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)