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   
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>> x=60*(cos(t)+t*sin(t))
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& b3 M, Q: `! P. dx =

60*cos(t) + 60*t*sin(t)

( Q$ A+ p) m: E/ `+ \>> y=60*(sin(t)-t*cos(t))

8 y( E& L7 q- _( N) H, xy =

# }# s3 n/ U$ Y8 `  f& P. f% ^60*sin(t) - 60*t*cos(t)
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>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)
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x1 =

1 ?9 n- |4 W3 m$ Z" P& }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)
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% j: }/ s/ @, yy1 =
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7 N7 @! t6 h! m8 ^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))

- G' F4 J' M7 Z  o>> diff(x1,t)
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5 }# G  [% T- ~6 Xans =

/ d% K8 N. z! `: w- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)

6 k1 n/ n4 s# ?, r( u% ?5 U" Q2 F>> diff(y1,t)
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ans =
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60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)
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1 M. ]( T- g4 u4 l- \1 C( O>> diff(x1,q)
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ans =
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  ~2 N( J3 m& W$ c7 E(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)

% t5 R" \7 b' d( [9 y>> diff(y1,q)
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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
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8 f: o' i0 M/ T7 j1 g& d: ]6 V>> 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))')

. p! J$ A, W; E. ~1 j6 mf1 =
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& X6 d  S0 |7 `(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)
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5 L% F# x/ c3 e8 g1 X- Z>> simplify(f1)

# v. C, V1 ?+ H7 _6 C0 z: Qans =

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

) g+ W& I1 y, t; H1 h7 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)  
( J  O. g7 T" o, G/ y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))
9 @* U+ j3 y! Q5 V8 _从方程上可以看出小齿轮的方程仍为渐开线