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Strings - Analytical Mechanics - Lecture Slides, Slides of Applied Mechanics
In these Lecture slides, the Lecturer has discussed the following key concepts of Analytical Mechanics : Strings, Masses on a String, Displacements, Separation, Constraints, Potential, Tension, Longitudinal Problem, Replace Tension, Elastic
Typology: Slides
2012/2013
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Strings
Masses
on
a
String
-^ Couple
n^ equal
masses
on^
a
massless
string.
-^ Displacements
i
-^ Separation
a
-^ Constraints
=^0
n+
=^0
-^ Potential
from
string
tension
-^ The
longitudinal
problem
is
similar^ –^
Displacements
in^ x
-^ Replace
tension
with
elastic
springs
x
^1
Transverse vibration,
n^ segments
^0
^2
^3
n
n+
a
Large
Matrix
-^ The
direct
solution
is^ not
generally
possible.
-^ If
there
is^ a
solution
it^ is
an
harmonic
oscillator.
-^ Each
row
related
to^ the
previous
one
-^ The
eigenvalue
equation
reduces
to^ three
terms.
2 2 2
a a a
a
a
0 1
^
ij i j j^
e e^
^
2
^
^
^
i
i^
e a
a e a
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Fixed
Boundaries
-^ The
eigenvalue
equation
gives
a^ result
based
on
-^ The
phase
difference
depends
on^
initial
conditions.
-^ Pick
sin^
for^0
-^ Find
the
other
end
point
-^ Requires
periodicity
-^ Substitute
to^ get eigenfrequencies.^ –^
Integer
m^ gives
values
for
^
^
cos 1 2 2
^
a
Im Im^
1 0
^
n
t A
^
sin 0
t ijAe j
^ sin
t
n A n
^
sin) 1 sin(
Im^
1
sin(
^ n
^
cos 1 2 2 )(
m n a m
Periodic
Boundaries
-^ To
simulate
an^
infinite
string,
use boundaries
that
repeat.
-^ Phase
^ repeats
after
n
intervals.^ –^
Require
whole
number
of
wavelengths – Integer
m^ for
solutions
with
that
period
-^ Substitute
to^ get eigenfrequencies
as^ before.
^
^
cos 1 2 2
^
n a
^
0 m^ ^ n
^
^
m n
a
^
cos 1 2 2
1 1
^
n
Traveling
Wave
-^ In
a^ traveling
wave
the
initial
point
is^ not
fixed.
-^ Other
points
derive
from
the
initial
point
as^ before.
-^ The
position
can
be^
expressed
in^ terms
of^ the
unit
length
and
wavenumber.
ti i ti
ee A Ae
^
^
^
^
t jm n A j
sin Im
^
^
t jmi n Aej
2
^
^
t mka A sin j Im
m na k
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