Transverse Displacement Mechanical Engineering Worksheet

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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