KTH Sound&Vibration BOOK

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Sound and Vibration

n

S

und Source

This material is protected by copyright and cannot be used or reproduced in any form without the written permission from:

The Marcus Wallenberg Laboratory, KTH, SE-100 44 Stockholm, SWEDEN

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Program JEP-31018-2003 based on the Swedish book “Ljud och Vibrationer” used at KTH.

Chapters 1-12 are translated from “Ljud och Vibrationer” by Hans Bodén, Ulf Carlsson,

Ragnar Glav, Hans-Peter Wallin, and Mats Åbom by Robert Hildebrand

Chapters 13, 15 and 16 are written by Hans Bodén Chapter 14 is written by Ulf Carlsson and Hans Boden Chapter 17 is written by Mats Åbom

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CHAPTER ONE

INTRODUCTION

Acoustics can be regarded as the science of sound and vibration. Sound today refers not only to those mechanical wave motions in air that give rise to sensations of hearing, but even to low-frequency (infrasonic) and high-frequency (ultrasonic) motions that cannot be sensed by hearing, as well as analogous wave motions in, for example, water (underwater acoustics). In solid materials, one speaks instead of vibrations or structure-borne sound.

The phenomenon of hearing has fascinated mankind all through the ages. The mathematical theory of sound propagation can be said to have begun with Isaac Newton (1642 - 1727), whose work Principia (1686) contained a mechanical interpretation of sound as pressure pulses propagating in a medium. A more solid mathematical and physical groundwork of theory was provided by Euler (1707 – 1783), Lagrange (1736 – 1813), and d’Alembert (1717 – 1783). That development took place just as continuum mechanics and field theory began to take form, and the wave equation was formulated for functions of space and time. The modern theory of sound and vibrations is the product of the efforts of these mathematical physicists.

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1.1 THE FIELD OF SOUND AND VIBRATION

In today’s well-developed, technological society, the number of systems that emit sound and vibrations is steadily increasing. Examples are machines, vehicles, and processes of all types, in which driving forces and engine power are constantly. Examples are machines, vehicles, and processes of all kinds, in which driving forces and engine power are continually being increased even as simultaneous efforts are made to hold down weight and materials usage. This implies the need for an ever more intensive research and development effort to identify and alleviate noise and vibration disturbances, and satisfy the needs of mankind for an acceptable environment.

The fundamentals of Sound and Vibrations are part of the broader field of mechanics, with strong connections to classical mechanics, solid mechanics, and fluid dynamics. The subject of Sound and Vibrations encompasses the generation of sound and vibrations, the distribution and damping of vibrations, how sound propagates in a free field, and how it interacts with a closed space, as well as its effect on man and measurement equipment.

Technical applications span an even wider field, from applied mathematics and mechanics, to electrical instrumentation and analog and digital signal processing theory, to machinery and building design. Several of the more important areas are worthy of particular mention:

Figure 1-1 Acoustics spans such

diverse areas as biology, art, the natural sciences and technology. The proposed subdivision of the field (adjacent) covers, accordingly, many aspects and fields of science. The unshaded area describes the activity at

a typical university sound and vibration laboratory. (Source: From

R.B. Lindsay, who published the "acoustic wheel" in the Journal of the Acoustical Society of America 1964, vol 36)

NATURAL

SCIENCES TECHNICAL _SCIENCES

LIFE

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Machinery and Vehicle Acoustics deals with constructive measures to bring about

machines, vehicles, and processes that are quieter and vibrate less; that requires knowledge of how sound and vibrations are generated, and how that generation relates to such physical parameters as flow velocities, masses, stiffnesses, losses, and geometry, for example. The most common reason that sound arises, in technical applications, is that a time-varying force excites vibrations of a mechanical structure, which then radiates sound. We can convince ourselves of that by knocking on a tabletop. The surrounding air is influenced by the vibrating structure, and responds with contractions and expansions. The mechanical form of energy we call sound has arisen. Analysis of the mechanisms of sound generation helps us to answer the question of why it sounds differently when we hit a tabletop with the soft part of the index finger, than it does when we hit it with the steel tip of a ball-point pen, and why it is louder when we knock the middle of the table with our knuckles than when we do the same thing at the edge of the table.

The dull rumble of a Harley – Davidson motorcycle has become a defining characteristic – so important that the factory sought a patent on the sound to prevent competitors from plagiarizing it. That cash register click of a Mercedes door closing shut is something that other auto manufacturers strive for. The concept of Sound Quality is becoming ever more entrenched. Sound should convey a sense of product quality and reliability. The automobile industry has made pioneering efforts in this area and sound quality is given a high priority, along with such other important vehicle characteristics as road performance, safety and design. To describe sound, such subjective designators as “sharpness”, “rawness”, and “boxiness” are used. Several manufacturers have developed software packages that make it possible to modify the sound of products in order to mimic different design variations. The modified sounds are then judged by a “panel of listeners”. In the future, many consumer, industrial, and transportation products will be “sound-designed”.

Force

Vibrations

Acoustic radiation

Source Response Acoustic radiation

Technology Model Mathematical structure Three stages Examples: Geometries Masses Stiffnesses Losses Surface area

Mode shape (oscillation pattern) Resonances

Structure – air coupling Roughness forces

Gear tooth forces Mechanical imbalances Aerodynamic forces

Force

Roughness

Figure 1-2 Forces that cause

vibrations appear in countless situations. It can be a matter of a roller in a bearing exhibiting out-of-roundness, or perhaps small surface irregularities at the contact between a railway wheel and rail. The ability of the forces to induce vibrations depends on mechanical laws in which such parameters as the structure’s geometry, mass distribution, stiffness and losses come into play. The structure’s effectiveness as an acoustic radiator depends on the surface area, its oscillation pattern, and the coupling between the structure and the air at the surface of separation. Sound generation in connection with mechanical structures can be described in three stages:

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pj Z v Sij i i Z Y F Sij ki k i k L i N i N = = = = =

∑

1 1 1 Figure 1-3 In a machine as

complex as a car, the three stages of sound generation occur in many parallel chains. The velocity v at a specific point, induced by a force F at another specific point, can be described by a so-called frequency response function Y. Similarly, other frequency response functions Z can describe the relationships between the velocities on the surfaces and the sound pressure

p at an interior point. The

frequency response functions can, moreover, be combined to

describe all three stages in the propagation chain. By adding the contributions from all significant forces, via the dominant radiating surfaces, the total sound pressure at a location of interest in the interior is obtained. The noise in the interior comes mainly from three sources:

• The driveline, i.e., the engine, transmission, and drive axles. • The contact zone between the tires and the roadway. • Airflow over the car body.

Sound passes from the source into the passenger compartment in two distinct ways: as structure-borne and as air-borne sound. Structure-air-borne sound has essentially propagated in the form of vibrations from the source to the receiver, whereas air-borne sound had already radiated as sound, e.g., in the engine compartment, before its transmission into the passenger compartment (Sketch: Volvo Technology Report, nr 1 1988)

Forces Car Vibration i l Passenger Sound pressure

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Flow Acoustics is the study of the direct generation of sound in an elastic gaseous or

liquid medium. The most common sound generation mechanisms are volume, force, and moment fluctuations in the medium, throughout which the elastic energy then spreads. Such phenomena are of great significance for noise from, for example, propellers, jet engines, fans, and vehicles, as well as for the propagation of sound in ducts.

Figure 1-4 Volvo’s wind

tunnel is used to find ways to reduce flow-induced sound. Mitigating such sound typically requires an even flow over the body, as well as fine tolerances and good fits, especially at doors and other seals. Airflow over projecting details can cause whistling. The small-scale turbulence around the car body gives a roaring type of noise, and if there are openings an airstream can sometimes create whistling sounds (Photos: Volvo Technology Report, nr 1 1988)

Room Acoustics, or Building Acoustics, deals with how sound fields are built up in

various types of rooms and enclosed spaces, as well as how sound is transmitted through different types of structures, such as walls and systems of joists.

Figure 1-5 Heavy

insulating surfaces effectively mitigate sound transmission from one enclosed space to another, as from, for example, the engine compartment to the cabin of a truck. For vehicles such as ships, cars, trains and aircraft, however, low weight is also a critical demand. Enormous efforts are therefore made in the transportation industry to bring about light, yet effective, designs for sound reduction. In the

picture, the roof of a passenger car has been mounted in the opening between a sound reflective “reverberant room” at MWL, KTH (seen), and an adjacent sound absorbing ”anechoic room” (not seen). Using microphones and mathematical models for the sound interaction with the measurement rooms, the sound insulating properties of the car roof can be deduced (Photo: HP Wallin, MWL)

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Signal analysis is an important part of the science of sound and vibrations. The associated

experimental methodology is primarily based on transducers, such as microphones and accelerometers, that convert sound and vibrations into equivalent electrical signals for further analysis. Until the 1960’s, most acoustic measurement instrumentation was analogue; then, digital computer technology arrived on a broad front. A numerical algorithm, FFT (Fast Fourier Transform), that transformed signals from the time domain (signal strength as a function of time) to the frequency domain (signal strength as a function of frequency), was a breakthrough in experimental methodology.

The Sound and Vibration subject area offers a wealth of opportunities for both the theoretically-inclined and the more practically / experimentally-oriented who are interested in applying themselves towards the improvement of machinery and vehicle designs, or towards quieter workplaces and societal environments.

Figure 1-6 To solve a

noise problem, in which roof vibrations are suspected as the direct source, the car is driven on a chassis dynamometer.. The roof motion is registered as a function of time with the help of a motion sensor - an accelerometer. Transforming the signal to the frequency domain, i.e., expressing its strength as a function of frequency, often greatly facilitates interpretation. Such a transformation is carried out by electrical filters or by a computer. That broad subject is called signal analysis. Its mathematical foundation is derived from mathematical statistics and from Fourier methods. In the case illustrated, a poorly made transmission is to blame. A shaft in the transmission, rotating at 1800 revolutions per minute, is bent or out-of-balance, giving rise to a dominant vibration disturbance at 30 Hz.

(Sketch: Brüel & Kjær, Structural Testing, Part 1)

Signal analysis Time Amplitude

Signal

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Figure 1-7 The strength of sound is often given in dB. A change of a single dB is hardly noticeable. An increase

of 8-10 dB is perceived as a doubling of the degree of disturbance. In order to adjust for variations in the sensitivity of the ear at different frequencies, corrected values are often used. The result is called Sound Level, and indicated in units of dB(A). (Diagram: ASF, ”Bullerbekämpning” [in Swedish], 1977, Illustrators: Anette Dünkelberg /Arne Karlsson)

1.2 JOB MARKET FOR SOUND AND VIBRATION ENGINEERS

The task of minimizing sound and vibration in mechanical constructions falls primarily upon those who are responsible for their design and analysis, i.e., primarily engineers with mechanical, vehicle, and machinery backgrounds.

Such issues have a great deal of relevance these days, and industry is therefore preoccupied with them. It is usually larger manufacturing companies that have the greatest need for engineers in this field. The vehicle industry employ many sound and vibration engineers but also other industrial sectors that manufacture products in which some type of energy conversion takes place (compressors, separators, turbines, fans, appliances, etc.) have a great need for this kind of knowledge.

There are, moreover, many consulting firms in the field. Others that have a need for specialists in this area include national and local authorities that have regulatory responsibilities, as well as national research organizations that have the task of dproducing new knowledge.

Normal conservation

Train 100 meters away

(100 km / h) Eccentric press Pain threshold

Manual grinding Air-cooled electric motor 50 kW Spray painting Weakest

perceivable Sound

Quiet bedroom Impact

riveting Coarse grinding Chainsaw

Near a jet plane starting up Highest sound level that can be attained Sound insulated lounge

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1.3 DEVELOPMENT

Figure 1-8 Swedish billboard demonstrating the marketing value of acoustic performance, describing the

“world’s quietest dishwasher” as “Unbelievably unhearable”) (Photo: HP Wallin, MWL)

Classical acoustics was summarized by Lord Rayleigh in his fundamental treatise "The theory of sound" 1894-96. That classical theory serves as the basis for the modern science of sound and vibrations. Lord Rayleigh received the Nobel prize in physics in 1904 for his discoveries in the field of optics.

The field of noise control engineering started to develop in in the 1930’s. Development on a larger scale came in thethe 1950’s in connection with for instance the effort to build quiet submarines and civil jet engine airplanes. The knowledge acquired spread to other areas of machinery acoustics. In Sweden Atlas Copco was one of the

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Today, technical specifications on sound and vibration performance are made for a large share of the products that are delivered by manufacturing industry. The EU has brought about further demands on products to be sold in Europe, with respect to sound and vibration performance. Quality oriented manufacturers consider quiet, vibration-free products as a crucial quality characteristic and a marketable feature, so that all larger manufacturers tend to have departments responsible for sound and vibration performance. In the future, we will see improved and more easily used computer-based computational tools, and a continued development of cheap, personal computer-based systems for measurement and diagnostics. An example of the development of new technology is the so-called active sound and vibration control. The method is based on the use of, for example, microphones registering a disturbing sound field. Making use of automatic control methodologies, an “anti-noise” (phase inverted sound) is emitted that, under some conditions, can eliminate the disturbing field in an effective way.

Commercial systems of that type exist already in, for example, ventilation systems and for automobile interiors and airplane cabins. In the future, we can expect corresponding systems on the vibration side as well. Today, vibration isolators, primarily of rubber, are inevitably used to prevent engine vibrations in an automobile from spreading into the car body. By installing parallel electrodynamic vibrators that generate reversed-phase forces, those disturbing forces that manage to pass through the isolators can be counteracted.

Kontrollenhet

Varvtalsgivare

Signal från motor Signal från motor

Högtalare Mikrofon

Figure 1-9 Cross section of the cabin of a turbo-prop SAAB 340 airplane with active noise control. The

microphones register the sound at a number of points and transfer the electrical signals to the control unit. The signals can be treated to drive the speakers in such a way (”antiphase”) as to reduce the noise level in the cabin. The tachometer signals from the engines help to fine tune the automatic control in the control unit. (Source: Ny Teknik)

Signal From motor Signal From motor

Tachometer Controller Microphone Loudspeaker

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1.4 PRINCIPLES, EXAMPLES AND COUNTERMEASURES

The following pages give some examples of principles and applications in the field of sound and vibrations. On the left side, the underlying principles are presented, and on the facing right side, examples and countermeasures. Source: Asf, Bullerbekämpning (in Swedish), 1977.

Rapidness of a process determines the high frequency content

Figure 1-10a (Sketch: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson).

The faster that force, pressure, or velocity changes occur, the higher the frequency content of the resulting sound. A fast ping pong ball gives a high frequency pop when it hits the table, while a slow handball bounces against the floor with a dull, low frequency sound.

Principle

Slow impact against the floor – low frequency sound

Rapid impact against the table – high frequency

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Figure 1-10b (Sketch: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson).

Sound level

Angular gear teeth

Rounded gear teeth High-pitched Low-pitched tones

Example - Countermeasure

In the primitive tooth configuration, the teeth slap together so that the forces between them rise and fall rapidly. The high-pitched tones are then strong. The more effort is made to modify the tooth configuration, the more softly they can be made to mesh together. Finally, the forces can be made to rise and fall again slowly. The high-pitched tones are then no longer so dominant. Since the peak force

Force on a tooth Angular gear Rounded gear teeth Force on a tooth Time Time

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Make vibrating surfaces as small as possible

Figure 1-11a (Sketches: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson).

An object with small surfaces can vibrate very strongly without radiating a great deal of noise. The lower the frequency of the disturbing tones, the greater the surface area before it becomes a disturbing noise source. Since there is practically always a risk of vibrations when dealing with machinery, the shells and housings used should be as small as possible.

Principle

The electric shaver vibrations are transmitted into the large glass shelf and the resulting noise level is high.

The vibrations are no longer transmitted and the noise diminishes.

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Figure 1-11b (Sketch: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson). Oil tank Pump Motor Instrument panel Instrument panel moved to the wall

Example:

The hydraulic aggregate was a powerful noise source. Since the wall vibrations of the oil tank were damped by the oil itself, most of the noise was radiated by the instrument panel.

Countermeasure:

The panel was separated from the aggregate,

reducing the radiating surface area, and thereby even the noise level.

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Densely perforated sheet emits little noise

Figure 1-12a (Sketch: Asf, Bullerbekämpning, 1977 Illustrator: Claes Folkesson).

Large vibrating shells cannot always be avoided. They give off a lot of noise. The reason for their sound radiation is that the vibrating sheet pumps the air to and from, like the piston of a pump. If the sheet is perforated, then it “leaks” and the pumping effect is weakened. The reduction in sound radiation is many times greater than the mere reduction in surface area. Alternatives to perforated sheet are nets and gratings.

Sound level Principle

Perforated sheet metal Unperforated sheet metal

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Shield over drive belt and flywheel, of unperforated sheet

Perforated sheet

Example:

The shield over the flywheel and drive belt constituted a strong noise source. The shield was fabricated of unperforated sheet metal.

Example:

The shield over the flywheel and drive belt constituted a strong noise source. The shield was fabricated of unperforated sheet metal.

Countermeasure:

A new shield was fabricated from perforated sheet and wire netting. One of the noise sources of the press was thereby eliminated.

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Wind tone is removed by a changed profile or a small disturbing element

When sound flows past an object at certain speeds, a strong pure tone called a Strouhal tone can arise. By extending the dimension of the object in the direction of the airflow, with a “tail” for example, or by disturbing the regularity of the object profile, the tone can be prevented. In a duct, a resonance can amplify a Strouhal tone so much that the duct can be damaged.

Air Flow

Regular pattern of vortices gives a strong tone

Length extension Small disturbing objects

Irregular vortex pattern _{Irregular vortex pattern}

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Wind

Sheet iron spiral Chimney

Example: The shield over the

flywheel and drive belt

constituted a strong noise

source. It was fabricated of

unperforated sheet metal.

Countermeasure: A new shield

was fabricated from perforated

sheet and wire netting. One of

the noise sources of the press

was thereby eliminated.

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Pure tones can be cancelled with sound in anti-phase

L2

L1

When the sound only consists of a single tone, or several such within a narrow frequency band, it can be completely or partially cancelled out in an interference muffler. It consists of a branched duct in which sound propagates through two separate paths that subsequently recombine with different time delays. In its simplest variation, shown in the figure, the path difference L1-L2 determines the frequencies at which sound reduction

occurs. The time-delayed sound behaves as if in anti-phase.

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Figure 1-14b (Sketch: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson).

Interference muffler

Inlet

Outlet Branches of varying lengths

Example:

When the frequency of the disturbing tone, or the temperature of the gas, varies in time, the effective frequency band of the muffler can be widened by a variation of the path length difference through multiple paths. The improvement obtained at the nominal frequency is, however, somewhat less than in the preceding variation. The interference damper is suitable for use

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The ASU Sound & Vibration Lab. At the Faculty of Engineering, Ain Shams University

The Sound and Vibration Laboratory at the Faculty of Engineering, Ain Shams University was established in 2004 by a European Union Grant (JEP-31018-2003) within the TEMPUS Program. This is the first laboratory in Egypt specialized in sound and vibration teaching and research. The establishment of the laboratory and the development of the courses were done in cooperation with:

- The Marcus Wallenberg Laboratory for Sound and Vibration Research (MWL) at the Royal Institute of Technology (KTH) in Sweden.

- Institute of Sound and Vibration Research (ISVR) at the University of Southampton in United Kingdom.

Our mission is to produce a new generation of Egyptian engineers and researchers in the field of sound and vibration. ASU-SVL welcomes research institutions, industry and others to cooperate with our staff and use the laboratory facilities.

The laboratory hosts the Acoustical Society of Egypt, whose mission is to group all engineers, researchers and students in Egypt interested in the field of sound and vibration. ASU-SVL has joined the X3-Noise Aircraft External Noise Network in Europe acting as the regional focal point for MEDA Countries in this network. X3-Noise is funded within the Sixth Framework Programme.

The activities in the lab are divided into three main domains: education, research and consultancy.

The lab is equipped with the state-of-the art sound and vibration measurement equipment including different types of analyzers, transducers, calibrators, and accessories. There is an anechoic room (80 m3), a reverberation room (80 m3), and an acoustic flow facility. There is a teaching lab of 13 computers working as data acquisition systems to be used in lab exercises in different courses.

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CHAPTER TWO

FUNDAMENTAL CONCEPTS

The aim of this chapter is, partly, to give an overview of the subject area of sound and vibrations, before the more detailed descriptions that follow, and partly to present and define a number of important concepts early enough that technically interesting problems may be treated in parallel with the main presentation.

In the introduction, we shall relate the subject of sound and vibration (sometimes expressed as "vibroacoustics") to the mechanical sciences as a whole, describe what types of mechanical elastic waves can arise, as well as the conditions for their existence, and define important quantities which specify sound and vibration fields. In the section “Diffraction”, we will examine the conditions under which a sound field can bend around different objects. Using models from the subject of room acoustics, we will make a short preliminary survey of different methods for analyzing how sound propagation and sound fields interact with various types of rooms.

Sound and vibration disturbances can only be tracked as functions of time, but are often more effectively analyzed and characterized as functions of frequency. Important concepts, such as the time and frequency domains, are defined, and filtering and frequency analysis are described.

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2.1 BASIC AND APPLIED MECHANICS

The science of sound and vibration is a part of applied mechanics. This can be subdivided as in Figure 2-1.

Figure 2-1 Proposed subdivision of mechanics.

In statics, systems are examined in equilibrium states, i.e., the object considered is neither accelerated nor decelerated, and often has, in practice, no velocity. In particle mechanics, one studies the motion of the center of gravity of an object. The object / particle has three degrees of freedom, i.e., translations which can be described in a Cartesian coordinate system. In order to describe rigid body movement, six degrees of freedom are needed: three translations and three rotations.

In applied mechanics one studies the behavior of solid, liquid and gaseous media, including deformation, wave propagation, tension, and fracture, when these media are subjected to physical agents of various types, such as forces and temperature shifts. In strength of materials one studies strain and deformation properties with the objective of properly dimensioning a construction for good strength and stability. In fluid mechanics one describes various aspects of the motions of, above all, fluids and gases. A typical problem from this field would be to improve the lift of an airplane wing, for example, or reduce the head loss (pressure drop) across a ventilation duct. Vibroacoustics covers everything from the basics of how sound and vibrations are generated and how they propagate to the question of how vibrations in a solid structure, a combustion engine for example, give rise to acoustic radiation.

The division of the fields of mechanics as in Figure 2-1 is a mere simplification which can be adjusted and supplemented in many ways. A model from particle mechanics can, for example, be applied in vibroacoustics to explain the appearance of a certain tone when we blow over the mouth of a bottle, as in Figure 2-2.

Figure 2-2 The tone that arises

when we blow over the bottle opening can be determined by methods from particle mech-anics. The phenomenon is call-ed a Helmholz resonator. It was already employed in the amphitheaters of ancient Greece where clay pots were used as resonators, see chapter 10. (Sketch (far left):Asf, Buller-bekämpning, 1977, Ill: Claes Statics _{m echanics}Particle bodyR igid

m echanics

Strength of M aterials Fluid M echanics

V ibroacoustics

Fundam ental M echanics A pplied m echanics

Mass Force

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Figure 2-4a A loudspeaker generates

longitudinal waves in which particles oscillate parallel to the direction of wave propagation.

Figure 2-4b The hand shaking the rope generates a

transverse wave in which particles oscillate perpendicular to the direction of wave propagation.

2.2 DEFINITION OF SOUND AND VIBRATION FIELDS

Sound and vibration waves are mechanical elastic waves, and thus the conditions for their existence are that the medium possess mass and elasticity. If a mass particle is displaced from its equilibrium position, the elastic forces will seek to return it to its original position. The particle influences the surrounding particles and, in this way, a disturbance propagates in the medium. Sound waves transport relatively little mechanical energy; thus, as is familiar to all from everyday life, speech is not a particularly exhausting activity. A speaking person sends out only a few thousandths of a Watt. Nobody is ever going to be able to cook potatoes by yelling at them. This means, on the other hand, that machines driven at high power levels have almost unlimited capacities to excite acoustic fields; small flaws can convert a share of the available mechanical power into acoustic disturbances. In vibroacoustics there are two classes of waves: longitudinal waves and transverse waves. Longitudinal waves have a particle motion which is parallel to the direction of wave propagation, see Figure 2-4a, while transverse waves have a particle motion which is perpendicular to the direction of propagation as in Figure 2-4b. These shall be described in further detail in later chapters.

Particle Displacement

Wave Propagation

Wave Propagation Particle displacement

In a gas or a fluid, at a certain point in space rG and time t, an acoustic wave can be

described by the acoustical field quantities sound pressure p( trG, )[Pa] and particle velocity uG( trG, ) [m/s]. Sound pressure variations are normally very small deviations of the

ambient pressure, and the particle velocity is small compared to the speed of sound. Figure 2-3 A disturbance in water on a pond gives rise

to a radial wave pattern. These water (or gravitation) waves are nevertheless driven, in contrast to sound and vibration waves, by a balance between the water’s inertia and gravitational effects. (Photo: Klas Persson)

pt _{Sound Pressure, p}

Atmospheric Pressure p0

5

Figure 2-5 An acoustic field ordinarily

implies only small disturbances. Normal speech at a distance of several meters, for example, gives a sound pressure p of a few hundredths Pa superimposed upon atmospheric pressure p, which is about

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2.3 PEAK VALUE, MEAN VALUE, RMS-VALUE, AND POWER

Before going any further, we recall some formulas from electrical engineering which are used to characterize signals, for example sound pressure p(t). A harmonic signal is a signal which can be described by a sine or cosine function as

p(t) = pˆsin(ω t + ϕ), (2-1)

where p(t) is the instantaneous, time-dependent sound pressure,

pˆ is the peak value or simply the amplitude,

ω = 2πf is the angular frequency, ϕ is the phase angle.

The time-averaged mean value of a signal, marked by an overbar, is dt t p T p T

∫

= 0 ) ( 1 , (2-2)

where T [s]is the time over which the average value is determined. The root mean square or RMS value is marked by p~ and is defined according to

∫

= T dt t p T p 0 2₍₎ 1 ~ _. _(2-3)

The rms-value is a very important, and oft-used, form, since it gives information about the time average of the signal power content.

T t peak-peak (t) p ~ = 2 ^ p p p ^ p

The relation between the peak value and the rms-value for an arbitrary signal is called the peak factor or crest factor

p p

TF = ˆ ~ . (2-4)

For a harmonic signal, the following relationship applies between the rms-value and the peak value:

2 ˆ

~ _p

p= . (2-5)

Figure 2-6 The amplitude

characteristics of a harmonic signal, of sound pressure for example, can be described in various ways. The period T is the time required for one complete oscillation.

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Exercise 2-1

Show that equation 2-5 is valid for a harmonic signal.

n

S

W

Sound Source

The instantaneous mechanical power is the product of the instantaneous force and the instantaneous velocity. In acoustics, consequently, the instantaneous acoustic power

) , ( tr

W G [W], which is transported through a surface S with a normal vector nG, as in Figure

2-7, is

∫

= S dS n t r u t r p t r W(G, ) (G, )G(G, )G . (2-6)

In acoustics, we are typically interested in the time average of power quantities W(rG), see chapter 3.

2.4 LONGITUDINAL WAVES IN GASES AND FLUIDS

In gases and fluids, shear stresses are, as a rule, small enough to be neglected. Consequently, only longitudinal mechanical elastic waves can exist in such media. Longitudinal waves are characterized by a particle velocity which is parallel to the direction of wave propagation (see Figure 2-4a). Two cases are considered here for longitudinal waves: plane waves and spherical waves. More complicated wave fields can be constructed from these simple special cases.

2.4.1 Longitudinal plane waves

Longitudinal plane waves are characterized by the condition that points with the same acoustical state, i.e., the same sound pressure and particle velocity, form parallel planes. Figure 2-8 shows the acoustic field of an infinite duct with a harmonically (i.e., sinusoid-ally) oscillating piston at one end.

Direction of

Propagation

λ

Harmonic vibrational velocity

v(t) = v sin(2 ft). π

Figure 2-8 A harmonically oscillating

piston in an infinitely long duct gives rise to a plane longitudinal acoustic wave that propagates in the duct. The frequency of the wave is the same as the piston oscillating frequency. The quantity 2π f is the so-called angular

frequency, designated by ω. The distance λ between two planes in the medium with the same acoustic state is the wavelength in the medium.

Figure 2-7 The acoustic power W over

an area S in an acoustic field is the product of the sound pressure, particle velocity, and area.

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When the piston begins to move forward in the cylinder, it compresses the air before it. That compression propagates at a speed completely independent of the particle velocity, hereafter referred to as the disturbance propagation speed c [m/s]. After a half period, the piston moves in the opposite direction and creates an expansion in the medium, which also moves at the speed c through the medium. After period T [s], the disturbance will have propagated the distance λ [m], so that the following elementary relation applies

f c cT = = λ (2-7) or λ f c= , (2-8)

where c is the disturbance’s propagation speed or the sound speed, f is (the disturbance’s) frequency,

λ is the wavelength.

In air at normal pressure and temperature, c ≈ 340 m/s.

For the case of a free plane longitudinal wave, i.e., wave propagation without reflections, there is a very simple relationship between the field quantities sound pressure p and particle velocity ux as well as the time average of the sound power W . Sound

pressure and particle velocity are always in phase, i.e., they attain their respective maxima and minima simultaneously. The relation between them can be expressed

) , ( ) , (xt 0cu xt p =ρ _x , (2-9)

where ρ0 [kg/m3] is the density in the undisturbed medium (in air at normal pressure and

temperature ρ0 ≈ 1.21 kg/m3) and

c is the sound speed.

The time-averaged sound power W becomes, according to (2-2)

dt t W T W T

∫

= 0 ) ( 1 _{. (2-10)}

Putting (2-6) and (2-9) in (2-10) gives

dt c S t x p T W T

∫

= 0 0 2₍ _, ₎ 1 ρ , (2-11)

which, in accordance with (2-3), gives

c S p

W =~2 ρ₀ . (2-12)

If we study the sound power per unit area, the so-called sound intensity IG [W/m2], the

time-average of the x-component is obtained as

c p S W

I_x = =~2 ρ₀ . (2-13)

Thus, for a free plane wave, the time average of the sound intensity is proportional to the square of the rms-value of the sound pressure.

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In linear vibroacoustics, in more general terms, the time-averaged sound power is proportional to the square of the rms-amplitude of the relevant field quantity, as

2 2 ~ ~ _p p C W = ∝ . (2-14)

This important relation implies that if the constant of proportionality C is known, we can determine the time-averaged sound power from a measurement of the rms sound pressure alone, using a pressure-sensitive microphone. That is usually considerably simpler than determining the particle velocity and making use of (2-6) directly. For a free plane wave, C = S/ρ0c according to (2-12). Note that if the losses in the medium are neglected, and

since the wave doesn’t spread through an expanding volume, then time-averaged quantities are independent of distance to the source, i.e., time-averaged sound intensity and sound pressure amplitude are independent of spatial position.

2.4.2 Spherical and cylindrical waves

In the preceding section on plane waves, we were able to show that when the wave does not suffer losses, certain quantities remain independent of the distance to the source. If the source is an arbitrarily placed sphere and all points on its surface oscillate radially at the same amplitude and phase, or if the source radius a is small compared to the sound wavelength, then the source will produce spherical waves, as shown in Figure 2-9.

The mechanical power W emitted into the medium by the pulsating sphere spreads over an

ever-expanding spherical area (fig 2-10); thus, the time-averaged sound intensity is

2 4 r W I_r = π . (2-15) r 2r 3r

λ λ

a

λ λ

Figure 2-9 A spherical source which

oscillates at the same amplitude and phase over its entire surface area gives rise to spherical wave propagation.

Figure 2-10 In spherical wave propagation, sound power is divided over an ever-increasing area. The intensity decreases to one fourth its original value for a doubling of the distance to the source, and to one ninth when the distance is tripled.

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Examples of real phenomena giving rise to spherical wave propagation are loudspeakers and the outlets of exhaust pipes, provided that the dimensions of the source are small compared to the wavelength, i.e., at sufficiently low frequencies as implied by (2-8). As is evident from Figure 2-9, the curvature of the wave fronts decreases with increasing radius. For engineering purposes, the waves can be considered (locally) plane for radii r > λ / 3, and (2-13) is then applicable even for this spherical wave case. The rms sound pressure can then be expressed in the form

2

0 4

~ _c_W _r

p= ρ π . (2-16)

If the source is, instead, and infinitely long cylinder, the entire surface of which oscillates with a uniform phase and amplitude, then cylindrical waves arise; see Figure 2-11.

r 2r 3r S 2S 3S

The mechanical power, often given in units of power per unit length in the case of a line source, is distributed over a cylindrical area, so that the sound intensity can be expressed

r W

I_r = '2π , (2-17)

where W' [W/m] is the sound power per unit length. Examples of sound sources that can be regarded as cylindrical are electrical distribution cables, pipes, ducts, transport belts at breweries, and heavily trafficked roads. A pre-requisite is that the distance from the source be small compared to the length of the source.

2.5 DIFFRACTION

Diffraction takes place in all types of wave propagation. Water waves are not noticeably affected by the presence of a thin mooring post in the water. The waves roll on as if the post did not exist. On the other hand, we know that behind a break wall in a protected harbor, or behind a narrow spit of land jutting into the water, a shadow zone free of waves develops. The relationship between the size of the hindrance and the wavelength determines how the wave motion is bent about it.

Visible light has an approximate wavelength of 10-7_{m, whereas typical human}

speech has a wavelength of about 1 m. For this reason, we can hear someone speaking from behind a pole without being able to see her at the same time. Thus, wave motions with wavelengths large in relation to the obstacle are little influenced by it and spread

Figure 2-11 An infinitely long cylindrical

source oscillating with uniform phase and amplitude over its entire surface gives rise to cylindrical waves.

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Exercise 2-2

Go to an open area, without any reflecting objects, such as buildings and the like, in the vicinity. Ask a companion to stand about 5 meters away, facing away from you, and take turns voicing an extended

(i) ooooooo... (“oo” as in “food”, not “foot”) (ii) ssssssss...

Note: the oo-sound has a frequency of about 250 Hz and the s-sound is a hiss with a frequency around 6000 Hz. Compare the wavelengths with the diameter of the head. These diffraction phenomena, as well as the

persistence of free plane waves as planar and of free spherical waves as spherical, can be explained with the help of Huygen’s principle: "Every point on a wave front can be described as the center of a secondary wave field, a so called elementary wave. The new position [a moment later] of the wave front is the tangent to the set of these elementary waves ", see Figure 2-12.

In Figure 2-13a, it is seen that at such low frequencies that the wavelength is large in relation to the opening in the wall, Huygen’s principle implies a spherical wave propagat-ion from the opening. If the frequency is high, and the wavelength thereby small in relatpropagat-ion to the opening, as in Figure 2-13b, then the wave propagates as a beam with shadow zones on each side of it.

a) b)

Figure 2-12 Huygen’s principle can be used to show that free plane waves remain plane, and spherical waves remain spherical as they propagate. In a), the tangent to the so-called elementary waves is seen to build a plane wave front, and in b) the corresponding principle for spherical waves is shown.

λ λ λ Shadow zone a) b) Spherical propagation λ Shadow zone

Figure 2-13 a) shows sound transmission through a hole, small in comparison to the sound wavelength. From the opening, spherical wave propagation occurs. b) shows the case of hole that is large with respect to the wavelength. In the middle, the wave passes relatively unhindered, while shadow zones are built up at the sides.

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Exercise 2-3

Noise from roads and railways is an enormous problem. A common attempt at a solution is to install noise walls or barriers.

How should noise barriers be placed?

a) Is it in the high or the low frequency range that noise barriers can be expected to provide the greatest benefit?

b) Is the greatest effect obtained from the barrier when the source is near or far from it? c) Where should the receiver be located with respect to the barrier, to receive the greatest benefit?

Figure 2-14 For highways with average vehicle speeds exceeding 70 km/h, the dominant noise is that generated at the contacts between tires and the road surface.

2.6 ROOM ACOUSTICS MODELS

In many situations, sound interacts with a closed space such as rooms in dwellings, concert halls, or industrial facilities, but even automobile or railway car interiors, or ship or airplane cabins. Knowledge of sound propagation and sound fields in such spaces is therefore an essential part of acoustics. When sound from a source reaches one of the room’s bounding surfaces, a share of the sound power is reflected back into the room and a share absorbed by the wall. In room acoustics, three distinct methods or models are used to describe sound propagation and the sound fields that arise.

At low enough frequencies that the wavelength of sound is of the same order of magnitude as the dimensions of the room, wave theoretical room acoustics is a powerful tool. It turns out that when some of the room’s dimensions are whole multiples of half the wavelength, the incident and reflected waves interact such as to bring about standing wave fields, which then dominate the sound field in the room. The particular frequencies at which that occurs are called eigenfrequencies, or resonance frequencies in ordinary speech. The wave pattern, with its characteristic sound maxima and minima, is called an eigenmode, or simply mode; see Figure 2-15.

Figure 2-15 When a dimension of a closed space, an airplane cabin or a vehicle interior for example, is a whole multiple of the half wavelength, then constructive interference arises between incident and reflected waves. A standing wave pattern with nodes and anti-nodes results. The frequencies at which that happens are called eigenfrequencies, and the standing wave patterns are called eigenmodes.

(Sketch: Brüel & Kjær, Technical Review).

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Direct wave

Reflected wave

Observation point

Sound source

At higher frequencies, the eigenfrequencies are so tightly spaced that, for practical reasons, we must choose another mathematical description. In chapter 8, we will study statistical energy methods for so called diffuse fields, see Figure 2-16. In practice, the demands for such a field are seldom fulfilled. The actual field is then called a reverberant field. The third method, geometrical room acoustics, is described in the next section.

2.6.1 Reflections and geometrical acoustics

A free wave, incident upon a reflecting surface with irregularities much smaller than a wavelength λ, changes its direction, i.e., is reflected, in a predictable way. In Figure 2-17, the wave fronts of the incident and reflected waves are marked. It is convenient, at times, to indicate a wave by an arrow. The arrow is perpendicular to the wave fronts and points in the direction that the wave propagates. Using Huygen’s principle, we can show that: (i) Against a plane surface, a plane wave is reflected as a plane wave and a spherical

wave as a spherical wave.

(ii) The direction of the incident wave, the normal to the reflecting surface, and the direction of the reflected wave, all lie in the same plane.

(iii) The angle of incidence θi is equal to the angle of reflection θr, see Figure 2-17.

Figure 2-16 Near a source, direct

unreflect-ed sound dominates; such sound comprises the so called direct, or free field in which the power from the source is distributed over an ever expanding volume, resulting in a halv-ing of sound pressure for a doublhalv-ing of dist-ance, according to relation (2-16) between sound power and sound pressure for spheric-al waves. Further away, sound that has been reflected at least once dominates; that sound constitutes the so-called reverberant field. With a stricter demand, that sound pressure be constant and independent of distance to the source, sound must arrive at an observat-ion point from all directobservat-ions, with random phase; it then fulfills the definition of an (ideal) diffuse field.

Figure 2-17 With the help of

Huygen’s principle, we can show that a plane wave, incident on a reflecting surface, is reflected as a plane wave, and that the

angle of incidence θi is

equal to the angle of

reflection θr. θ i θ r Incident plane wave n Wave fronts Reflected plane wave Wave fronts λ λ

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If the reflecting surface is rigid, the sound pressure amplitudes and phases of the incident and reflected waves are equal the point of reflection. An observer is influenced by both the direct sound from the source and the reflected sound. We can describe the sound field above the reflecting surface by a superimposition of the direct wave field direct from the actual source with the direct field of an identical, imagined mirror source, placed at the same distance from the surface, but on the opposite side; see fig 2-18.

Source Mirror source Wave fronts Rigid surface Observer d d Reflectedspherical wave Incident spherical wave

In geometrical acoustics, a sound wave is represented by an arrow in the direction of wave propagation, just as a light wave is represented in geometrical optics. We can regard a wave as a beam originating from the source. The method can be used to explain how a parabolic microphone functions; see Figure 2-19.

Geometrical acoustics is primarily applied in the design of auditoriums. The purpose is to bring about an even distribution of sound, without focuses and shadow zones; see Figure 2-20. The limitation of the method is that typically only the first, or possibly even the second, reflection can be studied before it becomes impractical to follow the “sound trail”.

Figure 2-19 In a parabolic microphone, the

reflected waves are focused into a focal point, at which the microphone is located, to maximize the amplification.

Figure 2-18 A spherical wave incident upon a reflecting surface is reflected as a spherical wave. If the surface is rigid, the reflected wave can be regarded as equivalent to a direct field from an imagined mirror source, identical to the actual source, at the same distance from the surface, but on the opposite side. Parabola Microphone Incident waves Stage Roof Parquet Balcony Figure 2-20 In geometrical

acoustics, sound waves are regarded as beams. The method is often used in the design of large musical auditoriums. The limitation is that it is typically only possible to follow the first few reflections of the sound waves.

(Source: Brüel & Kjær, Measurements in Building

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2.7 WAVE TYPES IN SOLID MEDIA

Solid media can sustain both normal and shear stresses, and thereby resist both volume and shape changes. That implies that in solids, in contrast to gases and liquids, not only longitudinal waves, but even transverse waves, can exist. Transverse and longitudinal waves can also, in combination with each other, build special types of waves, e.g., bending waves. Here, we will acquaint ourselves with some important special cases. Figure 2-21a illustrates a longitudinal wave and Figure 2-21b a transverse wave. When the medium is infinite in the direction of wave propagation, so that no reflections occur, we speak of free wave propagation.

Utbredningsriktning

Partikelrörelse Partikelrörelse Utbredningsriktning

a) b)

Figure 2-21 Important wave types in solid media.

a) Longitudinal wave: particle motion is parallel to the direction of disturbance propagation. b) Transverse wave: particle motion is perpendicular to the direction of disturbance propagation. In a bounded medium, reflections occur at the boundaries. At certain frequencies, the incident and reflected waves interfere, or interact, such that standing wave patterns arise. These frequencies are called, as in section 2.6, eigenfrequencies or resonance frequencies, and the oscillation pattern is called an eigenmode or mode.

Bending oscillations Longitudinal oscillations Tuning fork Bending oscillations Table-top

Figure 2-22 When a tuning fork is

stricken, bending oscillat-ions take place in the legs, essentially in the first eigenmode. That disturb-ance excites a longitudinal oscillation at the same frequency. That latter, in its turn, excites bending waves in the tabletop. As is well known, it isn’t until that large table surface has been put into contact with the tuning fork, and begun vibrating itself, that we begin to hear the tuning fork well.

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Exercise 2-4

Hold a long measuring stick, extended about 1.5 m from your grasp, and shake it transversally at about 1 Hz. It will now (hopefully) be oscillating in its first eigenmode. Increase the frequency until you find the second eigenmode. Observe the antinodes, where there is a large amount of bowing, i.e., deformation, and the nodes, at which there is none.

First eigenmode Second eigenmode

Node Anti- Node

2.8 CHARACTERIZATION OF SOUND ACCORDING TO FREQUENCY

Sound can be characterized by its frequency content. For mankind, audible sound is that which falls in the 20 – 20 000 Hz range. For frequencies lower that 20 Hz, we speak of infrasound, and over 20 000 Hz of ultrasound. Most acoustic phenomena are frequency-dependent. It is well known that different sound sources have different character, see Figure 2-23. Slowly changing and “soft” events mainly generate low frequency sound, while short and fast events mainly generate high frequency sound. We can be convinced of that by slapping the soft part of the thumb against the tabletop, and then do the same with a ballpoint pen.

As is evident from Figure 2-23, the audible region spans over a range of wavelengths from 17 m to 17 mm, which is of fundamental significance for the types of noise control measures that can be undertaken. Analysis of the frequency content of a sound is therefore of great importance in the field of sound and vibration.

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1 10 20 100 1000 10000 20000

17 17 10. -3

Frekvens [Hz] Våglängd [m]

Infraljud Hörbart ljud Ultraljud

Skärande bearbetning Fartygsmaskiner Luftblåsning Sågning Smältugn Vindbuller Ekolod Bilar, tåg etc Människa Hund Fladdermus Hörområden Ljudkällor 340 3,4 0,34 λ

Figure 2-23 Classification of sound according to frequency, and the relation between frequency and wavelength

in air with a sound speed c = 340 m/s. Frequency ranges of various sound generation mechanisms. (Source: Asf, Bullerbekämpning, 1977.)

2.8.1 Time and frequency domains

Figure 2-24a shows a harmonic signal of frequency f0 as a function of time. It can also be

described as a function of frequency, as in Figure 2-24b. All power in the signal is concentrated at the frequency f0. We have thus introduced the, for vibroacoustics

fundamental, concepts of the time and frequency domains. These are equivalent concepts, and we can choose to describe a signal in either of these domains. We can only record a signal as a function of time. It can thereafter be transformed into the frequency domain by frequency analysis, i.e., by methods from Fourier analysis.

T=1/f0 Signal Time Amplitude Frequency f0 a) b)

Figure 2-24 Harmonic signal in a) the time domain, and b) the frequency domain.

Infrasound Audible sound Ultrasound

Wavelength [m]

3.4

Frequency [Hz]

340 17 0.34 17*10-3

Audible range _Mankind

Dogs Bats Wind noise

Ship machinery

Cars, Trains, etc. Smelting oven Cutting tools Sawing Air blasting Sound Sources: Echo

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Jean Baptiste Fourier (1768-1830) demonstrated more than a century ago that all periodic signals can be described by a summation of harmonic signals. We can illustrate that by means of the three-dimensional Figure 2-25. A periodic signal, e.g., a sound pressure p(t) that repeats with period T, can be written

) ( )

(t p t nT

p = + , n = 1, 2, 3,... (2-18)

Periodic signals are very common in engineering applications. They can be sound pressures or vibrations arising from, for example, rotating machines whose sound and vibration generation is repeated in an identical fashion in cycle after cycle. In Figure 2-25, a signal is shown (heavy line) together with its harmonic components (thin line), at frequencies fn. Every periodic signal can be expressed as the sum of harmonic signals with

a so-called Fourier series (see chapter 3) according to

∑

= + = N n n n nf t p t p 1 0 ) 2 cos( ˆ ) ( π ϕ , (2-19)

where pˆ_n is the peak value of the n-th harmonic component,

f0 = 1/T and nf0 are the frequencies of the n-th harmonic component,

n

ϕ is the phase angle.

1/T 0 Frequency [Hz] Time [s] T 2T/3 T/3 0 2/T 3/T 4/T Amplitude

The description of a signal in the frequency domain is usually called a (frequency) spectrum and each line, frequency component, is called a tone or a spectral line. In order to obtain a complete description, the phase angle of each frequency component is also required. Typically, however, we are only interested in the amplitudes of the frequency components. The frequency spectrum is very useful when the signal is to be analyzed, e.g., it provides the opportunity to distinguish every frequency component, even in cases in which a single frequency is dominant in the signal.

Figure 2-25 A periodic signal

can be described in both the time and the frequency domains. In the time domain, the signal is represented by its variation in time, and in the frequency domain by the amplitude of the harmonic signals it is built up of. Here, the phase angles ϕn are

zero for all frequency components.

KTH Sound&amp;Vibration BOOK

Sound and Vibration

n

S

und Source

CONTENTS

CHAPTER ONE

INTRODUCTION

∑

∑

∑

Example: The shield over the

flywheel and drive belt

constituted a strong noise

source. It was fabricated of

unperforated sheet metal.

Countermeasure: A new shield

was fabricated from perforated

sheet and wire netting. One of

the noise sources of the press

was thereby eliminated.

CHAPTER TWO

FUNDAMENTAL CONCEPTS

∫

∫

n

S

W

∫

∫

∫

λ λ

λ λ

∑

KTH Sound&Vibration BOOK