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Some basic concept Vibration of Structures are Harmonic Waves, Influence of Axial Force, Initial Value Problem, Mathematical Modeling, Modal Analysis, Motion of Material Points, Orthogonality Relations, Projection Methods.Main points of this lecture are: Variational Formulation, Lagrangian Examples, Variational Procedure, Hamilton's Principle, Equation of Motion, Extended Hamilton's Principle, Potential Energy Expressions, Infinitesimal Variation, Constant Tension
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Contents:
Keywords: Variational formulation, Hamilton’s principle, Lagrangian, La-
grange’s equation of motion, Extended Hamilton’s principle
Consider the temporal evolution of the configuration of a one-dimensional
continuous system, recorded at two time instants t = t 1 and t = t 2 to be,
respectively, w(x, t 1 ) and w(x, t 2 ) with no record of the intermediate config-
urations. Now, the question is:
Can we determine the intermediate configurations through which the system
passed while going from w(x, t 1 ) to w(x, t 2
The answer to this question is provided by Hamilton’s principle which
states:
Of all the infinite paths available to a system between any two observed config-
urations, the system follows that path which extremizes the action A defined
by
t 2
t 1
L dt, (1)
where L = T − V is known as the Lagrangian, and T and V are, respectively,
the kinetic energy and potential energy expressions of the system at an arbi-
trary configuration.
2
Lagrangian:
l
0
(ρAw
2
,t
− T w
2
,x
) dx. (5)
Axial vibrations of a bar:
Kinetic energy:
l
0
ρAu
2
,t
dx. (6)
Potential energy:
l
0
σAdx =
l
0
2 dx =
l
0
EAu
2
,x
dx. (7)
Lagrangian:
l
0
(ρAw
2
,t
− T w
2
,x
) dx. (8)
Torsional vibrations of a circular bar:
Kinetic energy:
l
0
ρI p φ
2
,t
dx. (9)
Potential energy:
l
0
A
τ γdAdx =
l
0
A
Gr
2
2 dAdx =
l
0
p φ
2
,x
dx. (10)
Lagrangian:
l
0
(ρAw
2
,t
− T w
2
,x
) dx. (11)
4
The Lagrangian is, in general, a function of the field variable, its time and
space derivatives, and time. Thus, for a one-dimensional continuous system
l
0
L(w, w ,t , w ,x , w ,xt , w ,xx , t) dx,
where
L(·) is known as the Lagrangian density of the system. Note that we
have considered upto second order derivatives in space and mixed derivatives
in space-time for later reference.
Using the extremization condition (2), we obtain
t 2
t 1
l
0
δ
L(w, w ,t , w ,x , w ,xt , w ,xx , t) dx dt = 0,
t 2
t 1
l
0
∂w
δw +
∂w ,t
δw ,t
∂w ,x
δw ,x
∂w ,xt
δw ,xt
∂w ,xx
δw ,xx
dx dt = 0.
Integrating by parts and using the conditions δw| t 1
= δw| t 2
= 0 (initial and
final configurations on the trajectory are known), one can obtain
t 2
t 1
∂w ,xx
δw ,x
∂w ,x
∂t
∂w ,xt
∂x
∂w ,xx
δw
l
0
dt
t 2
t 1
l
0
∂w
∂t
∂w ,t
∂x
∂w ,x
2
∂t∂x
∂w ,xt
2
∂x
2
∂w ,xx
δw dx dt = 0. (12)
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The equation of motion in this case is obtained as
∂w
∂t
∂w ,t
∂x
∂w ,x
2
∂x∂t
∂w ,xt
2
∂x
2
∂w ,xx
+Q(x, t) = 0.
The boundary conditions, however, remain the same.
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