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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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The position of particle Pat any time tcan be describedby:
Rectangular coordinates (x, y, z)
(Cartesian coordinates)
Cylindrical coordinates (r, , z)
Choice of Coordinates
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The position of particle Pat any time tcan be describedby:
Rectangular coordinates (x, y, z)
(Cartesian coordinates)
Cylindrical coordinates (r, , z)
Spherical coordinates (R, , )
Choice of Coordinates
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The position of particle Pat any time tcan be describedby:
Rectangular coordinates (x, y, z)
(Cartesian coordinates)
Cylindrical coordinates (r, , z)
Spherical coordinates (R, , )
Tangential and normal to the curve
(t, n)
Choice of Coordinates
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The End
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