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
/ k7 P  O2 k9 P: }- V: i>> x=60*(cos(t)+t*sin(t))
* A/ q% C, @9 f$ C0 u% O, g
x =

5 q+ E( Q5 z: j0 v60*cos(t) + 60*t*sin(t)
( u0 B) n8 a7 b4 h
/ d& p: d8 g$ J0 q>> y=60*(sin(t)-t*cos(t))

y =
* R. A) o9 m- e
60*sin(t) - 60*t*cos(t)

( h0 I: Q+ {7 s& k! l, W4 }  k* x. D* O8 a# o>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)
. o' W3 ^! A9 y
0 N, F4 z# m) S1 t9 H, g0 Ox1 =
6 _& H5 b; o: r  q4 e* @
0 a9 t' N' g& o1 _' ~110*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)

( l) U, M( n; |) Xy1 =
5 s" {0 E7 g, [- g* P$ X
- m! R- I- d6 f0 l/ {7 V& J110*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)
3 C1 z! V& W; V8 Y1 @
8 n7 q9 a! V1 [; C- o, H2 hans =

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

! K! [4 n% A- F3 w1 U  l: |4 C>> diff(y1,t)
4 X, k8 [% ?/ T/ q2 K+ W
' o5 Y. D% `3 P( w6 {6 a# }ans =

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

>> diff(x1,q)

ans =

7 {) _; W; F0 z% Z(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)

>> diff(y1,q)
8 @; ^& b% U" p: _) [
% G; J9 v' P4 m: xans =
/ }- N2 s1 k$ ^" s  u' p* [: m
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
& T$ W  v# ^5 i0 Y
4 s6 q  G! U6 @) T- N>> 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))')
* c5 I+ F& i7 k. y3 j8 y" d
f1 =

; I2 _" M' U) \- ]1 O$ y(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)
1 @6 s  y: K0 L0 Y9 T
>> simplify(f1)
5 e0 r, B( l+ y# Q
  H; n; E: G2 L. pans =
: a, e* \+ x' T: o
-6600*t*(cos((5*q)/6 - t) - 1)

-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)      
1 J# j2 q+ V& y+ a  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