01-MENG212-Intro N Ch. 12-1

MENG 212 Fall 2014 – Dr. Jong B. Lee, ME @NYIT 1

MENG 212

Engineering Mechanics: Dynamics (#1)

Sept. 03, 2014

ⓒ 2014 Jong B. Lee, PhD

ME @NYIT

Introduction to

Dynamics

MENG 212 – Engineering Mechanics II: Dynamics. Jong B. Lee, PhD, All rights reserved.

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MENG 212

Engineering Mechanics: Dynamics – Fall

2014

Class Hour

Wednesday 5:45PM ~ 8:25PM

Class room

HSH #212

Office Address: HSH Room 224A

Office Phone: (516) 686 – 7955

Course web site: http://iris.nyit.edu/~jlee26

Email: [email protected]

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MENG 212

Office Hours

I adopt an open door policy

You are encouraged to come to my office and

ask questions, consult, provide feedback, or

give suggestions at anytime during the day

However, I may not be available all the time

Set times for offices hours are the office this

semester are:

Mon. ~ Wed. 02:30 PM – 03:30 PM

or by appointment via email or phone

can be changed without pre-notification

3 MENG 212 – Engineering Mechanics II: Dynamics. Jong B. Lee, PhD, All rights reserved.

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MENG 212

Description of Engineering Mechanics:

Dynamics

• This course teaches students how to apply Newtonian

physics to relatively simple physical situations. It follows

on from the Statics course, but considers systems that

are not in equilibrium i.e. with velocity and acceleration.

Some of the topics covered are pure kinematics (a

mathematical description of motion only), while others

are kinetic (determine motion in problems involving the

concepts of force and energy). The course restricts

itself to 2-D (planar) mechanisms.

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MENG 212

Objectives

• The student should understand the basic physical

concepts of dynamics.

• The student should understand and be able to relate the

kinematics of particles and rigid bodies to the solution of

dynamics problems in straight line and curvilinear

motion.

• The student should understand and be able to apply

Newton’s Laws to particles and rigid bodies to solve

problems related to dynamic behavior.

• The student should be able to apply the methods of

work, momentum and energy to particles and rigid

bodies associated with dynamic behavior.


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MENG 212 Course outcomes: After successful completion of this course, you will

have

1. Understand basic kinematics concepts – displacement, velocity

and acceleration (and their angular counterparts).

2. Understand basic dynamics concepts – force, momentum, work

and energy.

3. Understand and be able to apply Newton’s laws of motion.

4. Understand and be able to apply other basic dynamics concepts

- the Work-Energy principle, Impulse-Momentum principle and

the coefficient of restitution.

5. Learn to solve dynamics problems. Appraise given information

and determine which concepts apply, and choose an appropriate

solution strategy.

6. Gain an introduction to basic machine parts such as pulleys and

mass-spring systems.

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MENG 212

Class Discussion

• Communication is very important in achieving our

collective goals and objectives

• Feel free to voice your opinions and ask questions

anytime during a class period

• Remember you are here to learn and I am here to

teach and that teaching and learning are

intertwined

• So you can help me teach you as much as I can

help you learn

• I urge you to be an active participant in the learning

process and recognize that it takes a team effort to

realize meaningful things in life


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MENG 212

Major Courses in mechanical engineering

Engineering mechanics: The study of how

bodies react to forces acting on them

Statics: The study of bodies in equilibrium

Dynamics

Strength and Materials

Vibration

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Solid Mechanics

Thermodynamics

Fluid Mechanics

3 Major

Mechanics


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MENG 212

Engineering Mechanics

9

EngineeringMechanics

SolidMechanics

FluidMechanics

Rigid BodyMechanics

DeformableBody

Mechanics

Statics: F=ma,

Dynamics:

v = 0

a = 0 F = 0

F 0

F = ma

a = ?

v = ?

Mechanics of Materials

Elastics

Plastics

ExternalLoad

StressStrain


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Adv. Mach. Design

by Numerical Method

Statics &

Dynamics

Applied

Mach. Design

Stress Analysis

MENG 212

Mechanical Course Flow

10

Applied Solid

Mechanics

Strength and

Materials

Element

Mach. Design

Elastics

Senior

Mach. Design


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MENG 212

Topics covered

• Kinematics of Particle

• Kinetics of a Particle

• Force and Acceleration

• Work and Energy

• Impulse and Momentum

• Planar Kinematics of a Rigid Body

• Force and Acceleration

• Work and Energy

• Impulse and Momentum

• Three-Dimensional Kinematics of a Rigid Body

• Three-Dimensional Kinetics of a Rigid Body

• Vibrations


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MENG 212

Text Book

Engineering Mechanics: Dynamics, 13th edition,

R. C. Hibbeler

ISBN-10: 0132911272

Pearson

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References Engineering Mechanics: Dynamics, 7th ed., by by J. L.

Meriam and L. G. Kraige, Willey

Vector Mechanics for Engineers – Dynamics, 10th

Edition by BEER

Engineering Mechanics: Statics and Dynamic, Google

eBook, C. L. Rao, J. LAKSHINARASHIMAN, R.

SETHURAMAN, S. M. SIVAKUMAR

Engineering Dynamics: A Comprehensive Introduction

N. Jeremy Kasdin & Derek A. Paley, Princeton

University Press.

Engineering Mechanics: Dynamics, 5th ed. A. M.

Bedford and W. Fowler, Pearson

Engineering Mechanics: Dynamics, 4th ed., I. H.

Shames, Pearson


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MENG 212

Complete class syllabus and rule will be

given today

• Class syllabus can be changed without pre-

notification

• Please check at course website

PLEASE READ SYLLABUS CAREFULLY,

and let me know if you have any questions

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Exams and Quizzes Ideally, all quizzes and exams are closed book

There will be two midterm exams which constitute 20% and 20%

each of the grade

• Mid term Exam I: Oct. 08 (20%)

• Mid term Exam II: Nov. 12 (20%)

There will be one final exam which constitutes 20% of the grade

• Final Exam: Dec. 17

There will be number of quizzes which constitute 35% of grade

• Eight to Ten Quizzes: (Mostly every week)

Participation (Homework and attendance, etc): 5%

Total: 100%

This schedule and constitution of the grade can be changed


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Grade

Grading Policy

• The following straight scale will be used:

• Grade “I”, “IF”, “W” and “WF”: Please check on

University Catalog

• Remember final drop day or add/drop from the

University Academic Calendar

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A: 94-100

A-: 90-93

B+: 88-89

B: 83-87

B-: 80-82

C+: 78-79

C: 73-77

C-: 70-72

D+: 68-69

D: 61-67

F: 0-60


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Grade

How to curve an exam and assign grades

We have all given exams where the grades end up

lower than we hoped.

If the class does significantly lower than I think they

should have, I will consider curving the exam. How do I

do it?

What’s the goal of the curve?

How do I curve an exam?

Flat scale

Least squares regression

Linear scale


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Exams Grading Policy

Only neatly written problems will be graded

A correct answer without a correct outline of the work will not carry any grade

All incorrect work must be clearly crossed out on the page

In cases where more than one solution is presented for a problem, the solution with the most errors will be graded

Each solution must have proper units

No units or inappropriate units: 0 credit

Class attendance and participation in discussions are not strongly recommended, it is mandatory

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MENG 212 Quizzes

• Will cover theoretical aspects (definitions and derivations) and problem solving skills

• Will be closed book, closed notes, with no crib sheet

• Will be announced couple of days before the exam

• Will contain one to four problems

• No formula sheet

• No makeup quizzes will be given

Homework

Due of the homework will be after they are assigned

Problems will be graded only if they are written neatly

Late assignment is no credit


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MENG 370 Midterm and Final Exams

• Will contain four to eight problems

• Will be comprehensive

• Closed book and closed notes

• No formula sheet

• No makeup exams will be given

Remember, Make-up exams will not be available

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MENG 212

Class Rules

• Cheating will be dealt with

according to the rules of the

University

• Materials to be covered in an exam

will be announced at least one

week prior to the exam

• Cell phone is not allowed, only

calculator


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MENG 212

Student Code of Conduct

• It is the responsibility of each student to adhere to

the principles of academic integrity

• Academic integrity means that a student is honest

with him/herself, fellow students, instructors, and

the University in matters concerning his or her

educational endeavors

• Thus, a student should not falsely claim the work of

another as one's own, or misrepresent him/herself

so that the measures of one's academic

performance do not reflect his/her own work or

personal knowledge

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MENG 212


• In this regard, cheating will not be tolerated

• Cheating includes (but is not limited to) any

communication (written or oral) during

examinations and sharing of work, such as

using the same models or computer

programs or copying work

• All homework and projects must be an

individual effort unless specifically noted


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MENG 212


• Students who cheat on any assignment or during

any examination will be assigned a failing grade for

the course

• Therefore, avoid all appearance of improper

behavior!

• Students who witness cheating should report the

incident to the instructor as soon as possible.

• Students are also welcome to discuss any concerns

related to cheating with Dr. Lu, Chair of Mechanical

Engineering

• Dropping: Find the last day to drop this course form

the university academic calendar

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MENG 212

Attendance Sheet

• Please sign if your name, student

id number is corrected on the

attendance sheet, otherwise make

correction

• Please fill in the entire line on the

form if your name is missed


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MENG 212

Questions

• Please raise your hand or stop me at any

time when you have a question.

Please do not talk () to your classmates

during the lecture. If you absolutely need

to speak with someone, please feel free

to go out the classroom

• Please shut down your cell phone!!!

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MENG 212 Class Materials

• Lecture notes will be provided prior to or after the class time through course website

• User Name: students

• Password:

• Any changes will be mentioned in class

• Please visit and check here frequently for updates? Even though you are not able to attend class, please download lecture notes to catch up class.

• Detailed homework, quizzes and exam’s information will be noticed via course website so that problems caused by not visiting course website is your responsibility


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Course Schedule & Outline

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Class schedule, outline and office hours can be changed without pre-notification

Week # Date Course Contents

1 Sept. 03 Introduction to Dynamics

2 Sept. 10 Kinematics of Particle

3 Sept. 17 Kinetics of a Particle: Force and Acceleration

4 Sept. 24 Kinetics of a Particle: Work and Energy

5 Oct. 01 Kinetics of a Particle: Impulse and Momentum

6 Oct. 08 Mid Term Exam I

7 Oct. 15 Planar Kinematics of a Rigid Body

8 Oct. 22 Planar Kinematics of a Rigid Body: Force and Acceleration

9 Oct. 29 Planar Kinematics of a Rigid Body: Work and Energy

10 Nov. 05 Planar Kinematics of a Rigid Body: Impulse and Momentum

11 Nov. 12 Mid Term Exam II

12 Nov. 19 Three-Dimensional Kinematics of a Rigid Body

13 Nov. 26 No Class - Thanksgiving Holiday

14 Dec. 03 Three-Dimensional Kinetics of a Rigid Body

15 Dec. 10 Vibrations

16 Dec. 17 Final Exam


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Wrap Up

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Read the syllabus in detail!

Please visit course web site frequently

• http://iris.nyit.edu/~jlee26

• Course information is subject to change,

so always check here for the latest info

Office hours will not be held this week.

They begin next week.

Welcome, good luck, and enjoy!


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Dynamics

Dynamics is that branch of mechanics which

deals with the motion of bodies under the

action of forces

Kinematics

study of motion w/o reference to the forces causing motions

study of the geometry of motion. Kinematics is used to relate

displacement, velocity, acceleration, and time without reference

to the cause of motion.

Kinetics

relates the action of forces on bodies to their resulting

motions

study of the relations existing between the forces acting on a

body, the mass of the body, and the motion of the body.

Kinetics is used to predict the motion caused by given forces or

to determine the forces required to produce a given motion.

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Dynamics

Dynamics includes

Rectilinear motion: position, velocity, and

acceleration of a particle as it moves along a

straight line.

Curvilinear motion: position, velocity, and

acceleration of a particle as it moves along a

curved line in two or three dimensions.


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Dynamics

Basic Concepts

Space: geometric region occupied by bodies

Time: a measure of the succession of events

and is considered as absolute quantity in

Newtonian mechanics

Mass: quantitative measure of the inertia or

resistance to change in motion of a body

Force: vector action of one body on another

Particle: a body of negligible dimensions

Rigid body: a body whose changes in shape

are negligible compared w/ the changes in

position of a body as a whole

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Dynamics

Newton’s Laws

Law 1: A particle remains at rest or continues to

move w/ uniform velocity (in a straight line w/ a

constant speed) if there is no unbalanced force

acting on it

Law 2: The acceleration of a particle is

proportional to the resultant force acting on it

and is in the direction of this force (F=ma)

Law 3: The forces of action and reaction b/w

interacting bodies are equal in magnitude,

opposite in direction, and collinear


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Dynamics

Units

SI Units (US Customary Units)

Mass: kg (slug)

Length: m (ft)

Time: sec. (sec.)

Gravitation:

F: the mutual force of attraction between two particles

G: a universal constant called the constant of gravitation

m1,m2: the masses of the two particles

r: the distance b/w the centers of the particles

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2

21

r

mmGF


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Dynamics

Gravitation

Gravitational acceleration

M is the mass of the larger body, is a unit vector

directed from the large mass to the smaller mass.

Negative sign means the force is an attractive force

In the same way,

g =9.80665 m/s2 (32.1740 ft/s2)

Variation of g with altitude

go: gravitational acceleration at the sea level, h:

altitude, R: the radius of the earth, me: the mass of

the earth35

2

21

r

mmGF

rr

GMg ˆ

2

r̂

22

hR

Rgg o

2R

mGg e


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Kinematics of a Particle

Rectilinear Kinematics: Continuous

Motion

Find the kinematic quantities (position,

displacement, velocity, and acceleration) of a

particle traveling along a straight path.

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Applications

Relations between s(t),

v(t), and a(t) for general

rectilinear motion.

Relations between s(t),

v(t), and a(t) when

acceleration is constant.



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Applications

The motion of large objects,

such as rockets, airplanes,

or cars, can often be

analyzed as if they were

particles.

Why?

If we measure the altitude

of this rocket as a function

of time, how can we

determine its velocity and

acceleration?


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Applications

A sports car travels along a straight road.

Can we treat the car as a particle?

If the car accelerates at a constant rate, how

can we determine its position and velocity at

some instant?

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Overview of Mechanics

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Statics: The study of

bodies in equilibrium.Dynamics:1. Kinematics – concerned with

the geometric aspects of motion

2. Kinetics - concerned with

the forces causing the motion

Mechanics: The study of how bodies

react to forces acting on them.


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Rectilinear Kinematics: Continuous Motion

A particle travels along a straight-

line path defined by the coordinate

axis s.

The position of the particle at any

instant, relative to the origin, O, is

defined by the position vector r, or

the scalar s. Scalar s can be positive

or negative. Typical units for r and s

are meters (m) or feet (ft).

The displacement of the particle is

defined as its change in position.

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Vector form: r = r’ - r Scalar form: s = s’ - s

The total distance traveled by the particle, sT, is a

positive scalar that represents the total length of the

path over which the particle travels.


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Velocity

Velocity is a measure of the rate of change in the position of a

particle. It is a vector quantity (it has both magnitude and direction).

The magnitude of the velocity is called speed, with units of m/s or

ft/s.

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The average velocity of a particle during a

time interval t is

vavg = r / t

The instantaneous velocity is the time-derivative of position.

v = dr / dt

Speed is the magnitude of velocity: v=ds/dt

Average speed is the total distance traveled divided by elapsed

time: (vsp)avg = sT / tMENG 212 – Engineering Mechanics II: Dynamics. Jong B. Lee, PhD, All rights reserved.

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Acceleration

Acceleration is the rate of change in the velocity of a particle. It is a

vector quantity. Typical units are m/s2 or ft/s2.

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As the text indicates, the derivative equations for velocity and

acceleration can be manipulated to get a ds = v dv

The instantaneous acceleration is the time

derivative of velocity.

Vector form: a = dv / dt

Scalar form: a = dv / dt = d2s / dt2

Acceleration can be positive (speed

increasing) or negative (speed decreasing).



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Summary

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• Differentiate position to get velocity and acceleration.

v = ds/dt ; a = dv/dt or a = v dv/ds

• Integrate acceleration for velocity and position.

• Note that so and vo represent the initial position and

velocity of the particle at t = 0.

Velocity:

t

o

v

vo

dtadv s

s

v

v oo

dsadvvor t

o

s

so

dtvds

Position:


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Constant Acceleration

The three kinematic equations can be integrated for the

special case when acceleration is constant (a = ac) to

obtain very useful equations. A common example of

constant acceleration is gravity; i.e., a body freely falling

toward earth. In this case, ac = g = 9.81 m/s2 = 32.2 ft/s2

downward. These equations are:

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tavv coyields

t

o

c

v

v

dtadvo

2coo

s

t(1/2) a t vss yields t

os

dtvdso

)s-(s2a)(vv oc

2

o

2 yields s

s

c

v

v oo

dsadvv


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Example

A particle travels along a straight line to

the right with a velocity of v = (4t–3t2) m/s

where t is in seconds. Also, s = 0 when t =

0.

The position and acceleration of the

particle when t = 4s.


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Example: Solution

Take a derivative of the velocity to

determine the acceleration

(or in the direction) when t = 4s

Calculate the distance traveled in 4s by

integrating the velocity using so = 0:

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mttss

dtttdsvdtdsdt

dsv

o

s

so

322

34

4

0

32

4

0

2

22

/206434

smatdt

ttd

dt

dva


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Example

A particle is moving along a straight line

such that its velocity is defined as v = (-

4s2) m/s, where s is in meters.

The velocity and acceleration as

functions of time if s = 2 m when t = 0.

Since the velocity is given as a function

of distance, use the equation v=ds/dt.

Express the distance in terms of time.

Take a derivative of it to calculate the velocity

and acceleration.


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Example: Solution

Since v=-4s2

Determine the distance by integrating using so=2

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2

2 44s

dsdts

dt

dsv

18

21

2

14

2

11

2

114

14

4

22

1

0

2

2

2

0

ts

st

sst

sst

dsss

dsdt

sst

ss

s

t

o



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Example: Solution

Take a derivative of distance to calculate

the velocity and acceleration

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2

332

22

/18

256

18

8216

18

16

/18

16

18

812

18

2

18

2

smtttdt

d

dt

dva

smtttdt

d

dt

dsv

mt

s


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Rectilinear Kinematics: Erratic Motion

Determine position, velocity, and acceleration of

a particle using graphs.

Applications

s-t, v-t, a-t, v-s, and a-s diagrams

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Application

In many experiments,

a velocity versus

position (v-s) profile is

obtained.

If we have a v-s graph

for the tank truck, how

can we determine its

acceleration at

position, s=1,500 feet?


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Applications

The velocity of a car is

recorded from a

experiment. The car

starts from rest and

travels along a straight

track.

If we know the v-t plot,

how can we determine

the distance the car

traveled during the time

interval 0<t<30s or

15<t<25s? 52


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Rectilinear Kinematics: Erratic Motion Erratic motion

Graphing provides a good way

to handle complex motions that

would be difficult to describe

with formulas.

Graphs also provide a visual

description of motion and

reinforce the calculus concepts

of differentiation and integration

as used in dynamics.

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The approach builds on the facts that slope and

differentiation are linked and that integration can be

thought of as finding the area under a curve.


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s-t Graph

Plots of position vs. time can be

used to find velocity vs. time

curves. Finding the slope of the

line tangent to the motion curve

at any point is the velocity at

that point (or v = ds/dt).

Therefore, the v-t graph can be

constructed by finding the slope

at various points along the s-t

graph.

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Rectilinear Kinematics: Erratic Motion v-t Graph

Plots of velocity vs. time can be

used to find acceleration vs. time

curves. Finding the slope of the

line tangent to the velocity curve at

any point is the acceleration at that

point (or a = dv/dt).

Therefore, the acceleration vs.

time (or a-t) graph can be

constructed by finding the slope at

various points along the v-t graph.

Also, the distance moved

(displacement) of the particle is

the area under the v-t graph during

time t.


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a-t Graph

Given the acceleration vs.

time or a-t curve, the

change in velocity (v)

during a time period is the

area under the a-t curve.

So we can construct a v-t

graph from an a-t graph if

we know the initial velocity

of the particle.

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Rectilinear Kinematics: Erratic Motion a-s Graph

A more complex case is presented by

the acceleration versus position or a-

s graph. The area under the a-s

curve represents the change in

velocity

This equation can be solved for v1,

allowing you to solve for the velocity

at a point. By doing this repeatedly,

you can create a plot of velocity

versus distance.

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)vdvads(Recall

graph sa under theArea

2

1 2

1

2

0

2

1

s

s

adsvv


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Rectilinear Kinematics: Erratic Motion v-s Graph

Another complex case is presented by

the velocity vs. distance or v-s graph.

By reading the velocity v at a point on

the curve and multiplying it by the

slope of the curve (dv/ds) at this same

point, we can obtain the acceleration

at that point. Recall the formula

Thus, we can obtain an a-s plot from

the v-s curve

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ds

dvva


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Example

The s-t graph for a sports car moving

along a straight road

The v-t graph and a-t graph over the time

interval shown


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Example: Solution

The v-t graph can be constructed by

finding the slope of the s-t graph at key

points. What are those?

When 0<t<5s

When 5<t<10s

60

v (m/s)

t (s)

30

5 10

smtdt

d

dt

dsv

smttdt

d

dt

dsv

/307530

/63

105

2

50



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Example: Solution Similarly, the a-t graph can be constructed by finding

the slope at various points along the v-t graph. Using

the results of the first part where the velocity was

found:

When 0<t<5s

When 5<t<10s

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2

105

2

50

/030

/66

smdt

d

dt

dva

smtdt

d

dt

dva

a (m/s2)

t (s)

6

5 10


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Example

The v-t graph shown.

Find: The a-t graph, average speed, and

distance traveled for the 0-50s interval.

Find slopes of the v-t curve and draw the a-t graph. Find

the area under the curve. It is the distance traveled.

Finally, calculate average speed (using basic

definitions!).

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Example: Solution

Find the a–t graph:

When 0<t<30s

When 30<t<50s

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0.4

a (m/s²)

30 50t (s)0

a-t graph

2

5030

2

300

/0

/4.0

smdt

dva

smdt

dva


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Example: Solution

Now find the distance traveled:

64

v = 12

sm

v

m

sss

mdtvdts

mdttvdts

savg

/4.850

420

time

distancetotal

420240180

24030501212

1802

304.04.0

)500(

5030300500

50

30

5030

230

0

300


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Example The v-t graph shown.

Find: The a-t graph, average speed, and distance

traveled for the 0-48s interval.

Find slopes of the v-t curve and draw the a-t graph.

Find the area under the curve. It is the distance

traveled. Finally, calculate average speed (using basic

definitions!).


N

Y

I

T

Example: Solution

Find the a-t graph

When 0<t<30s

When 30<t<48s

66

2

4830

2

300

/333.0483

1

/2.05

1

smtdt

d

dt

dva

smtdt

d

dt

dva

-0.33

0.2

a(m/s²)

30 48 t(s)

a-t graph



N

Y

I

T

Example: Solution

Now find the distance traveled:

67

sm

v

m

sss

mtdttvdts

mdttvdts

savg

/348

144

time

distancetotal

1445490

54482

1

3

148

3

1

90302

1

5

1

5

1

)480(

4830300480

48

30

248

30

4830

230

0

300


N

Y

I

T

Announcement

Homework

Problems 12-2, 6, 7, 9, 13, 15, 18, 22, 40, 46,

48, 58, and 69

Due: Sept. 10, 2014

Quiz #1

Sept. 10

Two problems will be given.

Review lecture notes, examples, and

homeworks

68

01-MENG212-Intro N Ch. 12-1

Documents

Transcript of 01-MENG212-Intro N Ch. 12-1