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ME451 Kinematics and Dynamics of Machine Systems

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ME451 Kinematics and Dynamics of Machine Systems

Relative Constraints

Composite Joints

3.3

September 25, 2013

Radu Serban

University of Wisconsin-Madison

- 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

- 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:

- 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:

- Page 67 (sign)

- Page 68 (unbalanced parentheses)

- 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?

- 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.)

- 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

- For a revolute joint:

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

- 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

- 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: