Motion Relative to Point-Attached Reference Frame

Relative Motion Velocity

Relative Motion - Acceleration

Instantaneous Centre of Zero

Velocity

Wheel Rolling Without Slip

Sliding Contact Between Two

Bodies

2 KinematicsofaRigidBody

x i", "y j", "

Bodyattachedframe

Pointattachedframe

x i' ,

y j' ,

O x i,

y j,

Basicfixedframe

A( x yA A, )

Referencepoint

d.o.f = 3

Independent coordinates: xA yA

Planar motion of a rigid slab

(1) Translation

(IF = constant)(2) Rotation about A

(IF xA, yA are constant)

Independent coordinates: xA yA

A-xytranslates

with A

Slab rotatesabout A-xy

Absolute vel=Re + En

A

O

BrBA

x'

y '

x

y

v A

Relative vel.

Entrained vel.

AB vv '

v A

Entraining point B on A-xy:

B

vBvB A/

AO

BrBA

x'

y'

x

y

v A

AABB vvv /

v A

vB A/vB

RelativeMotionAnalysis:Velocity

Velocity vB/A hasamagnitudeof vB/A =rB/A anda

directionwhichisperpendiculartorB/AABAB /vvv

RelativeMotionAnalysis:VelocityVelocity vB isdeterminedbyconsideringtheentire

bodytotranslatewithavelocityofvA,androtateaboutA withanangularvelocity

Vectoradditionofthesetwoeffects,appliedtoB,yieldsvB

vB/A representstheeffectofcircularmotion,aboutA.Itcanbeexpressedbythecrossproduct

ABAB /rvv

RelativeMotionAnalysis:VelocityVelocity PointAonlinkABmustmovealonga

horizontalpath,whereaspointBmovesonacircularpath

Thewheelrolls withoutslippingwherepointAcanbeselectedattheground

AO

y

B

aB A/aB

x'

y'

a A

a A

x

AABB aaa /

RelativeMotionAnalysis:Acceleration

Thetermscanberepresentedgraphically

= +

ABABAB /2

/ rraa

v rB A BA/ a r rB A BA BA/ ( ) Relative motion: (Circular)

Entrained motion

Av

Aa

,

AABB vvv / AABB aaa /

Absolute = Relative + Entrained

A wheel rolling without slip

Example

Example

G0.45 m

0.3 m

i

j

P v aGiven

Determine

Rolling without slipping

C

A

C

G

C

Av

0.45 m

0.3 m

i

j

Rolling without slipping vC = vC = 0

ACr CCAA vvv /

0 v

A relative to C= 0.75=1.33v

(AC)

(ans)(CW)

For the instant

ia janA iaG (AG) jAG 2GGAA aaa

/i=

C

A

(AG)t GAa /AGan GA 2/

a 0.75=1.33a

(ans)(CW)

G iaG 45.0

? ??

?Ca

C

A G iaG 45.0

Instantaneous center of zero velocity

For a slab in general planar motion,

there exists a point C on the slab

at any instant

with vC = 0.

C : Instantaneous Center of Zero Velocity

GC

A

i

j

vC = 0

Av

ACr CCAA vvv /

0

Av (AC)

Instant Center of zero vel.

Pure rotationAbout I.C.

Proportional to AC

For the instant

GC

A

i

j

vC = 0

ACr CCAA vvv /

0

Av

Av (AC)Pure rotationAbout I.C.

Instant Center of vel.

GC

A

i

j

vC = 0

ACr CCAA vvv /

0

Av

Av (AC)Pure rotationAbout I.C.

Instant Center of vel.

InstantaneousCenterofZeroVelocity VelocityofanypointB locatedonarigidbodycan

beobtainedifbasepointA haszerovelocity SincevA =0,thereforevB = xrB/A. PointA iscalledtheinstantaneouscenterofzero

velocity (IC)anditliesontheinstantaneousaxisofzerovelocity

Magnitude ofvB isrB/IC Duetocircularmotion,

direction ofvB mustbeperpendicular torB/IC

InstantaneousCenterofZeroVelocity

Forwheelrollingwithoutslipping,thepointofcontactwiththegroundhaszerovelocity

HencethispointrepresentstheIC forthewheelLocationoftheIC Velocity ofapointonthebodyisalways

perpendicular totherelativepositionvectorextendingfromtheIC tothepoint

ICislocatedalongthelinedrawnperpendiculartovA,distancefromA totheICisrA/IC =vA/

InstantaneousCenterofZeroVelocityLocationoftheIC ConstructAand Blinesegmentsthatare

perpendiculartovA andvB. Extendingtheseperpendiculartotheirpointof

intersection asshownlocatesthe IC ICisdeterminedbyproportionaltriangles

f fixed P vP

P

Sliding contact of a point along the boundary of a rigid slab

P fixed

f moves P

vP

f moves f moves

Velocity of P tangential to the curve

Velocity of P tangential to the curve,tooNo penetration

BC

200 mm

600 mm

G 30

= 0.5 rad/s

700 mm

D on the wall

D on AB

Example

AB?A

BC

600 mm

G 30

= 0.5 rad/sD on AB

D on the wall

vDA

= ?

Example

P

f

P fixed

P vP

f

P movesvP

vP

vP/P

Condition of sliding contact

vP/P nvP/P tangential to the curve

nNo penetration

fP vP/P

P

If f and P are both moving

f f

vP/P tangential to the curve always

BC

600 mm

G 30

= 0.5 rad/sD on AB

Example 2-5

D on the wall

vDA

I.C.

Sliding contact

Example

AB

C

200 mm

400 mm

G

D

30

0.5 rad/s AB

vA

vD

E

45

15 D

27.4 27.4

700 mm

I.C. of rod AB

AEvA

AB

)(BEv ABB

Example

Motion Relative to a Body-Attached

Reference Frame

Relative Motion Velocity

Relative Motion -

Acceleration

2 KinematicsofaRigidBody

P belongs to the Body fA

O

x

y

x

y

Av

Pv

P Coincideswith P

at the instant

PfP vv /P

PAAP rvv '

f

PAr

P Entraining pointP'Pv

?

'/ PPv PfP vv / 'Pv

Relative to f Relative to P=

'/ PfPP vvv

Abs. = Rel. + Ent.

Principle of Velocity Combination

A point / particle relative to a body

Acceleration of

AO x

y

A body f

f

f

P A particle P

GivenPa

Pa

f

f

Aa

Aa

f

AO

P

x

y

f

Pa

fAa

APAP aaa /

P relative to A

point A attachedTrans. Frame A-xy

x

y

AO

f

x

y

Pa

What is accel. of Prelative to f ?

fPa /

x

y

Body attachedFrame A-xy

P

f

Aaf

AP aa

If f is in translation+ rotation

AO x

y

Pa

x

y

Body attachedFrame A-xy

Pf

Aa

Introduce

P Entraining pointP

ff

fPP vv / 'Pv

Abs = Rel + Ent

Do we also have a aP P f / 'Pa ?Abs = Rel + Ent

Since

xy

P

O

erererera PrPPrPP 2 2

fPa /

xUsing Body attached Ref.

(O- x,y)

yPr

Relative accel.

rPer

xy

P

O

erP

rP er 2

erererera PrPPrPP 2 2 aP'

tPa '

nPa 'Entrained Accel.

P entraining point

xy

P

O

erererera PrPPrPP 2 2

CPa

erP 2

Coriolis Accel. fPf v /2

Acceleration of P

a aP P f / 'PaAbs = Relative + Entrained

fPf v /2

+ Coriolis

Principle of Acceleration Combination