ME 012 Engineering Dynamics - CEMS Homejmmeyers/ME012/Lectures/ME 012 Lecture 26 FI… · ME 012...

ME 012 Engineering Dynamics: Lecture 26

J. M. Meyers, Ph.D. ([email protected])

ME 012 Engineering Dynamics

Lecture 26

FINAL EXAM REVIEW

(Chapter 12, 13, 14, 15, 16, and 17)

Tuesday,

Apr. 30, 2013


J. M. Meyers, Ph.D. ([email protected]) 2

OVERVIEW

Chapter 12: Kinematics of a Particle

• 12.1 Introduction

• 12.2 Rectilinear Kinematics: Continuous Motion

• 12.3 Rectilinear Kinematics: Erratic Motion

• 12.4 General Curvilinear Motion

• 12.5 Curvilinear Motion: Rectangular Components

• 12.6 Motion of a Projectile

• 12.7 Curvilinear Motion: Normal and Tangential

Components

• 12.8 Curvilinear Motion: Cylindrical Components

• 12.9 Absolute Dependent Motion Analysis of Two Particles

• 12.10 Relative-Motion of Two Particles Using Translating

Axes

Chapter 13: Kinematics of a Particle: Force and Acceleration

• 13.1 Newton’s Second Law of Motion

• 13.2 The Equation of Motion

• 13.3 Equation of Motion for a System of Particles

• 13.4 Equations of Motion: Rectangular Coordinates

• 13.5 Equations of Motion: Normal and Tangential Coordinates

• 13.6 Did not cover


• Chapter 12 and 13 material is, in a sense,

introductory material

• The material represents the building blocks

for all subsequent chapters.

• Each and every problem from chapter 14 to

chapter 17 requires the foundations

developed in these chapters

• Go over Exam 1 review



EXAM FORMAT

Chapter 14: Kinetics of a Particle: Work and Energy

• 14.1 The Work of a Force

• 14.2 Principle of Work and Energy

• 14.3 Principle of Work and Energy for a System of

Particles

• 14.4 Power and Efficiency

• 14.5 Conservative Forces and Potential Energy

• 14.6 Conservation of Energy

Chapter 15: Kinetics of a Particle: Impulse and Momentum

• 15.1 Principle of Linear Impulse and Momentum

• 15.2 Principle of Linear Impulse and Momentum for a System of Particles

• 15.3 Conservation of Linear Momentum for a System of Particles

• 15.4 Impact

• 15.5 Angular Momentum

• 15.6 Relation Between Moment of a Force and Angular Momentum

• 15.7 Angular Impulse and Moment Principles



• This material will not be reviewed as this has

already been done

• Go over Exam 2 review



EXAM FORMAT

Chapter 16: Planar Kinematics of a Rigid Body

• 16.1 Rigid-Body Motion

• 16.2 Translation

• 16.3 Rotation About a Fixed Axis

• 16.4 Absolute Motion Analysis

• 16.5 Relative Motion Analysis: Velocity

• 16.6 Instantaneous Center of Zero Velocity

• 16.7 Relative Motion Analysis: Acceleration


Chapter 17: Planar Kinematics of a Rigid Body: Force and Acceleration

• 17.1 Moment of Inertia

• 17.2 Planar Kinetic Equations of Motion

• 17.3 Equations of Motion: Translation

• 17.4 Equations of Motion: Rotation About a Fixed Axis

• 17.5 Equations of Motion: General Plane Motion

• Today’s Review will mainly cover material

not already covered by a review session



There are three types of planar rigid body motion:

1) Translation (Rectilinear and Curvilinear)

2) Rotation about a fixed axis 3) General plane motion

16.1 Rigid Body Motion



16.2 Rigid-Body Motion: Translation

The positions of two points � and � on a translating body

can be related by:

Note, all points in a rigid body subjected to translation move with the same velocity

and acceleration.

The velocity at B is:

POSITION

VELOCITY

�� = ��+ ��/�where �� & �� are the absolute position vectors defined

from the fixed x-y coordinate system, and ��/� is the

relative-position vector between � and �.

�� = �� + ��/��

Now drB/A/dt = 0 since rB/A is constant, ergo, vB = vA

By following similar logic, aB = aA.



16.3 Rigid-Body Motion: Rotation About a Fixed Axis

• Angular velocity ( ): obtained by taking the time derivative of angular

displacement:

ANGULAR MOTION

• Angular acceleration (�): if positive, accelerating, if negative, decelerating:

• Angular position (�): angular position of radial line �with units of either

radians or revolutions related by: 1 revolution = 2π radians

= ��

� = � = ��

��• Useful relation between the three quantities: �� =

When a body rotates about a fixed axis, any point P in the body travels along a

circular path.

If the angular acceleration of the body is constant, � = ��:

= � + �� = �� + �� + 12�� = � � + 2�� − ��

Note these equations are very similar to the constant acceleration relations

developed for the rectilinear motion of a particle.

CONSTANT ANGULAR ACCELERATION



The magnitude of the velocity of P is equal to r (the text provides the

derivation). The velocity’s direction is tangent to the circular path of P.

In the vector formulation, the magnitude and direction of v can be

determined from the cross product of � and rrrr�. Here rrrr� is a vector from

any point on the axis of rotation to P.

vvvv = �× rrrr� = �× rrrr

MOTION OF POINT P: VELOCITY

Scalar form:

Vector form:

v = � MOTION OF POINT P: ACCELERATION

Using the vector formulation, the acceleration of P can also be defined by

differentiating the velocity.

� = �� = �

� × �� +�× ��

� = � × �� − �� = aaaa! + aaaa"a! = �r a" = �rScalar form:

Vector form:

a = a! � + a" �

O a!

a"

��

P

MOTION OF POINT P: POSITION

The position of P is defined by the position vector r which extends from Oto P.

16.3 Rigid-Body Motion: Rotation About a Fixed Axis



16.4 Absolute Motion Analysis

Absolute motion analysis (also called the parametric method) can be used to study planar motion.

• Once a relationship in the form of sP = f(θ) is established, the velocity and acceleration of point Pare obtained in terms of the angular velocity, ω, and angular acceleration, α, of the rigid body by

taking the first and second time derivatives of the position function.

#� = $(�, �) &� = [$ � ] � = $ �, �, )� = �[$(�) ]

�� = $ �, �, , �

Book Example

16.3

* = 2� cos �

Using trigonometry, a relation between the rotational motion

of OA and rectilinear translation of rod R (measured from

fixed point O):

Using the chain rule:

*� = −2� (sin �) �� & = −2� (sin �)

&� = −2�

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



16.5 Relative Motion Analysis: Velocity

0� = 0� + 0�/�

= +

0� = 0� +�× 1�/�

When a wheel rolls without slipping, point

A is often selected to be at the point of

contact with the ground. Since there is no

slipping, point A has zero velocity and

&� = �

• This equation is formulated such that point � moves

along a circular path about point �.

• The relative velocity, 0�/�, acts perpendicular to the

relative displacement vector between � and �… thus



16.6 Instantaneous Center of Zero Velocity

If the location of this point can be determined, the velocity analysis can be

simplified because the body appears to rotate about this point at that

instant.

For any body undergoing planar motion, there always exists a point in the

plane of motion at which the velocity is instantaneously zero (if it were

rigidly connected to the body) known as the instantaneous center of zero

velocity (IC).

A third case is when the magnitude and direction of two parallel velocities

at � and� are known.

Here the location of the IC is determined by proportional triangles. As a

special case, note that if the body is translating only (vA = vB), then the 23would be located at infinity. Then equals zero, as expected.



16.6 Instantaneous Center of Zero Velocity

Being able to solve for using 23 analysis can help in solving more complicated relative-

motion analysis problems involving acceleration.



16.7 Relative Motion Analysis: Acceleration

The equation relating the accelerations of two points on the body is determined by

differentiating the velocity equation with respect to time:

Measured from a set of fixed

x and y axes.

Will develop tangential and

normal components because of

circular path motion!!!!

0�� = 0�

� + 0�/��

Absolute acceleration

of point �Absolute acceleration

of point �= + Acceleration of point �with respect to point �

4� = 4� + 4�/� = 4� + 4�/� ! + 4�/� "



B

A

B

A

B

A

4� = 4� + � × 1�/� − �1�/�

Shape doesn’t matter so long as the

body is rigid!

Analysis is still thought of as a rigid

bar connecting points A and B

B

A




4� = 4� + � × 1�/� − �1�/�

B

A

B

A

)�/� "

)�/� !

y’

x’


�4�

4�

1�/�

)�5)�6)�7=

)�5)�6)�7+

�5�6�7×

��/�5��/�6��/�7− �

��/�5��/�6��/�7

iiii jjjj kkkk�5 �6 �7��/�5 ��/�6 ��/�7

You can think of this cross-product

as the determinant of this relation



17.1 Mass Moment of Inertia

• The figures on the right show the mass moment of inertia formulations for two

flat plate shapes commonly used when working with three dimensional

bodies.

• The shapes are often used as the differential element being integrated over

the entire body.

• Moment of Inertia is a measure of the resistance of a body to angular

acceleration (: = 2�) in the same way that mass is a measure of the body’s

resistance to acceleration (; = <))

• In other words, the mass moment of inertia is a measure of an object’s

resistance to rotation… mathematically:

2 = =�� < = =�� >?< ?

• Center of gravity and mass moment of inertia of these and other homogenous solid geometries are

given inside the back cover of your text (will be supplied for exam)

• Most problems only rely on the analysis on a collective of simple geometries



PARALLEL-AXIS THEOREM

If the mass moment of inertia of a body about an axis passing

through the body’s mass center is known, then the moment of

inertia about any other parallel axis may be determined by using

the parallel axis theorem:

2� 7 = 2@ 7A +<�

17.1 Mass Moment of Inertia

If a body is constructed of a number of simple shapes, such as disks, spheres, or

rods, the mass moment of inertia of the body about any axis can be determined

by algebraically adding together all the mass moments of inertia, found about the

same axis, of the different shapes.

The mass moment of inertia of a body about a specific axis can be defined using the radius of

gyration (k). The radius of gyration has units of length and is a measure of the distribution of the

body’s mass about the axis at which the moment of inertia is defined

COMPOSITE BODIES

RADIUS OF GYRATION

2 = <B� ⟹ B = 2/<



17.2 Planar Kinetic Equations of Motion

• Translation (Section 17.3)

• Rectilinear (x-y coordinates)

• Curvilinear (n-t coordinates)

• Pure Rotation (Section 17.4)

• General Plane Motion ((Section 17.5)

Our study of planar kinetics covered three (actually four!) types of motion:



17.3 Equations of Motion: Translation

When a rigid body undergoes only translation, all

the particles of the body have the same

acceleration so )D = )and � = 0. The equations

of motion become:

RECTILINEAR TRANSLATION

Σ;5 = < )@ 5Σ;6 = < )@ 6Σ:@ = 0

1

2

3

When a rigid body is subjected to curvilinear

translation, it is best to use an n-t coordinate

system. Then apply the equations of motion,

as written below, for n-t coordinates.

CURVIILINEAR TRANSLATION

Σ;" = < )@ "Σ;! = < )@ !Σ:@ = 0Σ:� = Σ ℳG �Σ:� = H ∙ < )@ !−ℎ ∙ < )@ "

1

2

3



17.4 Equations of Motion: Rotation About a Fixed Axis

• When a rigid body rotates about a fixed axis perpendicular to the plane

of the body at point K, the body’s center of gravity D moves in a

circular path of radius rD.

• Thus, the acceleration of point D can be represented by a tangential

component and a normal component:

aaaa@ ! = rrrr@� aaaa@ " = rrrr@ �acceleration magnitudes

acts in direction

consistent with �acts in direction

from point D to K• Angular acceleration and velocity (� and ) are caused by the external

forces (FFFF) and couple moment system on the body



17.4 Equations of Motion: Rotation About a Fixed Axis

=

Free Body Diagram Kinetic Diagram

ΣF" = < )@ " = < ��@ΣF! = < )@ ! = <��@

ΣM@ = ID�

ΣF" = < )@ " = < ��@ΣF! = < )@ ! = <��@

ΣM� = Σ(ℳG)�= 2@� + �@< )@ != 2@ +< �@ � �=IK�

• It may be convenient to sum these moments

about the pin point K• Here the moment about pin point K will be

equal to the sum of kinematic moments about

pin point K (Σ(ℳG)�)

• The body experiences an angular acceleration so its

inertia creates a moment of ID� that is equal to the

moment of the external forces about point D.

• The body’s moment of inertia (ID) is calculated about

an axis which is perpendicular to the page and passes

through D

parallel-axis

theorem



17.5 Equations of Motion: General Plane Motion

17.4) Rotation About

a Fixed Axis

17.5) General Plane

Motion




P

ΣF6 = < )@ 6

ΣF5 = < )@ 5

ΣM� = Σ ℳG �

ΣF6 = < )@ 6

ΣF5 = < )@ 5

ΣM@ = ID�

Sometimes, it may be convenient to write the

moment equation about some point P other

than D. In this case, Σ ℳG �represents the

sum of the moments of ID� and <�@ about

point P.

Using an x-y inertial coordinate system, the

equations of motions about the center of

mass, D, may be written as:




FRICTIONAL ROLLING PROBLEMS

When analyzing the rolling motion of wheels, cylinders, or disks, it may not be known if the body rolls

without slipping or if it slides as it rolls.

• For example, consider a disk with mass < and radius �, subjected to a

known force PPPP.

• The equations of motion will be:

There are 4 unknowns in

these three equations:

F, N, �, and )DΣF6 = < )@ 6ΣF5 = < )@ 5 ⇒ T− U = <)@

⇒ V−<W = 0ΣM@ = ID� ⇒ U� = ID�

Case 1:

• Assume no slipping and use )@ = �� as the 4th equation

• DO NOT use static friction: FX = YZN• After solving, you will need to verify that the assumption was

correct by checking if FX ≤ YZNCase 2:

• Assume slipping and use FX = YGNas the 4th equation.

• In this case, )@ ≠ ��

Hence, we have to make an assumption to provide another equation.



EXAM FORMAT

• 20 Short Answer Word Problems (your quiz problems and previous short

answer problems and variations thereof may make another appearance)

• 6 HW Style Problems: if the following are not applied, you will lose points:

• Define what is given before you proceed

• Define what is to be found before you proceed

• Show your FBD and KD when applicable

• Show your relative motion diagrams when applicable (pulley problems)

• Show full version of equation used before you put in numbers

• Be clear about your approach… this helps the grading process but also

helps your thought process through the problem

• You will be supplied with:

• Equation Sheets (regular equations and formulations for mass moments of

simple geometries)

• Paper (you will not be using your own)

• You can bring one “cheat sheet” filled out on both sides

ME 012 Engineering Dynamics - CEMS Homejmmeyers/ME012/Lectures/ME 012 Lecture 26 FI… · ME 012...

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