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   
& {' U. g7 R& q& F> syms t q
>> x=60*(cos(t)+t*sin(t))
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x =
4 a+ h: `! ?; i; p. o9 c
60*cos(t) + 60*t*sin(t)

7 |! l" x1 s2 m9 [$ `$ x3 t; D>> y=60*(sin(t)-t*cos(t))

/ W0 ^+ t3 _* @& q1 P+ g7 N6 {y =

2 F! P$ _. y0 K8 ~" P/ [60*sin(t) - 60*t*cos(t)

' |) N- A/ _; y8 `; A>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)

x1 =

$ ^; N' l% y* d) Z; W! ]* f110*cos(q) - cos((11*q)/6)*(60*cos(t) + 60*t*sin(t)) - sin((11*q)/6)*(60*sin(t) - 60*t*cos(t))
& C' l( }0 V4 a5 o; I8 Y
" E& O$ n4 b" ~7 o+ f4 A* l& r>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)

y1 =

5 b+ L: ^% T9 o3 v& G9 ?9 q110*sin(q) + cos((11*q)/6)*(60*sin(t) - 60*t*cos(t)) - sin((11*q)/6)*(60*cos(t) + 60*t*sin(t))
% K. N+ G) G) J. x
>> diff(x1,t)

ans =

' c9 c: m) B0 E9 O: p- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)
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& _8 ^6 f, x% w* c( k9 M5 o5 y>> diff(y1,t)

ans =

0 ?# Q  Y* f% {' o/ q. x: ~4 t60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)

* }& e# ^2 H5 i9 `>> diff(x1,q)
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ans =
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(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)
; q3 U" |4 ^+ g' F. a) c$ t" c
>> diff(y1,q)
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+ ]6 N1 \& _( R. Tans =

% ^9 ?$ e# b) t( Y$ I110*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

' q! l1 A# w' L- J7 a$ w>> 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 =

0 H% k: t# t  @6 h(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 r- y) O* A1 \7 z4 `+ Q9 x* Z5 t7 X>> simplify(f1)

; a: f. S% _, L' uans =
0 K  v- N% O5 |
' Y7 W5 E# p  ~-6600*t*(cos((5*q)/6 - t) - 1)

-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)      
/ A8 [# e' H, a% L/ G  y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)  
- a6 l- m. @3 c; X7 y! M/ E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