Kinematics Lecture 1 Particle

7/31/2019 Kinematics Lecture 1 Particle

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KinematicsKinematics is the branch of mechanics,

which treats of motion as such, withoutregard to its cause, that is Kinematicsdeals only with geometrical aspect of

the motion.


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MotionMotion is change in the location or positionof an object in the time domain with respectto other body (reference body).

To study motion of any object we have tointroduce at least one reference system.

The reference system consists of:

Datum (reference point); Coordinate system;

A system of time reading


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z

y

x

O

Tree is of the

coordinate system


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z

y

x

O

Point O is

of the

coordinatesystem Oxyz

M

M0


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It is supposed that at least one motionless

frame of reference exists. This frame of

reference is called inertial. In engineering

practice the heliocentric referencesystems is used as inertial. The

geocentric reference systems can be

used as inertial if displacement of body ofinterest is substantially smaller then Earth

radius.


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Particle motion.

Trajectory, representationof particle motion

The motion of a particle is completelydescribed if the position of the particle isgiven as a function of the time.

Trajectory (Path) is the line along whichparticle travels. If path is straight line, themotion is called rectilinear; if path iscurved line the motion is curvilinear.


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Trajectory of VOYAGERs motion


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Trajectory of Space Shuttles crash

landing


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Descriptions of particle motion

There are three main methods of the

particle motion descriptions. They are:

1. vector,2. coordinate,

3. natural.


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Vector way of a particle motiondescription

The position of a particle in three-

dimensional space is specified by its

vector-position onnecting the origin

of reference, the point , with a point M,where the particle is situated. Vector

position is determined by its magnitude

(module) and direction. The motion iscompletely described when vector is

known as a function of the time.

r


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1r(t ) M3

M4

O

0r(t )

M0

M1

M2

4r(t )

2r(t )

3r(t )


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Coordinate methods of a particlemotion description

The position of a particle in three-

dimensional space is specified by

particles coordinates: x, y, z,or , or other.Motion of a particle is completely

described if particle coordinates areknown as a function of the time.

, , z


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z

y

x

O

M

xM

yM

zM

M

M

M

x f ( t )

y f ( t )

z f ( t )

1

2

3


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z

y

x

O

M

f ( t )

f ( t )

z f ( t )

1

2

3

zM


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Relation between vector and

coordinate methods of the particle

motion descriptions can be

expressed as

r xi yj zk

f ( t )i f ( t ) j f ( t )k 1 2 3


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Natural way of a particle motiondescription

It can be realized only if the particletrajectory is given (constrained

motion). Natural method supposes thatmotion is completely described if position

of particle on its trajectory is given as

function of the time.


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To realize the natural method of a

particle motion representation is to

introduce the reference point M0 (the

position of the particle on thetrajectory at the moment whent=0);

the positive direction ofcurvilinear coordinate reading;

time dependence of curvilinearcoordinate:

.The last expression is called

lawofmotion.

M

M0

0

f ( t )


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Relation between coordinate and

natural methods of the particlemotion description is:

d ( dx ) ( dy ) ( dz )

x y z dt

2 2 2

2 2 2

t

x y z dt 2 2 2

0


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Average velocity of a particle during

interval is ratio

Particle Velocity

t

av

rvt


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Particle Velocity

Instantaneous velocity is the vectorposition time derivative

0

limt

r drv r

t dt

,

.

x y z

drv v i v j v k

dt

v x i yj zk

2 2 2

x y zv v v v


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z

yx

O

M0

0r(t )

r

r(t+ t) M1

M1

r ( t + t )

r

drv

dt

M1

r

v

r ( t + t )


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Particle Acceleration

Instantaneous acceleration is the vectorvelocity time derivative or vector position

second time derivative

tv dv d r W lim r ,t dt dt

2

20

x y z

dv

W W i W j W k x i yj zk.dt

x y zW W W W .2 2 2


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z

yx

O

M1

1r(t )

M2r ( t ) 2

v1

v2

v


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z

yx

O

r(t)

r (t dt )

v ( t )

dv

v( t dt )

W

Acceleration points toward the side of curve concavity.

v(t dt )

tM

t dtM


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Velocity and Acceleration

in Terms of Path variables(natural way of particle motion

representation)


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To realize the natural method of a

particle motion representation is to

introduce the reference point M0

(the position of theparticle on the trajectory

at the moment when t=0); the positive direction ofcurvilinear coordinatereading;

time dependence of

curvilinear coordinate:

.

The last expression iscalled lawofmotion.

M

M0

0

f ( t )


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Natural coordinate system

(Frenet coordinate system)


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MM1

n

b

M00

tangent to

the trajectory

Principal normalto the trajectory

osculating plane

r

O

1

C C is center of curvature


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Unit vectors ofnatural coordinate system

is unit vector of the tangent to the trajectory

points to the direction of increasing;

is unit vector of the principal normal, pointsto the center of curvature of the trajectory;

is unit vector, forms the right hand triad withand

n

bn


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Particle Velocity

Instantaneous velocity is the vectorposition time derivative

tr drv lim r t dt0


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t t

t t

r rv lim lim

t tr

lim lim .

t

0 0

0 0

t

r drlim ;

d0

t

dv

t dt0

lim .


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Velocity is vector along the tangent to thetrajectory with magnitude equals .

Velocity is in the same direction with ifparticle moves in the direction of increasing

.

Velocity is opposite to the if particle moves

in the direction of decreasing .

v 1

0

0


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M v

M00

0

M

v

M00

0


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Particle Acceleration

2

2

dv d d W

dt dt dt d d d

. 2

dt dt dt


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d d d

. (3 )dt d dt

d ?d

d v, 4dt


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Mtangent tothe trajectory

C

C is center of curvature

osculating circle

tangent tothe circle

is radius of curvature


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d 1

v. 5

d

22

2d d 1W n. (7 )dt dt

d d d v v . (6 )

dt dt d


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Acceleration components:

tangent and normal

dW

dt

2

2;

The first component in eq. (7) is inthe direction of tangent to the path. Itis called tangential acceleration. Itcharacterizes variation of the velocity

magnitude.The second component in eq. (7) is inthe osculating plane at right angles tothe path tangent and pointing toward

the center of curvature. Itcharacterizes the variation of thevelocity direction.

n

v

W n

2


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W

M

M0 0

n

,n

WWW

22

nWWW

v

nW W


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Character of particle motion

andV W or

and

0 0

0 0

Motion is accelerated

Motion is decelerated

Motion is with uniform

velocity

andV W or

and

0 0

0 0

in fin W for t t t 0


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Conditions under which

acceleration components

equal zero

W

if

0

0

,

.

Speed is constant during a time

interval;

Speed runs up its extreme

value.

At the point of inflection;

On the straight trajectory.

nW

if

0,

.


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Normal and tangent accelerations in terms

of rectangular coordinates for 2-D case

x x y y

x y

v W v W v WW

v v v

2 2

x y y x n

x y

v W v W v W v W v W W

v v v v2 2

sin( , ).

x x y y n

x y z

x y

v W v W W W W W W W

v v

22 2 2 2

2 2.

Kinematics Lecture 1 Particle

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