6-IC - AME 352 INSTANT CENTERS Instant Center of Velocities Instant center of velocities is a simple graphical method for performing velocity

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AME 352 INSTANT CENTERS P.E. Nikravesh 1 Instant Center of Velocities Instant center of velocities is a simple graphical method for performing velocity analysis on mechanisms. The method provides visual understanding on how velocity vectors are related. In this lesson we review the fundamentals of instant centers. A Single Link and The Ground Assume that the link shown is pinned to the ground at O . The link has a known angular velocity ± (assume CCW). The magnitude of the velocity of any point on the link, such as A , B , or C , can be computed as V = R The direction of the velocity vector is determined by rotating the position vector R 90 ± in the direction of . Using the “rotated vector” notation, we have V = ± R (v.1) This equation represents both the magnitude and the direction of the velocity vector. Note that the position vector has a constant length. This problem can also be stated as: For a link that is pinned to the ground at O , the velocity of a point such as A is given. Determine the angular velocity of the link and the velocity of point B (or C ). The magnitude of the angular velocity is obtained by measuring the magnitude of vector R AO and then the above equation is used to obtain the angular velocity. After finding , the velocity of B is determined. 1, i I (i) A B C O R C,O V C B,O R B V A,O R A V The pin joint that connects link i to the ground can be viewed as two coinciding points: point O i on the link and point O 1 on the ground (the ground is always given index 1 ). Point O i has the same velocity as point O 1 . Obviously, since the velocity of O 1 is zero, the velocity of O i is zero as well. These two points that are on two different bodies but coincide have identical velocities—they form an instant center between the two bodies. This instant center is denoted as I 1, i (or I i ,1 ). In this example, the instant center between link i and the ground is an actual pin joint. As we will see next, an instant center may be an imaginary pin joint. A side note: In the example shown below, the given velocity is incorrect! The velocity of A must be perpendicular to R AO . O Another incorrect problem statement is shown below. Velocity of A suggests a CW rotation but velocity of B suggests the opposite. O C V C
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AME 352 INSTANT CENTERS P.E. Nikravesh 2 In the example shown here, link i is not pinned to the ground. Assume that the velocities of two or more points on the link are given. The following observations can be made: a) The normal axes to the velocity vectors, each passing through its corresponding point, intersect at one point, denoted as I 1, i . b) The distances of these points from I 1, i yield V A R A , I 1, i = V B R B , I 1, i = V C R C , I 1, i ± = ² c) All velocity vectors give the same sense of direction for the angular velocity (CCW in this illustration).
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This note was uploaded on 09/08/2010 for the course AME 352 taught by Professor Nikravesh during the Fall '08 term at University of Arizona- Tucson.

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6-IC - AME 352 INSTANT CENTERS Instant Center of Velocities Instant center of velocities is a simple graphical method for performing velocity

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