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   
* \1 _" q$ j& o7 U! Y> syms t q
>> x=60*(cos(t)+t*sin(t))
( M$ c' }; ^, |9 D% |$ Z: E; I- [% a
% A7 K: M3 G' u/ p# P0 ox =

+ G, q& N5 O5 i% n  j) `' v- b60*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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>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)

x1 =

* n& l; I# O! P& F3 P8 H. u110*cos(q) - cos((11*q)/6)*(60*cos(t) + 60*t*sin(t)) - sin((11*q)/6)*(60*sin(t) - 60*t*cos(t))
( i+ }/ E" n) W& T
>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)
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y1 =
0 ]( h1 U" g& b
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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9 p, k* J- ?: I% ?$ }ans =
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: \0 x1 s2 }4 T* y4 F5 _5 n- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)
1 H7 W2 k7 S' c- I
4 \" o4 a% I3 U: F9 \& ^>> diff(y1,t)

/ D; J2 a, n+ Xans =
0 v7 w: s) u2 Q, g
60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)
, N4 @$ \! ?  k8 d! ~8 d6 i, g( u, K
>> diff(x1,q)

1 ~# ]$ i+ n: y( _( R7 O5 ]* wans =
" }9 Y+ {7 x. b3 ~5 W
4 C% v" `; v$ r2 C6 @& ~(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)

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
( a! ~; ?: a: g7 @* `, s7 c4 ]
7 x5 Z0 C/ E0 Z& p, o0 I+ o>> 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))')
# x. }$ |1 S1 }/ h* m7 O0 l, Q
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)
( U* s# j, D" m
  L' L1 p; J! f# H>> simplify(f1)
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" [+ I: N0 s' @+ ~0 f; \' ians =
) I0 H* U* ]# C0 v$ L/ i$ t
-6600*t*(cos((5*q)/6 - t) - 1)
  F8 P5 E% b& d: @4 S
9 _: k6 B) n" w# W- 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))
! l3 k8 }) f- L/ B* v% x从方程上可以看出小齿轮的方程仍为渐开线