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   
; e2 c5 H* g, V& m' W+ v> syms t q
>> x=60*(cos(t)+t*sin(t))
$ W6 E/ Z5 l! [0 q, t
9 v1 A1 }! q) o9 M. x4 ~4 t% F9 K$ hx =
5 G/ E, b9 c" D3 N- M0 a9 [
60*cos(t) + 60*t*sin(t)

8 H  I! g# e* ], |/ Y>> y=60*(sin(t)-t*cos(t))

y =
  T% T5 K, N& r' X$ a1 I
9 t+ b; @2 @- y( h! q7 P60*sin(t) - 60*t*cos(t)

>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)

1 Z0 c% ^" s7 x  g" z+ o) h/ ~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))

>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)

y1 =

$ j  F* I4 n( C110*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)
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ans =
, ~2 p; O* S7 `8 q
- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)

7 `. g2 N  R" |/ M>> diff(y1,t)
# I3 h' O7 F& j- c1 D7 z6 e$ ]9 a
ans =
; |, C* D3 g1 ^
60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)

2 o8 ]' v* ?+ n9 y: J9 o>> diff(x1,q)
9 B7 I, X$ K* S
ans =
; {; y8 v% j; d  W7 Y
2 |3 N! B5 T$ {) u9 w; [(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)

>> diff(y1,q)
7 L4 B* e0 P/ a* W' X, I- Y1 {
6 W/ H3 D) e5 I; Mans =

! J2 a3 K5 P+ t110*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

>> 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 =
. ]: `4 T3 B& C& @1 z, f# a" A
(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)

/ x7 o- O6 |5 w& c' ]>> simplify(f1)
* b; `8 }) O& z) N) ]% H+ |
ans =
- W6 f- h) }( r/ P( E! R5 t
-6600*t*(cos((5*q)/6 - t) - 1)
8 Q. W5 |% z( B
, p# z7 }% M+ `/ X( y -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)  
1 U0 g- L6 @' x5 i. N( w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