RELATIVE MOTION ANALYSIS: ACCELERATION

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RELATIVE MOTION ANALYSIS: ACCELERATIONToday’s Objectives:Students will be able to:1. Resolve the acceleration of a pointon a body into components oftranslation and rotation.2. Determine the acceleration of apoint on a body by using a relativeacceleration analysis.In-Class Activities: Check Homework Reading Quiz Applications Translation and RotationComponents of Acceleration Relative AccelerationAnalysis Roll-Without-Slip Motion Concept Quiz Group Problem Solving Attention Quiz

READING QUIZ1.If two bodies contact one another without slipping, and the points incontact move along different paths, the tangential components ofacceleration will be and the normal components ofacceleration will be .A) the same, the sameB) the same, differentC) different, the sameD) different, different2. When considering a point on a rigid body in general planemotion,A) It’s total acceleration consists of both absoluteacceleration and relative acceleration components.B) It’s total acceleration consists of only absoluteacceleration components.C) It’s relative acceleration component is always normalto the path.D) None of the above.

APPLICATIONSIn the mechanism for a window, linkAC rotates about a fixed axis throughC, and AB undergoes general planemotion. Since point A moves along acurved path, it has two componentsof acceleration while point B, slidingin a straight track, has only one.The components of acceleration ofthese points can be inferred sincetheir motions are known.How can we determine theaccelerations of the links in themechanism?

APPLICATIONS (continued)In an automotive engine, theforces delivered to thecrankshaft, and the angularacceleration of the crankshaft,depend on the speed andacceleration of the piston.How can we relate theaccelerations of the piston,connection rod, and crankshaftto each other?

RELATIVE MOTION ANALYSIS: ACCELERATION(Section 16-7)The equation relating the accelerations of two points on thebody is determined by differentiating the velocity equationwith respect to time.dvBdvAdv B/ AdtdtdtThese are absolute accelerationsof points A and B. They aremeasured from a set of fixedx,y axes.This term is the accelerationof B with respect to A andincludes both tangential andnormal components.The result is aB aA (aB/A)t (aB/A)n

RELATIVE MOTION ANALYSIS: ACCELERATION(continued)aB aA (aB/A)t (aB/A)nGraphically:The relative tangential acceleration component (aB/A)t is ( rB/A)and perpendicular to rB/A.The relative normal acceleration component (aB/A)n is ( 2 rB/A)and the direction is always from B towards A.

RELATIVE MOTION ANALYSIS: ACCELERATION(continued)Since the relative acceleration components can be expressedas (aB/A)t rB/A and (aB/A)n - 2 rB/A, the relativeacceleration equation becomesaB aA rB/A 2 rB/ANote that the last term in the relative acceleration equation isnot a cross product. It is the product of a scalar (square ofthe magnitude of angular velocity, 2) and the relativeposition vector, rB/A.

APPLICATION OF THE RELATIVE ACCELERATIONEQUATIONIn applying the relative acceleration equation, the two points used in theanalysis (A and B) should generally be selected as points which have aknown motion, such as pin connections with other bodies.In this mechanism, point B is known to travel along a circular path, soaB can be expressed in terms of its normal and tangential components.Note that point B on link BC will have the same acceleration as point Bon link AB.Point C, connecting link BC and the piston, moves along a straight-linepath. Hence, aC is directed horizontally.

PROCEDURE FOR ANALYSIS1. Establish a fixed coordinate system.2. Draw the kinematic diagram of the body.3. Indicate on it aA, aB, , , and rB/A. If the points A and Bmove along curved paths, then their accelerations shouldbe indicated in terms of their tangential and normalcomponents, i.e., aA (aA)t

1. Resolve the acceleration of a point on a body into components of translation and rotation. 2. Determine the acceleration of a point on a body by using a relative acceleration analysis. In-Class Activities: Check Homework Reading Quiz Applications Translation and Rotation Components of Accel

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