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
* j( L( ~2 r" v: O. ]' B6 t>> x=60*(cos(t)+t*sin(t))

. b& _( n2 u1 Y- p9 O' _x =
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60*cos(t) + 60*t*sin(t)
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>> y=60*(sin(t)-t*cos(t))
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! e& _2 D! L1 G+ n. m% x* _y =
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60*sin(t) - 60*t*cos(t)

>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)
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- A9 L' \. }# ]7 C; K: J' x* xx1 =
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- k: r# Q1 R5 B3 R110*cos(q) - cos((11*q)/6)*(60*cos(t) + 60*t*sin(t)) - sin((11*q)/6)*(60*sin(t) - 60*t*cos(t))

/ Z; W5 S6 M1 K2 v1 k$ Z7 @" O>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)
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y1 =
9 Y8 Y  X$ J4 d% n+ n  Y
8 }, [& C3 h0 t  D* a2 i: m9 A$ B110*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 Q$ Q0 V! z- _( C2 w>> diff(x1,t)
# L+ K4 ]( k/ S+ M9 y( ]
9 K! D7 Y5 o; `ans =

- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)
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>> diff(y1,t)
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ans =
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$ D3 \+ W: H6 _; `6 V60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)
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" J) ?' s) a, I0 J: U* d7 \>> diff(x1,q)
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ans =

1 W+ E8 j9 W3 B: D(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)
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>> diff(y1,q)
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$ P- a3 E% {$ m7 e1 Nans =
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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

>> 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))')
5 y. T) u* t1 f
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)

! T1 f6 \0 |% i7 w( p  E9 K! S, _>> simplify(f1)
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ans =
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-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)      
5 T  u% R7 o, `8 y7 z2 Y* D  y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)  
+ f& ]/ g1 Y1 ?, p8 ^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))
" H! o% y6 N( Z4 W- Y; A7 w7 k: e从方程上可以看出小齿轮的方程仍为渐开线