DYNAMICS OF MACHINERY

B.TECH. DEGREE COURSE

SCHEME AND SYLLABUS

(2002-03 ADMISSION ONWARDS)

MAHATMA GANDHI UNIVERSITY

KOTTAYAM, KERALA

DYNAMICS OF MACHINERY

Module 1

Balancing: - Balancing of rotating masses, static balancing and dynamic balancing,

Balancing of several masses rotating in same plane, Balancing of several masses

rotating in several planes, Balancing machines.

Balancing of reciprocating masses: - The effect of inertia force of the reciprocating

mass on the engine. Partial primary balance. Partial balancing of locomotive,

Hammer blow, Variation of tractive effort, Swaying couple. Coupled locomotives,

Balancing of multi cylinder inline engines, v-engines, Radial engines, Direct and

Reverse cranks

Module 2

Vibrations: - Definitions, simple harmonic motion. Single degree freedom systems:

Undamped free vibrations: - Equations of motion Natural frequency, Energy

method, Equilibrium methods, Rayleigh’s methods, Equivalent stiffness of spring

combinations.

Damped free vibrations: - Viscous damping, Free vibrations with viscous damping,

over-damped system, critically damped system, under-damped system, Logarithmic

decrement, viscous dampers, coulomb damping.

Forced Vibrations: - Forced harmonic excitation Rotating unbalance, Reciprocating

unbalance. Energy dissipated by damping, vibration isolation and Transmissibility.

Vibration measuring instruments.

Module 3

Two degree freedom systems: - Principal modes of vibration, Rectilinear and

angular modes, systems with damping, vibration absorbers, centrifugal pendulum

damper, dry friction damper, untuned viscous damper.

Multi-degree of freedom system: - Free vibrations, equations of motion, Influence

coefficients method, lumped mass and distributed mass systems, Stodola method,

Dunkerly’s method, Holzer’s method, Matrix iteration method.

Torsional Vibrations: - Torsionally equivalent shaft, torsional vibration of two-

rotor, three-rotor, and geared systems.

Module 4

Critical speeds of shafts: - Critical speed of a light shaft having a single disc

without damping. Critical speeds of a light cantilever shaft with a large heavy disc at

its end.

Transient vibration: - Laplace transformation, response to an impulsive input,

response to a step input, response to a pulse input, phase plane method, shock

spectrum.

Non-linear vibrations: - Phase plane, undamped free vibration with non-linear

spring forces, hard spring, soft spring, Perturbation method, Forced vibration with

nonlinear forces, Duffings equation, self excited vibrations.

Module 5

Noise control: - Sound propagation, decibels, acceptance noise levels, Air columns,

Doppler effect, acoustic measurements, microphones and loud speakers, Recording

and reproduction of sound, fourier’s theorem and musical scale, Acoustics of

buildings, Acoustic impedence filters and human ear.

MODULE 1

Introduction

The high speed of engines and other machines is a common phenomenon

now-a-days. It is, therefore, very essential that all the rotating and reciprocating parts

should be completely balanced as far as possible. If these parts are not properly

balanced, the dynamic forces are set up. These forces not only increase the loads on

bearings and stresses in the various members, but also produce unpleasant and even

dangerous vibrations. In this chapter we shall discuss the balancing of unbalanced

forces caused by rotating masses, in order to minimize pressure on the main bearings

when an engine is running.

BALANCING OF ROTATING MASSES

Static and dynamic unbalance: A rotor can in general have two types of unbalance

viz., “static” and “dynamic”. It is of course to be appreciated that practical systems

will all have dynamic unbalance only and considering it as static unbalance is a

“good-enough” approximation for some cases.

Fig 1.1 A thin Rotor Disc - Illustration of Static Unbalance

If the rotor is thin enough (longitudinally) as shown in Fig. 1.1 the unbalance force

can be assumed to be confined to one plane (the plane of the disc). Such a case is

known as “static” unbalance. Such a system when mounted on a knife-edge as shown

in Fig. 1 will always come to rest in one position only – where the centre of gravity

comes vertically below the knife-edge point. Thus in order to “balance out”, all we

need to do is to attach an appropriate “balancing mass” exactly 1800 opposite to this

position.

Fig 1.2 Thin Rotor on a knife edge - Illustration of Static Unbalance

Thus we first mount the disc on a knife edge and allow it to freely oscillate. Mark the

position when it comes to rest. Choose a radial location (1800 opposite to this

position) where we can conveniently attach a balancing mass. By trial and error the

balancing mass can be found out. When perfectly balanced, the disc will exhibit no

particular preferred position of rest. Also when the disc is driven to rotate by a motor

etc., there will be no centrifugal forces felt on the system (for example, at the

bearings). Thus the condition for static balance is simply that the effective centre of

gravity lie on the axis.

Fig 1.3 A Case of Dynamic Unbalance

Consider the rotor shown in Fig. 1.3. It is easily observed that mass distribution

cannot be approximately confined to just one plane. So unbalance masses and hence

unbalance forces are in general present all along the length of the rotor. Such a case

is known as “dynamic unbalance”.

The fundamental difference between static and dynamic unbalance needs to be

clearly appreciated.

When a rotor as shown in Fig.1. 3 is mounted on a knife edge and allowed to

oscillate freely, it too may come to rest in one particular position all the time – the

position corresponding to the resultant unbalance mass (centre of gravity) vertically

below the knife edge. We could, like earlier, mount an appropriate balance mass

exactly 1800 opposite to this position. It would then have no preferred position of rest

when mounted on a knife-edge. Thus effective center of gravity lies on the axis.

Fig1.4 Example of unbalance masses leading to unbalance force that for a resultant

couple because of axial.

However, when mounted in bearings and driven by a motor etc., it could still wobble

due to the unbalanced moments of these forces as shown in Fig1.4. This becomes

apparent only when the rotor is driven to rotate and hence the name “dynamic

unbalance”. Thus it is not, in general, sufficient to do just static balance but

achieving good dynamic balance is more difficult. We will discuss one important

method of achieving dynamic balance in the next lecture.

Two-plane balancing technique

Consider the turbo-machine rotor that was discussed earlier wherein each stage

contains several blades around the circumference of a disk. Eventhough typically

each stage is balanced in itself to the extent possible, it has a likely net unbalance.

When the rotor is set to spin, it will cause dynamic forces and moments on the

bearings that support the shaft. Therefore it is of interest to achieve “good balance”

of this shaft so that the fluctuating forces on the bearings are reduced. Conceptually

our strategy can be simply stated as follows:

Step 1: Consider the shaft supported on its bearings. For each unbalance mass, there

will be a centrifugal force set-up when the rotor spins at some speed . This would

cause some reactions at the supports. Estimate these support reactions that would

come onto the bearings.

Step 2: Estimate the balancing mass that needs to be placed in the plane of bearings,

to counter this reaction force due to unbalance mass.

Repeat steps 1 and 2 for each unbalance mass in the system and each time add the

balancing masses obtained in step 2 vectorially to determine the resultant balancing

mass required.

Let us now understand the details of the technique mentioned earlier. Firstly we

choose to place “balancing or correcting” masses on the shaft (rotating along with the

shaft) to counter-act the unbalance forces. We understand that this is to be done on

the rotor on site, perhaps during a maintenance period. From the point of view of

accessibility, we therefore choose the balancing masses to be kept near the bearings.

Figure 1.5 Two plane balancing technique

The calculations proceed as shown in Fig1.5. For an unbalance mass mi situated at an

angular location in a plane at an axial distance from the left end bearing and

rotating at a radius as shown in the figure, the unbalance force is . It is

resolved into X and Y components as shown in the figure. These forces are

represented by EQUIVALENT FORCES in the balancing planes ( shown in blue

). These forces can be readily calculated (based on calculations similar to

those involved in finding support reactions for a simply supported beam). In order to

counterbalance this force, we need to place a balancing mass at a radius in the

balancing plane such that it creates an equal and opposite force.

Now we need to repeat the calculations for ALL the unbalance masses mi (i =

1,2,3,…..) and find the resultant equivalent force in the balancing plane as shown in

blue in Fig. 2.3.1. This resultant force is balanced out by placing a suitable balancing

mass creating an equal and opposite force (shown in red).

Since all the masses are rotating at the same speed along with the shaft, we can

drop in our calculations – i.e., a rotor balanced at one speed will remain balanced

at all speeds or in other words, our technique of balancing is independent of speed.

We will review this towards the end of the lecture.

While these calculations can be done in any manner perceived to be convenient, a

tabular form is commonly employed to organize the computations. While doing this,

it is also common practice to include the two balancing masses in the balancing

planes as indicated in the table.

Table 1.1 Tabular form of organizing the computations for two-plane balancing

technique

Sr. No

Cos(

)

Sin(

)

Cos(

)

Sin(

)

1

2

3

....

…..

Balancing

Plane 1

0

Balancing

Plane 2

L

TOTAL FORCES 0 0 0 0

It is observed in Table 1 that the balancing masses and their locations (radial as well

as angular) are unknowns while the location of the balancing plane itself is treated as

a known (any accessible location near the bearings etc). The resultant total forces and

moments must sum up to ZERO and therefore we have four equations but six

unknowns. Thus any two of the six unknowns can be freely chosen and the other four

determined from the computations given in the table. This method of balancing is

known as the “two-plane balancing technique” since balancing masses are kept in

two planes.

Balancing of reciprocating masses:

Figure 1.6 Slider-crank Mechanism of IC Engine

A typical crank-slider mechanism as used in an IC Engine is shown in Fig 1.6 It

essentially consists of four different parts viz., frame(i.e., cylinder) ,crank

,connecting rod and reciprocating piston. The frame is supposedly stationary; crank

is undergoing purely rotary motion while the piston undergoes to-and-fro rectilinear

motion. The connecting rod undergoes complex motion – its one end is connected to

the crank (undergoing pure rotation) and the other end is connected to the piston

(undergoing pure translation).

We know that the inertia forces are given by mass times acceleration and we shall

now estimate the inertia forces (shaking forces and moments) due to the moving

parts on the frame (cylinder block).

CONNECTING ROD

One end of the connecting rod is circling while the other end is reciprocating and any

point in between moves in an ellipse. It is conceivable that we derive a general

expression for the acceleration of any point on the connecting rod and hence estimate

the inertia forces due to an elemental mass associated with that point. Integration

over the whole length of the connecting rod yields the total inertia force due to the

entire connecting rod. Instead we try to arrive at a simplified model of the connecting

rod by replacing it with a “dynamically equivalent link” as shown in Fig 1.7

Figure 1.7 Dynamically Equivalent link for a connecting rod .

In order that the two links are dynamically equivalent, it is necessary that:

Total mass be the same for both the links

Distribution of the mass be also same i.e., location of CG must be same and the

mass moment of inertia also must be same.

Thus we can write three conditions:

For convenience we would like the equivalent link lumped masses to be located at

the big and small end of the original connecting rod and if its center of mass (G)

location is to remain same as that of original rod, distances AG and GB are fixed.

Given the mass m and mass moment of inertia of the original connecting rod, the

problem of finding dynamically equivalent link is to determine , and .

An approximate equivalent link can be found by simply ignoring and treating

just the two lumped masses and connected by a mass-less link as the

equivalent of original connecting rod. In such a case we take:

= (GB)/L

= (AG)/L

Thus the connecting rod is replaced by two masses at either end (pin joints A and B)

of the original rod. rotates along with the crank while purely translates along

with the piston. It is for this reason that we proposed use of crank's effective rotating

mass located at pin A, which can now be simply added up to part of connecting rod's

mass.

On the shop floor , can be immediately determined by mounting the existing

connecting rod on two weighing balances located at A and B respectively. The

readings of the two balance give and directly

Dynamic Model of a single cylinder IC Engine Mechanism

Figure 1.8 Dynamic Model of slider-crank Mechanism

Based on our discussion thus far, we can arrive at a simplified model of the crank-

slider mechanism for the purpose of our dynamic analysis as shown in Fig. 1.8 Thus

we have either purely rotating masses or purely translating masses and these are

given by:

where the first term in the rotating masses is due to the effective crank mass at pin A

and the second term is due to the part of equivalent connecting rod mass located at

pin A. Similarly the first term in reciprocating masses is due to the mass of the piston

and the second is due to the part of equivalent connecting rod mass located at pin B.

There are inertia forces due to .

The inertia forces due to can be nullified by placing appropriate balancing

masses. Thus the effective force transmitted to the frame due to rotating masses can

ideally be made zero.

Figure 1.9Counter balancing of rotating masses

Figure 1.10 Opposed position configuration

However it is not so straight forward to make the unbalanced forces due to

reciprocating masses vanish completely. As given in Equation and depicted in Fig.

1.9 there are components of the force which are at the rotational speed and those at

twice this speed. It is conceivable to use a configuration as shown in Fig.1.10 to

completely balance out these forces but the mechanism becomes too bulky. Thus a

single cylinder engine is inherently unbalanced. .

Partial Balancing of Locomotives

The locomotives, usually, have two cylinders with cranks placed at right

angles to each other in order to have uniformity in turning moment diagram. The two

cylinder locomotives may be classified as :

1. Inside cylinder locomotives ; and

2. Outside cylinder locomotives.

In the inside cylinder locomotives, the two cylinders are placed in between

the planes of two driving wheels whereas in the outside cylinder locomotives, the

two cylinders are placed outside the driving wheels, one on each side of the driving

wheel. The locomotives may be

(a) Single or uncoupled locomotives ; and (b) Coupled locomotives.

A single or uncoupled locomotive is one, in which the effort is transmitted

to one pair of the wheels only ; whereas in coupled locomotives, the driving wheels

are connected to the leading and trailing wheel by an outside coupling rod.

Effect of Partial Balancing of Reciprocating Parts of Two Cylinder Locomotives

We have discussed in the previous article that the reciprocating parts are

only partially lanced. Due to this partial balancing of the reciprocating parts, there is

an unbalanced primary force along the line of stroke and also an unbalanced primary

force perpendicular to the line of stroke. The effect of an unbalanced primary force

along the line of stroke is to produce;

1. Variation in tractive force along the line of stroke ; and 2. Swaying

couple.

The effect of an unbalanced primary force perpendicular to the line of stroke

is to produce variation in pressure on the rails, which results in hammering action on

the rails. The maximum magnitude of the unbalanced force along the perpendicular

to the line of stroke is known as a jammer blow. We shall now discuss the effects of

an unbalanced primary force in the following articles.

Variation of Tractive Force

The resultant unbalanced force due to the two cylinders, along the line of

stroke, is known as tractive force. Let the crank for the first cylinder be inclined at an

angle with the line of stroke. Since the crank for the second cylinder is at right

angle to the first crank, therefore the angle of inclination for the second crank will be

(90° + ).

Let m = Mass of the reciprocating parts per cylinder, and

c = Fraction of the reciprocating parts to be balanced.

We know that unbalanced force along the line of stroke for cylinder 1

21 c m. .rcos

Similarly, unbalanced force along the line of stroke for cylinder 2,

2 o1 c m. .rcos 90

As per definition, the tractive force,

TF = Resultant unbalanced force along the line of stroke

21 c m. .rcos

2 o1 c m. .rcos 90

21 c m. .r cos sin

The tractive force is maximum or minimum when (cos -sin ) is

maximum or minimum. For (cos -sin ) to be maximum or minimum,

d

cos sin 0d

or sin cos 0 or sin cos

tan 1 or = 135o or 315

o

Thus, the tractive force is maximum or minnimum when = 135o or 315

o.

Maximum and minimum value of the tractive force or the variation in

tractive force

2 o o 21 c m. .r cos135 sin135 2 1 c m. .r

Swaying Couple

The unbalanced forces along the line of stroke for the two cylinders

constitute a couple.This couple has swaying effect about a vertical axis, and tends to

sway the engine alternate in clockwise and anticlockwise directions. Hence the

couple is known as swaying couple.

Let a = Distance between the centre lines of the two cylinders.

Swaying couple

2 a1 c m. .r cos

2

2 o a1 c m. .r cos 90

2

2 a1 c m. .r cos sin

2

The swaying couple is maximum or minimum when (cos + sin ) to be

maximum or minimum.

d

cos sin 0d

or sin cos 0 or sin cos

tan 1 or = 45o or 225

o

Thus, the swaying couple is maximum or minimum when = 45o or 225

o.

Maximum and minimum value of the swaying couple

2 o o 2a a1 c m. .r cos45 sin 45 1 c m. .r

2 2

Note: In order to reduce the magnitude of the swaying couple, revolving

balancing masses are introduced. But, as discussed in the previous article, the

revolving balancing masses cause unbalanced forces to act at right angles to the line

of stroke. These forces vary the downward pressure of the wheels on the rails and

cause oscillation of the locomotive in a vertical plane about a horizontal axis. Since a

swaying couple is more. harmful than an oscillating couple, therefore a value of ‘c’

from 2/3 to 3/4, in two-cylinder locomotives with two pairs of coupled wheels, is

usually used. But in large four cylinder locomotives with three or more pairs of

coupled wheels, the value of ‘c’ is taken as 2/5.

Hammer Blow

We have already discussed that the maximum magnitude of the unbalanced

force along the perpendicular to the line of stroke is known as hammer blow.

We know that the unbalanced force along the perpendicular to the line of

stroke due to the balancing mass B, at a radius b, in order to balance reciprocating

parts only is B. 2 .b sin . This force will be maximum when sin is unity, i.e.

when = 90° or 270°.

Hammer blow = B. 2 .b (Substituting sin = 1)

The effect of hammer blow is to cause the variation in pressure between the

wheel and the rail.

Let P be the downward pressure on the rails (or static wheel load).

Net pressure between the wheel and the rail

2P B. .b

If 2P B. .b is negative, then the wheel will be lifted from the rails.

Therefore the limiting condition in order that the wheel does not lift from the rails is

given by

2P B. .b

and the permissible value of the angular speed,

P

Bb

Balancing of Coupled Locomotives

The uncoupled locomotives as discussed in the previous article, are obsolete

now-a-days. In a coupled locomotive, the driving wheels are connected to the leading

and trailing wheels by an outside coupling rod. By such an arrangement, a greater

portion of die engine mass is utilised by tractive purposes. In coupled locomotives,

the coupling rod cranks are placed diametrically opposite to the adjacent main cranks

(i.e. driving cranks). The coupling rods together with cranks and pins may be .treated

as rotating masses and completely balanced by masses in the respective wheels. Thus

in a coupled engine, the rotating and reciprocating masses must be treated separately

and the balanced masses for the two systems are suitably combined in the wheel.

It may be noted that the variation of pressure between the wheel and the rail

(i.e., hammer blow) may be reduced by equal distribution of balanced mass (B)

between the driving, leading and trailing wheels respectively.

Balancing of Primary Forces of Multi-cylinder In-line Engines

The multi-cylinder engines with the cylinder centre lines in the same plane

and on the same side of the centre line of the crankshaft, are known as In-line

engines. The following two conditions must be satisfied in order to give the primary

balance of the reciprocating parts of a multi-cylinder engine:

Figure 1.11 Typical Inline engine

1. The algebraic sum of the primary forces must be equal to zero. In

other words, the primary force polygon must *close; and

2. The algebraic sum of the couples about any point in the plane of the

primary forces must be equal to zero. In other words, the primary

couple polygon must close.

We have already discussed, that the primary unbalanced force due to the

reciprocating masses is equal to the component, parallel to the line of stroke, of the

centrifugal force produced by the equal mass placed at the crankpin and revolving

with it. Therefore, in order to give the primary balance of the reciprocating parts of a

multi-cylinder engine, it is convenient to imagine the reciprocating masses to be

transferred to their respective crankpins and to treat the problem as one of revolving

masses.

Notes: 1. For a two cylinder engine with cranks 180o, condition (1) may be

satisfied, but this will result in an unbalanced couple. Thus the above method of

primary balancing cannot be applied in this case.

2. For a three cylinder engine with cranks at 120o and if the reciprocating

masses per cylinder are same, then condition (1) will be satisfied because the forces

may be represented by the sides of an equilateral triangle. However, by taking a

reference plan through one of the cylinder centre lines, two couples with non-parallel

axes will remain and these cannot vanish vectorially. Hence the above method of

balancing fails in this case also.

3. For a four cylinder engine, similar reasoning will show that complete

primary balance is possible and it follows that

‘For a multi-cylinder engine, the primary forces may be completely

balanced by suitably arranging the crank angles, provided that the number of cranks

are not less than four'.

Balancing of Secondary Forces of Multi-cylinder In-line Engines

When the connecting rod is not too long (i.e. when the obliquity of the

connecting rod is considered), then the secondary disturbing force due to the

reciprocating mass arises.

We have the secondary force,

2

S

cos2F m. .r

n

This expression may be written as

2

S

rF m. 2 cos2

4n

As in case of primary forces, the secondary forces may be considered to be

equivalent to be component, parallel to the line of stroke, of the centrifugal force

produced by an equal mass placed at the imaginary crank of length r/4n and

revolving at twice the speed of the actual crank.Thus, in multi-cylinder in-line

engines, each imaginary secondary crank with a mass attached to the crankpin s

inclined to the line of stroke at twice the angle of the actual crank. The values of the

secondary forces and couples nay be obtained by considering the revolving mass.

This is done in the similar way as discussed for primary forces, the following two

conditions must be satisfied in order to give a complete secondary balance of an

engine :

1. The algebraic sum of the secondary forces must be equal to zero. In other

words, die secondary force polygon must close, and

2. The algebraic sum of the couples about any point in the plane of the

secondary forces must be equal to zero. In other words, the secondary

couple polygon must close.

Balancing of Radial Engines (Direct and Reverse Cranks Method)

The method of direct and reverse cranks is used in balancing of radial or V-

engines, in which the connecting rods are connected to a common crank. Since the

plane of rotation of the various cranks (in radial or V-engines) is same, therefore

there is no unbalanced primary or secondary couple.

Fig. 1.12 Typical Radial Engine (Not to scale)

Consider a reciprocating engine mechanism. Let the crank known as the

direct crank) rotates uniformly at radians per second in a clockwise direction. Let

at any instant the crank makes an angle with the line of stroke. The indirect or

reverse crank is the image of the direct crank when seen through the mirror placed at

the line of stroke. A little consideration will show that when the direct crank revolves

in a clockwise direction, the reverse crank will revolve in the anticlockwise direction.

We shall now discuss the primary and secondary forces due to the mass of the

reciprocating parts.

Considering the primary forces

We have already discussed that primary force is 2m. .rcos . This force is

equal to the component of the centrifugal force along the line of stroke, produced by

a mass placed at the crank pin. Now let us suppose that the mass of the reciprocating

parts is divided into two parts, each equal to m/2.

It is assumed that m/2 is fixed at the direct crank (termed as primary direct

crank) pin and m/2 at the reverse crank (termed as primary reverse crank) pin . We

know that the centrifugal force acting on the primary direct and reverse crank

2m.r

2

Component of the centrifugal force acting on the primary direct crank

2m.rcos

2

and, the component of the centrifugal force acting on the primary reverse crank

2m.rcos

2

Total component of the centrifugal force along the line of stroke

2 2m

2 .rcos m. .r cos2

= Primary force, FP

Hence, for primary effects, the mass m of the reciprocating parts may be

replaced by two masses each of magnitude m/2.

MODULE 2

Vibration (Oscillation)

Any motion which repeats itself after an interval of time is called vibration.

Eg: Swinging of simple pendulum.

Causes of vibration

Unbalanced forces in the machine.

External excitations applied on the system.

Elastic nature of the system

Winds, Earthquakes etc.

Effect of Vibration

Produces unwanted noise, high stresses, wear, poor reliability and premature

failure of one or more of the parts.

Inspite of these harmful effects, it is used in musical instruments, vibrating

conveyors etc.

Elimination of Vibrations

Using shock absorbers

Using vibration absorbers

Resting the machinery on proper type of isolation.

Definitions

Frequency

Number of cycles per unit time.

Natural Frequency (fn)

Frequency of free vibration of the system.

Expressed in Hz or rad/sec.

Amplitude

The maximum displacement of a vibrating body from its equilibrium

position.

Resonance

When the frequency of external excitation is equal to the natural frequency

of a vibrating body, the amplitude of vibration becomes excessively large. This

concept is known as resonance.

Periodic Motion

A motion which repeats itself after equal intervals of time.

Time Period

Time taken to complete one cycle.

Fundamental mode of vibration

The fundamental mode of vibration of a system is the mode having the lowest natural

frequency.

Degree of Freedom

The minimum number of independent co-ordinates required to specify the

motion of system at any instant is known as degrees of freedom.

It is equal to the number of independent displacements that are possible.

This number varies from zero to infinity.

Zero degree of freedom

The body at rest is said to have zero degree of freedom.

Single Degree of freedom

Here there is only one independent co-ordinate to specify the configuration.

Eg: A mass supported by a spring.

Two degree of freedom

There are two independent co-ordinates to specify the configuration.

Eg: Springs supported Rigid mass. (It can move in the direction of springs

and also have angular motion in one plane)

Multi degrees of freedom

A cantilever beam has inifinite degrees of freedom.

Types of Vibration

1. Free (Natural) Vibration

eg: Simple pendulum.

After disturbing the system the external excitation is removed, then the

system vibrates on its own. This type of vibration is known as free vibrations.

3 types:-

a) Longitudinal vibrations.

When the particles of the shaft or disc move parallel to the axis of the shaft,

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.

b) Transverse vibrations

When the particles of the shaft move approximately perpendicular to the

axis of the shaft, 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.

c) Torsional vibrations

When the particles of the shaft move in a circle about the axis of the shaft,

then the vibrations are known as torsional vibrations.

In this case the shaft is twisted and untwisted alternately and torsional shear

stresses are induced in the shaft to.

2. Forced Vibration

Eg: Machine tools, Electric bells.

The vibration which is under the influences of external force is called forced

vibration.

The external force applied to the body is periodic disturbing force created by

unbalance.

The vibrations have the same frequency as the applied force. Due to the

application of external forces the amplitude of these vibrations is maintained almost

constant.

3. Damped vibration

When there is a reduction in amplitude over every cycle of vibration, the

motion is said to be damped vibration.

That is if the vibrators system has a damper. The motion of the system will

be opposed by it and the energy of the system will be dissipated in friction.

4. Undamped vibration.

There is no damper. There is no loss of energy due to friction.

5. Deterministic vibration

If in the vibratory system the amount of external excitation is known in

magnitude it is deterministic vibration.

6. Random vibration

Non deterministic vibrations

7. Steady state vibrations

In ideal systems, the free vibrations continue indefinitely as there is no

damping. Such vibration is termed as steady state vibration.

8. Transient vibrations

In real systems, the amplitude of vibration decays continuously because of

damping and vanishes finally. Such vibration is real system is called transient

vibration.

9. Linear vibration

A vibratory system basically consists of there elements:

Mass

Spring

Pamper

Fig.

- If in a vibratory system mass, spring and damper behave in a linear

manner, the vibrations caused are known as linear vibrations.

- Linear vibrations are governed by linear differential equations.

- They follow the law for superposition.

10. Non linear vibrations

- if any of the basic components of a vibratory system behaves non linearly,

the vibration is called non-linear vibration.

- it does not follow the law of superposition.

Linear vibration becomes non linear for very large amplitude of vibration.

Forced vibrations are also known as excitations.

The excitation may be:

a) Periodic

b) Impulsive

c) Random

Vibrations because of impulsive forces are called transient.

Earth quake is because of random forces.

- External force keeps the system vibrating.

This force is called external excitation.

Harmonic motion :

Simplest form of periodic motion is harmonic motion and it is called simple

harmonic motion (SHM). It can be expressed as

where A is the amplitude of motion, t is the time instant and T is the period of

motion.

Harmonic motion is often represented by projection on line of a point that is

moving on a circle at constant speed.

Figure 2.1: The Simple Harmonic Motion

From Figure 2.1 , we have

where x is the displacement and is the circular frequency in rad/sec.

where T is the period (sec) and f is the frequency (cycle/sec) of the harmonic motion.

The SHM repeats itself in radians.

Displacement can be expressed as

So that the velocity and acceleration can be written as

FREE VIBRATIONS

In absence of damping, the system can be considered as conservative and

principle of conservation of energy offers another approach to the calculation

of the natural frequency.

The effect of damping is mainly evident in diminishing of the vibration

amplitude at or near the resonance

Undamped Free Vibration

A spring mass system as shown in Figure 2.2 is considered. For simplicity at present

the damping is not considered.

Figure 2.2

The direction of x in the downward direction is positive. Also velocity, ,

acceleration, , and force, F, are positive in the downward direction as shown in

Figure 2.2. From Figure 2.2(d) on application of Newton's second law, we have

or

From Figure 2.2(b), we have (i.e. spring force due to static deflection is

equal to weight of the suspended mass), so the above equation becomes

The choice of the static equilibrium position as reference for x axis datum has

eliminated the force due to the gravity. Equation can be written as

or

where is the natural frequency (in rads/sec).This Equation satisfies the simple

harmonic motion condition.

The undamped free vibration executes the simple harmonic motion as shown in

Figure 2.3.

Figure 2.3: Simple harmonic motion

Since sine & cosine functions repeat after 2 radians (i.e. Frequency Time

period = 2 ), we have

The time period (in second) can be written as

The natural frequency (in rads/sec or Hertz) can be written as

From Figure 2.2(b), we have

On substitution we get

Here T , f , are dependent upon mass & stiffness of the system, which are

properties of the system.

Above analysis is valid for all kind of SDOF system including beam or torsional

members. For torsional vibrations the mass may be replaced by the mass moment of

inertia and stiffness by stiffness of torsional spring. For stepped shaft an equivalent

stiffness can be taken or for distributed mass an equivalent lumped mass can be

taken.

The undamped free response can also be written as

where A & B are constants to be determined from initial conditions.

Equivalent Stiffness of Series and Parallel Springs :

For this system having springs connected in series or parallel, this equation is still

valid with the equivalent stiffness as shown in Figures 2.4 and 2.5.

Figure 2.4

Figure 2.5

Energy method :

In a conservative system (i.e. with no damping) the total energy is constant, and

differential equation of motion can also be established by the principle of

conservation of energy.

For the free vibration of undamped system: Energy=(partly kinetic energy +

partly potential energy).

Kinetic energy T is stored in mass by virtue of its velocity.

Potential energy U is stored in the form of strain energy in elastic

deformation or work done in a force field such as gravity, magnetic field etc.

Our interest is to find natural frequency of the system, writing this equation for two

positions

where, 1 & 2 represents two instants of time.

Let 1 represents a static equilibrium position (choosing this as the reference point of

potential energy, here U1=0 ) and 2 represents the position corresponding to

maximum displacement of mass and at this position velocity of mass will be zero and

hence T2 = 0.

Damped Free Vibration

Vibration systems may encounter damping of following types:

1. Internal molecular friction.

2. Sliding friction

3. Fluid resistance

Generally mathematical model of such damping is quite complicated and not suitable

for vibration analysis.

Simplified mathematical model (such as viscous damping or dash-pot) have been

developed which leads to simplified formulation.

A mathematical model of damping in which force is proportional to displacement

i.e., Fd = cx is not possible because with cyclic motion this model will encounter an

area of magnitude equal to zero as shown inFigure 2.1(a). So dissipation of energy is

not possible with this model.

The damping force (non-linearly related with displacement) versus displacement

curve will enclose an area, it is referred as the hysteresis loop (Figure 2.1(b)), that is

proportional to the energy lost per cycle.

(a): Linear relation (b): Non-linear relation

Figure 2.6: Variation of damping force vs displacement

Viscously damped free vibration :

Viscous damping force is expressed as,

c is the constant of proportionality and it is called damping co-efficient.

Figure 2.7 shows spring-damper-mass system with free body diagram.

From free body diagram, we have

(1)

Let us assume a solution of equation(1) of the following form

(2)

where s is a constant (can be a complex number) and t is time.

So that and , on substituting in equation (1), we get,

From the condition that equation (2) is a solution for all values of t , above equation

gives a characteristic equation (Frequency equation) as

(3)

Equation (3) has the following form

solution of which is given as

Solution of equation (3) can be written as

(4)

Hence the general solution of equation (1) from equations (2) and (4) is given by the

equation

(5)

where A and B are integration constants to be determined from initial conditions.

Substituting equation (4) into equation (5).

(6)

The term outside the bracket in RHS is an exponentially decaying function. The term

can have three cases.

(i) : exponents in equation (6) will be real numbers.

No oscillation is possible as shown in Figure 2.8.

This is an overdamped system (Figure 2.8).

Figure 2.8: Overdamped system

ii) : exponents in equation (6) are imaginary numbers :

we can write

Hence the equation (6) takes the following form

Let and , equation (6) can be written as

(7)

where

(iii) Critical case between oscillatory and non-oscillatory motion :

Damping corresponding to this case is called critical damping, cc

(8)

Any damping can be expressed in terms of the critical damping by a non-dimensional

number called the damping ratio

(9)

Response corresponding to the critical damping case is shown in Figure 2.9 for

various initial conditions.

Figure 2.9: Critical damping

Equation of motion for damped system can be expressed in terms of and as

(10)

This form of equation is useful in identification of natural frequency and damping of

system.

It is useful in modal summation of MDOF system also.

The roots of characteristic equation (10) can be written as

(11)

with

Depending upon value of damping ratio we can have the following cases

, overdamped condition

, underdamped condition

, critical damping

, undamped system

1) Oscillatory motion : [ , underdamped case]

General solution equation (1) becomes:

(12)

(3.30)

(13)

where and and , where C &

D and X, are arbitrary constants to be determined from initial conditions, x (0) and

(0).

From equation (13), we have

On application of initial conditions, we get

x(0)=C

and

; which gives

Hence, equation (13), becomes

(14)

Equation (14) indicates that the frequency of damped system is equal to,

(15)

It should be noted that for small ( which is the case of most engineering systems)

2) Non-oscillatory motion : ( over damped case)

Two roots remain real with one increases and another decreases.

The general solution becomes

(16)

so that

On application of initial conditions, we have

x(0)=A + B

and

or

which gives

and

3) Critically damped systems :

We obtained two roots

Two terms in solution combines to give one constant

From equation (14) for critically damped case (when ), we have

(17)

Hence the general solution will be

(18)

so that,

On application of initial conditions, we get

x(0)=A

and

The necessary and sufficient conditions for crossing once can be obtained as

(19)

or

is a necessary condition for crossing time axis once but sufficient conditions

is given by equation (19) as shown in Figure 2.9.

Logarithmic Decrement :

Rate of decay of free vibration is a measure of damping present in a system. Greater

is the decay, larger will be the damping.

Damped (free) vibration, general equation of the response is given as

Defining a term logarithmic decrement which is defined as the natural logarithm

of the ratio of any two successive amplitudes as shown Figure 2.11.

since

Td = damped period, where = damped natural frequency

We have damped period , we get logarithmic

decrement as

Since , the above equation reduces to

Experimental determination of natural frequency and damping ratio :

rad/sec, Td can be obtained from displacement-time free vibration

oscillations.

, where x1and x2 are two consecute amplitudes in the free vibration

displacement-time curve.

Figure 2.10

The above illustration shows for two successive amplitude. But in case, the

amplitude are recorded after "n" cycles, the formula is modified as

Taking log,

Forced Harmonic Vibration:

Steady State Response due to Harmonic Oscillation :

Consider a spring-mass-damper system as shown in figure 2.11. The equation of

motion of this system subjected to a harmonic force can be given by

where, m , k and c are the mass, spring stiffness and damping coefficient of the

system, F is the amplitude of the force, w is the excitation frequency or driving

frequency.

Figure 2.11 Harmonically excited system

Figure 2.12: Force polygon

The steady state response of the system can be determined by solving the above

equation in many different ways. Here a simpler graphical method is used which will

give physical understanding to this dynamic problem. From solution of differential

equations it is known that the steady state solution (particular integral) will be of the

form

As each term of equation (4.1) represents a forcing term viz., first, second and third

terms, represent the inertia force, spring force, and the damping forces. The term in

the right hand side of equation is the applied force. One may draw a close polygon as

shown in figure 2.12 considering the equilibrium of the system under the action of

these forces. Considering a reference line these forces can be presented as follows.

Spring force = (This force will make an angle with

the reference line, represented by line OA).

Damping force = (This force will be perpendicular to

the spring force, represented by line AB).

Inertia force = (this force is perpendicular to the

damping force and is in opposite direction with the spring force and is

represented by line BC) .

Applied force = which can be drawn at an angle with respect to

the reference line and is represented by line OC.

From equation the resultant of the spring force, damping force and the inertia force will

be the applied force, which is clearly shown in figure 2.12

It may be noted that till now, we don't know about the magnitude of X and which can

be easily computed from Figure 2. Drawing a line CD parallel to AB, from the triangle

OCD of Figure 2,

or

As the ratio is the static deflection of the spring, is known as the

magnification factor or amplitude ratio of the system

Figure 2.13 shows the magnification factor frequency ratio and phase angle

frequency ratio plot.

Following observation can be made from these plots.

For undamped system ( i.e. ) the magnification factor tends to infinity

when the frequency of external excitation equals natural frequency of the

system .

But for underdamped systems the maximum amplitude of excitation has a

definite value and it occurs at a frequency

For frequency of external excitation very less than the natural frequency of

the system, with increase in frequency ratio, the dynamic deflection ( X )

dominates the static deflection , the magnification factor increases till it

reaches a maximum value at resonant frequency .

For , the magnification factor decreases and for very high value of

frequency ratio ( say )

One may observe that with increase in damping ratio, the resonant response

amplitude decreases.

Irrespective of value of , at , the phase angle .

For , phase angle .

For, , phase angle approaches for very low value of .

Figure 2.13 : (a) Magnification factor ~ frequency ratio for different values of

damping ratio.

Figure 2.13 : (b) Phase angle ~frequency ratio for different values of damping ratio.

For a underdamped system the total response of the system which is the combination

of transient response and steady state response can be given by

The parameter will depend on the initial conditions.

It may be noted that as , the first part of equation tends to zero and second part

remains.

Rotating Unbalance:

One may find many rotating systems in industrial applications. The unbalanced force

in such a system can be represented by a mass m with eccentricity e , which is

rotating with angular velocity as shown in Figure 4.1.

Figure 2.14: Vibrating system with rotating unbalance

Figure 2.15 Freebody diagram of the system

Let x be the displacement of the nonrotating mass (M-m) from the static equilibrium

position, then the displacement of the rotating mass m is

From the freebody diagram of the system shown in figure 2.15, the equation of

motion is

or

This equation is same as equation (1) where F is replaced by . So from the

force polygon as shown in figure 2.16

or

or

Figure 2.16: Force polygon

or

and

So the complete solution becomes

Figure 2.17 : plot for system with rotating unbalance

Figure 2.18 : Phase angle ~ frequency ratio plot for system with rotating unbalance

From figure following observations may be made for a rotating unbalanced system.

For very low value of frequency ratio (say ) ), the response of the

system is very small.

For frequency ratio between 0.5 and1, there is a sharp increase in system

response with increase in frequency of excitation of the system.

At frequency ratio equal to 1, the phase angle is 900.

Maximum response amplitude occurs at a frequency slightly greater than

.

With increase in damping, the response of the system decreases.

For higher value of (say >2), the response amplitude approaches and

phase angle approaches 1800

Vibration Isolation & Transmissibility:

In many industrial applications, one may find the vibrating machine transmit forces

to ground which in turn vibrate the neighbouring machines. So in that contest it is

necessary to calculate how much force is transmitted to ground from the machine or

from the ground to the machine.

Figure 2.19 : A vibrating system

Figure.2.19 shows a system subjected to a force and vibrating with

.

This force will be transmitted to the ground only by the spring and damper.

Force transmitted to the ground

It is known that for a disturbing force , the amplitude of resulting

oscillation

Substituting these equations and defining the transmissibility TR as the ratio of the

force transmitted Force to the disturbing force one obtains

Comparing equations for support motion, it can be noted that

When damping is negligible

to be used always greater than

Replacing

To reduce the amplitude X of the isolated mass m without changing TR, m is often

mounted on a large mass M. The stiffness K must then be increased to keep ratio

K/(m+M) constant. The amplitude X is, however reduced, because K appears in the

denominator of the expression

Figure 2.20: Transmissibility ~frequency ratio plot

Figure 2.20 shows the variation transmissibility with frequency ratio and it can be

noted that vibration will be isolated when the system operates at a frequency ratio

higher than

Equivalent Viscous Damping :

In the previous sections, it is assumed that the energy dissipation takes place due to

viscous type of damping where the damping force is proportional to velocity. But

there are systems where the damping takes place in many other ways. For example,

one may take surface to surface contact in vibrating systems and take Coulomb

friction into account. Also in many cases energy is dissipated in joints also, which is

a form of structural damping. In these cases one may still use the derived equations

by considering an equivalent viscous damping. This can be achieved by equating the

energy dissipated in the original and the equivalent system.

The primary influence of damping on the oscillatory systems is that of limiting the

amplitude at resonance. Damping has little influence on the response in the

frequency regions away from resonance. In case of viscous damping, the amplitude

at resonance is

For other type of damping, no such simple expression exists. It is possible to

however, to approximate the resonant amplitude by substituting an equivalent

damping Ceq in the foregoing equation. The equivalent damping Ceq is found by

equating the energy dissipated by the viscous damping to that of the nonviscous

damping with assumed harmonic motion.

Where must be evaluated from the particular type of damping.

Structural Damping :

When materials are cyclically stressed, energy is dissipated internally within the

material itself. Experiments by several investigators indicate that for most structural

metals such as steel and aluminum, the energy dissipated per cycle is independent of

the frequency over a wide frequency range and proportional to the square of the

amplitude of vibration. Internal damping fitting this classification is called solid

damping or structural damping. With the energy dissipation per cycle proportional to

the square of the vibration amplitude, the loss coefficient is a constant and the shape

of the hysteresis curve remains unchanged with amplitude and independent of the

strain rate. Energy dissipated by structural damping can be written as

Where is a constant with units of force displacement.

By the concept of equivalent viscous damping

or

Coulomb Damping :

Coulomb damping is mechanical damping that absorbs energy by sliding friction, as

opposed to viscous damping, which absorbs energy in fluid, or viscous, friction.

Sliding friction is a constant value regardless of displacement or velocity. Damping

of large complex structures with non-welded joints, such as airplane wings, exhibit

coulomb damping.

Work done per cycle by the Coulomb force

For calculating equivalent viscous damping

From the above equation equivalent viscous damping is found

Summary

Some important features of steady state response for harmonically excited systems

are as follows-

The steady state response is always of the form . Where it

is having same frequency as of forcing. X is amplitude of the response, which

is strongly dependent on the frequency of excitation, and on the properties of

the spring—mass system.

There is a phase lag between the forcing and the system response, which

depends on the frequency of excitation and the properties of the spring-mass

system.

The steady state response of a forced, damped, spring mass system is

independent of initial conditions.

In this chapter response due to rotating unbalance, support motion, whirling of shaft

and equivalent damping are also discussed.

Magnification Factor (Dynamic magnifier) or Amplitude Ratio

The ratio of the maximum displacement of the forced vibration (xmax) to the

static deflection under the static force F0 (xo) is known as Magnification factor.

Denoted by M.F.

i.e, M.F = max

0

x

x

We have 0max

22 2

n n

F /sx

1 2

0max

22 2

n n

xx

1 2

2

2 2

n n

1M.F

1 2

From this equation, it is clear that the magnification factor depends upon

-The ratio of circular frequencies n

-The damping factor ( )

Energy Dissipated by damping

The energy lost per cycle due to a damping force Fd is calculated by

d d xw F .d

where damping force, Fd = dx

C.dt

Energy dissipated/cycle, 2

dW c x

Vibration Measuring Instruments

The instruments which are used to measure the displacement, velocity or

acceleration of a vibrating body are called vibration measuring instruments.

Widely used → vibrometers (low frequency transducer)

→Accelerometers (high frequency transducer)

Example: Accelerometers

a) Bonded strain gauge accelerometer

i) Cantilever beam type accelerometer

ii) Solid cylinder accelerometer

b) Piezoelectric accelerometer

c) Servo accelerometer (force balance accelerometer)

Piezo electric accelerometers

Certain crystals exhibit the property that they generate a charge across their

faces when a stress is applied to them. This property is made use of in piezoelectric

accelerometer.

The change generated to the device is given by q = f.d.

When f → applied force

d → Piezoelectric constant

When the device is subjected to acceleration the mass exerts a variable force

on the Piezoelectric disc, which is proportional to the acceleration. The charge

developed across the disc is in turn proportional to the acceleration of the mass.

MODULE 3

TWO-DEGREE-OF-FREEDOM-SYSTEMS

A single degree of freedom system has only one natural frequency and

requires only one independent co-ordinate to define the system completely.

But a Two Degree freedom system has two natural frequencies and the free

vibration of any point in the system, in general, is a combination of two harmonies of

these two natural frequencies respectively. Under certain conditions, any point in the

system may execute harmonic vibrations, at any of the two natural frequencies, and

these are known as the principal moles of vibrations.

In Two Degree freedom systems, there are two independent co-ordinates to

specify the configuration.

The vibrating systems, which require two coordinates to describe its motion,

are called two-degrees-of -freedom systems.

These coordinates are called generalized coordinates when they are

independent of each other and equal in number to the degrees of freedom of

the system.

Unlike single degree of freedom system, where only one co-ordinate and

hence one equation of motion is required to express the vibration of the

system, in two-dof systems minimum two co-ordinates and hence two

equations of motion are required to represent the motion of the system. For a

conservative natural system, these equations can be written by using mass

and stiffness matrices.

One may find a number of generalized co-ordinate systems to represent the

motion of the same system. While using these co-ordinates the mass and

stiffness matrices may be coupled or uncoupled. When the mass matrix is

coupled, the system is said to be dynamically coupled and when the stiffness

matrix is coupled, the system is known to be The set of co-ordinates for

which both the mass and stiffness matrix are uncoupled, are known as

principal co-ordinates. In this case both the system equations are independent

and individually they can be solved as that of a single-dof system.

A two-dof system differs from the single dof system in that it has two natural

frequencies, and for each of the natural frequencies there corresponds a

natural state of vibration with a displacement configuration known as the

normal mode. Mathematical terms associated with these quantities are

eigenvalues and eigenvectors.

Normal mode vibrations are free vibrations that depend only on the mass and

stiffness of the system and how they are distributed. A normal mode

oscillation is defined as one in which each mass of the system undergoes

harmonic motion of same frequency and passes the equilibrium position

simultaneously.

The study of two-dof- systems is important because one may extend the same

concepts used in these cases to more than 2-dof- systems. Also in these cases

one can easily obtain an analytical or closed-form solutions. But for more

degrees of freedom systems numerical analysis using computer is required to

find natural frequencies (eigenvalues) and mode shapes (eigenvectors).

Derivation of Equation of Motion

Few examples of two-degree-of-freedom systems ::

Figure 3.1(a) shows two masses m1 and m2 with three springs having spring stiffness

k1, k2 and k3 free to move on the horizontal surface. Let x1 and x2 be the displacement

of mass respectively.

Figure 3.1(a)

As described in the previous lectures one may easily derive the equation of motion

by using d'Alembert principle or the energy principle (Lagrange principle or

Hamilton 's principle)

Figure 3.1(b): Free body diagrams

Using d'Alembert principle for mass m1 from the free body diagram shown in figure

3.1(b)

and similarly for mass m2

Derivation of Equation of Motion and Coordinate Coupling

Noting , the above two equations in matrix form can be written as

Now depending on the position of point C, few cases can are studied below.

Case 1 : Considering , i.e., point C and G coincides, the equation of motion can

be written as

Figure 3.2

So in this case the system is statically coupled and if , this coupling

disappears, and we obtained uncoupled x and vibrations.

Case 2 : If, , the equation of motion becomes

Hence in this case the system is dynamically coupled but statically uncoupled.

Case 3: If we choose , i.e. point C coincide with the left end, the equation of

motion will become

Here the system is both statically and dynamically coupled.

Normal Mode Vibration

Again considering the problem of the spring-mass system in figure 6.1.1 with

, , , the equation of motion can be written as

We define a normal mode oscillation as one in which each mass undergoes harmonic

motion of the same frequency, passing simultaneously through the equilibrium

position. For such motion, we let

Hence,

or, in matrix form

Hence for nonzero values of and (i.e., for non-trivial response)

Now substituting , equation yields

Hence, and

So, the natural frequecies of the system are and

Now it may be observed that for these frequencies, as both the equations are not

independent, one can not get unique value of and . So one should find a

normalized value. One may normalize the response by finding the ratio of to .

From the first equation. the normalized value can be given by

and from the second equation the normalized value can be given by

Now, substituting in equation the same values, as both these

equations are linearly dependent. Here,

and similarly for

It may be noted

If one of the amplitudes is chosen to be 1 or any number, we say that

amplitudes ratio is normalized to that number.

The normalized amplitude ratios are called the normal modes and designated

by .

The two normal modes of this problem are:

In the 1st normal mode, the two masses move in the same direction and are said to be

in phase and in the 2nd

mode the two masses move in the opposite direction and are

said to be out of phase. Also in the first mode when the second mass moves unit

distance, the first mass moves 0.731 units in the same direction and in the second

mode, when the second mass moves unit distance; the first mass moves 2.73 units in

opposite direction.

VIBRATION ABSORBER

Tuned Vibration Absorber

Consider a vibrating system of mass , stiffness , subjected to a force .

As studied in case of forced vibration of single-degree of freedom system, the system

will have a steady state response given by

(1)

which will be maximum when Now to absorb this vibration, one may add a

secondary spring and mass system as shown in figure .

The equation of motion for this system can be given by

(2)

As we know for steady state vibration, the system will vibrate with a frequency of

the external excitation; we can assume the solution to be

(3)

Substituting Equation (3) in equation (2) one may write

(4)

Or, (5)

Using Cramer's rule one may write

(6)

(7)

where

Now

Here are the roots of the characteristic equation . One may note

that these roots are the normal mode frequency for this two-degrees of freedom

system. These free-vibration frequencies can be given by

From equation (6), it is clear that,

Hence, if a system called the primary system with a stiffness mass is subjected

to an exciting force or base motion to vibrate, it is possible to completely eliminate

the vibration of the primary system by suitably designing an attached spring-mass

system (secondary system) with stiffness and mass such that the natural

frequency of the secondary system coincide with the exciting frequency. .

This is the principle of dynamic vibration absorber.

From equation (1) it may be noted that the primary system will have resonance when

the natural frequency of the primary system coincide with that of the excitation

frequency.

Hence to reduce the vibration at resonance of the primary system one should design

the secondary system such that the natural frequency of both the components

coincides.

For this condition

Substituting and , the above equation reduces to

or,

For, ,

and

To keep the displacement of secondary mass small, the stiffness of the secondary

spring should be very large. To have this the secondary mass should also be large

which is not desirable from practical point of view.

Hence a compromise is usually made between the amplitude and the mass ratio. The

mass ratio is usually kept between 0.05 and 0.25.

Resonant frequency of the vibration absorber

Centrifugal Pendulum Vibration Absorber

The centrifugal pendulum vibration absorber was devised and patented in France

about 1935 and at the same time it was independently conceived and put into practice

by E. S. Taylor. Its purpose was to overcome serious torsional vibration problem

inherent in geared radial aircraft-engine �propeller system. Later it was modified

and incorporated into automobile IC engines in order to reduce the torsional

vibrations of the crankshaft. This was done by integrating the absorber mass with

crankshaft counter balance mass.

The tuned vibration absorber is only effective when the frequency of external

excitation equals to the natural frequency of the secondary spring and mass system.

But in many cases, for example in case of an automobile engine, the exciting torques

are proportional to the rotational speed �n' which may vary over a wide range. For

the absorber to be effective, its natural frequency must also be proportional to the

speed. The characteristics of the centrifugal pendulum are ideally suited for this

purpose.

Placing the coordinates through point O', parallel and normal to r, the line r rotates

with angular velocity ( ).

The acceleration of mass

Since the moment about is zero,

Assuming to be small, , so

If we assume the motion of the wheel to be a steady rotation plus a small

sinusoidal oscillation of frequency , one may write

Hence the natural frequency of the pendulum is

and its steady-state solution is

t may be noted that the same pendulum in a gravity field would have a natural

frequency of . So it may be noted that for the centrifugal pendulum the gravity

field is replaced by the centrifugal field .

Torque exerted by the pendulum on the wheel

With the component of equal to zero, the pendulum force is a tension along ,

given by times the component of .

Now assuming small angle of rotation

Now substituting

Hence the effective inertia can be written as

which can be at its natural frequency. This possesses some difficulties in the

design of the pendulum. For example to suppress a disturbing torque of frequency

equal to four times the natural speed n , the pendulum must meet the requirement

. Hence, as the length of the pendulum becomes very

small it will be difficult to design it. To avoid this one may go for Chilton bifilar

design.

MULTIDEGREE OF FREEDOM SYSTEMS

It must be appreciated that any real life system is actually a continuous or distributed

parameter system (i.e. infinitely many d.o.f). Hence to derive its equation of motion

we need to consider a small (i.e., differential) element and draw the free body

diagram and apply Newton 's second Law. The resulting equations of motion are

partial differential equations and more complex than the simple ordinary differential

equations we have been dealing with so far. Thus we are interested in modeling the

real life system using lumped parameter models and ordinary differential equations.

The accuracy of such models (i.e. how well they can model the behavior of the

infinitely many d.o.f. real life system) improves as we increase the number of d.o.f.

Thus we would like to develop mult-d.o.f lumped parameter models which still yield

ordinary differential equations of motion – as many equations as the d.o.f . We would

discuss these aspects in this lecture.

Derivation of Equations of Motion

Fig 3.3 Typical multi-d.o.f. system

Consider a typical multi-d.o.f system as shown in Fig. 3.3. As mentioned earlier our

procedure to determine the equations of motion remains the same irrespective of the

number of d.o.f of the system and is recalled to be: Step 1 : Consider the system in a

displaced Configuration Step 2 : Draw Free Body diagrams and Step 3 : Use

Newton 's second Law to write the equation of motion From the free body diagrams

shown in Fig. 3.3, we get the equations of motion as follows:

Rewriting the equations of motion in matrix notation, we get:

Or in compact form,

There are “n” equations of motion for an “n” d.o.f system. Correspondingly the mass

and stiffness matrices ([M] and [K] respectively) are square matrices of size (n x n).

If we consider free vibrations and search for harmonic oscillations,

. Substituting these in equation, we get,

ie

For a non-trivial solution to exist, we have the condition that the determinant of the

coefficient matrix must vanish. Thus, we can write,

In principle this (n x n) determinant can be expanded by row or column method and

we can write the characteristic equation (or frequency equation) in terms of ,

solution of which yields the “n” natural frequencies of the “n” d.o.f. system just as

we did for the two d.o.f system case.

We can substitute the values of in eqn and derive a relation between the

amplitudes of various masses yielding us the corresponding normal mode shape.

Typical mode shapes are schematically depicted in Figure for a d.o.f system.

Dunkerly's method of finding natural frequency of multi- degree of freedom

system

We observed in the previous lecture that determination of all the natural frequencies

of a typical multi d.o.f. system is quite complex. Several approximate methods such

as Dunkerly's method enable us to get a reasonably good estimate of the fundamental

frequency of a multi d.o.f. system. Basic idea of Dunkerly's method

Fig 3.4 A typical multi d.o.f. system

Consider a typical multi d.o.f. system as shown in fig 3.4 Dunkerly's

approximation to the fundamental frequency of this system can be obatined in two

steps:

Step1: Calculate natural

frequency of all the modified

systems shown in Fig 3.5

These modified systems are

obatined by considering one

mass/inertia at a time. Let

these frequencies be

Step2: Dunkerly's estimate of

fundamental frequency is

now given as:

Fig 3.5 Modified system considered in Dunkerly's

Method

Fig 3.6 A Typical two d.o.f. example

Consider a typical two d.o.f. system as shown in Fig 3.6 and the equations of

motion are given as:

For harmonic vibration, we can write:

Thus,

Inverting the stiffness matrix and re-writing the equations

The equation characteristic can be readily obtained by expanding the determinant as

follows:

As this is a two d.o.f. system, it is expected to have two natural frequencies viz

and ., Thus we can write Equation as:

Comparing coefficients of like terms on both sides, we have:

It would appear that these two equations can be solved exactly for and . While this is

true for this simpleexample, we can't practically implement such a scheme for an n-

d.o.f system, as it would mean similar computational effort as solving the original

problem itself. However, we could get an approximate estimate for the fundamental

frequency. If >> , then we can approximately write

Let us now study the meaning of and . It is easily verified that

These can be readily verified to be the reciprocal of the equivalent stiffness values

for the modified systems.

Thus, we can write:

Holzer's method of finding natural frequency of a multi-degree of freedom

system

Holzer's Method.

This method is an iterative method and can be used to determine any number of

frequencies for a multi-d.o.f system. Consider a typical multi-rotor system as shown

in Fig. 3.7

Fig 3.7 Typical multi-rotor system

The equations of motion for free vibration can be readily written as follows:

For harmonic vibration, we assume

Thus:

Summing up all the equations of motion, we get:

This is a condition to be satisfied by the natural frequency of the freely vibrating

system.

Holzer's method consists of the following iterative steps:

Step 1: Assume a trial frequency

Step 2: Assume the first generalized coordinate say

Step 3: Compute the other d.o.f. using the equations of motion as follows:

Step 4: Sum up and verify if this equation is satisfied to the prescribed degree of

accuracy.

If Yes, the trial frequency is a natural frequency of the system. If not, redo the steps

with a different trial frequency.

In order to reduce the computations, therefore one needs to start with a good trial

frequency and have a good method of choosing the next trial frequency to converge

fast. Two trial frequencies are found by trial and error such that is a

small positive and negative number respectively than the mean of these two trial

frequencies(i.e. bisection method) will give a good estimate of for which

.

Holzer's method can be readily programmed for computer based calculations.

TORSIONAL VIBRATIONS

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

shaft, then the vibrations are known as torsional vibrations. In this case the shaft is

twisted and untwisted alternately and torsional shear stresses are introduced in the

shaft.

Torsional vibrations may result in shafts from following forcings:

Inertia forces of reciprocating mechanisms (such as pistons in Internal

Combustion engines)

Impulsive loads occurring during a normal machine cycle (e.g. during

operations of a punch press)

Shock loads applied to electrical machineries (such as a generator line fault

followed by fault removal and automatic closure)

Torques related to gear tooth meshing frequencies, turbine blade passing

frequencies, etc.

For machines having massive rotors and flexible shafts (where system natural

frequencies of torsional vibrations may be close to, or within, the source frequency

range during normal operation) torsional vibrations constitute a potential design

problem area.

In such cases designers should ensure the accurate prediction of machine torsional

frequencies and frequencies of any of the torsional load fluctuations should not

coincide with torsional natural frequencies.

Hence, determination of torsional natural frequencies of a dynamic system is very

important.

Simple systems with a single disc mass: Consider a rotor system as shown in

Figure 4.1. The shaft is considered as massless and it provides the stiffness only. The

disc is considered as rigid and it has no flexibility. If we give a small initial

disturbance to the disc in the torsional mode and allow it to oscillate its own, it will

execute free vibrations. The oscillation will be simple harmonic motion (SHM) with

a unique frequency, which is called natural frequency of the rotor system.

From the theory of torsion of shaft, we have

where, kt is the torsional stiffness of shaft, Ip is the rotor polar mass moment of

inertia, J is the shaft polar second moment of area, l is the length of the shaft and q is

the angular displacement of the rotor.

From free body diagram of the disc

External torque on the disc

or

The free (or natural) vibration has the simple harmonic motion (SHM).

For the simple harmonic motion of the disc, we have

so that

where is the amplitude of the torsional vibration and is natural frequency of the

torsional vibration.

On substitution we get

or

Hence, the torsional natural frequency is given by square root of the ratio of torsional

stiffness to the polar mass moment of inertia.

A two-disc torsional system

In a two-discs torsional system as shown in Figure 4.3, whole of the rotor is free to

rotate i.e. the shaft being mounted on frictionless bearings.

From free body diagrams of discs in Figure 4.4, we can write

External torque on disc 1 and External torque on disc 2

and

or and

For SHM,

and

where is the torsional natural frequency.

On substitution we get

and

which can be written in the matrix form, as

with and

This equation is a homogeneous equation and the non-trial solution is obtained by

taking determinant of the matrix [k] as

|k| = 0

which gives frequency equation of the following form

which can be simplified as

Roots of equation are given as

and

From equation corresponding to first natural frequency for = 0 , we get

From equation it can be concluded that, the first root of equation represents the case

when both discs simply rolls together in phase with each other as shown in Figure

4.5 i.e. the rigid body mode, which is of a little practical significance.

From equation (4.9), for , we get

On substituting for in the above equation we get

which can be simplified as

or

which gives relative amplitudes of two discs as

The second mode represents the case when both masses vibrate in anti-phase with

one another. Figure 4.6 shows the second mode shape of two-rotor system, showing

two discs vibrating in opposite directions.

From the second mode shape, i.e. from Figure 4.6 and noting equation we have

Since both masses are always moving in the opposite direction, so there must be a

point on the shaft where the torsional vibration is not taking place i.e. a torsional

node. The location of the node may be established by treating each end of the real

rotor system as a separate single-disc cantilever system as shown in Figure 4.5. The

torsional node being treated as the point where the shaft is rigidly fixed.

Since the natural frequency of the system is known and the frequency of oscillation

of each of the single-disc system must be same, hence we write

and

Lengths l1 and l2 then can be obtained by, noting equation as

and

which must satisfy

Torsional Systems with a Stepped Shaft

Figure 4.7(a) shows a two-disc stepped shaft. The polar mass moment of inertia of

the shaft is negligible as compared to discs. In such cases the actual shaft should be

replaced by an unstepped equivalent shaft for the purpose of the analysis as shown in

Figure 4.7(b). The equivalent shaft diameter may be same as the smallest diameter of

the real shaft.

The equivalent shaft must have the same torsional stiffness as the real shaft, since for

the present case torsional springs are connected in series. The equivalent torsional

spring can be written as

we have

which gives

where

where , and are equivalent lengths of shaft segments having equivalent shaft

diameter d3 and le is the total equivalent length of unstepped shaft having diameter d3

as shown in Figure 4.7(b).

From Figure 4.7(b) and noting equations in the equivalent shaft the node

location can be obtained as

with and

From equations the node position a & b can be obtained in the equivalent shaft

length.

Now the node location in real shaft system can be obtained as follows:

we have

Since above equation is for shaft segment in which node is assumed to be present, we

can write

and

It can be combined as

So once a & b are obtained the location of the node in the actual shaft can be

obtained i.e. the final location of the node on the shaft in real system is given in the

same proportion along the length of shaft in the equivalent system in which the node

occurs.

MODULE 4

WHIRLING OF SHAFTS – CRITICAL SPEED

Consider a typical shaft, carrying a rotor (disk) mounted between two bearings as

shown in Figure.Let us assume that the overall mass of the shaft is negligible

compared to that of the rotor (disk) and hence we can consider it as a simple

torsional spring. The rotor (disk) section has a geometric centre i.e., the centre of the

circular cross-section and the mass centre due to the material distribution. These two

may or may not coincide in general, leading to eccentricity. The eccentricity could be

due to internal material defects, manufacturing errors etc. As the shaft rotates, the

eccentricity implies that the mass of the rotor rotating with some eccentricity will

cause in-plane centrifugal force. Due to the flexibility of the shaft, the shaft will be

pulled away from its central line as indicated in the figure. Let us assume that the air-

friction damping force is negligible. The centrifugal force for a given speed is thus

balanced by the internal resistance force in the shaft-spring and the system comes to

an equilibrium position with the shaft in a bent configuration as indicated in the

figure. Thus the shaft is rotating about its own axis and the plane containing the bent

shaft and the line of bearings rotates about an axis coinciding with the line of

bearings. We consider here only the case, wherein these two rotational speeds are

identical, called the synchronous whirl.

Centrifugal force

shaft resistance force

wherein, the shaft stiffness k is the lateral stiffness of a shaft in its bearings i.e., considering the rotor at mid-

span, this is the force required to cause a unit lateral displacement at mid-span of the simply supported shaft.

Thus

Where E is the Young's modulus, I is the second moment of area, and L is the length between the supports.

Thus, for equilibrium,

ie

where we have used to represent the natural frequency of the lateral vibration of the springy-

shaft-rotor system. Thus when the rotational speed of the system coincides with the natural frequency of

lateral vibrations, the shaft tends to bow out with a large amplitude. This speed is known as the critical speed

and it is necessary that such a resonance situation is avoided in actual practice. As discussed earlier in the

case of resonance, it takes some time for the amplitude to build up to a large value. Some of the turbine

rotors whose operating speeds go beyond the critical speed are able to use this fact and rush-through the

critical speed. It is necessary to observe that, in synchronous whirl. the heavier side remains all the time on

the outer side. Thus when the shaft bends, an inner fibre is under compressive stress and outer fibre is under

tensible stress but there is NO REVERSAL of stress.

Rayleighs Method

Fig 4.1 Multiple disks on a shaft

Rayleigh's method is based on the principle of conservation of energy. The energy in

an undamped system consists of the kinetic energy and the potential energy. The

kinetic energy T is stored in the mass and is proportional to the square of the

velocity. The potential energy U includes strain energy that is proportional to elastic

deformations and the potential of the applied forces. For a conservative system, the

total energy must remain constant. That is

Differentiating this expression, we get the equation of motion as follows.

Note that the amounts of kinetic and potential energy in the system may change with

time but their sum must remain constant. Thus if and are energies at time and

and are energies at time , then

For a shaft as shown in Fig.4.2 the potential energy is zero at the specific instant of

time when the mass is passing through its static equilibrium position and kinetic

energy is at its maximum . Similarly at the instant when the mass is at its

extreme position the kinetic energy is zero and the potential energy is at its

maximum . Thus we have the following relationship.

Therefore we have, considering all the disks on the shaft,

Where i=1, n represents summation over all the "n" disks.

So we get the frequency of natural vibration as,

Dunkerley's Empirical Method

When a shaft carries multiple disks it is always efficient to use this method.

Fig 4.2 Dunkerly's approximation for shaft

We consider only one force (wt of disk) acting on the shaft at a time. For each disk,

we find the corresponding natural frequency as . The Nutural

frequency of the shaft when all the loads(disks) act on the shaft simultaneously

can be found out by the using the formulae:

For each of the dub-systems ie shaft with only one disk, natural frequency is obtained

as where k is the lateral stiffness of the shaft in its bearings and m is the mass

of the disk.

To understand the basis of this method, we need to appreciate multi-d.o.f system

vibrations.

CRITICAL SPEEDS OF A LIGHT CANTILEVER SHAFT WITH A LARGE

HEAVY DISC AT ITS END

If a light shaft having two end supports has a central disc then the system

has been shown to have one critical speed. Even if the disc is not central, the system

will have one critical speed as long as we assume the mass of the disc to be

concentrated. If, however, the disc has mass as well as moment of inertia, and is not

central, then the system will have two critical speeds. The treatment given below is

for a light cantilever shaft having a disc which has mass as well as moment of inertia.

Since the critical speed is numerically equal to the natural frequency of lateral

vibrations, we will find the later for this system.

Consider the beam so as to be displaced from the equilibrium position as

shown in Fig. In this figure,

M = mass of the disc,

Mr2 = moment of inertia of the disc about an axis passing through

the CG of the disc and perpendicular to the plane of the paper.

y, y = displacement and acceleration of the CG of the disc,

, = angular displacement and angular acceleration of the axis of

the disc due to bending

Further let

a11 = deflection of the CG of the disc per unit force acting on it in

the lateral direction = l3/3EL

a22 = slope at the free end of the beam per unit moment acting on

the CG of the disc, in the plane of the paper = l/EI

a12 = deflection of the CG of the disc per unit moment acting on it

= slope at the free and of the beam per unit force acting on it

in a lateral direction = l2/2EI

where l = length of the beam,

I = moment of inertia of the section of the beam about the neutral

axis.

E = modulus of elasticity of the material of the beam.

The inertia force and the inertia torque on the disc in the displaced position

are shown in Fig.8.6.1, along with their directions. These are as follows:

2

2 2 2

Inertia force My M y

Inertia torque Mr Mr

(1 )

where is natural frequency for the principal mode of vibration of the

system.

Then the deflection at the CG of the disc and the rotation of the disc in the

plane of the paper are given by

2 2 211 12y a M y a Mr (2)

2 2 221 22a M y a Mr (3)

Eliminating y and from the above two equations, and putting

3

2

2

Mlg

3EI

3rh

l

(4)

we have hg4 – 4(h + 1)g

2 + 4 = 0 (5)

giving the two natural frequencies as

2 21,2

2g (h 1) (h 1) h

h

(6)

Figure (2) is a plot of the above equation and shows the variation of the two

natural frequencies of the system with the change in 2

2

3rh

l

; it may be recalled

that r is radius of gyration of the disc about an axis passing through its CG and

perpendicular to the axis of the disc.

h = 0, corresponds to the concentrated mass

h , corresponds to the disc having large radius of gyration with the

change in 2

2

3rh

l

.

Transient Vibrations

INTRODUCTION

A system subjected to periodic excitation has two components of motion,

the transient and the steady state. In most of such cases the transient part is not

important as it dies out soon, and the steady state part is the one that persists.

However, where the excitation is of the aperiodic nature like a shock pulse or a

transient excitation, the response of the system is purely transient. After the duration

of the excitation, the system undergoes vibrations with its natural frequency with an

amplitude depending upon the type and duration of the excitation. It is in such cases

that the transient vibrations have importance. The practical examples of shock

excited transient vibrations are rock explosions, gunfires, loading or unloading of

packages by dropping them on hard floors, punching operations, automobiles at high

speeds passing over pits or curbs on the road, etc.

The use of Laplace transform is introduced in this chapter for the analysis of

systems subjected to shock pulses. The usual differential equations method or the so-

called classical method becomes very lengthy and cumbersome with transient

excitations of different shapes.

LAPLACE TRANSFORMATION

Laplace transform is a powerful mathematical tool that is extremely useful

in the solution of differential equations, and especially so, where transients are

involved. It is that branch of operational calculus wherein a function is transformed

from t (time) domain to a new s domain. The original differential equation in t

domain, by use of Laplace tranform, changes itself into an algebraic equation in s

domain. The solution of an algebraic equation is very easy as compared to that of a

differential equation. Once the solution is s domain is obtained, the process of

inverse transformation gives the solution back in t domain. Manipulation with

transformation and inverse transformation is facilitated by the use of table of

transform pairs which is given later in this section.

Laplace transform F(s) of a function f(t) is defines as

st

0

F(s) f (t)d dt

In shorthand it is generally written as

L [f(t)] = F(s)

The use of the basic definition of Laplace transform is illustrated below by

actually transforming a few common functions.

RESPONSE TO AN IMPULSIVE INPUT

Consider a damped spring mass system subjected to an impulse F̂ (t) , the

strength of the impulse being F̂ . Since the impulse acts for an extremely small

duration, its effect is to give an initial velocity to the mass given by

F̂ mdv

where dv is the change in velocity of the mass due to the impulse F̂ . If the system is

initially at rest, impulse gives it a starting velocity of

F̂dv

m

The initial displacement of the mass form the equilibrium position is zero

because of the extremely small duration of the impulse.

Thus the initial conditions for the mass are

x(0) 0

F̂x(0)

m

(1)

The differential equation for the system can now be written as

mx cx kx 0 (2)

The forcing function on the right has been taken to be zero since the impulse

effectively gives only the initial conditions obtained in Eq. (1).

Dividing Eq. (2) by m through out, it can be written as

2n nx 2 x x 0 (3)

Taking the Laplace transform of the above equation, we have

2 2n n[s X(s) sx(0) x(0)] 2 [sX(s) x(0)] X(s) 0

Substituting the initial conditions of Eq. (1), and re-arranging, gives

2 2

n n

F̂ 1X(s)

m s 2 s

(4)

In order to obtain the inverse transformation fro the above equation, the

expression on the right has to be re-arranged in one of the forms corresponding to the

transform pairs , for direct inversion. If 1 , the above equation is re-written in the

following form

2n

222 2

nn n

1FX(s)

m 1 (s ) 1

(5)

The inverse transform is,

t 2nn

2n

F̂x(t) e sin 1 t

m 1

(6)

which is the response of the system to an impulsive input.

RESPONSE TO A STEP INPUT

Figure shows a spring-mass-dashpot system subjected to a step force F0 u(t).

The magnitude of the force is constant at a value F0 for all time greater than or equal

to zero. The force is zero for 1 < 0. The differential equation of motion can be written

as

0mx cx kx F u(t)

or 2 0n n

Fx 2 x x u(t)

m (1)

Taking the Laplace transform of the above equation, we have

2 2 0n n

F 1[s X(s) sx(0) x(0)] 2 [sX(s) x(0)] X(s) .

m s

A second order system subjected to a finite step cannot have any initial

velocity or displacement. So, putting all initial conditions zero in the above equation,

and re-arranging, we have

0

2 2n n

F 1X(s)

m s(s 2 s )

(2)

The inverse transform of the above equation cannot be obtained straightway

from the tables. Hence splitting the right hand side into partial fractions, we have

0 n

2 2 2n n n

F s 21 1X(s)

m s s 2 s

The right had expression in the bracket above is still not invertable directly.

Assuming an underdamped system, i.e. 1 , the above equation is written as

follows:

2n

2

0 n

2 2 22 2 2 2n

n n n n

. 11F (s )1 1

X(s)m s

(s ) 1 (s ) 1

(3)

Inverse the transform of Eq. (3) can now be obtained directly from the table

and is given below

t t2 20 n n

n n2 2n

Fx(t) 1 e cos 1 t e sin 1 t

m 1

Putting 2nm = k, we finally have

t 2 20 nn n

2

Fx(t) 1 e cos 1 t sin 1 t

k 1

(4)

For an undamped case, response equation can be written from the above

equation by putting 0 , or

0n

Fx(t) [1 cos t]

k (5)

RESPONSE TO A PULSE INPUT

Pulse applications in engineering practice are very common. An explosion

occurring on a system with a comparatively larger natural period will be an impulse

while the same explosion occurring on a system with a smaller natural period will be

pulse. In this section two important types of pulses, rectangular and half sinusoidal,

are treated. The method lends itself for the analysis of any type of pulse for which a

mathematical equation can be written.

The vibratory systems considered in this section have been taken as

undamped systems to make the response equations simpler. Further, since most

physical systems are lightly damped and in most cases we are interested in maximum

displacements and accelerations, we will be slightly erring on the safer side in

neglecting small amount of damping.

Rectangular pulse

Consider a spring-mass system subjected to a rectangular pulse to height F0

and duration as shown. The response equation can be written directly by

comparing the response of the system to a multi-step input by considering in this

case two equal and opposite steps, one at t = 0 and the other at t . Therefore,

0 0n n

F Fx(t) [1 cos t]u(t) [1 cos (t )]u(t )

k k (1)

The above equation can be written as the following two equations

0n

0n n

Fx(t) [1 cos t] for 0 t

k

F[cos (t ) cos t] for t

k

(2)

PHASE PLANE METHOD

A spring mass system with initial conditions X0 and V0, has its differential

equation written as

2nx x 0 (1)

Its solution may be written as

nx Asin( t )

where 2

2 00 2

n

VA X

and 1 n 0

0

Xtan

V

Differentiating equation (1) for velocity, we have

nx A ncos( t )

or n

n

xAcos( t )

(2)

Squaring and adding Eq. (1) and (2), we have

2

2 2

n

xx A

(3)

The above equation is a circle in a plane with coordinate axes x and n(x / )

Its radius is A and centre at the origin. This is shown in Figure.. The starting point on

this displacement velocity plot is marked P1. At t1 seconds later the displacement and

velocity of the system are represented by point P2 where 1 2 n 1P OP t radians.

From this diagram, the displacement and velocity phase of the motion are available

from the single point which corresponds to a particular time. This is the phase plane

plot. The horizontal projection of the phase trajectory on a time base gives the

displacement-time plot of the motion and is shown in Figure Similarly the vertical

projection on the time base will give the velocity-time plot of the motion.

It may be noted that the centre of the phase trajectory always lies on the x-

axis at a distance equal to the static equilibrium displacement of the system. In the

case discussed the static equilibrium displacement was zero and therefore the centre

of the circle was located at the origin. In case of a step force input F0, the static

equilibrium position suddenly changes through a distance F0/k. Thus the phase plane

plot for such a motion will be a circle whose centre lies F0/k above the centre. The

radius of this circle will be F0/k so that the trajectory starts from the origin

corresponding to zero initial conditions.

The use of the phase plane method is illustrated by the following examples

for systems subjected to multiple steps.

SHOCK SPECTRUM

The response of a spring-mass system to a particular pulse depends upon the

natural frequency of the system. The plot of the maximum response of the system

against the natural frequency of the system is called the shock spectrum of the

particular disturbance. The shock spectrum shows at a glance the natural frequencies

which cause large response amplitudes for the particular disturbance.

Non-Linear Vibrations

INTRODUCTION

Most physical systems can be represented by linear differential equations,

the types of which have been dealt with in the previous chapters. A general equation

of this type is

mx cx kx F(t) (1)

In this equation which is for a linear system, the inertia force, the damping

force and the spring force are linear functions of x,x and x respectively. This is not

so in the case of non-linear systems. A general equation for a non-linear system is

mx ( )x f (x) F(t) (2)

in which the damping force and the spring force are not linear functions of x and x.

There are quite some physical systems which have non-linear spring and damping

characteristics. Rubber springs and other similar isolators have spring stiffness which

increases with amplitude. Cast iron and concrete have spring stiffness which

decreases with amplitude. Examples of non-linear damping are dry friction damping

and material damping. Even so called linear systems tend to become non-linear with

larger amplitudes of vibration. The analysis of non-linear systems is comparatively

difficult. In certain cases there is no exact solution.

One major difference between the linear and non-linear systems is that the

law of superposition does not hold good for non-linear systems. Mathematically

speaking, if x1 is a solution of

1mx cx kx F (t)

and x2 is a solution of

2mx cx kx F (t)

then (x1 + x2) is a solution of

1 2mx cx kx F (t) F (t)

This is not so in the case of non-linear systems. Even for the case of free

vibration any two known solutions of the non-linear system cannot be superimposed

to obtain a general solution.

PHASE PLANE

Phase plane was introduced in Sec. 9.6 for the case of linear systems. Here

we extend it for the case of non-linear systems.

Consider the differential equation

mx f (x) 0 (1)

The acceleration x can also be written as

dv

x vdx

where v is the velocity of the particle. Substituting it in Eq. (1), we have

dv

mv f (x) 0dx

or mv dv = –f(x) dx (2)

The above equation is intergrable directly. If v = V0 when x = X0, then the

integration of Eq. (2) gives

v x

V X0 0

mvdv f (x)dx (2a)

or 22

00

mVmv[F(x) F(X )]

2 2

The above equation is in accordance with the Law of Conservation of

Energy. The left hand side is the increase in kinetic energy of the system and the

right hand side is the decrease in potential energy. The equation can also be written

as

22

00

mVmvF(x) F(X )

2 2 (3)

which states that the total energy of the system at any instant is equal to the total

initial energy of the system. Curve in x – v plane can be drawn from Eq. (3) and this

will be a curve of constant energy. A set of such curves can be drawn, each for

different total energy. These curves are known as Energy Curves or Integral Curves

in phase plane.

We had taken the phase with x along the ordinate and v along the abscissa,

the trajectory was always counter-clockwise. Here, for convenience we will take x

along the abscissa and v along the ordinate. The trajectories here will be clockwise.

Systems which have periodic motion would be represented on the phase

plane by means of a set of closed curves, each curve for different energy of the

system.

Consider the linear case when f(x) = kx. Equation (2a) then integrates to

2 22 20 0mV kXmv kx

2 2 2 2

2 22 20 0mV kXmv kx

E(say)2 2 2 2

(4)

The phase plane trajectories are clearly a set of ellipses with the origin as the

centre. The right hand side of Eq. (4) is the total initial energy of the system. As the

value of this initial energy (depending upon initial conditions) increases, the size of

the ellipse also increases.

In general a point P(x, v) in the phase plane, called the representative point

of the system, represents the state of the system at any instant t ad the trajectory

traced gives the history of the system. Through any and every point of the phase

plane passes one and only one trajectory and thus the trajectories in the plane do not

intersect one another.

With the passage of time the representative point moves along the trajectory

in a clockwise direction with what is known as the phase velocity given by

2 2u x v (5)

The velocity is always non-zero at points of equilibrium. Origin is a point of

stable equilibrium.Most non-linear equations cannot be solved explicitly. However,

their phase-plane plots can be drawn graphically and these diagrams give several

important conclusions regarding motion of the systems.

Consider the general case of a system with non-linear damping and non-

linear spring. Let the differential equation of motion be

mx (x) f (x) 0 (6)

Letting x v , the above equation can be written down in the form of the

following two equations.

dxv

dt

dv (v) f (x)

dt m

(7)

Differential dt can be eliminated from Eq. (7) to give

dv (v) f (x)

dx mv

(8)

Equation (8) gives the slope of the trajectory at any point (x, v) in the phase

plane and is useful for constructing the plots. The slope of the trajectory is directly

obtainable at all points except the ones where v = 0 and (v) f (x) 0 . At these

points the slope becomes indeterminate and these points are called singular points.

From Eq. (7), these points correspond to the conditions v = 0 and dv/dt = 0, i.e. the

points of equilibrium. Singular points exist on x-axis (v = 0) wheredv/dt = 0. Origin

is always a singular point. Other singular points may or may not be there for the

system. At singular points the phase velocity is zero.

UNDAMPED FREE VIBRATION WITH NON-LINEAR SPRING FORCES

Let the system be represented by

mx f (x) 0 (1)

The above equation can be written as

dv

mv f (x) 0dx

(2)

Integrating Eq. (10.4.2) we have

2mv

F(x) E2

(3)

where F(x) is the integral of f(x) and so represents the potential energy of the system,

and E is the total energy of the system and depends upon the initial conditions.From

Eq. (3), we have 2v [E F(x)]

m (4)

Let the system have cubic non-linearity represented by

3f (x) x x ( 0) (5)

then, 2 4x x

F(x)2 4

(6)

From Eq. (10.4.3) and (10.4.6), we get

2 2 4mv x x

E2 2 4

(7)

The above equation gives the plot in the phase plane for different values of

E. These are closed curves when 0 for any amplitude. For the case when 1 ,

the phase plane plots are closed curves upto a certain amplitude and beyond that they

are unstable. For the case of closed curves, the system has periodic motion. Let

maxx a be the amplitude of vibration. When maxx x a , v = 0; Eq. (7) becomes

4 2a aE 0

4 2

(8)

giving

22 4 E

a

(9)

In the above equation only positive sign before the radical is applicable

whether or 0 .

Writing v as dx

dt Eq. (4) is written as

dx m dx

dt22 [E F(x)]

[E F(x)]m

(10)

For the case of periodic motion of amplitude a, the time period per cycle of

vibration is given by integrating the above equation over a quarter of a cycle and

multiplying it by 4. Thus,

a

0

m dx4

2 E F(x)

(11)

Substituting for F(x) from Eq. (6) in the above equation, we have

a

2 40

m dx4

2 E ( x / 2) ( x / 4)

(12)

The quadratic expression in the radical sign of the above equation can easily

be factorized since comparing it with Eq. (8) it shows that (a2 – x

2) is one of the

factors of this quadratic function.

Therefore, let

2 4 42 2 2x x x

E (a x ) b2 4 4

(13)

Comparing the coefficient of x2 and constant terms in the above equation we

have

2 2

22

E a b

ab

2 4

(14)

The second of the above equations gives

2

2 xb

4 2

(15)

Equation (12) can now be written with the help of Eq. (13) and (15) as

a

2 20 2 2

m dx4

2 a x(a x )

4 2 4

or

a

2 2 2 20

m dx8

2 (a x )( a 2 x )

(16)

It is possible to convert the above complex integral into complete elliptic

integral of the firs kind, the value of which can be obtained from tables of Elliptic

integrals. Discussed below are the two cases falling under this type of non-linearity.

PERTURBATION METHOD

This is a very useful method for obtaining solutions of non-linear systems to

any degree of accuracy by successive approximations. Consider the system to be

represented by the differential equation

2 30x x x 0 (1)

where 0 is the natural frequency of the linear system. Assuming that the solution

can be written in the form of a Taylor series in terms of the parameter (known as

perturbation parameter), we can write

20 1 2x x x x ..... (2)

where all x’s are functions of time t. The only restriction in the above way of writing

is that be a small quantity.

Since the frequency of vibration which is dependent upon amplitude of

vibration is also unknown, we can write

20 1 2 .....

But as only 20 appears in the differential equation, it is more convenient to

write

2 2 20 1 2 ..... (3)

Substituting equations (2) and (3) in Eq. (1) we have

2 2 2 20 1 2 2 1 2 0 1 2(x x x ...) ( ...)(x x x ...)

30 1 2 2(x x x ...) 0 (4)

The above equation after expanding can be written as

2 2 30 0 1 1 1 0 0(x x ) (x x x x )

2 2 22 2 1 1 2 0 0 1(x x x x 3x x ) ... 0 (5)

Since the above equation must hold good for any small value of , it means

that each of the terms in the parenthesis must individually be zero, therefore.

20 0

2 31 1 1 0 0

2 22 2 1 1 2 0 0 1

x x 0

x x x x 0

x x x x 3x x

... ... ... ... ... ... ... ...

(6)

Now let the initial conditions be,

x a

x 0

at t = 0

Substituting these in Eq. (2) and its derivative, we have

20 1 2a x (0) x (0) x (0) ...

20 2 20 x (0) x (0) x (0) ...

Again, since these equations must be satisfied for any small value of , we

have

0 0

1 1

2 2

x (0) a x (0) 0

x (0) 0 x (0) 0

x (0) 0 x (0) 0

... ... ... ...

(7)

With the first set of initial conditions in Eq. (7), the solution of the first

differential equation in Eq. (6), is

0x acos t (8)

Substituting the above in the right side of the second differential equation in

Eq. (6), we get

2 3 31 1 1x x acos t a cos t (9)

Applying the relation

3 3 1cos t cos t cos3 t

4 4

to Eq. (10.5.9), it becomes

2 3 31 1 1

3 1x x ( a a )cos t a cos3 t

4 4 (10)

In the above equation the forcing function 31

3( a a )cos t

4 will cause

resonance to the system since the left hand side shows the natural frequency of the

system as , the same as that of the first part of excitation. In order to avoid this

absurdity, we must have

31

3a a 0

4 (11)

Therefore, 2 31 1

1x x a cos3 t

4

the solution of which is

3

1 1 2 2

ax (A cos t A sin t) cos3 t

32

(12)

Applying the zero initial conditions from Eq. (7), we get the constants as

3

1 2

aA

32

A2 = 0

Eq. (12) becomes

3

1 2

ax [cos t cos3 t]

32

(13)

Substituting Eq. (8) and (13) in the first two terms of Eq. (2), the solution

upto first order correction is obtained as

3

0 1 2

ax x x a cos t [cos t cos3 t]

32

(14)

with 2 given by Eq. (3) upto first order correction as

2 2 2 20 1 0

3a

4

(15)

[from Eq. (11)]

The above process of successive approximations can be continued to include

the higher order corrections.

FORCED VIBRATIONS WITH NON-LINEAR SPRING FORCES

(DUFFING’S EQN.)

Consider a system represented by the differential equation

2 3n 0x x x F cos t (1)

This equation is known as Duffings Equation after the name of the

mathematician who made an exhaustive study of this equation. Rewrite Eq. (1) as

2 3n 0x x x F cos t (2)

Considering only small values of and F0, it is known from experience that

the frequency of steady state vibration will be the same as that of excitation plus

some higher harmonics. So, the first approximate solution can be written as

1x acos t (3)

Substituting the above approximate solution in the right hand side of Eq.

(2), we have

2 32 n 0x acos t a cos3 t F cos t (4)

The double integration of the above equation will give x2 which will be a

better approximation than x1 given in Eq. (3). Using the relationship

3 3 1cos t cos t cos3 t

4 4

Eq. (4), is rewritten as

2 3 32 0 n

3 1x (F a a )cos t a cos3 t

4 4 (5)

Integrating the above equation twice and dropping out the constants of

integration to ensure periodic motion, we have

32 3

2 0 n2 2

1 3 ax {F a } a cos t cos3 t

4 36

(6)

If x1 were a good first approximation of the system then the coefficient of

cos t in Eq. (3) should be approximately the same as the coefficient of cos t in Eq.

(6). Equating the two

2 3n 02

1 3a ( a a F )

4

(7)

or 2 2 2 0n

F3a

4 a (7a)

Then the solution x(t) can be written as

3

2 2

ax(t) x a cos t cos3 t

36

(8)

Equation (7) is a cubic in a and therefore for any value of there are in

general three values of a; one root is always real, the other may be real or complex

conjugate.

In the non-linear systems there is no resonance like we have in linear

systems. The amplitude never becomes infinite. The dotted lines in these figures

show the relationship between the amplitude of vibration and the natural frequency.

This is obtained from Eq. (7a) by setting F0 is zero. These lines show that the natural

frequency of a hard spring system increases with the amplitude and for a soft spring

system the natural frequency decreases with increase in amplitude. Since the natural

frequency is different at different amplitudes, the resonance can not build up.

When damping is present in the non-linear systems, the skewed peaks wind

up at a certain stage.When the frequency of excitation is gradually increasing from

zero, the response varies along the points there being a sudden change in amplitude

from 2 to 3 at the corresponding frequency. The portion of the response curve is

never traced.This phenomenon of sudden change in amplitude from 2 to 3 while the

frequency is gradually increasing and the sudden change from 5 to 6 when the

frequency is gradually decreasing is known as Jump Phenomenon.

SELF EXCITED VIBRATIONS

The self excited vibrations differ from forced vibrations in that the

fluctuating force that sustains the motion is controlled by some part of the motion

itself. The exciting force may be a function of a displacement, velocity or

acceleration of the motion. When motion is stopped by some means, the fluctuating

force also disappears. The forcing function is thus dependent motion itself unlike

forced vibrations. Tool chatter and aeroplane wing flutter are some of the common

examples of self excited vibration.

It is better here to define what is known as Stability of Oscillations. If the

system is such that when disturbed from its equilibrium position it comes back there

after the transients die out, the system is known as dynamically stable. In case any

disturbance cause the amplitude to build up with time, the system is said to be

dynamically unsteady. Effectively, the system becomes unstable when negative

damping appears in its differential equation of motion. A more general definition of

stability is that the roots of the characteristic equation of the system should either be

negative real number or complex with negative real parts. Consider a third order

system with its roots of the characteristic equation as

s1 = a1

s2 = a2 + jb2

s3 = a2 – jb2

The solution of the equation would then be given by

a t a t1 21 2 2 2x C e C e cos(b t )

If either a1 or a2 is positive, the response would build up with time, giving

instability to the system. If on the other hand both a1 and a2 are negative, the response

would die out with time giving stability to the system.

Consider the system where a spring-mass system is supported on a

horizontal belt moving with a constant velocity V. The coefficient of friction

between the mass and the belt material is such that it decreases slightly with the

increase in relative velocity.

When mass is stationary, the friction coefficient between the mass and the

belt is a . When mass is moving towards the right, the relative velocity decreases

and the coefficient of friction increases. On the other hand when the mass is moving

towards the left, the relative increases and therefore the coefficient of friction

decreases. Since the friction force on the mass is always towards right, the helping

friction force when mass moves towards right is always greater than the opposing

friction force when mass moves towards left. That means a certain net energy is put

into the system in each cycle. The amplitude continues to increase. If however, the

mass is brought to rest in the equilibrium position, it stays in that position. The least

disturbance will set it vibrating with increasing amplitude. This is a case of self

excited vibration. The frequency of vibration in the cases of self excited system is

approximately equal to the natural frequency of the system provided damping is not

large.

This problem can also be tackled analytically. At any instant when the

displacement of mass is x and its velocity x , the relative velocity between the mass

and the belt is (V x) . At this instant coefficient of friction is a( x) . The normal

reaction on the mass is mg. The differential equation of motion is then written as

amx kx mg( x)

amx mg x kx mg (1)

Equation (1) gives an effective negative damping to the system which sends

it into large increasing amplitudes. The static displacement is amg / k .

In case there is viscous damping between the mass and the belt, the slope of

the equivalent friction line is no longer negative and there will not be any self excited

vibrations. There are numerous other examples of self excited vibrations caused by

dry friction, a few of which are:-

(i) Excitation of violin string by a bow.

(ii) Screeching of door joints when dry.

(iii) Shaft whirl due to dry friction.

MODULE 5

PROPOGATION OF SOUND

Sound is produced by a vibrating body. A material medium is necessary for

the propagation of sound waves. In wave motion momentum and energy are

transferred. Characteristics of wave motion.

1. Wave motion is a disturbance produced in the medium by the repeated

periodic motion of the particles of the medium. .

2. Only the wave travels forward whereas the particles of the medium

vibrate about their mean positions.

3. There is a regular phase change between the various particles of the

medium. The particle ahead starts vibrating a little later than the

particles preceding it.

4. The velocity of the wave is different from the velocity with which the

particles of the medium are vibrating about their mean positions. The

wave travels with a uniform velocity whereas the velocity of the

particles is different at different positions. It is maximum at the mean

position and zero at the extreme positions of the particles. There are

two types of wave motions.

a. Transverse wave.

b. Longitudinal wave

Sound wave are longitudinal waves and light waves are transverse waves.

Figure 1 shows the formation and propagation of transverse and longitudinal wave.

A sound wave is propagated and conveyed to the ear by means of the

intervening layers of air. Consider a vibrating tuning fork. Let us confine our

attention to the right hand prong only. When it moves towards the right, it

compresses the layer of air in front of it and as a consequence the pressure of this

layer will be greater than the adjacent layers. It tends to relieve the strain thus

created, by compressing them. These in turn hand on the compression. Thus a pulse

of compression will travel onwards to the right. Again, when the prong moves

towards the left rarefaction is started. These follow one another and as the fork

vibrates, compressions and rarefaction are sent out in regular succession. These

waves at last reach the car and set the tympanic membrane, which is ultimately

transmitted via a system of bones and cords to the brain and causes the mental

sensation called sound.

MEASUREMENT OF INTENSITY Of SOUND

The intensity of sound is defined as the quantity or energy propagating

through a unit area per unit time, the direction of propagation being perpendicular to

the area. The amount of power transmitted is measured in Watts/m2. A convenient

unit is microwatts /m2.

According to Weber Fechner law in psychology, the loudness of a sound as

judged by the ear is proportional to the logarithm of the intensity. If l1 and l0

represents the intensities of two sounds of a particular frequency and L1 and L0 the

corresponding measure of loudness then L1 = K log l1 and L0 = K log 10. The

difference in loudness technically known as Intensity Level, L between them is given

by

L = L1 - L2 = K[log1 - log 10]

1

2

lL Klog( )

l

Where K is a constant that depends on the units and I0 is some standard

reference intensity arbitrarily taken as 10-12

watt/m2, which corresponds to the

intensity of the sound which can be just heard at a frequency of about 1000 cycles

per second. This is threshold of audibility of a normal ear.

Where K in the equation above is taken as I the difference in loudness is

expressed in bels, a unit named in honour of Alexander Graham Bell, the inventor of

telephone. This unit of loudness is rather too large, one tenth of it, the decibel (db)

has become the standard. So in order to express the difference in loudness of a sound

of intensity I in decibel, the above relation should be written as

1

2

lL 10log( )

l

To build a scale of loudness, we have to fix its zero. The loudness

corresponding to the threshold of hearing is the zero of this scale. This occurs when

the intensity of sound wave equals 10 or 10-12

watt/m2. The maximum intensity which

the ear can tolerate without sensation of pain is about 10-2

watt/m2 and it corresponds

to the intensity level

2

12

10L 10log( ) 120db

10

The following table gives the approximate value of some sound measured in

decibels.

Source Intensity level in Decibel

Threshold of hearing 0

Rustle of leaves 10

Whisper 15-20

Ordinary conversation 60-65

Motor trucks and heavy street traffic 70-80

Roaring lion at 20 ft. 90

Thunder 100-110

Painful sounds 130 or above

In the above table we have expressed the loudness in decibels on the

assumption that the threshold of audibility is the same for all frequencies of the ear

and the limits of audibility vary over wide ranges of intensity and frequency, hence

the sound of same intensity but different frequencies seem to differ in loudness.

Therefore a different unit for measuring loudness is used. It is called the phon. The

measure of loudness in phons of any sound is equal to the loudness in decibels of an

equally loud pure tone of frequency 1000 cycles/second.

Acceptable noise levels:

Type of residential Area Acceptable noise level in dB

Rural 25-35

Suburban 30-40

Residential Urban 35-45

City 45-55

Industrial area 50-65

Outdoor noise levels in residential areas

Type of place/building Acceptable noise level in dB

Radio and TV studios 25-30

Music room 30-35

Hospital, classroom, Auditorium 35-40

Apartments, hotel, home 35-40

Conference Room, small office, concert

room

40-45

Private offices 40-45

Libraries, Large public office, banks,

stores

45-50

Restaurants 50-55

Indoor noise levels in public/private places

The above table is as per IS standards.

AIR COLUMNS

Stationary longitudinal waves can be produced in a column of air by any

periodical movement whose frequency coincides with one of the natural frequencies

of the column. All wind instruments are provided with a column of air called a

resonator, which may be in the form of a rectangular air chamber. The periodic

movement is caused by an important part of the musical instrument called the mouth-

piece, which is different in construction in different instruments: It is the mouth -

piece that acts as a generator and supplies the energy necessary to maintain the

vibrations in the column of air. In the theoretical treatment the following assumptions

are made:

1. The diameter of the pipe is small compared with the length of the pipe

and with the wave length of sound.

2. The diameter is sufficiently great so that the viscosity effects can be

neglected.

3. The walls of the pipe are rigid.

The organ pipes are classified into two groups: Flute or Flue pipes and Reed

pipes.

DOPPLER EFFECT

It is commonly observed that the pitch of a note apparently changes when

either the source or the observer are in motion relative to each other. When the

source approaches the observer or when the observer approaches the source or when

both approach each other the apparent pitch is higher than the actual pitch of the

sound produced by the source. Similarly when the source moves away from the

observer or when the observer moves away from the source or when both move away

from each other, the apparent pitch is lower than the actual pitch of the sound

produced by the source.

This apparent change in pitch due to relative motion between the source and

the observer is called Doppler effect.

Doppler effect in sound is asymmetric, when the source move towards the

observer with a certain velocity, the apparent pitch is different to the case when the

observer is moving towards the source with the same velocity. But it is not so in the

case of light. Doppler effect in light is symmetric.

The apparent pitch in different cases is calculated below

Let n - pith of sound

- wavelength

v - velocity of sound

n - apparent pitch

Case 1: When the source moves towards the stationary observer with a

velocity ‘a’

vn n

v a

Case 2: When the source moves away from the stationary observer with a

velocity ‘a’

vn n

v a

Case 3: When the observer moves towards a stationary source with a

velocity ‘b’

v bn n

v

Case 4: When the observer moves away from a stationary source.

v bn n

v

Case 5: When the source moves towards the observer and the observer

moves away from the source.

v bn n

v a

Case 6: When the source and the observer move towards each other

v bn n

v a

Case 7: When the source and the observer move away from each other.

v bn n

v a

Case 8: Source moving away from the observer and the observer moving

towards the source.

v bn n

v a

MUSICAL SCALES:

CHORD: When two notes of different frequencies are sounded

simultaneously, their combination is called a chord. In the case of a concord or

consonance the combination produce a pleasant or agreeable effect. While other

combinations produce a disagreeable or unpleasant effect and it is called dissonance.

HARMONY: When the two notes sounded together produce concord their

combination is called Harmony.

MELODY: If the two notes are sounded one after the other, combination

is called melody.

DIATONIC MUSICAL SCALE: A series of notes separated by definite

and simple intervals so as to produce a musical effect when played in succession is

said to constitute a musical scale. The most common musical scale is the Gamut or

the Diatonic scale. It consists of a series of eight notes, the interval between last and

the first note being 2/1. It is therefore called an octave. The series of note is denoted

as:

C D E F G A B C

sa re ga ma pa dha ni (sa)'

All these notes are arranged in increasing order of frequencies so that they

present a regular graduation in pitch and their vibration frequencies are represented

by

l 9

8

5

4

4

3

3

2

5

3

15

8 2

l 1.125 1.25 1.333 1.500 1.667 1.875 2

i.e., if the frequencies of first note C called the tonic or the key note be taken

as 24, the relative frequencies (ratio of successive frequencies) of the various notes

of the diatonic scale will be

(24) (27) (30) (32) (36) (40) (45) (48)

9

8

10

9

16

15

9

8

10

9

9

8

16

15

(Major (Minor (Semi (Major (Minor (Major (Semi

tone) tone) tone) tone) tone) tone) tone)

If the frequencies of the note C be taken as 256 and 264 respectively, the

various notes of the scale will be denoted by

256 288 320 341.3 384 426.7 480 512

264 297 330 352 396 440 495 528

The above scale consists of three main intervals Major tone, Minor tone and

semi major tone respectively, so that the sequence of interval in Diatonic scale is

major tone, minor tone, semi-tone, or neglecting the difference between major and

minor tones, tone, tone, semi-tone, tone, tone, semi-tone. Since the major tones occur

more frequently, this scale is called Major Diatonic scale. It could be extended both

above C’ and below C level. Intervals within an octave with small intervals viz.,4

3,

3

2, 2 are most consonant and are named as fourth, fifth and octave their names being

derived from the positions of these notes on the scale.

MICROPHONE

The microphone is essentially an arrangement for the conversion of sound

vibrations into vibrations of electrical current. In the telephone communication

system it is very successfully used as a transmitter and the vibrations of electrical

current thus produced are converted into sound by the receiver at the distant end. It is

also the first element of a loud speaking equipment or a broadcasting arrangement in

which electrical oscillations after proper amplification are reconverted into sound by

loud speaker. The general principle of modern carbon microphones is shown in fig.

It is based on the variation of the resistance of fine carbon granules when subjected

to pressure changes. Carbon granules are enclosed in between two plates one of

which is fixed and the other serves as a diaphragm which responds to rapid change in

pressure. The plates are placed in series with a key, a battery, and the primary of a

transformer, the secondary of the transformer is connected to the telephone receiver.

When the key is closed, a steady small current flows through the circuit. If R be

resistance of the circuit when there is no displacement of the diaphragm and Ka sin

t the varying resistance due to the displacement a sin t at any instant. K being

constant of the microphone then the total resistance of the circuit at any instant is

(R+Kasin t). Let V be the direct emf of the circuit, then the current at any instant is

given by

1V V Ka

i l sin tR Kasin t R R

2 22

2

V Ka K al sin t sin t .......

R R R

The first term indicates a steady current when the diaphragm is at rest, the

second an alternating current of the same frequency as is impressed on the diaphragm

and the rest of the terms denote its other harmonics. The current is thus modulated.

This modulated electrical current passes through P, the primary of a transformer and

produces by induction a corresponding varying current in S, its secondary. This

amplified current passes through the telephone T and excites its diaphragm. The

movements of the latter set the air in corresponding vibrations reproducing the

original sound.Other commonly used types of microphones are

The Electrodynamic Microphone: This microphone is base don the

principle of electromagnetic induction consisting of a small coil of wire attached to

the back of a freely moving light plate. The coil is situated in the magnetic field

between the central pole piece and the peripheral pole piece of a permanent "pot"

magnet. The sound wave cause the plate end of the coil to vibrate and varying

current thus induced in the coil are amplified and conveyed to the distant end.

Ribbon or Velocity Microphone: It is also based on the principle of

electromagnetic induction.

The Condenser Microphone: It is based on the principal that a charged

condenser connected to an electrical circuit be subjected to sound vibrations, they

will produce variations in the distance between the plates thus changing its capacity,

then an alternating current will be set up in the circuit.

The Crystal Microphone: This microphone is based on the principle of

piezoelectric effect, according to which certain crystals like quartz, Rochelle Salt

produce minute differences of potential between its opposite faces when subjected to

pressure.

The Hot Wire Microphone: It is based on the principle that resistance of a

metallic wire changes with change in temperature.

THE LOUDSPEAKER:

It is a device for converting electrical energy into sound energy and

therefore essentially a microphone worked backwards. It is provided with a horn or a

circular board called 'baffle' for effectively transferring the vibrations of the moving

part to the external air. The commonest and the most efficient type of speakers

nowadays is the moving coil type. It is based on the principle that when a variable

current is passed through a conductor in a magnetic field, the conductor is acted on

by a variable force in accordance with Fleming's Left Hand Rule and is f the current

is oscillatory the conductor is set in vibration.

The moving part of the apparatus consists of a small coil called the 'voice

coil' wound on a cylindrical strip to which the variable current output of the

microphone is fed. The voice coil is free to move in the annular gap between the

central S and the ‘peripheral’ pole piece N, of a 'pot' magnet designed to produce a

strong radial magnetic field in it. It is usually magnetized by a steady (DC) current

flowing in the coil wound round it. The coil is attached to a conical diaphragm made

of parchment with circular corrugation and supported round the periphery by a

flexible annular strip of leather or rubber. When the variable current passes through

the coil in the magnetic field, it causes varying movement of the coil along the axis

with the frequency of the current variations. The diaphragm is thus set into vibrations

which are communicated to the external air and the sound is reproduced. The greater

the energy supplied to the voice coil, the louder will be the sound emitted by the

diaphragm. Completely surrounding the cone and attached to it by silk threads is the

'baffle'. It prevents the air vibrating behind the cone from flowing round to its front.

The relation between the current and driving force is linear and force is

independent of the position of the coil in the gap for considerable movements. When

suitably designed a fairly uniform response of 80cps to 1000cps is secured. It is

capable of radiating large power without appreciable asymmetric distortion.

ACOUSTICAL MEASUREMENTS:

The measurement of airborne and waterborne sound is of increasing interest

to engineers. Airborne sound measurements are important in the development of less

noisy machinery and equipment, in diagnosis of vibration problems, and in the

design and test of sound recording and reproducing equipment. In large rocket and

jet engines, the sound pressures produced by the exhaust may be large enough to

cause fatigue failure of metal panels because of vibration. Waterborne sound has

been applied in underwater direction and range finding equipment like "sonar". Most

sound transducers are basically pressure measuring devices.

The basic definitions of sound are in terms of the magnitude of fluctuating

component of pressure in a fluid medium. The sound pressure level (SPL) is defined

by SPL = Sound pressure level =

P

.00021020Log decibels (dB) [1]

p - root mean square (rms) sound pressure, bar.

The rms value of fluctuating component of pressure is used in equation I,

because most sounds are random signals rather than pure sine waves. The value

0.0002 bar is an accepted standard reference value of pressure against which other

pressure s are compared by equation1. When p = 0.0002 bar, the sound pressure

level is 0dB

SOUND LEVEL METER

The most commonly used instrument for sound measurement is the sound

level meter. This actually made up of a number of interconnected components. The

sound pressure is transduced to a voltage by means of a microphone. Microphones

generally employ a thin diaphragm to convert pressure to motion. Microphone often

have an arrangement so that it will not respond to constant and slowly varying

responses. This is necessary because the hour-to-hour and day-to-day changes in

atmospheric pressure are much greater than the sound pressure fluctuations to which

the microphone must respond. The motion is then converted to voltage by some

suitable transducer, usually a capacitance, piezo-electric or moving coil type.

The output voltage of the microphone generally is quite small and at a high

impedance level. So an amplifier of high input impedance and gain is used at the

output of the microphone. This can be a relatively simple ac amplifier, since response

to static or slowly varying voltages is not required.

Following the first amplifier are the weighting networks. They are electrical

filters whose frequency response is tailored to approximate the frequency response of

the average human ear. Readings taken with a weighting network are called sound

level rather than sound pressure level.

The output of the weighting network is further amplified and an output jack

provided to lead this signal to an oscilloscope or to a spectrum analyzer. If only the

overall sound magnitude is desired, the rms value of e3 must be found. The average

value of e3 is determined by rectifying and filtering and then the meter scale is

calibrated to read rms values. This procedure is exact for pure sine waves since there

is a precise relation between the average value and rms value of a sine wave. For non

sinusoidal wave this is not true, but the error is generally small enough to be

acceptable for relatively unsophisticated work.

ACOUSTICS OF BUILDINGS

Factors affecting the acoustics of a building are: Reverberation, Loudness,

Focussing, Echelon effect, Extraneous noise, Resonance.

Reverberation

It is observed that for a listener in room or auditorium, whenever a sound

pulse is produced, he receives directly sound waves from the source, as well as sound

waves from the walls, ceiling and other materials present in the room. The waves

received by the listener are:

1. direct waves

2. Reflected waves due to multiple reflections at the various surfaces.

The quality of the note received by the listener will be the combined effect

of these two sets of waves. There is also a time gap between the direct wave received

by the listener and the direct waves received by the listener and the waves received

by successive reflection. Due to this, the sound persists for sometime even after the

source has stopped. This persistence of sound is termed as reverberation. The time

gap between the initial direct note and the reflected note up to the minimum

audibility level is called reverberation time. The reverberation time will depend on

the size of the room or the auditorium, the nature of the reflecting material on the

wall and the ceiling and the area of the reflecting surfaces.

In a good auditorium it is necessary to keep the reverberation time

negligibly small. The intensity of sound received by the listener, is shown

graphically in fig. When a source emits sound, the waves spread out and the listener

is aware of the commencement of the sound when the direct waves reach his ears.

Subsequently the listener receives sound energy due to reflected waves also. If the

note is continuously sounded, the intensity of sound at the listener's ear gradually

increases. After sometime a balance is reached between the energy emitted per

second by the source and the energy lost or dissipated by walls or other materials.

The resultant energy attains an average steady value, and to the listener the intensity

sound appears to be steady and constant. When the intensity of sound falls below the

minimum audibility level, the listener will not hear the sound.

When a series of notes are produced in the auditorium (say speech or music)

each note will give rise its own in intensity curve with respect to time. For clear

audibility of speech or music, it is necessary that (1) each separate note should give

sufficient intensity of sound in every part of the auditorium, and (2) each note should

die down rapidly before the maximum average intensity due to the next note is heard

by the listener as in fig. This is particularly important with speech. In case of music

comparatively more reverberation can be tolerated.

Loudness:

The speech of a person in a hall can be heard by an audience consisting of

about 100 persons. However, to ensure uniform distribution of sound intensity in the

hall, electrically amplified loud speakers are used. These speakers are kept at

different places in the auditorium and are located generally at a height higher than

the speaker's head. Amplifiers, however make the low frequency tones more

prominent and hence the amplification has to be kept low. The presence of low

artificial ceilings improves the audibility in general.

Focusing:

The presence of cylindrical or spherical surfaces on the walls or the ceiling

gives rise to undesirable focusing. In fig. the observer also receives the sound waves

after reflection from the ceiling. Thus the intensity of sound at O is comparatively

higher than other positions in the auditorium. It may also happen that the direct and

the reflected waves are in the opposite phase. This results in minimum intensity of

sound at O. Further the direct and the reflected waves may form a stationary wave

pattern. This causes distribution of sound intensity.

Echelon Effect

If there is regular structure similar to a flight of stairs or a set of railings in

the hall, the sound produced in front of such a structure may produce a musical note

due to regular successive echoes of sound reaching the observer. Such an effect is

called Echelon effect. If the frequency of this note is within the audible range, the

stair cases are covered with carpets to avoid reflection of sound.

Extraneous Noise

The extraneous noise may be due to (1) sound received from outside the

room (2) the sound produced by fans etc., inside the auditorium. The external sound

cannot be completely eliminated but can be minimized by using double or triple

windows and doors. Proper attention must also be paid to maximum permissible

speed of fans and rate of air circulation in the room. The air conditioning pipes

should be covered with cork and insulated acoustically from the main building.

Resonance

The acoustics of a building may also be affected by resonance. If there is

resonance for any audio frequency note, the intensity of the note will be entirely

different from the intensity desired. In halls of large size, the resonance frequency is

much below the audible limit and harmful effects due to resonance will not be

present.

Sound Distribution in an Auditorium

The design of an auditorium requires smooth decay and growth of sound. In

an auditorium, the sound must be distributed or diffused over the whole area. To

ensure these factors, acoustic treatment is given viz., scattering effect of objects,

irregularities on the wall surfaces, fixing absorptive material on the walls etc.

The first reflection of sound waves at different positions of ceiling. It is

clear from the figure that the reflected sound is distributed evenly in the auditorium

viz., the main floor and the balcony. This design enables an even distribution of

sound intensity.

Requisites for Good Acoustics

The reverberation of sound in an auditorium is due to multiple reflections

taking place at various surfaces present within the auditorium. The acoustics of an

auditorium can be improved by using the surfaces with high absorption coefficient.

This will reduce the reverberation time below the optimum value. This can be

achieved as follows.

1. By hanging Heavy Curtains

2. By hanging Picture and Maps

3. By having a few Open Windows.

4. By having Good Audience. Each person is equivalent to about 0.50m2

area of an open window.

5. The curved walls and comers bounded by two walls should be avoided.

This is done to avoid 1. Concentration of sound 2. Dead spaces.

6. Upholstered seats should be provided so that the absorption is

approximately the same with or without the audience.

7. the walls and the ceiling should be covered with the materials having

high absorption coefficient i.e., with perforated cardboards, felt

asbestos fibre glass etc.

8. The walls should be engraved and made rough with decorative

materials to increase absorption.

FOURIER THEOREM

Fourier theorem deals with the summation of a number of simple harmonic

vibrations, in which the vibrations are in the same straight line. The theorem also

helps in the synthesis and analysis of complex forms of vibrations. This theorem was

formulated by J.B.T. Fourier in 1828.

The theorem states that "any single valued periodic function can be

expressed as a sum of a number of simple harmonic terms which are multiples of the

given function." The theorem is generally referred in relation to the study of

transverse vibration of strings. However, the theorem has a wider scope.

The theorem is valid only if the following conditions are satisfied.

1. The displacement must be single valued function and continuous. This

condition is satisfied in all cases of mechanical vibrations because a

single particle cannot actually have two different displacements

simultaneously.

2. The displacement must always have a finite value. This is true in the

case of sound.

Fourier Series

The Fourier series can be expressed by the series.

0 1 1y f ( t) A A cos( t ) 2 2 m mA cos(2 t ) ......... A cos(m t )

This can be written in the form

m

m m

m 1

y f ( t) A cos(m t )

Here y represents the displacement of the complex periodic vibrations of

frequency 2

. A1, A2, ......., Am are amplitudes of the components of the 4 simple

harmonic vibrations 1 2 m,........,, ,........, represent their respective initial phases.

It may be mentioned that sometimes it is convenient to represent y as a sum

of sine and cosine series in the form.

m

m m

m 0

y f ( t) A sin m t B cosm t)

Here A0 = 0

m m

0 m m

m 1 m 1

y f ( t) B A sin m t B cosm t

The method of finding the amplitude of Fourier coefficients (B0, Am and Bm

for all values of m is called Fourier analysis.

1

0

0

lB ydt

T

1

m

0

2A ysin(m t)dt

T

1

m

0

2B ycos(m t)dt

T

Where T is time period of the function

Fourier series has been employed for the analysis of complex periodic

curves with a view to finding the various harmonic components of which they may

be built up together with their amplitudes and relative phases. It can be shown in

various ways that such components have an objective physical existence and are not

a mere mathematical fiction. A trained musical ear can easily resolve complex

musical sound hearing each simple harmonic component as a separate simple tone

and thus serve as a natural Fourier analyzer.

For practical purposes the-wavefonns of different sounds are recorded and

analysed by various methods. The coordinates of the complex curve under study at

different instants are determined. Then substitute the data obtained in the above

equations for obtaining the Fourier coefficients. Thus the various harmonic

components and their relative amplitudes are ascertained.

ACOUSTICAL IMPEDANCE AND FILTERS

An acoustic filter is a device which has been extensively used for analyzing

the quality of complex sound waves. It is so designed that it transmits certain

selected ranges of frequency with negligible attenuation, and suppresses other

frequencies almost entirely. The analogy between acoustic filters and electrical filters

used in ac circuits for a similar purpose is so close that the considerations and

equations operating in the functioning of the latter have helped a good deal in

designing the former.

If we consider the equation of motion of a body executing forced vibration

in a resisting medium

2

2

d y dym ky Fsin t

dtdt ................ 1

The corresponding equation in electricity is

2

2

d q dq qL R Esin t

dt cdt ................ 2

The following are mechanical analogues of the corresponding electrical

quantities.

Mechanical Electrical

Force (F) Emf (E0)

Mass (m) Inductance (L)

Resistance ( ) Resistance

Stiffness (k) l/Capacitance

Displacement (y) Charge (q)

Velocity (dy/dt) Current (dq/dt)

Current does not flow along a single line but is branched at several points

and so in the case of acoustic filters, where the main acoustic tube or conductor is

provided with several branch tubes of suitable dimensions to serve as guide for the

waves in the same way as electrical transmission line with its several branches guides

the current. Several other conditions must be satisfy before the analogy is complete

in every respect.

1. The length of any selected section of an acoustic transmission line must

be small as compared to the wave length so that no change of the phase

occurs within it.

2. The algebraic sum of the volume displacements at any junction of the

line is zero just as in Krichoff's law.

Corresponding to the electrical impedance there is the acoustical impedance

which is defined as the ratio of the applied pressure difference and the rate of change

of volume displacement.

There are two types of acoustic filters:

1. Low pass Filters: These are made up of two concentric cylinders

joined by walls equally spaced and perpendicular to the axes. Each chamber thus

formed had a row apertures in the inner cylinder which served as the transmission

tube.

2. High pass filter: These are made with a straight tube for transmission

and short side tubes 0.5cm long and 0.28cm diameter opening through a hole with

conductivity 0.08 into a tube 10cm long and 1 cm diameter. Six sections of such a

filter would transmit about 90 percent of sounds above 800hz but would refuse

transmission to sounds of lower frequency.