dissertação - wiliam alves diniz júnior - 2014

8/15/2019 Dissertação - Wiliam Alves Diniz Júnior - 2014

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11/120


12/120

1

1 a a 1

a b a b a

b c


13/120

n


14/120


15/120

25

3


16/120


17/120


18/120

π


19/120


20/120

•

•

•


21/120

•

•

2006

2006 2005

2011

3.000 525


22/120

2001 300

2011

2011

2012

2012

2011

200


23/120

1974

10

2012

•

•

2012 2013


24/120

P l l P 2013

1992

2012

2013

2012 1802−1860

1792−1852 2001

2013

2.300


25/120

4 2006

u

n AB n

AB n

AB u

AB

w

w

n

u m AB n m

w AB u AB u w n u

w 1/n AB m w AB m/n

u AB m/n w n


26/120

u m AB m n


27/120

p/q

pq

2

= 1 2 + 1 2

p2

q 2 = 2

p2 = 2 q 2.

p2 q 2 2q 2

2 2q 2 p2 q p

√ 2 1, 414

1, 415 AB

AB

n 1/n w m AB m

w m + 1 w

mn

< AB < m + 1

n m/n (m+1) /n

AB 1/n


28/120

1 4 n n2

n2


29/120

A = n2.

m/n m n Am/n

n2 1/n

A1/n 1/n 2

n2A 1n

= 1

A 1n

= 1n2

A 1n

=1

n

2

.

m2 1/n m/n

m/n m2 A1/n A m

n= m2A 1

n

A mn

= m2 1n2

A mn

= m2

n2


30/120

A mn

=mn

2.

l l

xk yk

xk < l < y k yk −xk < 1/k k xk yk l

x2k < A l < y2k

x2k < l 2 < y 2k l2 Al (x2k , y2k )

|Al −l2| < y 2k −x2k = ( yk −xk)(yk + xk). yk −xk < 1k

|Al −l2| < 1k

(yk + xk)


31/120

|Al −l2| < 1k

(yk −xk + 2xk)

|Al −l2| < 1k1k

+ 2xk .

k Al −l2

Al = l2.

Q a b AQ = ab

a b a a b

b

a + b (a + b)2 a b AQ

ab

(a + b)2 = a2 + b2 + 2 AQ


32/120

a2 + 2 ab + b2 = a2 + b2 + 2 AQ

2ab = 2AQ

AQ = ab.

ABCD S AD DC BC

DC ADD BCC

ABCD ABC D bh

ABC AA CB A B

AC AA BC ABC ABA (LLL )

ABC

AABC = a.h

2 .


33/120

F F r σ : F →F

F F

•

X

Y

F

X

σ(X )

Y = σ(Y )

F X Y = rXY

ABCD A B C D r

a b ra rb ab r2ab

r 1 r

r r2 1


34/120

1

1 π π = 3, 141592

r

A = πr 2 π 1

n n

C/ 2 r

rC 2

= πr 2

C = 2πr.


35/120

1 S

C

n

n

n3 C n3

q q 3 1/q


36/120

1/q

q 3V 1q

= V 1

V 1q

= 1q 3

V 1q

=1q

3

.

p/q p q p3 1/q

p/q V p

q= p3V 1

q

V pq

= p3 1q 3

V pq

= p3

q 3

V pq

= pq

3

.

C b b

x b3 b

r < b b x < r 3 < b3 C b D r vol(D) < vol(C )

vol(D) = r 3 x < vol (C ) y

b3 y > vol(C ) vol(C ) = b3


37/120

1 a a a

1 V V = a

1 1

a a

1 a b

ab V = ab

1 a

b a b

a

b

c

abc V = abc

B

a/q b/q c/q a b c q 1/q

abc 1/q 1/q 3


38/120

a b

c

vol(B) = abc 1q 3

= aq

bq

cq

.

vol(B) = abc

a b c

1598 1629

1647 1635


39/120

•

•


40/120

A B α α0 α

vol(A) vol(B)

a b

x 2a 2 +

y2

b2 = 1, a > b x2 + y2 = a2

y

y = ba

(a2 −x2)12

y = ( a2 −x2)12 .

b/a


41/120

b/a b/a

Ae = ba

Ac

Ae = ba

πa 2 = πab.

90


42/120

vol

P Q

P Q P Q g


43/120

vol

A1A2 . . . An α V α

V V A1A2 . . . An


44/120


45/120

k

V DV A = DE AB = V E V B = EF BC = V F V C = F DCA = k.

ABC DEF

DE AB

= EF BC

= FDCA

= k.

k X Y V Y h V X H Y E XB V Y E V XB

k = V E V B =

EY BX =

V Y V X =

hH .

DEF ABC k

h1 h2 DEF ABC EF BC

ADEF AABC =

1

2h1.EF

12 h2.BC =

h1.EF h2.BC .

EF BC

= hH

h1h2

= hH

ADEF AABC

= hH

. hH

=hH

2

.


46/120


47/120

C AB C A AB C

A A B C

vol 3 vol

vol 1

3

V

V

g V r h

V


48/120

H R α H

α h V A2 r

A2A

=rR

2.


49/120

V BC V DE

V DV B

= DE BC

= hH

= rR

.

A1 A

A1

A =

πr 2

πR2 =

r2

R2 =

r

R

2=

h

H

2

.

vol 1

3

13

πR 2H.

R

r

h

A R A r

H H −h

= Rr ⇒H =

(H −h).Rr

⇒

H = HR

r − hR

r


50/120

⇒HR −Hr = hR⇒H = hRR −r.

vol 1

3πR 2H −

13

πr 2(H −h)

vol 1

3πR 2H −

13

πr 2H + 13

πr 2h

vol π

3

(R2

−r 2)H +

π

3

r 2h

vol π

3(R −r )(R + r )

hRR −r

+ π3

r 2h

vol π

3(R + r )hR +

π3

r 2h

vol π

3R2h +

π3

Rrh + π3

r 2h

vol π

3h(R2 + Rr + r 2)

P O R

R 2R α R α


51/120

R R

α

α

α R πR 2

β α h h < R


52/120

β c

1 r R r h

R2 = h2 + r 2⇒r2 = R2 −h2.

A1 c1

A1 = πr 2 = π(R2 −h2). A2 β

R h

h

R R h h

A2 = πR2 −πh 2A2 = π(R2 −h2).

vol vol vol

V = πR2.2R −213

πR 2.R

V = π 2R3 − 23

R3

V = π6R3 −2R3

3

V = 43πR 3.


53/120

r h

2πr h

A = πr 2 + πr 2 + 2 πrh

A = 2πr 2 + 2 πrh

A = 2πr (r + h).

c = 2

hh

r

r

r

h r g

r h


54/120

g2 = r 2 + h2

g = √ r 2 + h2. g

AB O

h

r

g

g

O

A

B

α R

2π →πR2

α →A2πα

= πR 2

A ⇒A = αR 2

2 .

α AOB AB


55/120

2π →2πRα →AB2πα

= 2πRAB ⇒α =

ABR

.

Ac OAB

Ac = αg2

2 =

AB.g 2

2g =

AB.g2

.

AB 2πr

Ac = 2πrg

2 = πrg

Ac = πr√ h2 + r 2.

2006

A n A1, A2, . . . , A n n

V 1, V 2, . . . , V n A1, A2, . . . , A n R

vol V 1 + V 2 + · · ·+ V n43

πR 3 = 13

A1.R + 13

A2.R + · · ·+ 13

An .R

43

πR 3 = 13

R (A1 + A2 + · · ·+ An )A = 4πR 2.


56/120

R V 1 A

R + h h V 2 h

n B1, B 2, . . . , B n h

V 2 −V 1 = B1.h + B2.h + ·+ Bn .h43

π (R + h)3 − 43

πR 3 = h. (B1 + B2 + · · ·+ Bn )4

3π R3 + 3 R2.h + 3 R.h 2 + h3

− 4

3πR 3 = h.A

43

πh (3R2 + 3 Rh + h2) = h.A

A = 4πR 2 + 4 πRh + 43

πh 2.

h A 4πR 2

25/ 2014 25

25

15.802 11 2006

• 1


57/120

• 2

• 3

• 4

03/ 2014 03

37, 8

37, 8

< 37, 8

< 22, 8 < 37, 8

< 22, 8 ≥37, 8 22, 8 ≤ < 37, 8

25

•


58/120

37, 8 ≤ < 60 60 ≤ < 93

≥93

•

06

03/ 2014

10− 6 20


59/120

•

•

•

•

•

•

•

6.1.7.2 25


60/120


61/120

1, 5

15 3

0, 60

1

1.600 3

2.400 3

1.600 3


62/120

2

380 3

570 3

380

3

3

4 2

45

10%

5

45


63/120

6 .1 .8 .1

20 3

6 .1 .8 .2

2 3

6 .4 .4 .6

1, 5

15


64/120

6.1.4

25

≤45

1/ 6

1

1/ 6

1

1/ 6

1

> 45

6.1.7.1

1/ 6

1/ 4

1/ 6

> 45

6.1.7.2

1/ 4

1/ 3

1/ 4


65/120

V = π.52.8

V = 628 3

h h

h = h + 20 h = h + 40

π

π = 3 , 14


66/120

A 132

A 169 2

h

vol vol 169.h = 628

h ⇒h = 3, 92

h = V A

.

l A

A = 12

l.h.

h l/ 2


67/120

l2

= h2

+l2

2

⇒h2

= l2

− l2

4

h2 = 34

l2⇒h =√ 32

l.

A = 12

l.√ 32

l⇒A =√ 34

l2.

l

P A n n l

n

n l h

12

l.h = An

.

β 2π/n α (π −β )/ 2 α h

α = π −β

2

α = π − 2πn

2

α = π(n −2)

2n


68/120

tan( α) = h

l/ 2 ⇒h =

l

2 tan( α).

12

l.h = An

12

l.12

l. tan( α) = An

l2

4 tan( α) =

An

l2 = 4An tan( α)

l = 2 An. tan( α)l = 2 An. tan π (n − 2)2n .

n

a b 2 p A a.b

A = a.b

2 p = 2(a + b) ⇒A = a.( p−a)⇒A = ap −a2.


69/120

α

A a = p/ 2 A p2/ 4 b p/ 2

f (x) = ax 2 + bx + c ⇒xv = − b

2 a yv = − ∆

4 a


70/120

Ab.h = V t

Ab.1, 3 = 628

Ab = 483, 1 2

h

vol vol vol 222.h = 628 + π.52.0, 5

484.h = 667, 3


71/120

h = 1 , 38 ⇒ h = 1, 38 + 0, 2 = 1, 58 ht = 1, 78

10 18

V

V = Ab.h

V = π.92.10


72/120

V = 2.543, 4 3

d 1/ 6

d = 16

.18


73/120

d = 3

d

d

d

h

vol vol vol vol

Ab.h = 2543, 4 + π.92.h + 0

422.h −92.π.h = 2543, 41509, 53.h = 2543, 4

h = 1 , 68 ⇒h = 1, 88 ht h

h

d h

vol vol vol

422.h

h = 5.086, 8

1.764

h = 2 , 9 ⇒h = 3, 1


74/120

l

h = h −0, 4⇒h = 2 , 6 vol vol vol

l2.h

l2 = 5.086, 8

2, 6

l = 1.956, 46l = 44, 23

3 d

1/ 6

d = 16

(2, 4 + 2 , 4 + 2 , 4)

d = 1, 2

20 3


75/120


76/120

3

3

V 3 R h A


77/120

R V

3

3

2

h = V /(πR 2) V = πR2 A = 2πR 2 + 2 πR.h

3


78/120

3

f R + →R +

f (R) = 2πR 2 + 2 πR.h

f (R) = 2 πR2

+ 2V

R .

f (R) = 4 πR − 2V R2

= 0

4πR = 2V R2

R3 = V 2π

R = 3

V 2π . V 3

R h

1/ 6

d

d = 16

(3 + 3 + 3) = 1 , 5.


79/120

h

h

vol vol vol

6.6.h = π1, 52.2, 83 + π1, 52.0, 3

36h = 20 + 2 , 12

h = 22, 12

36 = 0, 62

h = 62 ⇒h = 62 + 20 = 82


80/120

d

1/ 6


81/120

40

vol vol vol vol vol vol

vol π.202.16 = 20.096 3


82/120


83/120

1202.h = 1 .256h + 22 .608 ⇒ h (1202 −1256) = 22.608h = 1 , 72

⇒h = 1, 92 h = 2, 12


84/120

vol vol vol vol vol

vol

vol π.202.16 = 20.096 3

vol π.202.1 = 1.256 3

vol = 4.π. 102.1 = 1.256 3

vol = 4.π. 102.h = 1 .256.hvol = 4.120.0, 2.0, 5 = 48 3

vol = 20.096 3 1.256 3 1.256 3 +1 .256.h 48 3

1202.h = 1 .256h + 22 .656 ⇒ h (1202 −1256) = 22.656h = 1 , 73 ⇒ h = 1, 93 h = 2, 13


85/120

• 2

•


86/120

2


87/120

(V ) 3

< V ≤ < V ≤

V > 120

≥

< 10

≥

< 10

≥

< 10

≥

< 10

•

•


88/120

>

1

2

≤ < ≤

>

≤ 1/ 2 > 2 1/ 3


89/120

1

3

≥ ≤ <

≤ < ≤ < ≤ <

≤ <

h h

vol vol

922.h = π212.15

h = 20.771, 1

8464


90/120

h = 2 , 46 ⇒h = 2, 46 + 0, 2+ , 02 = 2, 96

φ

2 Ac

Ac π 2

φ

φ

V r t

3

V r φ tV r 3


91/120


92/120

φ φT φv

φ = φT + φv

φT φv 2

φ = φT + φvφ = 3.956, 4 + 3.956, 4 = 7.912, 8

V r

V r φ t

t

3

V r 3

T 1 T 2 T 3


93/120

V B h

V T 1 V D 2 V D 3

h

V B = V T 1 + V D 2 + V D 3

h π252 π202h π152h16.445h −1.962, 5h = 29 .437, 5

h = 29.437, 514.482, 5

h = 2 , 04 ⇒h = 2, 04 + 0, 2 + 0 , 2 = 2, 44


94/120


95/120

h h

vol vol vol vol vol

352h = π52.8 + 122.0, 3 + 3 .122.0, 3 + 3 .π52h

1.225h −235, 5h = 628 + 43 , 2 + 129, 6

h = 800, 8989, 5

h = 0 , 81 ⇒h = 0, 81 + 0, 2 = 1, 01


96/120

φ1 2 A

c1

Ac1 = 2πRH Ac1 = 2π5 2

φ1 = 2 251, 2 = 502, 4

φ2

2 1/ 3 At Ac2

At = 3π52 = 235, 5 2

Ac2 = π5 2

φ2 = 513

(At + Ac2)

φ2 = 513

989, 1

φ2

φ φ1 φ2

φ = φ1 + φ2

φ = 502, 4 + 1 .648, 5 = 2.150, 9

3 t

V r

V r = φ tV r = 2.150, 9 120 = 258.108 3


97/120


98/120

φ

2 A

p

A p = 2

φ =

3 t

V r

V r = φ t

V r = 120 60


99/120

• Moeller

•

•

•

3

•


100/120


101/120


102/120

d ≤9 ≤6 9 < d ≤18

d > 18

d ≤9 > 6 9 < d ≤18

d > 18

2


103/120

2

d

d ≤24 24 < d ≤36

36 < d ≤42

42 < d ≤48 48 < d ≤54 54 < d ≤60

d > 60 2

3


104/120

d

d ≤20 20 < d ≤36

d > 36

d

d ≤10, 5 10, 5 < d ≤28, 5

d > 28, 5

T A

2

At

θT

θT = T A.AtAt = π212 2

θT = 4, 1 1384, 74

θT


105/120


106/120

V eb 3 V

ab

≈

3

V LGEb = 0, 03

212”

V etb

V atb V LGEtb

V etb π 0,1522

4 50 + π0,12

4 50 + π0,063 2

4 50 3

V atb ≈ 3 V LGEtb = 0, 03

V ae V LGE

V ae = V at + V ab + V atb 3 V LGE = V LGEt + V LGEb + V LGEtb

V etb =ni =1 π

d i2

4 ci di ci


107/120

T A

2

At

θT

θT = T A.AtAt = π212 2

θT = 6, 9 1.384, 74

V et θT t

V et = θT .tV et = 9.554, 7 30 ≈286, 64 3

V at V LGEt

V at = ≈ 3 V LGEt =

V eb V ab V LGEb

V eb 3 V ab ≈ 3

V LGEb = 0, 06


108/120


109/120

θT

V et θT t

V et = θT .tV et = 8.046, 25 25 ≈201, 16 3

V at V LGEt

V at = ≈ 3 V LGEt

θb V eb V ab V LGEb


110/120


111/120

2

θeb

θeb = T A.AbAb = 12 2

θeb = 6, 5 216


112/120

V eb θ

b

V eb = θb.tV eb = 1.404 30

V ab V LGEb

V ab =

≈

3

V LGEb =

•

•

•

•

V r V ae

V ri

V ri = V r + V aeV ri = 949, 54 + 184, 13 = 1.133, 67 3


113/120

V ri = V r + V aeV ri = 3.459 + 215, 03 = 3.674, 03 3

V ri = V r + V aeV ri = 7, 2 + 40, 9 = 48, 1 3

•

•

•

• Φ

φ θ

Φ = φ + θ

Φ

Φ = φ + θΦ = φ1 + φ2 + θT + θb θb

Φ = 4.710 + 4.898, 4 + 8.046, 25 + 600


114/120


115/120

− 17.505−7

− −


116/120


117/120

−

−


118/120

π

π 3, 1416

π

π 3, 1416. . .

π


119/120

π

1a

π

π

π


120/120

dissertação - wiliam alves diniz júnior - 2014

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