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ME451 Kinematics and Dynamics of Machine Systems

ME451 Kinematics and Dynamics of Machine Systems. Relative Constraints 3.3 September 23, 2013. Radu Serban University of Wisconsin-Madison. Before we get started…. Last time: Absolute constraints ( x , y , f , distance) Recall the drill: Identify and analyze the physical joint

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ME451 Kinematics and Dynamics of Machine Systems

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  1. ME451 Kinematics and Dynamics of Machine Systems Relative Constraints 3.3 September 23, 2013 Radu Serban University of Wisconsin-Madison

  2. Before we get started… • Last time: • Absolute 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 • Derive for the absolute distance constraint • Relative constraints (x, y, f, distance, revolute, translational) • Assignments: • HW 4 – due Wednesday, in class (12:00pm) • Matlab 2 and ADAMS 1 – due Wednesday, Learn@UW (11:59pm)

  3. Absolute distance-constraint • Step 1: the distance from a point Pon body i to a point C defined in the GRF stays constant and equal to some known value c3 • Step 2: Identify • Step 3: • Step 4: • Step 5:

  4. Example 3.2.1 • An example using absolute coordinate constraints:simple pendulum

  5. Example 3.2.2 • An example using an absolute angle constraint:slider along x-axis

  6. Attributes of a Constraint[it’ll be on the exam] • 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?

  7. [handout]Example 3.1.3

  8. Example 3.1.3 • Note that when passing through the origin, the algebraic constraintsfail to specify the actual kinematics of the mechanism • Translating plain English into equations is not always straightforward • Translating plain English into the right equations: Unexpected problem when passing through origin…

  9. 3.3 Relative Constraints

  10. Loose Ends:Switching representation between two Reference Frames with rotation matrices Ai and Aj, respectively 10

  11. Loose Ends, Continued:Notation (related to changing representation from Aj to Ai) 11

  12. Loose Ends, Final Slide(related to changing representation from Aj to Ai) • For later reference, it is useful to recall that, • Therefore 12

  13. Vector between Pi and Pj • Something that we’ll use a lot: the expression of the vector from Pi to Pj in terms of the generalized coordinates q

  14. Relative x-constraint • Step 1: The difference between the x coordinates of point Pj and point Pi should stay constant and equal to some known value C1 • Step 2: Identify • Step 3: • Step 4: • Step 5:

  15. Relative y-constraint • Step 1: The difference between the y coordinates of point Pj and point Pi should stay constant and equal to some known value C2 • Step 2: Identify • Step 3: • Step 4: • Step 5:

  16. Relative angle-constraint • Step 1: The difference between the orientation angles of the LRFs associated with bodies i and j should stay constant and equal to some known value C3 • Step 2: Identify • Step 3: • Step 4: • Step 5:

  17. Relative distance-constraint • Step 1: The distance between the points Pj (on body j) and Pi(on body i) should stay constant and equal to some known value C4 • Step 2: Identify • Step 3: • Step 4: • Step 5:

  18. 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:

  19. 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:

  20. Attributes of a Constraint (1)[it’ll be on the exam] • 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? Think about it…

  21. 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.)

  22. 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

  23. [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

  24. Example 3.3.4 • Consider the slider-crank below. Come up with the set of kinematic constraint equations to kinematically model this mechanism

  25. Composite Joints (1) • Revolute-Revolute • Also called a coupler • Practically eliminates need of connecting rod • Attributes: • Location of points Pi and Pj • Distance dij of the massless rod • Revolute-Translational • Eliminates the intermediate body • Attributes: • Distance c • Point Pj (location of revolute joint) • Axis of translation vi’ • Just a means to eliminate one intermediate body whose kinematics you are not interested in

  26. 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:

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