{"id":2362,"date":"2022-03-18T21:37:07","date_gmt":"2022-03-18T21:37:07","guid":{"rendered":"https:\/\/blog.metu.edu.tr\/eresmech\/?page_id=2362"},"modified":"2022-09-02T21:16:24","modified_gmt":"2022-09-02T21:16:24","slug":"translating_roller","status":"publish","type":"page","link":"https:\/\/blog.metu.edu.tr\/eresmech\/mechanisms\/ch8\/8-5\/translating_roller\/","title":{"rendered":"translating_roller"},"content":{"rendered":"
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Cam Profile for a Radial Cam With Translating Roller Follower<\/strong><\/h1>\n

Now let us apply the envelope theory to a basic cam-follower system: a radial cam with a translating roller follower. We shall assume that the follower sliding axis is offset by an amount c from the centre of rotation of the cam (when c = 0, it is an inline follower).<\/p>\n

As in the case of geometrical construction, let us use kinematic inversion and fix the cam and release the fixed link. For the same relative motion, if the cam rotates by a counter clockwise angle q, the fixed link will rotate by a clockwise angle\u00a0q<\/span>\u00a0and the follower\u00a0 will be displaced by an amount s(q<\/span>) with respect to the fixed link as shown. The roller takes different positions for each\u00a0q<\/span>\u00a0angle, hence it is the roller profile that is the generating curve and its equation for any cam angle\u00a0q<\/span>\u00a0is:<\/p>\n\n\n\n
f(x, y, \u03b8) = (x \u2212 xp<\/sub>)2<\/sup> + (y \u2212 yp<\/sub>)2<\/sup> \u2212 rr<\/sub> = 0<\/td>\n(1)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where rr<\/sub>\u00a0is the radius of the roller. P(xp<\/sub>, yp<\/sub>) are the coordinates of a point on the pitch curve (the centre of the roller), which is a function of the cam angle\u00a0q<\/span>. xp<\/sub>, yp<\/sub>\u00a0is given by:<\/p>\n\n\n\n
\n

xp<\/sub> = c cos\u03b8 + (k + s) sin\u03b8<\/p>\n

yp<\/sub> = \u2212c sin\u03b8 + (k + s) cos\u03b8<\/p>\n<\/td>\n

(2)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where\u00a0k = \\displaystyle \\sqrt{{{{{\\left( {{{\\text{r}}_{\\text{t}}}+{{\\text{r}}_{\\text{r}}}} \\right)}}^{2}}-{{\\text{c}}^{2}}}} <\/span>.\u00a0(One can also write the coordinates of point P in complex numbers as: z = xp<\/sub> + iyp<\/sub> = ce\u2212i\u03b8<\/sup> + i(k + s)e\u2212i\u03b8<\/sup>. The real and imaginary parts of this complex number will yield xp<\/sub>\u00a0and yp<\/sub>\u00a0coordinates as given in equation 2).<\/p>\n

\"\"<\/p>\n

Note that the equation for the roller profile is a generating curve which depends on constant parameters rb<\/sub>, rr<\/sub>, and c and on the follower displacement s and cam rotation angle\u00a0q<\/span>. Since s is also a function of\u00a0q\u00a0<\/span>defined by the motion curve, the generating curve (roller profile) is in the form f(x, y, q<\/span>). Where\u00a0q<\/span>\u00a0is a parameter (for each value of\u00a0q<\/span>\u00a0we obtain a member within the family of curves). Hence the generating curve is:<\/p>\n