Problemas De Tecnologia De Maquinas
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Parte I
Fundamentos del Diseño de Máquinas
3
x σnx = 12 σny = 6 τxy = −4 σnx = 16 σny = 9 τxy = 5 σnx = −4 σny = 12 τxy = 7 σnx = −42 σny = −81 τxy = −30 σnx = 20 σny = −10 τxy = −8
σ1 = 14 σ2 = 4 σ3 = 0 τ = 7 φ = 26,565o
σ1 = 18,603 σ2 = 6,397 σ3 = 0 τ = 9,302 φ = 27,504o
σ1 = 14,630 σ2 = 0 σ3 = −6,630 τ = 10,630 φ = 69,407o
σ1 = 0 σ2 = −25,719 σ3 = −97,281 τ = 48,640 φ = 90o
σ1 = 22 σ2 = 0 σ3 = −12 τ = 17 φ = 14,036o
! " # # σ1 = 14 σ2 = 4 σ3 = 0 $ % & ' # # # σ1 " σ3 τ = 7
4
( $ ( x " ) # # σ1
φ = −26,565o ⇒ 26,565o
1
! " # # σ1 = 18,603 σ2 = 6,397 σ3 = 0 $ % & ' # # # σ1 " σ3 τ = 9,302 $ ( x " ) # # σ1
φ = 27,504o
! " # # σ1 = 14,630 σ2 = 0 σ3 = −6,630 $ % & ' # # # σ1 " σ3 τ = 10,630 $ ( x " ) # # σ1
φ = −69,407o ⇒ 69,407o
! " # # σ1 = 0 σ2 = −25,719 σ3 = −97,281 $ % & ' # # # σ1 " σ3 τ = 48,640 $ ( x " # ) # # *
# φ = 90o
! " # # σ1 = 22 σ2 = 0 σ3 = −12
1 Sentido
contrario al del reloj.
5
$ % & ' # # # σ1 " σ3 τ = 17 $ ( x " ) # # σ1
φ = 14,036o
6
(
!
40 10 10 10 30 0 10 0 30
" " # $ " %&◦ x '(◦ y $
σx = 38,284 σn = 49,142
σ1 = 50 σ2 = 30 σ3 = 20 2 1 1 α1 = √ β1 = √ γ1 = √ 6 6 6 −1 1 α2 = 0 β2 = √ γ2 = √ 2 2 1 −1 −1 α3 = √ β3 = √ γ3 = √ 3 3 3
σy = 22,071 σz = 22,071 τ = 5
# α = 0,707 β = 0,5 γ = 0,5 $ # + ) ! # σx = 38,284 σy = 22,071 σz = 22,071
7
$ ) " ) (
σn = 49,142 τ =5
# # # σ1 = 50 σ2 = 30 σ3 = 20 $ # ) # # $ ( ) # # $ ) # #
2 1 1 α1 = √ β1 = √ γ1 = √ 6 6 6 −1 1 α2 = 0 β2 = √ γ2 = √ 2 2 1 −1 −1 α3 = √ β3 = √ γ3 = √ 3 3 3
8
(
) *( + ,'(( -· . / 0 ,(
,
+ %( -· . 1 ! " 2&( 3 4 ,((( -·
,) - n = 1,159 ,) n = 1,292 ) - n = 0,106
$ " -
σAo = 603,609 σBo = 407,437
$ % ) % & ) # ) ./
) 0/
0
Ktx = 1,8
" -
σA = 603,609 σB = 733,387
$ % ( ) - # " ) " # % + n = 1,159
9
$ % & "τA = 188,628 τB = 5092,958 $ % ) % & # % &
) -
Kts = 1,5
$ " σA = 603,609 τA = 188,628 σB = 733,387 τB = 7639,437 $ # # " ,) 1 σ1 = 657,707 σ2 = 0 σ3 = −54,098 ,) -1 σ1 = 8014,93 σ2 = 0 σ3 = −7281,54 $ % ( #
nA = 1,292 nB = 0,106
10
(
2
2((( &( # $ " 5((( $ " 5((
d > 1,001
# (
d > 2,157
# (
$ 2/// #
T = 1575,634
·# (
$ # % & # 3/// # d > 1,001 # (
$ 2// #
T = 15756,339
·# (
$ # % & # 3/// # d > 2,157 # (
2 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 2-28, p. 91, McGrawHill, Mexico, 1990
11
) 1
,5( HB / + 0,004838 " 4 0,14364 ! +
Sy = 263,782
$ 4 # ( & - Sut = 372 $ % ) + # % ) # 5
εy = 0,004826
$ 6 # % ) # ) 7 # # 4 σ0 = 567,511 5 ) 7 # # 5 Sy = 263,782
12
(
3
) + *6( !
# σx = 180 σy = 180 σx = 140 τxy = −80 σx = −80 τxy = 120 τxy = −200
n = 2,160 n = 2,160 n = 1,834 n = 1,980 n = 1,541 n = 1,751 n = 0,975 n = 1,126
# #
σ1 = 180
σ2 = 180 σ3 = 0
+ (4 " 8 σ = 180 σ = 180 3 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 6-3, p. 295, McGrawHill, Mexico, 1990
13
$ 6 ( % % # 5 n = 2,160 n = 2,160
# # σ1 = 176,300
σ2 = 0 σ3 = −36,300
+ (4 " 8 σ = 212,600 σ = 196,970 $ 6 ( % % # 5 n = 1,834 n = 1,980
# # σ1 = 86,490 σ2 = 0 σ3 = −166,490 + (4 " 8 σ = 252,980 σ = 222,708 $ 6 ( % % # 5 n = 1,541 n = 1,751
# # σ1 9 2// σ2 9 / σ3 9 2// + (4 " 8
σ = 400 σ = 346,410
$ 6 ( % % # 5 n = 0,975 n = 1,126
14
(
4
) /78 *( *( 9 ,(( 9 !
# σA = 20 9 σB = 20 9 τxy = 15 9 σA = −80 9 σB = −80 9 σA = 20 9 σB = −10 9
n = 1,5 n = 1,5 n = 2 n = 1,538 n = 1,25 n = 1,25 n = 1,5 n = 1,304
# # σ1
9 2/ 7# σ2 9 2/ 7# σ3 9 / 7#
+ (4 7 " !
σ = 20 σ = 20
7# 7#
4 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 6-5, p. 295, McGrawHill, Mexico, 1990
15
$ 6 ( % % # 5
n = 1,5 n = 1,5
# # σ1 9 0: 7# σ2 9 / 7# σ3 9 0: 7# + (4 7 " !
σ = 15 σ = 19,5
7# 7#
$ 6 ( % % # 5
n = 2 n = 1,538
# # σ1 9 / 7# σ2 9 3/ 7# σ3 9 3/ 7# + (4 7 " !
σ = 24 σ = 24
7# 7#
$ 6 ( % % # 5
n = 1,25 n = 1,25
# # σ1 9 2/ 7# σ2 9 / 7# σ3 9 0/ 7# + (4 7 " !
σ = 20 σ = 23
$ 6 ( % % # 5
7# 7#
n = 1,5 n = 1,304
16
(
5
! 4 / 1 " : 0 /;7; ,(('
4 F = 0,55 9- P = 8 9- T = 30 -·
1 n = 2,771 n = 3,266 -1 n = 6,707 n = 7,905
' , 0//; % Sut = 330 Sy = 280 5 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 6-6, p. 295, McGrawHill, Mexico, 1990
17
< % &
$ ) + (4 8
σxF = 70,028 σxP = 25,460 τ = 19,098
σ = 101,055
$ % ( 5 "
< % & -
$ ) + (4 8 $ % ( 5 "
n = 2,771 n = 3,266
σxF = 0 σxP = 25,460 τ = 19,098
σ = 41,747
n = 6,707 n = 7,905
18
(
6
! " /;7; ,(5( 12,5 9 " ! < *' 9
Sf = 40,04
7#
N = 44431
' " ## 4 , 0/2/ Sut = 55 7# $ % ( # +(
Se = Se = 27,72
$ % ( Sf # ) 12,5 · 103
Sf = 40,04
7# 7#
# ( % ( ) # # # % & .; 7# N = 44431
6 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 7-1, p. 357, McGrawHill, Mexico, 1990
19
) /;7; ,(*& *( 7 # ! + = + " 0 70·103 *&( ◦ 3 +
Se = 200,979
Se = 224,396
Sf = 270,662
' " ## 4 , 0/.: % Sut = 550 $ % ( (
Se = 277,2
$ % " % ( ( ka = 0,847 kb = 0,856 kc = kd = ke = 1 Se = 200,979
$ + % " % ( ( # 5') + ) % = & %+ ka = 0,847 kb = 0,958 kc = kd = ke = 1 Se = 224,396
20
(
$ 4 # # )
.:/ ◦ $ Sut,350 = 518,65 $ + % ( ( % % ( (
"
Se = 261,4 ka = 0,860 kb = 0,856 kc = kd = ke = 1 Se = 192,432
$ % ( # ) 70 · 103
Sf = 270,662
21
0 : : : 2,5 $ /;7; ,(5(
: 0 # 0 " P "
0 " ±P "
>
- 01 P
= 237,5
7> - 21 P
- 21 P
= 103,191
= 237,5
7> - 21 P
7>
= 52,112
7>
" ## " % ## , 0/2/ Sut = 380
22
( $ ( #+ % # % 0 " 2 # 4 - 01 P = 237,5 7> - 21 P = 237,5 7>
$ % ( ( # 0 Se = 191,52 ka = 0,934 kc = 0,923 Se = 165,106 $ % ( ( # 2 Se = 191,52 ka = 0,934 kc = 0,923 Ktn = 2,35 q = 0,725 Kf n = 1,979 ke = 0,505 Se = 83,379 $ " σa = 1,6 P
σm = 0
# P 7>
$ P ) ? - 01 P = 103,191 7> - 21 P = 52,112 7>
23
) 5( 4 ,((( + '(( 4 0 " ? " " 4 + &(( -· *(( -· 0 " *(
, 7 : # 0 0 " : > " + &( -· @ "
ny = 0,836 n = 0,248 ny = 0,775 n = 0,248
$ + ' ) 5 " #
# )
σx = 636,62 τxy = 190,986
24
( $ ) + 8
5
" % ( σeq = 717,434 ny = 0,836
5') " ) # " ) % ( (4 8
σeq,a = 636,62 σeq,m = 330,797 % ( ( % " % ( ( Se = 504 ka = 0,723 kb = 0,896 Ktx = 2,07 qx = 0,84 Kf x = 1,899 ke = 0,527 Se = 172,063 $ % ( ) ?
n = 0,248
$ + ' ) 5 ( " 5 " ) σx = 700,282 τxy = 190,986 $ ) + 8
5
" % ( σeq = 774,482 ny = 0,775
$ + " (4 8 σeq,a = 636,62 σeq,m = 336,868
$ % ( % (
n = 0,248
25
7 4 4 ,((( : ,A&( 0,225 !
5( B : *( ,&
* / " + ,5& *A& ,5& 5&( 5&(
3 + 4 ! " : " / C > 3
Sy = 1248,552
n = 0,941
N = 169,316
Su = 1250
7
$ % ) # % 2/ @
ε = 0,223
26
( $
Sy = 1248,552 Su = 1250
$ " 5') % % ) % & *# %( #' + Sy " Su σxm0 = 250 σxa0 = 125 Ktx = 1,42 qx = 0,93 Kf x = 1,391 σxm = 347,75 σxa = 173,875 ( # )
τm0 = 187,5 τa0 = 62,5 Kts = 1,22 qs = 0,95 Kf s = 1,209 τm = 226,688 τa = 75,563 A # ) # % & # σnm0 = 250 Ktn = 1,65 qn = 0,93 Kf n = 1,605 σnm = 401,25 $ " + 8 * # %( σm = 845,673 σa = 217,628 % ( ( # % 5') " ) # # Se = 630 ka = 0,862 kb = 1,037 Se = 563,153 $ % ( ) ?
n = 0,941
$ ) + ?
σa0 = 672,809
$ ) # ) + N = 169,316 7
27
7 /;7; ,(*( 5' * $ + ( A( 9- 60 000 ) > ? : " ?
, # Se∗ = 131,1
" ## " % 4 , 0/./ % Sut = 520 $ % ( ( # Se = 262,08 ka = 0,86 kc = 0,923 Ktn = 1,7 q = 0,8 ke = 0,641 Se = 133,35 $ "
σm = 111,41 σa = 111,41
28
( $ ) + ?
σa0 = 141,788 σa0 > Se ⇒ #
$ ) # #
N0 = 713493
$ ) +& ;//// " % ( N = 653493 Se∗ = 131,1
29
) 4 ,5((
*(( " " " ,(( '(( 7 "
5(( < " ? ! # 8 " -4 " ? ,(( 9 D E E
? 4 7 >
σa0,d = 200
N = 196887
n = 2,054
σa0,f
= 400
$ ) + # # ) ) ?
σa0,d = 200 σa0,f = 400
30
(
$ % ( ( % ( # # # *. ( 0,9Sut " *: ( σa0,d Se∗ = 86,086 $ ) # ) + % N = 15064 $ ) + # )
N0 = 211951
$ 4 # # % N = 196887
$ + " % ( N = 113507 Se∗ = 253,339 $ % ( # ) 105
Sf∗ = 410,78
% ( σm n = 2,054
31
) *( 4 %(( " ,&( + ,&( 3 :
σa = 2 σm − 150 ! -4 + ,&( 104 7 " > <
n = 0,687
N = 25021
N = 14268
$ % ( ( ka = 0,781 kb = 0,958 Se = 201,6 Se = 150,837 , ) # '#) % )
) ? % % ( " # =
n = 0,687
32
(
$ ) + ?
σa0 = 240
N = 25021
$ ) # )
$ ) + ( ( " + # % ( *( # (
( # ) # 104 σa0 = 240 Se2 = 106,667 $ % ( + # ( ( % ! # " % ( # # ( ka2 = 0,781 kc2 = 0,923 Se2 = 145,326 ked = 0,734 Se∗ = 110,714
$ ) # ) ( + # " * 4 # # ( # = # " N = 10753 $ # % 4 ' # # ( N = 14268
Parte II
Ejes, acoplamientos y apoyos
35
# " #"
7
) 4 ,(( 9 + A( 9 : , 1,5
0,125 7 4 + " 2(( · %(( · " 3 # 0 + : > 8 0 : 7 8 0 : > C 0 : /70 C 0 : 7 8 + 2(( · " '(( ·
ny = 7,683 ns = 2,856
ns = 2,837
ns = 2,608
ns = 2,842
ns = 4,018
7 Cfr. SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problemas 18-7 a 18-11, p. 817, McGraw-Hill, Mexico, 1990
36
(
$ % ( (
ka = 0,797 kb = 0,872 Ktx = 1,59 q = 0,84 ke = 0,669 Se = 23,433 7#
$ 5') " ) ) + " % ( 5 σx = 8,149 7# τ = 2,037 7# σeq = 9,111 7# ny = 7,683
$ ( # " " ! # α = 41,659 ( τm = 0,237 7# τa = 4,047 7# < % ( ) ? '# = ns = 2,856
$ ( # " " ! # α = 40,249 ( τm = 0,336 7# τa = 4,018 7# < % ( ) , ( '# =
ns = 2,837
$ " + 8 )
σm = 3,528 7# σa = 8,149 7# < % ( ) ?
ns = 2,608
37
# " #"
$ + " ( % ( (4 # ,
ns = 2,842
$ % ( (
ka = 0,797 kb = 0,872 Kts = 1,3 q = 0,87 ke = 0,793 Se = 27,777 7#
$ ( # " " ! # α = 13,941 ( τm = 1,905 7# τa = 2,701 7# < % ( ) , ( '# =
ns = 4,018
38
(
8
7 ? D d $ d : " r dR = d − 2r 7 ? " d = 0,75D r = D/20
7/0 5*%( < : Sut = 1226 1
*'2 0 + A( -· %& -· ) ? 2,5 : 7 >
dR = 21,563
r = 1,659
D = 33,174
d = 24,880
$ % #6 " % ) % & )
ka = 0,685 D/dR = 1,538 r/dR = 0,077 Ktx = 2,05
$ 5') " ) " " + 8 % ) dR
713014,145 229183,118 σx = τ = 3 3 dR 713014,145 σa = d3R
dR 396956,805 τ= d3R
8 Cfr. SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 18-12, p. 817, McGraw-Hill, Mexico, 1990
39
# " #" B !#) * r # q " dR # ke " kb ( Se ( " )
? dR 6 r = 1
dR = 13
$ % ( ( " ) ?
% #
kb = 0,941 q = 0,88 ke = 0,520 Se = 207,112 dR = 21,117
# # ) # + dR
dR = 21,117
r = 1,624
kb = 0,891 q = 0,90 ke = 0,514 Se = 193,844 dR = 21,548
# # ) # + dR
dR = 21,548
r = 1,658
kb = 0,889 q = 0,90 ke = 0,514 Se = 193,409 dR = 21,563
# # ) # + dR
dR = 21,563
r = 1,659
kb = 0,889 q = 0,90 ke = 0,514 Se = 193,409 dR = 21,563
$
dR = 21,563
r = 1,659 D = 33,174
d = 24,880
40
(
)
,&( ,( 9F ,5(( &( 0 .
,&( / ,((
,& ,(
, 7 /;7; ,(&( C # 0 + 0 " : " 4 > 0 " : " 4 > 0 " : " 4 7
ny = 2,702 ns = 1,979 ns = 1,964 ns = 1,766
- + 5 " 4 , 0/:/ % Sy = 580 Sut = 690
41
# " #" $ # " % & (
T = 79,577 >· F = 530,516 >
$ ) #" & " 5 '
RA = 353,678 > M = 17,684 >·
$ ( 5 " " # # % + ,) 01 ( * ' 5 ,) 21 ) * " # ) % & $ 5 " 5') " ) ' + " # " + ) 0 M1 = 17,684 >· T1 = 79,577 >· σx1 = 53,371 τ1 = 120,053 σmax1 = 214,678 σm1 = 207,938 σa1 = 53,371 $ 5 " 5') " ) ' + " # " + ) 2 M2 = 8,842 >· T2 = 0 σx2 = 90,064 τ2 = 0 σmax2 = 90,064 σm2 = 0 σa2 = 90,064 , ) % + # + 5 " % ( ny = 2,702 ) 0
$ % ( ) 0
$ % ( ) 2
ka1 = 0,798 kb1 = 0,926 Se1 = 256,977
ka2 = 0,798 kb2 = 0,97 Ktx2 = 1,68 q = 0,75 ke2 = 0,662 Se2 = 178,202
42
( , + # ) ? % " % ( # % ( " ns1 = 2,364 ns2 = 1,979 ns = 1,979 ) 2
, + = # ) ? " # ns1 = 1,964 ns2 = 1,979 ns = 1,964 ) 0
,( # ! ) , ( ns1 = 1,766 ns2 = 1,979 ns = 1,766 ) 0
43
# " #"
9
) 4 ,((( + 2(( 2((
5,5 -· %(( ! %6 9.
2,75 9- 0 *( ,,
,& 2
,(
1,5 3 C 0 + 0 " : 4 > 0 " : 4 >
ny = 1,087
ns = 1,288
ns = 1,472
$ #"
9 PEDRERO,
RA = RB = 1375
>
J. I., Fundamentos del Diseño por Fatiga, problema 15, p. 59, UNED, Madrid, 1996
44
( $ ( 5 " % & ' " # # % + ,) 01 ) 00
,) 21 ) ) *# ,) .1 ) 3
) $ 5 " % & ' 5') " ) ' + " #
" + ) 0 M1 = 15,125 >· T1 = 212 >· N1 = 0 σx1 = 45,648 τ1 = 319,913 σn1 = 0 σmax1 = 555,983 σm1 = 554,106 σa1 = 45,648 A ) 2
M2 = 550 >· T2 = 212 >· N2 = 49000 > σx2 = 207,491 τ2 = 39,989 σn2 = 69,321 σmax2 = 285,346 σm2 = 97,994 σa2 = 207,491
A ) . σmax3
M3 = 11 >· T3 = 0 N3 = 49000 > σx3 = 112,045 τ3 = 0 σn3 = −623,887 = 735,932 σm3 = −623,887 σa3 = 112,045
, ) % + # + 5 " % ( ny = 1,087 ) .
$ % ( ) 0
ka1 = 0,723 kb1 = 0,926 Ktx1 = 1,75 q = 0,86 ke1 = 0,608 Se1 = 205,156
$ % ( ) 2
$ % ( ) .
ka2 = 0,723 kb2 = 0,856 Se2 = 311,920 ka3 = 0,723 kb3 = 0,97
45
# " #" Ktx3 = 1,58 q = 0,86 ke1 = 0,667 Se3 = 235,758
, + # ) ? % " % ( # % ( " * σm3 #) ns1 = 1,288 ns2 = 1,310 ns3 = 2,104 ns = 1,288 ) 0
, + = # ) ? " # ns1 = 1,478 ns2 = 1,472 ns3 = 2,104 ns = 1,472 ) 2
46
(
10
) 5( 4 ,((( + '(( 4 0 " ? " " 4 + &(( -· *(( -· 0 " *(
, 3 # 0 + 0 " : > " + &( -· @ "
ny = 0,836 ns = 0,248 ny = 0,775 ns = 0,248
$ + ) 5') ) ) + 8 " % ( 5 σx = 636,62 τ = 190,986 σ = 717,434 ny = 0,836
10 PEDRERO,
J. I., Fundamentos del Diseño por Fatiga, problema 14, p. 58, UNED, Madrid, 1996
47
# " #"
$ # " )
σm = 330,797 σa = 636,62 $ + % ( ( # ) #6 & ka = 0,723 kb = 0,896 Ktx = 2,08 q = 0,83 ke = 0,527 Se = 172,063 , + ) ? % % ( " # =
ns = 0,248
$ + ) 5') ) ) + 8 " % ( 5 + # 5 σx = 700,282 τ = 190,986 σ = 774,482
ny = 0,775
$ # " )
σm = 336,868 σa = 636,62 , + ) ? % % ( " # =
ns = 0,248
48
(
11
) ,&( *( : 6( %& " $ 5(( ,&( 0 " " %(( - *(( 7 : 0,25
pa = 203,046
7 T
= 38,765
>·
$ ( % " # & # ' #) " ) " = )
% & θ1 = 8,13 ( θ2 = 98,13 ( θa = 90 ( a = 250
c = 500
$ % & " & % ) pa '# % (& " ) + pa MN = 1,041 · 10−3 pa >· Mf = 0,056 · 10−3 pa >·
Rf = 1 pa = 203,046 7
$ # %
T = 38,765
>·
11 Cfr. SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 16-5, p. 745, McGraw-Hill, Mexico, 1990
49
# " #"
12
) %((
5((
< " 0 ,5& $ %(( - 4 " 7 5&(
: 0,3 &( # $ 0
P2 = 4022,39
T = 342,799
> P1 = 1280 >
>·
$ # ( = ( & # θ = 218,682 ( $ ) ' ! # ) # " # ) '
) % & P1 = 1280 > P2 = 4022,39 >
$ # %
T = 342,799
>·
12 Cfr. SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 16-10, p. 746, McGraw-Hill, Mexico, 1990
50
(
7 ? : " ,' -· & 9- 7 : 0,2 A
d = 21,351
D = 42,649
< D " d # '# F " T
d (D − d) = 454,728 d D2 − d2 = 29102,618
, ( '#) # + " # + d
2
3
D + d = 64
d = 21,351 d = 10,649
+ D + d = 64
" D ! ( +
" d " D D = 42,649
51
# " #"
7 ? " ,' -· " : 5&( 9 7 :
: 0,2 ? # 0 $
d = 74,134
F = 1,58
D = 128,405
7>
'#) # % ( % # D2 " # + ) + d ! d = 74,134
$ D # + d
$ % &
D = 128,405
F = 1,58
7>
52
(
) *( *((
4 %(( "< / ,((
0
0 ,(( 12,5 9
" , $
" 4 P = 0,3ω (1 − 0,2 · 10−2 ω) P " 9F ω G 0 ,( G / &(
5(
5 3 : 0,2 :
,(( 9
" & 9- 7 " : C > ;
d = 559,557 d = 28,433
D = 616,443
D = 1147,557
= +
ns = 2,431 σm = 0,011σa2 − 2,085σa + 83,557
'#
53
# " #"
$ '#) # # # % ) + ( " + ' ! # T = 300 (1 − 0,002ω) >· T = 294 >· $ '#) # % + D + d " % & d (D − d) D + d = 1176
dD − d2 = 31830,989
2 + ( ( d = 559,557
D = 616,443
" = d = 28,433
D = 1147,557
$ # + ) + ( ! ' # ( " + # # + ( ω = 250 C T = 150 >· '# % & ( " % & % ( # # ' % ) # ( '# >· Fe = 10T > Fω = 3125 − 20,833T + 0,035T 2 > $ ) % & # # % & 5 # # % & % ( " 5 # # % & ( # % ) T Nm = 5000 > Mm = 78,125 − 0,521T + 0,85 · 10−3 T 2 >· Ma = 0,25T >· $ # ) "
" σnm = −15,915 σxm = 99,472 − 0,663T + 1,082 · 10−3 T 2 σxa = 0,318T σm = 83,557 − 0,663T + 1,082 · 10−3 T 2 σa = 0,318T
54
( T '# " # ) ( σm = 0,011σa2 − 2,085σa + 83,557
$ % ( (
ka = 0,922 kb = 0,896 Ktx = 1,67 q = 0,73 ke = 0,672 Se = 111,918
, '# σm " σa # T # + # )
# # % ( ) ?
" # ns ns = 2,431
55
# " #"
13
7 % 9- " N10 ,5(( " '(( $ " N10 *2(( &(( ! 7 " * 2 9- :
FR = 2,895
R = 0,97
7>
$ # ( %
$ ) N10 ! ##
N10 = 2715,021 !
FR = 2,895
7>
6 # ) 02// ! + N10 = 2715,021 ! R = 0,97
13 Cfr. SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problemas 11-2 y 11-3, p. 537, McGraw-Hill, Mexico, 1990
56
(
14
) ,2(( : 6' B !
N10 = 3373,509
!
$ ) N10 # # ) %) # R = 0,96
N10 = 3373,509
!
14 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 11-4, p. 537, McGraw-Hill, Mexico, 1990
57
# " #"
15
7 (5 2 9- % 9- $ N10 &(((
6(( 7
d = 80
D = 140
C = 70,2 7>
$ % ) + L10 " ) Fa /Fr
V =1 L10 = 270 Fa /Fr = 0,5
& # #' ) % & + Fe = V Fr " # ( Fe = 8 7> C = 51,706 7> , # (
# 51,706 7> " # + & !#) d = 65
D = 120
C0 = 34 7> C = 55,9 7> Fa /C0 = 0,118 e ≈ 0,309 < 0,5 X = 0,56 Y ≈ 1,426 0,5 > '(0,309, 0,309) + $ % & + # " # ( Fe = 10,184 7> C = 65,822 7> 15 Cfr. SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 11-6, p. 538, McGraw-Hill, Mexico, 1990
58
( , # (
# 65,822 7> % & + " # ( " # = d = 75
D = 130
C0 = 40,5 7> C = 66,3 7> Fa /C0 = 0,099 e ≈ 0,296 < 0,5 X = 0,56 Y ≈ 1,486 Fe = 10,424 7> C = 67,373 7> > 66,3 7> + ( d = 80
D = 140
C0 = 45 7> C = 70,2 7> Fa /C0 = 0,089 e ≈ 0,289 < 0,5 X = 0,56 Y ≈ 1,524 Fe = 10,576 7> C = 68,356 7> < 70,2 7> +
59
# " #"
16
0 " *( (5 " 5(( ((( " ,2 9- *( 9-
L = 266,551
7
< # (
C = 20,3
$ ) # ( 03 7>
L = 1,434
7>
$ ) " # ( #+ L = 1,234 C ∗ = 19,307 7> $ ) # + ( # ( L = 266,551 7
16 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 11-20, p. 541, McGraw-Hill, Mexico, 1990
60
(
17
)
,5 9- 7 " %((( " A&( 3
C = 57
7>
$ ) $ # (
L = 180
C = 56,985 ≈ 57
7>
17 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 11-8, p. 538, McGraw-Hill, Mexico, 1990
61
# " #"
18
)
&( && & 7/0 *( && ◦ 3 5(( 9 $
%5 µ " %2 "G 10 9- 3 " * ◦ 3 +
∆T = 68
◦
$ Qs = 3726,828
3 C h0 = 5,04 µ
# #
P = 4 r = 595,238 c 1 l = d 2 µ < µ (T1 ) = 33 ·
, # + + * % .. ·
( + # # ) "
# D # + " + " (6 " -" + ε " rc f D # " # ! + #' # + # 1 µ = 17,5 · Tm = 70 ◦ $ ∆T = 30 ◦ $
S = 0,074 r ε = 0,82 f = 2,6 c ∆T = 239,823 ◦ $ > 30 ◦ $
18 Cfr. SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 12-23, p. 594, McGraw-Hill, Mexico, 1990
62
( A + ! # ! " &
" # ,( 1 µ = 10 · Tm = 86 ◦ $ ∆T = 62 ◦ $
S = 0,043 r ε = 0,87 f = 1,8 c ∆T = 90,715 ◦ $ > 62 ◦ $
A + ! 4 1 µ = 5 · Tm = 112 ◦ $ ∆T = 114 ◦ $ A + ! " # µ = 10 ·
# 90,715 ◦ $ + # # " $ 1 µ = 7 · Tm = 98 ◦ $ ∆T = 86 ◦ $
S = 0,030 r ε = 0,89 f = 1,5 c ∆T = 51,489 ◦ $ < 86 ◦ $
A + ! " E 1 µ = 9 · Tm = 89 ◦ $ ∆T
= 68 ◦ $ S = 0,038 r ε = 0,88 f = 1,6 c ∆T = 70,422 ◦ $ < 68 + 3 ◦ $
# + # + S = 0,038
# # # # Qs = 3726,828
3 C h0 = 5,04 µ
63
# " #"
19
) &(
&& &
7/0 %( %( ◦ 3 5&( 9 7 %( µ ,5 9- " %2 "G 3 : + $
f = 0,003 Qs = 3428,325
3 C pa = 21,12 271,434 F Qs " H #
f = 0,005 Q = 10800
3 C p = 11,163 452,389 F #)
h0 = 5 µ H =
h0 = 16,8 µ H =
# #
P = 4,8 r = 625 c 1 l = d 2
, # + + ( + # # ) " # D # + " + " (6 " -" + 19 PEDRERO,
J. I., Fundamentos del Diseño por Fatiga, problema 10, p. 57, UNED, Madrid, 1996
64
( ε " rc f D # " # ! + #' # + # 1 µ = 10 · Tm = 93 ◦ $ ∆T = 106 ◦ $ S = 0,039 r ε = 0,88 f = 1,8 c ∆T = 93,668 ◦ $ < 106 ◦ $
A + ! " # ! " & " # " ,( 1 µ = 11 · Tm = 90 ◦ $ ∆T = 100 ◦ $
S = 0,043 r ε = 0,87 f = 1,9 c ∆T = 110,353 ◦ $ > 100 ◦ $
A + ! # ! " & " # 1 µ = 10,5 · Tm = 91 ◦ $ ∆T = 102 ◦ $ S = 0,041 r ε = 0,875 f = 1,85 c ∆T = 101,827 ◦ $ ≈ 102 ◦ $
# + # + S = 0,041 # # # # r f = 1,85 f = 0,003 c Qs = 3428,325
3 C P = 0,23 pa = 21,12 p ε = 0,875 h0 = 5 µ H = 271,434 F
# #
P = 4,8 r = 625 c l =1 d
, # + + ( + # # ) " # D #
65
# " #" + " + " (6 " -" + Q rc f rcnl " QQ D # " # ! + #' # + # 1 µ = 10 · Tm = 93 ◦ $ ∆T = 106 ◦ $ s
S = 0,039 r Q Qs f = 1,4 = 4,8 9 /3; c rcnl Q ◦ ∆T = 20,386 $ < 106 ◦ $
A + ! " # ! " & " # " ,( 1 µ = 20 · Tm = 73 ◦ $ ∆T = 66 ◦ $
S = 0,078 r Q Qs f = 2,14 = 4,62 9 /G; c rcnl Q ◦ ∆T = 29,208 $ < 66 ◦ $
A + ! " 4 1 µ = 30 · Tm = 64 ◦ $ ∆T = 48 ◦ $
S = 0,117 r Q Qs f = 3 = 4,5 9 /;3 c rcnl Q ◦ ∆T = 40,242 $ < 48 ◦ $
A + ! " " #' $ 1 µ = 34 · Tm = 61 ◦ $ ∆T = 42 ◦ $
S = 0,133 r Q Qs f = 3,1 = 4,5 9 /;G: c rcnl Q ◦ ∆T = 41,892 $ ≈ 42 ◦ $
# + # + S = 0,133 # # # # r f = 3,1 f = 0,005 c Q = 4,5 Q = 10800
3 C rcnl P = 0,43 p = 11,163
p
h0 /c = 0,42 h0 = 16,8 µ H = 452,389 F
66
(
20
) && &( ,& 9- 5( "G 5& µ 0 & %(( 9 &6 ◦ 3 / : 75,4 F ! :
+ 7 7/0
f = 0,0016 Ts = 120 pa = 30,4
◦
$ Qs = 1630,537
3 C h0 = 2,5 µ
, ./
# #
P = 6 r = 1000 c 1 l = d 2
$ 6 %) =( #
# # # %) ! ( # # = %( )
f = 0,0016
20 PEDRERO,
J. I., Fundamentos del Diseño por Fatiga, problema 11, p. 57, UNED, Madrid, 1996
67
# " #" $ +
r cf
" (
r f = 1,6 c S = 0,03
# '#) S + + "
5 µ = 9 · Qs = 1630,537
3 C $ + S ) ' # " # # # " #) ' ε = 0,9 ∆T = 61,038 ◦ $ ≈ 61 ◦ $ Ts = 120 ◦ $ h0 = 2,5 µ P = 0,2 pa = 30,4 p
$ # # )
Tm = 89,5
◦
$
, ( + # # # " + " " + # # = , ./
Parte III
Transmisiones por engranajes
71
# (
7 ? ? ,6 '*
αn = 25 , 1,25 0,3 $
! 2,5 H0 ? "
,(* " I 0 " < : ?
' # #)
) 1
x1 = 0,2 x2 = 0
"
)
Cnom = 102,5
$ & # + !( # ra1 = 26,75
ra2 = 81,75
$ # ) # & #) 4x π + tg αn + 2(tg αt − αt ) − 2 Z Z
ra2 − 1 − arctg rb2
ra2 −1 rb2
>
0,3mn ra
72
( # #) *0,02008 < 0,028037 > # *0,011488 > 0,00917 $ 4 # ' # )
#) Z ≥ 12,0576 > ' # ) #) $ 6 # & #) " # ) % xT = 0,2
# # & # *6 # &
" + & #) " ra1 = 26,75
ra2 = 81,25
$ # + ) # #) ! & # 6 # & ! # & " # ' # > # #) *0,04059 > 0,028037 $ #6 ++ # #) " #
re =
r02 +
rp − r0 tg αt
2
r0 = rp − mn b + mn rf (1 − sen αt ) + mn x re1 = 22,064 re2 = 76,277
$ 6 & ) ( #) "
2 rf2 in = ra + C 2 − 2C rb cos αt + sen αt ra 2 − rb 2
rf in1 = 22,2565 rf in2 = 76,8911
73
# ( $ # ) % #) " *rf in > re > % #) *rf in1 > re1 > % *rf in2 > re2
74
(
) ,( 9F ,5(( < $ ? 5& %( < 5(
0 ? ,6 %& 8 %
5(o 3 # 0 " 0 ? 0 ?
αt = 22,296
εα = 1,847
εα = 1,170
( n2 = 750 #
$ # " ( # Cnom1 = 130
Cnom2 = 128
# ( # #
75
# ( $ # + #) " #+ " ( #) % % rp1 = 38
rp2 = 90
rb1 = 35,708
rb2 = 84,572
αt = 22,296 ( $ +
n2 = 750
#
$ & + !( # ra1 = 44
ra2 = 96
$ ( % ( #
2 2 ra1 ra2 1 εα = − 1 + Z2 − 1 − ZT tg αt Z1 2π rb1 rb2 εα = 1,847
$ &
ra1 = 42 ra2 = 94
$ ( % ( #
εα = 1,170
76
(
! ? ? 5( '( $
& 5(o 1,25 " < 5&o 0 0,8 " "
? 7 ;7= ' D,&'( 7 " %( o 3 2( 7 $ '( 9F 5((( ! ? " : 3 1,243 3 ! ? 3 4 ?
x1 = 0,25 x2 = −0,25
C = 221,748
h = 0,8798
rus1 = 59,3507
rui1 = 54,2158
rus2 = 167,6529
rui2 = 163,1516
$ 4 + #)
Zv1 = 26,866
77
# ( < 6 # & & # 6 % ) 4 + #) " )
)
x1 = 0,25 x2 = −0,25
$ # + #) " " #+ rp1 = 55,1689
rp2 = 165,5067
rb1 = 51,1948
rb2 = 153,5846
$ & #) "
ra1 = 61,4189 ra2 = 169,2567
$ ( #) % % # ' #) ( % αt = 22,56 ( # ( #) % % C = 221,748
$ !( # )
h = 2C(tg αt − αt ) − 2C(tg αt − αt ) −
4C xT tg αn ZT h = 0,8798
$ 6 & ) ( #) "
rf in1 = 53,0596 rf in2 = 161,8757
$ 6 & ) ( 4 # #) " rus
2
2 2π rf in = rb − 1 + 1 + Z rb
78
( rus1 = 59,3507 rus2 = 167,6529
$ 4 % #) " rui
2
2 ra 2π = rb −1− +1 rb Z
rui1 = 54,2158 rui2 = 163,1516
79
# (
) < "< %#, ? " 82,5 4 ! J ? J 4 ? -
; 5(o J <
J 1,25 0,25 /
< " ±40o 7
< J4 J ! <
! : 4 K 3 " ? " ;
: "
?
80
(
D Z1 = 16 Z2 = 64
mn = 2 β = 14 ( αt = 20,562 ( αt = 20,656
x1 = 0,38 x2 = −0,38
xT = 0,025 ⇒ x1 = 0,405 x2 = −0,38
(
>
# 4 ' # # 4 # #)
Z1 = 16
# ) " ( != # # ) 82,5
mn = 2 β = 14 ( $ ( #) #
$ 4 + #)
αt = 20,562 αt = 20,656
Zv1 = 17,515
( (
(6 # ) # & # 6 * A % 21 #( .. ' 6 # & #) " x1 = 0,38 x2 = −0,38
21 LAFONT,
P., Cálculo de engranajes paralelos, E.T.S.I.I., Madrid, 1995
81
# (
# '#) !( # # & # = xT = 0,025
# # &
x1 = 0,405 x2 = −0,38
82
(
) " < *((( " 4 ' 9F 0,75 0
" %#, $ ? ,2 %&
5(o * *( $ 5( *5
< ,&o %( 8 D,&,'
;7= & # 7 ; ;; ! 3 : < &(
0,0001 7 " &(o &( 7 " , -G fsho 0,5 3 <
mn2 = 3,5
αt = 21,108
SH(0,01 %) = 2,293
( εγ = 2,406
83
# (
SF (0,01 %) = 4,088
$ # #
Cnom = 94,5
$ ) ( ( # #'
# # # " & ) #)' mn2 = 3,511 ⇒ mn2 = 3,5
$ ( #) ) % $ ( #) % %
αt = 20,647
(
αt = 21,108
(
# % ( = = ( != βb = 14,076 ( $ " & #) " rb1 = 33,907
rb2 = 54,252
ra1 = 39,735
ra2 = 61,475
$ ( ) % ( # + #
2 2 ra1 ra2 1 Z1 εα = − 1 + Z2 − 1 − ZT tg αt 2π rb1 rb2
εα = 1,464
$ ( εβ =
b sen β πmn
84
( b ! ( ( # ( H/
εβ = 0,942
$ (
εγ = 2,406
6 ( % % # # + #
SH =
2
σHP σH
σHP # #) B& 5 # σHP = σHlim ZN ZL ZR ZV ZW ZX
" σH #) B& 5 # σH =
KA KV KHβ KHα σHo
σHo #) B& # * A % #( 30
σHo = ZH ZE Zε Zβ
Ft u + 1 2rp1 b u
ZH % ( = ZE % Zε % ) " Zβ % ) $ % ( = ZH #
ZH =
2 cos βb cos αt cos2 αt sen αt ZH = 2,396
$ % ZE # 1 ZE = 1−ν 2 π E1 1 +
1−ν22 E2
2 ν1 = ν2 = 0,3 E1 = E2 = 207 · 10 >C
2
ZE = 60,169
>C
2
85
# ( $ % ) Zε # ! 0 < εβ < 1 * A % #( 30
Zε =
4 − εα εβ (1 − εβ ) + 3 εα Zε = 0,832
$ % ) Zβ # '#) # Zβ = √ cos β & # 5 # " & #) B& '# % =
Zβ = 0,983
$ # # # G: @ % ( + #) ( # " # + P = 8 7F n1 = 1200 # d1 = 72,469
$ % & ( #) # + > + # P (7F) 6 Ft ( >) = 106 π n1 (# ) d1 (
) Ft = 175,694 > $ ) ) ( # + #) B&
σHo = 37,003
u2 = 1,6
>C
2
< % # ) KA # ) * ! % " % ( * ( % KA = 1,5
$ % # ( ! εβ < 1 # KV = KV α − εβ (KV α − KV β )
86
( KV α " KV β + % # #+
" ! εβ > 1 % # ! * A % #( 0/; '#)
KV = 1 + K350 f
$ # Z1 v1 100
u2 1 + u2
v1 + ( #) "
% ) = % K350 # " # ! Z1 v1 u2 v1 = 4,567 C = 0,775 100 1 + u2 K = 0,0375 K = 0,027 $ % f * A % #( 0/G % ) # Ft /b·KA " > f = 2,30 f = 2,69 $ % KV α " KV β " % KV α = 1,086 KV β = 1,0783 KV = 1,078
$ % A % #( 0/31
KHβ
# ( *
2Cγ Fβy
Fm ≤ Fβy b
KHβ =
Fm ≥ Fβy b
KHβ = 1 +
Fm b
Cγ Fβy 2 Fbm
(KHβ ≥ 2) (KHβ ≤ 2)
Cγ ( & # + ( 2 * >C
Cµ ) + # Fβy = Fβx − Yβ
Fβx ) " Yβ ) ) # ) Fβx = fma + fsh
87
# ( fma ) # % ) " fsh )
# % ) ( KHβ $ ) # % ) fma # 6 ) ( # % ( 5 fHβ fma = 6,5 µ $ ) # % ) ( fsh Fm = 284,097 > fsho = 0,5 fsh = 3,5512 µ $ ) Fβx
Fβx = 10,0512 µ
$ ) ) # * A % #( 003 Yβ " ) Fβy Yβ = 1,50768 µ Fβy = 8,54352 µ $ ) Fm /b " # ) Fβy Fm /b = 7,1024 Fm /b ≤ Fβy $ 0
KHβ = 2,1935
$ % ) + ( KHα ( % ) 6 ( * A % #( 02/1 εγ ≤ 2 εγ > 2
KHα
εγ = 2
0,90 + 0,40
KHα = 0,90 + 0,40
Cγ (fpb − Yα )
FtH b
2(εγ − 1) Cγ (fpb − Yα ) FtH εγ b
$ # fpb # fpb = fpt cos αt cos βb fpt # * A % 6( ;: #( 022
fpt = T fpt = 0,400ϕp + 5 ϕp = 32,61 fpt = 18,044 fpb = 16,378
88
( $ ) # * A % #( 02. Yα " ) FtH /b Yα = 1,2283 µ FtH /b = 15,505 >C
+ % KHα
εγ = 2,406 > 2 KHα = 1,745
+ #) B& 5
σH = 92,057
>C
2
' % ( #) #6 # * A % #( 0.H σHlim = 163 >C
2 $ % ) ZN # #6 (4 ( # ( " ) # 5 · 107 * A % #( 0.; ZN = 1
$ % + ZL
$ % + ZV
CZL = 0,91 ZL = 0,956
CZV = 0,93 ZV = 0,98
$ % ) & ZW # #) " )
ZW = 1
$ % ZX # ( % " ) 2 " 0/ * A % #( 0H2 ZX = 1
$ % ( ZR
CZR = 0,08 Rtm100 = 3,0571 µ ZR ≈ 1
89
# ( + # #) B&
σHP = 152,711
>C
2
$ 6 ( % % # # #6
SH(1 %) = 2,752
$ 6 ( % % # # #6 # # % ////0 *//0 @ SH(1 %) = SH(0,01 %) KRH(0,01 %)
KRH(0,01 %) % 6 # # % ////0
KRH(0,01 %) = 1,2 SH(0,01 %) = 2,293
$ 6 ( % % # + # SF =
σF P σF
σF ) # σF = KA KV KF β KF α σF o
σF o ) # σF o =
Ft YF s Yε Yβ bmn
YF s % & #6 Yε % )
Yβ % ) " σF P # ) # σF P = σF lim YST YN T YδrelT YRrelT Yx
$ 4 + #) ( # "
= % & YF S * A % #( I; Zv1 = 22,192 YF S = 4,58
90
( $ % ) Yε
εαn = 1,556 Yε = 0,732
$ % ) Yβ
Yβ = 0,882
+ ) σF o
σF o = 3,711
>C
2
$ % ) ( ( KF β * A % #( 02/
N = 0,809 KF β = 1,888
$ % ) + ( KF α * A % #( 02/ " 020
KF α = 1,745
+ )
σF = 19,770
>C
2
' % ( # ) # 0:0; σF lim = 46 >C
2 $ % YST % ) ( " ( # # 4 # ( 2 YST = 2
$ % ) YN T # ) " # + # 3 · 106 * A % #( 0:/ YN T = 1
$ % YδrelT ( 3H * A % #( 0:. % ) % YSa )
# ( & YSa = 1,61 YδrelT = 0,99
91
# ( $ % ( + YRrelT # ( 3: * A % #( 0:H
YRrelT = 1,065
$ % ) YX ( 3; * A % #( 0:H
YX = 1
+ # ) σF P = 97 >C
2 $ 6 ( % % # % (
SF (1 %) = 4,906
$ 6 ( % % # % ( # # % ////0 *//0 @ SF (1 %) = SF (0,01 %) KRH(0,01 %)
KRH(0,01 %) % 6 # # % ////0
KRH(0,01 %) = 1,2 SF (0,01 %) = 4,088
92
(
! ? " * *'% 9F $ D,&'( " %%, 0
5(0 1,25 " 0,25 " 7 : ;7= & : 2 ,( 7 : : $ " '
Z1 = 22 Z2 = 66 mn = 10 d1 = 220 P = 527,423 7F P = 485,018 7F
b = 130
' 0:;/ * A % #( 0.H " 0H3 σHlim = 163 >C
2 σF lim = 50 >C
2 , + 4 # #) ( >
" 4 Z1 = 22 Z2 = 66 $ % + KB * A % #( 0;0 # 6 " 3 0/ ! " ! #
93
# (
" ! % #
2 KRH /ZN = 1 KA = 2 KB = 2
$ % C1 # C1 =
π u n1 (# ) 6 u+1 C1 = 173,18
< % C2 ( 3I * A % #( 0;. # αn = 20o " β = 0 C2 = 0,21
, # + + ( #) ! # Vt = 6,5 C $ % C3 # ! ( I/ * A % #( 0;:
C3 ≈ 0,935
$ % # ! " + % C3 # ! # % C3 # + % ) % Vt Z1 /100 C
Vt Z1 = 1,43 100 = C /1,013 = 0,923
$ % C5
C5 = 7,339
+ % % + C6 # #) " * A % #( 0G2 C6 = 1
$ # ) + % # % ( #6 (4 = #6 ,< KB P (7F) = C1 C2 C3 C4 C5 C6 ,
94
( % + ) C4 # .;H 7F
C4 = 2,955
# + # % + ) C4 + ! ( b # + b/d1 # (6 ( I: * A % #( 0G/ b/d1 = 0,75 ⇒ b = 130
b/d1 = 1 ⇒ b = 160
b/d1 = 1,25 ⇒ b = 190
! 4 ) # # >
# ) &) b/d1 * A % #( 0GH b/d1 ≤ 1,1 , # , ( # # +
b = 130 d1 = 173,333
$ ) #) # # + " &
# *#% mn = 7,878
⇒ mn = 8
$ + ( # ) & ! ( d1 = 176
b/d1 = 0,739
$ + ( # + " + % C3
Vt = 4,064 C C3 = 0,925
$ % ) ( ( KHβ " ) + ( KHα " % C4 KHβ = 1,516 KHα = 1 C4 = 2,656
% KHβ ! '# # ( # ,< : "
95
# ( # & 2/ @ # #6 # # + # # ( + % # % ( #6 Padm = 327,864 7F A # % ⇒ + # ) &
# #
mn = 10
$ + ( # ) &
! ( d1 = 220
b/d1 = 0,591
$ + ( # + " + % C3
Vt = 5,08 C C3 = 0,926
$ % ) ( ( KHβ " ) + ( KHα " % C4 KHβ = 1,474 KHα = 1
C4 = 4,268
+ # # ( + % # % ( #6 # mn = 10
Padm = 527,423 7F A # # ⇒ + $ # ) # = #6 6 ( + & Ft = 1102,3 > 350 >C
KA b , # ) # = #6 +
96
( $ % KBF ,< + % #' H/ @ # + % + % % # % ( #6 KB KBF = 2,8
$ % CB1 CB1 =
π · 10−6 · Z1 · m2n · n1 (# ) 6 CB1 = 0,508
$ ( ) % " = + % CB2 ( I3 * A % #( 03. rb1 = 103,366
rb2 = 310,099
x1 = 0,28 x2 = −0,28 Cnom = 440
ra1 = 122,8
ra2 = 337,2
εα = 1,63
CB2 = 1,41 $ % CB3 # ( II * A % #( 03H
CB3 = 0,965
$ % YF S # ( :H * A % #( I; " = % CB4 + # CB4 = 1/YF S
YF S = 4,37 CB4 = 0,228
$ % CB5 $ % CB6
CB5 = 92,659
CB6 = 100
>C
2
+ % CB7 # #6 # ( 0// * A % #( 03H
CB7 = 0,93
+ # ' + % # % (
97
# ( Padm = 485,018
A # # ⇒ +
98
(
7 '( 9F ,5(( < '(( : 2 $ ? *5 '% 5 & : b/d1 = 0,60 D,,&(
αn = 20o 1,25 " 0,25 " 0
:
: " D,5&5 ;
" 0 ;7= & 3 ? :
H0 : "
" I 0 " :
P = 6,61 7F A # )
P = 64,789 7F P = 74,895 7F +
99
# (
$ % # ) KA # )
H ! * A % #( 0/H " & # !
(= % ! * A % #( 0/: KA = 1,5
$ % & ( #) # + N # # ;/ 7F Ft = 11936,62 > $ ! # ) b/d1 /;/
b = 48
$ # ) # = #6
Ft = 373,02 > 350 b
>C
( & # = #6 # # KA
$ % + KB # 6 ) % # # 3 " 0/ ! 2 KRH /ZN = 1 KA = 1,5 KB = 1,5
$ % C1 $ % C2
C1 = 418,88
C2 = 0,21
$ + ( # + " % C3
$ % C4
Vt = 5,0265 C C3 = 0,899 KHα = 1 KHβ = 1,17
100
( C4 = 0,2625
$ % C5
$ % C6
σHlim = 46 >C
2 C5 = 0,5845
C6 = 1
+ # # & ) + % # % ( #6 Padm = 6,61 7F A # # # )
' 6 ) # + 02:2 σHlim = 136 >C
2 σF lim = 35 >C
2 $ # ) ) ! C = Cnom = 120
' # 4 # ) ## # >
+ # ) ) 2 Z1 = [19 ÷ 29] < # % ) 4 #) # 120
m = 2,5
⇒ Z1 = 32 m = 3
⇒ Z1 = 26,67 m = 4
⇒ Z1 = 20 m = 5
⇒ Z1 = 16 ,) + ) . " H , + + , ( m = 4 Z1 = 20 $ ! ( # = #6 b = 51,15
101
# ( $ % C1 $ % C2
C1 = 418,88
C2 = 0,21
$ + ( # + " % C3
$ % C4
$ % C5 $ % C6
Vt = 5,0265 C C3 = 0,935 KHα = 1 KHβ = 1,4154 C4 = 0,23128
C5 = 5,109
C6 = 1
+ # # & ) + % # % ( #6 Padm = 64,789 7F A # # ⇒ + $ % KBF $ % CB1
KBF = 2,1
CB1 = 0,201
$ ( ) % " = + % CB2 ( I3 * A % #( 03. rb1 = 37,588
rb2 = 75,175
ra1 = 44
ra2 = 84
εα = 1,636
CB2 = 1,41 $ % CB3
CB3 = 0,98
102
( $ % YF S # ( :H * A % #( I; " = % CB4 + # CB4 = 1/YF S
YF S = 4,67 CB4 = 0,214
$ % CB5 $ % CB6
CB5 = 37,8023
CB6 = 70
>C
2
+ % CB7 # #6 # ( 0// * A % #( 03H CB7 = 1
+ # ' + % # % ( Padm = 74,895 7F A # # ⇒ +
103
# (
) " < *((( " 4 ' 9F 0,75 0 " %#,
$ ? ,2 %&
5(o * *( $ 5( *5
< ,&o %( 8 D,&,'
;7= &
" "
%&o3
Ss = 0,577 λ = 0,4087 ⇒
# ( '+ #' H/ @
$ + (4 = # # # !22 1 ν40 =
7000 V 0,5
ν40 + H/0 $ , " V + %C # V = 0,0103 d · n 22 ERRICHELLO,
R., Friction, Lubrication, and Wear of Gears, Lubrication Engineering, 1990.
104
( d #)
" n + ( #) # ν40 = 233 , < = # ( # ( " + = % J # # % # ) # )
( % ) KKA/Vt " + .3o * A % ( 0/G #( 20/ Ft = 175,695 > K = 0,09849 >C
2 Vt = 4,567 C KA = 1,5 KKA = 0,032 ν38 ≈ 230 , VT
<+ # = , ## +
H/o 2..:. , ,< 8? .2/ 6 ( % % # (# + # Ss =
θSint θint
# (# # # # θSint " # ( # θint $ # 5 θMT " + # 5 ! θf la int T # ( 020 * A % #( 2.; # % ) ( ( " (# K? # * A % #( 22; T1 = 63,662 >· θMT = 94,64226 o $ θf la int T = 12,4396 o $ $ (# # # # θSint
θSint = 113,30166
o
$
+ # ( # θint # # 5 θM " + 5 !
105
# ( # θf la int # ( '#)
θint = θM + 1,5θf la int
# 5 θM # # θoil = 45o $ " + 5 ! # θf la int )
θM = (θoil + 0,7θf la int )XS
XS % # # ) 0/ # ) # " 02 # ) # ")
# ) # ) , # # 2 · 106 ≤ d(
) · n2 (# ) ≤ 108
" # ") % +
d1 · n21 = 104,355 · 106 d2 · n22 = 65,22 · 106
# # )
XS = 1,0
$ + # 5 ! θf la int + # '#) # 3/4
θf la int = µβ XM XBE
1/2
WtB Vt C 1/4
Xε Xca XQ
C
" Vt + ( C = 94,5
Vt = 4,567 C WtB
( >C
# '#)
WtB =
Ft KA KHβ KHα KBγ b
Ft = 1756,94 > b = 40
KA = 1,5 KHβ = 2,1935 KHα = 1,745 " % KBγ ( 003 * A % #( 2.H 2 < εγ ≤ 3,5 % KBγ = 1 + 0,2 (εγ − 2)(5 − εγ )
106
( εγ = 2,406 KBγ = 1,205 WtB = 303,8843 >C
% # 5 XM = 50 >−3/4 1/2 −1/2
$ 6 %) µβ " '#) # µβ = 0,045
KA Fbbt −0,05 (KHβ KHα )0,2 ηM XR 2vρ
$ % & + Fbt > # Fbt (>) =
60 6 P (7F) 10 π n1 (# ) db1 (
) db1 = 67,814
Fbt = 1877,5 >
$ + 2v ( Vt sen αt
2v = 1,6447
C
$ + #6 # '#
# '#)
u + 1 1 cos βb cos αt 1 = ρ u rp1 sen αt cos αt ρ = 8,304
$ + # 5 θM '# · # # ! + & # # # 5 + " 6 ! # " ( , # + θM ( ;: o $
107
# ( , ! # # + # #
& # ) ( & ) 8(1 b t + θ µ() = µ( ·) = k · e
k b " θ # # ( # k b " θ # # & # ) # µ # t ( ( ) ,< ,< 8? .2 ,< 8? H; ,< 8? ;3 ,< 8? 0// ,< 8? 0:/ ,< 8? 22/ ,< 8? .2/ ,< 8? H;/ ,< 8? ;3/ ,< 8? 0/// ,< 8? 0://
k
/0./I;0 /00H:3. /0/II:I //GI;2: /00IIH; /0/322: /0G0I/. /2:00IG /.30G33 /..;3:. /0G;..0
b
:3/.3./2; ;;I..:GHI G:/H2/;0; I0:HH2I.I 30:0//303 3332H;:0: 32:./I.:3 G:/2GH3.: ;G/0;H;/: GI.H32:;G I302H33G0
θ
G0.0I;02 GG0../00 32H/2:.. IHG3HGH. GI.H03./ 30G.33H3 GH.:H:/0 ;.3;.I2H :H0I:3H. ;...:2H; G0H2HI::
$ + # ,< 8? .2/ ;: o $
ηM = µ = 64,17
·
$ % 5 (
XR = 3,8
Ra (µ ) d1 (
)
1/4
Ra ( = 5 Ra =
Ra1 + Ra2 2
Ra1 " Ra2 ( = #) " # + # ( Rtm + #& ( = * ( $A +
# #' # " + #' Ra =
Rtm 6
108
( Rtm1 = Rtm2 = 3 µ Ra1 = Ra2 = 0,5 µ Ra = 0,5 µ XR = 1,09518
$ 6 %)
µβ = 0,2698
$ % ( = XBE % # + 5 6 ( , + XBE = 0,5(u + 1)1/2
1/2
ρE1 − (ρE2 /u)1/2 (ρE1 · ρE2 )1/4
1 d2a1 − d2b1 2 = C sen αt − ρE1
ρE1 = ρE2
ρE1 = 20,7163
ρE2 = 13,3157
XBE = 0,3297
$ % XQ # ) ' 6 ( ε1 " ε2 " #) A 6 ( # ( '# 1
2
ra1 1 − 1 − Z1 tg αt Z1 2π rb1
2 ra2 1 − 1 − Z2 tg αt ε2 = Z2 2π rb2
ε1 =
ε1 = 0,71607 ε2 = 0,7481 XQ = 1,0
$ % XCA # # Ca & 5 * A % #( 2.: XCA = 1 + 1,55 · 10−2 ε4max Ca
# % ( ' ( " # & " #
109
# ( # = ) 5 ! # # Ca # " 6 ( ε1 " ε2 Ca = Ca2
$ # %+ + + Ca2 " + # Cef f =
KA Fbbt Cγ
Cγ 6 ( & ( ( 2 >C*
·µ
Fbt2 = 187,75 > Ca2 = Cef f = 3,52 µ εmax = ε2 9 /GH30 XCA = 1,017
$ % Xε # ( % " 6 ( ε1 " ε2 # ( 02/ * A % #( 2.: εα = 1,464 ε1 = 0,71607 Xε = 0,30
$ + 5 ! # " # 5 θf la int = 65,45 o $ θM = 90,815 o $ 8+ 6 & µβ # 5 ! # θf la int " + # 5 θM ! ( +( θM = 90,815o$ ⇒ ηM = 25,429 · ⇒ µβ = 0,2826 ⇒ θf la int = o 68,556 $ ⇒ θM = 92,989o $ θM = 92,989o$ ⇒ ηM = 23,831 · ⇒ µβ = 0,2835 ⇒ θf la int = o 68,775 $ ⇒ θM = 93,142o $ θM = 93,142o$ ⇒ ηM = 23,724 · ⇒ µβ = 0,2836 ⇒ θf la int = o 68,799 $ ⇒ θM = 93,159o $ θM = 93,159o$ ⇒ ηM = 23,712 · ⇒ µβ = 0,2836 ⇒ θf la int = o 68,799 $ ⇒ θM = 93,159o $ $ # ( # # # 5 I.0:I o $ θint = 196,3575 o $
110
( $ 6 ( % % # (#
Ss = 0,577
# (# ( ( #
+ ( ( '+ 5 ( ! ) L " B(( ## + # # " ) 1 0,3
0,6 ηM Vt 0,7 σH −0,26 u 1 hmin = 1,26 sen αt αE C u + 1 cos βb EC E
+ ηM >·C
2 # 5 θM = 93,159 o $ ηM = 29,174 · 10−9 >·C
2 $ + (
C
Vt = 4567
C
$ # E # + # 1 1 = E 2
1 − ν1 1 − ν2 + E1 E2
ν1 = ν2 = 0,3 E1 = E2 = 2,06 · 105 E = 2,26 · 105
>C
2 >C
2
$ #) B& 5 >C
2 + # σH =
KA KV KHβ KHα σHo
σHo #) B& #
σHo = ZH ZE Zε Zβ
Ft u2 + 1 2rp1 b u2 σH = 920,57
>C
2
$ 6 + C#) α
2 C> # '#) F α = (0,6 + 0,965 log10 ηM )10−2
111
# ( α = 1,927 · 10−2
2 C>
# " C = 94,5
# # hmin = 0,222264 µ $ ( #6 σ=
σ12 + σ22
σ1 " σ2 ( #) " # + ( # ( = $A + # 0./ σ1 = 0,3846 µ σ2 = 0,3846 µ σ = 0,5439 µ # # 6 # λ 6 &) # # " ( #6 λ=
hmin σ λ = 0,4087
+ # ( # ( 00. * A % #( 22H #' H/ @
Fundamentos del Diseño de Máquinas
3
x σnx = 12 σny = 6 τxy = −4 σnx = 16 σny = 9 τxy = 5 σnx = −4 σny = 12 τxy = 7 σnx = −42 σny = −81 τxy = −30 σnx = 20 σny = −10 τxy = −8
σ1 = 14 σ2 = 4 σ3 = 0 τ = 7 φ = 26,565o
σ1 = 18,603 σ2 = 6,397 σ3 = 0 τ = 9,302 φ = 27,504o
σ1 = 14,630 σ2 = 0 σ3 = −6,630 τ = 10,630 φ = 69,407o
σ1 = 0 σ2 = −25,719 σ3 = −97,281 τ = 48,640 φ = 90o
σ1 = 22 σ2 = 0 σ3 = −12 τ = 17 φ = 14,036o
! " # # σ1 = 14 σ2 = 4 σ3 = 0 $ % & ' # # # σ1 " σ3 τ = 7
4
( $ ( x " ) # # σ1
φ = −26,565o ⇒ 26,565o
1
! " # # σ1 = 18,603 σ2 = 6,397 σ3 = 0 $ % & ' # # # σ1 " σ3 τ = 9,302 $ ( x " ) # # σ1
φ = 27,504o
! " # # σ1 = 14,630 σ2 = 0 σ3 = −6,630 $ % & ' # # # σ1 " σ3 τ = 10,630 $ ( x " ) # # σ1
φ = −69,407o ⇒ 69,407o
! " # # σ1 = 0 σ2 = −25,719 σ3 = −97,281 $ % & ' # # # σ1 " σ3 τ = 48,640 $ ( x " # ) # # *
# φ = 90o
! " # # σ1 = 22 σ2 = 0 σ3 = −12
1 Sentido
contrario al del reloj.
5
$ % & ' # # # σ1 " σ3 τ = 17 $ ( x " ) # # σ1
φ = 14,036o
6
(
!
40 10 10 10 30 0 10 0 30
" " # $ " %&◦ x '(◦ y $
σx = 38,284 σn = 49,142
σ1 = 50 σ2 = 30 σ3 = 20 2 1 1 α1 = √ β1 = √ γ1 = √ 6 6 6 −1 1 α2 = 0 β2 = √ γ2 = √ 2 2 1 −1 −1 α3 = √ β3 = √ γ3 = √ 3 3 3
σy = 22,071 σz = 22,071 τ = 5
# α = 0,707 β = 0,5 γ = 0,5 $ # + ) ! # σx = 38,284 σy = 22,071 σz = 22,071
7
$ ) " ) (
σn = 49,142 τ =5
# # # σ1 = 50 σ2 = 30 σ3 = 20 $ # ) # # $ ( ) # # $ ) # #
2 1 1 α1 = √ β1 = √ γ1 = √ 6 6 6 −1 1 α2 = 0 β2 = √ γ2 = √ 2 2 1 −1 −1 α3 = √ β3 = √ γ3 = √ 3 3 3
8
(
) *( + ,'(( -· . / 0 ,(
,
+ %( -· . 1 ! " 2&( 3 4 ,((( -·
,) - n = 1,159 ,) n = 1,292 ) - n = 0,106
$ " -
σAo = 603,609 σBo = 407,437
$ % ) % & ) # ) ./
) 0/
0
Ktx = 1,8
" -
σA = 603,609 σB = 733,387
$ % ( ) - # " ) " # % + n = 1,159
9
$ % & "τA = 188,628 τB = 5092,958 $ % ) % & # % &
) -
Kts = 1,5
$ " σA = 603,609 τA = 188,628 σB = 733,387 τB = 7639,437 $ # # " ,) 1 σ1 = 657,707 σ2 = 0 σ3 = −54,098 ,) -1 σ1 = 8014,93 σ2 = 0 σ3 = −7281,54 $ % ( #
nA = 1,292 nB = 0,106
10
(
2
2((( &( # $ " 5((( $ " 5((
d > 1,001
# (
d > 2,157
# (
$ 2/// #
T = 1575,634
·# (
$ # % & # 3/// # d > 1,001 # (
$ 2// #
T = 15756,339
·# (
$ # % & # 3/// # d > 2,157 # (
2 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 2-28, p. 91, McGrawHill, Mexico, 1990
11
) 1
,5( HB / + 0,004838 " 4 0,14364 ! +
Sy = 263,782
$ 4 # ( & - Sut = 372 $ % ) + # % ) # 5
εy = 0,004826
$ 6 # % ) # ) 7 # # 4 σ0 = 567,511 5 ) 7 # # 5 Sy = 263,782
12
(
3
) + *6( !
# σx = 180 σy = 180 σx = 140 τxy = −80 σx = −80 τxy = 120 τxy = −200
n = 2,160 n = 2,160 n = 1,834 n = 1,980 n = 1,541 n = 1,751 n = 0,975 n = 1,126
# #
σ1 = 180
σ2 = 180 σ3 = 0
+ (4 " 8 σ = 180 σ = 180 3 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 6-3, p. 295, McGrawHill, Mexico, 1990
13
$ 6 ( % % # 5 n = 2,160 n = 2,160
# # σ1 = 176,300
σ2 = 0 σ3 = −36,300
+ (4 " 8 σ = 212,600 σ = 196,970 $ 6 ( % % # 5 n = 1,834 n = 1,980
# # σ1 = 86,490 σ2 = 0 σ3 = −166,490 + (4 " 8 σ = 252,980 σ = 222,708 $ 6 ( % % # 5 n = 1,541 n = 1,751
# # σ1 9 2// σ2 9 / σ3 9 2// + (4 " 8
σ = 400 σ = 346,410
$ 6 ( % % # 5 n = 0,975 n = 1,126
14
(
4
) /78 *( *( 9 ,(( 9 !
# σA = 20 9 σB = 20 9 τxy = 15 9 σA = −80 9 σB = −80 9 σA = 20 9 σB = −10 9
n = 1,5 n = 1,5 n = 2 n = 1,538 n = 1,25 n = 1,25 n = 1,5 n = 1,304
# # σ1
9 2/ 7# σ2 9 2/ 7# σ3 9 / 7#
+ (4 7 " !
σ = 20 σ = 20
7# 7#
4 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 6-5, p. 295, McGrawHill, Mexico, 1990
15
$ 6 ( % % # 5
n = 1,5 n = 1,5
# # σ1 9 0: 7# σ2 9 / 7# σ3 9 0: 7# + (4 7 " !
σ = 15 σ = 19,5
7# 7#
$ 6 ( % % # 5
n = 2 n = 1,538
# # σ1 9 / 7# σ2 9 3/ 7# σ3 9 3/ 7# + (4 7 " !
σ = 24 σ = 24
7# 7#
$ 6 ( % % # 5
n = 1,25 n = 1,25
# # σ1 9 2/ 7# σ2 9 / 7# σ3 9 0/ 7# + (4 7 " !
σ = 20 σ = 23
$ 6 ( % % # 5
7# 7#
n = 1,5 n = 1,304
16
(
5
! 4 / 1 " : 0 /;7; ,(('
4 F = 0,55 9- P = 8 9- T = 30 -·
1 n = 2,771 n = 3,266 -1 n = 6,707 n = 7,905
' , 0//; % Sut = 330 Sy = 280 5 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 6-6, p. 295, McGrawHill, Mexico, 1990
17
< % &
$ ) + (4 8
σxF = 70,028 σxP = 25,460 τ = 19,098
σ = 101,055
$ % ( 5 "
< % & -
$ ) + (4 8 $ % ( 5 "
n = 2,771 n = 3,266
σxF = 0 σxP = 25,460 τ = 19,098
σ = 41,747
n = 6,707 n = 7,905
18
(
6
! " /;7; ,(5( 12,5 9 " ! < *' 9
Sf = 40,04
7#
N = 44431
' " ## 4 , 0/2/ Sut = 55 7# $ % ( # +(
Se = Se = 27,72
$ % ( Sf # ) 12,5 · 103
Sf = 40,04
7# 7#
# ( % ( ) # # # % & .; 7# N = 44431
6 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 7-1, p. 357, McGrawHill, Mexico, 1990
19
) /;7; ,(*& *( 7 # ! + = + " 0 70·103 *&( ◦ 3 +
Se = 200,979
Se = 224,396
Sf = 270,662
' " ## 4 , 0/.: % Sut = 550 $ % ( (
Se = 277,2
$ % " % ( ( ka = 0,847 kb = 0,856 kc = kd = ke = 1 Se = 200,979
$ + % " % ( ( # 5') + ) % = & %+ ka = 0,847 kb = 0,958 kc = kd = ke = 1 Se = 224,396
20
(
$ 4 # # )
.:/ ◦ $ Sut,350 = 518,65 $ + % ( ( % % ( (
"
Se = 261,4 ka = 0,860 kb = 0,856 kc = kd = ke = 1 Se = 192,432
$ % ( # ) 70 · 103
Sf = 270,662
21
0 : : : 2,5 $ /;7; ,(5(
: 0 # 0 " P "
0 " ±P "
>
- 01 P
= 237,5
7> - 21 P
- 21 P
= 103,191
= 237,5
7> - 21 P
7>
= 52,112
7>
" ## " % ## , 0/2/ Sut = 380
22
( $ ( #+ % # % 0 " 2 # 4 - 01 P = 237,5 7> - 21 P = 237,5 7>
$ % ( ( # 0 Se = 191,52 ka = 0,934 kc = 0,923 Se = 165,106 $ % ( ( # 2 Se = 191,52 ka = 0,934 kc = 0,923 Ktn = 2,35 q = 0,725 Kf n = 1,979 ke = 0,505 Se = 83,379 $ " σa = 1,6 P
σm = 0
# P 7>
$ P ) ? - 01 P = 103,191 7> - 21 P = 52,112 7>
23
) 5( 4 ,((( + '(( 4 0 " ? " " 4 + &(( -· *(( -· 0 " *(
, 7 : # 0 0 " : > " + &( -· @ "
ny = 0,836 n = 0,248 ny = 0,775 n = 0,248
$ + ' ) 5 " #
# )
σx = 636,62 τxy = 190,986
24
( $ ) + 8
5
" % ( σeq = 717,434 ny = 0,836
5') " ) # " ) % ( (4 8
σeq,a = 636,62 σeq,m = 330,797 % ( ( % " % ( ( Se = 504 ka = 0,723 kb = 0,896 Ktx = 2,07 qx = 0,84 Kf x = 1,899 ke = 0,527 Se = 172,063 $ % ( ) ?
n = 0,248
$ + ' ) 5 ( " 5 " ) σx = 700,282 τxy = 190,986 $ ) + 8
5
" % ( σeq = 774,482 ny = 0,775
$ + " (4 8 σeq,a = 636,62 σeq,m = 336,868
$ % ( % (
n = 0,248
25
7 4 4 ,((( : ,A&( 0,225 !
5( B : *( ,&
* / " + ,5& *A& ,5& 5&( 5&(
3 + 4 ! " : " / C > 3
Sy = 1248,552
n = 0,941
N = 169,316
Su = 1250
7
$ % ) # % 2/ @
ε = 0,223
26
( $
Sy = 1248,552 Su = 1250
$ " 5') % % ) % & *# %( #' + Sy " Su σxm0 = 250 σxa0 = 125 Ktx = 1,42 qx = 0,93 Kf x = 1,391 σxm = 347,75 σxa = 173,875 ( # )
τm0 = 187,5 τa0 = 62,5 Kts = 1,22 qs = 0,95 Kf s = 1,209 τm = 226,688 τa = 75,563 A # ) # % & # σnm0 = 250 Ktn = 1,65 qn = 0,93 Kf n = 1,605 σnm = 401,25 $ " + 8 * # %( σm = 845,673 σa = 217,628 % ( ( # % 5') " ) # # Se = 630 ka = 0,862 kb = 1,037 Se = 563,153 $ % ( ) ?
n = 0,941
$ ) + ?
σa0 = 672,809
$ ) # ) + N = 169,316 7
27
7 /;7; ,(*( 5' * $ + ( A( 9- 60 000 ) > ? : " ?
, # Se∗ = 131,1
" ## " % 4 , 0/./ % Sut = 520 $ % ( ( # Se = 262,08 ka = 0,86 kc = 0,923 Ktn = 1,7 q = 0,8 ke = 0,641 Se = 133,35 $ "
σm = 111,41 σa = 111,41
28
( $ ) + ?
σa0 = 141,788 σa0 > Se ⇒ #
$ ) # #
N0 = 713493
$ ) +& ;//// " % ( N = 653493 Se∗ = 131,1
29
) 4 ,5((
*(( " " " ,(( '(( 7 "
5(( < " ? ! # 8 " -4 " ? ,(( 9 D E E
? 4 7 >
σa0,d = 200
N = 196887
n = 2,054
σa0,f
= 400
$ ) + # # ) ) ?
σa0,d = 200 σa0,f = 400
30
(
$ % ( ( % ( # # # *. ( 0,9Sut " *: ( σa0,d Se∗ = 86,086 $ ) # ) + % N = 15064 $ ) + # )
N0 = 211951
$ 4 # # % N = 196887
$ + " % ( N = 113507 Se∗ = 253,339 $ % ( # ) 105
Sf∗ = 410,78
% ( σm n = 2,054
31
) *( 4 %(( " ,&( + ,&( 3 :
σa = 2 σm − 150 ! -4 + ,&( 104 7 " > <
n = 0,687
N = 25021
N = 14268
$ % ( ( ka = 0,781 kb = 0,958 Se = 201,6 Se = 150,837 , ) # '#) % )
) ? % % ( " # =
n = 0,687
32
(
$ ) + ?
σa0 = 240
N = 25021
$ ) # )
$ ) + ( ( " + # % ( *( # (
( # ) # 104 σa0 = 240 Se2 = 106,667 $ % ( + # ( ( % ! # " % ( # # ( ka2 = 0,781 kc2 = 0,923 Se2 = 145,326 ked = 0,734 Se∗ = 110,714
$ ) # ) ( + # " * 4 # # ( # = # " N = 10753 $ # % 4 ' # # ( N = 14268
Parte II
Ejes, acoplamientos y apoyos
35
# " #"
7
) 4 ,(( 9 + A( 9 : , 1,5
0,125 7 4 + " 2(( · %(( · " 3 # 0 + : > 8 0 : 7 8 0 : > C 0 : /70 C 0 : 7 8 + 2(( · " '(( ·
ny = 7,683 ns = 2,856
ns = 2,837
ns = 2,608
ns = 2,842
ns = 4,018
7 Cfr. SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problemas 18-7 a 18-11, p. 817, McGraw-Hill, Mexico, 1990
36
(
$ % ( (
ka = 0,797 kb = 0,872 Ktx = 1,59 q = 0,84 ke = 0,669 Se = 23,433 7#
$ 5') " ) ) + " % ( 5 σx = 8,149 7# τ = 2,037 7# σeq = 9,111 7# ny = 7,683
$ ( # " " ! # α = 41,659 ( τm = 0,237 7# τa = 4,047 7# < % ( ) ? '# = ns = 2,856
$ ( # " " ! # α = 40,249 ( τm = 0,336 7# τa = 4,018 7# < % ( ) , ( '# =
ns = 2,837
$ " + 8 )
σm = 3,528 7# σa = 8,149 7# < % ( ) ?
ns = 2,608
37
# " #"
$ + " ( % ( (4 # ,
ns = 2,842
$ % ( (
ka = 0,797 kb = 0,872 Kts = 1,3 q = 0,87 ke = 0,793 Se = 27,777 7#
$ ( # " " ! # α = 13,941 ( τm = 1,905 7# τa = 2,701 7# < % ( ) , ( '# =
ns = 4,018
38
(
8
7 ? D d $ d : " r dR = d − 2r 7 ? " d = 0,75D r = D/20
7/0 5*%( < : Sut = 1226 1
*'2 0 + A( -· %& -· ) ? 2,5 : 7 >
dR = 21,563
r = 1,659
D = 33,174
d = 24,880
$ % #6 " % ) % & )
ka = 0,685 D/dR = 1,538 r/dR = 0,077 Ktx = 2,05
$ 5') " ) " " + 8 % ) dR
713014,145 229183,118 σx = τ = 3 3 dR 713014,145 σa = d3R
dR 396956,805 τ= d3R
8 Cfr. SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 18-12, p. 817, McGraw-Hill, Mexico, 1990
39
# " #" B !#) * r # q " dR # ke " kb ( Se ( " )
? dR 6 r = 1
dR = 13
$ % ( ( " ) ?
% #
kb = 0,941 q = 0,88 ke = 0,520 Se = 207,112 dR = 21,117
# # ) # + dR
dR = 21,117
r = 1,624
kb = 0,891 q = 0,90 ke = 0,514 Se = 193,844 dR = 21,548
# # ) # + dR
dR = 21,548
r = 1,658
kb = 0,889 q = 0,90 ke = 0,514 Se = 193,409 dR = 21,563
# # ) # + dR
dR = 21,563
r = 1,659
kb = 0,889 q = 0,90 ke = 0,514 Se = 193,409 dR = 21,563
$
dR = 21,563
r = 1,659 D = 33,174
d = 24,880
40
(
)
,&( ,( 9F ,5(( &( 0 .
,&( / ,((
,& ,(
, 7 /;7; ,(&( C # 0 + 0 " : " 4 > 0 " : " 4 > 0 " : " 4 7
ny = 2,702 ns = 1,979 ns = 1,964 ns = 1,766
- + 5 " 4 , 0/:/ % Sy = 580 Sut = 690
41
# " #" $ # " % & (
T = 79,577 >· F = 530,516 >
$ ) #" & " 5 '
RA = 353,678 > M = 17,684 >·
$ ( 5 " " # # % + ,) 01 ( * ' 5 ,) 21 ) * " # ) % & $ 5 " 5') " ) ' + " # " + ) 0 M1 = 17,684 >· T1 = 79,577 >· σx1 = 53,371 τ1 = 120,053 σmax1 = 214,678 σm1 = 207,938 σa1 = 53,371 $ 5 " 5') " ) ' + " # " + ) 2 M2 = 8,842 >· T2 = 0 σx2 = 90,064 τ2 = 0 σmax2 = 90,064 σm2 = 0 σa2 = 90,064 , ) % + # + 5 " % ( ny = 2,702 ) 0
$ % ( ) 0
$ % ( ) 2
ka1 = 0,798 kb1 = 0,926 Se1 = 256,977
ka2 = 0,798 kb2 = 0,97 Ktx2 = 1,68 q = 0,75 ke2 = 0,662 Se2 = 178,202
42
( , + # ) ? % " % ( # % ( " ns1 = 2,364 ns2 = 1,979 ns = 1,979 ) 2
, + = # ) ? " # ns1 = 1,964 ns2 = 1,979 ns = 1,964 ) 0
,( # ! ) , ( ns1 = 1,766 ns2 = 1,979 ns = 1,766 ) 0
43
# " #"
9
) 4 ,((( + 2(( 2((
5,5 -· %(( ! %6 9.
2,75 9- 0 *( ,,
,& 2
,(
1,5 3 C 0 + 0 " : 4 > 0 " : 4 >
ny = 1,087
ns = 1,288
ns = 1,472
$ #"
9 PEDRERO,
RA = RB = 1375
>
J. I., Fundamentos del Diseño por Fatiga, problema 15, p. 59, UNED, Madrid, 1996
44
( $ ( 5 " % & ' " # # % + ,) 01 ) 00
,) 21 ) ) *# ,) .1 ) 3
) $ 5 " % & ' 5') " ) ' + " #
" + ) 0 M1 = 15,125 >· T1 = 212 >· N1 = 0 σx1 = 45,648 τ1 = 319,913 σn1 = 0 σmax1 = 555,983 σm1 = 554,106 σa1 = 45,648 A ) 2
M2 = 550 >· T2 = 212 >· N2 = 49000 > σx2 = 207,491 τ2 = 39,989 σn2 = 69,321 σmax2 = 285,346 σm2 = 97,994 σa2 = 207,491
A ) . σmax3
M3 = 11 >· T3 = 0 N3 = 49000 > σx3 = 112,045 τ3 = 0 σn3 = −623,887 = 735,932 σm3 = −623,887 σa3 = 112,045
, ) % + # + 5 " % ( ny = 1,087 ) .
$ % ( ) 0
ka1 = 0,723 kb1 = 0,926 Ktx1 = 1,75 q = 0,86 ke1 = 0,608 Se1 = 205,156
$ % ( ) 2
$ % ( ) .
ka2 = 0,723 kb2 = 0,856 Se2 = 311,920 ka3 = 0,723 kb3 = 0,97
45
# " #" Ktx3 = 1,58 q = 0,86 ke1 = 0,667 Se3 = 235,758
, + # ) ? % " % ( # % ( " * σm3 #) ns1 = 1,288 ns2 = 1,310 ns3 = 2,104 ns = 1,288 ) 0
, + = # ) ? " # ns1 = 1,478 ns2 = 1,472 ns3 = 2,104 ns = 1,472 ) 2
46
(
10
) 5( 4 ,((( + '(( 4 0 " ? " " 4 + &(( -· *(( -· 0 " *(
, 3 # 0 + 0 " : > " + &( -· @ "
ny = 0,836 ns = 0,248 ny = 0,775 ns = 0,248
$ + ) 5') ) ) + 8 " % ( 5 σx = 636,62 τ = 190,986 σ = 717,434 ny = 0,836
10 PEDRERO,
J. I., Fundamentos del Diseño por Fatiga, problema 14, p. 58, UNED, Madrid, 1996
47
# " #"
$ # " )
σm = 330,797 σa = 636,62 $ + % ( ( # ) #6 & ka = 0,723 kb = 0,896 Ktx = 2,08 q = 0,83 ke = 0,527 Se = 172,063 , + ) ? % % ( " # =
ns = 0,248
$ + ) 5') ) ) + 8 " % ( 5 + # 5 σx = 700,282 τ = 190,986 σ = 774,482
ny = 0,775
$ # " )
σm = 336,868 σa = 636,62 , + ) ? % % ( " # =
ns = 0,248
48
(
11
) ,&( *( : 6( %& " $ 5(( ,&( 0 " " %(( - *(( 7 : 0,25
pa = 203,046
7 T
= 38,765
>·
$ ( % " # & # ' #) " ) " = )
% & θ1 = 8,13 ( θ2 = 98,13 ( θa = 90 ( a = 250
c = 500
$ % & " & % ) pa '# % (& " ) + pa MN = 1,041 · 10−3 pa >· Mf = 0,056 · 10−3 pa >·
Rf = 1 pa = 203,046 7
$ # %
T = 38,765
>·
11 Cfr. SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 16-5, p. 745, McGraw-Hill, Mexico, 1990
49
# " #"
12
) %((
5((
< " 0 ,5& $ %(( - 4 " 7 5&(
: 0,3 &( # $ 0
P2 = 4022,39
T = 342,799
> P1 = 1280 >
>·
$ # ( = ( & # θ = 218,682 ( $ ) ' ! # ) # " # ) '
) % & P1 = 1280 > P2 = 4022,39 >
$ # %
T = 342,799
>·
12 Cfr. SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 16-10, p. 746, McGraw-Hill, Mexico, 1990
50
(
7 ? : " ,' -· & 9- 7 : 0,2 A
d = 21,351
D = 42,649
< D " d # '# F " T
d (D − d) = 454,728 d D2 − d2 = 29102,618
, ( '#) # + " # + d
2
3
D + d = 64
d = 21,351 d = 10,649
+ D + d = 64
" D ! ( +
" d " D D = 42,649
51
# " #"
7 ? " ,' -· " : 5&( 9 7 :
: 0,2 ? # 0 $
d = 74,134
F = 1,58
D = 128,405
7>
'#) # % ( % # D2 " # + ) + d ! d = 74,134
$ D # + d
$ % &
D = 128,405
F = 1,58
7>
52
(
) *( *((
4 %(( "< / ,((
0
0 ,(( 12,5 9
" , $
" 4 P = 0,3ω (1 − 0,2 · 10−2 ω) P " 9F ω G 0 ,( G / &(
5(
5 3 : 0,2 :
,(( 9
" & 9- 7 " : C > ;
d = 559,557 d = 28,433
D = 616,443
D = 1147,557
= +
ns = 2,431 σm = 0,011σa2 − 2,085σa + 83,557
'#
53
# " #"
$ '#) # # # % ) + ( " + ' ! # T = 300 (1 − 0,002ω) >· T = 294 >· $ '#) # % + D + d " % & d (D − d) D + d = 1176
dD − d2 = 31830,989
2 + ( ( d = 559,557
D = 616,443
" = d = 28,433
D = 1147,557
$ # + ) + ( ! ' # ( " + # # + ( ω = 250 C T = 150 >· '# % & ( " % & % ( # # ' % ) # ( '# >· Fe = 10T > Fω = 3125 − 20,833T + 0,035T 2 > $ ) % & # # % & 5 # # % & % ( " 5 # # % & ( # % ) T Nm = 5000 > Mm = 78,125 − 0,521T + 0,85 · 10−3 T 2 >· Ma = 0,25T >· $ # ) "
" σnm = −15,915 σxm = 99,472 − 0,663T + 1,082 · 10−3 T 2 σxa = 0,318T σm = 83,557 − 0,663T + 1,082 · 10−3 T 2 σa = 0,318T
54
( T '# " # ) ( σm = 0,011σa2 − 2,085σa + 83,557
$ % ( (
ka = 0,922 kb = 0,896 Ktx = 1,67 q = 0,73 ke = 0,672 Se = 111,918
, '# σm " σa # T # + # )
# # % ( ) ?
" # ns ns = 2,431
55
# " #"
13
7 % 9- " N10 ,5(( " '(( $ " N10 *2(( &(( ! 7 " * 2 9- :
FR = 2,895
R = 0,97
7>
$ # ( %
$ ) N10 ! ##
N10 = 2715,021 !
FR = 2,895
7>
6 # ) 02// ! + N10 = 2715,021 ! R = 0,97
13 Cfr. SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problemas 11-2 y 11-3, p. 537, McGraw-Hill, Mexico, 1990
56
(
14
) ,2(( : 6' B !
N10 = 3373,509
!
$ ) N10 # # ) %) # R = 0,96
N10 = 3373,509
!
14 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 11-4, p. 537, McGraw-Hill, Mexico, 1990
57
# " #"
15
7 (5 2 9- % 9- $ N10 &(((
6(( 7
d = 80
D = 140
C = 70,2 7>
$ % ) + L10 " ) Fa /Fr
V =1 L10 = 270 Fa /Fr = 0,5
& # #' ) % & + Fe = V Fr " # ( Fe = 8 7> C = 51,706 7> , # (
# 51,706 7> " # + & !#) d = 65
D = 120
C0 = 34 7> C = 55,9 7> Fa /C0 = 0,118 e ≈ 0,309 < 0,5 X = 0,56 Y ≈ 1,426 0,5 > '(0,309, 0,309) + $ % & + # " # ( Fe = 10,184 7> C = 65,822 7> 15 Cfr. SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 11-6, p. 538, McGraw-Hill, Mexico, 1990
58
( , # (
# 65,822 7> % & + " # ( " # = d = 75
D = 130
C0 = 40,5 7> C = 66,3 7> Fa /C0 = 0,099 e ≈ 0,296 < 0,5 X = 0,56 Y ≈ 1,486 Fe = 10,424 7> C = 67,373 7> > 66,3 7> + ( d = 80
D = 140
C0 = 45 7> C = 70,2 7> Fa /C0 = 0,089 e ≈ 0,289 < 0,5 X = 0,56 Y ≈ 1,524 Fe = 10,576 7> C = 68,356 7> < 70,2 7> +
59
# " #"
16
0 " *( (5 " 5(( ((( " ,2 9- *( 9-
L = 266,551
7
< # (
C = 20,3
$ ) # ( 03 7>
L = 1,434
7>
$ ) " # ( #+ L = 1,234 C ∗ = 19,307 7> $ ) # + ( # ( L = 266,551 7
16 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 11-20, p. 541, McGraw-Hill, Mexico, 1990
60
(
17
)
,5 9- 7 " %((( " A&( 3
C = 57
7>
$ ) $ # (
L = 180
C = 56,985 ≈ 57
7>
17 SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 11-8, p. 538, McGraw-Hill, Mexico, 1990
61
# " #"
18
)
&( && & 7/0 *( && ◦ 3 5(( 9 $
%5 µ " %2 "G 10 9- 3 " * ◦ 3 +
∆T = 68
◦
$ Qs = 3726,828
3 C h0 = 5,04 µ
# #
P = 4 r = 595,238 c 1 l = d 2 µ < µ (T1 ) = 33 ·
, # + + * % .. ·
( + # # ) "
# D # + " + " (6 " -" + ε " rc f D # " # ! + #' # + # 1 µ = 17,5 · Tm = 70 ◦ $ ∆T = 30 ◦ $
S = 0,074 r ε = 0,82 f = 2,6 c ∆T = 239,823 ◦ $ > 30 ◦ $
18 Cfr. SHIGLEY, J. E., MISCHKE, C. R., Diseño en Ingeniería Mecánica, problema 12-23, p. 594, McGraw-Hill, Mexico, 1990
62
( A + ! # ! " &
" # ,( 1 µ = 10 · Tm = 86 ◦ $ ∆T = 62 ◦ $
S = 0,043 r ε = 0,87 f = 1,8 c ∆T = 90,715 ◦ $ > 62 ◦ $
A + ! 4 1 µ = 5 · Tm = 112 ◦ $ ∆T = 114 ◦ $ A + ! " # µ = 10 ·
# 90,715 ◦ $ + # # " $ 1 µ = 7 · Tm = 98 ◦ $ ∆T = 86 ◦ $
S = 0,030 r ε = 0,89 f = 1,5 c ∆T = 51,489 ◦ $ < 86 ◦ $
A + ! " E 1 µ = 9 · Tm = 89 ◦ $ ∆T
= 68 ◦ $ S = 0,038 r ε = 0,88 f = 1,6 c ∆T = 70,422 ◦ $ < 68 + 3 ◦ $
# + # + S = 0,038
# # # # Qs = 3726,828
3 C h0 = 5,04 µ
63
# " #"
19
) &(
&& &
7/0 %( %( ◦ 3 5&( 9 7 %( µ ,5 9- " %2 "G 3 : + $
f = 0,003 Qs = 3428,325
3 C pa = 21,12 271,434 F Qs " H #
f = 0,005 Q = 10800
3 C p = 11,163 452,389 F #)
h0 = 5 µ H =
h0 = 16,8 µ H =
# #
P = 4,8 r = 625 c 1 l = d 2
, # + + ( + # # ) " # D # + " + " (6 " -" + 19 PEDRERO,
J. I., Fundamentos del Diseño por Fatiga, problema 10, p. 57, UNED, Madrid, 1996
64
( ε " rc f D # " # ! + #' # + # 1 µ = 10 · Tm = 93 ◦ $ ∆T = 106 ◦ $ S = 0,039 r ε = 0,88 f = 1,8 c ∆T = 93,668 ◦ $ < 106 ◦ $
A + ! " # ! " & " # " ,( 1 µ = 11 · Tm = 90 ◦ $ ∆T = 100 ◦ $
S = 0,043 r ε = 0,87 f = 1,9 c ∆T = 110,353 ◦ $ > 100 ◦ $
A + ! # ! " & " # 1 µ = 10,5 · Tm = 91 ◦ $ ∆T = 102 ◦ $ S = 0,041 r ε = 0,875 f = 1,85 c ∆T = 101,827 ◦ $ ≈ 102 ◦ $
# + # + S = 0,041 # # # # r f = 1,85 f = 0,003 c Qs = 3428,325
3 C P = 0,23 pa = 21,12 p ε = 0,875 h0 = 5 µ H = 271,434 F
# #
P = 4,8 r = 625 c l =1 d
, # + + ( + # # ) " # D #
65
# " #" + " + " (6 " -" + Q rc f rcnl " QQ D # " # ! + #' # + # 1 µ = 10 · Tm = 93 ◦ $ ∆T = 106 ◦ $ s
S = 0,039 r Q Qs f = 1,4 = 4,8 9 /3; c rcnl Q ◦ ∆T = 20,386 $ < 106 ◦ $
A + ! " # ! " & " # " ,( 1 µ = 20 · Tm = 73 ◦ $ ∆T = 66 ◦ $
S = 0,078 r Q Qs f = 2,14 = 4,62 9 /G; c rcnl Q ◦ ∆T = 29,208 $ < 66 ◦ $
A + ! " 4 1 µ = 30 · Tm = 64 ◦ $ ∆T = 48 ◦ $
S = 0,117 r Q Qs f = 3 = 4,5 9 /;3 c rcnl Q ◦ ∆T = 40,242 $ < 48 ◦ $
A + ! " " #' $ 1 µ = 34 · Tm = 61 ◦ $ ∆T = 42 ◦ $
S = 0,133 r Q Qs f = 3,1 = 4,5 9 /;G: c rcnl Q ◦ ∆T = 41,892 $ ≈ 42 ◦ $
# + # + S = 0,133 # # # # r f = 3,1 f = 0,005 c Q = 4,5 Q = 10800
3 C rcnl P = 0,43 p = 11,163
p
h0 /c = 0,42 h0 = 16,8 µ H = 452,389 F
66
(
20
) && &( ,& 9- 5( "G 5& µ 0 & %(( 9 &6 ◦ 3 / : 75,4 F ! :
+ 7 7/0
f = 0,0016 Ts = 120 pa = 30,4
◦
$ Qs = 1630,537
3 C h0 = 2,5 µ
, ./
# #
P = 6 r = 1000 c 1 l = d 2
$ 6 %) =( #
# # # %) ! ( # # = %( )
f = 0,0016
20 PEDRERO,
J. I., Fundamentos del Diseño por Fatiga, problema 11, p. 57, UNED, Madrid, 1996
67
# " #" $ +
r cf
" (
r f = 1,6 c S = 0,03
# '#) S + + "
5 µ = 9 · Qs = 1630,537
3 C $ + S ) ' # " # # # " #) ' ε = 0,9 ∆T = 61,038 ◦ $ ≈ 61 ◦ $ Ts = 120 ◦ $ h0 = 2,5 µ P = 0,2 pa = 30,4 p
$ # # )
Tm = 89,5
◦
$
, ( + # # # " + " " + # # = , ./
Parte III
Transmisiones por engranajes
71
# (
7 ? ? ,6 '*
αn = 25 , 1,25 0,3 $
! 2,5 H0 ? "
,(* " I 0 " < : ?
' # #)
) 1
x1 = 0,2 x2 = 0
"
)
Cnom = 102,5
$ & # + !( # ra1 = 26,75
ra2 = 81,75
$ # ) # & #) 4x π + tg αn + 2(tg αt − αt ) − 2 Z Z
ra2 − 1 − arctg rb2
ra2 −1 rb2
>
0,3mn ra
72
( # #) *0,02008 < 0,028037 > # *0,011488 > 0,00917 $ 4 # ' # )
#) Z ≥ 12,0576 > ' # ) #) $ 6 # & #) " # ) % xT = 0,2
# # & # *6 # &
" + & #) " ra1 = 26,75
ra2 = 81,25
$ # + ) # #) ! & # 6 # & ! # & " # ' # > # #) *0,04059 > 0,028037 $ #6 ++ # #) " #
re =
r02 +
rp − r0 tg αt
2
r0 = rp − mn b + mn rf (1 − sen αt ) + mn x re1 = 22,064 re2 = 76,277
$ 6 & ) ( #) "
2 rf2 in = ra + C 2 − 2C rb cos αt + sen αt ra 2 − rb 2
rf in1 = 22,2565 rf in2 = 76,8911
73
# ( $ # ) % #) " *rf in > re > % #) *rf in1 > re1 > % *rf in2 > re2
74
(
) ,( 9F ,5(( < $ ? 5& %( < 5(
0 ? ,6 %& 8 %
5(o 3 # 0 " 0 ? 0 ?
αt = 22,296
εα = 1,847
εα = 1,170
( n2 = 750 #
$ # " ( # Cnom1 = 130
Cnom2 = 128
# ( # #
75
# ( $ # + #) " #+ " ( #) % % rp1 = 38
rp2 = 90
rb1 = 35,708
rb2 = 84,572
αt = 22,296 ( $ +
n2 = 750
#
$ & + !( # ra1 = 44
ra2 = 96
$ ( % ( #
2 2 ra1 ra2 1 εα = − 1 + Z2 − 1 − ZT tg αt Z1 2π rb1 rb2 εα = 1,847
$ &
ra1 = 42 ra2 = 94
$ ( % ( #
εα = 1,170
76
(
! ? ? 5( '( $
& 5(o 1,25 " < 5&o 0 0,8 " "
? 7 ;7= ' D,&'( 7 " %( o 3 2( 7 $ '( 9F 5((( ! ? " : 3 1,243 3 ! ? 3 4 ?
x1 = 0,25 x2 = −0,25
C = 221,748
h = 0,8798
rus1 = 59,3507
rui1 = 54,2158
rus2 = 167,6529
rui2 = 163,1516
$ 4 + #)
Zv1 = 26,866
77
# ( < 6 # & & # 6 % ) 4 + #) " )
)
x1 = 0,25 x2 = −0,25
$ # + #) " " #+ rp1 = 55,1689
rp2 = 165,5067
rb1 = 51,1948
rb2 = 153,5846
$ & #) "
ra1 = 61,4189 ra2 = 169,2567
$ ( #) % % # ' #) ( % αt = 22,56 ( # ( #) % % C = 221,748
$ !( # )
h = 2C(tg αt − αt ) − 2C(tg αt − αt ) −
4C xT tg αn ZT h = 0,8798
$ 6 & ) ( #) "
rf in1 = 53,0596 rf in2 = 161,8757
$ 6 & ) ( 4 # #) " rus
2
2 2π rf in = rb − 1 + 1 + Z rb
78
( rus1 = 59,3507 rus2 = 167,6529
$ 4 % #) " rui
2
2 ra 2π = rb −1− +1 rb Z
rui1 = 54,2158 rui2 = 163,1516
79
# (
) < "< %#, ? " 82,5 4 ! J ? J 4 ? -
; 5(o J <
J 1,25 0,25 /
< " ±40o 7
< J4 J ! <
! : 4 K 3 " ? " ;
: "
?
80
(
D Z1 = 16 Z2 = 64
mn = 2 β = 14 ( αt = 20,562 ( αt = 20,656
x1 = 0,38 x2 = −0,38
xT = 0,025 ⇒ x1 = 0,405 x2 = −0,38
(
>
# 4 ' # # 4 # #)
Z1 = 16
# ) " ( != # # ) 82,5
mn = 2 β = 14 ( $ ( #) #
$ 4 + #)
αt = 20,562 αt = 20,656
Zv1 = 17,515
( (
(6 # ) # & # 6 * A % 21 #( .. ' 6 # & #) " x1 = 0,38 x2 = −0,38
21 LAFONT,
P., Cálculo de engranajes paralelos, E.T.S.I.I., Madrid, 1995
81
# (
# '#) !( # # & # = xT = 0,025
# # &
x1 = 0,405 x2 = −0,38
82
(
) " < *((( " 4 ' 9F 0,75 0
" %#, $ ? ,2 %&
5(o * *( $ 5( *5
< ,&o %( 8 D,&,'
;7= & # 7 ; ;; ! 3 : < &(
0,0001 7 " &(o &( 7 " , -G fsho 0,5 3 <
mn2 = 3,5
αt = 21,108
SH(0,01 %) = 2,293
( εγ = 2,406
83
# (
SF (0,01 %) = 4,088
$ # #
Cnom = 94,5
$ ) ( ( # #'
# # # " & ) #)' mn2 = 3,511 ⇒ mn2 = 3,5
$ ( #) ) % $ ( #) % %
αt = 20,647
(
αt = 21,108
(
# % ( = = ( != βb = 14,076 ( $ " & #) " rb1 = 33,907
rb2 = 54,252
ra1 = 39,735
ra2 = 61,475
$ ( ) % ( # + #
2 2 ra1 ra2 1 Z1 εα = − 1 + Z2 − 1 − ZT tg αt 2π rb1 rb2
εα = 1,464
$ ( εβ =
b sen β πmn
84
( b ! ( ( # ( H/
εβ = 0,942
$ (
εγ = 2,406
6 ( % % # # + #
SH =
2
σHP σH
σHP # #) B& 5 # σHP = σHlim ZN ZL ZR ZV ZW ZX
" σH #) B& 5 # σH =
KA KV KHβ KHα σHo
σHo #) B& # * A % #( 30
σHo = ZH ZE Zε Zβ
Ft u + 1 2rp1 b u
ZH % ( = ZE % Zε % ) " Zβ % ) $ % ( = ZH #
ZH =
2 cos βb cos αt cos2 αt sen αt ZH = 2,396
$ % ZE # 1 ZE = 1−ν 2 π E1 1 +
1−ν22 E2
2 ν1 = ν2 = 0,3 E1 = E2 = 207 · 10 >C
2
ZE = 60,169
>C
2
85
# ( $ % ) Zε # ! 0 < εβ < 1 * A % #( 30
Zε =
4 − εα εβ (1 − εβ ) + 3 εα Zε = 0,832
$ % ) Zβ # '#) # Zβ = √ cos β & # 5 # " & #) B& '# % =
Zβ = 0,983
$ # # # G: @ % ( + #) ( # " # + P = 8 7F n1 = 1200 # d1 = 72,469
$ % & ( #) # + > + # P (7F) 6 Ft ( >) = 106 π n1 (# ) d1 (
) Ft = 175,694 > $ ) ) ( # + #) B&
σHo = 37,003
u2 = 1,6
>C
2
< % # ) KA # ) * ! % " % ( * ( % KA = 1,5
$ % # ( ! εβ < 1 # KV = KV α − εβ (KV α − KV β )
86
( KV α " KV β + % # #+
" ! εβ > 1 % # ! * A % #( 0/; '#)
KV = 1 + K350 f
$ # Z1 v1 100
u2 1 + u2
v1 + ( #) "
% ) = % K350 # " # ! Z1 v1 u2 v1 = 4,567 C = 0,775 100 1 + u2 K = 0,0375 K = 0,027 $ % f * A % #( 0/G % ) # Ft /b·KA " > f = 2,30 f = 2,69 $ % KV α " KV β " % KV α = 1,086 KV β = 1,0783 KV = 1,078
$ % A % #( 0/31
KHβ
# ( *
2Cγ Fβy
Fm ≤ Fβy b
KHβ =
Fm ≥ Fβy b
KHβ = 1 +
Fm b
Cγ Fβy 2 Fbm
(KHβ ≥ 2) (KHβ ≤ 2)
Cγ ( & # + ( 2 * >C
Cµ ) + # Fβy = Fβx − Yβ
Fβx ) " Yβ ) ) # ) Fβx = fma + fsh
87
# ( fma ) # % ) " fsh )
# % ) ( KHβ $ ) # % ) fma # 6 ) ( # % ( 5 fHβ fma = 6,5 µ $ ) # % ) ( fsh Fm = 284,097 > fsho = 0,5 fsh = 3,5512 µ $ ) Fβx
Fβx = 10,0512 µ
$ ) ) # * A % #( 003 Yβ " ) Fβy Yβ = 1,50768 µ Fβy = 8,54352 µ $ ) Fm /b " # ) Fβy Fm /b = 7,1024 Fm /b ≤ Fβy $ 0
KHβ = 2,1935
$ % ) + ( KHα ( % ) 6 ( * A % #( 02/1 εγ ≤ 2 εγ > 2
KHα
εγ = 2
0,90 + 0,40
KHα = 0,90 + 0,40
Cγ (fpb − Yα )
FtH b
2(εγ − 1) Cγ (fpb − Yα ) FtH εγ b
$ # fpb # fpb = fpt cos αt cos βb fpt # * A % 6( ;: #( 022
fpt = T fpt = 0,400ϕp + 5 ϕp = 32,61 fpt = 18,044 fpb = 16,378
88
( $ ) # * A % #( 02. Yα " ) FtH /b Yα = 1,2283 µ FtH /b = 15,505 >C
+ % KHα
εγ = 2,406 > 2 KHα = 1,745
+ #) B& 5
σH = 92,057
>C
2
' % ( #) #6 # * A % #( 0.H σHlim = 163 >C
2 $ % ) ZN # #6 (4 ( # ( " ) # 5 · 107 * A % #( 0.; ZN = 1
$ % + ZL
$ % + ZV
CZL = 0,91 ZL = 0,956
CZV = 0,93 ZV = 0,98
$ % ) & ZW # #) " )
ZW = 1
$ % ZX # ( % " ) 2 " 0/ * A % #( 0H2 ZX = 1
$ % ( ZR
CZR = 0,08 Rtm100 = 3,0571 µ ZR ≈ 1
89
# ( + # #) B&
σHP = 152,711
>C
2
$ 6 ( % % # # #6
SH(1 %) = 2,752
$ 6 ( % % # # #6 # # % ////0 *//0 @ SH(1 %) = SH(0,01 %) KRH(0,01 %)
KRH(0,01 %) % 6 # # % ////0
KRH(0,01 %) = 1,2 SH(0,01 %) = 2,293
$ 6 ( % % # + # SF =
σF P σF
σF ) # σF = KA KV KF β KF α σF o
σF o ) # σF o =
Ft YF s Yε Yβ bmn
YF s % & #6 Yε % )
Yβ % ) " σF P # ) # σF P = σF lim YST YN T YδrelT YRrelT Yx
$ 4 + #) ( # "
= % & YF S * A % #( I; Zv1 = 22,192 YF S = 4,58
90
( $ % ) Yε
εαn = 1,556 Yε = 0,732
$ % ) Yβ
Yβ = 0,882
+ ) σF o
σF o = 3,711
>C
2
$ % ) ( ( KF β * A % #( 02/
N = 0,809 KF β = 1,888
$ % ) + ( KF α * A % #( 02/ " 020
KF α = 1,745
+ )
σF = 19,770
>C
2
' % ( # ) # 0:0; σF lim = 46 >C
2 $ % YST % ) ( " ( # # 4 # ( 2 YST = 2
$ % ) YN T # ) " # + # 3 · 106 * A % #( 0:/ YN T = 1
$ % YδrelT ( 3H * A % #( 0:. % ) % YSa )
# ( & YSa = 1,61 YδrelT = 0,99
91
# ( $ % ( + YRrelT # ( 3: * A % #( 0:H
YRrelT = 1,065
$ % ) YX ( 3; * A % #( 0:H
YX = 1
+ # ) σF P = 97 >C
2 $ 6 ( % % # % (
SF (1 %) = 4,906
$ 6 ( % % # % ( # # % ////0 *//0 @ SF (1 %) = SF (0,01 %) KRH(0,01 %)
KRH(0,01 %) % 6 # # % ////0
KRH(0,01 %) = 1,2 SF (0,01 %) = 4,088
92
(
! ? " * *'% 9F $ D,&'( " %%, 0
5(0 1,25 " 0,25 " 7 : ;7= & : 2 ,( 7 : : $ " '
Z1 = 22 Z2 = 66 mn = 10 d1 = 220 P = 527,423 7F P = 485,018 7F
b = 130
' 0:;/ * A % #( 0.H " 0H3 σHlim = 163 >C
2 σF lim = 50 >C
2 , + 4 # #) ( >
" 4 Z1 = 22 Z2 = 66 $ % + KB * A % #( 0;0 # 6 " 3 0/ ! " ! #
93
# (
" ! % #
2 KRH /ZN = 1 KA = 2 KB = 2
$ % C1 # C1 =
π u n1 (# ) 6 u+1 C1 = 173,18
< % C2 ( 3I * A % #( 0;. # αn = 20o " β = 0 C2 = 0,21
, # + + ( #) ! # Vt = 6,5 C $ % C3 # ! ( I/ * A % #( 0;:
C3 ≈ 0,935
$ % # ! " + % C3 # ! # % C3 # + % ) % Vt Z1 /100 C
Vt Z1 = 1,43 100 = C /1,013 = 0,923
$ % C5
C5 = 7,339
+ % % + C6 # #) " * A % #( 0G2 C6 = 1
$ # ) + % # % ( #6 (4 = #6 ,< KB P (7F) = C1 C2 C3 C4 C5 C6 ,
94
( % + ) C4 # .;H 7F
C4 = 2,955
# + # % + ) C4 + ! ( b # + b/d1 # (6 ( I: * A % #( 0G/ b/d1 = 0,75 ⇒ b = 130
b/d1 = 1 ⇒ b = 160
b/d1 = 1,25 ⇒ b = 190
! 4 ) # # >
# ) &) b/d1 * A % #( 0GH b/d1 ≤ 1,1 , # , ( # # +
b = 130 d1 = 173,333
$ ) #) # # + " &
# *#% mn = 7,878
⇒ mn = 8
$ + ( # ) & ! ( d1 = 176
b/d1 = 0,739
$ + ( # + " + % C3
Vt = 4,064 C C3 = 0,925
$ % ) ( ( KHβ " ) + ( KHα " % C4 KHβ = 1,516 KHα = 1 C4 = 2,656
% KHβ ! '# # ( # ,< : "
95
# ( # & 2/ @ # #6 # # + # # ( + % # % ( #6 Padm = 327,864 7F A # % ⇒ + # ) &
# #
mn = 10
$ + ( # ) &
! ( d1 = 220
b/d1 = 0,591
$ + ( # + " + % C3
Vt = 5,08 C C3 = 0,926
$ % ) ( ( KHβ " ) + ( KHα " % C4 KHβ = 1,474 KHα = 1
C4 = 4,268
+ # # ( + % # % ( #6 # mn = 10
Padm = 527,423 7F A # # ⇒ + $ # ) # = #6 6 ( + & Ft = 1102,3 > 350 >C
KA b , # ) # = #6 +
96
( $ % KBF ,< + % #' H/ @ # + % + % % # % ( #6 KB KBF = 2,8
$ % CB1 CB1 =
π · 10−6 · Z1 · m2n · n1 (# ) 6 CB1 = 0,508
$ ( ) % " = + % CB2 ( I3 * A % #( 03. rb1 = 103,366
rb2 = 310,099
x1 = 0,28 x2 = −0,28 Cnom = 440
ra1 = 122,8
ra2 = 337,2
εα = 1,63
CB2 = 1,41 $ % CB3 # ( II * A % #( 03H
CB3 = 0,965
$ % YF S # ( :H * A % #( I; " = % CB4 + # CB4 = 1/YF S
YF S = 4,37 CB4 = 0,228
$ % CB5 $ % CB6
CB5 = 92,659
CB6 = 100
>C
2
+ % CB7 # #6 # ( 0// * A % #( 03H
CB7 = 0,93
+ # ' + % # % (
97
# ( Padm = 485,018
A # # ⇒ +
98
(
7 '( 9F ,5(( < '(( : 2 $ ? *5 '% 5 & : b/d1 = 0,60 D,,&(
αn = 20o 1,25 " 0,25 " 0
:
: " D,5&5 ;
" 0 ;7= & 3 ? :
H0 : "
" I 0 " :
P = 6,61 7F A # )
P = 64,789 7F P = 74,895 7F +
99
# (
$ % # ) KA # )
H ! * A % #( 0/H " & # !
(= % ! * A % #( 0/: KA = 1,5
$ % & ( #) # + N # # ;/ 7F Ft = 11936,62 > $ ! # ) b/d1 /;/
b = 48
$ # ) # = #6
Ft = 373,02 > 350 b
>C
( & # = #6 # # KA
$ % + KB # 6 ) % # # 3 " 0/ ! 2 KRH /ZN = 1 KA = 1,5 KB = 1,5
$ % C1 $ % C2
C1 = 418,88
C2 = 0,21
$ + ( # + " % C3
$ % C4
Vt = 5,0265 C C3 = 0,899 KHα = 1 KHβ = 1,17
100
( C4 = 0,2625
$ % C5
$ % C6
σHlim = 46 >C
2 C5 = 0,5845
C6 = 1
+ # # & ) + % # % ( #6 Padm = 6,61 7F A # # # )
' 6 ) # + 02:2 σHlim = 136 >C
2 σF lim = 35 >C
2 $ # ) ) ! C = Cnom = 120
' # 4 # ) ## # >
+ # ) ) 2 Z1 = [19 ÷ 29] < # % ) 4 #) # 120
m = 2,5
⇒ Z1 = 32 m = 3
⇒ Z1 = 26,67 m = 4
⇒ Z1 = 20 m = 5
⇒ Z1 = 16 ,) + ) . " H , + + , ( m = 4 Z1 = 20 $ ! ( # = #6 b = 51,15
101
# ( $ % C1 $ % C2
C1 = 418,88
C2 = 0,21
$ + ( # + " % C3
$ % C4
$ % C5 $ % C6
Vt = 5,0265 C C3 = 0,935 KHα = 1 KHβ = 1,4154 C4 = 0,23128
C5 = 5,109
C6 = 1
+ # # & ) + % # % ( #6 Padm = 64,789 7F A # # ⇒ + $ % KBF $ % CB1
KBF = 2,1
CB1 = 0,201
$ ( ) % " = + % CB2 ( I3 * A % #( 03. rb1 = 37,588
rb2 = 75,175
ra1 = 44
ra2 = 84
εα = 1,636
CB2 = 1,41 $ % CB3
CB3 = 0,98
102
( $ % YF S # ( :H * A % #( I; " = % CB4 + # CB4 = 1/YF S
YF S = 4,67 CB4 = 0,214
$ % CB5 $ % CB6
CB5 = 37,8023
CB6 = 70
>C
2
+ % CB7 # #6 # ( 0// * A % #( 03H CB7 = 1
+ # ' + % # % ( Padm = 74,895 7F A # # ⇒ +
103
# (
) " < *((( " 4 ' 9F 0,75 0 " %#,
$ ? ,2 %&
5(o * *( $ 5( *5
< ,&o %( 8 D,&,'
;7= &
" "
%&o3
Ss = 0,577 λ = 0,4087 ⇒
# ( '+ #' H/ @
$ + (4 = # # # !22 1 ν40 =
7000 V 0,5
ν40 + H/0 $ , " V + %C # V = 0,0103 d · n 22 ERRICHELLO,
R., Friction, Lubrication, and Wear of Gears, Lubrication Engineering, 1990.
104
( d #)
" n + ( #) # ν40 = 233 , < = # ( # ( " + = % J # # % # ) # )
( % ) KKA/Vt " + .3o * A % ( 0/G #( 20/ Ft = 175,695 > K = 0,09849 >C
2 Vt = 4,567 C KA = 1,5 KKA = 0,032 ν38 ≈ 230 , VT
<+ # = , ## +
H/o 2..:. , ,< 8? .2/ 6 ( % % # (# + # Ss =
θSint θint
# (# # # # θSint " # ( # θint $ # 5 θMT " + # 5 ! θf la int T # ( 020 * A % #( 2.; # % ) ( ( " (# K? # * A % #( 22; T1 = 63,662 >· θMT = 94,64226 o $ θf la int T = 12,4396 o $ $ (# # # # θSint
θSint = 113,30166
o
$
+ # ( # θint # # 5 θM " + 5 !
105
# ( # θf la int # ( '#)
θint = θM + 1,5θf la int
# 5 θM # # θoil = 45o $ " + 5 ! # θf la int )
θM = (θoil + 0,7θf la int )XS
XS % # # ) 0/ # ) # " 02 # ) # ")
# ) # ) , # # 2 · 106 ≤ d(
) · n2 (# ) ≤ 108
" # ") % +
d1 · n21 = 104,355 · 106 d2 · n22 = 65,22 · 106
# # )
XS = 1,0
$ + # 5 ! θf la int + # '#) # 3/4
θf la int = µβ XM XBE
1/2
WtB Vt C 1/4
Xε Xca XQ
C
" Vt + ( C = 94,5
Vt = 4,567 C WtB
( >C
# '#)
WtB =
Ft KA KHβ KHα KBγ b
Ft = 1756,94 > b = 40
KA = 1,5 KHβ = 2,1935 KHα = 1,745 " % KBγ ( 003 * A % #( 2.H 2 < εγ ≤ 3,5 % KBγ = 1 + 0,2 (εγ − 2)(5 − εγ )
106
( εγ = 2,406 KBγ = 1,205 WtB = 303,8843 >C
% # 5 XM = 50 >−3/4 1/2 −1/2
$ 6 %) µβ " '#) # µβ = 0,045
KA Fbbt −0,05 (KHβ KHα )0,2 ηM XR 2vρ
$ % & + Fbt > # Fbt (>) =
60 6 P (7F) 10 π n1 (# ) db1 (
) db1 = 67,814
Fbt = 1877,5 >
$ + 2v ( Vt sen αt
2v = 1,6447
C
$ + #6 # '#
# '#)
u + 1 1 cos βb cos αt 1 = ρ u rp1 sen αt cos αt ρ = 8,304
$ + # 5 θM '# · # # ! + & # # # 5 + " 6 ! # " ( , # + θM ( ;: o $
107
# ( , ! # # + # #
& # ) ( & ) 8(1 b t + θ µ() = µ( ·) = k · e
k b " θ # # ( # k b " θ # # & # ) # µ # t ( ( ) ,< ,< 8? .2 ,< 8? H; ,< 8? ;3 ,< 8? 0// ,< 8? 0:/ ,< 8? 22/ ,< 8? .2/ ,< 8? H;/ ,< 8? ;3/ ,< 8? 0/// ,< 8? 0://
k
/0./I;0 /00H:3. /0/II:I //GI;2: /00IIH; /0/322: /0G0I/. /2:00IG /.30G33 /..;3:. /0G;..0
b
:3/.3./2; ;;I..:GHI G:/H2/;0; I0:HH2I.I 30:0//303 3332H;:0: 32:./I.:3 G:/2GH3.: ;G/0;H;/: GI.H32:;G I302H33G0
θ
G0.0I;02 GG0../00 32H/2:.. IHG3HGH. GI.H03./ 30G.33H3 GH.:H:/0 ;.3;.I2H :H0I:3H. ;...:2H; G0H2HI::
$ + # ,< 8? .2/ ;: o $
ηM = µ = 64,17
·
$ % 5 (
XR = 3,8
Ra (µ ) d1 (
)
1/4
Ra ( = 5 Ra =
Ra1 + Ra2 2
Ra1 " Ra2 ( = #) " # + # ( Rtm + #& ( = * ( $A +
# #' # " + #' Ra =
Rtm 6
108
( Rtm1 = Rtm2 = 3 µ Ra1 = Ra2 = 0,5 µ Ra = 0,5 µ XR = 1,09518
$ 6 %)
µβ = 0,2698
$ % ( = XBE % # + 5 6 ( , + XBE = 0,5(u + 1)1/2
1/2
ρE1 − (ρE2 /u)1/2 (ρE1 · ρE2 )1/4
1 d2a1 − d2b1 2 = C sen αt − ρE1
ρE1 = ρE2
ρE1 = 20,7163
ρE2 = 13,3157
XBE = 0,3297
$ % XQ # ) ' 6 ( ε1 " ε2 " #) A 6 ( # ( '# 1
2
ra1 1 − 1 − Z1 tg αt Z1 2π rb1
2 ra2 1 − 1 − Z2 tg αt ε2 = Z2 2π rb2
ε1 =
ε1 = 0,71607 ε2 = 0,7481 XQ = 1,0
$ % XCA # # Ca & 5 * A % #( 2.: XCA = 1 + 1,55 · 10−2 ε4max Ca
# % ( ' ( " # & " #
109
# ( # = ) 5 ! # # Ca # " 6 ( ε1 " ε2 Ca = Ca2
$ # %+ + + Ca2 " + # Cef f =
KA Fbbt Cγ
Cγ 6 ( & ( ( 2 >C*
·µ
Fbt2 = 187,75 > Ca2 = Cef f = 3,52 µ εmax = ε2 9 /GH30 XCA = 1,017
$ % Xε # ( % " 6 ( ε1 " ε2 # ( 02/ * A % #( 2.: εα = 1,464 ε1 = 0,71607 Xε = 0,30
$ + 5 ! # " # 5 θf la int = 65,45 o $ θM = 90,815 o $ 8+ 6 & µβ # 5 ! # θf la int " + # 5 θM ! ( +( θM = 90,815o$ ⇒ ηM = 25,429 · ⇒ µβ = 0,2826 ⇒ θf la int = o 68,556 $ ⇒ θM = 92,989o $ θM = 92,989o$ ⇒ ηM = 23,831 · ⇒ µβ = 0,2835 ⇒ θf la int = o 68,775 $ ⇒ θM = 93,142o $ θM = 93,142o$ ⇒ ηM = 23,724 · ⇒ µβ = 0,2836 ⇒ θf la int = o 68,799 $ ⇒ θM = 93,159o $ θM = 93,159o$ ⇒ ηM = 23,712 · ⇒ µβ = 0,2836 ⇒ θf la int = o 68,799 $ ⇒ θM = 93,159o $ $ # ( # # # 5 I.0:I o $ θint = 196,3575 o $
110
( $ 6 ( % % # (#
Ss = 0,577
# (# ( ( #
+ ( ( '+ 5 ( ! ) L " B(( ## + # # " ) 1 0,3
0,6 ηM Vt 0,7 σH −0,26 u 1 hmin = 1,26 sen αt αE C u + 1 cos βb EC E
+ ηM >·C
2 # 5 θM = 93,159 o $ ηM = 29,174 · 10−9 >·C
2 $ + (
C
Vt = 4567
C
$ # E # + # 1 1 = E 2
1 − ν1 1 − ν2 + E1 E2
ν1 = ν2 = 0,3 E1 = E2 = 2,06 · 105 E = 2,26 · 105
>C
2 >C
2
$ #) B& 5 >C
2 + # σH =
KA KV KHβ KHα σHo
σHo #) B& #
σHo = ZH ZE Zε Zβ
Ft u2 + 1 2rp1 b u2 σH = 920,57
>C
2
$ 6 + C#) α
2 C> # '#) F α = (0,6 + 0,965 log10 ηM )10−2
111
# ( α = 1,927 · 10−2
2 C>
# " C = 94,5
# # hmin = 0,222264 µ $ ( #6 σ=
σ12 + σ22
σ1 " σ2 ( #) " # + ( # ( = $A + # 0./ σ1 = 0,3846 µ σ2 = 0,3846 µ σ = 0,5439 µ # # 6 # λ 6 &) # # " ( #6 λ=
hmin σ λ = 0,4087
+ # ( # ( 00. * A % #( 22H #' H/ @
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