ME451 Kinematics and Dynamics of Machine Systems (Gears) Cam-Followers and Point-Follower 3.4.1, September 27, 2013 Radu Serban University of Wisconsin-Madison

2 Before we get started… Last time: Relative constraints (revolute, translational) Composite joints (revolute-revolute, revolute-translational) Today: Gears Cam – Followers Point – Follower Assignments: HW 5 – due September 30, in class (12:00pm) Matlab 3 – due October 2, (11:59pm)

Gears (convex-convex, concave-convex, rack and pinion) 3.4.1

4 Gears Convex-convex gears Gear teeth on the periphery of the gears cause the pitch circles shown to roll relative to each other, without slip First Goal: find the angle , that is, the angle of the carrier What’s known: Angles  i and  j The radii R i and R j You need to express  as a function of these four quantities plus the orientation angles  i and  j Kinematically: P i P j should always be perpendicular to the contact plane

5 Gears - Discussion of Figure (Geometry of gear set)

6 Note: there are a couple of mistakes in the book, see Errata slide

7 Gear Set Constraints

8 Example: Gear 1 is fixed to ground Given to you:  1 = 0,  1 =  /6,  2 =7  /6, R 1 = 1, R 2 = 2 Find  2 as gear 2 falls to the position shown (carrier line P 1 P 2 becomes vertical)

9 Gears (Convex-Concave) Convex-concave gears – we are not going to look into this class of gears The approach is the same, that is, expressing the angle  that allows on to find the angle of the Next, a perpendicularity condition using u and P i P j is imposed (just like for convex-convex gears)

10 Rack and Pinion Preamble Framework: Two points P i and Q i on body i define the rack center line Radius of pitch circle for pinion is R j There is no relative sliding between pitch circle and rack center line Q i and Q j are the points where the rack and pinion were in contact at time t=0 NOTE: A rack-and-pinion type kinematic constraint is a limit case of a pair of convex-convex gears Take the radius R i to infinity, and the pitch line for gear i will become the rack center line

11 Rack and Pinion Kinematics Kinematic constraints that define the relative motion: At any time, the distance between the point P j and the contact point D should stay constant (this is equal to the radius of the gear R j ) The length of the segment Q i D and the length of the arc Q j D should be equal (no slip condition) Rack-and-pinion removes two DOFs of the relative motion between these two bodies

12 Rack and Pinion Constraints

13 Errata: Page 73 (transpose and signs) Page 73 (perpendicular sign, both equations)

Cam – Followers 3.4.2

15 Cam – Follower Pair Setup: Two shapes (one on each body) that are always in contact (no chattering) Contact surfaces are convex shapes (or one is flat) Sliding is permitted (unlike the case of gear sets) Modeling basic idea: The two bodies share a common point The tangents to their boundaries are collinear Source: Wikipedia.org

16 Interlude: Boundary of a Convex Shape (1)

17 In the LRF: where and therefore [handout] Interlude Boundary of a Convex Shape (2) In the GRF:

18 Cam – Follower Pair Step 1 The two bodies share the contact point: (2 constraints) The two tangents are collinear: (1 constraint)

19 Cam – Follower Constraints

20 Example Determine the expression of the tangents g 1 and g 2

21 Cam – Flat-Faced Follower Pair A particular case of the general cam-follower pair Cam stays just like before Flat follower Typical application: internal combustion engine Not covered in detail, HW touches on this case

22 Errata: Page 83 (Q instead of P) Page 80 (subscript ‘j’ instead of ‘i’)

Point – Follower 3.4.3

24 Point – Follower Pair

25 Point – Follower Constraints