ME451 Kinematics and Dynamics of Machine Systems

1. ME451 Kinematics and Dynamics of Machine Systems Relative Constraints Composite Joints 3.3 September 25, 2013 Radu Serban University of Wisconsin-Madison

2. Before we get started… • Last time: • Relative constraints (x, y, f, distance) • Recall the drill: • Identify and analyze the physical joint • Derive the constraint equations associated with the joint, (q)=0 • Compute constraint Jacobian, q • Get (RHS of velocity equation) • Get (RHS of acceleration equation, this is challenging in some cases) • Today • Relative constraints (revolute, translational) • Composite joints (revolute-revolute, revolute-translational) • Assignments: • HW 5 – 3.3.4, 3.3.5 – due September 30, in class (12:00pm) • Matlab3 – due October 2, Learn@UW (11:59pm)

3. 3.3 Relative Constraints

4. 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: Identify • Step 3: • Step 4: • Step 5:

5. Translational Joint • Step 1: Physically, it allows relative translation between two bodies along a common axis. No relative rotation is allowed. • Step 2: Identify • Step 3: • Step 4: • Step 5:

6. Errata • Page 67 (sign) • Page 68 (unbalanced parentheses)

7. Attributes of a Constraint(1) • What do you need to specify to completely specify 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 particular point P on that body, and the value that xiP should assume • For absolute-y constraint: you need to specify the body “i”, the particular point P on that body, and the value that yiPshould 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 coordinates c1 and c2of point C (in the GRF). • How about an absolute angle constraint?

8. Attributes of a Constraint(2) • 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.)

9. Attributes of a Constraint(3) • 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 C4

10. [handout]Example 3.3.2 / Problem 3.3.3 Four different ways of modeling the same mechanism for Kinematic Analysis • Approach 1: bodies 1, 2, and 3 • Approach 2: bodies 1 and 3 • Approach 3: bodies 1 and 2 • Approach 4: body 2

11. Composite Joints (1) • Revolute-Revolute • Eliminates need of connecting rod • Attributes: • Points Pi and Pj: and • Length of the massless rod: • Revolute-Translational • Eliminates the intermediate body • Attributes: • Distance c • Point Pj (location of revolute joint) • Axis of translation: • Just a means to eliminate one intermediate body (a.k.a. coupler) whose kinematics you are not interested in

12. Composite Joints (2) • One follows exactly the same steps as for any other joint: • Step 1: Physically, what type of motion does the joint allow? • Step 2: Identify • Step 3: • Step 4: • Step 5: