exercícios lagrange
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8/17/2019 Exercícios Lagrange
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Sec. 3.21
The vibration of structures with more than two degrees of freedom
109
If it is assumed that h, = h,
---
m , m(- 1
-
w:
k o k
m, 3m
o
= h, = h, I I = I, = l3
=
Iandm, =
m2
=
m =
m,
)
24
E l
m
(
G
m
k
g
mh3
4EI
In a particular case, m, = 2
x
lo6 kg,
m =
200
x lo3
kg,
h
=
4
m,
E =
200 x lo9N/m2,
I = 25 x 10 m4 and k, = lo7 N/m.
Thus
1
0:
(2 x 10') (3 x 200 x 10')
10'
200 x 10, x
4 ,
4 x 200 x io9 x 25 x
io
-
= 0.26 0.64,
so that w , = 1.05 rad/s,f, = 0.168 Hz and the period of the oscillation is 5.96
s.
3.2.2 The Lagrange equation
Consideration of the energy in a dynamic system together with the use of the Lagrange
equation is a very powerful method
of
analysis for certain physically complex systems.
This is an energy method that allows the equations of motion to be written in terms of any
set of
generalized coordinates.
Generalized coordinates are a set of independent para-
meters that completely specify the system location and that are independent of any
constraints. The fundamental form of Lagrange's equation can be written in terms of the
generalized coordinates
q1
as follows:
where T is the total kinetic energy of the system, V is the total potential energy of the
system, D E is the energy dissipation function when the damping
is
linear (it is half the
rate at which energy is dissipated so that for viscous damping
D E
=
;cA?* ,
Qi is a
generalized external force
(or
non-linear damping force) acting on the system, and q is a
generalized coordinate that describes the position of the system.
The subscript i denotes n equations for an n degree of freedom system, so that the
Lagrange equation yields as many equations of motion
as
there are degrees
of
freedom.
For a free conservative system Q, and
D E
are both zero,
so
that
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110 The vibrat ion
of
structures w ith m or e than o ne degree o f freedom
[Ch
3
The full derivation of the Lagrange equation can be found in Vibration Theory and
Applications by W.
T.
Thomson (Allen Unw in, 1989).
xample 24
A
solid cylinder has a mass
M
and radius
R.
Pinned to the axis of the cylinder is an arm
of length 1 which carries a bob of mass
rn
Obtain the natural frequency of free vibration
of the bob. The cylinder is free to roll on the fixed horizontal surface shown.
The generalized coordinates are
xl
and
x,.
They completely specify the position of the
system and are independent of any constraints.
T
=
hxi
f(-hR2)82
mi;
= -hi: ; -hi:);mi;.
v = rngl(1
-
cos @)= rng1/2) bZ=
(rng/21)(x2
for small values of 4 Apply the Lagrange equation with qi = xl:
(d/dt)(aT/dil) =
Mi l
I.?,
av/aX = (rng/21)(-2x2 + k, .
Hence ' XI (rng/l)(xl
x,)
=
0
is an equation of m otion.
Apply the Lagrange equation with
q
=
x,:
(d/dt)(aT/ ,) = m i 2
av/ax,
= ( rng /21)(h2-k, ,
Hence
m f ,
(rng/f)(x ,
x,)
=
0
is an equa tion of motion.
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8/17/2019 Exercícios Lagrange
3/5
Sec. 3.21
The vibrat ion of structures with m ore than two degrees of f reedom
111
These equations of motion can be solved by assuming that
x ,
= XI
inm and
x , =
X,sinm. Then
X1
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8/17/2019 Exercícios Lagrange
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112
The vibration of structures w ith m or e than on e degree of freedom
[Ch.
3
Now
apply the Lagrange equation with q = q .
aT
~-
-
0.
and
Thus the first equation of motion is
M q l
2kIql k, b-d)q, =
0.
Similarly by putting qi = q2 and qi = q3, he other equations of motion are obtained as
Mq2 2kIq2 - 2ak2q3 = 0
and
I G q 3
k , ( b
- d)ql -
2ak2q2 b2 )k,
2a2k2q3= 0.
The system therefore has three coordinate-coupled equations of motion. The natural
frequencies can be found by substituting
qi
=
A i
sin
a,
nd solving the resulting
frequency equation. It is usually desirable to have all natural frequencies low so that the
transmissibility is small throughout the range of frequencies excited.
E x a m p l e 26
A two-storey building which has its foundation subjected to translation and rotation is
modelled by the system shown. Write down expressions for
T
and V and indicate how the
natural frequencies of free vibration may be found using the Lagrange equation.
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Sec. 3.21
The vibration of structures with more than two degrees of freedom
11
3
For small-amplitude vibration,
T
= ~m ’ i J o 8
i m l ioh b
+i, ’ ;.IJ,
m, i0 2 h b
i2)‘
i J 8
and
v
=
ik,x,2
4,8‘
2(;kk,x:
h2
X’ XI)’>
where, x,,
0
x, and x 2 are the generalized coordinates. Substituting in the Lagrange
equation with each coordinate gives four equations of motion to be solved for the
frequency equation and hence the natural frequencies of free vibration and their associated
mode shapes.
3.2.3 Receptance analysis
Some simplification in the analysis of multi-degree of freedom undamped dynamic
systems can often be gained by using receptances, particularly if only the natural
frequencies are required. If a
harmonic force F
sin vt acts at some point in a system so that
the system responds at frequency v, and the point of application of the force has a
displacement
x
=
X
sin vt, then if the equations of motion are linear, x =
a F
sin vt where
a which is a function of the system parameters and v, but not a function of
F,
is known
as the direct receptance at x. If the displacement is determined at some point other than
that at which the force is applied,
a
is known as the transfer or cross receptance.
The analogy with influence coefficients (section 3.2.1) is obvious.
It can be seen that the frequency at which a receptance becomes infinite is a natural
frequency of the system. Receptances can be written for rotational and translational
coordinates in a system, that is, the slope and deflection at a point.
Thus, if a body of mass
m
is subjected to a force
F
sin vt and the response of the body
is x
= X
sin
vt,
F
sin
vt = m s
=
m(- XV’
sin vt)
=
- mv’x.
Thus
x
=
F
sin vt
mv’
and
a =
mv”
This is the direct receptance of a rigid body.
For a spring, a = Ilk. This is the direct receptance of a spring.
In an undamped single degree of freedom model of a system, the equation of motion
is
mi‘
kr
= F sin vt.
If x = X sin vt, a = l/(k mv’ . This is the direct receptance of a single degree of
freedom system.
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