Lec 1 Dynamic

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Lecture # 1

Introduction to Dynamics

Prepared By:Courtesy of Engr. Abdul Wahab

HITEC University

Taxila Cantt.

ME 201 - Engineering Dynamics


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Definition

Mechanics is a branch of the physical sciencesthat is concerned with the state of rest or motionof bodies subjected to the action of forces.

Engineering mechanics is divided into two areas

of study, namely, statics and dynamics.

Statics is the branch of mechanics which dealswith equilibrium of bodies at rest

Dynamics is that branch of mechanics whichdeals with the motion of bodies under the actionof forces.


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Dynamics

Dynamics has two distinct parts: kinematics, which is the study of motion without

reference to the forces which cause motion, and

kinetics, which relates the action of forces on

bodies to theirresulting motions.


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History of Statics

Archimedes (287-212B.C.) on the principle of Liverand principle of Bouyancy

Stevinus (1548-1642) vector combination of

forces


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History of Dynamics

Newton (1642-1727), guided byGalileo's work, was able to make an

accurate formulation of the laws of

motion

Following Newton's time, important

contributions to mechanics were made

by Euler, D'Alembert, Lagrange, Laplace,

Poinsot, Coriolis, Einstein, and others.

The beginning of a rational understanding of dynamics iscredited to Galileo (1564-1642) who made careful observations concerning bodies in free fall,

motion on an inclined plane, and motion of the pendulum.


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Applications of Dynamics Analysis and Design of:

moving structures

fixed structures subject to shock loads

robotic devices

automatic control systems

Rockets

Missiles

Spacecraft

ground and air transportation vehicles

machinery of all types Turbines

Pumps

reciprocating engines

etc.


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Basic Concepts

Timeis a measure of the succession of events

Mass is the quantitative measure of the inertia or resistance tochange in motion of a body.

Mass may also be considered as the quantity of matter in a body as well asthe property which gives rise to gravitational attraction.

Force is the vector action of one body on another.

A particle is a body of negligible dimensions.

When the dimensions of a body are irrelevant to the description of itsmotion or the action of forces on it, the body may be treated as a particle.

An airplane, for example, may be treated as a particle for the description ofits flight path.

A rigid body is a body whose changes in shape are negligiblecompared with the overall dimensions of the body or with thechanges in position of the body as a whole.


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Review: Vector Analysis

Vector is a quantity which has magnitude anddirection, and adds according to the

parallelogram law.

As Shown in Fig., A = B + C, where A is theresultant vector and B and C are component

vectors.


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Unit Vector

A unit vector, UA , has a magnitude of onedimensionless unit and acts in the same

direction as A.

It is determined by dividing A by its magnitude A, i.e,


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Cartesian Vector Notation

The directions of the positive x, y, z axes are defined by theCartesian unit vectors i, j, k, respectively.

As shown in Fig., vector A is

formulated by the addition of its

x, y, z components as

The magnitude ofAis determined from


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Cartesian Vector Notation (Contd.)

The direction of A can be defined in terms of its coordinatedirection angles, , , , measured from the tail ofA to the positivex, y, z axes

These angles are determined from the

direction cosines which represent the i, j, k

components of the unit vector UA ; i.e.,

so that the direction cosines are

Hence, Ua = cos i + cos j + cos k

Since:


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The Cross Product

The cross product of two vectors A and B, which yields the

resultant vector C, is written as

C = A x B

The magnitude ofC is given by:

C = ABsin

where is the angle made between

the tails ofA and B(0 180).

The direction ofC is determined by the right-hand rule perpendicular to the plane containing vectors A and B.


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Cross Product (Contd.)

The vector cross product is not commutative,

i.e., AxBBxA. Rather,

A x B = - B x A

The distributive law is valid; i.e., A x(B + D )= A x B + A x D

And the cross product may be multiplied by a scalar m in any manner; i.e.,

m(A x B) = (mA)x B = A x(mB) = (A x B)m

EquationC

=A

xB

can be used to find the cross productof any pair of Cartesian unit vectors, using the scheme

shown in the fig., e.g.,

k x i = j

i x k = -j


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Cross Product (Contd.)

IfA and B are expressed in Cartesian componentform, then the cross product can be evaluated by

expanding the determinant

which yields


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The Dot Product

The dot product of two vectors A and B , which yields a scalar, is

defined as

If A and B are expressed in Cartesian component form, then the

dot product can be expressed as

The dot product may be used to determine the angle formed

between two vectors.


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The Dot Product (Contd.)

It is also possible to find the component of a vectorina given direction using the dot product.

From Figure, A cos defines the magnitude of thecomponent (or projection) of vector A in the direction ofB.

UB represents a unit vector acting in the direction ofB


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For Further Review

Check out: Appendix A

Appendix B

Appendix C From the Book:

Engineering Mechanics: Dynamics

ByR.C. Hibbeler


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Solving Problems In Dynamics

Method of Attack

1. Formulate the problem:a) State the given data.

b) State the desired result.

c) State your assumptions and approximations.

2. Develop the solution:a) Draw any neededdiagrams, and include coordinates which are appropriate

for the problem at hand (Establish a coordinate system).

b) State the governing principles to be applied to your solution.

c) Make your calculations.

d) Ensure that your calculations are consistent with the accuracyjustified by the

data.e) Be sure that you have used consistent units throughout your calculations.

f) Ensure that your answers are reasonable in terms of magnitudes, directions,common sense, etc.

3. Draw conclusions.


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Units

The four fundamental quantities of mechanics, and their units and symbols for the

two systems, are summarized in the following table

SI Units U.S. Customary Units

QUANTITYDIMENSIONAL

SYMBOLUNITS SYMBOLS UNITS SYMBOLS

Mass M kilogram kg slug -

Length L meter m foot ft

Time T second s second sec

Force F newton n pound lb


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Dimensions

The principle of dimensional homogeneity statesthat all physical relations must be dimensionallyhomogeneous; that is, the dimensions of all terms inan equation must be the same.

It is customary to use the symbols L, M, T, andF tostand forlength, mass, time, and force, respectively.

A given dimension such as length can be expressed ina number of different units such as meters,millimeters, or kilometers. Thus, a dimension is different from a unit.


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Dimensional Consistency

In SI units force is a derived quantity

F = ma

From above equation force has the dimensions of mass

times acceleration or

[F] = [M][L][T]-2


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The position of particle Pat any time tcan be describedby:

Rectangular coordinates (x, y, z)

(Cartesian coordinates)

Choice of Coordinates


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Cylindrical coordinates (r, , z)



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Spherical coordinates (R, , )



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Spherical coordinates (R, , )

Tangential and normal to the curve

(t, n)



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The End

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