vibrations, sound measurement and noise control

8/9/2019 Vibrations, Sound Measurement and Noise Control

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THEORY OF VIBRATIONS, SOUND MEASUREMENT AND

NOISE CONTROL

INTRODUCTION....................................................................................................................................2

HARMONIC MOTION ...........................................................................................................................2

VIBRATION MODEL..............................................................................................................................3

EQUATION OF MOTION: NATURAL FREQUENCY.......................................................................4

VISCOUSLY DAMPED FREE VIBRATION.........................................................................................6

FORCED HARMONIC VIBRATION................................................................................................... 10

ROTATING UNBALANCE.................................................................................................................... 13

SOUND MEASUREMENT AND NOISE CONTROL........................................................................ 16

SOUND LEVEL METER......................................................................................................................... 16

TYPES OF SOUND LEVEL METER..................................................................................................... 18

COMMUNITY NOISE ......................................................................................................................... 22

STATISTICAL DESCRIPTION OF COMMUNITY NOISES....................................................... 23

TYPES OF COMMUNITY NOISES.......................................................................................... ......... 27

NOISE CONTROL................................................................................................................ ................ 28

SOURCE-PATH-RECEIVER CONCEPT.............................................................................................. 30



2

Introduction

There are two general classes of vibrations - free and forced. Free vibration takes place when a

system oscillates under the action of forces inherent in the system itself, and when external

impressed forces are absent. The system under free vibration will vibrate at one or more of its

natural frequencies , which are properties of the dynamic system established by its mass and

stiffness distribution.

Vibration that takes place under the excitation of external forces is called forced vibration. When the excitation is oscillatory, the system is forced to vibrate at the excitation frequency. If

the frequency of excitation coincides with one of the natural frequencies of the system, a

condition of resonance is encountered, and dangerously large oscillations may result. Thefailure of major structures such as bridges, buildings, or airplane wings is an awesome

possibility under resonance. Thus, the calculation of the natural frequencies of major importance

in the study of vibrations.

Vibrating systems are all subject to damping to some degree because energy is dissipated by

friction and other resistances. If the damping is small, it has very little influence on the natural

frequencies of the system, and hence the calculation for the natural frequencies are generallymade on the basis of no damping. On the other hand, damping is of great importance in limiting

the amplitude of oscillation at resonance.

The number of independent coordinates required to describe the motion of a system is called

degrees of freedom of the system. Thus, a free particle undergoing general motion in space will

have three degrees of freedom, and a rigid body will have six degrees of freedom, i.e., three

components of position and three angles defining its orientation. Furthermore, a continuous

elastic body will require an infinite number of coordinates (three for each point on the body) to

describe its motion; hence, its degrees of freedom must be infinite. However, in many cases,

parts of such bodies may be assumed to be rigid, and the system may be considered to be

dynamically equivalent to one having finite degrees of freedom. In fact, a surprisingly large

number of vibration problems can be treated with sufficient accuracy by reducing the system to

one having a few degrees of freedom.

Harmonic Motion

Oscillatory motion may repeat itself regularly, as in the balance wheel of a watch, or displayconsiderable irregularity, as in earthquakes. When the motion is repeated in equal intervals of

time T, it is called period motion. The repetition time t is called the period of the oscillation,

and its reciprocal, ,is called the frequency. If the motion is designated by the time

function x(t), then any periodic motion must satisfy the relationship .



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armon c mot on s o ten represente as t e pro ect on on a stra g t ne o a po nt t at s

moving on a circle at constant speed, as shown in Fig. 1. With the angular speed of the line o-p

designated by w , the displacement x can be written as

(1)

Figure 1 Harmonic Motion as a Projection of a Point Moving on a Circle

The quantity w is generally measured in radians per second, and is referred to as the angular

frequency. Because the motion repeats itself in 2p radians, we have the relationship

(2)

where t and f are the period and frequency of the harmonic motion, usually measured in seconds

and cycles per second, respectively.

The velocity and acceleration of harmonic motion can be simply determined by differentiation

of Eq. 1. Using the dot notation for the derivative, we obtain

(3)

(4)

Vibration Model

The basic vibration model of a simple oscillatory system consists of a mass, a massless spring,

and a damper. The spring supporting the mass is assumed to be of negligible mass. Its force-

deflection relationship is considered to be linear, following Hooke's law, , where thestiffness k is measured in newtons/meter.



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e v scous amp ng, genera y represente y a as pot, s escr e y a orce proport ona to

the velocity, or . The damping coefficient c is measured in newtons/meter/second.

Figure 2 Spring-Mass System and Free-Body Diagram

Equation of Motion: Natural Frequency

Figure 2 shows a simple undamped spring-mass system, which is assumed to move only along

the vertical direction. It has one degree of freedom (DOF), because its motion is described by a

single coordinate x.

When placed into motion, oscillation will take place at the natural frequency f n which is aproperty of the system. We now examine some of the basic concepts associated with the free

vibration of systems with one degree of freedom.

Figure 2 Spring-Mass System and Free-Body Diagram

Newton's second law is the first basis for examining the motion of the system. As shown in Fig.

2 the deformation of the spring in the static equilibrium position is D , and the spring force kD is

equal to the gravitational force w acting on mass m

(5)

By measuring the displacement x from the static equilibrium position, the forces acting on m are

and w. With x chosen to be positive in the downward direction, all quantities - force,velocity, and acceleration are also positive in the downward direction.



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e now app y ewton s secon aw o mot on to t e mass m :

and because kD = w, we obtain :

(6)

It is evident that the choice of the static equilibrium position as reference for x has eliminated w,

the force due to gravity, and the static spring force kD from the equation of motion, and the

resultant force on m is simply the spring force due to the displacement x.

By defining the circular frequency w n by the equation

(7)

Eq. 6 can be written as

(8)

and we conclude that the motion is harmonic. Equation (8), a homogeneous second order lineardifferential equation, has the following general solution :

(9) where A and B are the two necessary constants. These

constants are evaluated from initial conditions , and Eq. (9) can be shown toreduce to

(10)

The natural period of the oscillation is established from , or

(11) and the natural frequency is

(12)



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ese quant t es can e expresse n terms o t e stat c e ect on y o serv ng q. ,

. Thus, Eq. (12) can be expressed in terms of the static deflection D as

(13)

Note that , depend only on the mass and stiffness of the system, which are

properties of the system.

Viscously Damped Free Vibration

Viscous damping force is expressed by the equation

(14)

where c is a constant of proportionality.

Symbolically. it is designated by a dashpot, as shown in Fig. 3. From the free body diagram, the

equation of motion is .seen to be

(15)

The solution of this equation has two parts. If F(t) = 0, we have the homogeneous differential

equation whose solution corresponds physically to that of free-damped vibration. With F(t) ¹ 0,

we obtain the particular solution that is due to the excitation irrespective of the homogeneous

solution. We will first examine the homogeneous equation that will give us some understanding

of the role of damping.

Figure 3 Viscously Damped Free Vibration



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With the homogeneous equation :

(16)

the traditional approach is to assume a solution of the form :

(17)

where s is a constant. Upon substitution into the differential equation, we obtain :

which is satisfied for all values of t when

(18)

Equation (18), which is known as the characteristic equation, has two roots :

(19)

Hence, the general solution is given by the equation:

(20)

where A and B are constants to be evaluated from the initial conditions and .

Equation (19) substituted into (20) gives :

(21)

The first term, , is simply an exponentially decaying function of time. The behavior of

the terms in the parentheses, however, depends on whether the numerical value within the

radical is positive, zero, or negative.

When the damping term (c/2m)2 is larger than k/m, the exponents in the previous equation are

real numbers and no oscillations are possible. We refer to this case as overdamped .



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en t e amp ng term c m)2 s ess t an m, t e exponent ecomes an mag nary num er,

. Because

the terms of Eq. (21) within the parentheses are oscillatory. We refer to this case as

underdamped .

In the limiting case between the oscillatory and non oscillatory motion , and the

radical is zero. The damping corresponding to this case is called critical damping, c c.

(22)

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

number z , called the damping ratio:

(23)

and

(24)

(i) Oscillatory Motion (z < 1.0) Underdamped Case :

(25)

The frequency of damped oscillation is equal to :

(26)

Figure 4 shows the general nature of the oscillatory motion.



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Figure 4 Damped Oscillation z < 1

(ii) Non oscillatory Motion (z > 1.0) Overdamped Case :

(27)

The motion is an exponentially decreasing function of time as shown in Fig. 5.

Figure 5 Aperiodic Motion z > 1

(iii) Critically Damped Motion (z = 1.0) :

(28)

Figure 6 shows three types of response with initial displacement x(0).



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Figure 6 Critically Damped Motion z = 1

Forced Harmonic Vibration

Harmonic excitation is often encountered in engineering systems. It is commonly produced by

the unbalance in rotating machinery. Although pure harmonic excitation is less likely to occurthan periodic or other types of excitation, understanding the behavior of a system undergoing

harmonic excitation is essential in order to comprehend how the system will respond to more

general types of excitation. Harmonic excitation may be in the form of a force or displacement

of some point in the system.

We will first consider a single DOF system with viscous damping, excited by a harmonic force

, as shown in Fig. 7. Its differential equation of motion is found from the free-body

diagram.

(29)

Figure 7 Viscously Damped System with Harmonic Excitation

The solution to this equation consists of two parts, the complementary function , which is thesolution of the homogeneous equation, and the particular integral . The complementary

function. in this case, is a damped free vibration.



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e part cu ar so ut on to t e prece ng equat on s a stea y-state osc at on o t e same

frequency w as that of the excitation. We can assume the particular solution to be of the form :

(30)

where X is the amplitude of oscillation and f is the phase of the displacement with respect to theexciting force.

The amplitude and phase in the previous equation are found by substituting Eqn. (30) into the

differential equation (29). Remembering that in harmonic motion the phases of the velocity and

acceleration are ahead of the displacement by 90° and 180°, respectively, the terms of thedifferential equation can also be displayed graphically, as in Fig. 8.

Figure 8 Vector Relationship for Forced Vibration with Damping

It is easily seen from this diagram that

(31)

and

(32)

We now express Eqs (31) and (32) in non-dimensional term that enables a concise graphical

presentation of these results. Dividing the numerator and denominator of Eqs. (31) and (32) by

k , we obtain :

(33)

and



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(34)

These equations can be further expressed in terms of the following quantities:

The non-dimensional expressions for the amplitude and phase then become

(35)

and

(36)

These equations indicate that the non dimensional amplitude , and the phase f are

functions only of the frequency ratio , and the damping factor z and can be plotted as

shown in Fig 9.



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Figure 9 Plot of Eqs. (35) and (36)

Rotating Unbalance

Unbalance in rotating machines is a common source of vibration excitation. We consider here a

spring-mass system constrained to move in the vertical direction and excited by a rotating

machine that is unbalanced, as shown in Fig. 10. The unbalance is represented by an eccentric

mass m with eccentricity e that is rotating with angular velocity w . By letting x be the

displacement of the non rotating mass (M - m) from the static equilibrium position, thedisplacement of m is :



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Figure 10 Harmonic Disturbing Force Resulting from Rotating Unbalance

The equation of motion is then :

which can be rearranged to :

(37)

It is evident that this equation is identical to Eq. (29), where is replaced by , and hence

the steady-state solution of the previous section can be replaced by :

(38)

and

(39)

These can be further reduced to non dimensional form :



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(40)

and

(41)

Example

A counter rotating eccentric weight exciter is used to produce the forced oscillation of a spring-

supported mass as shown in Fig. 11. By varying the speed of rotation, a resonant amplitude of

0.60 cm was recorded. When the speed of rotation was increase considerably beyond the

resonant frequency, the amplitude appeared to approach a fixed value of 0.08 cm. Determine the

damping factor of the system.

Figure 11

Solution :

From Eqn. (40), the resonant amplitude is :

When w is very much greater than , the same equation becomes



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By solving the two equations simultaneously, the damping factor of the system is

Sound Measurement and Noise control

Section A

Sound Level Meter

A sound level meter is an instrument designed to respond to sound in approximately by the

same way as the human ear and to give objective, reproducible measurements of sound pressurelevel. There are many different sound measuring systems available. Although different in detail,each system consists of a microphone, a processing section and a read-out unit. Figure A1 illustrates a functional schematic diagram of the sound level meter.

Figure A1 Functional Schematic Diagram of a Sound Level Meter

(a) Microphone (b) Processing Section (i) Weighting Networks



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e s gna may pass t roug a we g t ng networ an resu te n erent we g t ngs, say

"B" "C' or "D". Sound level meters usually also have a linear or "Lin." network. This does not

weight the signal but enables the signal to pass through without modification. (ii) Filters The frequency range of the sound from 20 Hz to 20 kHz is divided into sections or bands bymeans of electronic filters which reject all signal with frequencies outside the selected band.

These bands usually have a bandwidth of either one octave or one third octave. (See Figure A2)

The above process of dividing complex sound is termed frequency analysis and the results are

presented on a chart called a spectrogram.

Figure A2 Sound Filtering

(iii) Root mean square detector



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ter t e s gna as een we g te an or v e nto requency an s t e resu tant s gna s

amplified, and the Root Mean Square (RMS) value determined in an RMS detector. The RMS

value is used because it is directly related to the amount of energy in the sound being measured. (c) Read-out Unit The read-out unit displays the sound level in dB, or some other derived unit such as dB(A)(which means that the measured sound level has been A-weighted). The signal may also be

available at output sockets, in either AC or DC form, for connection to external instruments

such as level or tape recorders to provide a record and/or for further processing. Sound level meters should be calibrated in order to provide precise and accurate results. This is

best done by placing a portable acoustic calibrator, such as a sound level calibrator or a Piston

Phone, directly over the microphone (See Figure A3). It is good measurement practice to

calibrate sound level meters immediately before and after each measurement session.

Figure A3 Calibration of Sound Level Meter Types of Sound Level Meter

2.1 Detector Response



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ost soun s t at nee to e measure uctuate n eve . o measure t e soun proper y we

want to be able to measure these variations as accurately as possible. For this reason, two

detector response characteristics were standardized. These are known as "F" (for Fast) and "S"

(for Slow) (See Figure A4). (a) "F" Characteristic provides a fast reacting display response enabling us to follow and

measure not too rapidly fluctuating sound levels. (b) "S" Characteristic provides a slower response which helps average-out the display

fluctuations on an analogue meter, which would otherwise be impossible to read using the "F"

characteristic.

Figure A4 "F" and "S" Detector Response Characteristics



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2.2 Impulse Sound Level Meter The impulse sound level meter (Figure A5) should be used to measure isolated impulses or

sound containing a high proportion of impact noise as the normal "F" and "S" time responses of the simple sound level meter are not sufficiently short to give a measurement which is

representative of the subjective human response. (See Figure A6) Since sound level meters include a circuit for measuring the peak value of the impulse sound,

independent of it's duration as the peak value may cause the risk of damage to hearing. It is

known as a Hold Circuit which stores either the peak value or the maximum RMS value. 2.3 Noise Dose Meters The need to ascertain, for the purpose of occupational hearing conservation, the noise exposure

of employees during their normal working day, has lead to the development of a new type of

specialised integrating sound level meter from which the noise dose can be determined directly.



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Figure A6 "I" Impulse Characteristics of Impulse Sound Level Meter

The noise dose is a measure of the total A-weighted sound energy received by an employee, and

is expressed as a proportion of the allowed daily noise dose. It therefore depends not only on the

level of the noise but also on the length of time that the employee is exposed to it. This

instrument is portable and can be carried in a person's pocket. The microphone can be separated

from the dose meter body and should preferably be mounted close to the individuals more noise

exposed ear. Figure A5 shows a typical application of the dose meter.



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Community Noise

In Hong Kong, with its large population within the limited coastal plains, noise pollution

imposes a severe pollution problem. In order to cater for such a high population density, tall

buildings are erected, and the crowded environment implies that any pollution such as noise will

affect a greater number of people than would a more open environment. In this respect, Hong

Kong is more susceptible than most cities to all forms of pollution.



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eop e n ong ong su er rom many types o no se, o w c a rcra t no se, tra c no se,

construction noise, and industrial noise are the prominent examples. The areas affected cover

nearly the whole of Hong Kong. Other less serious or more localized noises found are air-

conditioner noise, helicopter noise, noise from the railway and mass transit railway, and noise

from power-supply equipment such as transformers. Apart from the above, noises from radio,

television, hi-fi, mahjong, parties, people, and so forth are less serious. However, the persistent

of these noises at night hours can be very disturbing. Equipment in offices and appliances in homes can also be noise producers. Fortunately, the

duration of operation of home appliances is usually not long enough to arouse disgust.

Nevertheless the noises from office equipment can become a nuisance because of long hours of

operation. On the whole, the high population density, the tall buildings and their proximity and economical

design, the lack of open spaces, the narrow roads, and finally the hilly nature of Hong Kong are

the factors which amplify the problems of noise pollution in this place.

Statistical Description of Community Noises

Figure B1 shows the A-weighted sound levels for two different time periods in a typical

suburban environment. These curves show the following characteristics of most community

noises: (a) The noises comprise two parts: a fairly steady 'residual noise level' which comes from

distant, unidentifiable sources, together with some 'discrete' noise events of identifiable origins. (b) The residual noise level varies slowly with time, usually displaying diurnal, or weekly cycle

but with maximum deviations rarely exceeding about 10 dB(A). (c) The individual noise events vary in magnitude and duration, rising as much as 40 dB(A)

above the residual level for seconds, minutes, or even longer.



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Figure B1 Typical Community A-weighted Sound Levels in (a) Daytime (b) Night-time



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Figure B2 Statistical Representation of Community Noise (a) Equivalent Continuous Sound Level, LAeq This is the steady-state A-weighted sound level that has the same acoustic energy as that of the

time-varying sound averaged over the specified time interval. See Figure B3.



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Figure B3 LAeq and SEL LAeq can be estimated from a record of A-weighted sound level verse time by using the

definition :

(1) where L A(t) = instantaneous A-level of sound T = specified time period during which sound is sampled By breaking the sound-level record into n nos. of equal increments of time , equation (1) can

be approximated by :

(2) where L Ai = average A-level over the ith increment of time



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Types of Community Noises

The following paragraphs discuss some of the sources of community noises, with particular

reference to the Hong Kong situations. Ambient Level The day-time ambient sound pressure levels measured in rural and urban areas in Hong Kong

are shown in Figure B4. These values are the minimum levels at the locations shown. The levels

for rural areas, 44 to 55 dB(A), indicate the lowest level which exist in Hong Kong. Regrettably,

the levels in housing estates are high. Depending on the location, the difference between rural

and urban areas shown can be 10 to 20 dB(A).

Figure B4 Some Daytime Ambient Noise in HK

Construction Noise Construction noise in Hong Kong comes from a number of sources, notably the pile driver,

pneumatic drill, air compressor, concrete mixer and other diesel power construction machinery.

Although the area affected by piling noise is far less than that of aircraft, the hazardous effect

sometimes could be higher. This is because of the impulsive nature of the piling noise and the

long duration of operation which may be more than eight hours a day. This means that the noise

dose to people around a construction site may exceed the safety limit and hence may result intemporary and permanent shifting of thresholds of hearing.



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Traffic Noise Hong Kong has a very high vehicle density of about 200 vehicles per kilometers of road. Due to

this high vehicle density, many main r ads in Hong Kong have daytime traffic volume of more

than 3,000 vehicles per hour, and sometimes may reach 4,000 vehicles per hour in the rushhours. Figure B5 shows the time variation of the indoor sound pressure level due to the traffic at

an adjacent road over a 24-hour period.

Figure B5 24-Hour Indoor Traffic Noise Survey Aircraft Noise In Hong Kong, the presence of the airport imposes very severe noise pollution on the highly

populated areas under the flight path. Because of the large number of people severely affected

by aircraft noise, and because of the tremendous costs involved to reduce the impact of this kindof noise, aircraft noise has received more attention than any other environmental noise. Noise Control

A noise usually has influence on two groups of people: (a) The occupants of a site, house or factory (b) Third party people like persons living and working in the locality.



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en ea ng w t pro ems cause y no se, one must cons er t ree erent aspects o ts

effect on people and they are annoyance, communication interference, and hearing-damage risk.

Although there is no single widely accepted standard for indoor noise environments, there have

been numerous studies which provide recommended dB(A) levels for specific situations. Table

Cl lists dB(A) level indoor design goals and NC criteria recommended in the ASHRAE system

guides for steady background noise in various indoor spaces. Duration per day Sound level dBA (slow)

8 90

6 92

4 95

3 97

2 100

1.5 102

1 105

0.5 110

0.25 or less 115

Table C2 OSHA Standard on Permissible Noise Exposure

Table C2 gives the OSHA (Occupational Safety and Health Act) Noise Standards which are

designed to protect workers exposed to hazardous noise environments from incurring permanent

hearing loss. The control of noise abuse in the community is strengthened by the enforcement of the Noise

Control Ordinance No. 75 of 1988. Technical memorandum have been published by

Environmental Protection Department for the assessment of noise from (i) Percussive pilling. (ii) Construction work.



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aces ot er t an omest c prem ses, pu c p aces or construct on s tes. Source-Path-Receiver Concept

A straightforward approach to solve a noise problem is to examine the problem in terms of its

three basic elements: the source, the conveying medium and the receiver (see Figure C1).

Unless a free field is present, both direct and reflected (reverberent) sound waves that reach thereceiver must be carefully looked into. The most appropriate solution to a given noise problem

requires alternation or modification of any, if or all of the three basic elements. For instance: (a) to modify the source to reduce its noise output, (b) to alter or control the transmission path and the environment so as to reduce noise level

reaching the recipient, (c) to provide the receiver with personal protective device. The most appropriate solution is the one that can achieve the desired amount of noise reduction

by using the lowest expenses and causing least inconvenience to the occupant activities.

Figure C1 Noise Control Procedures are Applied to Source, Path and Receiver

2.1 Control of Noise Source by Design The source may be a single or multiple of mechanical devices that emit acoustical energy. Due

consideration on acoustic matter during the design stage of the mechanical equipment will

definitely minimize the noise problem. The improvement techniques include the following: (a) reduction of impact forces, (b) reduction of speed and pressure,



c re uct on o r ct ona res stance, (d) reduction of noise leakage, and, (e) isolation of vibrating elements. 2.2 Noise Control in Transmission Path The next line of defense is to set up devices in the transmission path to block or reduce the flow

of sound energy before it reaches the receiver. This can be done in several ways: (a) to absorb the sound along the path, (b) to deflect the sound in some other directions by placing a reflecting barrier in its path, (c) to contain the sound by placing the source inside a sound-insulating box or enclosure. 2.3 Protecting the Receiver When physical exposure to intense noise fields is unavoidable and none of the measures

mentioned above is practical, then measures must be taken to protect the receiver as a final

resort. The following two techniques are commonly employed. (a) Alter work schedule In order to limit the amount of continuous exposure to high noise levels, it is preferable to

schedule an intensive noisy operation for a short'. interval of time each day over a period of

several days rather than a continuous 8-hour run for a day or two. Inherently noisy operations, such as street repair, factory operation and aircraft traffic should be

curtailed at night or early in morning.

(b) Ear Protection Ear plugs and other ear protectors are commercially available. They may provide noise

reductions ranging from 15 to 35 dB. It should be aware that protective ear devices do interferewith speech communication and can be hazardous when warning calls from a routine part of the

operation.

vibrations, sound measurement and noise control

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