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Net6 - Kinematics of Rigid Bodies Applications Analysis of cams gears shafts linkages connecting rods etc Definition A rigid body is a special case

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Kinematics of Rigid Bodies Applications: Analysis of cams, gears, shafts, linkages, connecting rods, etc. Definition: A rigid body is a special case of a system of particles wherein the distances between all particles remain unchanged. For a system of particles with general three-dimensional motion prescribed for each particle, there are up to 3 N possible degrees of freedom for a system of N particles. With the restriction on distance between two points A and B so that | ~ r A/B | = constant, the only possible change with respect time for ~ r A/B is due to rotation. Since the rotational degrees of freedom can also be represented by a vector, the general description of rigid body motion can be reduced to (i) 3 translations and 3 rotations for three-dimensional problems and (ii) 2 translations and 1 rotation for two-dimensional (plane) problems. Motion of an Arbitrary Point on a Rigid Body If the translation components of a rigid body is given at a reference point B and the rotation vector is also known for the rigid body, then the motion at any arbitrary point A on the rigid body can be determined completely. Consider first the relationship ~ r A = ~ r B + ~ r A/B , in which ~ r B is the known motion of the reference point B while ~ r A is the motion of the arbitrary point A to be determined. The relative displacement ~ r A/B has the additional restriction of | ~ r A/B | = constant for a rigid body. A B O ~r B A / To obtain the velocity at point A , take derivative of the position vector ~ r A with respect to time to yield d~r A dt = B dt + A/B dt or ~v A = B + A/B where the relative velocity A/B is defined as A/B = A/B dt = × ~ r A/B for a rigid body because | ~ r A/B | = constant and the change occurs only due to rotation. Hence, for a rigid body A = B + × ~ r A/B . The above equation indicates that once B and are known, the velocity everywhere on the rigid body can be calculated easily. To obtain the acceleration vector, perform another time derivative to yield d~v A dt = B dt + d~ω dt × ~ r A/B + × A/B dt or ~a A = B + × ~ r A/B + × ( × ~ r A/B ) . –6/1–
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Example – Plane Rigid Body Kinematics A rigid bar AP is connected to sliders A and B as shown in the figure. Slider A is restricted to move only in the horizontal direction while slider B can move only in the vertical direction. If at the instant shown, slider A is moving to the left with a velocity of 4 mm/sec, find ~v B , and P . P 300 mm 200 mm B 45 o A Solution: Since points A , B and P are all on the same rigid body, we can write B = A + × ~ r B/A and P = A + × ~ r P/A . Of course, we can also use P = B + × ~ r P/B if the velocity at B is known. The angular velocity, , is the same at any location along the rigid bar. With A given as {- 4 , 0 , 0 } T , let B = 0 v B 0 ,~ ω = 0 0 ω and ~ r B/A = - 200 2 / 2 200 2 / 2 0 .
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This note was uploaded on 04/16/2008 for the course CE 325 taught by Professor Docwong during the Spring '08 term at USC.

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Net6 - Kinematics of Rigid Bodies Applications Analysis of cams gears shafts linkages connecting rods etc Definition A rigid body is a special case

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