Relative-Motion Analysis: Velocity

• Translation onlyKinematics– Position

– Velocity

– Acceleration

A

B

ABAB rrr /

AB vv

AB aa

x

y

rA

rB

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

Review of Cross Products• See Section 4.2 for full details

zyx

zyx

BBBAAAkji

BA

ˆˆˆ

or

To use, must define right-hand x, y, z coordinate system

Example ProblemIf rod AB slides along the horizontal slot with a velocity of 60 ft/s, determine the angular velocity of link BC at the instant shown. (F16-11, 48 rad/s) What about the velocity of the pin at

C, and the angular velocity of wheel OC at that instant?(104 ft/s up)

Special Case for Velocity Solution

Rolling without slip

Can also have slip, in that instance direction of vA is at least known but magnitude unknown

Example ProblemA bowling ball is cast on the “alley” with a backspin of ω = 10 rad/s while its center O has a forward velocity of vO = 8 m/s. Determine the velocity of the contact point A in contact with the alley. (16-58, 9.20 m/s to the right)

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)