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   
# h4 G$ g6 _1 }$ ^+ u> syms t q
>> x=60*(cos(t)+t*sin(t))
% B6 b9 q  t% I$ ]
x =
& N$ V; M! t5 w) J0 a, i) t9 B
60*cos(t) + 60*t*sin(t)
6 g3 ~# L( O" c) v! t% ~( e
' s8 W1 q0 p" {/ n( ^* |>> y=60*(sin(t)-t*cos(t))
- X# E: u( o8 b/ a' B$ k' v! O% `3 u
, X# t# g" `# x8 _$ Ty =

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

>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)
/ u$ K& N8 c, |- P/ p  D, d
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))

7 }- Q2 \  O8 q' Q3 O>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)

/ {2 Y9 c$ c% l! uy1 =
$ z! n/ K. k& D1 d5 V  f5 Q/ m* l' k
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))
; N* n3 S" ]$ x0 h
# Z; ^3 B, n. l# ?5 [- R5 @$ G>> diff(x1,t)
- ?+ t: p$ G) X( t- l3 T* c
, Q# Y7 z/ D3 r) ?ans =
; h# `% D0 m! r1 `/ n
; M% ?( M8 ?' e- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)
+ x7 @* _. }8 ]2 d
" `" f" f$ o7 d+ T) s! i>> diff(y1,t)
( X9 c* Z2 D3 L7 p) X- _
ans =

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

; W3 L/ i5 y" o! a>> diff(x1,q)

% S  g/ k$ d0 l7 Mans =
( E7 _( Y4 A% v5 b3 w! f' Z$ ~
8 R; J0 j# I! {) M(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)
+ W7 u. u$ A$ x: T) s8 N
>> diff(y1,q)

+ J3 s! ~7 _- mans =
1 S2 o+ R" F# L  e  Y  D1 a& t
5 }- V: v& j9 P. e8 @3 }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
0 B# S/ u( e2 O; m% p- E0 W' d
0 _; m0 z* T% P, H+ ]) X>> 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))')

f1 =
% Z: ^: [* {: X- J- K) K" b
+ @0 V7 Z3 D( F- {+ B7 W(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)

: s  L1 V, Z; ?3 B>> simplify(f1)

ans =
1 @! V$ A2 r4 X5 {
-6600*t*(cos((5*q)/6 - t) - 1)

9 _8 a: C3 v/ a -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)      
6 L6 k' |4 k# l8 @# m+ k2 G  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))
' S( X5 j6 s. [/ J6 U! u5 N( w从方程上可以看出小齿轮的方程仍为渐开线