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ME321 Kinematics and Dynamics of Machines

ME321 Kinematics and Dynamics of Machines. Steve Lambert Mechanical Engineering, U of Waterloo. Kinematics and Dynamics. Position Analysis Velocity Analysis Acceleration Analysis Force Analysis. We will concentrate on four-bar linkages. q. l. s. p. Four-Bar Linkages.

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ME321 Kinematics and Dynamics of Machines

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  1. ME321 Kinematics and Dynamics of Machines Steve Lambert Mechanical Engineering, U of Waterloo

  2. Kinematics and Dynamics • Position Analysis • Velocity Analysis • Acceleration Analysis • Force Analysis We will concentrate on four-bar linkages

  3. q l s p Four-Bar Linkages • What type of motion is possible?

  4. q l s p Grashof’s Criteria • Used to determine whether or not at least one of the links can rotate 360o • the sum of the shortest and longest links of a planar four-bar mechanism cannot be greater than the sum of the remaining two links if there is to be continuous relative rotation between the two links. • s + l < p + q

  5. l q l s s q p p Grashof’s Criteria s + l > p + q Non-Grashof Mechanism s + l < p + q Grashof Mechanism

  6. l q s p l q s p Grashof Mechanisms (s+l < p+q) Crank-Rocker Rocker-Crank Shortest link pinned to ground and rotates 360o

  7. s q q p l p l s Grashof Mechanisms (s+l < p+q) Drag-Link - Both input and output links rotate 360o Double-Rocker - Coupler rotates 360o

  8. l s q p Change-Point Mechanism S+l = p+q

  9. s q p l Non-Grashof Mechanisms • Four possible triple- rockers • Coupler does not rotate 360o

  10. coupler  output link input link Transmission Angle • One objective of position analysis is to determine the transmission angle,  • Desire transmission angle to be in the range: • 45o <  < 135o

  11. coupler output link input link in Position Analysis • Given the length of all links, and the input angle,in, what is the position of all other links? • Use vector position analysis or analytical geometry

  12. Vector Position Analysis • ‘Close the loop’ of vectors to get a vector equation with two unknowns • Three possible solution techniques: • Graphical Solution • Vector Components • Complex Arithmetic

  13. Graphical Solution • Draw ground and input links to scale, and at correct angle • Draw arcs (circles) corresponding to length of coupler and output links • Intersection points represent possible solutions

  14. R y, i 3  x, j 4 2 O4 O2 Vector Component Solution ‘Close the loop’ to get a vector equation:

  15. Vector Component Solution (con’t) Rewrite in terms of i and j component equations: • These represent two simultaneous transcendental equations in two unknowns:  3 and 4 • Must use non-linear (iterative) solver

  16. iy R  x Complex Arithmetic • Represent (planar) vectors as complex numbers • Write loop equations in terms of real and imaginary components and solve as before

  17. 3 4 2 O4 O2 B  A   Analytical Geometry • Examine each mechanism as a special case, and apply analytical geometry rules • For four-bar mechanisms, draw a diagonal to form two triangles • Apply cosine law as required to determine length of diagonal, and remaining angles

  18. Limiting Positions for Linkages • What is the range of output motion for a crack-rocker mechanism?

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