
i
=0
2
− b⋅
Xi − b ⋅N
∑ N−1
i
=0
0
0
⎛ a⎞ ⎜ ⎟ := Find( a , b) ⎝ b⎠ x := 1 , 1.1 .. 5
Xi
a = −0.743
b = 4.543
y( x) := a ⋅x + b
4 y ( x) Y
2
0
0
2
x, X
4
6
approximation exptriment_point
4.1.3. a
i := 0 .. N − 1
b
m := 0 .. 1
k := 0 .. 1 Ak , m :=
∑
N−1
i
=0
( Xi)
k+ m
Dk :=
⎛ 4 11 ⎞ ⎟ ⎝ 11 39 ⎠
A=⎜
⎛ 10 ⎞ ⎟ ⎝ 21 ⎠
i
D=⎜
⎛ b⎞ −1 ⎜ ⎟ := A ⋅D a ⎝ ⎠ x := 1 , 1.1 .. 5
∑
N−1
a = −0.743
=0
Yi⋅( Xi)
k
b = 4.543
y( x) := a ⋅x + b
4 y ( x) Y
2
0
0
2
x, X
4
6
approximation exptriment_point
4.1.4. a b
"intercept"
b := intercept ( X , Y )
"slope"
a := slope ( X , Y )
a = − 0.743 b = 4.543
x := 1 , 1.1 .. 5
y ( x) := a ⋅x + b 4 y ( x) Y
2
0
0
2
x, X
4
x, X
4
6
approximation exptriment_point 4.1.5. a b
"line"
⎛ b ⎞ ⎜ ⎟ := line( X , Y ) ⎝ a ⎠ a = − 0.743 b = 4.543
x := 1 , 1.1 .. 5
y ( x) := a ⋅x + b 4 y ( x) Y
2
0
0
2
approximation exptriment_point
6
4.1.6.
[4]
p := 1
N = 4
∑
N−1
SSf :=
=0
∑ i
N−1
SSr :=
=0
∑ i
N−1
SS :=
=0
i
( y ( X i) − mean ( Y) ) 2 ( Yi − y ( Xi) ) 2 ( Yi − mean ( Y) ) 2
SS = 6.5
R_2 :=
R :=
SSf + SSr = 6.5
R_2 = 0.743
SSf SS
a ⋅corr ( Y , X ) ⋅stdev ( X ) stdev ( Y )
R = 0.743 2
0,683 F :=
R ⋅( N − p − 1 )
( 1 − R 2) ⋅ p
2
F-
F = 5.778
qF ( 0.683 , p , N − p − 1 ) = 1.749
F > qF ( 0.683 , p , N − p − 1 ) = 1 F-
, -
,
4.2.
.
X
Y
X Y
X [15].
Y
,
X i :=
Y i :=
1 2 3 4 5 6 7 8 9 10 11 12
145 111 135 130 122 98 100 85 90 79 15 68
i := 0 .. 11
x := min( X ) , min( X ) + 0.01 .. max ( X )
p := 1
n := length ( X ) 4.2.1. Y
b ⋅a
log ( Y ) b1
X
log ( b ) + X ⋅ log ( a )
(
log ( b )
)
→ ⎯ → b1 := intercept X , log ( Y ) b := 10 a1
(
(
)
log ( a ) → ⎯ → a1 := slope X , log ( Y )
a := 10
)
→ ⎯ → intercept X , log ( Y )
(
)
→ ⎯ → slope X , log ( Y )
a = 0.891
b = 185.732
y( x) := b ⋅a
x
2.5
2
log( y ( x) ) log( Y)
1.5
1
5
x, X
10
∑
( log( y( Xi) ) − mean(log(Y))) 2
∑
( log( Yi) − mean(log(Y))) 2
n −1
SSf :=
i
SS :=
0
=0
n −1
i
R_2 :=
=0
15
SSf SS
R_2 = 0.478
F :=
0,9
R_2⋅( n − p − 1) ( 1 − R_2) ⋅p
FF = 9.151
qF ( 0.9 , p , n − p − 1) = 3.285
F > qF ( 0.9 , p , n − p − 1) = 1 F-
, -
X
Y
,
4.2.2.
"linfit"
⎜⎛ ⎜ f ( x) := ⎜ ⎜ ⎜ ⎝
1 ⎞
⎟
x ⎟
⎟
x ⎟ 2
⎟
x ⎠ 3
⎛ ⎜ a = ⎜ ⎜ ⎜ ⎝
a := linfit( X , Y , f )
y1 ( x) := f ( x) ⋅ a 150
130.091 ⎞
⎟ ⎟ − 1.944 ⎟ ⎟ 0.079 ⎠ 5.103
100
Y y1 ( x)
50
0
0
5
10
X,x
15
∑ (a ⋅corr (Y , X ) ⋅stdev (X )) 3
R :=
j
=1
j
j
j
stdev ( Y )
R = 0.873
R = 0.761 2
F :=
R ⋅( n − p − 1 )
( 1 − R 2) ⋅ p
0,9
2
F-
F = 31.89
qF ( 0.9 , p , n − p − 1 ) = 3.285
F > qF ( 0.9 , p , n − p − 1 ) = 1 F-
, -
X
Y
,
⎛ ⎜ ⎜ ⎜ ff ( x) := ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 ⎞
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
1 x 1 x
2
1 x
3
⎛ − 33.131 ⎜ ⎜ 1.253 × 10 3 a = ⎜ 3 ⎜ − 2.795 × 10 ⎜ 3 ⎝ 1.721 × 10
a := linfit( X , Y , ff )
y2 ( x) := ff ( x) ⋅ a
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
150
100
Y y2 ( x)
50
0
0
5
X,x
10
15
∑ (a ⋅corr (Y , X ) ⋅stdev (X )) − j
3
R :=
j
=1
j
− j
stdev ( Y )
R = 0.896
R = 0.802 2
F :=
0,9
R ⋅( n − p − 1 )
( 1 − R 2) ⋅ p
2
F-
F = 40.542
qF ( 0.9 , p , n − p − 1 ) = 3.285
F > qF ( 0.9 , p , n − p − 1 ) = 1 F-
, -
X
Y
⎛ ⎜ ⎜ ⎜ fff( x) := ⎜ ⎜ ⎜ ⎜ ⎝
,
1 ⎞
⎟
1 ⎟ x ⎟
⎟
x ⎟ 1 ⎟ 2
⎛ 141.471 ⎜ ⎟ a := linfitX ( , Y , fff) 3 − 44.111 x ⎠ a = ⎜ ⎜ − 0.676 ⎜ y3 ( x) := fff( x) ⋅a ⎝ 47.454
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
150
Y
100
y3 ( x) 50
0
R :=
0
5
X, x
10
15
(a1⋅corr(Y , X− 1) ⋅stdev(X− 1)) + (a2⋅corr(Y , X2) ⋅stdev(X2)) + (a3⋅corr(Y , X− 3) ⋅stdev(X− 3)) stdev( Y)
R = 0.876
R = 0.767 2
F :=
R ⋅( n − p − 1 )
(1 − R2) ⋅p
0,9
2
F-
F = 32.908
qF ( 0.9 , p , n − p − 1) = 3.285
F > qF ( 0.9 , p , n − p − 1) = 1 F-
, -
X
4.3.
.
,
Y
4.1
y = φ( x )
y = a1 ⋅ a0 x .
4.3.1.
(
)
n −1 ⎡ n−1 2⎤ X X Xi ⎢ ⎡Yi − (a1 ⋅ a0 Xi)⎤ ⎥ → −2 ⋅ Yi − a1 ⋅ a0 i ⋅ a1 ⋅ a0 i ⋅ ⎣ ⎦ ⎥ ⎢ a0 da0 ⎣ i =0 ⎦ i =0
∑
d
.
∑
(
)
n −1 2⎤ Xi Xi ⎡Yi − (a1 ⋅ a0 Xi)⎤ ⎥ → −2 ⋅ Yi − a1 ⋅ a0 ⋅ a0 ⎣ ⎦ ⎥ ⎣ i =0 ⎦ i =0
∑
⎡ n−1
d ⎢ da1 ⎢
⎛ ⎜ X := ⎜ ⎜ ⎜ ⎝
1 ⎞
⎟ 2 ⎟ 3 ⎟ ⎟ 5 ⎠
∑
⎛ ⎜ Y := ⎜ ⎜ ⎜ ⎝
3 ⎞
⎟
4 ⎟ 2.5 ⎟
x := min( X) , min( X) + 0.01 .. max( X) n := length( X) n=4
⎟
0.5 ⎠
a
a0 := 1
∑
Given n −1
i =0 n −1
∑ i
=0
a1 := 2
(
−2 ⋅ Yi − a1 ⋅ a0
Xi
−2 ⋅ Yi − a1 ⋅ a0
Xi
(
b
) ⋅ a1 ⋅ a0X ⋅ Xa0i i
) ⋅ a0X
i
⎛ a0 ⎞ ⎜ ⎟ := Find( a0 , a1) ⎝ a1 ⎠
0
0
a0 = 0.773
y( x) := a1 ⋅ a0
x
a1 = 4.824
4
Y 2
y ( x)
0
0
2
4.3.2.
4
X, x
"genfit" a1 ⋅ a0
x
y( x)
"
/
/
". :
a0
a1 ⋅ a0 ⋅ x
a1 x a0
x
a0
6
x ⎡ ⎤ ⎢ a1 ⋅ ( a0) ⎥ ⎢ ⎥ x x f ( x , a) := ⎢ a1 ⋅ ( a0) ⋅ ⎥ a0 ⎢ ⎥ x ⎢ ⎥ ( a0) ⎣ ⎦ ⎛ a0 ⎞ ⎜ ⎟ := genfit( X , Y , v , f ) ⎝ a1 ⎠
y ( x) := a1 ⋅ ( a0)
⎛ 1 ⎞ ⎟ ⎝ 2 ⎠
v := ⎜ a0 = 0.775
a1 = 4.799
x 4
Y 2
y ( x)
0
4.3.3.
0
2
X, x
"Minimize" a0 := 1 f ( a0 , a1 ) :=
∑ n −1
i
=0
4
6
a1 := 1
⎡Yi − ( a1 ⋅ a0 Xi)⎤ ⎣ ⎦
2
Given 0 < a0 < 10
0 < a1 < 10
⎛ a0 ⎞ ⎜ ⎟ := Minimize( f , a0 , a1 ) ⎝ a1 ⎠ y ( x) := a1 ⋅ a0
a0 = 0.773
a1 = 4.824
x 4
Y 2
y ( x)
0
4.4.
0
2
X, x
4
6
.
298 K . 4.2 [9]: 4.2
1 2
0.0363 0.0184
0.0668 0.0504
0.0940 0.0977
c1 – ,
/ .
0.126 0.146
0.210 0.329
, n
K
/ , c2 –
0.283 0.553
c1n =K. c2
0.558 0.650
0.756 2.810
0.912 4.340
c1i :=
c2i :=
0.0363 0.0668 0.0940 0.126 0.210 0.283 0.558 0.756 0.912
0.0184 0.0504 0.0977 0.146 0.329 0.553 0.650 2.810 4.340
i := 0 .. 8
C1 := min( c1) , min( c1) + 0.01 .. max( c1) 4.4.1.
c1
K K i :=
K
c2
c1i c2i 1/ 2 = K
1/ 2
,
2 Ki
1
0
5
10
i n
4.4.2.
n
K
⎛ 1 ⎞ + n ⋅ log( c1) ⎟ ⎝K⎠ ⎛1⎞ log⎜ ⎟ ⎝K⎠ 1
10 b1 K :=
(
b1
)
→ ⎯ → ⎯ intercept log( c1) , log( c2)
(
1
)
→ ⎯ → ⎯ intercept log ( c1) , log ( c2)
(
)
→ ⎯ → ⎯ n := slope log( c1) , log( c2) 10
C2 ( C1 ) :=
1 K
⋅ C1
n = 1.572
6
log ( C2( C1) )
C2( C1) 1.5
1
K = 0.278
n
2
log ( c2)
K
log⎜
log( c2) b1
c1 c2
K
0.5
0
2 log ( C1) , log ( c1)
c2
4 2
0
0.5
C1, c1
1
4.4.3.
"genfit" n
c1 c2
K 1
a0
K
a1
n
C2
a0 ⋅ C1
"
a1
/
/
". :
a0 C1
a1
a1
a1 ⎛ a0 ⋅ C1 ⎜ ⎜ a1 f ( C1 , a) := C1 ⎜ a1 ⎜ ⎝ a0 ⋅ C1 ⋅ ln( C1 )
⎛ a0 ⎞ ⎜ ⎟ := genfit( c1 , c2 , v , f ) ⎝ a1 ⎠
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
a0 ⋅ C1
a1
⋅ ln( C1 )
⎛ 1 ⎞ ⎟ ⎝ 1 ⎠
v := ⎜
n := a1 K :=
1 a0
C2 ( C1 ) :=
1 K
⋅ C1
n
6
4
c2 C2( C1)
2
0
4.5.
.
0
0.5
1
c1 , C1
. 4.3 [1].
– 11.3
.
– 100
. 4.3
№№
ΔL,
P, 1 2 2 4 5 6 7 8 9
1.5 2.0 2.5 3.0 3.5 3.7 3.9 4.1 4.3 σ– (
-
0.0 0.2 0.3 1.9 1.7 2.6 3.8 5.6 9.2 ).
.
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ΔL := ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
0.0 ⎞ 0.2 0.3 0.9 1.7 2.6 3.8 5.6 9.2
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
,
d := 0.0113 d
ε :=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ σ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
F ⋅10
L
146.728 ⎞ 244.547 293.456 342.366 361.929 381.493 401.057 420.621
3.5 3.7 3.9 4.1
,
−4
1000 ⋅9.81
195.638
^2
3
2
4
ΔL
,
2.5
,
F = 1.003 × 10 σ := P ⋅
4.3
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
2
,
L := 100
F := π ⋅
1.5 ⎞
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ P := ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
,
6
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ε = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
0
2 × 10
3 × 10
9 × 10
−3 −3 −3
0.017 0.026 0.038 0.056 0.092
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
4.5.1.
N
N := length(ε ) N =9
i := 0 .. N − 1
m := 0 .. N − 1
k := 0 .. N − 1
Ak , m :=
∑ N−1
∑ i
Dk :=
N−1
i
=0
=0
(ε i)k+ m
σ i⋅(ε i)
a := lsolve( A , D)
k
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ a= ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
⎞ ⎟ −3.052 × 10 ⎟ ⎟ 7 4.139 × 10 ⎟ 9 ⎟ −8.76 × 10 ⎟ 11 7.789 × 10 ⎟ ⎟ 13 ⎟ −3.441 × 10 ⎟ 14 7.827 × 10 ⎟ 15 ⎟ −8.639 × 10 ⎟ 16 ⎟ 3.6 × 10 ⎠ 147.222
4
x := min(ε ) , min(ε ) + 0.0001 .. max( ε ) y( x) :=
∑ N−1
k
ak ⋅x
k
=0
5 .10
5
σ
0
y ( x)
0.05
0.1
5 .10
5
1 .10
6
ε, x
N
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ y(ε ) = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ σ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
147.222 ⎞ 193.062 246.902 293.087 342.494 361.893 381.498 401.057 420.621
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
146.728 ⎞ 195.638 244.547 293.456 342.366 361.929 381.493 401.057 420.621
4.5.2.
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
3
i := 0 .. N − 1 N := 3
m := 0 .. N − 1
k := 0 .. N − 1 Ak , m :=
∑
N−1
i
=0
(ε i)
a := lsolve( A , D)
k+ m
Dk :=
∑
N−1
i
=0
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ a= ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
σ i⋅(ε i)
k
141.083
9.693 × 10
3
−7.442 × 10 5.183 × 10
8
−7.355 × 10 3.952 × 10
10
12
−9.945 × 10 1.163 × 10
4
13
15
−5.016 × 10
15
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
y( x) :=
∑ N−1
k
=0
ak ⋅x
k
600 σ
400
y ( x) 200
0
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ y( ε ) = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
0.05
141.083 ⎞ 160.172 169.494 222.296 284.363 342.802 401.965 450.521 402.953
4.5.3.
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ σ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
-
⎛ ⎜ ⎜ ⎜ u := ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
146.728 ⎞ 195.638 244.547 293.456 342.366 361.929 381.493 401.057 420.621
0 0.004 0.005 0.015 0.02 0.03 0.092
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
, B-
y( x) := interp(W , ε , σ , x) 600
400
y ( x) y( u)
200
0
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
B-
. W := bspline(ε , σ , u , 3)
σ
0.1
ε, x
0.05
ε , x, u
0.1
4.6.
.
– –
298 K
. 4.4 [9]: 4.4
XC2H5OC2H5 PC2H5OC2H5 ·10-4, PCHCl2 ·10-4,
0 0.000 1.933
0.2 0.460 1.480
0.4 1.287 0.920
0.6 2.666 0.460
0.8 4.093 0.165
1.0 5.333 0.000 .
, ,
. ,
. . .
.
, (ΔH < 0).
xC2H5OC2H5 := i 0 0.2 0.4 0.6 0.8 1.0
;
i := 0 .. 5
PC2H5OC2H5 := i
0.000 0.460 1.287 2.666 4.093 5.333
PCHCl3 := i
1.933 1.480 0.920 0.460 0.165 0.000
6
PC2H5OC2H5
4
PCHCl3 PC2H5OC2H5+ PCHCl3
2
0
0
0.5
1
xC2H5OC2H5
4.6.1.
(
) + 0.01 .. max(xC2H5OC2H5 ) Z1 ( x) := linterp( xC2H5OC2H5 , P C2H5OC2H5 , x) Z2 ( x) := linterp( xC2H5OC2H5 , P CHCl3 , x) Z3 ( x) := linterp( xC2H5OC2H5 , P C2H5OC2H5 + P CHCl3 , x) "linterp"
)
(
x := min xC2H5OC2H5 , min xC2H5OC2H5
6 PC2H5OC2H5 PCHCl3 PC2H5OC2H5+ PCHCl3
4
Z1 ( x) Z2 ( x)
2
Z3 ( x)
0
0
0.2
0.4
0.6
0.8
xC2H5OC2H5 , xC2H5OC2H5 , xC2H5OC2H5 , x, x, x
1
"linterp"
xC2H5OC2H5 = i 0
Z1 ( 0.101 ) = 0.232
0.2 0.4
Z1 ( 0.3 ) = 0.873
Z2 ( 0.101 ) = 1.704
0.6
Z2 ( 0.3 ) = 1.2
Z3 ( 0.101 ) = 1.937
0.8 1
4.6.2.
(
"interp"
W1 := lspline xC2H5OC2H5 , P C2H5OC2H5
(
Z3 ( 0.3 ) = 2.073
)
)
Z4 ( x) := interp W1 , xC2H5OC2H5 , P C2H5OC2H5 , x
(
W2 := cspline xC2H5OC2H5 , P C2H5OC2H5
(
)
)
Z5 ( x) := interp W2 , xC2H5OC2H5 , P C2H5OC2H5 , x
(
W3 := pspline xC2H5OC2H5 , P C2H5OC2H5
(
)
)
Z6 ( x) := interp W3 , xC2H5OC2H5 , P C2H5OC2H5 , x
"linterp" "interp"
0..0.2
0.4 Z1 ( x) Z4 ( x)
0.2
0
0
0.1
linterp interp
(
"cspline", "pspline"
)
0.2
x
(
"interp" "lspline",
)
(
)
x := min xC2H5OC2H5 − 1 , min xC2H5OC2H5 − 1 + 0.01 .. max xC2H5OC2H5 + 1 20
Z1( x)
10
Z4( x) Z5( x) Z6( x)
1
0
1
2
10
20 x
4.7.
.
. 4.5
, (
αHCOOH – pH)
pH, pH 1.00 6.00. pH 1.00 – 6.00. ( αHCOO¯ – pH) [5]. 4.5
pH 1.00 2.00 3.00 4.00
HCOOH 100 98 85 36
HCOO¯ 0 2 15 64
5.00 6.00
5 0 i := 0 .. 5
pH i :=
95 100
HCOO_ i :=
HCOOH i := 100 98 85 36 5 0
1.00 2.00 3.00 4.00 5.00 6.00
0 2 15 64 95 100
100
HCOOH HCOO_
50
0
2
4
6
pH
"interp"
"lspline"
x := min( pH ) , min( pH ) + 0.01 .. max ( pH )
W1 := lspline( pH , HCOOH )
Z1 ( x) := interp ( W1 , pH , HCOOH , x)
W2 := lspline( pH , HCOO_ )
Z2 ( x) := interp ( W2 , pH , HCOO_ , x)
150
HCOOH 100 HCOO_ 50
Z1( x) Z2( x)
0
50
0
2
pH , pH , x, x
"
4
6
/ / " : (3.75 , 51.225 )
"interp" Z1( 3.75) = 48.775
Z2( 3.75) = 51.225 "Find" x := 0 Given Z1( x)
Z2( x)
Find( x) = 3.727
5. 5.1.
( .
M ,
) (R = M·k2·v2,
k–
. ) [3].
:
V0 := 0
, /
k := 0.16
g :=
g
, / −2
m⋅sec
, / ^2
M := 1
, :
t := 0 .. 15 v( t) :=
, 2⋅ k⋅ t ⋅ g
g e −1 ⋅ k e2⋅ k⋅ t⋅ g + 1
5.1.1.
I
"odesolve"
Given M ⋅V' ( t) V ( 0)
M ⋅g − M ⋅k ⋅V ( t) 2
2
, /
0
vv := odesolve( t , 15 , 100)
15 -
t := 0 , 1 .. 15
20 vv ( t) v ( t)
10
0
5
10
15
t
5.1.2.
V0 := V0
I
N := 15
D( t , V) := g − k ⋅( V0) 2
t1 := 0
vv( t) =
v( t) =
0 9.061 14.923 17.727 18.874 19.313 19.477 19.537 19.559 19.567 19.57 19.572 19.572 19.572 19.572 19.572
0 9.061 14.923 17.727 18.874 19.313 19.477 19.537 19.559 19.567 19.57 19.572 19.572 19.572 19.572 19.572
"rkfixed"
t2 := 15
2 0
V_ := rkfixed( V , t1 , t2 , N , D)
20 v ( t) V_
〈1〉
V_ =
10
0
0
1
1
9.055
2
2
14.9
3
3 17.696
4
4 18.851
5
5 19.299
6
6
7
7 19.534
8
8 19.558
9
0
( v⎡⎣( V_
〈〉 v⎡⎣ V_ 0 〈0〉
5
)10⎤⎦ = 19.57 )15⎤⎦ = 19.572
t , V_
〈0〉
10
(V_ (V_
〈1〉
〈1〉
15
)10 = 19.57 )15 = 19.572
1
0
19.47
9 19.567
10
10
11
11 19.571
19.57
12
12 19.572
13
13 19.572
14
14 19.572
15
15 19.572
5.1.3.
I
N = 15
T0 := t1
V0 = 0
[18]
TN := t2 i := 1 .. N
F ( T , V) := g − k ⋅V 2
2
TN − T0
h :=
N Ti := T0 + h⋅i
Vi := Vi−1 + F ( Ti−1 , Vi−1) ⋅h
20 v( t) V
VN = 19.572
10
0
5
t,T
10
15
v( T3) = 17.727
T3 = 3
V3 = 19.427
v( T10) = 19.57
T10 = 10
V10 = 19.572
v( T15) = 19.572
T15 = 15
V15 = 19.572
5.2.
[13]. –
k,
Q,
.
.
Q –
, ,
–
y_0. .
,
. –
, – ,
-k·y,
k –
(«
»). ,
,
Q ,
(
− λ ⋅ v = −λ ⋅
,
). Q⋅
k, ≥ 0.
II
2
d y dt 2
= −k ⋅ y − λ ⋅
: dy , dt
: Q := 20
λ := 40
k := 100
y_0 := 0.01
V_0 := 0
, , / , / , , /
dy , dt
= const ≥ 0
: 5.2.1.
λ
y''( t) +
Q
II ⋅y'( t) +
k Q
⋅y( t)
0
5.2.1.1. II
p +
λ
2
"
Q
/
⋅p +
k
/
" p
0
Q
.
⎛ 1⎞⎤ ⎡ ⎡ ⎜ ⎟ ⎢ 1 ⎢ ⎝ 2⎠⎥ 2 ( ) ⎢ ⋅⎣ −λ + λ − 4 ⋅Q ⋅k ⎦ ⎢ ( 2 ⋅Q ) ⎢ ⎛ 1⎞⎤ ⎡ ⎜ 2⎟ ⎢ 1 ⎢ ⎝ ⎠⎥ 2 ( ) ⋅⎣ −λ − λ − 4 ⋅Q ⋅k ⎢ ⎦ ⎣ ( 2 ⋅Q )
⎤ ⎥ ⎥ ⎥ = ⎜⎛ −1 + 2i ⎞⎟ ⎥ ⎝ −1 − 2i ⎠ ⎥ ⎥ ⎦
5.2.1.2.
, ,
y( t)
II
exp(α ⋅t) ⋅(C1 ⋅cos (β ⋅t) + C2 ⋅sin(β ⋅t))
α = -1 5.2.1.3.
β =2 II
"
.
/
/
"
t
d dt
y( t)
α ⋅exp( α ⋅t) ⋅(C1 ⋅cos( β ⋅t) + C2 ⋅sin(β ⋅t) ) + exp(α ⋅t) ⋅(−C1 ⋅sin(β ⋅t) ⋅β + C2 ⋅cos (β ⋅t) ⋅β )
5.2.1.4.
C1
y( 0)
y_0
y'( 0)
V_0
exp(α ⋅0) ⋅(C1 ⋅cos (β ⋅0) + C2 ⋅sin(β ⋅0) )
C2
y(0)
y'(0)
Given
y_0
α ⋅exp(α ⋅0) ⋅( C1 ⋅cos ( β ⋅0) + C2 ⋅sin(β ⋅0)) + exp(α ⋅0) ⋅(−C1 ⋅sin(β ⋅0) ⋅β + C2 ⋅cos (β ⋅0) ⋅β )
V_0
⎡ 1.0000000000000000000 ⋅10-2 ⎢ Find( C1 , C2) → ⎢ -2 α ⎢ −(1.0000000000000000000 ⋅10 ) ⋅ β ⎣ 5.2.1.5.
II
α := −1
C1 := 1 ⋅10
β := 2 -2
C2 := −1 ⋅10 ⋅ -2
t := 0 , 0.1 .. 6 α⋅t
y( t) := e
α β
⋅(C1 ⋅cos (β ⋅t) + C2 ⋅sin(β ⋅t)) 0.01
0.005 y( t) 0
0.005
0
2
4 t
6
⎤ ⎥ ⎥ ⎥ ⎦
5.2.2.
II
Given
Q ⋅y'' ( t)
"odesolve"
− k ⋅y ( t) − λ ⋅y' ( t) ,
y( 0)
y_0
y' ( 0 )
V_0
, /
y_ := odesolve ( t , 6 , 100 )
6-
0.01
y_ ( t )
0.005
y ( t) 0
0.005
0
2
4
6
t
5.2.3.
II
Y0 := y_0
"rkfixed"
Y1 := V_0 II
dY 0
I
Y1
dt
λ −k ⋅ Y0 − ⋅ Y 1 Q Q
dY 1 dt
⎛ Y0 ⎞ ⎟ ⎝ Y1 ⎠
Y := ⎜
t1 := 0
II
t2 := 6
Y1 ⎛ ⎜ D ( t , Y) := ⎜ − k ⋅ Y0 − λ ⋅ Y 1 Q ⎝ Q Z := rkfixed( Y , t1 , t2 , N , D)
⎞ ⎟ ⎟ ⎠
N := 100
0
0.01
Z
〈1〉
0
0.005
Z=
y ( t) 0
0.005
0 Z
(Z (Z
〈0〉 〈1〉
)10 = 0.6
〈0〉
)10 = 4.546 × 10 − 3
,t
5
(
〈〉 y ⎡⎣ Z 0
⎤ ⎡⎛ 0.01 ⎞ ⎟ , t1 , t2 , N , D⎥ ⎣⎝ 0 ⎠ ⎦
Z_01 := rkfixed⎢⎜
⎤ ⎡⎛ 0.02 ⎞ ⎟ , t1 , t2 , N , D⎥ ⎣⎝ 0 ⎠ ⎦
Z_02 := rkfixed⎢⎜
⎡⎛ 0.04 ⎞ ⎤ ⎟ , t1 , t2 , N , D⎥ ⎣⎝ 0 ⎠ ⎦
Z_04 := rkfixed⎢⎜
⎤ ⎡⎛ 0.08 ⎞ ⎟ , t1 , t2 , N , D⎥ ⎣⎝ 0 ⎠ ⎦
Z_08 := rkfixed⎢⎜
1 0
2
0.01
0
1
0.06 914·10
-3
819·10
-3
2
0.12 669·10
-3
271·10
-3
3
0.18 288·10
-3
356·10
-3
4
0.24 794·10
-3
081·10
-3
5
0.3 206·10
-3
-0.01
6
0.36 545·10
-3
-0.012
7
0.42 832·10
-3
-0.012
8
0.48 083·10
-3
-0.013
9
0.54 316·10
-3
-0.013
10
0.6 546·10
-3
-0.013
11
0.66 786·10
-3
-0.013
12
0.72 048·10
-3
-0.012
13
0.78 341·10
-3
-0.011
14
0.84 675·10
-3
15
0.9 056·10
-3
-0.011 898·10
-3
)10⎤⎦ = 4.546 × 10 − 3
Z_01 Z_02 Z_04 Z_08
〈2〉 〈2〉 〈2〉 〈2〉
0
0.05
0.1
0.15
0.02
0 Z_01
D ( t , Y) :=
⎡⎛ ⎣⎝ ⎡⎛ Z_02 := rkfixed⎢⎜ ⎣⎝ ⎡⎛ Z_04 := rkfixed⎢⎜ ⎣⎝ ⎡⎛ Z_08 := rkfixed⎢⎜ ⎣⎝
0.04 〈〉 , Z_04 1
0.06 〈〉 , Z_08 1
0.08
⎞ ⎟ ⎟ ⎠
⎛ Y1 ⎜ ⎜ − k ⋅Y0 ⎝ Q
Z_01 := rkfixed⎢⎜
0.02 〈〉 , Z_02 1
〈1〉
0.01 ⎞
⎤ ⎟ , t1 , t2 , N , D⎥ ⎦ ⎤ ⎟ , t1 , t2 , N , D⎥ 0 ⎠ ⎦ 0.04 ⎞ ⎤ ⎟ , t1 , t2 , N , D⎥ 0 ⎠ ⎦ 0.08 ⎞ ⎤ ⎟ , t1 , t2 , N , D⎥ 0 ⎠ ⎦ 0 ⎠ 0.02 ⎞
0.2
Z_01 Z_02 Z_04 Z_08
〈2〉 〈2〉 〈2〉
0
〈2〉
0.2
5.3. V
α
. (
0.1 Z_01
0.05 〈1〉
, Z_02
0
〈1〉
, Z_04
〈1〉
0.05
, Z_08
〈1〉
0.1
,
. 5.1).
[14].
F=kV
y(t)
N(x, y)
V0 α
P=mg
0
x(t)
. 5.1. II
N(x, y) :
5.3 :
P = M·g
F = k·V.
x( 0) M⋅ M⋅
2
d
2
x
dt
d2 2
y
dt
d −k ⋅ x dt
d
d −k ⋅ y − M ⋅g dt
y( 0)
dt
d dt
0
x( 0) 0
y( 0)
V⋅cos (α )
V⋅sin(α )
:
α := 45 ⋅deg
, ,
V0 := 10
, /
k := 0.16
,
M := 1
,
g := 9.81
, / ^2
x ( 0) M⋅ M⋅
d
2 2
x
dt d
2 2
y
dt
d −k ⋅ x dt
d
d −k ⋅ y − M ⋅g dt
y ( 0)
dt
d dt
: II I dx dt
z1
dz1
−k
dt
M
dy dt
⋅z1
z3
dz3
−k
dt
M
⋅z3 − g⋅
M M
0
x( 0)
y( 0)
0
V⋅cos (α )
V⋅sin(α )
/
II "rkfixed"
"Bulstoer"
0 ⎛ ⎜ V0⋅cos (α ) z := ⎜ ⎜ 0 ⎜ ⎝ V0⋅sin(α )
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
z1 ⎛ ⎞ ⎜ ⎟ ⎜ −k ⎟ ⎜ M ⋅z1 ⎟ D( t , z) := ⎜ ⎟ z3 ⎜ ⎟ ⎜ −k ⎟ ⋅z3 − g ⎟ ⎜ ⎝ M ⎠
Z := rkfixed( z , 0 , 1.5 , 15 , D)
Z1 := Bulstoer( z , 0 , 1.5 , 15 , D) t := 0 , 0.1 .. 1.5
⎛
−k
⎞
⋅t ⎜ M M ⎟ ⋅V0⋅cos (α ) ⋅⎝ 1 − e x( t) := ⎠ k
− k ⎞⎤ ⎡ ⎛ ⋅t ⎢M ⎜ g⋅M M ⎟⎥ y( t) := ⎢ ⋅(g⋅M + k ⋅V0⋅sin(α )) ⋅⎝ 1 − e ⋅t ⎠⎥ − 2 k ⎣k ⎦
4 y( t) Z Z1
〈3〉
2
〈3〉
0
5
2 x( t) , Z
〈1〉
10
, Z1
〈1〉
(Z1 〈0〉 〈0〉 x⎡⎣(Z (Z1 〈1〉 〈〉 y⎡⎣(Z 0 (Z1 〈3〉
)10 = 1 )10⎤⎦ = 6.534 )10 = 6.534 )10⎤⎦ = 1.881 )10 = 1.881
(Z 〈0〉 )10 = 1 x( 1) = 6.534
(Z 〈1〉 )10 = 6.534 y( 1) = 1.881
(Z 〈3〉 )10 = 1.881
(Z 〈1〉 )10 − x⎡⎣(Z 〈0〉 )10⎤⎦ −8 = −5.104 × 10 % 〈0〉 ) ⎤ ⎡ ( x⎣ Z 10⎦ 〈 〉 1 (Z1 )10 − x⎡⎣(Z1 〈0〉 )10⎤⎦ −9 = −1.186 × 10 % 〈 〉 0 )10⎤⎦ x⎡⎣(Z1 (Z 〈3〉 )10 − y⎡⎣(Z 〈0〉 )10⎤⎦ −6 = −1.715 × 10 % 〈0〉 ) ⎤ ⎡ ( y⎣ Z 10⎦ 〈 〉 3 (Z1 )10 − y⎡⎣(Z1 〈0〉 )10⎤⎦ −8 = −3.983 × 10 % 〈 〉 0 )10⎤⎦ y⎡⎣(Z1 5.4.
.
1
, /( · ),
2
= 0.23
/( · ),
3
: 1,3– 2 = 60 = 45 /( · ).
2
, –
–
1 3
T1 = 780 °C,
=8
α1 = 70
= 100
, 2 –
. /( 2· ), α2 = 12
/( 2· ).
– : 1 = 0.81 T2 = 20 °C.
: α1 := 70 ⋅ α2 := 12 ⋅
W
m ⋅K 2
W
m ⋅K 2
δ1 := 100 ⋅mm δ2 := 60 ⋅mm δ3 := 8 ⋅mm
λ1 := 0.81 ⋅ λ2 := 0.23 ⋅ λ3 := 45 ⋅
W
m⋅K W
m⋅K
W
m⋅K
T1 := ( 780 + 273) ⋅K
T2 := ( 20 + 273) ⋅K
F := 1
: k :=
⎛ 1 δ1 δ2 δ3 1 ⎞ + + + + ⎜ ⎟ ⎝ α1 λ1 λ2 λ3 α2 ⎠ 1
ΔT := T1 − T2
Q := k ⋅F ⋅ΔT
k = 2.074
W
m ⋅K 2
Q = 1.576 × 10
3
W 2
m
Tw1 := T1 − Q ⋅
1
α1
T1_2 := Tw1 − Q ⋅
δ1 λ1
T2_3 := T1_2 − Q ⋅ Tw2 := T2_3 − Q ⋅
⎛ ⎜ ⎜ ⎜ T := ⎜ ⎜ ⎜ ⎜ ⎝
δ2 λ2
δ3 λ3
T1 ⎞
⎟
Tw1 ⎟ T1_2 ⎟
i := 0 .. 5
⎟ T2_3 ⎟ Tw2 ⎟ ⎟ T2 ⎠
1000 Ti K Ti
⎛⎜ ⎜ ⎜ ⎜ T= ⎜ ⎜ ⎜ ⎜ ⎝
3 1.053 × 10 ⎟⎞
1.03 × 10 835.868 424.644 424.363 293
3
⎟ ⎟ ⎟K ⎟ ⎟ ⎟ ⎟ ⎠
800
K Ti K Ti
600
K Ti K
400
0
2
i, 1, 4, 2, 3
4
6
Tw2 Tw2_ := T2 + Q ⋅
t2-t1
q0 := 0
Tw2 = 424.363 K
1
α2
Tw2_ = 424.363 K
N := 100
t1 := 0
t2 := 100
D( t , q) := k ⋅F ⋅ΔT
Q_ := rkfixed( q , t1 , t2 , N , D) 0
2 .10
5
Q_
〈1〉 1 .105
Q_ = 0
0 Q_
(Q_ (Q_ 5.5.
〈1〉
〈1〉
50 〈0〉
100
)1 = 1.576 × 103 ) N = 1.576 × 105
1
85
85 .34·10 5
86
86356·10 5
87
87371·10 5
88
88387·10 5
89
89403·10 5
90
90419·10 5
91
91434·10 5
92
92 .45·10 5
93
93466·10 5
94
94482·10 5
95
95498·10 5
96
96513·10 5
97
97529·10 5
98
98545·10 5
99
99561·10 5
100
100576·10 5
. [16]. , – γ = 10 ,
L=3
, h = 0.16 –
,
/ 3, –
: (
.
.
. 5.2). ν = 1/6.
x h R
L
p
. 5.2.
5.5
R = 2 ,
,
, x. , x d 4ω dx k=4
3 ⋅ ( 1 − ν2 ) R2 ⋅ h2
ω– x– q– D–
4
+ 4⋅k ⋅ω =
q , D
(5.1)
, ,
, ,
D= ν– R– h–
E ⋅ h3
12 ⋅ ( 1 − ν2 )
,
, , . (5.1)
ω ( x ) = e k ⋅x ⋅ (C1 ⋅ sin(k ⋅ x ) + C 2 ⋅ cos(k ⋅ x )) + + e −k ⋅x ⋅ (C 3 ⋅ sin(k ⋅ x ) + C 4 ⋅ cos(k ⋅ x )) +
ϕ ( x) =
M ( x) = −D ⋅
–
Q( x ) = − D ⋅
–
d 2ω dx 2 d 3ω
dx 3 E ⋅h⋅ω N( x ) = , R
– E– γ–
dω , dx
γ ⋅ ( L − x) ⋅ R 2 E ⋅h
.
, ,
, ,
. 1 –
4
. ω(0)=0
ϕ(0)=0
,
, x=0.
,
M(x)= 0 ω(0)=0
, Q(x)= 0 ϕ(0)=0
(
x=L. 1 = 2 = 0.
x=0,
). R=2000
.
q, :
ϕ(x),
4
, L=3000
, h=160 . q = γ ⋅( L − x ) (
.
. 5.2).
, M(x), Q(x), N(x). ,
1 –
,
(5.1), MathCAD. MathCAD: «sbval», «load», «score»
«rkfixed».
MathCAD
E⋅h⋅ω ( x)
N ( x)
R
φ ( x)
d dx
ω ( x)
−D ⋅
M ( x)
−D ⋅
Q ( x)
d2 2
dx d3
3
dx
ω ( x)
ω ( x)
ω ( x)
e
k⋅ x
+ γ⋅
-
− k⋅ x
⋅( C1 ⋅sin( k ⋅x) + C2 ⋅cos ( k ⋅x) ) + e ( L − x) ⋅R E ⋅h
⋅( C3 ⋅sin( k ⋅x) + C4 ⋅cos ( k ⋅x) ) ...
2
h k⋅ x − k⋅ x E⋅ ⋅⎡ e ⋅( C1 ⋅sin( k ⋅x) + C2 ⋅cos ( k ⋅x) ) + e ⋅( C3 ⋅sin( k ⋅x) + C4 ⋅cos ( k ⋅x) ) ... ⎤ ⎥ R⎢ 2 ⎢ + γ ⋅ ( L − x) ⋅R ⎥ E ⋅h ⎣ ⎦
N ( x)
φ(x), M(x)
" -
φ ( x)
/ x:
Q(x)
/
"
k ⋅exp( k ⋅x) ⋅( C1 ⋅sin( k ⋅x) + C2 ⋅cos ( k ⋅x) ) ... + exp( k ⋅x) ⋅( C1 ⋅cos ( k ⋅x) ⋅k − C2 ⋅sin( k ⋅x) ⋅k) ... + −k ⋅exp( −k ⋅x) ⋅( C3 ⋅sin( k ⋅x) + C4 ⋅cos ( k ⋅x) ) ... + exp( −k ⋅x) ⋅( C3 ⋅cos ( k ⋅x) ⋅k − C4 ⋅sin( k ⋅x) ⋅k) −
γ ⋅R ( E⋅h) 2
M ( x)
−2 ⋅D⋅k ⋅exp( k ⋅x) ⋅C1 ⋅cos ( k ⋅x) + 2 ⋅D⋅k ⋅exp( k ⋅x) ⋅C2 ⋅sin( k ⋅x) ... 2
2
+ 2 ⋅D⋅k ⋅exp( −k ⋅x) ⋅C3 ⋅cos ( k ⋅x) − 2 ⋅D⋅k ⋅exp( −k ⋅x) ⋅C4 ⋅sin( k ⋅x) 2
2
-
2 ⋅D ⋅k ⋅exp ( k ⋅x) ⋅C1 ⋅sin( k ⋅x) − 2 ⋅D ⋅k ⋅exp ( k ⋅x) ⋅C1 ⋅cos ( k ⋅x) ... 3
Q ( x)
+ − 2 ⋅D ⋅k ⋅exp ( − k ⋅x) ⋅C3 ⋅cos ( k ⋅x) ...
3
3
+ 2 ⋅D ⋅k ⋅exp ( k ⋅x) ⋅C2 ⋅sin( k ⋅x) − 2 ⋅D ⋅k ⋅exp ( − k ⋅x) ⋅C3 ⋅sin( k ⋅x) ... 3
+ − 2 ⋅D ⋅k ⋅exp ( − k ⋅x) ⋅C4 ⋅cos ( k ⋅x) ...
3
3
+ 2 ⋅D ⋅k ⋅exp ( − k ⋅x) ⋅C4 ⋅sin( k ⋅x) + 2 ⋅D ⋅k ⋅exp ( k ⋅x) ⋅C2 ⋅cos ( k ⋅x) 3
3 ω ( 0)
3
4
0
φ ( 0)
0
C1
0
C2
0
Given L ⋅R
0
C4 + γ ⋅
0
− k ⋅C4 + C3 ⋅k −
2
E ⋅h
γ ⋅R
2
E ⋅h
⎡ − γ ⋅R2 ⋅ ( k ⋅L − 1 ) ⎢ ( k ⋅E ⋅h) Find ( C3 , C4 ) → ⎢ 2 ⎢ R ⎢ − γ ⋅L ⋅ ( E ⋅h) ⎣ ORIGIN ≡ 1 L := 3 ⋅m ν :=
1 6
h := 0.16 ⋅m
E := 2 ⋅10 ⋅MPa 3
R := 2 ⋅m
γ := 10 ⋅
kN 3
m
:
kN ≡ 10 ⋅newton 3
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
MPa ≡ 10 ⋅Pa 6
: D D := E⋅
(
3
h
12 ⋅ 1 − ν
)
k D = 702.171 kN ⋅m
2
1
4 ⎛ E ⋅h ⎞ ⎟ 2 ⎝ D⋅R ⋅4 ⎠
k := ⎜
k = 2.31 m
-1
1, 2, 3 C1 := 0 ⋅m C3 := −γ ⋅R ⋅ 2
( k ⋅L − 1) ( k ⋅E⋅h)
C3 = −3.209 × 10
−4
m
x := 0 ⋅mm, 10 ⋅mm.. L
4 C2 := 0 ⋅m C4 := −γ ⋅L⋅
2
R ( E⋅h)
C4 = −3.75 × 10
−4
m
k⋅ x − k⋅ x ⋅( C3 ⋅sin( k ⋅x) + C4 ⋅cos ( k ⋅x) ) ... ⎤ ω ( x) := ⎡ e ⋅( C1 ⋅sin( k ⋅x) + C2 ⋅cos ( k ⋅x) ) + e
⎢ 2 ⎢ + γ ⋅ ( L − x) ⋅R E ⋅h ⎣
⎥ ⎥ ⎦
h k⋅ x − k⋅ x N ( x) := E⋅ ⋅⎡ e ⋅( C1 ⋅sin( k ⋅x) + C2 ⋅cos ( k ⋅x) ) + e ⋅( C3 ⋅sin( k ⋅x) + C4 ⋅cos ( k ⋅x) ) ... ⎤ R⎢ ⎥ 2 ⎢ + γ ⋅ ( L − x) ⋅R ⎥ E⋅h ⎣ ⎦
⎤ ⎥ ⎥ 2 ⎥ ⎢ γ ⋅R ⎢ + exp( −k ⋅x) ⋅( C3 ⋅cos ( k ⋅x) ⋅k − C4 ⋅sin( k ⋅x) ⋅k) − ⎥ ( E⋅h) ⎦ ⎣
φ ( x) := ⎡ k ⋅exp( k ⋅x) ⋅( C1 ⋅sin( k ⋅x) + C2 ⋅cos ( k ⋅x) ) ... ⎢ + exp( k ⋅x) ⋅( C1 ⋅cos ( k ⋅x) ⋅k − C2 ⋅sin( k ⋅x) ⋅k) ... ⎢ + −k ⋅exp( −k ⋅x) ⋅( C3 ⋅sin( k ⋅x) + C4 ⋅cos ( k ⋅x) ) ...
2 2 M ( x) := ⎛ −2 ⋅D ⋅k ⋅exp( k ⋅x) ⋅C1 ⋅cos ( k ⋅x) + 2 ⋅D ⋅k ⋅exp( k ⋅x) ⋅C2 ⋅sin( k ⋅x) ... ⎞
⎜ ⎟ 2 2 ⎝ + 2 ⋅D⋅k ⋅exp( −k ⋅x) ⋅C3 ⋅cos ( k ⋅x) − 2 ⋅D⋅k ⋅exp( −k ⋅x) ⋅C4 ⋅sin( k ⋅x) ⎠
3 3 Q ( x) := ⎛ 2 ⋅D⋅k ⋅exp( k ⋅x) ⋅C1 ⋅sin( k ⋅x) − 2 ⋅D⋅k ⋅exp( k ⋅x) ⋅C1 ⋅cos ( k ⋅x) ...
⎞ ⎜ ⎟ 3 ⎜ + −2 ⋅D⋅k ⋅exp( −k ⋅x) ⋅C3 ⋅cos ( k ⋅x) ... ⎟ ⎜ + 2 ⋅D⋅k3⋅exp( k ⋅x) ⋅C2 ⋅sin( k ⋅x) − 2 ⋅D⋅k3⋅exp( −k ⋅x) ⋅C3 ⋅sin( k ⋅x) ... ⎟ ⎜ + −2 ⋅D⋅k3⋅exp( −k ⋅x) ⋅C4 ⋅cos ( k ⋅x) ... ⎟ ⎜ ⎟ 3 3 ⎝ + 2 ⋅D⋅k ⋅exp( −k ⋅x) ⋅C4 ⋅sin( k ⋅x) + 2 ⋅D⋅k ⋅exp( k ⋅x) ⋅C2 ⋅cos ( k ⋅x) ⎠
IV d
4 4
dx
ω + E⋅
h
D⋅R
2
⋅ω
(
d dx
Qx( 0) LL :=
−D ⋅ −D ⋅
d
D )
ω ( 0)
Mx( 0)
q
ω ( 0)
2 2
dx d3
3
dx
L m
−D ⋅
Mx( L)
0 φ
0
Qx( L)
−D ⋅
d
ω ( L)
0
ω ( L)
0
2 2
dx d
3 3
dx
ω ( 0)
ω ( 0) LL = 3
⎛ 0⎞ ⎟ ⎝ 0⎠
v := ⎜
⎛ ⎜ load( x1 , v) := ⎜ ⎜ ⎜ ⎝
0 ⎞
x=0
⎟
0 ⎟ v1 ⎟
⎟
v2 ⎠
ω2 ⎤ ⎡ ⎢ ⎥ ω3 ⎢ ⎥ ⎢ ⎥ DD(x , ω ) := ω4 ⎢ ⎥ ⎢ ( LL − x) ⎥ h ⎢ γ ⋅ D − E⋅ 2 ⋅ω 1 ⎥ D⋅R ⎣ ⎦
⎛ ω3 ⎞ ⎟ ⎝ ω4 ⎠
score(x2 , ω ) := ⎜
x=L
B := sbval( v , 0 , LL, DD , load, score)
⎛ 3.425 × 10− 3 ⎞ ⎟ ⎝ −0.017 ⎠
B=⎜
bnd := load( 0 , B)
IV
"rkfixed"
y := bnd
t1 := 0
y1 = 0 y2 = 0
t2 := LL
NN := 100 x=0
−3
y3 = 3.425 × 10 y4 = −0.017
Z := rkfixed( y , t1 , t2 , NN , DD)
(
〈〉 x_ := Z 1
(
〈〉 ω_ := Z 2
⎡ 〈2〉 Nθ_ := ⎢Z
(
⎣
〈〉 φ_ := Z 3
〈4〉 Mx_ := ⎡⎣Z
〈〉 Qx_ := ⎡⎣Z 5
Z
) ⋅m
) ⋅m
)
⎛ E⋅h ⎞⎤ ⋅m ⎟⎥ ⎝ R ⎠⎦
⋅⎜
⋅( −D)⎤⎦ ⋅m
−1
⋅( −D)⎤⎦ ⋅m
−2
, mm
, mm
3000
2000
1000
0.1
0
0.1
0.2
"MathCAD" ω ( 0 ⋅mm) = 0 mm
ω ( L) = −4.815 × 10
−4
L = 3 × 10 mm 3
max(ω_) = 0.252 mm
ω_1 = 0 mm
mm
ω_101 = −1.021 × 10
−3
mm
x_101 = 3 × 10 mm 3
ω_33 = 0.252 mm
ω ( 960 ⋅mm) = 0.252 mm
x_90 = 2.67 × 10 mm
ω ( x_90) = 0.041 mm
x_33 = 960 mm
3
ω_90 = 0.04 mm
, Q*m/kN
, mm
3000
2000
1000
10
0
10
20
30
40
50
"MathCAD" ,
, mm
3000
x_
2000
mm x mm 1000
2 .10
4
0
2 .10 4 .10 φ_ , φ( x) 4
4
"MathCAD"
, M*m/(kN*m)
, mm
3000
x_ 2000
mm x mm
1000
3
2 Mx_⋅
1 m
kN⋅ m
, M ( x) ⋅
1
m
kN⋅ m
"MathCAD"
, Q*m/kN
3000
, mm
0
2000
1000
5
0
5
10
15
MathCAD
5.6.
.
,
XX [8].
,
( . «0»
. , 1966),
«1»: 0 → 1 ( 0.1), «2»: 1 + 1 → 2 + 1 (103). , «2» «1»: 1 + 2 → 0 + 2 (102). , D(t, y).
) , [8].
«1» ,
(
: y0 := 1
0
y1 := 0
1
y2 := 0
−0.1 ⋅ y0 + 10 ⋅ y1 ⋅ y2
d y0 dt d dt
2 2
0
0.1 ⋅ y0 − 10 ⋅ y1 − 10 ⋅ y1 ⋅ y2 3
y1
2
1
10 ⋅ y1 + 10 ⋅ y1 ⋅ y2 − 10 ⋅ y1 ⋅ y2
d y2 dt
3
2
2
2
: 5.6.1.
t1 := 0
I
t2 := 100
Stiffb
N := 100
2 ⎛ −0.1 ⋅ y0 + 10 ⋅ y1 ⋅ y2 ⎜ D( t , y) := ⎜ 0.1 ⋅ y0 − 103 ⋅ y1 − 102 ⋅ y1 ⋅ y2 ⎜ ⎜ 103 ⋅ y + 102 ⋅ y ⋅ y − 102 ⋅ y ⋅ y 1 1 2 1 2 ⎝
Stif
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
⎡ d( d ( d ( d ( 2 2 2 2 − 0.1 ⋅ y0 + 10 ⋅ y1 ⋅ y2) − 0.1 ⋅ y0 + 10 ⋅ y1 ⋅ y2) −0.1 ⋅ y0 + 10 ⋅ y1 ⋅ y2) −0.1 ⋅ y0 + 10 ⋅ y1 ⋅ y2) ⎢ dt dy0 dy1 dy2 ⎢ ⎢ d d ( d ( d ( 3 2 3 2 3 2 3 2 0.1 ⋅ y0 − 10 ⋅ y1 − 10 ⋅ y1 ⋅ y2) 0.1 ⋅ y0 − 10 ⋅ y1 − 10 ⋅ y1 ⋅ y2) 0.1 ⋅ y0 − 10 ⋅ y1 − 10 ⋅ y1 ⋅ y2) ⎢ (0.1 ⋅ y0 − 10 ⋅ y1 − 10 ⋅ y1 ⋅ y2) dy0 dy1 dy2 ⎢ dt ⎢ d( 3 d ( 3 d ( 3 d ( 3 10 ⋅ y1) 10 ⋅ y1) 10 ⋅ y1) 10 ⋅ y1) ⎢ dt dy0 dy1 dy2 ⎣
"
/
/
"
0 − .1 100 ⋅ y2 100 ⋅ y1 ⎜⎛ ⎜ 0 .1 − 1000 − 100 ⋅ y2 − 100 ⋅ y1 ⎜ 0 0 1000 0 ⎝
100 ⋅ y1 100 ⋅ y2 ⎛ 0 − .1 ⎜ J ( t , y) := ⎜ 0 .1 − 1000 − 100 ⋅ y2 − 100 ⋅ y1 ⎜ 0 1000 ⎝ 0 0
Z := Stiffb( y , t1 , t2 , N , D , J)
⎟⎞ ⎟ ⎟ ⎠
⎞ ⎟ ⎟ ⎟ ⎠
Z1 := Stiffr( y , t1 , t2 , N , D , J)
1
Z Z Z
〈1〉 〈2〉 〈3〉
⋅5000 0.5
0 Z
50 〈0〉
100
0 1 2
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
5.6.2.
I stiffb
stiffr
acc := 0.000000001
kmax := 5
s := 10
ZZ := stiffb( y , t1 , t2 , acc , D , J , kmax , s)
ZZ1 := stiffr( y , t1 , t2 , acc , D , J , kmax , s)
1 ⎛ 0 ⎜ ⎜ 12.718 0.295 ⎜ ZZ = ⎜ 36.521 0.033 ⎜ −4 ⎜ 92.637 2.01 × 10 ⎜ −4 ⎝ 100 1.029 × 10
0
2.757 × 10
3.019 × 10
1.827 × 10
9.355 × 10
1 ⎛ 0 ⎜ ⎜ 10.107 0.377 ⎜ ZZ1 = ⎜ 20.295 0.146 ⎜ 0.058 ⎜ 30.327 ⎜ −4 ⎝ 100 1.029 × 10
(Z
(Z1 (Z
(Z1
〈0〉
〈0〉 〈2〉 〈2〉
−5 −6 −8 −9
0
3.549 × 10
1.347 × 10
5.326 × 10
9.355 × 10
)100 = 100
)100 = 100
(Z
〈1〉
(Z
〈3〉
(Z1
)100 = 9.355 × 10 − 9
)100 = 9.356 × 10 − 9
(Z1
⎞ ⎟ 0.705 ⎟ ⎟ 0.967 ⎟ ⎟ 1 ⎟ ⎟ 1 ⎠ 0
−5 −5 −6 −9
〈1〉
〈3〉
⎞ ⎟ 0.623 ⎟ ⎟ 0.854 ⎟ ⎟ 0.942 ⎟ ⎟ 1 ⎠ 0
)100 = 1.029 × 10 − 4
)100 = 1.029 × 10 − 4
)100 = 1
)100 = 1
6. 6.1.
( .
∂ u 2
∂t 2 x=0
L, F, x=L
= v2
. ∂ 2u ∂x 2
. ,
x=L. .
, F –S
E,
[11].
: E := 2.1 ⋅10
11
,
ρ := 7850
⎛ E⎞ ⎟ ⎝ρ⎠
v := ⎜
,
/ ^3
1 2
, /
v = 5.172 × 10
3
F := 1000
S := 0.0001
,
)
, , ^2
: N := 20
N1 := 40
i := 0 .. N
i1 := 1 .. N − 1 j := 0 .. N1
j1 := 1 .. N1 − 1
t_end := 0.00005 L_max := 1
t_begin := 0 L_min:= 0
L_max − L_min
h_x :=
h_t :=
N
h_x
N1 −6
h_x = 0.05 v⋅h_t
t_end − t_begin
h_t = 1.25 × 10
≤ 1 xi := 0 + h_x⋅i
u1( x) := F ⋅
u2( x) := 0
t j := 0 + h_t⋅ j
x
E ⋅S
Ui, 0 := u1( xi)
Ui, 1 := u1( xi) + h_t⋅u2( x)
U0 , j := 0 r :=
v⋅h_t h_x
(
)
Ui1, j1+ 1 := 2 − 2 ⋅r ⋅Ui1, j1 + r ⋅( Ui1+ 1 , j1 + Ui1−1 , j1) − Ui1, j1−1 2
2
x20 = 1
U
t20 = 2.5 × 10
−5
u( x , t)
∑ ∞
( −1) 8 ⋅F ⋅L_max ⎡ ( 2 ⋅n − 1) ⋅π ⋅v⋅t⎤ ⎡ ( 2 ⋅n − 1) ⋅π ⋅x⎤ ⋅cos ⎢ ⋅ ⎥ ⋅sin⎢ ⎥ 2 2 ⎣ 2 ⋅L_max ⎦ ⎣ 2 ⋅L_max ⎦ π ⋅E⋅S n = 0 ( 2 ⋅n − 1) n+ 1
∑
Mathcad
8 ⋅F ⋅L_max ( −1) ⎡ ( 2 ⋅n − 1) ⋅π ⋅v⋅t⎤ ⎡ ( 2 ⋅n − 1) ⋅π ⋅x⎤ u1( x , t) := ⋅cos ⎢ ⋅ ⎥ ⋅sin⎢ ⎥ 2 2 ⎣ 2 ⋅L_max ⎦ ⎣ 2 ⋅L_max ⎦ π ⋅E⋅S n = 0 ( 2 ⋅n − 1) n+ 1
20
U1i, j := u1( xi , t j)
U1
6.2.
U
.
u( 0,t ) = u( L ,t ) = 0,
.
.
∂u ∂ 2u =k 2 , ∂t ∂x x ⋅( L − x ) [13]. u( x ,0 ) = L2 : λ := 45.4
, /( * )
Cp := 460
,
ρ := 7850
k :=
,
λ
Cp ⋅ρ
/ ^3
−5
k = 1.257 × 10
: N := 100
N1 := 20
i := 0 .. N
j := 0 .. N1
i1 := 1 .. N − 1 t_end := 4
t_begin := 0
L_max := 2 h_x :=
L_max − L_min N h_x = 0.02 h_t ≤
2
h_x
2 ⋅k
=1
L_min:= 0
h_t :=
t_end − t_begin N1
h_t = 0.2
/( * )
xi := 0 + h_x⋅i
t j := 0 + h_t⋅ j
x⋅( L_max − x)
u1( x) :=
2
Ui, 0 := u1( xi)
L_max
U0 , j := 0
UN , j := 0
⎛
Ui1, j+ 1 := ⎜ 1 −
⎝
⋅( Ui1−1 , j + Ui1+ 1 , j) ⎟ ⋅Ui1, j + 2 h_x h_x ⎠
2 ⋅k ⋅h_t ⎞
k ⋅h_t
2
0.26 0.24 0.22 0.2 Ui , 0 0.18 0.16 U N1 0.14 i, 0.12 2 0.1 0.08 Ui , N1 0.06 0.04 0.02 0 − 0.02 0.02
U
0 0
u( x , t)
20
40
60
80
i
100 100
2 2 ⎡⎢ ⎤⎥ − ( 2⋅ n + 1) ⋅ π ⋅ k⋅ t 2 ( 2 ⋅ n + 1 ) ⋅ π ⋅ x 8 1 ⎡ ⎤ L_max ⎢ ⎥ ⋅ ⋅e ⋅sin⎢ ⎥⎥ 3 3 ⎢ L_max ⎣ ⎦ π n = 0 ⎣ ( 2 ⋅n + 1 ) ⎦
∑ ∞
Mathcad
2 2 ⎡⎢ − ⎡⎣( 2⋅ n + 1) ⋅ π ⋅ k⋅ t⎤⎦ ⎥⎤ 2 1 8 ( 2 ⋅ n + 1 ) ⋅ π ⋅ x ⎡ ⎤ L_max ⎢ ⎥ u1( x , t) := ⋅ ⋅e ⋅sin⎢ ⎥⎥ 3 3 ⎢ L_max ⎣ ⎦ π n = 0 ⎣ ( 2 ⋅n + 1 ) ⎦
∑ 20
U1i, j := u1( xi , t j)
U1
6.3.
.
.
. -
∂u ∂ 2u =d 2 , ∂t ∂x u( 0,t ) = u( L ,t ) – u( x ,0 ) = 1
: 0 ≤ x ≤ 0.5
, [9].
1
0
,
^2/c
: d := 5.9 ⋅10
−5
: N1 := 10
N := 100
i := 0 .. N
j := 0 .. N1
i1 := 1 .. N − 1 t_end := 2
t_begin := 0
L_max := 0.5 h_x :=
L_min:= 0
L_max − L_min
h_x = 5 × 10 h_t ≤
h_t :=
N
−3
t_end − t_begin N1
h_t = 0.2
2
h_x
2 ⋅d
xi := 0 + h_x⋅i
t j := 0 + h_t⋅ j
u1 ( x) := 1
Ui , 0 := u1 ( xi)
U0 , j := 1 −
h_t ⋅j t_end
U N , j := 1 −
t_end
⎛
h_t
Ui1, j+ 1 := ⎜ 1 −
⎝
⋅j
⋅( Ui1−1 , j + Ui1+ 1 , j) ⎟ ⋅Ui1, j + 2 h_x ⎠ h_x
2 ⋅d ⋅h_t ⎞
d ⋅h_t
2
1.1 Ui, N1 U N1 i,
2
Ui, 0 0
U
1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
50
100
0
i
100
0≤t≤2
,
6.4.
.
.
.
∂x 2
u( x , y ) = x 2 − y 2
: -1 =x= 1, -1 =y= 1.
: u( x , y) := x − y 2
2
: 6.4.1.
. .
"relax" N := 10
i := 0 .. N
N1 := 10
j := 0 .. N1
i1 := 1 .. 9
j1 := 1 .. 9
L_max := 2
L_min:= 0
h_x :=
h_y :=
L_max − L_min N
h_x = 0.2
xi := −1 + h_x⋅i
∂ 2u
L_max − L_min N1
h_y = 0.2
yj := −1 + h_y⋅ j
+
∂ 2u ∂y 2
= 0,
yj
0
1
0
1
xi
U1i, j := u( xj , yi)
U0 , j := U10 , j
UN , j := U10 , j
Ui, 0 := U1i, 0
Ui, N := U1i, 0 "relax"
ai, j := 1 b := a c := a
d := a
e := −4 ⋅a
f i, j := 0
v0 , j := U10 , j
vN , j := U10 , j
vi, 0 := U1i, 0
vi, N := U1i, 0
U := relax( a , b , c , d , e , f , v , 0.95)
U
U − U1
6.4.2.
. .
N := 4
N1 := 4
i := 0 .. N
j := 0 .. N1
i1 := 1 .. N − 1
j1 := 1 .. N1 − 1
L_max := 2
L_min:= 0
h_x :=
h_y :=
L_max − L_min N
h_x = 0.5
h_y = 0.5
xi := −1 + h_x ⋅ i
yj
L_max − L_min N1
yj := −1 + h_y ⋅ j
0
1
0
1
xi
U1i, j := u( xj , yi)
U0 , j := U10 , j
UN , j := U10 , j
u1 := 0
u6 := 0
u2 := 0
u7 := 0
Ui, 0 := U1i, 0
Ui, N := U1i, 0
u3 := 0
u8 := 0
u4 := 0
u9 := 0
u5 := 0
Given 0.75 − 4 ⋅ u1 + u2 − 0.75 + u4
1 − 4 ⋅ u4 + u5 + u1 + u7
0
0
0.75 − 4 ⋅ u7 + u8 + u4 − 0.75 u1 − 4 ⋅ u2 + u3 − 1 + u5
0
0
u4 − 4 ⋅ u5 + u6 + u2 + u8
0
u7 − 4 ⋅ u8 + u9 + u5 − 1
0
u2 − 4 ⋅ u3 + 0.75 − 0.75 + u6 u5 − 4 ⋅ u6 + 1 + u3 + u9
0
0
u8 − 4 ⋅ u9 + 0.75 + u6 − 0.75
0
Z := Find( u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8 , u9) Ui1, j1 := Z3⋅( i1−1) + j1−1
⎛ ⎜ ⎜ U=⎜ ⎜ ⎜ ⎝
0
−0.75
0.75
0
−0.25
1
0.25
0.75
0
0
−0.75
−1
−0.75
0
0
0.75
0
0.25
1
−1
0
0.75
−0.25
−0.75
0
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⎛ ⎜ ⎜ U − U1 = ⎜ ⎜ ⎜ ⎝
0 0 0 0 0 ⎞
⎟ ⎟ 0 0 0 0 0 ⎟ ⎟ 0 0 0 0 0 ⎟ 0 0 0 0 0 ⎠ 0 0 0 0 0
U
6.5.
.
.
∂ u 2
∂x
u(x,y)
2
+
∂ u 2
∂y 2
= − f ( x, y ) ,
.
,
f(x,y) 8 8.
1000,
:
N := 8
i := 0 .. N
j := 0 .. N
Mi, j := 0
M4 , 4 := 1000
: A := multigrid( − M , 2 )
A
A
7.
(
7.1.
.
)
,
(
. 7.1) [7].
n y
p
Vi (vi)
E1 A1 xi
C α i
L
Ui (ui) yi
x . 7.1.
C
, E1 – , ui , vi –
p–
, A1 –
,L– , i, n –
, Ui , Vi –
, xi , yi –
, .
, .
. 7.2
, 1
n.
, .
,
Y4
{R4} p
4
3
X4
(c) y (a)
(d) (b) 5
2
1
6
x
V3
p y
3
(a)
U3
2 ы
1
x
ч ы э е е . 7.2.
,
, .
,
1–3
,
a, .
p , U, V
-
u, v
,
,
{F }ap
. .
(
),
a,
{F }a
⎧U1 ⎫ ⎪V ⎪ 1 ⎧ F1 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪⎪U 2 ⎪⎪ = ⎨ F2 ⎬ = ⎨ ⎬ , ⎪ F ⎪ ⎪V2 ⎪ ⎩ 3 ⎭ ⎪U ⎪ 3 ⎪ ⎪ ⎪⎩ V3 ⎪⎭
{δ}a
⎧ u1 ⎫ ⎪v ⎪ 1 ⎧ δ1 ⎫ ⎪ ⎪ ⎪ u ⎪ ⎪ ⎪ ⎪⎪ = ⎨δ 2 ⎬ = ⎨ 2 ⎬ . ⎪ δ ⎪ ⎪ v2 ⎪ ⎩ 3 ⎭ ⎪u ⎪ 3 ⎪ ⎪ ⎪⎩ v3 ⎪⎭
,
{F }a = [k ]a {δ}a + {F }ap + {F }εa , {F }aε – , 0
–
,
0
(7.1) ,
,
,
,
.
,
.
{σ}a ,
.
{σ}a = [S ]a {δ}a + {σ}ap + {σ}εa
0
–
[k ]a
,
[S ]a
, (7.2) .
(7.2)
,
.
(7.1)
,
–
. ,
,
.
b .
, ,
, ,
.
.
{F }a Fi
δi
⎧ F1 ⎫ ⎪ . ⎪ ⎪⎪ ⎪⎪ =⎨ . ⎬ ⎪ . ⎪ ⎪ ⎪ ⎪⎩ Fm ⎪⎭
⎧ δ1 ⎫ ⎪ . ⎪ ⎪⎪ ⎪⎪ =⎨ . ⎬, ⎪ . ⎪ ⎪ ⎪ ⎪⎩δm ⎪⎭
.
,
[k ]
e
kii
{δ}a
,
⎡ kii ⎢ . ⎢ =⎢ . ⎢ ⎢ . ⎢k mi ⎣
kij . . . kmj
l ×l , l –
. .–
kim ⎤ . ⎥⎥ . ⎥, ⎥ . ⎥ kmm ⎥⎦
. C
E (
A
. 7.1).
p
xi , y i
xn , y n ,
L=
{(x
ε 0 = αT .
n
}
− x i )2 + ( y n − y i )2 , ⎛ y n − yi ⎞ ⎟⎟ . ⎝ x n − xi ⎠
α = arctg ⎜⎜
. ,
,
,
{F }ap
ε0
⎧U i ⎫ ⎧− sin α ⎫ ⎪ ⎪ ⎪ cosα ⎪ ⎧F ⎫ ⎪V ⎪ ⎪ ⎪ pL =⎨ i⎬ =⎨ i ⎬ =⎨ . ⎬ F U sin α − ⎩ n ⎭p ⎪ n ⎪ ⎪ ⎪ 2 ⎪⎩Vn ⎪⎭ ⎪⎩ cosα ⎪⎭ p , EαTA ,
pL 2 .
{F }εa
0
⎧U i ⎫ ⎧− cosα ⎫ ⎪ ⎪ ⎪ − sin α ⎪ ⎛ F ⎞ ⎪V ⎪ ⎪ ⎪ = ⎜⎜ i ⎟⎟ = ⎨ i ⎬ = − ⎨ ⎬(EαTA). F U cos α ⎝ n⎠ ⎪ n⎪ ⎪ ⎪ ⎪⎩Vn ⎪⎭ ⎪⎩ sin α ⎪⎭ ε0
,
{δ}a (un − ui )cos α + (vn − vi ) sin α .
,
,
{F }δa
⎧ ui ⎫ ⎪ ⎪ ⎧δ ⎫ ⎪v ⎪ =⎨ i⎬=⎨ i⎬ ⎩δn ⎭ ⎪un ⎪ ⎪⎩vn ⎪⎭ EA L , .
EαTA
⎧U i ⎫ ⎪V ⎪ ⎧ Fi ⎫ ⎪ ⎪ =⎨ ⎬ =⎨ i ⎬ = F ⎩ n ⎭δ ⎪U n ⎪ ⎪⎩Vn ⎪⎭ δ
⎡ cos 2 α ⎢ EA ⎢ sin α cos α = L ⎢ − cos 2 α ⎢ ⎣⎢− sin α cos α
sin α cos α
− cos 2 α
sin α
− sin α cos α
− sin 2 α
sin α cos α
2
− sin α cos α
cos 2 α
− sin α cos α ⎤ ⎥ − sin 2 α ⎥ × sin α cos α ⎥ ⎥ sin 2 α ⎦⎥
,
⎧ ui ⎫ ⎪v ⎪ ⎪ ⎪ × ⎨ i ⎬ = [k ]a {δ }a . ⎪u n ⎪ ⎪⎩v n ⎪⎭
(7.1). .
,
(7.2)
, ,
, I–
− sin α − sin α
cos α cos α
sin α ⎤ a ⎧ 1 ⎫ pL2 d ⎧1⎫ {δ } + ⎨ ⎬ − ⎨ ⎬ E αT , sin α ⎥⎦ ⎩− 1⎭ 8 I ⎩1⎭
.
(7.2). ,
:
L := 1000 ⋅mm d := 20 ⋅mm h := 40 ⋅mm
α := 45 ⋅deg
E := 2.1 ⋅10 ⋅MPa 5
a := 1.1 ⋅10
−5
T := 30 ⋅K
p := 10 ⋅
⎛ ⎜ ⎜ δ := ⎜ ⎜ ⎝
,
C,
⎧σ 1 ⎫ E ⎡ − cos α ⎨ ⎬ = ⎢ ⎩σ 2 ⎭ C L ⎣ − cos α
d–
,
⋅
1 K
N 1 ⎞
mm
⎟
2 ⎟ ⋅mm 1.5 ⎟
⎟
0.5 ⎠
.
:
A := 2 ⋅d ⋅h
−sin(α ) ⎞
⎛ ⎜ Fp := ⎜ ⎜ ⎜ ⎝
⎟
cos (α ) ⎟ p ⋅L ⋅ −sin(α ) ⎟ 2
,
⎟
⎛ ⎜ ⎜ Fε0 := ⎜ ⎜ ⎝
cos (α ) ⎠
−cos (α ) ⎞
⎟
−sin(α ) ⎟ ⋅E⋅a ⋅T⋅A cos (α ) ⎟
,
⎟
sin( α ) ⎠
: 2 2 ⎛ −sin( α ) ⋅cos ( α ) −cos (α ) cos ( α ) sin( α ) ⋅cos ( α ) ⎜ 2 2 ⎜ sin(α ) ⋅cos (α ) sin(α ) −sin( α ) −sin(α ) ⋅cos ( α ) k := ⎜ 2 ⎜ −cos (α )2 −sin( α ) ⋅cos ( α ) sin( α ) ⋅cos (α ) cos (α ) ⎜ 2 2 sin(α ) ⋅cos ( α ) −sin( α ) sin(α ) ⎝ −sin(α ) ⋅cos (α )
Fδ := k ⋅δ
,
F := Fδ + Fp + Fε0
⎛ ⎜ F=⎜ ⎜ ⎜ ⎝
7.2.
,
86.06 ⎞
⎟
93.132 ⎟ kN −93.132 ⎟
⎟
−86.06 ⎠
.
(
. 7.3),
,
,
E– ,L– L = 2 , h = 2 , E = 2·1011
⎞ ⎟ ⎟ E⋅A ⎟⋅ ⎟ L ⎟ ⎠
p⋅L ( . 7.3), 2⋅E ,h– , p = 1·108 , = 0.3.
p
ΔL
. ΔL =
.
,μ–
p
L . 7.3.
, 100
[7]:
.
,
,
,p–
h
100
.
1)
(
)
,
,
. 2)
,
,
; 3)
. (
),
. e,
, . 7.4
i, j, m. .
i, j, m,
{δi } = ⎨
⎧ui ⎫ , ⎬ ⎩vi ⎭
⎧ δi ⎫ . {δ} = ⎪⎨ δ j ⎪⎬ ⎪δ ⎪ ⎩ m⎭ e
.
y
m
vi(Ui) i
ui(Ui)
xi yi
j x
. 7.4.
u = α1 + α2 x + α3 y ,
v = α4 + α5 x + α6 y .
αi
,
,
(7.3)
,
(7.3) ,
. ui = α1 + α2 xi + α3 yi ,
u j = α1 + α 2 x j + α3 y j ,
(7.4)
um = α1 + α2 xm + α3 ym ,
α1 ,α 2 ,α 3
{
(
ui ,u j ,um
u=
)
}
1 (ai + bi x + ci y )ui + a j + b j x + c j y u j + (am + bm x + cm y )um , (7.5 ) 2Δ ai = x j y m − x m y j ,
bi = y j − ym = y jm ,
(7.5 )
ci = xm − x j = xmj ;
i , j ,m ,
⎡1 xi 2Δ = det ⎢1 x j ⎢ ⎢⎣1 xm v
yi ⎤ y j ⎥ = 2⋅ ⎥ ym ⎥⎦
2Δ
щ
ь
ь
ijm
.
(7.5 )
v=
(7.5 )
(
(7.6)
–
,
)
}
[
u⎫ e ⎬ = [N ]{δ} = IN i′ IN ′j v ⎩ ⎭
{ f } = ⎧⎨
2×2 ,
I – N i′
{
1 (ai + bi x + ci y )vi + a j + b j x + c j y v j + (am + bm x + cm y )vm . 2Δ
N i′ =
4)
ai + bi x + ci y 2Δ
]
IN m′ {δ}e ,
(7.6)
(7.7)
. .
(7.8).
. , :
(7.7)
⎧ ∂u ⎫ ⎪ ⎪ ⎧ ε x ⎫ ⎪ ∂x ⎪ . {ε} = ⎪⎨ ε y ⎪⎬ = ⎪⎨ ∂v ⎪⎬ ⎪ε ⎪ ⎪ ∂y ⎪ ⎩ xy ⎭ ⎪ ∂u ∂v ⎪ + ⎪ ∂y ∂x ⎪ ⎩ ⎭
(7.5 )
(7.6), ⎡ ∂N i′ ⎢ ⎢ ∂x {ε} = ⎢⎢ 0 ⎢ ⎢ ∂N i′ ⎢⎣ ∂y
[B] .
[B]
{ε0 }
⎡bi 1 ⎢ 0 = 2Δ ⎢ ⎢ci ⎣
0
∂N i′ ∂y ∂N i′ ∂x
,
я
∂x
0 cj
bm 0
bi
cj
bj
cm
,
в
∂y
∂y ∂N ′j
bj 0
0
∂N m′ ∂y
а
а
.
(7.9)
0⎤ ⎥ c m ⎥{δ}e , bm ⎥⎦
.
.
⎧ ε x0 ⎫ . {ε0 } = ⎪⎨ ε y 0 ⎪⎬ ⎪γ ⎪ ⎩ xy 0 ⎭
я
∂N ′j
∂N ′j
⎤ ⎧ ui ⎫ 0 ⎥⎪ v ⎪ i ⎥⎪ ⎪ ∂N m′ ⎥ ⎪⎪ u j ⎪⎪ ⎨ ⎬= ∂y ⎥ ⎪ v j ⎪ ⎥ ∂N m′ ⎥ ⎪u m ⎪ ∂x ⎥⎦ ⎪⎪v ⎪⎪ ⎩ m⎭
⎞ ⎛ ⎧ εx ⎫ ⎧ σx ⎫ ⎟, ⎜⎪ ⎪ ⎪ ⎪ {σ} = ⎨ σ y ⎬ = [D ]⎜ ⎨ ε y ⎬ − {ε 0 }⎟ ⎟⎟ ⎜⎜ ⎪ ⎪ ⎪σ ⎪ ε ⎩ xy ⎭ ⎠ ⎝ ⎩ xy ⎭ {σ0 }). (7.11)
,
а
0
∂N m′ ∂x
,
( –
∂x
0
0 ci
, ,
[D] ,
∂N ′j
(7.10)
σx ν ⋅ σ y + ε x0 , − E E ν ⋅ σx σ y εy = − + + ε y0 , E E 2 ⋅ ( 1 + ν ) ⋅ τ xy γ xy = + γ xy 0 . E
εx =
,
[D]
⎡ ⎤ 0 ⎥ ⎢1 ν , ⎢ν 1 [D ] = 0 ⎥ 2 1− ν ⎢ 1− ν⎥ ⎢0 0 ⎥ 2 ⎦ ⎣ E
E–
, ф
ν – ва
. я
σz.
,
в
εz = 0 = −
{δi }
.
⎡ ⎢ 1 ⎢ [D ] = E (1 − ν ) ⎢ ν ( 1 + ν )( 1 − 2ν ) ⎢ 1 − ν ⎢ ⎢ 0 ⎣
.
,
.
ν ⋅ σx ν ⋅ σ y σz − + + α ⋅ θe . E E E
[D]
(7.10)
{F }e
а
σx ν ⋅ σ y ν ⋅ σz − + α ⋅ θe , − E E E ν ⋅ σx σ y ν ⋅ σz εy = − + − + α ⋅ θe , E E E 2 ⋅ ( 1 + ν ) ⋅ τ xy . γ xy = E
σz ,
(
а
εx =
,
5)
ν 1− ν 1 0
⎤ ⎥ ⎥. 0 ⎥ ⎥ 1 − 2ν ⎥ 2( 1 − ν ) ⎥⎦ 0
,
{Fi }
,
,
.
)
, . -
e
{δ}e
–
{F }e = [k ]e {δ}e + {F }ep + {F }oe
-
[k ]e
(7.11)
,
, , –
[k ]e = ∫ [B]T [D] [B] dV .
ijm
[k ]e = ∫ [B]T [D] [B] t dx dy ,
,
t–
,
.
,
,
Δ–
,
y,
x
[k ]e = [B]T [D] [B] t Δ ,
[B] ,
.
,
,
[
(7.5 )]. (7.9),
[B ] = [Bi , B j ,B m ], ⎡ kii
[k ]e = ⎢⎢ k ji
⎢kmi ⎣
2×2
⎧bi [Bi ] = ⎪⎨ 0 ⎪c ⎩ i kij k jj k mj
0⎫ ⎪ ci ⎬ 2 Δ bi ⎪⎭
kim ⎤ ⎥, k jm ⎥ kmm ⎥⎦
. .
(7.12)
(7.13)
[k rs ] = [Br ] [D ] [Bs ] t Δ . :
T
6)
[K ]
7)
[Kij ] = ∑[kij ]
.
(7.14)
{F } = [K ]{δ} {δ} = [K ]−1{F }
.
,
,
x, y
.
{p} = ⎧⎨
X⎫ ⎬ ⎩Y ⎭
.
{F }ep = − ∫ [N ]T ⎧⎨
X⎫ , ⎬ dx dy ⎩Y ⎭
(7.7)
,
X⎫ ⎬∫ [N ]i dx dy ⎩Y ⎭
{Fi }p = − ⎧⎨ X
Y
.
,
Ni
.
∫ x dx dy = ∫ y dx dy = 0 , ,
,
(7.8),
{Fi }p = − ⎧⎨
X⎫ ⎧X ⎫ , ⎬ ∫ ai dx dy 2Δ = − ⎨ ⎬ ai 2 ⎩Y ⎭ ⎩Y ⎭
.
{ }p = {Fm }p .
{Fi }p = − ⎧⎨
X⎫ ⎬ Δ 3 = Fj Y ⎩ ⎭
{F }
e p
,
,
x
⎧X ⎫ ⎪Y ⎪ ⎪ ⎪ . ⎪X ⎪ = −⎨ ⎬ Δ 3 ⎪Y ⎪ ⎪X ⎪ ⎪ ⎪ ⎩Y ⎭
(7.15)
.
y,
8) ,
. Mathcad .
. :
μ := 0.3
E := 2 ⋅10
11
,
q_x := 10
8
X,
q_y := 0
Y, ,
a := 1
L := a t := 1
:
ORIGIN ≡ 1
. (7.11)
⎛ 1 μ 0 ⎜ E ⎜ μ 1 0 ⋅ D_N := 2 ⎜ 1−μ 1−μ ⎜ 0 0 2 ⎝
X
σ_x := q_x
ΔLmax :=
σ_x⋅L E
−4
ΔLmax = 5 × 10
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
1 - 4 (3
X (1 )
)
Y (2
)
0 a a 0 ⎞ ⎜⎛ ⎟ Nodes := ⎜ a a 0 0 ⎟ ⎜ 1 2 3 4⎟ ⎝ ⎠
( -
)
-
⎛ 1 3 2⎞ ⎟ ⎝ 1 4 3⎠
Top := ⎜
⎛ n_element ⎞ ⎛ rows( Top) ⎞ ⎜ ⎟ := ⎜ ⎟ ⎝ nn_point ⎠ ⎝ cols( Top) ⎠ ie := 1 .. n_element j := 1 .. nn_point
M_ie , 2⋅ j−1 := 2 ⋅Topie , j − 1
M_ie , 2⋅ j := 2 ⋅Topie , j
,
X, Y
⎛ Xnie , j ⎞ ⎡ Nodes1, (Top ie , j) ⎤ ⎜ ⎟ := ⎢ ⎥ ⎝ Ynie , j ⎠ ⎣ Nodes2, (Top ie , j) ⎦ j =
⎛⎜ 2 3 ⎞⎟ l := ⎜ 3 1 ⎟ ⎜ 1 2⎟ ⎝ ⎠
1 2 3
y1_2ie := ymedie :=
Ynie , 1 + Ynie , 2 2
Ynie , 3 + 2 ⋅y1_2ie 1+ 2
xmed1 = 0.667
ymed1 = 0.667
x1_2ie := xmedie :=
ymed2 = 0.333
bnie , j := Ynie , ( lj , 1) − Ynie , ( lj , 2) anie , j := Xnie , ( lj , 1) ⋅Ynie , ( lj , 2) − Xnie , ( lj , 2) ⋅Ynie , ( lj , 1)
2
Xnie , 3 + 2 ⋅x1_2ie
xmed2 = 0.333
(7.5 )
cnie , j := Xnie , ( lj , 2) − Xnie , ( lj , 1)
Xnie , 1 + Xnie , 2
. (7.5 )
1+ 2
1
0
0.5
0
0.5
1
1.5
. (7.5 )
⎛ 1 Xnie, 1 Ynie , 1 ⎜ Δ2nie := ⎜ 1 Xnie , 2 Ynie , 2 ⎜ 1 Xn ie , 3 Ynie , 3 ⎝
⎞ ⎟ ⎟ ⎟ ⎠ -
. (7.12)
⎛ bnie , j 0 ⎜ Bnie , j := ⋅⎜ 0 cnie , j Δ2nie ⎜ ⎝ cnie , j bnie , j 1
(
(
⎞ ⎟ ⎟ ⎟ ⎠
BBn( ie) := stack Bnie , 1 , Bnie , 2 , Bnie , 3 T
T
K_ie := BBn( ie) ⋅D_N ⋅BBn( ie) ⋅t⋅ T
ii := 1 .. 2 ⋅nn_point
T
))T
Δ2nie
-
. (7.13)
2
jj := 1 .. 2 ⋅nn_point
(7.14)
K( M_ ie , ii , M_ ie , jj) := K( M_ ie , ii , M_ ie , jj) + ( K_ie) ii, jj
Kii, jj := 0
A := K NN
⎛ 1 ⎞ ⎟ ⎝ 4 ⎠
NN := ⎜
n := 1 .. 2
k := 1 .. cols( Nodes ) ⋅2
Ak , 2⋅ NN n−1 := 0
A2⋅ NN n−1 , 2⋅ NN n−1 := ∞ Ak , 2⋅ NN n := 0
A NN n⋅ 2 , NN n⋅ 2 := ∞
⎛ ⎜ Uzl := ⎜ ⎜ ⎜ ⎝
2 ⎞
⎟
1 ⎟ 2 ⎟
⎟
1 ⎠
p := 1 .. cols( Nodes ) P2⋅ p −1 := P2⋅ p :=
-
. (7.15)
Δ2nie 2 − q_x ⋅ ⋅ ⋅Uzl( Top ie , j) cols( Top) 2 L
Δ2nie 2 − q_y ⋅ ⋅ ⋅Uzl( Top ie , j) cols( Top) 2 L
U := − A
−1
⋅P
U_xp := Up ⋅ 2−1
U_yp := Up ⋅ 2 uie_xie :=
⎡
⋅⎢
Δ2nie ⎢ 1
∑
-
3
⎣ j =1 ⎡ 3 1 ⎢ uie_yie := ⋅ Δ2nie ⎢ ⎣ j =1
∑
. (7.6)
⎤
( anie , j + bnie , j⋅xmedie + cnie , j⋅ymedie) ⋅U_x(Top ie , j) ⎥⎥
⎦ ⎤ ( anie , j + bnie , j⋅xmedie + cnie , j⋅ymedie) ⋅U_y(Top ie , j) ⎥⎥ ⎦ . mno := 1000
⎛ xdie , j ⎞ ⎢⎡ Nodes 1 , ( Top ie , j) + (mno⋅U_xTop ie , j) ⎜ ⎟ := ⎝ ydie , j ⎠ ⎢⎣ Nodes 2 , ( Top ie , j) + ⎡⎣( mno⋅U_y) Top ie , j⎤⎦
⎤ ⎥ ⎥ ⎦
1
0
1
0.5
0
ΔLmax = 5 × 10
0.5
−4
uie_x1 − uie_x2 = 2.266 × 10
1
−4
1.5
2
s := 1 .. max( Top)
⎡(
x_s := ⎣ Nodes
⎡(
xs := ⎣ Nodes
T
T
)
)
X δ_u2 := δ_u3 :=
⎡(
〈1〉 ⎤
⎦ s + U_xs
ys := ⎣ Nodes
⎡(
〈1〉 ⎤
⎦s
2
y_s := ⎣ Nodes
3
( x_2 − x2) − ΔLmax ΔLmax ( x_3 − x3) − ΔLmax ΔLmax
δ_u2 = 35.943 %
δ_u3 = 17.941 %
1
0
0
0.5
T
1
1.5
2
) T
)
〈2〉 ⎤
⎦s
〈2〉 ⎤
⎦ s + U_ys
σ
UUii, ie := U( M_ ie , ii )
εε
σσ
〈ie〉
ε
:= BBn( ie) ⋅UU
〈ie〉
:= D_N ⋅εε
〈ie〉
〈ie〉
1
σσ
εε
〈1〉
〈1〉
2
⎛ 1.371 × 108 ⎜ = ⎜ 3.728 × 106 ⎜ 6 ⎝ −3.728 × 10
⎞ ⎟ 〈2〉 ⎟ σσ ⎟ ⎠
⎛ 1.296 × 108 ⎜ = ⎜ 3.888 × 107 ⎜ 6 ⎝ 3.728 × 10
⎛ 6.797 × 10− 4 ⎜ = ⎜ −1.87 × 10− 4 ⎜ −5 ⎝ −4.847 × 10
⎞ ⎟ 〈2〉 ⎟ εε ⎟ ⎠
⎛ 5.897 × 10− 4 ⎜ 0 =⎜ ⎜ −5 ⎝ 4.847 × 10
σ_x = 1 × 10 ΔLmax L
1. . 2. , 1985. – 240 3.
. . ., . . .,
−4
. .,
.: .
. . . .
σy
σxy
⎞ ⎟ ⎟ ⎟ ⎠
εx
εy
εxy
εx
.– . .,
σx
σx
8
= 5 × 10
⎞ ⎟ ⎟ ⎟ ⎠
.
, 1954. – 288 . . . I: . . II:
.– .–
.:
.:
, 1985. –
496 . 4. ё . . . . – .: , 1994. – 268 . 5. . ., . . . – .: , 2001. – 267 . 6. . . MathCAD PLUS 6.0 PRO. – .: , – 1997 7. . . . . – .: , 1975. – 541 . 8. . MathCAD 2001. – .: , 2001. – 544 . 9. . ., . . . – .: . , – 1991 10. . . Excel: . – .: , 2002. – 336 . 11. . ., . . . – .: . . , 1996. – 368 . 12. . . . . 1. – .: , 1985. – 432 13. . . . . 2. – .: , 1985. – 560 14. . . .– : . ., 1988. – 367 . 15. . . Microsoft Excel .– .: , 2005. – 368 . 16. . . . – .: . ., 1982. – 264 . 17. . . . – .: . ., 1988. – 367 . 18. . . : . . – .: , 1987. – 320 . 19. . . . . – .: - , 1977. – 120 .
. . .
.