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P. Nikravesh, AME, U of A Instant Centers of Velocities Introduction Instant Centers of Velocities for a Six-bar Mechanism Part 1: Finding Instant Centers O2O2 O4O4 O6O6 In this presentation we find the instant centers for a six-bar mechanism. As shown this six- bar is made of two four-bar mechanisms that share one moving link and the ground.

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P. Nikravesh, AME, U of A Instant Centers of VelocitiesBook keeping We note that this system has three pin joints connected to the ground and four pin joints can move. We first number the bodies 1, …, 6 where 1 is the ground. The numbering order is totally up to us. There are 6 (6 − 1) ∕ 2 = 15 instant centers. For book keeping, we construct a circle with six link numbers on its circumference. 1 2 4 3 5 6 (2) (3) (4) (5) (6) (1) O2O2 O4O4 O6O6 ► ►

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P. Nikravesh, AME, U of A Instant Centers of Velocities (2) First four-bar We first consider the four-bar mechanism on the left highlighted in red. There are six IC’s for this four-bar. We locate them following the procedure we have reviewed in a previous presentation: First four-bar ► 1 2 4 3 5 6 (3) (4) (5) (6) (1) I 1,2 I 2,3 I 3,4 I 4,1 I 2,4 I 1,3

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P. Nikravesh, AME, U of A Instant Centers of Velocities (2) Second four-bar Next we consider the four-bar mechanism on the right highlighted in red. There are six IC’s for this four-bar. However, one of the six IC’s is already found since the two four bars share links 1 and 4. We locate the other five IC’s: Second four-bar ► 1 2 4 3 5 6 (3) (4) (5) (6) (1) I 1,2 I 2,3 I 3,4 I 4,1 I 2,4 I 1,3 I 4,5 I 5,6 I 6,1 I 4,6 I 1,5

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P. Nikravesh, AME, U of A Instant Centers of Velocities (2) Eleven instant centers So far we have found eleven out of fifteen IC’s. In the next few slides we locate the remaining four IC’s one at a time. Eleven instant centers 1 2 4 3 5 6 (3) (4) (5) (6) (1) I 1,2 I 2,3 I 3,4 I 4,1 I 2,4 I 1,3 I 4,5 I 5,6 I 6,1 I 4,6 I 1,5

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P. Nikravesh, AME, U of A Instant Centers of Velocities (2) IC number 12 Next we locate I 2,6. The line associated with this center on the circle can assist us in deciding what other centers to use. This line makes a triangle with the centers I 1,6 and I 1,2. We construct a line between I 1,6 and I 1,2. The line for I 2,6 on the circle also makes a triangle with the centers I 2,4 and I 4,6. We construct a line between I 2,4 and I 4,6. The intersect is I 2,6. IC number 12 ► 1 2 4 3 5 6 (3) (4) (5) (6) (1) I 1,2 I 2,3 I 3,4 I 4,1 I 2,4 I 1,3 I 4,5 I 5,6 I 6,1 I 4,6 I 1,5 ► ► ► ► ► I 2,6

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P. Nikravesh, AME, U of A Instant Centers of Velocities (2) IC number 13 Next we locate I 2,5. This line makes a triangle with I 1,5 and I 1,2. We construct a line between I 1,5 and I 1,2. The line for I 2,5 on the circle also makes a triangle with I 2,4 and I 4,5. We construct a line between I 2,4 and I 4,5. The intersect is I 2,5. Note that we could have used the line going through I 2,6 and I 5,6. IC number 13 ► 1 2 4 3 5 6 (3) (4) (5) (6) (1) I 1,2 I 2,3 I 3,4 I 4,1 I 2,4 I 1,3 I 4,5 I 5,6 I 6,1 I 4,6 I 1,5 I 2,6 ► ► ► ► I 2,5 ►

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P. Nikravesh, AME, U of A Instant Centers of Velocities (2) IC number 14 Next we locate I 3,5. This line makes a triangle with the centers I 3,4 and I 4,5. We construct a line between I 3,4 and I 4,5. The line for I 3,5 on the circle also makes a triangle with I 1,3 and I 1,5. We construct a line between I 1,3 and I 1,5. The intersect is I 3,5. Note that there are other choices to find this center. IC number 14 ► 1 2 4 3 5 6 (3) (4) (5) (6) (1) I 1,2 I 2,3 I 3,4 I 4,1 I 2,4 I 1,3 I 4,5 I 5,6 I 6,1 I 4,6 I 1,5 I 2,6 ► ► ► ► I 2,5 ► I 3,5

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P. Nikravesh, AME, U of A Instant Centers of Velocities (2) IC number 15 Next we locate I 3,6. This line makes a triangle with I 3,4 and I 4,6. We construct a line between I 3,4 and I 4,6. The line for I 3,6 on the circle also makes a triangle with I 1,3 and I 1,6. We construct a line between I 1,3 and I 1,6. The intersect is I 3,6. Note that there are other choices to find this center. IC number 15 ► 1 2 4 3 5 6 (3) (4) (5) (6) (1) I 1,2 I 2,3 I 3,4 I 4,1 I 2,4 I 1,3 I 4,5 I 5,6 I 6,1 I 4,6 I 1,5 I 2,6 ► ► ► ► I 2,5 ► I 3,5 I 3,6

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P. Nikravesh, AME, U of A Instant Centers of Velocities (2) All the IC’s Now we have all the fifteen instant centers. In the next presentation we use these centers to find any velocity of interest. All the IC’s 1 2 4 3 5 6 (3) (4) (5) (6) (1) I 1,2 I 2,3 I 3,4 I 4,1 I 2,4 I 1,3 I 4,5 I 5,6 I 6,1 I 4,6 I 1,5 I 2,6 I 2,5 I 3,5 I 3,6

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