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))

, |, f/ Q1 z+ `, i4 \- \! v9 Vx =
' d! o3 ^0 z- V; C
60*cos(t) + 60*t*sin(t)

7 j0 O+ ^5 G8 r  c2 t>> y=60*(sin(t)-t*cos(t))
' o! _8 m" d" y
4 x* X3 A3 @$ Y; }1 ?1 {y =

60*sin(t) - 60*t*cos(t)
' w) R  U7 A0 _4 i
6 }$ q) r! X5 z) p5 ^( i6 o>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)

1 |4 x4 v( e+ s6 }. |x1 =

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))
1 g+ D+ \6 s! w# S; I! o/ G# Q
9 g; M  M+ s2 Y- N( L>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)

y1 =
- z7 w2 }8 c, n( ]: S7 `
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)

0 F! T- y% B; G/ tans =

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

>> diff(y1,t)
& j: b! g, i  f3 q
, r" ~# f3 ^- i! U5 rans =

60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)
3 n9 Q0 V; P% y6 [, d3 l0 x, H
>> diff(x1,q)

ans =
6 ]0 F3 F- A5 g
. J, K8 f8 q+ ^8 P. A$ F9 d, g0 p(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)
& O2 _" \5 o- c
>> diff(y1,q)
2 V/ C4 x+ N! t
ans =

) C, r: w: Z: I: n, a# k) Y110*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

( H2 ~7 G! {& p>> 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))')
- J, h, U# V5 F, ?( B9 \
f1 =
" x  L7 Y7 o4 y; i
(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)

0 b- n- M' G; O>> simplify(f1)
1 Y/ _4 p# l- e4 e9 W9 I. I. q* N
ans =
5 [* v2 |) d. E' F
- z5 M* P* d3 n1 w-6600*t*(cos((5*q)/6 - t) - 1)

& Z2 L$ T9 O0 T  W' {' 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* i7 }3 O; Y  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