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   
0 n& M( W, l& P7 p) J8 L0 ^% {" O> syms t q
>> x=60*(cos(t)+t*sin(t))

* K& m# X% z: u! dx =
( ?. J9 f. ?4 b
! z0 f2 S) h( L: p) P60*cos(t) + 60*t*sin(t)
$ _+ t) C' B7 m  ^5 x
>> y=60*(sin(t)-t*cos(t))

y =

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

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

& o% i* \; F( Q/ Yx1 =

% U* |; s$ T3 j" o5 v+ \8 n110*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)

5 k$ e, A% }3 ?4 ]# |y1 =
9 M8 f2 x. d9 v4 u* q  d
8 M; m/ [+ I( k, s  S110*sin(q) + cos((11*q)/6)*(60*sin(t) - 60*t*cos(t)) - sin((11*q)/6)*(60*cos(t) + 60*t*sin(t))

* ~2 |9 g) W! h* E" t  o, w7 m>> diff(x1,t)
2 b) S* u3 h+ ?, Z( w4 s
ans =
. M1 [3 V* _. ?7 L- V5 j7 {- P
- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)

/ c# ]# D; C: m3 k! C" q>> diff(y1,t)

: U+ `& ^& q* t) E& g7 k7 Oans =

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

7 L/ d" ^. K3 e0 z>> diff(x1,q)
) ~! f, `  o& ]% b. O/ G
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)

7 e1 n) j3 R7 ]>> diff(y1,q)

ans =
( i: p: h+ O+ L  U9 b- ]
8 G' y" v/ E% E; F& I. q5 N2 R110*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))')

! x; R: I" }" H) qf1 =
9 O3 d; x* R( M2 n- T8 J
  a8 G+ ~0 W, {3 W! l(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)

6 B2 d1 L8 L! K2 w! F5 X# }, l>> simplify(f1)
) e& k( L# e. }4 S2 P
* ^$ z) x; @( C! Qans =
+ y& C" Z. {* C% Y: y
-6600*t*(cos((5*q)/6 - t) - 1)

: o; L2 e1 q" _! M$ X6 O! R -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)      
  F/ p0 h/ {' a2 D! l  y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)  
5 P$ Z% S2 N+ Y3 L7 A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