Introduction and Planar Kinematics
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Transcript of Introduction and Planar Kinematics
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MECH2350 Dynamics 2
Lecturer and Course Coordinator:-
Dr Chris Wensrich ES319, ph #16203
Tutors:-
Dr Chris Wensrich
Alex Smith
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MECH2350 Dynamics 2 Whats this course all about?
This course will start from where GENG1001 finished, and introduce;
Planar kinematics and kinetics of rigid bodies,
3D kinematics and kinetics of rigid bodies (brief intro)
Modeling of dynamic mechanical systems,
Vibration analysis in single and multiple degree of freedom systems.
How does it fit in my program?
its only the beginning
MCHA2000, MECH3110, MCHA3000, MCHA3900, ELEC4400
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MECH2350 Dynamics 2 Prerequisites and Assumed Knowledge
MATH1110 and MATH1120 or MATH1210 and MATH1220 (Calculus)
MATH2310 Calculus of Science and Engineering (ordinary differential equations)
GENG1001 Introduction to Engineering Mechanics (Basic mechanics)
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MECH2350 Dynamics 2 Text books
Two text books are required for this course;
Engineering Mechanics Dynamics (12th SI Ed), P Schiavone, and R Hibbeler, Prentice Hall, 2010.
(Same book as GENG1001)
Engineering Vibration (3rd Ed), D Inman, Prentice Hall, 2007.
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MECH2350 Dynamics 2 Assessment
Assessment will take the following form;
Mid Semester Quiz 40% of total Carried out in Week 8
Final Quiz 40% of total Carried out in Week 12
Super-eta Report 20% of total Due in Week 13 (details to follow)
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GENG1001 Dynamics A review
GENG1001 Dynamics provided an overview of the kinematics and Kinetics of a point mass:-
Kinematics How things move Kinetics Why things move
Question: What is special about a point mass?
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GENG1001 Dynamics A review
GENG1001 Dynamics provided an overview of the kinematics and Kinetics of a point mass:-
Kinematics How things move Kinetics Why things move
Question: What is special about a point mass?
Answer: It doesnt rotate (or at least if it does, we dont have to care)!
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GENG1001 Dynamics A review
GENG1001 Dynamics provided an overview of the kinematics and Kinetics of a point mass:-
Kinematics How things move (Mathematics of motion) Kinetics Why things move (Newtons Laws, Energy, etc.)
Question: What is special about a point mass?
Answer: It doesnt rotate (or at least if it does, we dont have to care)!
In this course we will begin to examine the mechanics of Rigid Bodies, which can and do rotate.
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GENG1001 Dynamics A review
The net force that acts on an
object is equal to the rate of
change of the objects linear
momentum
2
21 kmghU 2
21 mvT
Energy cannot be created or
destroyed!
2
1
12~~ rdFW
am
ma
ma
ma
F
F
F
F
z
y
x
z
y
x
~~
extWUT
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MECH2350 Dynamics 2
GENG1001 considered only point masses the real world is more complicated
This course is focused on developing understanding of rigid bodies.
3D is complicated so we will start in 2D.
In This Lecture;
Planar (2D) Kinematics of rigid bodies Types of motion Velocity and acceleration analysis Translating reference frames Instantaneous centre of rotation Moving reference frames general case
http://www.youtube.com/watch?v=NDkVeFMAp_w
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Planar Motion of Rigid Bodies Definition and Types of Motion
An object undergoing planar motion can have one of three different
behaviours;
Pure translation Pure rotation A general combination of translation and rotation
Translational motion is when all points on the object follow paths that always
have the same radius of curvature as each other.
http://www.youtube.com/watch?v=ErDx0A7TF18
Planar motion describes the motion of an object where all points on the object follow paths that remain equidistant from a fixed plane
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Planar Motion of Rigid Bodies Translation
Translation is fully described by the same kinematics as point masses (ie.
GENG1001)
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Planar Motion of Rigid Bodies Pure Rotation
Pure rotation describes the motion of an object that has a fixed axis of
rotation of points that do not move. All other points move around circular
paths.
All points have the same angular
displacement, angular velocity and
angular acceleration about the axis
of rotation.
dt
d
dt
d
dd
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Planar Motion of Rigid Bodies Pure Rotation
Note that angular velocity and angular acceleration are vector quantities!
(direction given by the right hand rule)
In GENG1001 we used a scalar version
to calculate velocity;
Now that we are more comfortable with
vectors, we can use the proper version;
rv ~~~
rv
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Planar Motion of Rigid Bodies Pure Rotation
rr
rr
dt
rdr
dt
d
dt
rd
dt
vda
~~~
~~~~~
~~~
~
)~~(~~
2
What about acceleration?
We can differentiate the velocity to calculate acceleration:-
errerra r~)2(~)(~ 2
Compare:-
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Planar Motion of Rigid Bodies General Planar Motion
Translation and rotation combined:-
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Planar Motion of Rigid Bodies Kinematic Analysis
There are 2 basic approaches in a general sense;
1. Absolute motion analysis:-
Describe the geometry of the system using a fixed
coordinate system. (see example)
2. Relative motion analysis:-
Describe the system using a reference frame that moves
with the body (known as a body fixed coordinate system)
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Planar Motion of Rigid Bodies Translating reference frames
Ar~We use 2 reference frames:-
1. One to describe the translation (eg, specifying the position of ).
2. One to describe the rotation.
ABAB
ABAB
ABAB
aaa
vvv
rrr
/
/
/
~~~
~~~
~~~
Motion of
coordinate
system
Motion of B
in the
coordinate
system
Motion
of B = +
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Planar Motion of Rigid Bodies Translating reference frames Velocity analysis
ABAB vvv /~~~
The relative component of velocity is entirely due to rotation about A
ABAB rvv /~~~~
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Planar Motion of Rigid Bodies Instant center of rotation
The Instant Center of Rotation defines the point on an object that has zero velocity;
If we choose our translating reference
frame to follow the instant center, the
velocity of any other point can be written
as:-
ICBB rv /~~~
0~~ ICv
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Planar Motion of Rigid Bodies Instant Center Location
There are several methods to locate the instant center of an object;
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Planar Motion of Rigid Bodies Instant Center Location
For example:-
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Planar Motion of Rigid Bodies Translating reference frames Acceleration Analysis
In a similar fashion, acceleration can be viewed as separate translating and
rotating components:-
ABABAB
nABtABAB
ABAB
rraa
aaaa
aaa
/
2
/
//
/
~~~~~
~~~~
~~~
errerra r~)2(~)(~ 2
Compare:-
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Planar Motion of Rigid Bodies Moving reference frames General case
Now consider the general case of a moving reference frame that may be
translating and rotating;
The most common version of this is a Body Fixed Coordinate System that moves with an object
Useful when two moving objects slide against each
other. (i.e. B may be a point
on another object that slides
relative to A)
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Planar Motion of Rigid Bodies Moving reference frames General case
In the XY coordinate system;
So velocity is;
ABAB rrr /~~~
Unit vectors:
XY:- and
xy :- and
I~
J~
i~
j~
??~
~~~
/
/
AB
ABAB
rdt
d
rdt
dvv
In the xy coordinate system;
jyixr AB~~~
/
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Planar Motion of Rigid Bodies Moving reference frames General case
i.e. we need to differentiate the unit vectors
So,
jdt
d
dt
id ~~
j
dt
d
dt
id ~~
ABxyAB
xyABAB
rv
iyjxvrdt
d
//
//
~~~
~~~~
ABxyABAB rvvv //~~~~~
dt
jdy
dt
idxj
dt
dyi
dt
dxr
dt
dAB
~~~~~
/
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Planar Motion of Rigid Bodies Moving reference frames General case
Differentiate again to get velocity
ABABxyABA
ABxyABABB
rdt
drv
dt
da
rvvdt
dv
dt
da
///
//
~~~~~~
~~~~~~
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Planar Motion of Rigid Bodies Moving reference frames General case
Differentiate again to get velocity
ABABxyABA
ABxyABABB
rdt
drv
dt
da
rvvdt
dv
dt
da
///
//
~~~~~~
~~~~~~
??
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Planar Motion of Rigid Bodies Moving reference frames General case
Differentiate again to get velocity
ABABxyABA
ABxyABABB
rdt
drv
dt
da
rvvdt
dv
dt
da
///
//
~~~~~~
~~~~~~
xyABxyABxyAB
vavdt
d///
~~~~ (Same process as )
xyAB
rdt
d/
~
ABxyABAB rvrdt
d///
~~~~~~~
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Planar Motion of Rigid Bodies Moving reference frames General case
Put it all together
xyABABABxyABAB
vrraaa ////~~2~~~~~~~~
ABr /~~
ABr /~~~
xyBA
v /~~2
xyAB
a /~
Accounts for the angular acceleration of xy frame
Centripetal component of acceleration from the rotation of xy frame
Coriolis acceleration (due to motion in the rotating frame)
Apparent acceleration in the xy frame
errerra r~)2(~)(~ 2
Compare:-