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
: z$ X& w5 i/ C) X>> x=60*(cos(t)+t*sin(t))
' g% V; E% @& U% b2 ^7 V
. F$ T5 y7 `6 _8 Y0 o- k* Kx =
2 C* w" v( q: A8 o2 {
60*cos(t) + 60*t*sin(t)
( G% X) _, ]- f, ~& k/ k$ w
>> y=60*(sin(t)-t*cos(t))

! U2 j! H( b# j$ |y =
# C( C& _% o* [" A( e
  ~/ W0 d' M' Y( S) I60*sin(t) - 60*t*cos(t)

; D- C+ \4 N8 a  {: @>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)
+ \" u3 ?& B! f; Z. s( c
5 [1 K3 t' Y- S# \4 I/ bx1 =
" f  W' |" _  f+ }3 l
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))
  f4 u1 q1 }% v. B
>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)
1 E! Z) s+ s$ t/ b% t1 N
/ b5 f/ C# K7 Ey1 =

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))
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>> diff(x1,t)
1 L  k! _- m! i
! r2 B  g9 e' i0 c7 q( oans =
/ G) z) q, d* k* j7 P
+ n7 ^+ k" X) Y& @3 I1 t6 f2 h- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)

* R  K0 s8 Y# G. P, ]! D>> diff(y1,t)
& z. f/ U* a( M
) `( g* c  h4 Oans =

60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)
( l% ^" \5 a* F$ T
5 @; p3 R  U9 Q. K, [5 J. N>> diff(x1,q)
' N2 j: A" k+ I2 O# s9 y0 E1 V
ans =
: R! K; N# i4 \0 g0 Y
(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)
1 J$ _4 r$ T. T8 B
ans =

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
: W' g7 d2 i1 Q
7 j( B6 I2 Q9 t+ t$ _>> 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))')
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f1 =

(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)

% @: Q5 p- M0 a2 U' ]# k' r>> simplify(f1)
4 {6 k0 y' z1 H) \. n* M
ans =

-6600*t*(cos((5*q)/6 - t) - 1)

! e/ }2 P$ Z8 `6 j! w) j* r, f -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