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   
( e# M1 j+ E0 {! T4 M> syms t q
>> x=60*(cos(t)+t*sin(t))

x =
9 }7 C1 y) p  r2 K
: w9 y. t, Z3 s$ i) g60*cos(t) + 60*t*sin(t)
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>> y=60*(sin(t)-t*cos(t))
/ i* ^& e8 m4 N$ e+ Q7 C
8 m" ]; c) }* n, ^. F* _3 Ky =

3 n, A& |( |# S5 O$ b2 K60*sin(t) - 60*t*cos(t)

  i3 e. z3 n+ y>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)

x1 =
5 r; k7 [# `- b
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)

, ]0 R% q( V! @- Z% n1 r  Ny1 =

# y8 @. L% O6 U0 }* n$ I+ @& }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))
4 N! h/ P2 l: ]: w! g) w
>> diff(x1,t)

ans =
* ^0 T  X. G9 h" k4 P5 V$ S/ j8 ^% z! }
- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)

  P6 [! q2 Y4 ?5 ]5 h" A' [>> diff(y1,t)

ans =

60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)
0 p0 F# `1 s0 g2 E" N
. b$ q7 k/ q8 n+ {; f0 {- j! n>> diff(x1,q)
! v& y: ~$ c% s6 @$ I) _' [3 H
$ }: Z1 b' J: R% \- O' l- G' dans =
+ K: ]* r3 a+ F: x
(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)
7 F! [" C6 ?. n. ?: Q/ o
>> diff(y1,q)

/ H- M/ g5 ]& Y8 ~1 e+ j+ Lans =

' i8 \8 v( Y  d! K110*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

* v4 p5 }% n) X+ m7 S: G>> 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))')
3 c7 A- D& i3 R  s# b! J
1 S$ T$ j2 R: l, e6 cf1 =
9 t. i6 L" |1 Q  u; y
; }- x& k# e# f(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)
2 ?4 e+ H1 g+ B
>> simplify(f1)
2 @% R; v3 @5 H3 ~3 g7 C! \9 x
5 @& I. a2 y: h( Y9 dans =

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

8 x( m4 b% R3 Z( X% z -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)      
2 K4 C4 Y0 E0 Q% w) \/ @  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