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Determine “how fast” parts of a machine are moving. Important when concerned with the timing of a mechanism. First step in acceleration analysis. Velocity Analysis Section 4

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Velocity Linear –Straight line, instantaneous speed of a point. v A = 30 in/s 30 0 Rotational –Instantaneous speed of the rotation of a link. 2 = 72 rad/s, ccw 2

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Linear and Angular Velocity For points on the same link v = r Points have linear velocity (v) Links have rotational velocity ( ) B vAvA vBvB A 22 2 rBrB rArA

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Relative Velocity Two points on a rigid body can only have a relative velocity: Perpendicular to the line that connects them. A B A B The motion of B relative to A (v B/A )

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Relative Velocity Method Relative velocity equation is used to form vector polygons, and determine velocities of key points. v i = v j +> v i/j vivi v i/j vjvj

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Problem 4-1 Determine the velocity of the piston, as the crank rotates at 600 rpm, cw. 2 in 8 in 65 0

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Problem 4-7 Determine the rotational velocity of the crushing ram, as the crank rotates clockwise at 60 rpm. 60 mm 400 mm 180 mm 360 mm

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Point on a Floating Link Must use simultaneous velocity equations v x = v i +> v x/i v x = v j +> v x/j X i j v X/i vivi vjvj v j/i v X/j vXvX Use the Velocity Image

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Determines the amount that parts of a machine are “speeding-up” or “slowing down”. Important because a force is required to produce accelerations. Acceleration Analysis

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Acceleration of a point, a, is caused by a change in velocity. Velocity can change its: Magnitude tangential acceleration In direction of velocity if part is accelerating. Opposite direction of velocity if part is decelerating. Acceleration of a Point

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Velocity can also change its: Direction normal acceleration directed towards center of rotation (or relative rotation). Acceleration of a Point

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Angular Acceleration Angular acceleration of a link, is influenced by the tangential acceleration. B atBatB vBvB A 22 2 22 anBanB

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Relative Acceleration Two points on a rigid body can only have a relative tangential acceleration: Therefore, the relative normal acceleration is: Perpendicular to the line that connects them. Parallel to the line that connects them.

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Relative Acceleration Method Relative acceleration equation is used to form vector polygons, and determine the acceleration of key points. a i = a j +> a i/j Breaking each component into normal and tangential components gives: a i n +> a i t = a j n +> a j t +> a i/j n +> a i/j t

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Acceleration Analysis Reminders Points on translating links have no normal acceleration. Points on links that rotate at constant speed have no tangential acceleration.

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Problem 4-31 Determine the acceleration of the piston, as the crank rotates clockwise, at a constant rate of 600 rpm. 2 in 8 in 65 0

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Problem 4-37 Determine the angular acceleration of the crushing ram, as the crank rotates clockwise at a constant rate of 60 rpm. 60 mm 400 mm 180 mm 360 mm

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Point on a Floating Link Must use simultaneous acceleration equations a x = a i n +> a i t +> a x/i n +> a x/i t a x = a j n +> a j t +> a x/j n +> a x/j t i j X vivi vjvj vXvX v X/i v j/i v X/j a n X/j atjatj aXaX atiati aniani a t X/j a n X/i a t X/i a n j/i a t j/i

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Problem 4-72 For the windshield wiper linkage shown, determine the acceleration of the cg of the connecting link. The motor is running at 30 rpm clockwise. 14 in 2 in 13 in 3.5 in 45 0 7.25 in 6.7 in

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Acceleration Image Must use total acceleration atjatj aXaX a n X/j atiati aniani a t X/j a n X/i a t X/i a n j/i a t j/i aiai a X/j a j/i a X/i

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Kinematics of Rigid Bodies

Kinematics of Rigid Bodies

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