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   
6 A2 v, O  p5 m1 Z& p> syms t q
>> x=60*(cos(t)+t*sin(t))

x =

& P* L6 K$ |3 A) E, q* ^60*cos(t) + 60*t*sin(t)
' G* L( j- K+ H9 O9 ^$ S2 s& T
>> y=60*(sin(t)-t*cos(t))

y =

& R" m0 u! Z7 ]5 P# @5 b) Q* O, f60*sin(t) - 60*t*cos(t)

>> x1=-x*cos(11/6*q)-y*sin(11/6*q)+110*cos(q)
5 V& ^( ?3 N: @0 g8 C1 b
x1 =
( \, k, e5 d( k  c' ~
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)

y1 =
3 z/ I" B3 d$ j; M: P
* o% L8 b! e9 ~4 e110*sin(q) + cos((11*q)/6)*(60*sin(t) - 60*t*cos(t)) - sin((11*q)/6)*(60*cos(t) + 60*t*sin(t))
* Q7 b6 H4 B& @* D: c6 g9 c
; O$ d% w& o6 ~$ Y( @: n>> diff(x1,t)

" P+ j4 u6 }3 g% u- Bans =
: D, K4 E4 c5 K/ Y3 L- p
- 60*t*cos((11*q)/6)*cos(t) - 60*t*sin((11*q)/6)*sin(t)
7 Z' g7 M; @/ t3 f3 [0 B
>> diff(y1,t)

: l1 N2 e- q# {( j. ^9 g0 jans =
6 p, g; _$ y+ R, u; y8 [: o2 A
60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)
& G1 c/ x+ K- I3 A! M
6 `' Q( U4 N; x6 z- Q+ c0 c>> diff(x1,q)
3 ^2 X( U; \# D! [- C
ans =

2 l3 u' j4 B5 o2 k& E(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)

; y. B0 b5 q3 |4 D6 g>> diff(y1,q)

* o4 d+ X( I( m' E; }" B+ x" qans =

4 C) M1 ~8 K  _+ z; h! i( r110*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
: H4 V1 h! u' y5 N* q" I
>> 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))')
8 s, V) F: y2 ?9 \! K6 u
f1 =
) v. a9 Z, G) b" f* M& [$ k0 p
* J8 A6 z) U# E(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)

( c3 l$ O9 ^( ^! ?; A- m8 ?4 O>> simplify(f1)
) t% `( R1 L  ^0 ]9 C' z1 \
- E: P( a! k* r! S& Yans =
! N3 F. n  h( c- H5 i$ \
- u$ R: G' z" `-6600*t*(cos((5*q)/6 - t) - 1)
4 S! w8 m3 S1 N0 c! `" L. C
-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)      
4 p, |2 y5 w/ O3 g5 ]9 y  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