vezbe iz fundiranja pavle dimitrijevic
TRANSCRIPT
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Vezba 1
zadatak 1 a) k 4.437242 MN/m^3
zadatak 2 geometrijske karakteristike nosacaL 10 m
B 2 m
fizicke karakteristike nosaca
Eb 31500 Ma fiksno
mod!" reakcije t"a
ko #3 kN/m^3
$rednosti si"a koje de"!j! na nosac
V1 %&0 kNV2 1000 kN
V3 &00 kN
10 kN/m
ozicije si"a ! resek!
a1 1 mL1 4 mL2 4 ma2 1 m
' nosac
( 1)2 m
b 0)5 m fiksno
d 0)# m fiksno
* 0)+ m
,t- 0.4 m
.- 0.128 m^#
λ= 0.153158
1.531583 m
reda je srednje d!zine Vesic
Proracun sile i momenta u tacki A
λ= 0)15315+ λ= 0)15315+ λ= 0)15315+X= 1 X= 5 X= 9
λ*L=
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- 0)%+201 - 0)&552%2 - 0)23%5#
B- 0)130+1 B- 0)321212 B- 0)2#5+&5
4- 0)%1&5+2 4- 0)012+# 4- 0)1%%5
(- 0)+#%32 (- 0)33#0&1 (- 0)0#%+
'a- 523.0947 kN'b- -459.1346 kN
Ma- 738.9065 kNm
Mb- 500.0537 kNm
6s"o$ne jednacine za kraje$e nosaca B s"obodni kraje$i
7ocetna matrica
1)&322+ 0)33&5&1 0)5 0)00#21
0)33&5&1 1)&322+ 0)00#21 0)5
0)5 0)00#21 0)0%&5% 0)01%0%
0)00#21 0)5 0)01%0% 0)0%&5%
.n$erzna matrica matrica B
0)++&135 0)5#1%+ 5)23+1#1 2)321%1# %3+)0&5 'oa
0)5#1%+ 0)++&135 2)321%1# 5)23+1#1 500)053% 'ob
5)2%%+3 2)#1025% 1+)%5+0# 11)2&33 523)0#% Moa
2)#1025% 5)2%%+3 11)2&33 1+)%5+0# #5)13#& Mob
7rorac!n !giba 89 momenta M9 trans$erza"ne si"e ' i kontaktnog naona :
A!"A 1
6gib ! tacki 1#1= 0.031247 m #1= 3.124745 cm
;ontaktni naon za tack! 1
$1= 277.3049 k%&m'3
Momenat ! tacki 1
(1= 13%)&1#& k%m
'ransferza"ne si"e
1 leo= 2%2)3#33 k%
1 *esno= -487.6567 k%
kontro"a<
'1 "='1 d V1
760 760 1
A!"A 2
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6gib ! tacki 2
#2= 0.027558 m #2= 2.755825 cm
;ontaktni naon za tack! 2
$2= 244.5653 k%&m'3
Momenat ! tacki 2(2= 23+)#2+& k%m
'ransferza"ne si"e
2 leo= #22)+#2 k%
2 *esno= 5%%)015+ k%
kontro"a<
'2 "='2 d V2
1000 1000 1
A!"A 3
6gib ! tacki 3
#3= 0.023808 m #3= 2.38076 cm
;ontaktni naon za tack! 2
$3= 211.2802 k%&m'3
Momenat ! tacki 3
(3= 101)132# k%m
'ransferza"ne si"e
3 leo= 3&)5052 k%3*esno= 203)##+ k%
kontro"a<
'3 "='3 d V3
600 600 1
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λ= 0)15315+ λ= 0)15315+ λ= 0)15315+ λ=
X= 0 X= 10 X= 4 X=
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- 1 - 0)22302+ - 0)%52% -
B- 0 B- 0)21#&0+ B- 0)310%+ B-
4- 1 4- 0)20&1+ 4- 0)131#01 4-
(- 1 (- 0)00+#2 (- 0)##21 (-
'razene $rednosti
205%)+1+ kN
1233)1% kN
%335)5# kNm
35%)012 kNm
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0)15315+
8
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0)3%3+1#
0)2%#+&
0)1%52#
0)0+#5
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Podaci za proračun matrice uticajnih funkcija za podelu nosača na n=10 delova
L = 10.00 m c =L/n= 1.00 ### ##B = 2.00 m 0.50 ### 0.
a) k= $&$'(2 MN/m^3
λ= 0.153158
1.5316 m
.- +&12 m-$
Eb*I= ...
kt= ... * Dij +&++22 3
0 1 2 3 4 5 6 7 8
2 -4 2 0 0 0 0 0 0
-2 5 -4 1 0 0 0 0 0
1 -4 6 -4 1 0 0 0 0
0 1 -4 6 -4 1 0 0 0
0 0 1 -4 6 -4 1 0 0
0 0 0 1 -4 6 -4 1 0
0 0 0 0 1 -4 6 -4 1
0 0 0 0 0 1 -4 6 -4
0 0 0 0 0 0 1 -4 6
0 0 0 0 0 0 0 1 -4
0 0 0 0 0 0 0 0 2
0 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
&+('4+5 ... &+5$4+5 +&+++4++ +&+++4++ +&+++4++ +&+++4++ ... ... +&+++4
... 2&+1(4+( ... $&+'24+5 +&+++4++ +&+++4++ +&+++4++ ... ... +&+++4
$&+'24+5 ... 2&$2+4+( ... $&+'24+5 +&+++4++ +&+++4++ ... ... +&+++4
+&+++4++ $&+'24+5 ... 2&$2+4+( ... $&+'24+5 +&+++4++ ... ... +&+++4
+&+++4++ +&+++4++ $&+'24+5 ... 2&$2+4+( ... $&+'24+5 ... ... +&+++4
+&+++4++ +&+++4++ +&+++4++ $&+'24+5 ... 2&$2+4+( ... ... ... +&+++4
+&+++4++ +&+++4++ +&+++4++ +&+++4++ $&+'24+5 ... 2&$2+4+( ... ... +&+++4
+&+++4++ +&+++4++ +&+++4++ +&+++4++ +&+++4++ $&+'24+5 ... ... ... $&+'24
+&+++4++ +&+++4++ +&+++4++ +&+++4++ +&+++4++ +&+++4++ $&+'24+5 ... ... .
+&+++4++ +&+++4++ +&+++4++ +&+++4++ +&+++4++ +&+++4++ +&+++4++ ... ... 2&+1(4
+&+++4++ +&+++4++ +&+++4++ +&+++4++ +&+++4++ +&+++4++ +&+++4++ ... ... .
0 1 2 3 4 5 6 7 8
2&'''46+% '&146+% '&11%46+% 2&'+46+% 1&5+46+% 1&+1$46+% '&(5746+5 ... ... 61&$24
1&7$146+% '&'+%46+% 2&(2%46+% 2&1%$46+% 1&5++46+% 1&+5$46+% %&$%$46+5 $&1%46+( ... 67&$124
1&%%46+% 2&(2%46+% 2&''246+% 1&72546+% 1&%146+% 1&11'46+% (&1'%46+5 '&2+(46+5 ... 6$&%2$4
1&17+46+% 2&1%$46+% 1&72546+% 1&546+% 1&$2746+% 1&1%(46+% &(7546+5 5&++246+5 '&2+(46+5 $&1%4
&'746+5 1&5++46+% 1&%146+% 1&$2746+% 1&'2$46+% 1&17146+% 1&+$+46+% &(7546+5 (&1'%46+5 %&$%$4
E b=
kN/m
2 Es=
c/# = I= m4 vs=
λ*L=
kNm2
D! =
K t! =
K t!"1 =
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6$&1'46+5 6$&%2$46+5 65&(2746+( '&2+(46+5 (&1'%46+5 1&11'46+% 1&%146+% 1&72546+% 2&''246+%
6(&1$246+5 67&$1246+5 6$&%2$46+5 $&1%46+( %&$%$46+5 1&+5$46+% 1&5++46+% 2&1%$46+% 2&(2%46+%
61&++746+% 61&$246+% 6&'5546+5 62&'(246+5 '&(5746+5 1&+1$46+% 1&5+46+% 2&'+46+% '&11%46+%
)Eib nosaca ?ea
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Ugib nosaca
+ 1 2 ' $ % 5 ( , 7
+&++4++
%&++46+'
1&++46+2
1&%+46+2
2&++46+2
2&%+46+2
'&++46+2
'&%+46+2
'&1(46+2 '&+746+2
'&+146+2 2&7'46+2 2&,546+2 2&,+46+2 2&('46+2
2&5546+2 2&5+46+2 2&%$46+2
odstojanje x(m)
u g i b n o s a c a w
( m )
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Reaktivno opterecenje
+ 1 2 ' $ % 5 ( , 7
+&++4++
%&++4+1
1&++4+2
1&%+4+2
2&++4+2
2&%+4+2
'&++4+2
2&,24+2 2&(%4+2
2&5(4+2 2&5+4+2
2&%$4+2 2&$,4+2
2&$24+2 2&'54+2
2&'14+2 2&254+
odstojanje x(m)
r e a k t i v n o
o p t e r e c e n j e
q (
n ! m
)
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Momenti savijanja
+ 1 2 ' $ % 5 ( , 7 1
6$++&++
6'++&++
62++&++
61++&++
+&++
1++&++
2++&++
'++&++
+&++
1'%&27
621%&,(
6'1+&(7
61%5&'2
2$1&',
61''&%1
62((&'2
6175&12
1+$&55
odstojanje x(m)
m o m e n t i s a v i j a n j a n o
s a c a
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Transverzalne sile
+ 2 $ 5 , 1+
6,++
65++
6$++
62++
+
2++
$++
5++
+
2$2&%
6%+(&%
62(2
6$'&%
1(,&%
'7$&%
651%&%
6$+$&%
6177
2
177
6$21
+
odstojanje x(m)
t r a n s v e r " a # n e s i # e
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atrice
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+ 7 ( 5 % $ 3 2 1 0
1
2
'
$
%
5
(
7
1+
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$ransver"a#ne si#e
+ +1 2$2&%
1 6%+(&%
2 62(2
' 6$'&%
$ 1(&%
% '7$&%
% 651%&%
5 6$+$&%
( 6177
2&++
7 177&++
7 6$21&++
1+ +
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Podaci za proračun matrice uticajnih funkcija za podelu nosača na n=10 delova
L = 10.00 m c =L/n= 1.00 ### ###
B = 2.00 m 0.50 ### 0.45
a) k= $&$'(2 MN/m^3
λ= 0.153158
1.5316 m
.- +&12 m-$
Eb*I= ...
kt= ... * Dij +&++5
0 1 2 3 4 5 6 7 8 9
2 -4 2 0 0 0 0 0 0 0
-2 5 -4 1 0 0 0 0 0 0
1 -4 6 -4 1 0 0 0 0 0
0 1 -4 6 -4 1 0 0 0 0
0 0 1 -4 6 -4 1 0 0 0
0 0 0 1 -4 6 -4 1 0 0
0 0 0 0 1 -4 6 -4 1 0
0 0 0 0 0 1 -4 6 -4 1
0 0 0 0 0 0 1 -4 6 -4
0 0 0 0 0 0 0 1 -4 5
0 0 0 0 0 0 0 0 2 -4
0 1 2 3 4 5 6 7 8 9
1.20303 0.92945 0.48984 0.33027 0.24870 0.19933 0.16628 0.14261 0.12484 0.11100
0.55972 2.40606 0.92945 0.48984 0.33027 0.24870 0.19933 0.16628 0.14261 0.12484
0.27358 0.92945 2.40606 0.92945 0.48984 0.33027 0.24870 0.19933 0.16628 0.14261
0.17848 0.48984 0.92945 2.40606 0.92945 0.48984 0.33027 0.24870 0.19933 0.16628
0.13199 0.33027 0.48984 0.92945 2.40606 0.92945 0.48984 0.33027 0.24870 0.19933
0.10459 0.24870 0.33027 0.48984 0.92945 2.40606 0.92945 0.48984 0.33027 0.24870
0.08658 0.19933 0.24870 0.33027 0.48984 0.92945 2.40606 0.92945 0.48984 0.33027
0.07384 0.16628 0.19933 0.24870 0.33027 0.48984 0.92945 2.40606 0.92945 0.48984
0.06436 0.14261 0.16628 0.19933 0.24870 0.33027 0.48984 0.92945 2.40606 0.92945
0.05703 0.12484 0.14261 0.16628 0.19933 0.24870 0.33027 0.48984 0.92945 2.40606
0.05120 0.11100 0.12484 0.14261 0.16628 0.19933 0.24870 0.33027 0.48984 0.92945
0 1 2 3 4 5 6 7 8 9
1&+224++ 6'&52746+1 6$&$(%46+2 6'&2(%46+2 62&1$(46+2 61&%7746+2 61&25746+2 ... ... 6&5746+'
62&2'546+1 %&(2246+1 61&5%'46+1 61&$$(46+2 61&11%46+2 65&72146+' 6%&+2146+' ... ... 62&7$146+'
61&71+46+2 61&5'46+1 %&%%46+1 61&55746+1 61&%5(46+2 61&17'46+2 6(&%+746+' ... ... 6'&'+46+'
61&%$46+2 61&51$46+2 61&5546+1 %&%5'46+1 61&55%46+1 61&%$'46+2 61&1((46+2 ... ... 6$&%%146+'
67&''46+' 61&2'546+2 61&%5$46+2 61&55%46+1 %&%5%46+1 61&55$46+1 61&%'746+2 ... ... 6%&((146+'
6(&2$'46+' 6(&$546+' 61&17146+2 61&%$$46+2 61&55$46+1 %&%5546+1 61&55$46+1 ... ... 6(&$546+'
6%&5746+' 6%&((146+' 6(&$7$46+' 61&1((46+2 61&%'746+2 61&55$46+1 %&%5%46+1 ... ... 61&2'546+2
6$&(1(46+' 6$&%%146+' 6%&$746+' 6(&$1(46+' 61&1((46+2 61&%$'46+2 61&55%46+1 %&%5'46+1 ... 61&51$46+2
6$&11$46+' 6'&'+46+' 6$&'546+' 6%&%1546+' 6(&%+746+' 61&17'46+2 61&%5(46+2 ... %&%%46+1 61&5'46+16'&2''46+' 62&7$146+' 6'&2%746+' 6'&72146+' 6%&+2146+' 65&72146+' 61&11%46+2 ... ... %&(2246+1
E b= kN/m2 Es=
c/#
= I= m4 vs=
λ*L=
kNm2
% f !"1 &
D! =
f ! =
f !"1 =
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67&'$46+' 6&5746+' 67&2$46+' 61&+%746+2 61&25746+2 61&%7746+2 62&1$(46+2 ...
0 1 2 3 4 5 6 7
&+724+5 ... &+5'4+5 ... ... ... ... ...
... 2&+14+( ... $&+'24+5 ... ... ... ...
$&+'14+5 ... 2&$214+( ... $&+'24+5 ... ... ...
... $&+'24+5 ... 2&$214+( ... $&+'24+5 ... ...
... ... $&+'24+5 ... 2&$214+( ... $&+'24+5 ...
... ... ... $&+'24+5 ... 2&$214+( ... ...
... ... ... ... $&+'24+5 ... 2&$214+( ...
... ... ... ... ... $&+'24+5 ... ...
... ... ... ... ... ... $&+'24+5 ...
... ... ... ... ... ... ... ...
... ... ... ... ... ... ... ...
0 1 2 3 4 5 6 7
2&55246+% $&5+(46+% '&7+$46+% '&22$46+% 2&%5746+% 1&7$146+% 1&''(46+% (&%1(46+5 1
2&'+%46+% $&+7+46+% '&%5146+% '&+'%46+% 2&%1546+% 2&++746+% 1&%1%46+% 1&+'146+% %
1&7%%46+% '&%5$46+% '&21146+% 2&'746+% 2&$%46+% 2&+($46+% 1&57146+% 1&'1+46+% 7
1&51%46+% '&+'46+% 2&$+46+% 2&5246+% 2&'746+% 2&1'146+% 1&5'46+% 1&%46+% 1
1&246+% 2&%2+46+% 2&$5+46+% 2&'746+% 2&2746+% 2&1('46+% 2&+2%46+% 1&5'46+% 1
7&('546+5 2&+1'46+% 2&+(546+% 2&1'246+% 2&1('46+% 2&17146+% 2&1('46+% 2&1'246+% 2
5&(+746+5 1&%146+% 1&57'46+% 1&5'46+% 2&+2%46+% 2&1('46+% 2&2746+% 2&'746+% 2
'&(((46+5 1&+''46+% 1&'1146+% 1&%46+% 1&5'46+% 2&1'146+% 2&'746+% 2&5246+% 2
7&1%%46+( %&%(246+5 7&'2546+5 1&'1+46+% 1&57146+% 2&+($46+% 2&$%46+% 2&'746+% '
61&7+'46+5 &5$746+( %&%5146+5 1&+'146+% 1&%1%46+% 2&++746+% 2&%1546+% '&+'%46+% '
6$&(+'46+5 6'&146+5 1&+(46+5 (&%1(46+5 1&''(46+% 1&7$146+% 2&%5746+% '&22$46+% '
)Eib nosaca ?ea
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Ugib nosaca
Reaktivno opterecenje
+ 1 2 ' $ % 5 ( , 7
$&$+46+2
$&5+46+2
$&,+46+2
%&++46+2
%&2+46+2
%&$+46+2
%&5+46+2
%&$+46+2 %&',46+2
%&'$46+2%&'+46+2
%&2546+2%&2+46+2
%&1'46+2
%&+$46+2
$&7%46+2
$&,54
odstojanje x(m)
u g i b n o s a c a w
( m )
+ 1 2 ' $ % 5 ( , 7
+&++4++
1&++4+2
2&++4+2
'&++4+2
$&++4+2
%&++4+2
5&++4+2
(&++4+2
,&++4+2
(&$(4+2
1&774+22&224+2 2&1+4+2
2&+%4+2 2&+24+2 1&7,4+2 1&754+2 2&++4+2 1&(%4
odstojanje x(m)
r e a k t i v n o
o p
t e r e c e n j e
q
( n ! m
)
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Momenti savijanja
Transverzalne sile
+ 1 2 ' $ % 5 ( , 7
+&++
1++&++
2++&++
'++&++
$++&++
%++&++
5++&++
(++&++
,++&++
7++&++
+&++
'5(&,'
1($&++171&17
$+(&(,
,1,&'%
$+7&$,
1,(&(21%1&',
'+'&,(
odstojanje x(m)
m o m e n t i s a v i j a n j a n o
s a c a
+ 2 $ 5 , 1+
6,++
65++
6$++
62++
+
2++
$++
5++
+
$11&%
6'',&%
61%(&%
27
2+7
',5
652$
6$%+
62(7&%
61+7&%
%+
6%(+
+
odstojanje x(m)
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odr!"je #$1 do L%1
i elementi inerzne matrice
ica 00
edeći na"in:
#2:L'3
C8B 4B4?
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1+7
(
5
%
$
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2
1
+
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+ +
1 $11&%
1 6''&%
2 61%(&%
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( 62(7&%
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1+ +
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Podaci za proračun matrice uticajnih funkcija za podelu nosača na n=10 delova Elasticni poluprostor
L = 12.00 m c =L/n= 1.20 2.1E+07 7.0E+03
B = 2.00 m 0.60 0.12800 0.45
a) k= + MN/m^3
λ= 0
% m
.- +&12++ m-$
Eb*I= 2&(4+5
kt= ... * Dij +&+212(2
0 1 2 3 4 5 6 7 8 9
2 -4 2 0 0 0 0 0 0 0
-2 5 -4 1 0 0 0 0 0 0
1 -4 6 -4 1 0 0 0 0 0
0 1 -4 6 -4 1 0 0 0 0
0 0 1 -4 6 -4 1 0 0 0
0 0 0 1 -4 6 -4 1 0 0
0 0 0 0 1 -4 6 -4 1 0
0 0 0 0 0 1 -4 6 -4 1
0 0 0 0 0 0 1 -4 6 -4
0 0 0 0 0 0 0 1 -4 5
0 0 0 0 0 0 0 0 2 -4
0 1 2 3 4 5 6 7 8 9
1.33910 0.96596 0.49574 0.33210 0.24949 0.19974 0.16652 0.14276 0.12494 0.11107
0.58705 2.67820 0.96596 0.49574 0.33210 0.24949 0.19974 0.16652 0.14276 0.12494
0.27750 0.96596 2.67820 0.96596 0.49574 0.33210 0.24949 0.19974 0.16652 0.14276
0.17961 0.49574 0.96596 2.67820 0.96596 0.49574 0.33210 0.24949 0.19974 0.16652
0.13245 0.33210 0.49574 0.96596 2.67820 0.96596 0.49574 0.33210 0.24949 0.19974 0.10482 0.24949 0.33210 0.49574 0.96596 2.67820 0.96596 0.49574 0.33210 0.24949
0.08671 0.19974 0.24949 0.33210 0.49574 0.96596 2.67820 0.96596 0.49574 0.33210
0.07392 0.16652 0.19974 0.24949 0.33210 0.49574 0.96596 2.67820 0.96596 0.49574
0.06441 0.14276 0.16652 0.19974 0.24949 0.33210 0.49574 0.96596 2.67820 0.96596
0.05707 0.12494 0.14276 0.16652 0.19974 0.24949 0.33210 0.49574 0.96596 2.67820
0.05123 0.11107 0.12494 0.14276 0.16652 0.19974 0.24949 0.33210 0.49574 0.96596
E b= kN/m2 Es= kN/m
c/# = I= m4 vs=
λ*L=
kNm2
% f !"1 &
D! =
f ! =
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0 1 2 3 4 5 6
&7$146+1 62&75546+1 6$&++$46+2 62&$+46+2 61&(%46+2 61&$++46+2 61&11246+2 67&246+
61&$$46+1 $&7$+46+1 61&'%$46+1 61&'%(46+2 67&7'246+' 65&2%+46+' 6$&%%$46+' 6'&%5746+
61&(%246+2 61&'(46+1 $&1246+1 61&'5(46+1 61&$%$46+2 61&+%546+2 65&(246+' 6$&7%46+
61&'%246+2 61&$7$46+2 61&'5546+1 $&1(46+1 61&'5$46+1 61&$'$46+2 61&+$246+2 65&5%+46+
6&55146+' 61&+7246+2 61&$%146+2 61&'5$46+1 $&1746+1 61&'5'46+1 61&$'+46+2 61&+$'46+
65&'746+' 6(&+1$46+' 61&+%$46+2 61&$'$46+2 61&'5'46+1 $&1746+1 61&'5'46+1 61&$'$46+
6%&+''46+' 6%&1(%46+' 65&(146+' 61&+$'46+2 61&$'+46+2 61&'5'46+1 $&1746+1 61&'5$46+
6$&1(+46+' 6$&+7146+' 6$&7$%46+' 65&5%+46+' 61&+$246+2 61&$'$46+2 61&'5$46+1 $&1(46
6'&5''46+' 6'&$$'46+' 6'&7%146+' 6$&7%46+' 65&(246+' 61&+%546+2 61&$%$46+2 61&'5(46+
62&746+' 62&5((46+' 62&7(246+' 6'&%5746+' 6$&%%$46+' 65&2%+46+' 67&7'246+' 61&'%(46+
6&$1146+' 6(&%$46+' 6&1+46+' 67&246+' 61&11246+2 61&$++46+2 61&(%46+2 62&$+46+
0 1 2 3 4 5 6
'&1$14+5 65&2'24+5 '&11+4+5 67&'774+2 65&2+%4+2 6$&5'14+2 6'&514+2 6'&+($4+
6'&11(4+5 (&(7$4+5 65&22(4+5 1&%%%4+5 6'&254+2 62&+54+2 61&%+(4+2 61&114+
1&%%%4+5 65&22(4+5 7&'$74+5 65&22(4+5 1&%%%4+5 6'&$7$4+2 62&2254+2 61&5$+4
6$&$('4+2 1&%%%4+5 65&22(4+5 7&'$74+5 65&22(4+5 1&%%%4+5 6'&$$74+2 62&2+14+
62&554+2 6'&51%4+2 1&%%%4+5 65&22(4+5 7&'$74+5 65&22(4+5 1&%%%4+5 6'&$%14+
62&11$4+2 62&'214+2 6'&$74+2 1&%%%4+5 65&22(4+5 7&'$74+5 65&22(4+5 1&%%%4+
61&55%4+2 61&(124+2 62&22'4+2 6'&$%14+2 1&%%%4+5 65&22(4+5 7&'$74+5 65&22(4+
61&'+4+2 61&'%$4+2 61&5'54+2 62&2+14+2 6'&$$74+2 1&%%%4+5 65&22(4+5 7&'$74+
61&2+24+2 61&1'74+2 61&'+(4+2 61&5$+4+2 62&2254+2 6'&$7$4+2 1&%%%4+5 65&22(4+
67&%%4+1 6&%(4+1 67&'54+1 61&114+2 61&%+(4+2 62&+54+2 6'&254+2 1&%%%4+
62&('4+2 62&%+74+2 62&5'4+2 6'&+($4+2 6'&514+2 6$&5'14+2 65&2+%4+2 67&'774+
0 1 2 3 4 5 6
2&$7$46+% $&1746+% '&$$'46+% 2&($%46+% 2&1+746+% 1&%'%46+% 1&+1(46+% %&$2246+
2&1+246+% '&(+$46+% '&1+46+% 2&55+46+% 2&15146+% 1&57+46+% 1&2$746+% &''246+
1&(246+% '&1%46+% 2&7546+% 2&%5146+% 2&2+'46+% 1&'746+% 1&$(46+% 1&12$46+
1&'(746+% 2&5546+% 2&%5$46+% 2&$2%46+% 2&21746+% 1&7(+46+% 1&5746+% 1&$1%46+
1&+5146+% 2&1(+46+% 2&2+46+% 2&22+46+% 2&1$46+% 2&+546+% 1&7++46+% 1&(++46+
(&($+46+5 1&5746+% 1&$'46+% 1&7(246+% 2&+5746+% 2&11146+% 2&+5746+% 1&7(246+
%&1'546+5 1&2%546+% 1&$246+% 1&(++46+% 1&7++46+% 2&+546+% 2&1$46+% 2&22+46+
2&(%+46+5 &'146+5 1&12(46+% 1&$1%46+% 1&5746+% 1&7(+46+% 2&21746+% 2&$2%46+
%&21(46+( $&$+'46+5 (&(7246+5 1&12$46+% 1&$(46+% 1&'746+% 2&2+'46+% 2&%5146+
61&51146+5 %&%+%46+( $&'(46+5 &''246+5 1&2$746+% 1&57+46+% 2&15146+% 2&55+46+
6'&(+'46+5 6'&2$%46+5 7&7'%46+( %&$2246+5 1&+1(46+% 1&%'%46+% 2&1+746+% 2&($%46+
f !"1 =
K t! =
K t!"1 =
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@leEanje
0 0
+&++4++ 0 $&2+46+% 0
1&++4++ 1 '&(+46+% 1
+&++4++ 2 '&146+% 2
+&++4++ 3 2&5(46+% 3
+&++4++ 4 2&1(46+% 4
+&++4++ 5 1&(+46+% 5
+&++4++ 6 1&2546+% 6
+&++4++ 7 &'46+5 7
+&++4++ 8 $&$+46+5 8
+&++4++ 9 %&%+46+( 9
+&++4++ 10 6'&2$46+5 10
@leEanje
0 0
+&++4++ 0 1&+246+% 0
+&++4++ 1 1&2%46+% 1
+&++4++ 2 1&$46+% 2
+&++4++ 3 1&(+46+% 3
+&++4++ 4 1&7+46+% 4
+&++4++ 5 2&+(46+% 5
1&++4++ 6 2&146+% 6
+&++4++ 7 2&2246+% 7
+&++4++ 8 2&2+46+% 8
+&++4++ 9 2&1546+% 9
+&++4++ 10 2&1146+% 10
@leEanje
0 0
+&++4++ 0 6'&2$46+5 0
+&++4++ 1 %&%+46+( 1
+&++4++ 2 $&$+46+5 2
+&++4++ 3 &'46+5 3
+&++4++ 4 1&2546+% 4
+&++4++ 5 1&(+46+% 5
+&++4++ 6 2&1(46+% 6
+&++4++ 7 2&5(46+% 7
+&++4++ 8 '&146+% 8
1 ++4++ 9 ' (+4 +% 9
R 1=1
#P$ = #w$ =
R 2=1
#P$ = #w$ =
R =1
#P$ = #w$ =
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'()0E"0* 1(+*E"0* *(*0E"0)
1(+,E"0* +(1-E"0* +(1)E"0*
*(*0E"0) +(1,E"0* '()0E"0*
.(-0E/0. ",(')E/0. '(,,E/0.
",(.0E/0. 1(.E/0* 61&1'4+%
'(,,E/0. "1(1+E/0* 7&214+$
"+(-'E/0' )(*)E/0' 6$&(24+' 1 + +
)(*)E/0' "+(0+E/0. 1&254+$ !"# = + 1 +
".()+E/0' 1(+,E/0. 6(&'4+' + + 1
!=k! /t!
.(*+E/0. "*(,1E/0. '&174+$ 75+
!K# = "*(,*E/0. 1().E/0* 61&++4+% 172+
'(1E/0. "(*E/0. &$'4+$ ('+
'()0.+E"00* 1(+'''E"00* 5&+(46++( *()E"0+
0(00001+.1 +(+10-E"00* 2&1%46++% )(01E"0+
,(0).0E"00) +(1.'0E"00* '&(+46++% ,(--E"0+
+'(-+ *(E"0+
+01.(0-'1- ,(-E"0+
,,)())*00' ,(1E"0+
!$t# =
!K t& =!$
t#%1=
!K k # =
#R &$ =
#2$ =! "1%#3 0$
!K# "1 = #U$ =
#R $ =4E!/ k !%5
t!&"1%#3
0$ G)H* =9F t
61*G?+H =9
t*G?
+H
#R $ = #U$% =
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'7&77 ''&27 6&'%1
''&27 25&'5 $&$(
k 6&'%1 $&$( 1'7&15
2)%%E02 3)22E03 1)3#E0#k1 3)22E03 3)+%E03 2)#0E05
1)3#E0# 2)#0E05 5)##E0#
>!?
>6?-
GIH
2)%%E02 3)22E03 1)3#E0# 2)5 +&1' 0)325
3)22E03 3)+%E03 2)#0E05 2)5 +& 2
1)3#E0# 2)#0E05 5)##E0# 2)5 6+&' 0)%5
s 6JsinKzcosK
t sinK 6cosK
I + + 1
s + 1 ' 2)#&E03 5)#%E03
t 1 + + &)%0E03 2)#&E03
I1 + + 1 #)13E0# #)13E0#
s +&1751 +&7+5 + 2)#&E03 %)0&E03
t +&7+5 6+&1751 + &)%0E03 1)10E03
I2 + + 1 #)13E0# #)13E0#
s +&1751 +&7+5 62&7$2 2)#&E03 +)2%E03
t +&7+5 6+&1751 +&% &)%0E03 +)5#E0#
I' + + 1 #)13E0# #)13E0#
@ 7(&557+ +&++++ +&++++
' +&++++ 11&11$+ 1$&+$++
+&++++ 1$&+$++ '5&$5'+
@ 7(&557+ +&++++ +&++++ 5)#%E03 5)3#E01
' +&++++ 11&11$+ 1$&+$++ 2)#&E03 2)15E02
>A?-k "1!>7?
cosK sinK
JcosKzsinK
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1 +&++++ 1$&+$++ '5&$5'+ #)13E0# 1)5E02
@ 7(&557+ +&++++ +&++++ %)0&E03 &)+E01
' +&++++ 11&11$+ 1$&+$++ 1)10E03 &)3E03
2 +&++++ 1$&+$++ '5&$5'+ #)13E0# 3)##E0#
@ 7(&557+ +&++++ +&++++ +)2%E03 +)0+E01
' +&++++ 11&11$+ 1$&+$++ +)5#E0# 3)&E03
' +&++++ 1$&+$++ '5&$5'+ #)13E0# 3)0%E03
s!m C 1)%E05
s!m z 2)20E05
s!m m #)#3E05
C +&25
, 1&5
mo +&5
0 0)1&1 0)1&1 :1 0)325
1 0)+0& 0)+0& :2 2
3 0 2)#1+ :3 0)%5
5)00050# 1 %)#015E01%
10)1++%+12 1)01%+# 0)332%35
5)0#30& 1)01%+# 0)332%35
:1 3)%5E01 5)3#E01
:2 1)53E=00 &)+E01
:3 1)2%E01 +)0+E01
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2)#&E03
&)%0E03
#)13E0#
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