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ME451 Kinematics and Dynamics of Machine Systems

ME451 Kinematics and Dynamics of Machine Systems. Driving Constraints 3.5 February 24, 2009. © Dan Negrut, 2009 ME451, UW-Madison. Before we get started…. Last Time We looked at some kinematic pairs Rack and pinion Cam-Follower Cam Flat-follower In each case, we followed five steps:

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ME451 Kinematics and Dynamics of Machine Systems

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  1. ME451 Kinematics and Dynamics of Machine Systems Driving Constraints 3.5February 24, 2009 © Dan Negrut, 2009ME451, UW-Madison

  2. Before we get started… • Last Time • We looked at some kinematic pairs • Rack and pinion • Cam-Follower • Cam Flat-follower • In each case, we followed five steps: • Analyze the physical joint and identify the relative motion allowed • Derive the constraint equations that imply the relative motion allowed • Compute constraint Jacobian q • Get  (RHS of velocity equation) • Get  (RHS of acceleration equation, this is complicated) • Today: • Wrap up the last kinematic constraints • Cam – flat follower • Point-curve constraint • Start talking about driving constraints 2

  3. 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 3

  4. Point-Follower Pair • Framework (Step 1): • Pin P is attached to body i and can move in slot attached to body j. • NOTE: the book forgot to mention what gj is (pp.85, eq. 3.4.32) • It represents the tangent to the slot in which P is allowed to move • The location of point P in slot attached to body j is captured by angle j that parameterizes the slot. • Therefore, when dealing with a point-follower we’ll be dealing with the following set of generalized coordinates: • Body i: xi, yi, fi, • Body j: xj, yj, fj, j 4

  5. Point-Follower Pair • Step 2: Constraint Equations (q,t) = ? • Step 3: Constraint Jacobian q = ? • Step 4:  = ? • Step 5:  = ? 5

  6. End Kinematic ConstraintsBegin Driving Constraints 6

  7. Driving Constraints • The context • Up until now, we only expressed time invariant kinematic constraints • These constraints describe the physical structure of a machine/mechanism • Normally the mechanism has a certain number of DOFs • Some additional time dependent constraints (“drivers”) are added to control these “unoccupied” DOFs • You thus control the motion of the mechanism 7

  8. Kinematic Drivers Absolute Coordinate Drivers • Absolute x-coordinate driver • Absolute y-coordinate driver • Absolute angle driver Relative Coordinate Drivers • You see these more often… • Relative distance driver • Revolute-rotational driver • Translational-distance driver 8

  9. Absolute Driving Constraints • Indicate that the coordinate of a point expressed in the global reference frame assumes a certain value that changes with time • X-position • Y-position • Orientation angle 9

  10. Absolute Driver Constraints • Step 2: Constraint Equations (q,t) = ? • Step 3: Constraint Jacobian q = ? • Step 4:  = ? • Step 5:  = ? Absolute Coordinate Drivers • Very simple to compute this information 10

  11. End Absolute Driving ConstraintsBegin Relative Driving Constraints 11

  12. Relative Distance Driver • The distance between Pi and Pj is prescribed as a function of time: ||Pi Pj||=C(t) 12

  13. Relative Distance Driver • Step 2: Constraint Equations • Step 3: Constraint Jacobian q = ? (see Eq. 3.3.8) • Step 4:  = ? • Step 5:  = ? 13

  14. Example: Specifying Relative Distance Drivers • Generalized coordinates: • Motions prescribed: • Derive the constraints acting on system • Derive linear system whose solution provides velocities 14

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