1.4 Oscillations in Fluid Systems

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Fluid systems can undergo oscillations (vibrations) quite analogous to mechanical and electrical

systems. Again, the reason for their natural oscillation is the ability to store and repeatedly

interchange two types of energy — kinetic energy and potential energy. The kinetic energy comes from

the velocity of fluid particles during motion. The potential energy arises primarily from the following

three main sources:

1. Gravitational potential energy

2. Compressibility of the fluid volume

3. Flexibility of the fluid container

A detailed analysis of these three effects is not undertaken here. However, we have seen from the

example in Figure 1.5(e) how a liquid column can oscillate due to repeated interchange between kinetic

energy and gravitational potential energy. Now, let us consider another example.

Example 1.3

Consider a cylindrical wooden peg of uniform cross section and height h; floating in a tank of water,

as in Figure 1.13(a). It is pushed by hand until completely immersed in water, in an upright orientation.

When released, the object will oscillate up and down while floating in the tank. Let r b and r l be the mass

Mass

m

Velocity

v

ms, k

Heavy Spring

l

x

FIGURE 1.12 A heavy spring connected to a rolling

stock.

1-14 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

densities of the body (peg) and the liquid (water), respectively. The natural oscillations and the stability

of this system may be studied as below.

Suppose that, under equilibrium in the upright position of the body, the submersed length is l: The

mass of the body is

m ¼ Ahr b ðiÞ

where A is the area of cross section (uniform).

By the Archimedes principle, the buoyancy force R is equal to the weight of the liquid displaced by the

body. Hence,

R ¼ Alr l g ðiiÞ

For equilibrium, we have

R ¼ mg ðiiiÞ

or

Alr l g ¼ Ahr bg

Hence,

l ¼

r b

r l

h for r b , r l ðivÞ

For a vertical displacement y from the equilibrium position, the equation of motion is (Figure 1.13(b))

my€ ¼ mg 2 Aðl þ yÞr l g

Substitute Equation ii and Equation iii. We obtain

my€ ¼ 2Ar l gy

Substitute Equation i:

Ahr by€ þ Ar l gy ¼ 0

y

C

M

mg

R = mg

C = body centroid

(b) (c) M = metacenter

(a)

FIGURE 1.13 (a) A buoyancy experiment; (b) upright oscillations of the body; (c) restoring buoyancy couple due

to a stable metacenter.

Time-Domain Analysis 1-15

© 2005 by Taylor & Francis Group, LLC

or

y€ þ

r l g

r bh

y ¼ 0

The natural frequency of oscillations is

vn ¼

ffiffiffiffiffiffiffi

r l g

r b h

r

Note that this result is independent of the area of the cross section of the body.

Assumptions made:

1. The tank is very large compared to the body. The change in liquid level is negligible as the body

is depressed into the water.

2. Fluid resistance (viscous effects, drag, etc.) is negligible.

3. Dynamics of the liquid itself is negligible. Hence, “added inertia” due to liquid motion is

neglected.

To study stability of the system, note that the buoyancy for R acts through the centroid of the volume of

displaced water (Figure 1.13(c)). Its line of action passes through the central axis of the body at point M.

The point is known as the metacenter. Let C be the centroid of the body.

If M is above C; then, when tilted, there will be a restoring couple that will tend to restore the body to

its upright position. Otherwise the body will be in an unstable situation, and the buoyancy couple will

tend to tilt it further towards a horizontal configuration.

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