1 - Intro and Kinematics of Particles (1)

8/12/2019 1 - Intro and Kinematics of Particles (1)

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Engineering Mechanics deals with the state of a

body at rest or motion of a body caused by the

action of forces.

Statics deals with the action of forces on bodiesat rest.

Dynamics deals with the motion of bodies under theaction of forces.

Kinematics study of motion without reference to

the forces which cause the motion. Deals with

position, velocity, and acceleration in terms of time

Kinetics relates the action of forces on bodies

to their resulting motions

Ken Youssefi 1MAE


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A body , not necessarily small, where its motion can be

characterized without considering its size and orientation.Rotation of the body about its own axis is neglected.

Particle

Rigid bodyThe body is called a rigid body If its rotation about its

own axis cannot be neglected.

Ken Youssefi 2MAE


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Problem Solving Technique

Identify all given data

Identify the desired results

Draw the necessary diagrams

Identify the dynamics principles applied to the

problem

Clearly state all assumptions

Use your technical judgment (common sense) to

determine if the answer is reasonable.

Report the answer with the same accuracy as the

given data

Careless solutions that cannot be read easily are of

little or no valueKen Youssefi 3MAE


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Position of a point in space

coordinate systems

Rectangular (Cartesian)

coordinates,x, y, andz

Cylindricalcoordinates, r, , andz

Spherical

coordinates,R, ,and

Ken Youssefi 4MAE


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Planar (2D) Motion

Rectangular (Cartesian) coordinates,x andy

Polar coordinates, rand

xx

y

y

P

O

r

Ken Youssefi 5MAE

x = r cos

y = r sin


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t

ttt

dt

d

t

}{}{lim

0

vvva

Position

Velocity

Acceleration

Kinematics

2

2

dt

d

dt

d rva

Ken Youssefi 6MAE

r = r{t}


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Graphical representation

Velocity at time t is the slope of theline tangent to displacement curve.

Acceleration at time t is the slope of

the line tangent to velocity curve.

Given the displacement curve,

velocity and acceleration curves

could be obtained graphically.

Ken Youssefi 7MAE


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Rectilinear Motion

Particle moving along a straight line from P to P

+s-s

O PP

s s

At time t +t the particle has moved to P

v = lims

t

t 0

}{tv

dt

ds

}{tadt

dv v

ds

dv

dt

ds

ds

dv

dt

dva

Ken Youssefi 8MAE


9/43

Bdttvts }{}{

t

t

v

v oo

dttadv }{ t

to

o

dttavtv }{}{

t

t

s

s oo

dttvds }{ t

to

o

dttvsts }{}{

Rectilinear Motion

}{tvdt

ds

Relationship between displacementandvelocity

Adttatv }{}{

}{tadt

dv

Relationship between velocity and acceleration

Ken Youssefi 9MAE


10/43

10

Special Cases: Constant Velocity or Acceleration

Ifs =s{t} and v is constant: t

oo dtvsts 0}{tvsts oo}{

ooo ssavv 222

s

s

v

voo odvvdsa

If v = v{s} and a is constant:

t

o dttvsts 0}{}{

221}{ tatvsts ooo

Ifs =s{t}and a is constant:

t

oo dtavtv 0}{

tavtvoo

}{ If v = v{t} and a is constant:

Ken Youssefi MAE

t

to

o

dttavtv }{}{ t

to

o

dttvsts }{}{


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11

Example

Ken Youssefi MAE

The acceleration of a train (at least during the interval from t= 2

to t= 4) is a = 2t, and it is known that at t= 2 its velocity v = 50.

What is the trains velocity at t= 4? (Assume all units are already

consistently in SI units).


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12

Example

The acceleration of a trainduring the interval from t= 2to t= 4 is a = 2t, and it isknown that at t= 2 itsvelocity vis 50.

What is the trains velocity att= 4? (Assume all units arealready consistently in SIunits).

Image(s) from Engineering Mechanics: Dynamics, 3rd ed.,

by A. Bedford & W. T. Fowler, Prentice Hall, 2002.

Ken Youssefi MAE


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13

Example

After deploying its drag parachute, the airplaneshown has an acceleration a = -.004v2 (m/s).

Determine the time required for thevelocity to decrease from 80 m/s to 10 m/s.

What distance does the plane cover duringthat time?

Ken Youssefi MAE


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14

Example

After deploying its drag parachute, the airplaneshown has an acceleration a = -.004v2 (m/s).

Determine the time required for thevelocity to decrease from 80 m/s to 10m/s.

What distance does the plane coverduring that time?

Image(s) from Engineering Mechanics: Dynamics, 3rd ed.,

by A. Bedford & W. T. Fowler, Prentice Hall, 2002.

Ken Youssefi MAE


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Ken Youssefi MAE 15

Curvilinear Motion 3D

Particle moving along a curved path.

Rectangular coordinates is particularly useful for describing thecurvilinear motion

kjir zyx

kjivzyx

vvv

kjia zyx aaa

kjirvdtdz

dtdy

dtdx

dtd

dt

dxvx

dt

dyvy

dt

dzvz

kjiv

adt

dv

dt

dv

dt

dv

dt

dzyx

dt

dva x

x dt

dva

y

y dt

dva z

z


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17/43

Ken Youssefi MAE 17

t

t xxox odttavtv }{)(}{

xox vv )(

t

t xo odttvxtx }{}{

tvxtx xoo )(}{

t

t yyoy odttavtv }{)(}{

gtvtv yoy )(}{

t

t yo odttvyty }{}{

2

21)(}{ gttvyty yoo

Curvilinear Motion Projectile Motion

Vertical motionHorizontal motion

Velocity

Displacement


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18

Example

The skier leaves the 20ledge with a speed of 10

m/s (along the upper

inclined surface).

How far along the lower

incline does she land?

(What is the distance d?)

Ken Youssefi MAE


19/43

Example

Ken Youssefi MAE 19

(vo )x = vo cos20

(vo )y = -vo sin20

tvxtx xoo )(}{

2

21)(}{ gttvyty yoo

(vo ) = 10 m/s

xo = 0

yo = 0

x = vo cos20 t = 10 cos20 t= 9.4 t

y = - vo sin20 t - gt2 = 10 cos20 t= - 3.42 t - 4.9 t 2

y = - 3 -x

t= 1.6 s

x = 15 m

y = -18 m

The skier leaves the 20 ledge witha speed of 10 m/s (along the upperinclined surface).

How far along the lower inclinedoes she land? (i.e. What is thedistance d?)

x

x

d= 21.3 m

(vo )x

(vo

)y vo


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Ken Youssefi MAE 20

Curvilinear Motion Normal and Tangential coordinates

These coordinates provide a natural description for curvilinear

motion and are frequently the most convenient coordinates to use.

Unit vector ut is tangent to the curve at P,

and points in the direction of increasings.

Unit vector un is always perpendicular to

ut, and points toward the concave side.

Coordinates measures positionof point P along its path with

respect to origin O.

Velocity vis always tangent tounit vector ut.

tuvdt

ds


21/43


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Ken Youssefi MAE 22

Radius of Curvature and Normal Component of Acceleration

dds )(

vdt

d

dt

ds

nt

dt

dv

dt

dvuua

nt

v

dt

dvuua

2

dt

dvat

2v

an

Tangential component Normal component


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Ken Youssefi MAE 23

Circular Path

tuvdtds nt

dtdv

dtdv uua

Velocity and acceleration

Rs

Rdt

dR

dt

dsv

dt

d

dt

d

Rdt

dR

dt

dvat

Magnitude of tangentialacceleration

2

Rdt

dvan

Magnitude of normal acceleration


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24

Example

A motorcycle starts from

rest on a circular trackof radius 400 m, and

accelerates with a

tangentialacceleration

of at= 2 + 0.2tm/s.

How much distance

does the motorcycle

cover in the first 10

seconds?

What is the magnitudeof totalacceleration |a|

at the 10-second mark?

Ken Youssefi MAE


25/43


26/43

Ken Youssefi MAE 26

Curvilinear Motion Polar Coordinates

dt

drv

rrur

dt

dva

u

u

dt

d

dt

d r rdt

d

dt

du

u


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Ken Youssefi MAE 27

dt

dr

dt

dr

dt

d rr

uu

rv

uuv

dt

dr

dt

drr

rrur

dt

drvr

dt

drv


Velocity

u

u

dt

d

dt

dr


28/43

Ken Youssefi MAE 28

dt

d

dt

dr

dt

dr

dt

d

dt

dr

dt

d

dt

dr

dt

rd

dt

d rr

uuu

uu

va

2

2

2

2

uua

dt

d

dt

dr

dt

dr

dt

dr

dt

rdr 22

22

2

2

uuvdtdr

dtdr r


Acceleration

r

dt

d

dt

du

u

u

u

dt

d

dt

d r


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29

Example

A point P moves along the spiral path r= (0.1)

meters, where is in radians. The angular position

= 2trad, where tis in seconds, and r= 0 at t= 0.Determine the magnitudes of the velocity and

acceleration of P at t= 1 s.

Ken Youssefi MAE


30/43


31/43

31

Coupled-Motion

Some problems are disguised as being

more complicated, even though there is

really only one degree of freedom.

These problems can be simplified by using

geometry to relate the motions of each

moving point, then taking derivates to

relate velocity and acceleration.

To express the dependency properly,

position coordinate s needs to be defined

with respect to a fixed datum, with

deliberately assigned positive direction.

Questions:

How are sA, sB, and h related?

How are vA and vB related?

How are aA and aB related?

Ken Youssefi MAE

l


32/43

32

Relative Motion

If pointA is a moving particle, it

can be assigned a local

reference frame that translateswith respect to a fixed

reference frame with origin O.

From the perspective of point

A, the relative motion of other

points such as B is often moreuseful than motion with

respect to a fixed reference

frame.

Vector analysis provides an

effective way to express therelative motion between

particles, based on the motion

of each with respect to a fixed

reference frame.

ABAB /rrr ABAB /vvv

ABAB /aaa Ken Youssefi MAE


33/43

Example


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Ken Youssefi MAE 34

ExampleUse polar coordinate system

aP =

=

d2

rOP

dt2

+ (u)OPd

2

OPdt

2

d OPdt

rOP +d rOP

dt2

OP length is constantd

2

rOP

dt2 = 0and

d rOP

dt= 0

rOPd OP

dt( )

2(ur)OP

rOP d OP

dt( )2(ur)OPaP =

rQPd Qp

dt( )

2(ur)QPaQ/P =

= rOPOP2 i

rQPQP2

(cos45) i rQPQP2 (sin45)j

aQ = aP + aQ/P = rOPOP2 rQPQP

2(cos45) i rQPQP

2 (sin45)j


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Ken Youssefi MAE 35

aQ = - (2)(4)2 ( 2 )(8)2(cos45) i -( 2 )(8)2(sin45)j

Example

aQ = 96 i 64j

aQ = (96)2

+ (64)2

= 115.38 ft/s2

= tan-1 6496

= 33.7 O

33.7O

aQ

aQ = aP + aQ/P = rOPOP2 rQPQP

2(cos45) i rQPQP

2 (sin45)j

P bl F l ti d R i


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Ken Youssefi MAE 36

Problem Formulation and Review

The ability to select the most appropriate method is a key

to the successful formulation and solution of engineering

problems.

It is important to recognize the following breakdowns:

1. Type of motion Rectilinear motion

(one coordinate needed to describe the motion)

Plane curvilinear motion

(two coordinate needed to describe the motion)

space curvilinear motion

(three coordinate needed to describe the motion)

P bl F l ti d R i


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Ken Youssefi MAE 37


2. Reference axes

Fixed reference axesAbsolute motion - axes attached to the surface of the

earth are sufficiently fixed for most engineering problems

Moving reference axes

Relative motion - axes attached to a moving body that is notattached to a fixed reference

Fixed axes



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Ken Youssefi MAE 38

Problem Formulation and Review3. Coordinates

A very important step in formulation of a problem

The options: Rectangular (Cartesian) coordinates

x, y 2D motion

x, y, z 3D motion

Normal and tangential coordinatesn, t 2D curvilinear motion

Polar coordinates

r, 2D motion

Cylindrical coordinatesr, , z 3D motion

Spherical coordinates

R, , 3D motion



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Ken Youssefi MAE 39


The choice is dictated by how the motion is generated or

measured

Straight line (rectilinear) motion

Type of Motion Type of Coordinates

Curvilinear motion

Curvilinear motion, when

measurements are made

along the curve

Normal and tangential (n, t)

Sliding and rotating motion

at the same time

Polar (r, )

Space curvilinear motion Rectangular (x, y, z)

Cylindrical (r, , z)

Spherical (R, , )

Polar (r, )

Rectangular (x, y)

Rectangular, x(or s)

Examples


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Ken Youssefi MAE 40

Examples


Polar (r, )

Rectangular (x, y)

Normal and tangential (n, t)Spherical (R, , )

Examples


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Ken Youssefi MAE 41

ExamplesPolar (r, )

Polar (r, )

Rectangular (x, y)


Examples


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Ken Youssefi MAE 42

Examples


Polar (r, )

Rectangular (x, y)


Polar (r, )

Examples


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Examples

Spherical (R, , )


Spherical (R, , )

1 - Intro and Kinematics of Particles (1)

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