mer212: kinematics of rigid bodies - union...

MER212:

Rotation (16 3)

Kinematics of Rigid Bodies Rotation (16.3) Absolute Motion Analysis (16.4)

Union CollegeMechanical Engineering

MER212: Rigid Body Mechanics 1

Rotation A state of motion of a body for which A state of motion of a body for which

different points of the body exert different displacementsp



Planar Motion Each particle in the Each particle in the

body remains in a single planesingle plane

Motion of the Rigid Body is determined byBody is determined by Location of one point Orientation of one line Orientation of one line

in the plane



Rotation About a Fixed Axis

• Angular Motion

• Angular Position• Angular Position

• Angular Displacement

• Angular Velocity

• Angular Acceleration


ESC020: Rigid Body Mechanics 4

Orientation of a Body Angle that a line makes with fixed Angle that a line makes with fixed

direction L ti f t i t Location of any two points



Angular Velocity The angular motion is The angular motion is

the same of every straight line on the gbody

CDCDABAB consttt

),(),(

ABCDABCDABCD

ABCD

dtd

dtd

)(



dtdt

Angular Acceleration Diff ti ti Differentiating

angular velocity with respect to with respect to time

dddd )()()()(

ABCDABCDABCD

ABCDABCD dtdtdtdt

)()()()(



ABCDABCDABCD

Example

A turntable attains its operating speed of 33 1/3 RPM in 5 revolutions after being turned on. Determine the initial angular acceleration o of the turntable if:angular acceleration o of the turntable if:

a) The angular acceleration is constant, =o = constant

b) Th l l ti d li l ith l b) The angular acceleration decreases linearly with angular velocity from o when =0 to o/4 when =33 1/3 RPM


ESC020: Rigid Body Mechanics 8

Omega TheoremThis theorem shows how to find the This theorem shows how to find the time derivative of any vector of constant magnitude whose direction constant magnitude whose direction changes as a result of rotation with a body in which it is fixed.body in which it is fixed.

This leads to the development of a This leads to the development of a method for taking the derivative of a UNIT VECTOR


R.B. Bucinell 9

UNIT VECTOR

Configuration Description:3D B d R t ti Ab t P i t3D Body Rotating About a Point

ˆ

y

j

Body Rotation Axis

Path of Particle on Body

A body is rotating about a fixed point O with an absolute angular velocity coinciding with the axis or rotation.

• The axis of rotation will move or precess

vd t

• The axis of rotation will move or precess

• Any point P located by describes a circular path perpendicular to the axis of rotation.

• With only one point fixed in space as in a top

rdr te

sinr y p p p

the axis of rotation will move or precess the path of P will be circular only for dt during this time the radius of the path is

th l f i

sinr

x i

r the angle of sweep is

the displacement of P is (arc length)dt

ˆsin tdr dt r e z

k

O


10

t

R.B. Bucinell

Velocity:3D B d R t ti Ab t P i t3D Body Rotating About a Point

ˆThe velocity is found by dividing the di l t b th ti i t l d dt

y

j

Body Rotation Axis

Path of Particle on Body

displacement by the time interval .

is perpendicular to the plane formed by the vectors and .

dr dt

v r

ˆsinˆsintt

dt r edrv r edt dt

1e

sinr

d t

vectors and .

• The right hand side of Equation 1 represents the definition of the cross-product.

r

v r r 2

v

dr

te

• may be the position vector to P from any origin on the axis of rotation regardless of whether the axis of rotation is fixed or moving.

If the axis has constant spatial

v r r 2

r

e

x i

r

zk

If the axis has constant spatial orientation (fixed), can change only in magnitude.

(will be collinear with ) For a moving axis

o will always change direction

e

e

O


11

o will always change directiono will possibly be changing

magnitude.o will have different direction

from

R.B. Bucinell

Si ifi f E ti 2Significance of Equation 2

ˆEquation 2 ( ) represents the

t t hi h t t l th t fi d v r r

y

j

rate at which a constant length vector fixed in a rotating body changes its direction

• is fixed in the rotating body joining particles A and B

e

BBody Rotation Axis

Path of Particle B

3

• Taking the time derivative of Equation 3

r

y

Path of Particle A

B Ar r 3

( ) ( )B Ad d r rdt dt

• Substituting in Equation 2

Br

A

B Ar r

( ) ( )( )B Ar r

x iAr

zk

( )B Ar r

4O


12R.B. Bucinell

THE OMEGA THEOREMTHE OMEGA THEOREM

ˆ Th ti d i ti f y

j

The time derivative of a constant length vector fixed in a rotating body is simply the cross product of the angular

e

BBody Rotation Axis

Path of Particle B

p gvelocity with the vector

r

y

Path of Particle A 5

Br

A

For a generic UNIT VECTOR ,The omega theorem gives it’s time derivative as

x iAr

zk

ˆ ˆe e 6O


13R.B. Bucinell

Derivatives of a Particle Traveling i Pl Ci l M tiin Planar Circular Motion

e ( )dv r r r

y

j ˆrer

eθ

v r r

ta r

d

( )v r r rdt

( )na r

r

( )

( )

da v vdt

d r

x i

( )r ( )rdt

r r

kz

k

k

( )

t n

r ra a


14

k

R.B. Bucinell

Alternate Derivation of the Derivative of a Unit Vector

Th it t f th The unit vectors for the rotating reference frame can be written in terms the inertial reference

y

ˆ

θ

ˆ ˆˆ cos ( ) sin ( )ˆ ˆˆ i ( ) ( )

re t i t j

t i t j

frame as:j

ˆre

r

ˆecosθ(t)

cosθ(t)

sinθ(t)

sinθ(t)

θ(t)

sin ( ) cos ( )e t i t j x

i

sinθ(t)θ(t)

The derivatives of these vectors can not be taken directly from the equations above.


15R.B. Bucinell

Directly Taking The Derivative ˆof the Unit Vector (2-D)ˆre

ˆ ˆˆ ˆ( ) cos ( ) sin ( )

r r

d de e t i t jdt dt

d d d d ˆ ˆ ˆ ˆcos ( ) cos ( ) sin ( ) sin ( )

ˆ ˆ( ) sin ( ) ( ) cos ( )

d d d dt i t i t j t jdt dt dt dt

t t i t t j0 0

( ) sin ( ) ( ) cos ( )

ˆ ˆ( ) sin ( ) cos ( )

ˆˆ ˆ ˆ( )

t t i t t j

t t i t j

t e e k e S R lt i E ti 6( ) rt e e k e Same Result as in Equation 6


16R.B. Bucinell

Directly Taking The Derivative ˆof the Unit Vector (2-D)ˆe

ˆ ˆˆ ˆ( ) sin ( ) cos ( )

d de e t i t jdt dt

d d d d ˆ ˆ ˆ ˆsin ( ) sin ( ) cos ( ) cos ( )

ˆ ˆ( ) cos ( ) ( ) sin ( )

d d d dt i t i t j t jdt dt dt dt

t t i t t j0 0

( ) cos ( ) ( ) sin ( )

ˆ ˆ ˆ ˆ( ) cos ( ) sin ( ) ( ) cos ( ) sin ( )

t t i t t j

t t i t j t t i t ˆˆ ˆ ˆ( )

j

t e e k e Same Result as in Equation 6( ) r rt e e k e Same Result as in Equation 6


17R.B. Bucinell

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