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标题:
渐开线齿轮啮合方程推导
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作者:
computer001
时间:
2015-3-27 11:36
标题:
渐开线齿轮啮合方程推导
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
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> syms t q
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>> x=60*(cos(t)+t*sin(t))
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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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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)
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x1 =
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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))
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>> y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)
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y1 =
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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))
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>> diff(x1,t)
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ans =
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- 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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60*t*cos((11*q)/6)*sin(t) - 60*t*sin((11*q)/6)*cos(t)
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>> diff(x1,q)
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ans =
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(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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ans =
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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
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>> 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))')
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f1 =
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(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)
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>> simplify(f1)
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ans =
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-6600*t*(cos((5*q)/6 - t) - 1)
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-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)
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y1=-x*sin(11/6*q)+y*cos(11/6*q)+110*sin(q)
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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))
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从方程上可以看出小齿轮的方程仍为
渐开线
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