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   
> syms t q
>> x=60*(cos(t)+t*sin(t))

x =

! m( o6 U1 |6 R60*cos(t) + 60*t*sin(t)
' q! c& m9 |) R4 o6 o; M+ V
7 b3 |$ p$ ~1 x: V6 `% c! x6 Z* Z>> y=60*(sin(t)-t*cos(t))

5 S. w' \# _; _' Y3 [) [y =
# @' U1 Y- E! x9 A' s; e0 ~( ~
/ |, ]- ^5 o# V; v9 K60*sin(t) - 60*t*cos(t)

1 x# }# K9 X# L8 _>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)
' g0 F8 F0 ]' q$ Q: w
x1 =
1 z) @. a. Q0 ]2 b+ p
9 R, Q( w) s8 S110*cos(q) - cos((11*q)/6)*(60*cos(t) + 60*t*sin(t)) - sin((11*q)/6)*(60*sin(t) - 60*t*cos(t))

>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)

: e/ J$ I5 P" _2 H6 ay1 =
  t0 ?& X+ W5 O/ P. S* F8 h2 Z+ q2 ^
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))
0 ~1 J9 P0 L/ V8 e
>> diff(x1,t)

ans =
) f6 E4 Q- A9 p1 e" N  _7 T2 t$ a) X
/ E6 A" D# s* Z& q5 ~2 T- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)

( h. y3 r/ ?: m2 U0 _>> diff(y1,t)

ans =

" X$ Q- u. y1 o8 W* X! y2 b60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)

+ ^" {, i6 y- k, P4 l>> diff(x1,q)

ans =

(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)
9 L/ J( _1 a2 @, c1 r5 c
1 A- i5 j5 L$ Z) h0 i6 D& n>> diff(y1,q)

$ a3 T' W4 }) x; n4 U5 F5 R8 oans =
- ~7 a7 p  k% ?0 ^/ a' O% z) _: d
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))')

! d0 U1 y) x' Y3 e& N5 Qf1 =

; t! B4 O, j  K" W9 E/ t$ ?" U(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)

/ a! v: c7 a% W  [7 j>> simplify(f1)
- [+ U7 G: ?5 D- G$ w$ p
ans =

, ^2 i# v" u  [+ U+ D0 T  l-6600*t*(cos((5*q)/6 - t) - 1)
2 g. H. A: ]' m
4 `, z' a4 @) W; j+ f -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)      
' f0 K4 ~% F* g  y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)  
  A: D3 H. J+ w- h( n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