introduction and planar kinematics

MECH2350 Dynamics 2

Lecturer and Course Coordinator:-

Dr Chris Wensrich ES319, ph #16203

([email protected])

Tutors:-

Dr Chris Wensrich

([email protected])

Alex Smith

([email protected])

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

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)

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.

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)

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?



Kinematics How things move Kinetics Why things move


Answer: It doesnt rotate (or at least if it does, we dont have to care)!



Kinematics How things move (Mathematics of motion) Kinetics Why things move (Newtons Laws, Energy, etc.)


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.


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

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

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

Planar Motion of Rigid Bodies Translation

Translation is fully described by the same kinematics as point masses (ie.

GENG1001)

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


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


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:-

Planar Motion of Rigid Bodies General Planar Motion

Translation and rotation combined:-

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)

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 = +

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 /~~~~

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

Planar Motion of Rigid Bodies Instant Center Location

There are several methods to locate the instant center of an object;

Planar Motion of Rigid Bodies Instant Center Location

For example:-

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:-

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)


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~~~

/


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

~~~~~

/


Differentiate again to get velocity

ABABxyABA

ABxyABABB

rdt

drv

dt

da

rvvdt

dv

dt

da

///

//

~~~~~~

~~~~~~



ABABxyABA

ABxyABABB

rdt

drv

dt

da

rvvdt

dv

dt

da

///

//

~~~~~~

~~~~~~

??



ABABxyABA

ABxyABABB

rdt

drv

dt

da

rvvdt

dv

dt

da

///

//

~~~~~~

~~~~~~

xyABxyABxyAB

vavdt

d///

~~~~ (Same process as )

xyAB

rdt

d/

~

ABxyABAB rvrdt

d///

~~~~~~~


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:-

introduction and planar kinematics

Documents

review geng1001 dynamics

elec4400 mech2350 dynamics

extwut mech2350 dynamics

planar kinematics

point mass

geng1001 engineering

mechanics of rigid bodies

mathematics of motion