1. Learning with Purpose February 4, 2013 Learning with Purpose February 4, 2013 22.322 Mechanical Design II Spring 2013

2. Learning with Purpose February 4, 2013 Lecture 5 Example 5-6 If  =15 rad/s and  =180 0 – 45 0 = 135 0 ; h=1”, What is the velocity and acceleration at the beginning of the return?

3. Learning with Purpose February 4, 2013 Lecture 5 Example 5-6

4. Learning with Purpose February 4, 2013 Once the s v a j functions have been defined, the next step is to size the cam. Two major factors that affect cam size: Pressure Angle Radius of Curvature Both involve either the base circle radius (R b ) when using flat- faced followers, or the prime circle radius (R p ) when using roller or curved followers. Base circle – smallest circle that can be drawn tangent to the physical cam surface Prime circle - smallest circle that can be drawn tangent to the locus of the centerline of the follower. Cams with roller followers are defined for manufacture with respect to the pitch curve rather than to the cam surface Cams with flat-faced followers must be defined with respect to their physical surface (no pitch curve) Lecture 6 Sizing the Cam

5. Learning with Purpose February 4, 2013 Pressure angle – angle between the direction of motion (velocity) of the follower and the direction of the axis of transmission (common normal) Force can only be transmitted from cam to follower or vice versa along the axis of transmission (perpendicular to axis of slip) Recall that 0-30 o is typically desired Eccentricity,  – perpendicular distance between the follower’s axis of motion and the center of the cam If a small enough cam cannot be obtained having an acceptable pressure angle, then eccentricity can be introduced to change the pressure angle (misaligning the center of the cam and axis of follower motion). The eccentricity may decrease the pressure angle on the rise but it will increase it on the return (and vice versa) Lecture 6 Sizing the Cam

6. Learning with Purpose February 4, 2013 For a flat-faced follower, the pressure angle is always zero However, the force located away from the axis of follower travel will induce a moment that can generate friction and jam the follower in its guides. In this case, we would like to keep the cam as small as possible in order to minimize the moment arm of the force. Eccentricity will affect the average value of the moment, but the peak- to-peak variation of the moment about that average is unaffected by eccentricity. Lecture 6 Sizing the Cam

7. Learning with Purpose February 4, 2013 Mathematical property of a function: Straight line = infinity everywhere Circle = constant value Parabola = constantly changing radius of curvature that approaches infinity Cubic curve = radii of curvature that are sometimes positive (convex) and sometimes negative (concave). The higher the degree of a function, in general, the more potential variety in its radius of curvature. Cam contours are usually functions of high degree. Will have portions that are concave, convex, or flat. Lecture 6 Radius of Curvature

8. Learning with Purpose February 4, 2013 Here is a problem in which the radius of curvature of the follower is larger than the minimum concave radius of the cam: Lecture 6 Radius of Curvature Follower cannot generate the motion that the cam dictates because it is too big.

9. Learning with Purpose February 4, 2013 When the follower radius is larger than the smallest positive (convex) local radius on the cam (or pitch curve), undercutting will occur. As a rule of thumb, keep the minimum radius of curvature of the cam pitch curve at least 2 to 3 times as large as the radius of the roller follower. Lecture 6 Radius of Curvature In figure (a), a cusp (sharp corner) is created and the cam will not run smoothly (stress factor). In figure (b), there is a cusp and material missing  the cam will not be able to reproduce the desired motion of the pitch curve.

10. Learning with Purpose February 4, 2013 Consider the torque required to perform a dumbbell curl exercise. When the weight is fully lowered, the amount of torque or force is low. Then, when the elbow reaches an angle of 90 o, the torque or force is maximum. Finally, when the elbow is at approximately the fully upright position, the torque or force is low again. Lecture 6 Dumbbell Curl Example

11. Learning with Purpose February 4, 2013 Lecture 6 Dumbbell Curl Example We want our workout to be like this

12. Learning with Purpose February 4, 2013 Lecture 6 Dumbbell Curl Example The goal is to match the required force to the human strength curve for a bicep curling exercise. mg Varying human strength mg

13. Learning with Purpose February 4, 2013 Lecture 6 Dumbbell Curl Example Strength in relation to maximum strength Arbitrary

14. Learning with Purpose February 4, 2013 Lecture 6 Dumbbell Curl Example Note: The design variables are the base circle and the distance between the cam and the idler sprocket. The sketch is not drawn to scale! Position of sprocket