渐开线齿轮啮合方程推导

computer001

设一对中心距为110的齿轮 齿数比为6：5  则大齿轮的参数方程为：x=60*(cos(t)+t*sin(t)) y=60*(sin(t)-t*cos(t))   根据啮合原理 利用matlab编程   
+ A9 Q( z2 Z. N& U> syms t q
% n9 c$ Q3 Q: q4 T>> x=60*(cos(t)+t*sin(t))
) R. A  G& |2 a6 Y* m
0 m( b4 p' |1 P! V6 M* U3 V) X6 nx =

60*cos(t) + 60*t*sin(t)
' `6 M" w/ V* O$ g
>> y=60*(sin(t)-t*cos(t))
8 h4 X/ |# O( O$ F3 P+ f
* J0 N( _2 F" r9 {4 w; J, Oy =

60*sin(t) - 60*t*cos(t)

>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)

! d% l6 H  p3 f) c2 l  \' Zx1 =
9 {& {7 L! y7 o: J! H2 R
* B4 Z6 n& ~. I% X" O110*cos(q) - cos((11*q)/6)*(60*cos(t) + 60*t*sin(t)) - sin((11*q)/6)*(60*sin(t) - 60*t*cos(t))

, _( a7 c/ _/ K8 A& U- ^7 P; t$ {>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)

y1 =

110*sin(q) + cos((11*q)/6)*(60*sin(t) - 60*t*cos(t)) - sin((11*q)/6)*(60*cos(t) + 60*t*sin(t))

>> diff(x1,t)

ans =
# I. t$ o1 ?! B4 L0 S/ s/ W* D
- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)
  g9 j$ n6 Z  M! c! _
>> diff(y1,t)

ans =
  A2 C2 S8 W  N% e' J
, g" b9 M6 o5 u  v4 k' u; p( s) V60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)
, _- E% a# z, N/ n+ f5 H: q# _3 ^
7 Z( u- {3 g; ^. \  D0 ^: d>> diff(x1,q)
  A% d; R' ^1 Q! l, `7 T) ~! Q
9 _( p1 ~' H: s$ [/ Uans =

: u) _  ^: ~  a% v, d  N7 N7 O(11*sin((11*q)/6)*(60*cos(t) + 60*t*sin(t)))/6 - (11*cos((11*q)/6)*(60*sin(t) - 60*t*cos(t)))/6 - 110*sin(q)
4 P6 P$ S: Z) Z  W; ]" K
3 S0 E; g' T9 C7 ~! R4 I" L! e>> diff(y1,q)
8 E- E* n: [# Q2 j: o6 V; u5 ~
+ \0 D$ }: n0 t" j. Hans =
! h, \6 k' {8 a% ?
110*cos(q) - (11*cos((11*q)/6)*(60*cos(t) + 60*t*sin(t)))/6 - (11*sin((11*q)/6)*(60*sin(t) - 60*t*cos(t)))/6

>> f1=sym('(110*cos(q) - (11*cos((11*q)/6)*(60*cos(t) + 60*t*sin(t)))/6 - (11*sin((11*q)/6)*(60*sin(t) - 60*t*cos(t)))/6)*(- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t))-((11*sin((11*q)/6)*(60*cos(t) + 60*t*sin(t)))/6 - (11*cos((11*q)/6)*(60*sin(t) - 60*t*cos(t)))/6 - 110*sin(q))*(60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t))')

f1 =

# @* r/ ~7 q: `; O# q2 |9 t(60*t*cos((11*q)/6)*cos(t) + 60*t*sin((11*q)/6)*sin(t))*((11*cos((11*q)/6)*(60*cos(t) + 60*t*sin(t)))/6 - 110*cos(q) + (11*sin((11*q)/6)*(60*sin(t) - 60*t*cos(t)))/6) + (60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t))*(110*sin(q) + (11*cos((11*q)/6)*(60*sin(t) - 60*t*cos(t)))/6 - (11*sin((11*q)/6)*(60*cos(t) + 60*t*sin(t)))/6)

# V7 t$ `( i6 ]* {>> simplify(f1)

ans =

-6600*t*(cos((5*q)/6 - t) - 1)
4 e) x. b: x- q2 {
-6600*t*(cos((5*q)/6 - t) - 1)=0     解得q=6/5*t   代入   x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)      
6 T- K  C- L4 E7 h: v4 I( p. h% N2 U  y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)  
化简后得X1=50*(cos(1.2*t)+1.2*t*sin(1.2*t))    y=50*(sin(1.2*t)-1.2*t*cos(1.2*t))
从方程上可以看出小齿轮的方程仍为渐开线