TP 2 y 3: Diseño de Engranajes Rectos y Helicoidales

c

c c

<

 
   

  

c c c c

c

Ú
   

        


       
 


<


 

 c
c 
c c c  c c  c c c c c c   c c

 c c c cc c 
c£
   
 -cc
 cc


c
  
c
ccc  cc cc‰
 -cc c  
c c ‰
  c c c  c c  c c c cc c
c-c c  
 c 
‰ 


   c   
 
 c  c c  c  c c  c c cc
c-c c  c
c c c  c 
    
   c  c c 

c  c 
c 
c  
c 
c c  c c c c 
 
 -c !c 

c c 
c 
c
c
ccc cc c  cc c
c c 
c -c!c cc  c  c
c c c
c  c c " c  
c 
 
 -c #c c 

c  c c $  c
c  c % c c c c 
 
 c c
c 
c  
cc  c c  -c c cc& cc
cc—®"'cc
c c (  c
-c)ccc 
c& c c c c cc
c  cc c c  c
c cc  cc  
c  c c *c c  c + c c   c c  c 
   
   c c c
c cc 
cc"'c
c  cc cc cc" ",*c
  
 -c c c c c   cc  -cc 
c
c
c  cc c c
c c 

c c - c  
c c c  c . c "/0c
c    c  c  -c
c   c c 
c
 
c
c
  
 c c 1
c 
 c }

   
 ! " #  c c )-c2 c$%
 c

c c

)ccc c  cc!3
c
c  c c
c c+-c c -


!& '

c )cc

) * }

(




+ ,

c

( c c c  cc c c c+c ) cc cc  cc%c * c4 cc c + c5( cc  
cc  c c+c ,c5( cc 
c c+c

c c  c &  c c 5c
c c c c  c 
c   
c c 
c    
c c   c c c
 c c  cc c+ c
c  
c6 
c
cc cc
c$  
c" ,7c cc  
c c$
c
c" ,*c 8 c
ccc" *0c cc
c
c" ,,c cc  
c 
c c+c" ,*c c c cc(
$.  !      
$. c c c  c  c  c c) c c
c c 

c 
 
c
c 
c c c
 c
c  /  /  -cc
c
+ c
c ccc  cc) c c%ccc cc$c
ccc
cccc  c

c c  c +c c ) c c  c c c
c   c 
c  
cc 
 
c c 
c c  cc c c c$c c+cc c-c c !
c &
c  
ccc
c   -c c -


!& '

&
#
 c     ! 0  #   

c c cc c  c  c c   1


 #
  c 2  c c  c c  c   -c  c c c c c c /c

-c #c
c c   c  c 
cc
c c  c c 
1 2
 
 c %c c

5c 
c 
c  -c  c &  c  c 
c c !3
c c c c c  c 
 c c c c(cc c
c c cc -cc c cc c  c c 
c
c cc cc c  c
   cc
c -c c  c 
c  c c c 
c 
 c  c 
c 9 c  
  c 
c c  c  c 
c  cc c+cc
 c 
 cc 
c
-c c c c c c  cc c cc 
c c+c
c (cc c&  cc 
c&  
cc 
c
c+cc-c


c c 
cc c 
&ccc c    cc$c cc-c®c:c c $c ccc9-c®c 0:c c )&  c   c c c 
1 ,
 345+65
! c )&  c6 c 

   1
!   
 c )&  c c c 
    1
!    
 c 8cc c
c ! 7
 89: ;


89: <
 c  )&  c
c c 7
 
! c

©
c c  c c cc 

c
ccc 
c c cc 
cc cc
c  c cc c+-c c )&  c   c c c 
1 ,
 345+65
 c )&  c6 c 

   1
   
 c )&  c c c 
    1
    
 c 8cc c
c   c   89: <
!! c 7
 89: ;
  )&  cc c
c c 7
! 
 c /c c

cc 
c  
c  
c
c c c   c c-c  c 
 c4 c c c -c  c 
 c4 c c ( $. ˆ
!


!


" c +   # c c¦ c ccc cc   c c+-c c ˆ
 *  }

#   #  ! " # 
  " c c c0c c$c c cc
c
c
) 
 #
!# c
c  cc$cc% *c -c c c
c
c cc c-c  c 
 c
c cc c 
 -c c c



0c c

  
  c

c c

c

c

ˆ
"c ˆ ˆ


"c 89: < ˆ
ˆ
:=> <
!"c

c

c

c c

%
 c 
 c c c$ c 
c
c
c  c 
c
  
cc   
-c

c

? 1 @A B
ˆ %    CA %
  CA


c c

c c

c c

c

ˆ %   !   

"c  %

? 1 @A C
ˆ   BA %
  BA


? 1 @D B
ˆ %    CD %
  CD


c

ˆ  ! 

"c % 

ˆ %      

"c % 

? 1 @D C
ˆ   BD %
  BD


  ˆ 

!"c  %

c

7c c


 


   
        
 
  


  9 
 
   
 

  
 


     


  

   

      
    
 
      
 

 
    


  





 

 
 
   
    
 


ó

<


 



 c c cc c  cc c c+c$  c//"c  c c%c; cc  cc 

c%-c c cc& cc$ c Í
E ccc cc 
c& 
c 
c
c c  c c
c c ccc 

 ccc& cc
c F
E -c)c cc
c
c
 cc( cc 
cc c+c ccc
   
  cccc  c c -c c c cc 

c
cc% c cc &c%c 
-c ccc   cc c+ c c( c   cc   cc 
c< cc 
c c( cc 
c cc c   cc c c  -c c  @

345+65 H  345+65c @
 89: G  c c 
c  c c c  c& c c 
c
c  c c  c c  c c®c " c 
   
 -c c c ccc. c"/0c
  
 -cc c 1
c 
 c}

  
 ! "  #  c c )-c2 c$%
 c c c

c 
c
ccc  cccc 
c

c
 
-c c 
c  c
c
c c
c c 
c$ 
c  
c
c c c   c c 
c  
c $  
c c 
c  
c 
  
c c
c c   c c 
c



-c -



!& '

)cc

c

( 89: G c ) * }
 + @

( c c c  c ) cc cc  cc/ 0c * c4 cc c + c5( cc  
cc  c c+c ,c5( cc 
c c+c

,c c

c  c c³c c 
c c  c  c c c  c c
c 
 c c
&c  c $c c cc &
c
c&c
c c
cc ccc
c
cc
c c cc c
-c -



!& '

$. 89: <
#
 c    ! " #   

c c  c c  c  -c 1






 c  2

c c  cc  c  -ccÿ
-c#c
c  c 
c
 
c

cc
c  c  -c 

1 2
 2
! c c 
c  c c 

 c
c 
c c c c c
c c c    ccc
c  cc
c
c c
cc cc  c   -c ! 


 c 
89: G 89: < c 
c6 c
c
c c
c c 
c 
c
 
c
c 
c 
c

ccc6 -c  

 c 
IJ> < IJ> G c ! c c  c c
c   1K


! c 2  c c c
c c c
 c&  c   -c    
! 
! c 1L
 @ c c$c cc
&c c 
) 
 
 c c #c
c  cc$cc/ 0c

-c c c c

c

"c c

3
 c c ccc  c cc  c &6 c 
 c4c c ˆ
 * 
}

#  !#  ! 0 # 
 " c c
c -c c
c c c c -c  c c
&c
c 
 c  c c4 -c c $. (
!


!" c ˆ
!

 # +  c c cc c-c 
 c
c cc c 
 -c

3
 c c

0




  ‰

c

c )c ‰ = > ?c
c c  cc cc   -c c ‰
2 +
2  
4+ c c  0

c     c ˆ
 * 
}
 0
c
#  !#  ! 0 #  
 " c c c  cc cc c-c 
 c
c cc c 
 c
c $c c c&  c
cˆ
!" M ˆ
 " -c

    c c
c  cc c&  cc
c
c
cc cc&  cc c 9  cc  c c   ccc c
-cc

-c c c 
-c &6 c   c 
 &c
c  c   c c 
-c &  c 
&c 
 cc cc 
 c &
cc  cc-c c .

ˆ  . c c c c

c .


 ‰ 89: G ˆ   N 89:  G   ‰  ˆ   N 89:  G

c

c )c c 
c c  c c c c c  c  c c c c 
c  
c c  
cc 
c
cO*OP"#Q-c " O O N
c O  O c c™cc cc  cc   c 
c c c ccc
c c c  
c &
c 
-cc  c"
 5 c c   cc"'-c c c c c  c cc 
  c
c c c
 
 c c c  c‰
4+ c
c cc c cc
c c c  c ccc cc

  -c c
c
c cc  c5 H # -c c !c c  c 
  
  c
c  c c  c  c  c  c 1

  cc
cc$
c c cc 
c  
cc c c cc
cc
c c   --c  c  c c   c
c  c 
9c  c  c  c c  c 5 H # -c
cc
c cc
c
cc  c™ c
 cc
c c c cc  c c "
 #
# c c #c  c 
c 
c 
c
c c  c

c  
cc  
c   c
c  
cO
O
O
 "# -c c   "  # 
"#  c N
   " #  c "

89:  
89:  < #
" c c .

!"  "
" c c .c 
c c c -c c  (c
c 
c  
c  c c 
c 
  c c c cc
cccc 
c
c  cc c  -cc cc c
c c
c  c  c 
c c  c  c 9  c c  c c c  6c } c
   -c !c c & c
9c   c
c  c c c   c c !3
c 
 c  c 
cc 

c c .
}
M }
 c  * 
c c

c

}
! $%
! $%
!"#  c }

c

}








}
"#  c 

"
!!"#  c #  !#

c  }
 M } c c
c c  c  c c 2@$ -c :&
c   c
c  -&cc   cc
 c
-c c

è c 
   c c
c  c   c
c  c c c
 c 
c 
-
c
 
cc c c
c 9c
  c c
  c c 
c-c 
c c
c c  c cc 
 c c 

c43cc cc& c-c c  c
c  cc
-c 
 cc c c
 c
c  cc c


c  
-c –c 
c c  c  
 c
c 
c c  c c c  c 

c c
c  c
cc c  

 cc c& cc
c cc cc c-c
 -cc ccc. c"/0cc-c
 cc0c;2 cccc& cc
c —®c "' c c   c c c  c
c  c c  c c 0T H    
 "  #  -c c  cc–c
c cc c
 c  -c c T @  


c Û
     T @  @ c c   Û 0T U .
c ˆ
89:  G c c !# #   " # ˆ

" c 89:  < c 8c c  c  c  c c .

" c c
c c c c c  
 c c

c
c cc cc& -c c c c

%c c

©
 

c c c
+cc!3
ccc
c 9 cc c!3
cc!3
A2 $ c cc c
cc c 
c 
c
 
cc2@$ cc c
 -cc
c

  c c c
c 
c c c c c &  c c  c c c c 
c  
ccc c$9-c . -c  
c 
c
c c c c  
c c c  c c &  c c  c   c c c 
c 
c  
c
c 0"c

c  0c

c 
    -cc c c c c&  c   c cc
c  cc c c&  c c c c   cc
(c
c c c  cc cc&  cc%0c

cc cc  c&  c c+c
c
c
cc
c6   cc0"c

ccc
c
cc cc c+c
&c cc c c c  -c c c c  c  c
c  c c
c  -c c 

1 2
 2
! c c 
! 


 c 89: G 89: < c c  c c  c c c  

! c 1K
2 2 c  

 345+65 H  345+65 c @
 ! 1K c c
c,c 
cc
c  c c&  c   -c c 
1K @
! 
 c c c #c c$c cc
&-c c 
) 
 
 c c c cc$cc0c

-c c c

c

/c c

3
 c c c c c( c   cc 
c< c c  @

345+65 H !345+65c @
  89: G c c 
c  c c c  c & c c 
c
c  c c  c c  c *
 c 
  
 -c c ccc  c cc  c &6 c 
 c4c c }

 ! " #  c c ˆ
 * 
}

#  !#  ! " # 
!" c c
c -c c
c c c c -c  c c
&c
c 
 c  c c4 -c c ( $. ˆ
!


!


 " c +   # c c cc c-c 
 c
c cc c 
 -c c

3
 c c

0




  ‰

c

c )c ‰ = > ?c
c c  cc cc   -c c ‰
2 +
2  
!4+ c c 
!c 0
  ! c ˆ
 * 
}
 0
c
#  !#  0 #  ! c c ˆ 
" c c 0c c

c  cc cc c-c 
 c
c cc c 
 c
c $c c c&  c
cˆ
 " M ˆ
" -c

    c cc

.

ˆ  . c

c .


 ‰ 89: G ˆ   N 89:  G   ‰  ˆ   N 89:  G

c

c )c c 
c c  c c c c c  c  c c c c 
c  
c c  
cc 
c
c cc c=B>  ?-c c " O O c N
O  O c@cc cc  cc   c 
c c c ccc
c cc

 
c &
c 
-cc  c"
 5 c   cc"'-c c c c  c cc 
  c
c c c
 
 c c c  c‰
!4+ c
c cc c cc
c c c  c ccc cc

  -c c
c
c cc  c5 H # -c !c c  c 
  
  c
c  c c  c  c  c 1

  cc
cc$
c c cc 
c  
cc c c cc
cc
c c   --c  c  c c   c
c  c 
9c  c  c  c c  c 5 H # -c
cc
c cc
c
cc  c™ c
 cc
c c c cc  c c c "
 #
# c c #c  c 
c 
c 
c
c c  c

c  
c c  
c   c
c  
cO
O
O
 "# -c c   " #   N

"# c   "  #  c "

89:  
89:  < # c .
" c c .

 "  "
  " c c c

.c 
c c c -c c  (c
c 
c  
c  c c 
c 
  c c c cc
cccc 
c
c  cc c  -cc cc c
c c
c  c c 
cc c c 9  cc cc c 6c} c
   
  -c !c c & c 
9c   c
c  c c c   c c !3
c 
 c c
cc 

c .
}
M }
 c  * 
c }
! $%
! $%
!"#  c c }
"#  c }

  c   "
!!"#  c }
#  !# c  }
 M } c
c c c  cc2@$ -cc

è c 
   c c ccc. c"/0cc-c
 cc0c;2 cccc& cc
c F
E cc c  c
c cc  cc 0T H    
 " #  -c c  ccCc
c cc c
 c  c c c T @   Û



c   T @  @    c c   Û 0T U .
c ˆ
89:  G c c # #   " #  ˆ

 " c 89:  < c c c 8cc c  c cc.

  " c
c cc cc 
 c  c

c
c cc cc& -c c c c

7c c

) 
cc 
c 
c : c  c 
c  c 
c c 
c6 c 8 c c$cc c
c .$c cc D cc
c

 0c

c / 7c

c 0 //c

c , /c

c / 0c 0c

c "'c +c$  c

<c )&  c   c )&  c6 c )&  c c 4c :  c D cc cc$ cNc

,c /% 0c

c / 0c

c %, 70c

c c$ 
cEc+c .c"/0c %"'c 8c$  c

<c )&  c   c )&  c6 c )&  c c 4c :  c D cc cc$ cNc c c c c c

*7c %" 0c

c %% 0c

c  70c

c c$ 
cEc+c .c"/0c %"'c

*c c

  
  c c cD cc
c  ®cFc®c"Gc cD cc$ ®cHcc®c%"Iccc c c c ˆ
 " c c ˆ
!" c ˆ



89: N 89: F c ˆ
ˆ
89: F :=> N
" c c ˆ
ˆ
:=> F
 " c c c c c

c

c ,c

c

%
 c 
c c

c

c

c

c

c c

? ˆ@
 ˆ@  C@
  C@
ˆ@
" c ? 1 @A B
ˆ %  CA %    ˆ@ 
c

 CA


ˆ %  ˆ@      

" c

%   

   

? 1 @D B
ˆ %  CD %  
  CD


ˆ %   

 !" c

%      

? ˆD
ˆ  BD  CD
  BD
CD  ˆ
 !   c  BD
 !" 5+643VVW56V%34WX%3Vc

? ˆA
 ˆ  BA  CA
  BA
ˆ  CA
   c  BA
 " c

!cc6 c
&c
cc c
-cc cc2-c c

"c c

À
 c


 J

  J 

 



  
  

J 


     J  
 c c c

 
 

c c



8   J 


 

8 
 
 

  Jó

 J


8 
 
 




 c
c

c



c c

 c c c c

c

 c
c



c

   

    



/c c

    

    



0c c

    

c c