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
/ p9 s. h* G) {>> x=60*(cos(t)+t*sin(t))

" H- p. K/ G. U) Sx =

60*cos(t) + 60*t*sin(t)
5 Y+ S8 x  I" ]2 R/ v6 _
>> y=60*(sin(t)-t*cos(t))

y =

60*sin(t) - 60*t*cos(t)

>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)
+ `0 r+ Z) n9 m& H1 ^
! _1 `1 p+ T6 d) u) @- _& vx1 =

4 H+ l) {; _- h7 V6 |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))
$ U# F+ e5 \' \7 \. e; X+ C' n+ ]
>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)

y1 =

% y( t; |$ H  g" E1 f110*sin(q) + cos((11*q)/6)*(60*sin(t) - 60*t*cos(t)) - sin((11*q)/6)*(60*cos(t) + 60*t*sin(t))
8 `3 U, [% {& v: C# T2 u
>> diff(x1,t)

/ A' N" L3 {$ `8 }( Y$ Eans =
4 ~* W+ P6 x/ V7 a% `- X
- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)

>> diff(y1,t)
6 h; A1 c- c2 D! l
ans =

60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)
+ H- L; y, U( p8 Q% T* c
>> diff(x1,q)

$ j; B: U/ |: d+ x9 qans =

3 R* ~1 P7 x$ Y(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)

, f( Y% ]4 ^5 H/ w>> diff(y1,q)
) _  v5 @' U( W( v2 a3 {1 T
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
0 ]. [( ?; D, Q( ?& S- r
>> 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 =

) |: C3 t8 C9 P3 D' a/ Q(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)
+ S1 B4 O- O: ^, Q) t* R& ^
>> simplify(f1)
6 u% m4 Y* M8 K9 {
! M4 k) |  r. uans =

-6600*t*(cos((5*q)/6 - t) - 1)
2 G$ h' d0 O' ~" C) ~/ L+ T% Z
7 e1 C' w1 U7 _$ ?/ G -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)      
1 L$ q2 ]9 q5 p! |3 H; S* v  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))
从方程上可以看出小齿轮的方程仍为渐开线