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ESPN Sports Science Blake Griffin. Relative-Motion Analysis: Velocity. d r A. d θ. y. Transl. & Rotation (General Plane Motion) Position Velocity (time deriv ) where and ( ω is rotation of member about A). r B /A. r B /A (new). d r A. d r B /A. d r B. r A. r B. x. A. B. - PowerPoint PPT Presentation
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ESPN Sports Science Blake Griffin
Relative-Motion Analysis: Velocity
• Transl. & Rotation(General Plane Motion)– Position
– Velocity (time deriv)
where
and (ω is rotation of member about A)
A
B
ABAB rrr /
ABAB vvv /
x
y
rA
rB
dθdrA
drB
drA
rB/A rB/A (new)drB/A
ABAB rv //
For our problems, we will just need to plug in for each of these variables to get vB.and often ω
Instantaneous Center of Zero Velocity
Rolling without slip (not always)• Relate velocity of two points on right body
• What if choose a point A which is instantaneously stationary (i.e. vA = 0)
• Can we find an instant point with this property to relate to?
ABAB rvv /
rvrv BABB ;/
What if we want velocity at each point on rim?
(each point will instantaneously rotate about axis fixed to that point)
Instantaneous Center of Zero Velocity
• Does an I.C. always exist?– At some instant, yes– Consider curvilinear motion in particle
mech.– For rigid body?– I.C. need not be ON the body?
• To find I.C.– Identify instantaneous direction
of velocity for each point– Draw perpendicular lines from each– Intersection is I.C. at that instant
• To solvevPoint = ωrPoint/IC
Graphic Examples
To find I.C.Identify instantaneous directionof velocity for each pointDraw perpendicular lines from eachIntersection is I.C. at that instant
To solvevPoint = ωrPoint/IC
Example ProblemIf link CD has an angular velocity of 6 rad/s, determine the velocity of point E on link BC and the angular velocity of link AB at the instant shown. (16-89, 6 rad/s CCW, 4.76 m/s, 40.9° above – x)
Relative-Motion Analysis: Acceleration
• Transl. & Rotation(General Plane Motion)– Position
– Velocity (dr/dt)
– Acceleration (dv/dt)
A
B
ABAB rrr /
x
y
rA
rB
dθdrA
drB
drA
rB/A rB/A (new)drB/A
nABtABAB aaaa //
ABAB rvv /
ABAB vvv /
Rotation about A
ABtAB ra //
ABnAB ra /2
/ Magnitude of Components:
Relative-Motion Analysis: Acceleration
BCBCBC rraa /2
/ How do we relate motion of
these?
Example ProblemThe hydraulic cylinder extends with a velocity of 1.5 m/s and an acceleration of 0.5 m/s2. Determine the angular acceleration of link ABC and the acceleration of end C at the instant shown. Point B is pin connected to the slider block. (16-122, 41.6 rad/s2, 38.2 m/s2, 39.4° below x)