1. ME451 Kinematics and Dynamics of Machine Systems Relative Kinematic Constraints, Composite Joints – 3.3 October 6, 2011 © Dan Negrut, 2011 ME451, UW-Madison “I want to put a ding in the universe.” Steve Jobs TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AAAAA

2. Before we get started… Last Time Discussed relative constraints x, y,  relative constraints distance constraint For each kinematic constraint, recall the procedure that provides what it takes to carry out Kinematics Analysis  Identify and analyze the physical joint  Derive the constraint equations associated with the joint  Compute constraint Jacobian  q  Get  (RHS of velocity equation)  Get  (RHS of acceleration equation, this is challenging in some cases) Today Covering relative constraints: Revolute, translational, composite joints, cam-follower type constraints Skipping gears Assignment, due in one week 3.3.2, 3.3.4, 3.3.5 + MATLAB + ADAMS 2

3. Revolute Joint Step 1: Physically imposes the condition that point P on body i and a point P on body j are coincident at all times Step 2: Constraint Equations  (q,t) = ? Step 3: Constraint Jacobian  q = ? Step 4: = ? Step 5:  = ? 3

4. Translational Joint Step 1: Physically, it allows relative translation between two bodies along a common axis. No relative rotation is allowed. Step 2: Constraint Equations  (q,t) = ? Step 3: Constraint Jacobian  q = ? Step 4: = ? Step 5:  = ? 4 NOTE: recall notation

5. Attributes of a Constraint [it’ll be on the exam] What do you need to specify to completely define a certain type of constraint? In other words, what are the attributes of a constraint; i.e., the parameters that define it? For absolute-x constraint: you need to specify the body “i”, the point P that enters the discussion, and the value that x P should assume For absolute-y constraint: you need to specify the body “i”, the point P that enters the discussion, and the value that y P should assume For a distance constraint, you need to specify the “distance”, but also the location of point P in the LRF, the body “I” on which the LRF is attached to, as well as the point of coordinates c 1 and c 2 How about an absolute angle constraint? Think about it… 5

6. The Attributes of a Constraint [Cntd.] Attributes of a Constraint: That information that you are supposed to know by inspecting the mechanism It represents the parameters associated with the specific constraint that you are considering When you are dealing with a constraint, make sure you understand What the input is What the defining attributes of the constraint are What constitutes the output (the algebraic equation[s], Jacobian, ,, etc.) 6

7. The Attributes of a Constraint [Cntd.] Examples of constraint attributes: For a revolute joint: You know where the joint is located, so therefore you know For a translational join: You know what the direction of relative translation is, so therefore you know For a distance constraint: You know the distance C 4 7

8. Example 3.3.4 Consider the slider-crank below. Come up with the set of kinematic constraint equations to kinematically model this mechanism Use the Cartesian (absolute) generalized coordinates shown in the picture 8

9. Example 3.3.2 – Different ways of modeling the same mechanism for Kinematic Analysis 9 Approach 1: bodies 1, 2, and 3 Approach 2: bodies 1 and 3 Approach 3: bodies 1 and 2 Approach 4: body 2

10. Errata: Page 67 (sign) 10 Page 68 (unbalanced parentheses, and text) Page 73 (transpose and sign) Page 73 (perpendicular sign, both equations)

11. Composite Joints (CJ) Just a means to eliminate one intermediate body whose kinematics you are not interested in 11 Revolute-Revolute CJ Also called a coupler Practically eliminates need of connecting rod Given to you (joint attributes): Location of points P i and P j Distance d ij of the massless rod Revolute-Translational CJ Given to you (joint attributes): Distance c Point P j (location of revolute joint) Axis of translation v i ’

12. Composite Joints One follows exactly the same steps as for any joint: Step 1: Physically, what type of motion does the joint allow? Step 2: Constraint Equations  (q,t) = ? Step 3: Constraint Jacobian  q = ? Step 4: = ? Step 5:  = ? 12

13. We will skip Gears (section 3.4) 13

14. 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 14 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

15. Gears - Discussion of Figure 3.4.2 (Geometry of gear set) 15

16. Gears - Discussion of Figure 3.4.2 (Geometry of gear set) 16 Note: there are a couple of mistakes in the book, see Errata slide before

17. Example: 3.4.1 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) 17

18. 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) 18

19. Example: 3.4.1 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) 19

20. 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 20

21. 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 21

22. Rack and Pinion Pair Step 1: Understand the physical element Step 2: Constraint Equations  (q,t) = ? Step 3: Constraint Jacobian  q = ? Step 4: = ? Step 5:  = ? 22

23. End Gear Kinematics Begin Cam-Follower Kinematics 23

24. Preamble: Boundary of a Convex Body Assumption: the bodies we are dealing with are convex To any point on the boundary corresponds one value of the angle  (this is like the yaw angle, see figure below) 24 The distance from the reference point Q to any point P on the convex boundary is a function of  : It all boils down to expressing two quantities as functions of  The position of P, denoted by r P The tangent at point P, denoted by g

25. Cam-Follower Pair Assumption: no chattering takes place The basic idea: two bodies are in contact, and at the contact point the two bodies share: The contact point The tangent to the boundaries 25 Recall that a point is located by the angle  i on body i, and  j on body j. Therefore, when dealing with a cam- follower, in addition to the x,y,  coordinates for each body one needs to rely on one additional generalized coordinate, namely the contact point angle  : Body i: x i, y i,  i,  i Body j: x j, y j,  j,  j

26. Cam-Follower Constraint Step 1: Understand the physical element Step 2: Constraint Equations  (q,t) = ? Step 3: Constraint Jacobian  q = ? Step 4: = ? Step 5:  = ? 26