Peaucellier

Peaucellier - O 4 PS(5 cos 2 2 2 2 = − − d b bx x where...

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The Peaucellier Mechanism The Peaucellier Mechanism is shown below. We will now prove that if 4 2 2 O O Q O = , ( 1 ) S O R O 4 4 = , ( 2 ) and SP RP QS QR = = = , ( 3 ) then point P will move along a straight line. 2 O 4 O 2 3 4 5 6 7 8
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We define the lengths , the angles d b a , , β α , , and the point T , as indicated in the figure below. In fact the motion of point P is along a vertical. It is thus suffice to prove that cos 4 P O is a constant, independent of the angle . 2 O 4 O 2 3 4 5 6 7 8 1 1 Q R S P T a a b b d d d d Firstly note that the lines Q O 4 and QP are collinear. This fact follows from the triangular similarity QR O QS O 4 4 and QSP QRP , and hence the angle . o 180 4 = QP O Secondly note that C P O Q O = 4 4 , ( 4 ) where C is a constant. This follows from the fact that the cosine rule applied to the triangles and O yields an identical expression 2 2 d b =
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Unformatted text preview: O 4 PS (5) cos 2 2 2 2 = − + − d b bx x where Q O x 4 = when the triangle is considered, and QS O 4 P O x 4 = when considering . It thus follows that PS O 4 ∆ Q O 4 and P O 4 are two solutions of the quadratic equation (5). But the product of the roots of a quadratic equation equals to the free term, i.e., C d b P O Q O = − = ⋅ 2 2 4 4 . Thirdly note that β cos 2 4 a Q O = , (6) since is isosceles and hence 2 4 QO O ∆ T O 2 is the perpendicular bisector of Q O 4 . Finally, it follows from (4) and (6) that a d b a C Q O C P O 2 cos cos 2 cos cos 2 2 4 4 − = = = , (7) is indeed independent of , and hence P moves along a vertical line....
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Peaucellier - O 4 PS(5 cos 2 2 2 2 = − − d b bx x where...

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