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162A 2 - Position and Displacement

162A 2 - Position and Displacement - 162A Position and...

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162A. Position and Displacement in Mechanisms Chapter 2: Position & Displacement
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Chapter 2 square6 Position and Displacement
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Notation square6 Consider a fixed reference coordinate system O xyz or simply O xy if we have a 2-d reference system. O is the origin from which positions are measured square6 The position of a point P in the coordinate system is represented as R which is a vector PO verbalized as “R P to O”. Remember: tip-to-tail. This R PO vector is sometimes simply written as R P , when it is w.r.t. the origin. square6 Vector R P is written with R in bold (or with a line above or below the letter) to connote it is a vector. Without bold (or a line), it connotes just the magnitude
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Terms square6 Absolute Position of a point is its position with respect to the origin of the coordinate system square6 Relative Position between two points P and Q can be expressed as position difference between those two points R PQ = R PO - R QO square6 Apparent Position of a point P is the position of the point P relative to the observer square6 Absolute and Relative Positions are related as : R PO = R QO + R PQ
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We’ll employ two methods for position analysis square6 Graphical Method box2 Intuitive box2 Inexact square6 Complex Algebra Method box2 Analytic box2 Exact
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Before that…. The Loop Closure Equation square6 Loop Closure Equation: A vector equation representing all or part of mechanism formulated according to the fact: The vector sum of a closed polygon equals zero. square6 That is, the sum of vectors to close the loop is 0 , or,
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Loop-Closure Equation square6 Applying the loop closure equation to the hand-operated clamp R BA + R CB + R DC + R AD = 0
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The above loop closure equation can be derived… y X R B = R A + R BA R C = R B + R
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