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))

7 O* e. O/ y8 e- t6 sx =
/ m$ R% G% m" N1 k0 y+ p; X( m
  f7 D% D2 @* x, j% o; r, x! e60*cos(t) + 60*t*sin(t)

>> y=60*(sin(t)-t*cos(t))

y =
6 |% f" R2 m; R& p- q$ V( ?
60*sin(t) - 60*t*cos(t)

9 `6 O- t7 f2 O, e: j5 `>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)

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))

3 p4 |2 F7 V* \5 O>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)
+ B: P0 J) G7 s
9 k" ?7 z4 S+ ?/ g' ly1 =
1 G( H: a) a! f3 P4 a9 H- n
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))
  l0 I9 m! \% S1 l, T2 `0 ?
>> diff(x1,t)
8 L+ v6 ?/ h1 F/ w' o+ y
" R8 A) d5 S0 ]; L+ V0 Z/ Zans =
4 G1 y* R$ ^5 c& B* I
. [0 s  R) m8 o$ V- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)
' c, Q' \% Q. p6 Z2 n
>> diff(y1,t)

ans =
$ V0 p! `7 s6 f+ L, p4 t( p
, u7 Q$ ]9 x! |* V( ?( e: ?60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)
; o7 e3 d# l7 X( }; K9 T
>> diff(x1,q)

+ C/ O% v1 M! l5 Vans =

9 f: J2 ^+ l8 d% ?) v  @5 F5 |(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)
% g( X9 F4 y* _, Q
$ [) j; D* C5 ~) q* e3 e+ B>> diff(y1,q)
( {2 D8 P/ U( x9 ~- N) M/ @
ans =
3 l6 R3 r) s$ T4 T, b, `; M
& f' g- v; W( p% K) x. w  I; V110*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

. y% d* }7 u1 R/ v' R( i>> 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))')
; y$ Q& z7 g2 Y7 z/ M0 S! k3 t
/ R2 h  C' d' Hf1 =
# L, h, Y) l4 w! I  V) S0 e- |
& {1 w+ T8 ~0 f* r) Z9 F& E(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)

>> simplify(f1)
! I, G. y  f" O$ B+ u
( A7 O' \, s+ f) X% L- K! Zans =

-6600*t*(cos((5*q)/6 - t) - 1)
$ n: a* l+ m: h( ?7 j; T0 g
# K; J# a  C: Q, b' T -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)      
  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