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   
3 T4 n  ~4 n8 _# O> syms t q
& u1 r, T( ~5 [8 _" M>> x=60*(cos(t)+t*sin(t))

$ ~- I% z8 s3 H0 ^% Nx =

" {' q& Z. Q6 [, A5 t7 J9 q60*cos(t) + 60*t*sin(t)

>> y=60*(sin(t)-t*cos(t))
' u; {8 \. U" u0 H* O  M5 N+ d
y =
) z* V: u3 g5 o$ q: a
60*sin(t) - 60*t*cos(t)
+ _5 e& |% Y8 P3 N& `( r( B
! Y4 ~, D8 x& ?; d1 d>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)

x1 =
$ ~3 s& |7 Z. v
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))
( [" \! g7 Q9 o
>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)

* a4 q8 |  Q$ U; v8 F; U' }+ Jy1 =

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))

>> diff(x1,t)
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ans =
  \( y3 g0 n! Q' _) `- o! d% L
- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)

  B. P" a/ P, b* i& F>> diff(y1,t)
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ans =
% v0 F0 k) D. h1 O
60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)

>> diff(x1,q)

) e8 R, k: D  o' ^ans =
6 X( S. [; ^0 }9 q6 L+ C8 i
& I& }9 W& v# n/ T. R  @(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)

! Q/ U6 Z6 ~+ p3 `, M  Y) d: n>> diff(y1,q)

& V. \- I  U! y* nans =
1 m3 Q, r* ~7 E, L
( ~) x8 ~9 d; @" w7 g- l110*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))')
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/ t8 j" o6 Y" O5 c6 _- u3 E# u1 jf1 =
* j7 }+ ]3 \% c3 m
. G; a2 ~/ r+ ~; b' \4 ]0 Z(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)

' Z5 [+ c+ L  N5 _/ f3 u>> simplify(f1)

/ G, i6 I8 m1 L6 Z2 K! h( Qans =
) O: a. L; D, M
% \, S! a' s. K- P2 f2 s) c$ R: N4 }-6600*t*(cos((5*q)/6 - t) - 1)

* B: _, t) R8 D1 n -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)  
# W% ]& o' r( d" b' ^* A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/ H; ]% Q) o0 Q, }从方程上可以看出小齿轮的方程仍为渐开线