KINEMATICS OF PARTICLES Kinematics of Particles This lecture introduces Newtonian (or classical)...

KINEMATICS OF

PARTICLES

Kinematics of Particles

This lecture introduces Newtonian (or classical) Mechanics. It concentrates on a body that can be considered as a single particle. This simply means that the size and shape of the body will not affect the solution of the problem.

The lecture discusses the motion of a particle in 3-D space

After this lecture, the student should understand the following concepts:•Newtonian Mechanics in terms of statics and dynamics•Understand the logical division of Dynamics into kinematics and kinetics•Solve problems in kinematics of particles

What is Mechanics?

•Science that describes and predicts the conditions of rest or motion of bodies under the action of forces

Mechanics

Rigid Bodies Deformable Bodies Fluids

Statics: concerns the equilibrium of bodies under the action of forcesDynamics: concerns the motion of bodies

Kinematics: concerns the geometry of motion independent of the forces that produce the motionKinetics: concerns the relationship between motion, mass, and forces

Review of Basic Concepts

Space = geometric region in which events take place. For general engineering applications, most machines will operate and move in 3-D space. Other examples:•Motion in a straight line, e.g. along the the x-axis 1-D space•Motion in a plane, e.g. along the x-y plane 2-D space

The concept of space is associated with position and orientation. Any point in 3-D space can be defined by 3 coordinates: x, y, and z measured from a certain reference point. The orientation of a machine can be defined by 3 rotational angles , , and about : x, y, and z axes respectively. These coordinates refer to a “System for Referencing” the position and orientation.•Both position/oritnation and time have to be used to define an event in space•Mass is used to characterize the bodies in space

Newtonian Mechanics Assumptions

•There exists a primary inertial frame of reference fixed in space•Measurements made w.r.t. this reference is absolute•Time, space and mass are absolute•Interaction between particles is instantaneous

The assumptions are invalid if velocities involved are of the same order as the speed of light!For most engineering problems of machines on earth’s surface, the assumptions are valid.For rockets and space-flight trajectories, using the assumptions may result in large errors

Inertial Frame

An inertial frame is one in which the principles of Galileo and Newton holds true, i.e. a body will remain at rest or continue with uniform velocity in a straight line unless it is compelled to change its state of rest or uniform rectilinear motion by some external influence.

A frame of reference is NOT inertial if a body not acted upon by outside influences accelerates on its own accord.

All frames moving with rectilinear velocity with respect to the inertial frame are also inertial frames.

Frame {a} = (X,Y,Z,t) is fixed and frame {b} = (e1,e2,e3,t1) is moving at constant velocity (v) in the direction of the positive X-axis, i.e. Tvv 00

1e 2e

3e

At t=t1=0, the two frames are together.

1e 2e

3e

At t1= t=t*, frame {b} has moved away from frame {a}

X-axis Y-axis

Z-axis

vt*

Galileo’s relative principle

Let the position of particle “P” referenced w.r.t frame {a} be (x, y, z) and w.r.t. frame {b} be (x’, y’, z’)


1e 2e

3e

1e 2e

3e

At t1= t1=t*

X-axis Y-axis

Z-axis

vt*

','*,' zzyyvtxx zzyyvtxx ','*,'

P


zzyyvtxx ','*,'

zzyyvxx ',','

zzyyvxx abapbp ',',///

abapbp vvv ///

Velocity of P in frame {b}

Velocity of P in frame {a}

Velocity of frame {b} relative to {a}

zzyyxx ',','

apbp aa //

Acceleration unchanged


zzyyvxx ',','

A body un-accelerated in the frame {a} is also un-accelerated in all frames moving with constant velocity w.r.t. frame {a}

Galileo’s relative principle:

We will use the right handed system of Cartesian coordinates to define a frame of reference:

X-axis Y-axis

Z-axis

}ˆ,ˆ,ˆ{}ˆ,ˆ,ˆ{ 321 kjieee :are the unit vectors for X,Y and Z-axis respectively

1e 2e

3e

X Y

Z

x

y

z

Right-handed system

O

Note that the unit vectors for the right handed Cartesian reference frame are orthonormal basis vectors, i.e.

,2,1,0,20)sin(

,2,1,22

0)cos(

nnbaba

nnbaba

The cross and dot products are defined as follow:

Right-handed system

X

ZY

Note the cyclic cycle of the right hand system

1ˆˆˆ kji

Unit vectors

jikikjkji ˆˆˆˆˆˆ,ˆˆˆ

0ˆˆˆˆˆˆ ikkjjiPerpendicular to each other

Position

X-axis Y-axis

Z-axis

With a frame of reference established, we can define the position of a particle “A” w.r.t. the frame at any instance of time using vectors:

Particle “A”

The vector (e.g. ) defines the position of particle “A” w.r.t. frame {a}. In the e.g., the particle is 1 unit along the positive x-axis, 2 units along the positive y-axis and 3 units along the positive z-axis

TR ]3,2,1[

kjiR T ˆ3ˆ2ˆ1]3,2,1[

This is called the parametric description of the position vector

O

Frame {a}=(X,Y,Z)

Path

X-axis Y-axis

Z-axis

The changes in position of a particle “A” with time w.r.t. the frame of reference can be described by a path:

Particle “A”at time t1


Path of particle “A”

The position of particle “A” along the path at any instance of time can be represented by

Tzyx tRtRtRR )](),(),([

E.g. : at time t=t1=1, the particle is at the point [1, 1, 1]T and at time t=t2=2, the particle is at the point [2, 4, 8]T

TtttR ],,[ 32

Average Velocity


Given

Average velocity between two points is defined as a vector w.r.t. the reference frame

Let and denote the position at time t1 and t2 respectively. The AVERAGE velocity of a particle “A” between time t1 and t2 w.r.t. the frame of reference can be defined as:

tR

tRtRttt

tRtRvav

)()(1)()(

1212

12

)( 1tR

)( 2tR

E.g. : At time t1=1, At time t2=2,

TtttR 32

TavT

T

vtR

tR]731[

842)(

111)(

2

1

Instantaneous Velocity

The instantaneous velocity of a particle “A” at any point along the path w.r.t. the frame of reference can be defined as:

T

zyx

dttRd

dt

tRd

dt

tRdRv

))((

,))((

,))((


Given

E.g. : The velocity is At time t1=1, the instantaneous velocity is [1, 2, 3]T At time t2=2, the instantaneous velocity is [1, 4, 12]T

TtttR 32

ktjtittRvT ˆ)3(ˆ)2(ˆ1321 22

Instantaneous velocity at any point is a vector defined w.r.t. the reference frame. It is tangential to the path at that point, i.e. along t

Average vs. Instantaneous Velocity

Average and instantaneous velocity are not the same. Below shows the path of a particle between two position vectors at time t1 and t2:

X-axis Y-axis

Z-axis

Direction of the instantaneous velocity at time t1 (tangential to the path)

Direction of instantaneous velocity at time t2 (tangential to the path)

)( 2tR)( 1tR

Rtvav

Direction of the average velocity

t Changes with time

Average vs. Instantaneous Velocity

Notice that if the time interval between t1 and t2 becomes smaller, i.e. t0, then 0R

t becomes smaller

In this case, the average velocity will approaches the instantaneous velocity at t1: i.e.

avtt

vtR

v

00limlim

Notice that the direction of will be tangential to the curve in the limit as t0, i.e. the instantaneous velocity is tangential to the path. The tangential vector is called

R

t

)( 1tR

)( 2tR R

)( 1tR

)( 2tR

0R

Speed and Velocity

Speed and velocity are not the same! Velocity is a vector (it has both magnitude an direction). Speed “v” is a scalar. Speed only refers to the magnitude of the velocity, i.e..

Example: The instantaneous velocity of a particle is Tv ]123[

The instantaneous speed of the particle has no direction :

14123 222 v

Just as there are instantaneous and average velocities, there are also instantaneous and average speed.

RRRRRv

PRRRvRRRR

zyx

zyxzyx

222

Instantaneous Acceleration

The instantaneous acceleration of a particle “A” at any point along the path w.r.t. the frame of reference can be defined as:

T

zyx

dttvd

dt

tvd

dt

tvdPva

))((

,))((

,))((

Given

Instantaneous acceleration at any point is a vector defined w.r.t. the reference frame.

Tzyx tvtvtvv )](),(),([

E.g. : The acceleration is At time t1=1, the particle acceleration is [0, 2, 6]T At time t2=2, the particle velocity is [0, 2, 12]T

Tttv ]3,2,1[ 2

ktjitva T ˆ)6(ˆ2ˆ0]6,2,0[

Arc Length

X-axis Y-axis

Z-axis

The total distance traveled by the particle “A” between time t1 and t2 is described by the arc length “s”:



“s” is the distance traveled

2

1

t

tdtRRs

E.g.

TtttR )cos()sin(

22

2

21)(sin)(cos

1)sin()cos(

1

0

1

0

22

2

1

t

t

t

t

T

ts

dtdtRRs

ttRR

ttR

Find the distance traveled between t1=0 and t2=1 sec.

Arc Length, Speed and Velocity

The arc length is given as: 2

1

t

tdtRRs

The instantaneous speed is defined as: RRv

Therefore, it is obvious that instantaneous speed is:dtds

v

But the instantaneous speed is the magnitude of the instantaneous velocity, which is tangential along the path, i.e. along the vector In this case, we can also define the instantaneous velocity as

t

tvv ˆ

where

dssRd

R

Rvv

t)(ˆ

Tangential, Normal and Binormal vectors

tGiven we can defined the tangential vector

Using the arc length “s”, we can defined the normal vector as

ds

td

dstd

dstd

nˆ

ˆ

ˆˆ

where is called the curvature

The binormal vector is defined as ntb ˆˆˆ

The three unit vectors are orthonormal basis vectors and form a right handed reference frame. Together, they are called the trihedron.

bnt ˆ,ˆ,ˆ

R

Trihedron

The trihedron can be determined as follow:

R

Rt

ˆ

RR

RRb

ˆ tbn ˆˆˆ

The curvature can be found using

The radius of curvature is defined as

1

The torsion is defined as

2

)(

RR

RRR

dstd

R

RR ˆ3

Trihedron Example

Given find the trihedron at time t T

tttR

32

32

Solution:

TttttR

Rt 2

421

1

1ˆ

TttR 21

14

12ˆ24

2

tt

tt

RR

RRb

T

TtR 210

1

2

210

1

ˆˆˆ

det

2

2 t

t

t

tt

kji

RR

Trihedron Example

tt

t

tt

tt

tt

kji

tbn

2

1

2

2

ˆˆˆ

detˆˆˆ3

4

3

2

2

Story so far:

1

1,1

1

1ˆ24

2

42

2

tttt

tt

ttt

TT

14

1,12

14

12ˆ24

2

24

2

tttt

tt

ttb

TT

Cylindrical Coordinates

X-axis Y-axis

Z-axis

A position vector can be defined using a Cartesian reference frame as

kajaiaR zyxˆˆˆ

ax

ay

az

We let

karrR

rrjaia

z

yx

ˆˆ

ˆˆˆ

x

y

y

x

a

a

ar

ar

)tan(

)sin(

)cos(

Note that:

X-axis Y-axis

Z-axis

r

az

Quick Review of circular motion

A quick review of velocity in planar circular motion: consider a particle that moves in a circle with a fixed angular velocity

r

rv

The direction of the velocity is always tangential to the curveThe magnitude of the velocity is

rv

r

dtds

v

rs


kr ˆˆˆ define a new coordinate system called the cylindrical coordinates. If we look at “r” and in the x-y plane:

X-axis Y-axis

Z-axis

r

az

karrrR

karrR

z

z

ˆˆˆ

ˆˆ

X-axis

Y-axis

r

rv

rvr

Similarly, replacing r with vr and v where

karrrrrR

kaaaraaR

z

zrr

ˆˆ)2(ˆ)(

ˆˆ)(ˆ)(2

2121


vr

rva r 1

rva rr 1

X-axis

Y-axis

v22 rvar

rrva 2

X-axis

Y-axis

rv

rv

r


karrrrrR

karrrR

karrR

z

z

z

ˆˆ)2(ˆ)(

ˆˆˆ

ˆˆ

2

Summary:

Spherical Coordinates

X-axis Y-axis

Z-axis

A position vector can be defined using Cylindrical coordinates as karrR z

ˆˆ

We let

RRR

RRkarr z

ˆ

ˆˆˆ

)sin(

)cos(

Rr

Rz

Note that:

X-axis Y-axis

Z-axis

r

az R


If we look at “R” and in the ‘r-R’ plane:

ˆˆR define a new coordinate system called the spherical coordinates. We know that:

)sin(Rrv

X-axis Y-axis

Z-axis

R

r

r-axis

Z-axis

R

Rv

RvR

RRRRR

RRR

ˆˆ)sin(ˆ

ˆ

Similarly we can replace “R” with vR, v and v and repeat the analysis to get:

)(sin

)cos(2)sin(2

)cos()sin(2

ˆˆˆ

222

2

RRRa

RRRa

RRRa

RaaaR

R

R

RRRRR

RRR

ˆˆ)sin(ˆ

ˆ


Summary

This lecture concentrates on a body that can be considered as a particle and discusses the motion of a particle in 3-D space

The following concepts were covered:•Newtonian Mechanics in terms of statics and dynamics•The logical division of Dynamics into kinematics and kinetics•Problems in kinematics of particles

In the treatment of a body as a particle, the shape and size of the body is not considered.

KINEMATICS OF PARTICLES Kinematics of Particles This lecture introduces Newtonian (or classical)...

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