{"id":2312,"date":"2022-03-15T22:51:13","date_gmt":"2022-03-15T22:51:13","guid":{"rendered":"https:\/\/blog.metu.edu.tr\/eresmech\/?page_id=2312"},"modified":"2022-09-02T20:51:13","modified_gmt":"2022-09-02T20:51:13","slug":"8-4","status":"publish","type":"page","link":"https:\/\/blog.metu.edu.tr\/eresmech\/mechanisms\/ch8\/8-4\/","title":{"rendered":"8-4"},"content":{"rendered":"
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8.4 Cam Size Determination<\/strong><\/h1>\n

Cam size determination is related to the determination of the base circle of the cam. In almost all applications we would like to minimize the size of the cam being used. Large cams are not desired due to the following reasons:<\/p>\n

1. More space is required.
\n2. Unbalanced mass increases
\n3. Follower has a longer path to follow for each cycle. Hence the angular velocity of the follower and the surface velocity increases.<\/p>\n

However as we decrease the cam size, the following factors take into effect:<\/p>\n

1. The force transmission characteristics deteriorate. The cam profile steepness increases.
\n2. The curvature of the cam profile decreases (sharp curves)
\n3. Strength requirements due to the forces and moments acting on the cam.<\/p>\n

In practice the cam size is determined by considering two factors: the pressure angle and the minimum radius of curvature..<\/p>\n

8<\/span>.4<\/span>.1. Pressure Angle<\/em><\/strong><\/h1>\n

The\u00a0pressure angle<\/em><\/strong>, which is the reciprocal of the transmission angle \u03bc, i.e. \u03b1 = \u03c0\/2 –<\/span> \u03bc, is defined as:<\/p>\n

\\displaystyle {\\tan \\text{\u03b1}=\\frac{\\text{Force component tending to apply pressure on the follower bearings}}{\\text{Force component tending to move the follower}}}<\/span><\/p>\n

In cams there is point contact between the two profiles. The force is transmitted along the common normal of the two contacting curves. Pressure angle for oscillating and translating roller follower radial cams are shown below. For flat faced followers the pressure is apparently zero at all times (the normal to the profile is normal to the flat face, which is always constant). Curvature characteristics is used for the determination of cam size.\u00a0 In force closed cams, the pressure angle is important during the rise portion where cam is driving the follower, in return motion it is the spring force (or any other force used for forced closure) that lowers the follower.<\/p>\n

\"\"<\/p>\n

The pressure angle is a function of cam rotation angle. We can express the force ratio expression for the pressure angle using the kinematics of the mechanism.<\/p>\n

\"\"Consider an inline translating roller follower radial cam . The pressure angle is a function of cam rotation and the amount of rise, s (which is also a function of the cam rotation angle). \u00a0At the instant considered, velocity of point B on link 3 (follower) will be\u00a0v<\/strong>B3<\/sub>\u00a0along the slider axis (vertical). The velocity of a point B on link 2 (cam) at this instant is\u00a0v<\/strong>B2<\/sub> perpendicular to the line OB. These two velocities are related by the equation (see Chapter 4):<\/p>\n

V<\/strong>B3<\/sub>\u00a0= V<\/strong>B2<\/sub>\u00a0+ V<\/strong>B3\/B2<\/sub><\/p>\n

In this equation:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong><\/p>\n

V<\/strong>B3<\/sub>: Follower speed with magnitude = ds\/dt<\/p>\n

V<\/strong>B2<\/sub>: Velocity of point B on the cam with magnitude = r\u03c9 (r = |OB| = s + rb<\/sub>\u00a0+ rr<\/sub>, rb<\/sub>\u00a0is the base circle radius, rr<\/sub> is the roller radius).<\/p>\n

The relative velocity direction is along the path tangent. Hence we have:<\/p>\n

\\displaystyle \\tan {\\text{\u03b1}}=\\frac{{{{\\text{V}}_{{\\text{B}3}}}}}{{{{\\text{V}}_{{\\text{B}2}}}}}=\\frac{{\\text{ds\/dt}}}{{\\text{r\u03c9}}}=\\frac{\\text{ds\/d\u03b8}}{{\\text{r}}} <\/span><\/p>\n

Here s\u2032 is the derivative of s with respect to \u03b8. If the motion curve is known, the pressure angle variation can be determined (at the dwell periods note that tan\u03b1\u00a0<\/span>= 0 or \u03b1\u00a0<\/span>= 0, no motion is transmitted). During the rise or return portion s(\u03b8) is monotonically increasing or decreasing. Except the double harmonic, for symmetric curves, s\u2032(\u03b8) is maximum at the midpoint (s = H\/2, \u03b8 = \u03b2\/2). Hence the pressure angle is maximum or minimum at half the rise or return motion for inline roller followers. It will be given by the equation:<\/p>\n

\\displaystyle \\tan {{\\text{\u03b1}}_{\\max}}=\\frac{{\\left( {\\text{ds\/d\u03b8}} \\right)}_{\\max}}{\\text{H\/2}+{\\text{r}}_{\\text{r}}+{\\text{r}}_{\\text{b}}} <\/span><\/p>\n

Since (ds\/d\u03b8)max<\/sub> = Cv<\/sub>H\/\u03b2, this value can easily be determined for a given motion curve. Usually rr<\/sub>\u00a0is determined from strength considerations. The designer selects an acceptable maximum pressure angle and solves the above expression for the base circle radius.<\/p>\n

Maximum pressure angle usually depends on the speed, load and place of application of the cam mechanism. In the literature as a rule of thumb:
\nIf the cam speed is less then 30 rpm:\u00a0 \u03b1max<\/sub> \u2264 45\u00b0
\nIf the cam speed is greater then 30 rpm:\u00a0 \u03b1max<\/sub> \u2264 30\u00b0
\nIf the follower is eccentric as shown in the figure, then the pressure angle is given by the equation:<\/p>\n

\\displaystyle \\tan {\\text{\u03b1}}=\\frac{{\\text{ds\/d\u03b8}}-{\\text{e}}}{{\\text{s(\u03b8)}}+{\\text{c}}} <\/span><\/p>\n

where e is the eccentricity and \\displaystyle \\text{c}=\\sqrt{{{\\text{r}}_{\\text{b}}}^{2}-{\\text{e}}^{2}} <\/span>\u00a0 or\u00a0 \\displaystyle {\\text{r}}_{\\text{b}}=\\sqrt{{\\text{c}}^{2}+{\\text{e}}^{2}} <\/span><\/p>\n

In force closed cams the follower is driven by the cam during the rise portion. In the return motion it is the spring force (or weight) that drives the follower towards the cam surface, hence the pressure angle is not that critical. Using eccentricity, for the same can size one can reduce the pressure angle during the rise while there is some increase of pressure angle during the return, or for the same pressure angle a smaller cam size can be used.<\/p>\n