22.322 Mechanical Design II

1. 22.322 Mechanical Design II Spring 2013

2. Lecture 5 Example 5-6 If w=15 rad/s and b=1800 – 450 = 1350; h=1”, What is the velocity and acceleration at the beginning of the return?

3. Lecture 5 Example 5-6

4. Lecture 6 Sizing the Cam • 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 (Rb) when using flat-faced followers, or the prime circle radius (Rp) 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)

5. Lecture 6 Sizing the Cam • 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 0-30o is typically desired • Eccentricity, e– 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)

6. Lecture 6 Sizing the Cam • 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.

7. Lecture 6 Radius of Curvature • 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.

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

9. Lecture 6 Radius of Curvature • 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. 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. Lecture 6 Dumbbell Curl Example • 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 90o, the torque or force is maximum. • Finally, when the elbow is at approximately the fully upright position, the torque or force is low again.

11. Lecture 6 Dumbbell Curl Example We want our workout to be like this

12. Lecture 6 Dumbbell Curl Example • The goal is to match the required force to the human strength curve for a bicep curling exercise. Varying human strength mg mg

13. Lecture 6 Dumbbell Curl Example Arbitrary Strength in relation to maximum strength

14. Lecture 6 Dumbbell Curl Example Position of sprocket 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!