DescriptionEMCH-532 INTERM DYNA

FINAL EXAM

NOME:___________________

1. A particle is constrained to travel along the path y 2 = 4x, where x = (4t 4)m. Assuming

Cartesian coordinates x = x1, y = x 2, z = x3 with right-handed orthonormal vectors

de ned by E1, E2, E3, determine the following:

1.1. (10 points) Position vector of the particle.

1.2. (10 points) Velocity vector of the particle.

1.3. (10 points) Acceleration vector of the particle.

1.4. (10 points) The magnitude of the particle’s velocity and acceleration when t = 0.5 s.

2. A particle of mass m is supported by two elastic cords, each having a nominal length l and

tension T. Assuming that the transverse displacement y is much smaller than either l or

T /k, where the sti ness coe cient of each cord is k and the angle of de ection is θ, do the

following (neglect gravity):

2.1. (15 points) Draw the free-body diagram of the system above and derive a linear

equation for the particle’s motion.

Hint: assuming y ≪ l, sinθ can be replaced by the linear term

2.2. (15 points) Determine the natural frequency ωn of the system.

y

.

l

y + ω 2 y = 0.

Hint: consider the equation of motion for free vibration ··

3. Given an underdamped system with ωn = 5 rad/s, ζ = 0.1, x0 = 1 cm, v0 = 0,

fl

ffi

ff

fi

3.1. (30 points) Obtain the response amplitude A and phase angle ϕ.

Hint: For the case of free motion wherein F (t) = 0, the arbitrary constants of

the transient solution can be evaluated immediately from the above initial

conditions (see lecture notes on the transient solution for the underdamped

case).

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Work

Energy

É me

m

É de

É

II.IE

I

É E at

mi de

d th

2

É

t

Id

I

i

m

B

f

F df

s

T

I mus Im Vi

Lm

Wy

IW TB

WEIL

Assumption

nd

Pr

w

Etta

ter

m rat

sential

m

gg

f

rer

tire

WE

Ir wet

t

t

rivet rwÉ

her

Finery

dW

T

Pr dr

mriddr

Imu

fat

Im

e

V

IT

mrutdr tmrw dw

2

Neglecting the radial velocity

dw

Substituting

dt

2

in

i

2nfÉIdr MIÉdw

2ft 1

Integrating

ro

21nF

Solving for

E

w

w

Note that

In

we

Ftw

can

also

lmiw mr.ae

Principle of conservation of angular

momentum

Katie

Force

1

F dp

F’s function of

position only

a

2

The line integral

independent

of the path AB

is

A

From 1

ftp.df I

B

ft

di

F di

fW

o

o

Force F is conservative

F df

F df

W

A

Hence

do

s

B

y

du

VA

WB

a

fwatta

we.tt

Principle of conservation of mechanical

Potential Energy

dU

dx

d f

dx É

Since

t d xzÉz t d x

Therefore

É

du

Egg

F df

dx

de t

t

t Px dx

Pyd x

t Ptsd x

Px

Pxz

Px

F

É

Px É t RzÉtPxÉ

É

EYÉ

OW

EE

É

I

É

I É

or

15 17

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Kinematics of a Particle – Part 3

Dynamics of a Single Particle

Dr. Léo A. Carrilho

1

Contents

• Review: Coordinate Systems

• Cartesian

• Cylindrical

• Spherical

• Areal Velocity Vector

• Curvilinear Coordinate System

2

Coordinate Systems

Kinematics of a Particle

3

Kinematics of a Particle

Cartesian Coordinate System

• Right-handed orthonormal vectors

! , ” , #

• Given any vector b in 3

#

= & $ $

$%!

4

Kinematics of a Particle

Cartesian Coordinate System

• Position Vector

#

= & $ $

$%!

• Cartesian Coordinates of the Particle

! , ” , #

5

Kinematics of a Particle

Cartesian Coordinate System

• Position Vector:

= ! ! + ” ” + # #

• Velocity Vector:

= ̇ ! ! + ̇ ” ” + ̇ # #

• Acceleration Vector: = ̈ ! ! + ̈ ” ” + ̈ # #

6

Kinematics of a Particle

Cylindrical Coordinate System

• Position Vector:

= & + #

• Velocity Vector:

̇ ‘ +

=

̇ & +

̇ #

• Acceleration Vector: = ̈ − ̇ ” & + ̈ + 2 ̇ ̇ ‘ +

̈ #

7

Kinematics of a Particle

Spherical Coordinate System

• Position Vector:

= (

• Velocity Vector:

̇ ) + sin ̇ ‘

̇ ( +

=

• Acceleration Vector:

= ̈ − ̇ ” − sin ” ̇ ” ( + ̈ + 2 ̇ ̇ − sin cos ̇ ” )

+ sin ̈ + 2 ̇ ̇ sin + 2 ̇ ̇ cos ‘

8

Areal Velocity Vector

Kinematics of a Particle

9

Kinematics of a Particle

Areal Velocity Vector A(t)

The magnitude of the areal velocity

vector is the rate at which the position

vector r of the particle sweeps out an

area about the fixed-point O.

1

= ×

2

10

Example

11

Curvilinear Coordinate System

Kinematics of a Particle

12

Kinematics of a Particle

Curvilinear Coordinate System

• The curvilinear coordinate system is defined by the functions

q1 = q1(x1, x2, x3)

q2 = q2(x1, x2, x3)

q3 = q3(x1, x2, x3)

• Given the curvilinear coordinates of any point in 3, there is a unique set of

Cartesian coordinates for this point and vice versa; that is,

x1 = x1(q1, q2, q3)

x2 = x2(q1, q2, q3)

x3 = x3(q1, q2, q3)

13

Kinematics of a Particle

a3

Curvilinear Coordinate System

q2 coo surf

• We write Cartesian coordinates as

x = x1, y = x2, z = x3.

• And curvilinear coordinates as

q1, q2, q3.

• Through each point P there pass three

coordinate surfaces

q1 = const., q2 = const., q3 = const.

q3

q1 coo surf

P

q1

q3 coo surf

a1

q2

a2

• They intersect along coordinate curves.

• We assume the three coordinate curves

through P to be orthogonal.

0

14

Kinematics of a Particle

a3

Curvilinear Coordinate System

• The surface corresponding to a

constant value of a coordinate qj

is known as a qj coordinate

surface.

• The curve corresponding to a

varying coordinate qk while the

remaining two curvilinear

coordinates are fixed is known as

a qk coordinate curve.

q3

q2 coo surf

(q2 = const)

P

q1 coo surf

(q1 = const)

q1

q3 coo surf

(q3 = const)

a1

q2

a2

0

15

Kinematics of a Particle

Curvilinear Coordinate System

• Example of a q1 coordinate

surface S.

• At a point on this surface, a1 is

normal to the surface, and a2 and

a3 are tangent to the surface.

• The q1 coordinate surface S is

foliated by curves of constant q2

and q3.

16

Kinematics of a Particle – Part 4

Dynamics of a Single Particle

Dr. Léo A. Carrilho

1

Contents

• Review:

• Areal Velocity Vector

• Curvilinear Coordinate System

• Curvilinear Coordinate System

• Position, Velocity, and Acceleration Vectors

2

Areal Velocity Vector

Kinematics of a Particle

3

Kinematics of a Particle

Areal Velocity Vector A(t)

The magnitude of the areal velocity

vector is the rate at which the position

vector r of the particle sweeps out an

area about the fixed-point O.

1

= ×

2

4

Example

5

Curvilinear Coordinate System

Kinematics of a Particle

6

Kinematics of a Particle

Curvilinear Coordinate System

• The curvilinear coordinate system is defined by the functions

q1 = ! 1(x1, x2, x3)

q2 = ! 2(x1, x2, x3)

q3 = ! 3(x1, x2, x3)

• Given the curvilinear coordinates of any point in 3, there is a unique set of

Cartesian coordinates for this point and vice versa; that is,

x1 = ! 1(q1, q2, q3)

x2 = ! 2(q1, q2, q3)

x3 = ! 3(q1, q2, q3)

7

Kinema0cs of a Par0cle

q3

Curvilinear Coordinate System

• Cartesian coordinates are

x = x1, y = x2, z = x3

• Curvilinear coordinates are

q1, q2, q3

P

q1

q2

0

8

Kinematics of a Particle

q3

Curvilinear Coordinate System

• S1, S2, S3 coordinate surfaces

pass through point P at

coordinates q1 = q2 = q3 = const,

respecHvely.

S2

S1

P

q1

S3

• S1, S2, S3 intersect along

coordinate curves.

q2

• Here, the three coordinate

curves through P are orthogonal.

0

9

Kinematics of a Particle

Curvilinear Coordinate System

• Fixing the q1 curvilinear

coordinate allows us to

determine the corresponding

Cartesian coordinates x1, x2, x3.

q1 = ( 1(x1, x2, x3)

• The union of all points

represented by there Cart

coordinates defines the q1

coordinate surface.

10

Kinematics of a Particle

Curvilinear Coordinate System

• Moving on the q1 coordinate

surface varies the coordinates q2

and q3.

• Varying q2 while keeping q3 fixed

generates the q2 coordinate

curves.

• Likewise, varying q3 while

keeping q2 fixed generates the q3

coordinate curves.

11

Kinematics of a Particle

Curvilinear Coordinate System

At a point on the surface,

• a1 is normal to the surface

(contravariant basis vector), and

• a2 and a3 are tangent to the

surface (covariant basis vectors).

12

q3

Curvilinear Coordinate System

• Position Vector:

$

#

%

$

= * (! , , !

P

q1

!”#

q2

0

13

Curvilinear Coordinate System

• Covariant Basis Vectors:

$

(&

! = ! = * ! &

&”#

ai is tangent to a qi coo curve

14

Curvilinear Coordinate System

a3

q3

• Contravariant Basis Vectors:

$

#

(

#

=*

!

!

!”#

$

%

(

% = *

!

!

!”#

P

q1

a1

q2

a2

$

$

(

$ = *

!

!

& = ∇ &

!”#

ai is normal to a qi coo surface

0

15

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Kinematics of a Particle

Dynamics of a Single Particle

Dr. Léo A. Carrilho

1

Contents

• Introduction

• Reference Frames

• Kinematics of a Particle

• Coordinate Systems

• Representations of Particle Kinematics

• Constraints

2

Introduction

Kinematics of a Particle

3

Kinematics of a Particle

Introduction: The Main Goal is

• Enabling the Engineer to

• Define a physical system

• Model it by using particles or rigid bodies

• Interpret the results of the model

• For this to happen, the Engineer needs to

• Be equipped with the necessary array of tools and techniques.

• Basis

• This foundation enables the Engineer to precisely formulate the kinematics of a particle.

• Otherwise, the future conclusions on which they are based either do not hold up or lack

conviction.

4

Kinematics of a Particle

Introduction: Major Topics

• We start with a discussion of coordinate systems for a particle

moving in a three-dimensional space.

• Then we discuss its application to particle mechanics.

• Outcome: We can establish expressions for gradient and acceleration

vectors in any coordinate system.

• The other major topics related to kinematics of particles pertain to

constraints on the motion of particles.

5

Reference Frames

Kinematics of a Particle

6

Kinematics of a Particle

Definitions

• To describe the kinematics of particles and rigid bodies, we presume

on the existence of a space with a set of three mutually perpendicular

axes that meet at a common point P.

• The set of axes and the point P constitute a reference frame.

• In Newtonian mechanics, we also assume the existence of an inertial

reference frame. In this frame, the point P moves at a constant speed.

7

Kinematics of a Particle

Considerations: When it works

For ballistics problems, the Earth’s rotation and the translation of its

center are ignored, and one assumes that a point, say E, on the Earth’s

surface can be considered as fixed.

The point E, along with three orthonormal vectors that are fixed to it

(and the Earth), is then taken to approximate an inertial reference

frame.

8

Kinematics of a Particle

Considerations: When it does not work

When the motion of the Earth about the Sun is explained, it is standard

to assume that the center S of the Sun is fixed and to choose P to be

this point.

The point S is then used to construct an inertial reference frame.

Other applications in celestial mechanics might need to consider the

location of the point P for the inertial reference frame as the center of

mass of the solar system with the three fixed mutually perpendicular

axes defined by use of certain fixed stars.

9

Kinematics of a Particle

Reference Frames

• For the purposes of our course,

we assume the existence of a

fixed-point O and a set of three

mutually perpendicular axes that

meet at this point.

• The set of axes is chosen to be

the basis vectors for a Cartesian

coordinate system.

• The axes and the point O are an

inertial reference frame.

10

Kinematics of a Particle

Reference Frames

• The space that this reference

frame occupies is a threedimensional space.

• Vectors can be defined in this

space, and an inner product for

these vectors is easy to

construct with the dot product.

• As such, we refer to this space as

a 3D Euclidean space and we

denote it by 3.

11

Kinematics of a Particle

Dynamics of a Single Particle

12

Kinematics of a Particle

Kinematics of a Particle

• Position Vector

=

• Velocity Vector

=

= ̇

• Acceleration Vector

=

= ̈

13

Kinematics of a Particle

• Linear Momentum

=

• Angular Momentum

! = ×

14

Kinematics of a Particle

• Areal Velocity Vector

1

= ×

2

• Kinetic Energy of the Particle

1

= 1

2

15

Coordinate Systems

Kinematics of a Particle

16

Kinematics of a Particle

Cartesian Coordinate System

• Right-handed orthonormal vectors

” , # , $

• Given any vector b in 3

$

= 5 % %

%&”

17

Kinematics of a Particle

Cartesian Coordinate System

• Position Vector

$

= 5 % %

%&”

• Cartesian Coordinates of the Particle

” , # , $

18

Kinematics of a Particle

Cylindrical Coordinate System

• Position Vector

= ‘ + $

• Unit Vectors

‘ = cos ” + sin #

( = cos # − sin ”

) = $

19

Kinematics of a Particle

Spherical Coordinate System

• Position Vector

= *

• Unit Vector

* = sin cos ” + sin sin # + cos $

20

Kinematics of a Particle

Dynamics of a Single Par2cle

Dr. Léo A. Carrilho

1

Contents

• Review

• Reference Frames

• Kinematics of a Particle

• Coordinate Systems

• Position Vector

• Velocity Vector

• Acceleration Vector

2

Reference Frames

Kinematics of a Particle

3

Kinematics of a Particle

Deﬁni*ons

• To describe the kinematics of particles and rigid bodies, we presume

on the existence of a space with a set of three mutually perpendicular

axes that meet at a common point P.

• The set of axes and the point P constitute a reference frame.

• In Newtonian mechanics, we also assume the existence of an inertial

reference frame. In this frame, the point P moves at a constant speed.

4

Kinematics of a Particle

Considerations: When it works

For ballistics problems, the Earth’s rotation and the translation of its

center are ignored, and one assumes that a point, say E, on the Earth’s

surface can be considered as fixed.

The point E, along with three orthonormal vectors that are fixed to it

(and the Earth), is then taken to approximate an inertial reference

frame.

5

Kinematics of a Particle

Considera*ons: When it does not work

When the motion of the Earth about the Sun is explained, it is standard

to assume that the center S of the Sun is fixed and to choose P to be

this point.

The point S is then used to construct an inertial reference frame.

Other applications in celestial mechanics might need to consider the

location of the point P for the inertial reference frame as the center of

mass of the solar system with the three fixed mutually perpendicular

axes defined by use of certain fixed stars.

6

Kinema0cs of a Par0cle

Reference Frames

• For the purposes of our course,

we assume the existence of a

fixed-point O and a set of three

mutually perpendicular axes that

meet at this point.

• The set of axes is chosen to be

the basis vectors for a Cartesian

coordinate system.

• The axes and the point O are an

inertial reference frame.

7

Kinematics of a Particle

Reference Frames

• The space that this reference

frame occupies is a threedimensional space.

• Vectors can be defined in this

space, and an inner product for

these vectors is easy to

construct with the dot product.

• As such, we refer to this space as

a 3D Euclidean space and we

denote it by 3.

8

Kinematics of a Particle

Dynamics of a Single Particle

9

Kinematics of a Particle

Kinematics of a Particle

• Position Vector

=

• Velocity Vector

=

= ̇

• Acceleration Vector

=

= ̈

10

Kinema*cs of a Par*cle

• Linear Momentum

=

• Angular Momentum

! = ×

11

Kinematics of a Particle

• Areal Velocity Vector

1

= ×

2

• Kinetic Energy of the Particle

1

= 1

2

12

Coordinate Systems

Kinematics of a Particle

13

Kinematics of a Particle

Cartesian Coordinate System

• Right-handed orthonormal vectors

” , # , $

• Given any vector b in 3

$

= 5 % %

%&”

14

Kinematics of a Particle

Cartesian Coordinate System

• PosiRon Vector

$

= 5 % %

%&”

• Cartesian Coordinates of the ParRcle

” , # , $

15

Kinematics of a Particle

Cartesian Coordinate System

• Position Vector:

= ” ” + # # + $ $

• Velocity Vector:

= ̇ ” ” + ̇ # # + ̇ $ $

• Acceleration Vector: = ̈ ” ” + ̈ # # + ̈ $ $

16

Kinematics of a Particle

Cylindrical Coordinate System

• Position Vector:

= ‘ + $

• Velocity Vector:

̇ ( +

=

̇ ‘ +

̇ $

• Acceleration Vector: = ̈ − ̇ # ‘ + ̈ + 2 ̇ ̇ ( +

̈ $

17

Kinematics of a Particle

Spherical Coordinate System

• Position Vector:

= )

• Velocity Vector:

̇ * + sin ̇ (

̇ ) +

=

• Acceleration Vector:

= ̈ − ̇ # − sin # ̇ # ) + ̈ + 2 ̇ ̇ − sin cos ̇ # *

+ sin ̈ + 2 ̇ ̇ sin + 2 ̇ ̇ cos (

18

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