Exercícios Lagrange

8/17/2019 Exercícios Lagrange

1/5

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


2/5

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.


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


4/5

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.


5/5

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.

Exercícios Lagrange

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