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ME 3109

Engineering Mechanics III

Department of Mechanical Engineering

KUET

Reference

1.Theory of Machines By R.S. Khurmi and J.K. Gupta

2.Mechanical Vibration By A.H. Church

What is Vibration?

Vibration can be considered to be the oscillation or repetitive motion of an

object around an equilibrium position.

How Vibration Occurs?

When elastic bodies such as a spring, a beam and a shaft are displaced

from the equilibrium position by the application of external forces, and

then released, they execute a ** vibratory motion. **This is due to the

reason that, when a body is displaced, the internal forces in the form of

elastic or strain energy are present in the body. At release, these forces

bring the body to its original position. When the body reaches the

equilibrium position, the whole of the elastic or strain energy is converted

into kinetic energy due to which the body continues to move in the

opposite direction. The whole of the kinetic energy is again converted into

strain energy due to which the body again returns to the equilibrium

position. In this way, the vibratory motion is repeated indefinitely.

Terms Used in Vibratory Motion

** Period of Vibration or Time Period: **It is the time interval
after which the motion is repeated itself. The period of
vibration is usually expressed in seconds.

** Cycle: **It is the motion completed during one time period.

** Frequency: **It is the number of cycles described in one
second. In S.I. units, the frequency is expressed in hertz
(briefly written as Hz) which is equal to one cycle per
second.

Types of Vibratory Motion

*1. Free or natural vibrations*.

**2. Forced vibrations.
**

**3. Damped vibrations.**

Free or natural vibrations When no external force acts on the body, after giving it an initial displacement, then the body is said to

be under ** free or natural vibrations. **The frequency of the free vibrations is called

*free or natural**frequency*.

Forced vibrations

When the body vibrates under the influence of external force, then the body is said to be under *forced
*

** vibrations. **The external force applied to the body is a periodic disturbing force created by unbalance.

The vibrations have the same frequency as the applied force.

Damped vibrations

When there is a reduction in amplitude over every cycle of vibration, the motion is said to be *damped
*

** vibration. **This is due to the fact that a certain amount of energy possessed by the vibrating system is

always dissipated in overcoming frictional resistances to the motion.

Types of Free Vibration

*1. Longitudinal vibrations
*

*2. Transverse vibrations
*

*3. Torsional vibrations*

Longitudinal Vibrations

When the particles of the shaft or disc moves parallel to the axis of the

shaft, as shown in Fig. then the vibrations are known as *longitudinal
*

*vibrations*.

In this case, the shaft is elongated and shortened alternately and thus

the tensile and compressive stresses are induced alternately in the

shaft.

Transverse Vibrations

When the particles of the shaft or disc move approximately

perpendicular to the axis of the shaft, as shown in Fig. then the

vibrations are known as *transverse vibrations*.

In this case, the shaft is straight and bent alternately and bending

stresses are induced in the shaft.

Torsional Vibrations

When the particles of the shaft or disc move in a circle about

the axis of the shaft, as shown in Fig. then the vibrations are

known as *torsional vibrations*.

In this case, the shaft is twisted and untwisted alternately and

the torsional shear stresses are induced in the shaft.

Natural Frequency of Free Longitudinal Vibrations

*1. Equilibrium Method
*

*2. Energy Method
*

*3. Rayleigh’s Method*

Equilibrium Method

*s *= Stiffness of the constraint. It is the force required to

produce unit displacement in the direction of vibration. It is

usually expressed in N/m.

*m *= Mass of the body suspended from the constraint in kg

*W *= Weight of the body in newtons = *m.g
*

= Static deflection of the spring in metres due to weight *W
*Newtons

*x *= Displacement given to the body by the external force, in
metres

Equilibrium Method(Contd.) In the equilibrium position,

the gravitational pull *W *= *m.g
*

*The spring force, W *= *s *

Since the mass is now displaced from its equilibrium

position by a distance *x*, as shown in Fig. and is then

released, therefore after time *t*,

Restoring force

*sx
*

*sxss
*

*sxsW
*

*xsW
*

)(

(1)

Equilibrium Method(Contd.) Accelerating force = Mass × Acceleration

2

2

*dt
*

*xd
m* (2)

Equating equation (1) and (2),

0

0

2

2

2

2

2

2

*x
m
*

*s
*

*dt
*

*xd
*

*sx
dt
*

*xd
m
*

*sx
dt
*

*xd
m
*

(3)

Equilibrium Method(Contd.)

0 2

2

*x
m
*

*s
*

*dt
*

*xd
*(3)

the fundamental equation of simple harmonic motion is

02 2

2

*x
dt
*

*xd
* (4)

Comparing equations (3) and (4), we have

*m
*

*s
*

Time period,
*s
*

*m
t p *

2

2

Equilibrium Method(Contd.)

and natural frequency,

*g
f
*

*m
*

*s
*

*t
f
*

*n
*

*p
*

*n
*

2

1

2

11

*smg *

Taking the value of *g *as 9.81 m/s2 and in metres,

*Hzf
*

*f
*

*n
*

*n
*

4985.0

81.9

2

1

Equilibrium Method(Contd.)

The value of static deflection may be found out from the

given conditions of the problem. For longitudinal

vibrations, it may be obtained by the relation,

*AE
*

*Wl
*

*l
E
*

*A
*

*W
*

*E
*

*Wl
*

*AE
fn *4985.0

Energy Method

0)...( *EPEK
dt
*

*d
*

We know that kinetic energy, 2)(

2

1 ..

*dt
*

*dx
mEK *

and potential energy, 2

2

1 )

2

0
(.. *sxx
*

*sx
EP *

0) 2

1 )(

2

1
( 22 *sx
*

*dt
*

*dx
m
*

*dt
*

*d
*Now,

0 2

2

*x
m
*

*s
*

*dt
*

*xd*

Rayleigh’s Method

Self Study

Rayleigh’s Method In this method, the maximum kinetic energy at the mean position is equal to the maximum potential

energy (or strain energy) at the extreme position. Assuming the motion executed by the vibration to be

simple harmonic, then

Natural Frequency of Free Transverse Vibrations

Natural Frequency of Free Transverse Vibrations(Contd.)