Zadanie 1
Wyznaczyć główne, centralne osie i momenty bezwładności
33
4
6
R 2
y
A
x
A
10 6
⋅
1
2
6
⋅ 3
⋅
−
π 2
2
⋅
2
+
:=
A
57.283
=
Sx
1
2
− 6
⋅ 3
⋅
2
−
(
)
π 2
2
⋅
2
3
4
3
2
π
⋅
+
⎛⎜
⎝
⎞⎟
⎠
⋅
+
:=
Sx 42.183
=
Sy
10 6
⋅ 1
⋅
1
2
6
⋅ 3
⋅ 2
⋅
−
π 2
2
⋅
2
2
−
(
)
⋅
+
:=
Sy 29.434
=
xc
Sy
A
:=
xc 0.514
=
yc
Sx
A
:=
yc 0.736
=
α
1
α
2
+
90.000
=
α
2
5.246
−
=
I2c 200.556
=
α
2
atan
Ixcyc
Iyc I2c
−
⎛
⎜
⎝
⎞
⎟
⎠
180
π
⋅
:=
I2c
Ixc Iyc
+
2
Ixc Iyc
−
2
⎛
⎜
⎝
⎞
⎟
⎠
2
Ixcyc
2
+
−
:=
α
1
84.754
=
I1c 525.004
=
α
1
atan
Ixcyc
Iyc I1c
−
⎛
⎜
⎝
⎞
⎟
⎠
180
π
⋅
:=
I1c
Ixc Iyc
+
2
Ixc Iyc
−
2
⎛
⎜
⎝
⎞
⎟
⎠
2
Ixcyc
2
+
+
:=
Ixcyc
29.54
−
=
Iyc 522.292
=
Ixc 203.269
=
Ixcyc
10 6
⋅
1
xc
−
(
)
⋅
yc
−
( )
⋅
6
2
3
2
⋅
72
−
1
2
6
⋅ 3
⋅
2
xc
−
(
)
⋅
2
−
yc
−
(
)
⋅
+
⎡
⎢
⎣
⎤
⎥
⎦
−
π 2
2
⋅
2
2
−
xc
−
(
)
⋅
3
4
3
2
π
⋅
+
yc
−
⎛⎜
⎝
⎞⎟
⎠
⋅
+
:=
Iyc
10
3
6
⋅
12
10 6
⋅
1
xc
−
(
)
2
⋅
+
6
3
3
⋅
36
1
2
6
⋅ 3
⋅
2
xc
−
(
)
2
⋅
+
⎡
⎢
⎣
⎤
⎥
⎦
−
π 2
4
⋅
8
+
π 2
2
⋅
2
2
−
xc
−
(
)
2
⋅
+
:=
Ixc
10 6
3
⋅
12
10 6
⋅
yc
−
( )
2
⋅
+
6 3
3
⋅
36
1
2
6
⋅ 3
⋅
2
−
yc
−
(
)
2
⋅
+
⎡
⎢
⎣
⎤
⎥
⎦
−
π 2
4
⋅
8
+
π 2
2
⋅
2
4
3
2
π
⋅
⎛⎜
⎝
⎞⎟
⎠
2
⋅
−
π 2
2
⋅
2
3
4
3
2
π
⋅
+
yc
−
⎛⎜
⎝
⎞⎟
⎠
2
⋅
+
:=
Sprawdzenie:
Ixcyc
29.54
−
=
Ixcyc
Ixy A xc
⋅
yc
⋅
−
:=
Iyc 522.292
=
Iyc
Iy A xc
2
⋅
−
:=
Ixc 203.269
=
Ixc
Ix A yc
2
⋅
−
:=
Ixy
7.866
−
=
Ixy
6
2
3
2
⋅
72
−
1
2
6
⋅ 3
⋅ 2
⋅
2
−
(
)
⋅
+
⎡
⎢
⎣
⎤
⎥
⎦
−
π 2
2
⋅
2
2
−
(
)
⋅
3
4
3
2
π
⋅
+
⎛⎜
⎝
⎞⎟
⎠
⋅
+
:=
Iy 537.416
=
Iy
10
3
6
⋅
12
10 6
⋅ 1
2
⋅
+
6
3
3
⋅
12
−
π 2
4
⋅
8
+
π 2
2
⋅
2
2
−
(
)
2
⋅
+
:=
Ix 234.332
=
Ix
10 6
3
⋅
12
6 3
3
⋅
36
1
2
6
⋅ 3
⋅
2
−
(
)
2
⋅
+
⎡
⎢
⎣
⎤
⎥
⎦
−
π 2
4
⋅
8
+
π 2
2
⋅
2
4
3
2
π
⋅
⎛⎜
⎝
⎞⎟
⎠
2
⋅
−
π 2
2
⋅
2
3
4
3
2
π
⋅
+
⎛⎜
⎝
⎞⎟
⎠
2
⋅
+
:=
Zadanie 1 - s2
Zadanie 1 - s3
33
4
6
R 2
yc
x
x
x
y
xc
A
x
C
1c
2c
α1
α2
Zadanie 2
Wyznaczyć główne, centralne osie i momenty bezwładności
4
6
42
3
R 2
y
A
x
x
x
x
A
10 6
⋅
1
2
6
⋅ 3
⋅
+
π 2
2
⋅
2
−
:=
A
62.717
=
Sx
10 6
⋅
1
−
(
)
⋅
1
2
6
⋅ 3
⋅ 3
⋅
+
:=
Sx
33
−
=
Sy
10 6
⋅ 1
⋅
1
2
6
⋅ 3
⋅ 2
⋅
+
π 2
2
⋅
2
4
−
4
3
2
π
⋅
+
⎛⎜
⎝
⎞⎟
⎠
⋅
−
:=
Sy 97.799
=
xc
Sy
A
:=
xc 1.559
=
yc
Sx
A
:=
yc
0.526
−
=
α
1
α
2
+
90.000
=
α
2
20.312
=
I2c 286.692
=
α
2
atan
Ixcyc
Iyc I2c
−
⎛
⎜
⎝
⎞
⎟
⎠
180
π
⋅
:=
I2c
Ixc Iyc
+
2
Ixc Iyc
−
2
⎛
⎜
⎝
⎞
⎟
⎠
2
Ixcyc
2
+
−
:=
α
1
69.688
−
=
I1c 412.507
=
α
1
atan
Ixcyc
Iyc I1c
−
⎛
⎜
⎝
⎞
⎟
⎠
180
π
⋅
:=
I1c
Ixc Iyc
+
2
Ixc Iyc
−
2
⎛
⎜
⎝
⎞
⎟
⎠
2
Ixcyc
2
+
+
:=
Ixcyc 40.96
=
Iyc 397.346
=
Ixc 301.853
=
Ixcyc
10 6
⋅
1
xc
−
(
)
⋅
1
−
yc
−
(
)
⋅
6
2
3
2
⋅
72
−
1
2
6
⋅ 3
⋅
2
xc
−
(
)
⋅
3
yc
−
(
)
⋅
+
π 2
2
⋅
2
4
−
4
3
2
π
⋅
+
xc
−
⎛⎜
⎝
⎞⎟
⎠
⋅
yc
−
( )
⋅
−
:=
Iyc
10
3
6
⋅
12
10 6
⋅
1
xc
−
(
)
2
⋅
+
6
3
3
⋅
36
+
1
2
6
⋅ 3
⋅
2
xc
−
(
)
2
⋅
+
π 2
4
⋅
8
π 2
2
⋅
2
4
3
2
π
⋅
⎛⎜
⎝
⎞⎟
⎠
2
⋅
−
π 2
2
⋅
2
4
−
4
3
2
π
⋅
+
xc
−
⎛⎜
⎝
⎞⎟
⎠
2
⋅
+
⎡
⎢
⎣
⎤
⎥
⎦
−
:=
Ixc
10 6
3
⋅
12
10 6
⋅
1
−
yc
−
(
)
2
⋅
+
6 3
3
⋅
36
+
1
2
6
⋅ 3
⋅
3
yc
−
(
)
2
⋅
+
π 2
4
⋅
8
π 2
2
⋅
2
yc
−
( )
2
⋅
+
⎡
⎢
⎣
⎤
⎥
⎦
−
:=
Sprawdzenie:
Ixcyc 40.96
=
Ixcyc
Ixy A xc
⋅
yc
⋅
−
:=
Iyc 397.346
=
Iyc
Iy A xc
2
⋅
−
:=
Ixc 301.853
=
Ixc
Ix A yc
2
⋅
−
:=
Ixy
10.5
−
=
Ixy
10 6
⋅ 1
⋅
1
−
(
)
⋅
6
2
3
2
⋅
72
−
1
2
6
⋅ 3
⋅ 2
⋅ 3
⋅
+
:=
Iy 549.853
=
Iy
10
3
6
⋅
12
10 6
⋅ 1
2
⋅
+
6
3
3
⋅
12
+
π 2
4
⋅
8
π 2
2
⋅
2
4
3
2
π
⋅
⎛⎜
⎝
⎞⎟
⎠
2
⋅
−
π 2
2
⋅
2
4
−
4
3
2
π
⋅
+
⎛⎜
⎝
⎞⎟
⎠
2
⋅
+
⎡
⎢
⎣
⎤
⎥
⎦
−
:=
Ix 319.217
=
Ix
10 6
3
⋅
12
10 6
⋅
1
−
(
)
2
⋅
+
6 3
3
⋅
36
+
1
2
6
⋅ 3
⋅ 3
2
⋅
+
π 2
4
⋅
8
−
:=
Zadanie 2 - s2
Zadanie 2 - s3
4
6
42
3
R 2
y
C
yc
A
α2
α1
1c
2c
x
x
x
x
xc
Zadanie 3
Wyznaczyć główne, centralne osie i momenty bezwładności
4
6
333
R 2
x
x
x
x
y
A
A
10 6
⋅
1
2
6
⋅ 3
⋅
+
π 2
2
⋅
2
−
:=
A
62.717
=
Sx
1
2
6
⋅ 3
⋅ 4
⋅
π 2
2
⋅
2
1
−
(
)
⋅
−
:=
Sx 42.283
=
Sy
10 6
⋅ 1
⋅
1
2
6
⋅ 3
⋅ 2
⋅
+
π 2
2
⋅
2
6
4
3
2
π
⋅
−
⎛⎜
⎝
⎞⎟
⎠
⋅
−
:=
Sy 45.634
=
xc
Sy
A
:=
xc 0.728
=
yc
Sx
A
:=
yc 0.674
=
α
1
α
2
+
90.000
=
α
2
23.948
=
I2c 256.737
=
α
2
atan
Ixcyc
Iyc I2c
−
⎛
⎜
⎝
⎞
⎟
⎠
180
π
⋅
:=
I2c
Ixc Iyc
+
2
Ixc Iyc
−
2
⎛
⎜
⎝
⎞
⎟
⎠
2
Ixcyc
2
+
−
:=
α
1
66.052
−
=
I1c 443.007
=
α
1
atan
Ixcyc
Iyc I1c
−
⎛
⎜
⎝
⎞
⎟
⎠
180
π
⋅
:=
I1c
Ixc Iyc
+
2
Ixc Iyc
−
2
⎛
⎜
⎝
⎞
⎟
⎠
2
Ixcyc
2
+
+
:=
Ixcyc 69.1
=
Iyc 412.318
=
Ixc 287.427
=
Ixcyc
10 6
⋅
1
xc
−
(
)
⋅
yc
−
( )
⋅
6
2
3
2
⋅
72
−
1
2
6
⋅ 3
⋅
2
xc
−
(
)
⋅
4
yc
−
(
)
⋅
+
π 2
2
⋅
2
6
4
3
2
π
⋅
−
xc
−
⎛⎜
⎝
⎞⎟
⎠
⋅
1
−
yc
−
(
)
⋅
−
:=
Iyc
10
3
6
⋅
12
10 6
⋅
1
xc
−
(
)
2
⋅
+
6
3
3
⋅
36
+
1
2
6
⋅ 3
⋅
2
xc
−
(
)
2
⋅
+
π 2
4
⋅
8
π 2
2
⋅
2
4
3
2
π
⋅
⎛⎜
⎝
⎞⎟
⎠
2
⋅
−
π 2
2
⋅
2
6
4
3
2
π
⋅
−
xc
−
⎛⎜
⎝
⎞⎟
⎠
2
⋅
+
⎡
⎢
⎣
⎤
⎥
⎦
−
:=
Ixc
10 6
3
⋅
12
10 6
⋅
yc
−
( )
2
⋅
+
6 3
3
⋅
36
+
1
2
6
⋅ 3
⋅
4
yc
−
(
)
2
⋅
+
π 2
4
⋅
8
π 2
2
⋅
2
1
−
yc
−
(
)
2
⋅
+
⎡
⎢
⎣
⎤
⎥
⎦
−
:=
Sprawdzenie:
Ixcyc 69.1
=
Ixcyc
Ixy A xc
⋅
yc
⋅
−
:=
Iyc 412.318
=
Iyc
Iy A xc
2
⋅
−
:=
Ixc 287.427
=
Ixc
Ix A yc
2
⋅
−
:=
Ixy 99.866
=
Ixy
6
2
3
2
⋅
72
−
1
2
6
⋅ 3
⋅ 2
⋅ 4
⋅
+
π 2
2
⋅
2
6
4
3
2
π
⋅
−
⎛⎜
⎝
⎞⎟
⎠
⋅
1
−
(
)
⋅
−
:=
Iy 445.522
=
Iy
10
3
6
⋅
12
10 6
⋅ 1
2
⋅
+
6
3
3
⋅
12
+
π 2
4
⋅
8
π 2
2
⋅
2
4
3
2
π
⋅
⎛⎜
⎝
⎞⎟
⎠
2
⋅
−
π 2
2
⋅
2
6
4
3
2
π
⋅
−
⎛⎜
⎝
⎞⎟
⎠
2
⋅
+
⎡
⎢
⎣
⎤
⎥
⎦
−
:=
Ix 315.934
=
Ix
10 6
3
⋅
12
6 3
3
⋅
36
+
1
2
6
⋅ 3
⋅ 4
2
⋅
+
π 2
4
⋅
8
π 2
2
⋅
2
1
−
(
)
2
⋅
+
⎡
⎢
⎣
⎤
⎥
⎦
−
:=
Zadanie 3 - s2
Zadanie 3 - s3
4
6
33
3
R 2
y
A
x
xc
yc
α1
α2
2c
C
1c
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