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))
, }3 ^+ M( l6 D, e+ a' N. {3 \
x =
% @5 k6 t) y# \; L
; k( }2 Q9 y! ]- ^60*cos(t) + 60*t*sin(t)
) x$ i! {6 q4 J$ ~9 \
>> y=60*(sin(t)-t*cos(t))

6 r, X; h3 H# p  _) My =
8 e( T% t" p  f  D  o  Y
60*sin(t) - 60*t*cos(t)

>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)
) g0 G0 ^( J1 G3 B/ a$ l. V0 d; N
4 s$ G/ |9 l4 ix1 =

$ N# n6 k+ q9 f  r2 X7 L110*cos(q) - cos((11*q)/6)*(60*cos(t) + 60*t*sin(t)) - sin((11*q)/6)*(60*sin(t) - 60*t*cos(t))

* t; O( l$ I2 c2 `( u: s2 D3 D>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)

y1 =
2 O: _! {  h, V9 Q! S$ \# m4 j+ }8 `
# R3 P6 N$ V# z: o! h5 l; {- b1 l110*sin(q) + cos((11*q)/6)*(60*sin(t) - 60*t*cos(t)) - sin((11*q)/6)*(60*cos(t) + 60*t*sin(t))
, [5 L" H* w. I9 A. V
>> diff(x1,t)

ans =
, n" W9 q5 w) o8 O& [
0 j+ Z, V& S- C) o% z+ P2 c- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)
: u! R" a  Z- R) w. D
>> diff(y1,t)
( Z+ J0 m8 W$ W( L; Q5 h
3 Y/ c# t9 j1 e+ k' N, S/ e# v/ [ans =

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

" H1 H7 B; X  ^, q+ ?+ z8 h, Z>> diff(x1,q)

8 {/ d' w! S7 x% L" Z8 R" z0 ians =
! f) a4 p/ \( |" J+ H1 W$ e* A$ L+ o/ [
( [+ S/ H7 s6 j# D: K' c; b(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)

* E' D1 ?, G3 a! S' ^, X& {>> diff(y1,q)
' _, S4 {' h6 ]& k9 f9 X
. D# B+ k) e" ~6 C6 I+ hans =
- e$ {8 @+ Y8 I
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% D( T& Y" t% S( a# D7 `>> 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 =

3 q, M' K  A; O4 @; ~6 c. 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)
9 v" d5 U: X  n; |8 [3 f
>> simplify(f1)

, Q2 |! e5 g  G( P+ P4 jans =

5 S, M: l3 E9 K2 r5 Q5 L4 A$ i-6600*t*(cos((5*q)/6 - t) - 1)

; ~) A( q' d$ f0 n! p! S -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))
  d3 o2 U" P- J  i从方程上可以看出小齿轮的方程仍为渐开线