226-Lect 2 - Grashof condition Week 2 (1) 4 bar mechanism...

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1 Week 2 Week 2 (1) 4 bar mechanism (cont’): Grashof condition ( 2.13, 3.3-3.5 ) (2) Cams ( 8.1-8.4 of Norton, and ch ) Week 2 Grashof condition • 4 bar linkage is the simplest possible pin- jointed mechanism for 1 DOF controlled motion. Thus, it should be among the first solutions to motion control problems to be investigated • The Grashof condition is a very simple relationship that predicts the rotation behavior based only on the link lengths Week 2 Classes of kinematic chain Let: s=length of shortest link L=length of longest link P=length of one remaining link Q=length of other remaining link Then: s+L < P+Q: class I case s+L > P+Q: class II case s+L = P+Q: class III case Nb: class I and III cases are both Grashof, which means at least one link is capable of making a full revolution (crank); class II is non-Grashof and no link is capable of a complete revolution relative to any other link Week 2 Class I case: s+L<P+Q • Crank-rocker : Ground either link adjacent to the shortest • Double-crank : Ground the shortest link. Both links pivoted to ground make complete revolutions • Double-rocker : Ground the link opposite the shortest. Both links pivoted to ground oscillate and only the coupler makes a full revolution
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2 Week 2 Class I case: s+L<P+Q Week 2 Class II case: s+L>P+Q • Triple-rockers : no link can fully rotate Week 2 Class III case: s+L=P+Q • Double-cranks or crank-rockers : “change points” twice when the links all become colinear Week 2 Limiting conditions • Toggle positions : they are determined by the colinearity of two of the moving links • In a toggle (triangular) position, it doesn’t allow further input motion in one direction from one of its rocker links • A fourbar triple-rocker mechanism has four and a Grashof double-rocker mechanism has two toggle positions. • A Grashof fourbar crank-rocker linkage has two stationary (toggle) positions
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3 Week 2 Transmission angle • Transmission angle μ : then angle between the output link and the coupler. It is taken as the acute angle (<=90 0 ). It varies continuously from some minimum to some maximum value as the linkage goes through its range of motion • Most machine designers try to keep the minimum transmission angle above about 40 0 to promote smooth running and good force transmission Week 2 Add a dyad to control motion •C a s e 1 : r o c k e r o u t p u t : for a Grashof crank- rocker mechanism, the output function is defined as two discrete angular position of the rocker • Case 2: coupler output : two positions of a line in the plane are defined as the output. This solution will frequently lead to a triple-rocker. the fourbar triple-rocker can be motor driven by an addition of a dyad (twobar chain), which results in a Watt sixbar
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226-Lect 2 - Grashof condition Week 2 (1) 4 bar mechanism...

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