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
9 }0 J5 A& M! h( I% ]>> x=60*(cos(t)+t*sin(t))

x =
' g4 n, v+ n& b- m# _
0 Q5 b/ G8 a' `7 q( `$ E60*cos(t) + 60*t*sin(t)
  H6 J* i# ^. T* C# R$ w
>> y=60*(sin(t)-t*cos(t))
# M1 b) Q8 {% b5 E# ?! L1 g
y =
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60*sin(t) - 60*t*cos(t)
4 D* @% o- s" O& y
6 J8 k& Z4 `  k+ R>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)

# h  L* m: q7 @# Bx1 =

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))
( }5 i  B; Z& O8 j& C
* f2 T* T+ t9 A: [3 \4 _- q>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)

2 V' P. x1 j/ A7 }* }4 F. A5 _y1 =

, j3 e8 U2 z7 ?7 ^% ~8 O0 O3 N110*sin(q) + cos((11*q)/6)*(60*sin(t) - 60*t*cos(t)) - sin((11*q)/6)*(60*cos(t) + 60*t*sin(t))
3 @8 B( J) @2 ?+ a
>> diff(x1,t)

9 @1 D+ e/ F' C. a4 K5 ^ans =

- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)
( q* [7 s) h; Q; Z" H
>> diff(y1,t)
2 V3 j* j4 Q# y  v! [+ ]5 e) s) e
ans =
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60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)

4 v$ P, o4 ]! c& K. U>> diff(x1,q)
* @: E0 Q) U4 ?* z6 M& ~5 L6 B. G- o
ans =

(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)
& ?& N/ e+ [) a" ~6 m4 m3 W
+ u1 ]. I7 ]# C* V>> diff(y1,q)

% `! s  N; Z% t  _2 A& H. f7 b' c: kans =

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
" e/ a% {9 [- b. L$ ~# r1 @2 k
>> 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))')

$ k' w" M$ P3 H0 r  ]1 Gf1 =

(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)
+ w7 n5 J* n0 Y1 }& y+ i2 C  v
>> simplify(f1)

. {" c7 P1 x3 c3 Lans =
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-6600*t*(cos((5*q)/6 - t) - 1)

' R/ j* o8 K3 N7 B -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)  
3 F8 Q& {- 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))
  Y& s2 C7 C: J3 M从方程上可以看出小齿轮的方程仍为渐开线