209-2020 Engineering Mechanics I

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Manner Guideline in this Lecture Course

❑ Be reasonable and act politely .

❑ Turn off your mobile phone. If you have urgent calls

to make or answer, kindly leave the room.

❑ No noisy chat and all other activities that can distract

the lecture should be avoided.

❑ No food. Only water are allowed.

❑ Do not disturb your classmates.

❑ Dress properly.

❑ No smoking

Course Syllabus• Engineering Mechanics I 3 (2-0-3) Credit

• Lecture Hour: Sunday Group1 08.00-10.00 -• Group2 10.00 - 12.00

• Grading Policy: Total Score: 125 point , • Homework 10 point

– Midterm Exam 15 point– Final 90 point– Class Activity 10 point

Contents

• 1- Introduction, force in level, vector algebra,Forces in a plane, Forces in space

• 2- Statics of particles

• 3- Statics of Rigid bodies

• 4- Equilibrium of Rigid bodies (2 and 3dimensions)

• 5- Analysis of structures (Trusses, Frames

• and Machines)

• 6- Friction

• 7- Centroids and Centers of gravity

• 8- Moments of inertia of areas and masses

• 9- Internal forces

Textbook

“Engineering Mechanics STATICS”

R.C. Hibbeler, Engineering Mechanics

“Engineering Mechanics, STATICS”

Meriam and Kraige

Mechanics ?

Mechanics

Statics

Dynamics-Equilibrium

-Selected Topics

Kinematics Kinetics

-Particles

-Rigid Bodies

-Particles

- Rigid Bodies

A branch of physical science

which deals with ( the states of

rest or motion of ) bodies under

action of forces

Dynamics: Motion of bodies

Statics:

Equilibrium of bodies

(no accelerated motion)

under action of Forces

PARTS OF MECHANICS

Introduction to Statics

MechanicsMechanics is the physical science which deals with the

effects of forces on objects. Mechanics plays a greaterrole in engineering analysis. The principles of mechanicshave wide applications in engineering; although they arefew. The principles of mechanics are central to researchand development in the fields of vibrations, stability andstrength of structures and machines, robotics, rocket andspacecraft design, automatic control, engineperformance, fluid flow, electrical machines andapparatus, and molecular, atomic, and subatomicbehavior.

Structures

Machines

Robotics

Introduction to Statics

Mechanics

The subject of mechanics is logically divided intotwo parts: statics, which concerns the equilibrium ofbodies under action of forces, and dynamics, whichconcerns the motion of bodies.

Mechanics #2

Mechanics

Statics

Dynamics

Mech of Materials

Fluid Mechanics

Vibration

Fracture Mechanics

Etc.

Structures

Automotives

Robotics

Spacecrafts

MEMs

Etc.

Basic Concepts

Basic Concept

The following concepts and definitions are basic to study of mechanics,

and they should be at the outset.

Space is the geometric region occupied by bodies whose positions are

described by linear angular measurements relative to coordinate system. For

three-dimensional problems, three independent coordinates are needed. For

two-dimensional problems , only two coordinates are required.

Time is the measure of the succession of events and is a basic quantity in

dynamics. Time is not directly involved in the analysis of static problems.

Mass is a measure of the inertia of a body, which is its resistance to a

change of velocity. Mass can also be thought of as the quantity of matter in a

body. The mass of a body affects the gravitational attraction force between it

and other bodies. This force appears in many applications in statics.

Force is the action of one body on another. A force tends to move a

body in the direction of its action. The action of a force is characterized by its

magnitude, by the direction of its action, and by its point of application.

Basic Concept continuedA particle is a body of a negligible dimensions. In the mathematical

sense, a particle is a body whose dimensions are considered to be near zero

so that we may analyze it as a mass concentrated at a point.

Rigid body: A body is considered rigid when the change in distance

between any two of its points is negligible.

Scalars and VectorsWe use two kinds of quantities in mechanics; scalars and vectors.

Scalar quantities are those with which only a magnitude is associated.

Examples of scalar quantities are time, volume, density, speed, energy, and

mass. Vector quantities possess direction as well as magnitude, and must

obey the parallelogram law. Examples of vector quantities are displacement,

velocity, acceleration, force, moment, and momentum.

Vectors can be classified as free, sliding, or fixed

• In dynamics, force is an action that tends to cause acceleration of an object.

• The SI unit of force magnitude is the newton (N). One newton is equivalent to one kilogram-meter per second squared (kg·m/s2 or kg·m · s – 2)

Basic Concept - Force

Force: Vector quantity that describes an action of

one body on another [Statics]

SCALARS AND VECTORS

Scalars: associated with “Magnitude” alone

Vectors: associated with “Magnitude” and “Direction”

- mass, density, volume, time, energy, …

- force, displacement, velocity, acceleration, …

: Direction

or V| |V

Magnitude:

V or V

Vector :

free vector

(“math” vector)

Vector’s Point of Application

Vectors: “Magnitude”, “Direction”

F

Free Vector

rotating motion, coupleE.g.) Force on

non- rigid body

Fixed Vector

F

F

Sliding Vector

F

FRigid Body

E.g.) Force on

rigid-body

F

F

=?

line of action

“Point of Application”

The external

consequence

of these two

forces will

be the same

if ….

- Rigid Body

Rotational motion occurs at every point in the object.point of action

rotation

vector

Principle of

Transmissibility

Internal Effect –stress

Externaleffect

The Principle of Transmissibility

“A force may be applied at any point on its given line of action

without altering the resultant effects external to the rigid body on

which it acts.”

We can slide the force along its line

of action.(force can be considered as sliding vector)

F

F

=?

The two force can be

considered equivalent if

……

If we concerns only about the external

resultant effects on rigid body.

UnitsIn mechanics we use four fundamental quantities called

dimensions. These are length, mass, force, and time. Although there are a number of different systems of units, only the two systems most commonly used in science and technology will be used in this course.

QuantityDimensional

Symbol

SI UnitsU.S. Customary

Units

Unit Symbol Unit Symbol

Mass M kilogram kg slug -

Length L meter m foot ft

Time T second s second sec

Force F newton N pound Ib

NEWTON’S LAWS OF MOTION (1st Law)

The study of rigid body mechanics is

formulated on the basis of Newton’s laws of

motion.

= 0F

First Law:

An object at rest tends to stay at rest and an object in motion

tends to stay in motion with the same speed and in the same

direction, unless acted upon by an unbalanced force.

NEWTON’S LAWS OF MOTION (2nd Law)

Second Law:

The acceleration of a particle is proportional to the vector sum of

forces acting on it, and is in the direction of this vector sum.

mF

a

amF

=

NEWTON’S LAWS OF MOTION

Third Law:

The mutual forces of action and reaction between two

particles are equal in magnitude, opposite in direction,

and collinear.

F F−F F−

Confusing? Concept of FBD (Free Body Diagram)

Point: Isolate the body

Forces always occur in pairs – equal and

opposite action-reaction force pairs.

Law of GravitationIn statics as well as dynamics we often need to compute the weight of a body, which is the gravitational force acting on it. This computation depends on the law of gravitation, which was also formulated by Newton. The law of gravitation is expressed by the following equation:

𝐹 = 𝐺.𝑚1𝑚2

𝑟2Where: F = the mutual force of attraction between two particles,

G = a universal constant known as the constant of gravitation,m1, m2 = the masses of the two particles, andr = the distance between the centers of the particles.

The mutual forces F obey the law of action and reaction, since they are equal and opposite and are directed along the line joining the centers of the particles, as shown in the following Figure.

𝑟

By experiment the gravitational constant is found to be 𝐺 = 6.673 10−11 𝑚3/(𝑘𝑔. 𝑠2)

FFm1 m2

Gravitational Attraction of the EarthFor a body of mass m near the surface of the earth, the gravitational attraction F on the body is specified by:

𝐹 = 𝐺.𝑚1𝑚2

𝑟2After replacing m1by m, m2 by the earth mass, and r by the radius of the earth, the gravitational attraction F is:

𝐹 = 𝑚. 𝑔

Where: m is the mass of the body

- g = GM/R2 = 9.81 m/s2 (32.2 ft/s2)

𝑔 = 9.80665 𝑚/𝑠2 in SI units𝑔 = 32.1740 𝑓𝑡/𝑠𝑒𝑐2 in U.S. units

Mass of the earth = 𝑀 = 5.976 1024 𝑘𝑔equatorial diameter of the earth = 12756 km

We usually denote the magnitude of the gravitational force or weight with the symbol W. because the body falls with an acceleration g i.e.

𝑊 = 𝑚.𝑔The standard values for g of 9.81 𝑚/𝑠2 and 32.2 𝑓𝑡/𝑠𝑒𝑐2 will be sufficiently accurate for our calculations in statics.

x

y

(0, 0)

(x, y)

x

y

r is standard notation for a length that can contain x, y and z coordinates.

Transforming from one coordinate system to another

Remember that positive angles are measured counterclockwise from 0o

(usually from the positive x-axis).

How can we relate r or to x and y?

222 yxr += 22 yxr +=

Pythagorean Theorem

x

y

r

y=sin

r

x=cos

x

y=tan

sinry =

cosrx =

Converts from Cartesian to polar coordinates.

r

These definitions are based on the angle shown in the diagram. If you use a different angle you may have to modify these expressions.

You must know how to use these trigonometric functions!!

Vector Properties

BA

= - Only when both magnitude and direction are exactly the same!

BA

+ - These must be added vectorally!

A

BBAR

+=

1. Draw first vector.2. Draw second vector from tip of first vector

(repeat for each additional vector present)3. Draw resultant (R) from tail of first vector to

tip of last vector.

ABBA

+=+ - Cumulative law of addition.

( ) ( ) CBACBA

++=++ - Associative law of addition.

A

− - Negative of a vector. Same magnitude but opposite direction.

( )BABA

−+=− - Subtraction of a vector. Same rules as for addition.

Am

- Multiply a vector by a scalar. - Positive scalar changes magnitude.- Negative scalar changes magnitude and direction.

Vector Components

Vectors can be broken up into components. Components are used to describe part of the vector along each of the coordinate directions for your chosen coordinate system.

- The number of components is determined by the number of dimensions.

A 2D vector in the x-y plane has 2 components

- 1 in the x direction and 1 in the y direction.

A 3D vector in spherical coordinates has 3 components

- 1 in the r direction, 1 in the direction and 1 in the f direction.

- All vector components are defined to be orthogonal (perpendicular) to each other.

- Vector components are also vector quantities.

x

y

A

xA

yA

yA

222

yx AAA

+=

A

Ay

=sin

A

Ax

=cos

x

y

A

A

=tan

yx AAA

+=Vectors can be moved as long as their length and orientation do not change!

When you square a vector, the result is a square of the magnitude. Directional information is removed.

sinAAy

=

cosAAx

=

These are only valid for the specified angle.

xA

Vector notation using unit vectors

A unit vector is a dimensionless vector with a magnitude of 1 that is used to describe direction.

Cartesian coordinate unit vectors:

x – direction =

y – direction =

z – direction =

i

j

k

- The hat ‘^’ is used to distinguish unit vectors from other vectors.

The following notation is also used:

x – direction =

y – direction =

z – direction =

x

y

z

kAjAiAA zyxˆˆˆ ++=

Ax = magnitude of A along x – directionAy = magnitude of A along y – directionAz = magnitude of A along z – direction

Examples of vectors using unit vector notation:

kzjyixr ˆˆˆ ++= x = length along x – direction

y = length along y – directionz = length along z – direction

Example: Adding vectors using vector notation

kAjAiAA zyxˆˆˆ ++=

kBjBiBB zyxˆˆˆ ++=

BAR

+= ( ) ( )kBjBiBkAjAiA zyxzyxˆˆˆˆˆˆ +++++=

( ) ( ) ( )kBAjBAiBA zzyyxxˆˆˆ +++++= ( ) ( ) ( )kRjRiR zyx

ˆˆˆ ++=

You add together magnitudes along similar directions. Remember to include appropriate sign.

Rx Ry Rz

You can determine magnitude and direction for 3D vectors in a similar fashion to what was done for 2D.

( ) ( ) ( )2222

zyx RRRR ++= R

Rxx =cos

R

Ry

y =cos

R

Rzz =cos

x, y and z are the angles measured between the vector and the specified coordinate direction.

The magnitude can be found using the Pythagorean Theorem.

Working with Vectors Continued

Vector Addition. Two vectors A and B may be added to form a“resultant” vector R = A + B by using the parallelogram law or using atriangle construction, which is a special case of the parallelogram law.

A

B

Vector Subtraction. Two vectors A and B may be subtracted to form a“resultant” vector R’ = A – B = A + (-B) by using the parallelogram lawor using a triangle construction, which is a special case of theparallelogram law.

RA

B B

R

A

B

R’A

B

AR’

A

B

Working with Vectors Continued

Resolution of a Vector. A vector may be resolved into two“components” having known lines of action by using the parallelogram law.For example, if R is to be resolved into components acting along the lines aand b, one starts at the head of R and extends a line parallel to a until itintersects b. Likewise, a line parallel to b is drawn from the head of R to thepoint of intersection with a. The two components A and B are then drawnsuch that they extended from the tail of R to the points of intersection.

R

b

a

R

b

a

B

A

Unit vectors. A vector V may be expressed mathematically by multiplyingits magnitude V by a vector n whose magnitude is one and whose directioncoincides with that of V. The vector n is called a unit vector. Thus,

V = Vn

Procedure For Analysis

Problems that involve the addition of two forces can be solved as follows:

Parallelogram Law• Make a sketch showing the vector addition using the parallelogram law.• Two “component” forces add according to the parallelogram law, yielding a

resultant force that forms the diagonal of the parallelogram.• If a force is to be resolved into components along two axes directed from the tail

of the force, then start at the head of the force and construct lines parallel to the axes, thereby forming the parallelogram. The sides of the parallelogram represent the components.

• Label all the known and unknown force magnitudes and the angles on the sketch and identify the two unknowns.

Trigonometry• Redraw a half portion of the parallelogram to illustrate the triangular head-to-tail

addition of the components.• The magnitude of the resultant force can be determined from the law of cosines,

and its direction is determined from the law of sines.• The magnitude of two force components are determined from the law of sines.

𝐵𝐴

𝐶

𝑐

𝑎𝑏

Sine Law:𝐴

sin 𝑎=

𝐵

sin 𝑏=

𝐶

sin 𝑐

Cosine Law:

𝐶 = 𝐴2 + 𝐵2 − 2𝐴𝐵𝑐𝑜𝑠𝑐

𝑈𝑥= 𝑈 𝑐𝑜𝑠𝜃𝑥 , 𝑈𝑦

= 𝑈 𝑐𝑜𝑠𝜃𝑦 , 𝑈𝑧= |𝑈| 𝑐𝑜𝑠𝜃𝑧

cos2 θx + cos2 θy + cos2 θz = 1.

U = |U|e.

𝑈𝑥𝑖 + 𝑈𝑦𝑗 + 𝑈𝑧𝑘 = 𝑈 (𝑒𝑥𝑖 + 𝑒𝑦𝑗 + 𝑒𝑧𝑘)

𝑈𝑥 = 𝑈 𝑒𝑥 , 𝑈𝑦 = 𝑈 𝑒𝑦, 𝑈𝑧 = 𝑈 𝑒𝑧𝑐𝑜𝑠𝜃𝑥 = 𝑒𝑥 , 𝑐𝑜𝑠𝜃𝑦 = 𝑒𝑦 , 𝑐𝑜𝑠𝜃𝑧 = 𝑒𝑧

• Position Vectors in Terms of Components

• 𝑟𝐴𝐵 = 𝑥𝐵 − 𝑥𝐴 𝑖 + 𝑦𝐵 − 𝑦𝐴 𝑗 + 𝑧𝐵 − 𝑧𝐴 𝑘

Components of a Vector Parallel to a Given Line

U= |U|eAB

Figure 1-8

Dot (scalar) product

• A • B = AB cos θ (0 < θ < 180°)

• A • B = B • A

• A • (B + C) = A • B + A • C

• i • i = j • j = k • k = 1

• i • j = j • k = k • i = 0

• A • B = (Ax i + Ay j + Az k) • (Bx i + Byj + Bz k)

• A • B = AxBx + Ay By + AzBz

Fig. 1.13

Cross (vector) product

• 𝑪 = 𝑨 × 𝑩

• C is C = AB sin θ

• Ax (B+C)=(Ax B) +(Ax C)• Ax (Bx C) = (Ax B) x C• Ax B ≠ Bx A• A x B= - B x A.• i × i = 0, j × j = 0, k × k = 0

(1.22)• i×j=k, j×k=i, k×i=j

• A×B = (Axi + Ayi + Azi )×(Bxi + Byi + Bzi)

• A×B = (AyBz - AzBy)i + (AzBx - AxBz)j + (AxBy - AyBx)k

• 𝐴 × 𝐵 =

𝑖 𝑗 𝑘𝐴𝑥 𝐴𝑦 𝐴𝑧𝐵𝑥 𝐵𝑦 𝐵𝑧

• 𝐴 × 𝐵. 𝐶 =

𝐴𝑥 𝐴𝑦 𝐴𝑧𝐵𝑥 𝐵𝑦 𝐵𝑧𝐶𝑥 𝐶𝑦 𝐶𝑧