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   
0 F  W/ G# _7 S4 ]( \9 \> syms t q
>> x=60*(cos(t)+t*sin(t))
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x =
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* C( o! {& _" A! E* ^* y60*cos(t) + 60*t*sin(t)

>> y=60*(sin(t)-t*cos(t))

y =
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60*sin(t) - 60*t*cos(t)
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3 }* D  |! y2 z, F& @& |>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)

' ~+ F3 h  [" E4 `x1 =

; c+ T! l3 g. d; O% w0 j0 e110*cos(q) - cos((11*q)/6)*(60*cos(t) + 60*t*sin(t)) - sin((11*q)/6)*(60*sin(t) - 60*t*cos(t))

5 Y. F! h- Z% ]9 n) V( W- K. _1 H* e>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)

y1 =
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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)
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ans =
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- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)
/ X" g4 T. {# j+ D' {7 ]- d  W
>> diff(y1,t)
$ |7 S5 P$ e0 U  E% o1 |5 M
9 B( p- z' M( q9 b5 t8 D2 bans =

  a8 C. \. O0 `0 V" [' [60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)
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5 g/ a! h& _" `* L>> diff(x1,q)

3 ~0 }* R$ z6 K6 F9 g$ ^2 ^, Wans =

. D+ F( X% A6 F% c(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)

: _) i- y8 Y2 c" c2 B% L! _>> diff(y1,q)
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( @3 c" U) s9 y. o% v# pans =
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& D- G, d5 ?: U0 S110*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
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; [5 N7 _- i7 s2 O! W; h$ Z>> 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 =
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(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)

3 e2 H7 g8 H* F% C>> simplify(f1)
  |3 ?: Q6 b) M& `! a" F5 r
$ i; D5 ~2 j& t7 kans =
( ?8 F( |, k: O; V3 H5 q
-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)      
  y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)  
5 F( X( e1 Y0 _$ v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