Engineering Acoustics

Engineering Acoustics
Edition 1.0 30th April 2006
Engineering Accoustics from Wikibooks, the open-content textbooks collection
Note: current version of this book can be found at
http:/en.wikibooks.org/wiki/Engineering_Acoustics
CONTRIBUTORS .........................................................................................................................................................6
PART 1: LUMPED ACOUSTICAL SYSTEMS ...............................................................................................................7
SIMPLE OSCILLATION ................................................................................................................................................7
Solving for the Position Equation ........................................................................................................................7
Alternate Position Equation Forms ......................................................................................................................9
FORCED OSCILLATIONS(SIMPLE SPRING-MASS SYSTEM)........................................................................................10
MECHANICAL RESISTANCE ......................................................................................................................................19
Mechanical Resistance .......................................................................................................................................19
Dashpots.............................................................................................................................................................19
Modeling the Damped Oscillator .......................................................................................................................20
Mechanical Impedance for Damped Oscillator..................................................................................................21
CHARACTERIZING DAMPED MECHANICAL SYSTEMS ...............................................................................................22
Characterizing Damped Mechanical Systems ....................................................................................................22
Calculating the Mechanical Resistance..............................................................................................................22
Critical Damping................................................................................................................................................22
Damping Ratio ...................................................................................................................................................22
Quality Factor ....................................................................................................................................................23
ELECTRO-MECHANICAL ANALOGIES .......................................................................................................................24
Why analogs to circuits? ....................................................................................................................................24
Two possible analogies ......................................................................................................................................24
The equivalent spring.........................................................................................................................................24
The equivalent Mass ..........................................................................................................................................25
The equivalent resistance ...................................................................................................................................25
Review of Circuit Solving Methods...................................................................................................................25
ADDITIONAL RESOURCES FOR SOLVING LINEAR CIRCUITS:......................................................................................26
METHODS FOR CHECKING ELECTRO-MECHANICAL ANALOGIES ..............................................................................27
1. Low-Frequency Limits:..................................................................................................................................27
2. Dot Method: (Valid only for planar network) ................................................................................................27
EXAMPLES OF ELECTRO-MECHANICAL ANALOGIES ................................................................................................28
Example 1 ..........................................................................................................................................................28
Example 1 Solution ............................................................................................................................................28
Example 2 ..........................................................................................................................................................29
Example 2 Solution ............................................................................................................................................30
Example 3 ..........................................................................................................................................................31
PRIMARY VARIABLES OF INTEREST ..........................................................................................................................34
Basic Assumptions .............................................................................................................................................34
Variables of interest ...........................................................................................................................................35
ELECTRO-ACOUSTIC ANALOGIES .............................................................................................................................37
Electro-acoustical Analogies..............................................................................................................................37
TRANSDUCERS - LOUDSPEAKER ..............................................................................................................................48
Acoustic Transducer...........................................................................................................................................48
Magnet Motor Drive System..............................................................................................................................48
Loudspeaker Cone System.................................................................................................................................48
Loudspeaker Suspension....................................................................................................................................48
MOVING RESONATORS ............................................................................................................................................48
Moving Resonators ............................................................................................................................................48
Example .............................................................................................................................................................50
PART 2: ONE-DIMENSIONAL WAVE MOTION .......................................................................................................51
TRANSVERSE VIBRATIONS OF STRINGS ....................................................................................................................51
Introduction........................................................................................................................................................51
What is a wave equation?...................................................................................................................................51
One dimensional Case........................................................................................................................................51
2
Characterization of the mechanical system ........................................................................................................53
TIME-DOMAIN SOLUTIONS ......................................................................................................................................55
d'Alembert Solutions..........................................................................................................................................55
Example of Time Domain Solution ...................................................................................................................55
BOUNDARY CONDITIONS AND FORCED VIBRATIONS ...............................................................................................57
Boundary Conditions .........................................................................................................................................57
Wave Properties .................................................................................................................................................65
Forced Vibrations...............................................................................................................................................66
PART 3: APPLICATIONS .........................................................................................................................................70
ROOM ACOUSTICS AND CONCERT HALLS ................................................................................................................70
Introduction........................................................................................................................................................70
Sound Fields.......................................................................................................................................................70
Room Coefficients .............................................................................................................................................70
Sound Decay and Reverberation Time...............................................................................................................72
Great Halls in the World ....................................................................................................................................73
References..........................................................................................................................................................73
BASS REFLEX ENCLOSURE DESIGN .........................................................................................................................74
Introduction........................................................................................................................................................74
Effects of the Port on the Enclosure Response...................................................................................................74
Quantitative Analysis of Port on Enclosure .......................................................................................................76
Development of Low-Frequency Pressure Response.........................................................................................78
Alignments .........................................................................................................................................................79
Butterworth Alignment ......................................................................................................................................79
Quasi-Butterworth Alignment............................................................................................................................80
Equating the system response | H(s) | 2 with | HQB3(s) | 2, the equations guiding the design can be found [1]: ..81
Chebyshev Alignment........................................................................................................................................81
Thus, the design equations become [1]: .............................................................................................................83
Choosing the Correct Alignment........................................................................................................................83
References..........................................................................................................................................................84
Appendix A: Equivalent Circuit Parameters ......................................................................................................85
Appendix B: Enclosure Parameter Formulas .....................................................................................................86
NEW ACOUSTIC FILTER FOR ULTRASONICS MEDIA .................................................................................................88
Introduction........................................................................................................................................................88
Changes in Media Properties Due to Sound Wave Characteristics....................................................................88
Why Coupled Acoustic Media in Acoustic Filters? ...........................................................................................89
Effects of High-Intensity, Ultrasonic Waves in Acoustic Media in Audio Frequency Spectrum ......................91
An Application of Coupled Media in Acoustic Filters.......................................................................................92
References..........................................................................................................................................................94
NOISE IN HYDRAULIC SYSTEMS...............................................................................................................................95
Noise in Hydraulic Systems ...............................................................................................................................95
Sound in fluids ...................................................................................................................................................95
Source of Noise..................................................................................................................................................95
Fluidborne Noise (FBN) ....................................................................................................................................95
Structure borne Noise (SBN) .............................................................................................................................96
Transmission ......................................................................................................................................................97
Airborne noise (ABN)........................................................................................................................................98
Noise reduction ..................................................................................................................................................99
Hydraulic System noise....................................................................................................................................100
References........................................................................................................................................................100
BASIC ACOUSTICS OF THE MARIMBA.....................................................................................................................101
Introduction......................................................................................................................................................101
Components of Sound ......................................................................................................................................101
Why would the marimba need tuning? ............................................................................................................104
Tuning Myths...................................................................................................................................................105
3
Conclusions......................................................................................................................................................106
Links and Referneces .......................................................................................................................................106
HOW AN ACOUSTIC GUITAR WORKS ......................................................................................................................107
Introduction......................................................................................................................................................107
The Strings .......................................................................................................................................................107
The Body..........................................................................................................................................................108
The Air .............................................................................................................................................................109
SPECIFIC APPLICATION-AUTOMOBILE MUFFLER .....................................................................................................110
Introduction......................................................................................................................................................110
The Configuration of A automobile muffler ....................................................................................................110
How Does automobile muffler function?.........................................................................................................111
Absorptive muffler ...........................................................................................................................................112
BESSEL FUNCTIONS AND THE KETTLEDRUM..........................................................................................................114
What is a kettledrum ........................................................................................................................................114
The math behind the kettledrum: the brief version ..........................................................................................114
The math behind the kettledrum: the derivation...............................................................................................115
The math behind the kettledrum:the entire drum .............................................................................................116
Sites of interest.................................................................................................................................................116
REFERENCES ..........................................................................................................................................................117
FILTER DESIGN AND IMPLEMENTATION .................................................................................................................118
Introduction......................................................................................................................................................118
Basic Wave Theory..........................................................................................................................................118
Basic Filter Design...........................................................................................................................................119
Actual Filter Design .........................................................................................................................................123
Links ................................................................................................................................................................128
References........................................................................................................................................................128
FLOW-INDUCED OSCILLATIONS OF A HELMHOLTZ RESONATOR AND APPLICATIONS ..............................................129
Introduction......................................................................................................................................................129
FEEDBACK LOOP ANALYSIS ...................................................................................................................................129
ACOUSTICAL CHARACTERISTICS OF THE RESONATOR ............................................................................................130
Lumped parameter model ................................................................................................................................130
Production of self-sustained oscillations..........................................................................................................133
APPLICATIONS TO SUNROOF BUFFETING ...............................................................................................................133
How are vortices formed during buffeting? .....................................................................................................133
How to identify buffeting.................................................................................................................................135
USEFUL WEBSITES.................................................................................................................................................136
REFERENCES ..........................................................................................................................................................136
ACOUSTICS IN VIOLINS ..........................................................................................................................................137
Acoustics of the Violin.....................................................................................................................................137
How Does A Violin Make Sound?...................................................................................................................137
References And Other Links ............................................................................................................................140
MOVING COIL LOUDSPEAKER................................................................................................................................141
MOVING COIL TRANSDUCER .................................................................................................................................141
The Magnet Motor Drive System.....................................................................................................................142
The Loudspeaker Cone System........................................................................................................................144
The Loudspeaker Suspension...........................................................................................................................145
Modeling the Loudspeaker as a Lumped System.............................................................................................147
References........................................................................................................................................................148
Links ................................................................................................................................................................148
ATTENUATION OF SOUND WAVES .........................................................................................................................149
Introduction......................................................................................................................................................149
Types of Attenuation........................................................................................................................................149
Modeling of losses ...........................................................................................................................................151
References........................................................................................................................................................151
4
CAR MUFFLERS .....................................................................................................................................................153
Introduction......................................................................................................................................................153
The absorber muffler........................................................................................................................................153
The reflector muffler ........................................................................................................................................154
Back pressure ...................................................................................................................................................156
Muffler Modeling by Transfer Matrix Method ................................................................................................156
Links ................................................................................................................................................................158
NOISE FROM COOLING FANS...................................................................................................................................159
Proposal............................................................................................................................................................159
Introduction......................................................................................................................................................159
Noise Generation Mechanisms.........................................................................................................................159
Installation Effects ...........................................................................................................................................163
Closing Comment ............................................................................................................................................163
Links to Interesting Sites about Fan Noise.......................................................................................................163
References........................................................................................................................................................164
HUMAN VOCAL FOLD ............................................................................................................................................165
Physiology of Vocal Fold.................................................................................................................................165
Voice Production..............................................................................................................................................165
Model ...............................................................................................................................................................166
Simulation of the Model...................................................................................................................................167
The Acoustic Output ........................................................................................................................................168
Related Links ...................................................................................................................................................169
References........................................................................................................................................................169
MICROPHONE DESIGN AND OPERATION.................................................................................................................170
Introduction......................................................................................................................................................170
Condenser Microphones...................................................................................................................................171
Conclusion .......................................................................................................................................................174
References........................................................................................................................................................174
Microphone Manufacturers Links....................................................................................................................174
PIEZOELECTRIC TRANSDUCERS .............................................................................................................................175
INTRODUCTION ......................................................................................................................................................175
VIBRATIONS & DISPLACEMENTS ...........................................................................................................................175
DYNAMIC PERFORMANCE ......................................................................................................................................175
Equivalent Electric Circuit ...............................................................................................................................176
Frequency Response.........................................................................................................................................176
RESONANT DEVICES ..............................................................................................................................................176
APPLICATIONS .......................................................................................................................................................177
Mechanical Measurement ................................................................................................................................177
Ultrasonic .........................................................................................................................................................177
MORE INFORMATION AND SOURCE OF INFORMATION............................................................................................178
MICROPHONE TECHNIQUE .....................................................................................................................................179
General Technique ...........................................................................................................................................179
Working Distance ............................................................................................................................................179
Stereo and Surround Technique .......................................................................................................................180
Placement for Varying Instruments..................................................................................................................182
Sound Propagation ...........................................................................................................................................183
Sources.............................................................................................................................................................183
SEALED BOX SUBWOOFER DESIGN ........................................................................................................................184
Introduction......................................................................................................................................................184
Closed Baffle Circuit........................................................................................................................................184
Driver Parameters ............................................................................................................................................185
Acoustic Compliance .......................................................................................................................................187
Sealed Box Design ...........................................................................................................................................188
ACOUSTIC GUITARS ...............................................................................................................................................189
5
Introduction......................................................................................................................................................189
Strings, Neck, and Head...................................................................................................................................189
Bridge...............................................................................................................................................................190
Soundboard ......................................................................................................................................................190
Internal Cavity..................................................................................................................................................190
BASIC ROOM ACOUSTIC TREATMENTS ..................................................................................................................191
ROOM ACOUSTIC TREATMENTS FOR "DUMMIES" ..................................................................................................191
Introduction......................................................................................................................................................191
Room Sound Combinations .............................................................................................................................191
Good and Bad Reflected Sound .......................................................................................................................191
How to Find Overall Trouble Spots In a Room ...............................................................................................195
References Sound.............................................................................................................................................195
BOUNDARY CONDITIONS AND WAVE PROPERTIES ................................................................................................196
Boundary Conditions .......................................................................................................................................196
Wave Properties ...............................................................................................................................................197
ROTOR STATOR INTERACTIONS .............................................................................................................................199
Noise emission of a Rotor-Stator mechanism ..................................................................................................199
Optimization of the number of blades..............................................................................................................199
Determination of source levels.........................................................................................................................200
Directivity ........................................................................................................................................................200
External references...........................................................................................................................................201
LICENSE .................................................................................................................................................................202
GNU Free Documentation License ..................................................................................................................202
0. PREAMBLE ................................................................................................................................................202
1. APPLICABILITY AND DEFINITIONS.....................................................................................................202
2. VERBATIM COPYING ..............................................................................................................................203
3. COPYING IN QUANTITY .........................................................................................................................203
4. MODIFICATIONS ......................................................................................................................................203
5. COMBINING DOCUMENTS.....................................................................................................................204
6. COLLECTIONS OF DOCUMENTS...........................................................................................................204
7. AGGREGATION WITH INDEPENDENT WORKS .................................................................................205
8. TRANSLATION..........................................................................................................................................205
9. TERMINATION..........................................................................................................................................205
10. FUTURE REVISIONS OF THIS LICENSE .............................................................................................205
External links ...................................................................................................................................................205
Contributors
Students in ME 513: Engineering Acoustics (http://widget.ecn.purdue.edu/~me513/Index.html)
started this Wikibook, Engineering Acoustics, during the fall semester 2005. Some pages of this
book contain author credits.
Since then, other and anonymous users have contributed to this book.
6
Part 1: Lumped Acoustical Systems
Simple Oscillation
Solving for the Position Equation
For a simple oscillator consisting of a mass m to one end of a spring with a spring constant s, the
restoring force, f, can be expressed by the equation
where x is the displacement of the mass from its rest position. Substituting the expression for f
into the linear momentum equation,
where a is the acceleration of the mass, we can get
or,
Note that
To solve the equation, we can assume
The force equation then becomes
Giving the equation
7
Solving for λ
This gives the equation of x to be
Note that
and that C1 and C2 are constants given by the initial conditions of the system
If the position of the mass at t = 0 is denoted as x0, then
and if the velocity of the mass at t = 0 is denoted as u0, then
Solving the two boundary condition equations gives
The position is then given by
This equation can also be found by assuming that x is of the form
And by applying the same initial conditions,
This gives rise to the same postion equation
8
Alternate Position Equation Forms
If A1 and A1 are of the form
Then the position equation can be written
By applying the initial conditions (x(0)=x0, u(0)=u0) it is found that
If these two equations are squared and summed, then it is found that
And if the difference of the same two equations is found, the result is that
The position equation can also be written as the Real part of the imaginary position equation
Due to euler's rule (ejφ = cosφ + jsinφ), x(t) is of the form
9
Forced Oscillations(Simple Spring-Mass System)
Recap of Section 1.3
In the previous section, we discussed how adding a damping component (e. g. a dashpot) to an
unforced, simple spring-mass system would affect the response of the system. In particular, we
learned that adding the dashpot to the system changed the natural frequency of the system from
to a new damped natural frequency , and how this change made the response of the system
change from a constant sinusoidal response to an exponentially-decaying sinusoid in which the
system either had an under-damped, over-damped, or critically-damped response.
In this section, we will digress a bit by going back to the simple (undamped) oscillator system of
the previous section, but this time, a constant force will be applied to this system, and we will
investigate this system's performance at low and high frequencies as well as at resonance. In
particular, this section will start by introducing the characteristics of the spring and mass
elements of a spring-mass system, introduce electrical analogs for both the spring and mass
elements, learn how these elements combine to form the mechanical impedance system, and
reveal how the impedance can describe the mechanical system's overall response characteristics.
Next, power dissipation of the forced, simple spring-mass system will be discussed in order to
corroborate our use of electrical circuit analogs for the forced, simple spring-mass system.
Finally, the characteristic responses of this system will be discussed, and a parameter called the
amplification ratio (AR) will be introduced that will help in plotting the resonance of the forced,
simple spring-mass system.
Forced Spring Element
Taking note of Figs. 1, we see that the equation of motion for a spring that has some constant,
external force being exerted on it is...
where
is the mechanical stiffness of the spring.
10
Note that in Fig. 1(c), force flows constantly (i.e. without decreasing) throughout a spring, but
the velocity of the spring decrease from
to
as the force flows through the spring. This
concept is important to know because it will be used in subsequent sections.
In practice, the stiffness of the spring
, also called the spring constant, is usually expressed as
, or the mechanical compliance of the spring. Therefore, the spring is very stiff if
is large
is small. Similarly, the spring is very loose or "bouncy" if
is small
is large. Noting that force and velocity are analogous to voltage and current,
respectively, in electrical systems, it turns out that the characteristics of a spring are analogous to
the characteristics of a capacitor in relation to, and, so we can model the "reactiveness" of a
spring similar to the reactance of a capacitor if we let
as shown in Fig. 2 below.
11
Forced Mass Element
Taking note of Fig. 3, the equation for a mass that has constant, external force being exerted on it
is...
12
If the mass
can vary its value and is oscillating in a mechanical system at max amplitude
such that the input the system receives is constant at frequency , as
increases, the
harder it will be for the system to move the mass at
at
until, eventually, the mass
doesn?tm)t oscillate at all . Another equivalently way to look at it is to let vary and hold
constant. Similarly, as increases, the harder it will be to get
to oscillate at and keep
the same amplitude
until, eventually, the mass doesn?tm)t oscillate at all. Therefore, as
increases, the "reactiveness" of mass
decreases (i.e.
starts to move less and less).
Recalling the analogous relationship of force/voltage and velocity/current, it turns out that the
characteristics of a mass are analogous to an inductor. Therefore, we can model the
"reactiveness" of a mass similar to the reactance of an inductor if we let
as shown in
Fig. 4.
13
Mechanical Impedance of Spring-Mass System
As mentioned twice before, force is analogous to voltage and velocity is analogous to current.
Because of these relationships, this implies that the mechanical impedance for the forced, simple
spring-mass system can be expressed as follows:
In general, an undamped, spring-mass system can either be "spring-like" or "mass-like". "Springlike" systems can be characterized as being "bouncy" and they tend to grossly overshoot their
target operating level(s) when an input is introduced to the system. These type of systems
relatively take a long time to reach steady-state status. Conversely, "mass-like" can be
characterized as being "lethargic" and they tend to not reach their desired operating level(s) for a
14
given input to the system...even at steady-state! In terms of complex force and velocity, we say
that " force LEADS velocity" in mass-like systems and "velocity LEADS force" in spring-like
systems (or equivalently " force LAGS velocity" in mass-like systems and "velocity LAGS
force" in spring-like systems). Figs. 5 shows this relationship graphically.
Power Transfer of a Simple Spring-Mass System
From electrical circuit theory, the average complex power
expressed as ...
dissipated in a system is
where and
represent the (time-invariant) complex voltage and complex conjugate current,
respectively. Analogously, we can express the net power dissipation of the mechanical system
in general along with the power dissipation of a spring-like system
system
as...
15
or mass-like
In equations 1.4.7, we see that the product of complex force and velocity are purely imaginary.
Since reactive elements, or commonly called, lossless elements, cannot dissipate energy, this
implies that the net power dissipation of the system is zero. This means that in our simple springmass system, power can only be (fully) transferred back and forth between the spring and the
mass. Therefore, by evaluating the power dissipation, this corroborates the notion of using
electrical circuit elements to model mechanical elements in our spring-mass system.
Responses For Forced, Simple Spring-Mass System
Fig. 6 below illustrates a simple spring-mass system with a force exerted on the mass.
16
This system has response characteristics similar to that of the undamped oscillator system, with
the only difference being that at steady-state, the system oscillates at the constant force
magnitude and frequency versus exponentially decaying to zero in the unforced case. Recalling
equations 1.4.2b and 1.4.4b, letting be the natural (resonant) frequency of the spring-mass
system, and letting
be frequency of the input received by the system, the characteristic
responses of the forced spring-mass systems are presented graphically in Figs. 7 below.
Amplification Ratio
The amplification ratio is a useful parameter that allows us to plot the frequency of the springmass system with the purports of revealing the resonant freq of the system solely based on the
force experienced by each, the spring and mass elements of the system. In particular, AR is the
magnitude of the ratio of the complex force experienced by the spring and the complex force
experienced by the mass, i.e.
17
If we let
, be the frequency ratio, it turns out that AR can also be expressed as...
.
AR will be at its maximum when
An example of an AR plot is shown below in Fig 8.
18
. This happens precisely when
.
Mechanical Resistance
Mechanical Resistance
For most systems, a simple oscillator is not a very accurate model. While a simple oscillator
involves a continuous transfer of energy between kinetic and potential form, with the sum of the
two remaining constant, real systems involve a loss, or dissipation, of some of this energy, which
is never recovered into kinetic nor potential energy. The mechanisms that cause this dissipation
are varied and depend on many factors. Some of these mechanisms include drag on bodies
moving through the air, thermal losses, and friction, but there are many others. Often, these
mechanisms are either difficult or impossible to model, and most are non-linear. However, a
simple, linear model that attempts to account for all of these losses in a system has been
developed.
Dashpots
The most common way of representing mechanical resistance in a damped system is through the
use of a dashpot. A dashpot acts like a shock absorber in a car. It produces resistance to the
system's motion that is proportional to the system's velocity. The faster the motion of the system,
the more mechanical resistance is produced.
19
As seen in the graph above, a linear realationship is assumed between the force of the dashpot
and the velocity at which it is moving. The constant that relates these two quantities is RM, the
mechanical resistance of the dashpot. This relationship, known as the viscous damping law, can
be written as:
Also note that the force produced by the dashpot is always in phase with the velocity.
The power dissipated by the dashpot can be derived by looking at the work done as the dashpot
resists the motion of the system:
Modeling the Damped Oscillator
In order to incorporate the mechanical resistance (or damping) into the forced oscillator model, a
dashpot is placed next to the spring. It is connected to the mass (MM) on one end and attached to
the ground on the other end. A new equation describing the forces must be developed:
It's phasor form is given by the following:
20
Mechanical Impedance for Damped Oscillator
Previously, the impedance for a simple oscillator was defined as
the impedance of a damped oscillator can be calculated:
. Using the above equations,
For very low frequencies, the spring term dominates because of the
relationship. Thus, the
phase of the impedance approaches
for very low frequencies. This phase causes the velocity
to "lag" the force for low frequencies. As the frequency increases, the phase difference increases
toward zero. At resonance, the imaginary part of the impedance vanishes, and the phase is zero.
The impedance is purely resistive at this point. For very high frequencies, the mass term
dominates. Thus, the phase of the impedance approaches
high frequencies.
and the velocity "leads" the force for
Based on the previous equations for dissipated power, we can see that the real part of the
impedance is indeed RM. The real part of the impedance can also be defined as the cosine of the
phase times its magnitude. Thus, the following equations for the power can be obtained.
21
Characterizing Damped Mechanical Systems
Characterizing Damped Mechanical Systems
Characterizing the response of Damped Mechanical Oscillating system can be easily quantified
using two parameters. The system parameters are the resonance frequency ('''wresonance''' and
the damping of the system '''Q(qualityfactor)orB(TemporalAbsorption'''). In practice, finding
these parameters would allow for quantification of unkwnown systems and allow you to derive
other parameters within the system.
Using the mechanical impedance in the following equation, notice that the imaginary part will
equal zero at resonance.
(Zm = F / u = Rm + j(w * Mm ? s / w))
Resonance case:(w * Mm = s / w)
Calculating the Mechanical Resistance
The decay time of the system is related to 1 / B where B is the Temporal Absorption. B is related
to the mechancial resistance and to the mass of the system by the following equation.
B = Rm / 2 * Mm
The mechanical resistance can be derived from the equation by knowing the mass and the
temporal absorption.
Critical Damping
The system is said to be critically damped when:
Rc = 2 * M * sqrt(s / Mm) = 2 * sqrt(s * Mm) = 2 * Mm * wn
A critically damped system is one in which an entire cycle is never completed. The absorbtion
coefficient in this type of system equals the natural frequency. The system will begin to oscillate,
however the amplitude will decay exponentially to zero within the first oscillation.
Damping Ratio
DampingRatio = Rm / Rc
The damping ratio is a comparison of the mechanical resistance of a system to the resistance
value required for critical damping. Rc is the value of Rm for which the absorbtion coefficient
equals the natural frequency (critical damping). A damping ratio equal to 1 therefore is critically
22
damped, because the mechanical resistance value Rm is equal to the value required for critical
damping Rc. A damping ratio greater than 1 will be overdamped, and a ratio less than 1 will be
underdamped.
Quality Factor
The Quality Factor (Q) is way to quickly characterize the shape of the peak in the response. It
gives a quantitative representation of power dissipation in an oscillation.
Q = wresonance / (wu ? wl)
Wu and Wl are called the half power points. When looking at the response of a system, the two
places on either side of the peak where the point equals half the power of the peak power defines
Wu and Wl. The distance in between the two is called the half-power bandwidth. So, the
resonant frequency divided by the half-power bandwidth gives you the quality factor.
Mathematically, it takes Q/pi oscillations for the vibration to decay to a factor of 1/e of its
original amplitude.
23
Electro-Mechanical Analogies
Why analogs to circuits?
Since acoustic devices contain both electrical and mechanical components, one needs to be able
to combine them in a graphical way that aids the user's intuition. The method that is still used in
the transducer industry is the Impedance and Mobility analogies that compare mechanical
systems to electric circuits.
Two possible analogies
i) Impedance analog
ii) Mobility analog
Mechanical
Electrical equivalent
i)impedance analog
Potential
Flux
Force
Velocity
F(t)
u(t)
Voltage
Current
Velocity
Force
u(t)
F(t)
Velocity
Current
V(t)
i(t)
ii)Mobility analog
Potential
Flux
u(t)
i(t)
Impedance analog is often easier to use in most accoustical systems while mobility analog can be
found more intuitively for mechanical systems. These are generalities, however, so it is best to
use the analogy that allows for the most understanding. A circuit of one analog can be switched
to the equivalent circuit of the other analog by using the dual of the circuit. (more on this in the
next section).
The equivalent spring
Mechanical spring
Impedance analogy of the mechanical spring
24
Mobility analogy of the mechanical spring
The equivalent Mass
Mechanical mass
Impedance analogy of the mechanical mass
Mobility analogy of the mechanical mass
The equivalent resistance
Mechanical resistance
F = RmU
Impedance analogy of the mechanical resistance
U = Rmi
Mobility analogy of the mechanical resistance
Review of Circuit Solving Methods
Kirchkoff's Voltage law
"The sum of the potential drops around a loop must equal zero."
This implies that the total potential drop around a series of elements is equal to the sum of the
25
individual voltage drops in the series.
etotal = drop1 + drop2 + drop3
Kirchkoff's Current Law
"The Sum of the currents at a node (junction of more than two elements) must be zero"
Using the pipe flow analogy of circuits, this can be thought of as the continuity equation.
For example if there was a node with three elements connected to it (numbered 1,2 and 3) i1 + i2
+ i3 = 0 From the current law, their sum would equal zero.
Hints for solving circuits:
-Remember that certain elements can be combined to simplify the circuit (the combination of
like elements in series and parallel)
-If solving a circuit that involves steady-state sources, uses impedances! (This reduces the circuit
down to a bunch of complex domain resistor elements that can be combined to simplify the
circuit.)
Additional Resources for solving linear
circuits:
Thomas & Rosa, "The Analysis and Design of Linear Circuits", Wiley, 2001
Hayt, Kemmerly & Durbin, "Engineering Circuit Analysis", 6th ed., McGraw Hill, 2002
26
Methods for checking Electro-Mechanical
Analogies
After drawing the electro-mechanical analogy of a mechanical system, it is always safe to check
the circuit. There are two methods to accomplish this:
1. Low-Frequency Limits:
This method looks at the behavior of the system for very large or very small values of the
parameters and compares them with the expected behavior of the mechanical system. The basic
formula to spot an error in the electro-mechanical circuit is as follows:
Capacitor (C)
Resistor (R)
Inductor (L)
Very large value:
Short circuit
Open circuit
Open circuit
Very small value:
Open circuit
Short circuit
Short circuit
2. Dot Method: (Valid only for planar network)
This method helps obtain the dual analog (one analog is the dual of the other). The steps for the
dot product are as follows: 1) Place one dot within each loop and one outside all the loops. 2)
Connect the dots. Make sure that only there is only one line through each element and that no
lines cross more than one element. 3) Draw in each line that crosses an element its dual element,
including the source. 4) The circuit obtained should have the same configuration as the dual
analog of the original electro-mechanical circuit.
27
Examples of Electro-Mechanical Analogies
Example 1
Draw the mobility analog representation of the mechanical system shown below.
Example 1 Solution
Using the fact that flux is equivalent to force and potential to velocity, the following is the
mobility analog representation of the mechanical system given in example 1.
28
Using the Low-frequency limits method to check the accuracy of the mobility analog circuit
drawn, we have:
i) If we make the Cms (inductor) very small, the Cms becomes a short circuit. This agrees with
the mechanical system.
ii) If we make the Mm2 (capacitor) very large, the Mm2 becomes a short circuit. No motion is
transmitted to the rest of the system and this agrees by inspecting the mechanical system given.
Example 2
Draw the mobility analog representation of the following axisymmetric device. Does your circuit
make sense if you consider behavior at low-frequency?
29
Example 2 Solution
The mobility analog representation of this system would be as follows:
30
Example 3
Draw the mobility analog representation of the mechanical system below. Consider the behavior
of the circuit at low frequency to check for validity. Then draw the impedence equivalent circuit
using the dot method.
31
The mobility analog representation of the mechanical system is shown as:
32
The impedance analog representation of the same mechanical system is shown as:
33
Primary variables of interest
Basic Assumptions
Consider a piston moving in a tube. The piston starts moving at time t=0 with a velocity u=up.
The piston fits inside the tube smoothly without any friction or gap. The motion of the piston
creates a planar sound wave or acoustic disturbance traveling down the tube at a constant speed
c>>up. In a case where the tube is very small, one can neglect the time it takes for acoustic
disturbance to travel from the piston to the end of the tube. Hence, one can assume that the
acoustic disturbance is uniform throughout the tube domain.
Assumptions
1. Although sound can exist in solids or fluid, we will first consider the medium to be a fluid at
rest. The ambient, undisturbed state of the fluid will be designated using subscript zero. Recall
that a fluid is a substance that deforms continuously under the application of any shear
(tangential) stress.
2. Disturbance is a compressional one (as opposed to transverse).
3. Fluid is a continuum: infinitely divisible substance. Each fluid property assumed to have
definite value at each point.
4. The disturbance created by the motion of the piston travels at a constant speed. It is a function
of the properties of the ambient fluid. Since the properties are assumed to be uniform (the same
at every location in the tube) then the speed of the disturbance has to be constant. The speed of
the disturbance is the speed of sound, denoted by letter c0 with subscript zero to denote ambient
34
property.
5. The piston is perfectly flat, and there is no leakage flow between the piston and the tube inner
wall. Both the piston and the tube walls are perfectly rigid. Tube is infinitely long, and has a
constant area of cross section, A.
6. The disturbance is uniform. All deviations in fluid properties are the same across the tube for
any location x. Therefore the instantaneous fluid properties are only a function of the Cartesian
coordinate x (see sketch). Deviations from the ambient will be denoted by primed variables.
Variables of interest
Pressure (force / unit area)
Pressure is defined as the normal force per unit area acting on any control surface within the
fluid.
For the present case,inside a tube filled with a working fluid, pressure is the ratio of the surface
force acting onto the fluid in the control region and the tube area. The pressure is decomposed
into two components - a constant equilibrium component, p0, superimposed with a varying
disturbance p'(x). The deviation p'is also called the acoustic pressure. Note that p' can be positive
or negative. Unit: kg / ms2. Acoustical pressure can be meaured using a microphone.
35
Density
Density is mass of fluid per unit volume. The density, ρ, is also decomposed into the sum of
ambient value (usually around ρ0= 1.15 kg/m3) and a disturbance ρ?tm)(x). The disturbance can
be positive or negative, as for the pressure. Unit: kg / m3
Acoustic volume velocity
Rate of change of fluid particles position as a funtion of time. Its the well known fluid mechanics
term, flow rate.
In most cases, the velocity is assumed constant over the entire cross section (plug flow), which
gives acoustic volume velocity as a product of fluid velocity and cross section S.
36
Electro-acoustic analogies
Electro-acoustical Analogies
Acoustical Mass
Consider a rigid tube-piston system as following figure.
Piston is moving back and forth sinusoidally with frequency of f. Assuming
(where c is sound velocity
), volume of fluid in tube is,
Then mass (mechanical mass) of fluid in tube is given as,
For sinusoidal motion of piston, fluid move as rigid body at same velocity as piston. Namely,
every point in tube moves with the same velocity.
Applying the Newton's second law to the following free body diagram,
37
Where, plug flow assumption is used.
"Plug flow" assumption:
Frequently in acoustics, the velocity distribution along the normal surface
of fluid flow is assumed uniform. Under this assumption, the acoustic volume
velocity U is simply product of velocity and entire surface. U = Su
Acoustical Impedance
Recalling mechanical impedance,
acoustical impedance (often termed an acoustic ohm) is defined as,
where, acoustical mass is defined.
Acoustical Mobility
Acoustical mobility is defined as,
38
Impedance Analog vs. Mobility Analog
Acoustical Resistance
Acoustical resistance models loss due to viscous effects (friction) and flow resistance
(represented by a screen).
39
rA is the reciprocal of RA and is referred to as responsiveness.
Acoustical Generators
The acoustical generator components are pressure, P and volume velocity, U, which are analogus
to force, F and velocity, u of electro-mechanical analogy respectively. Namely, for impedance
analog, pressure is analogus to voltage and volume velocity is analogus to current, and vice versa
for mobility analog. These are arranged in the following table.
Impedance and Mobility analogs for acoustical generators of constant pressure and constant
volume velocity are as follows:
40
41
Acoustical Compliance
Consider a piston in an enclosure.
When the piston moves, it displaces the fluid inside the enclosure. Acoustic compliance is the
measurement of how "easy" it is to displace the fluid.
Here the volume of the enclosure should be assumed to be small enough that the fluid pressure
remains uniform.
Assume no heat exchange 1.adiabatic 2.gas compressed uniformly , p prime in cavity
everywhere the same.
from thermal equitation
it is easy to get the relation between disturbing
pressure and displacement of the piston
where U is
volume rate, P is pressure according to the definition of the impedance and mobility, we can
42
get
Mobility Analog VS Impedance Analog
43
Examples of Electro-Acoustical Analogies
Example 1: Helmholtz Resonator
Assumptions - (1) Completely sealed cavity with no leaks. (2) Cavity acts like a rigid body
inducing no vibrations.
44
Solution:
- Impedance Analog -
45
Example 2: Combination of Side-Branch Cavities
46
Solution:
- Impedance Analog -
47
Transducers - Loudspeaker
Acoustic Transducer
The purpose of the acoustic transducer is to convert electrical energy into acoustic energy. Many
variations of acoustic transducers exists, although the most common is the moving coilpermanent magnet tranducer. The classic loudspeaker is of the moving coil-permanent magnet
type.
The classic electrodynamic loudspeaker driver can be divided into three key components:
1) The Magnet Motor Drive System
2) The Loudspeaker Cone System
3) The Loudspeaker Suspension
This illustration shows a cut-away of the moving coil-permanent magnet loudspeaker.
Woofer Picture here
Magnet Motor Drive System
Loudspeaker Cone System
Loudspeaker Suspension
An equivalent circuit can be used to model all three loudspeaker components as a lumped
system. This circuit provides a model of the loudspeaker and its separate sub-components and
can be used to provide insight into what parameters alter the loudspeaker's performance.
Moving Resonators
Moving Resonators
Consider the situation shown in the figure below. We have a typical Helmholtz resonator driven
by a massless piston which generates a sinusoidal pressure PG, however the cavity is not fixed in
this case. Rather, it is supported above the ground by a spring with compliance CM. Assume the
cavity has a mass MM.
48
Recall the Helmholtz resonator (see Module #9). The difference in this case is that the pressure
in the cavity exerts a force on the bottom of the cavity, which is now not fixed as in the original
Helmholtz resonator. This pressure causes a force that acts upon the cavity bottom. If the surface
area of the cavity bottom is SC, then Newton's Laws applied to the cavity bottom give
In order to develop the equivalent circuit, we observe that we simply need to use the pressure
(potential across CA) in the cavity to generate a force in the mechanical circuit. The above
equation shows that the mass of the cavity and the spring compliance should be placed in series
in the mechanical circuit. In order to convert the pressure to a force, the transformer is used with
a ratio of 1:SC.
49
Example
A practical example of a moving resonator is a marimba. A marimba is a similar to a xylophone
but has larger resonators that produce deeper and richer tones. The resonators (seen in the picture
as long, hollow pipes) are mounted under an array of wooden bars which are struck to create
tones. Since these resonators are not fixed, but are connected to the ground through a stiffness
(the stand), it can be modeled as a moving resonator. Marimbas are not tunable instruments like
flutes or even pianos. It would be interesting to see how the tone of the marimba changes as a
result of changing the stiffness of the mount.
For
more
information
about
the
http://www.mostlymarimba.com/techno1.html
50
acoustics
of
marimbas
see
Part 2: One-Dimensional Wave Motion
Transverse vibrations of strings
Introduction
This section deals with the wave nature of vibrations constrained to one dimention. Examples of
this type of wave motion are found in objects such a pipes and tubes with a small diameter (no
transverse motion of fluid) or in a string stretched on a musical instrument.
Streched strings can be used to produce sound (e.g. music instruments like guitars). The streched
string constitutes a mechanical system that will be studied in this chapter. Later, the
characteristics of this system will be used to help to understand by analogies accoustical systems.
What is a wave equation?
There are various types of waves (i.e. electromagnetic, mechanical, etc)that act all around us. It
is important to use wave equations to describe the time-space behavior of the variables of interest
in such waves. Wave equations solve the fundamentals equations of motion in a way that
eliminates all variables but one. Waves can propagate longitudinal or parallel to the propagation
direction or perpendicular (transverse) to the direction of propagation. To visualize the motion of
such waves click here (Acoustics animations provided by Dr. Dan Russell,Kettering University)
One dimensional Case
Assumptions :
- the string is uniform in size and density
- stiffness of string is negligible for samll deformations
- effects of gravity neglected
- no dissipative forces like frictions
- string deforms in a plane
- motion of the string can be described by using one single spatial coordinate
Spatial representation of the string in vibration:
51
The following is the free-body diagram of a string in motion in a spatial coordinate system:
From the diagram above, it can be observed that the tensions in each side of the string will be the
same as follows:
Using Taylor series to expand we obtain:
52
Characterization of the mechanical system
A one dimentional wave can be described by the following equation (called the wave equation):
where,
is a solution,
With
and
This
is
the
D'Alambert
solution,
for
more
http://en.wikibooks.org/wiki/Acoustic:Time-Domain_Solutions
information
see:
[1]
Another way to solve this equation is the Method of separation of variables. This is useful for
modal analysis. This assumes the solution is of the form:
The result is the same as above, but in a form that is more conveniant for modal anaylsis.
For more information on this approach see: Eric W. Weisstein et al. "Separation of Variables."
From MathWorld--A Wolfram Web Resource. [2]
http://mathworld.wolfram.com/SeparationofVariables.html
Please see Wave Properties
http://en.wikibooks.org/wiki/Acoustic:Boundary_Conditions_and_Forced_Vibrations for
information on variable c, along with other important properties.
For more information on wave equations see: Eric W. Weisstein. "Wave Equation." From
MathWorld--A Wolfram Web Resource. [3] http://mathworld.wolfram.com/WaveEquation.html
53
Example with the function f(?/4) :
This image has been released into the public domain by the copyright holder, its
copyright has expired, or it is ineligible for copyright. This applies worldwide.
This image has been released into the public domain by the copyright holder, its
copyright has expired, or it is ineligible for copyright. This applies worldwide.
Example: Java String simulation http://www.kw.igs.net/~jackord/bp/n1.html
This show a simple simulation of a plucked string with fixed ends.
54
Time-Domain Solutions
d'Alembert Solutions
In 1747, Jean Le Rond d'Alembertpublished a solution to the one-dimensional wave equation.
The general solution, now known as the d'Alembert method, can be found by introducing two
new variables:
and
and then applying the chain rule to the general form of the wave equation.
From this, the solution can be written in the form:
where f and g are arbitrary functions, that represent two waves traveling in opposing directions.
A more detailed look into the proof of the d'Alembert solution can be found here.
http://mathworld.wolfram.com/dAlembertsSolution.html
Example of Time Domain Solution
If f(ct-x) is plotted vs. x for two instants in time, the two waves are the same shape but the
second displaced by a distance of c(t2-t1) to the right.
55
The two arbitrary functions could be determined from initial conditions or boundary values.
56
Boundary Conditions and Forced Vibrations
Boundary Conditions
The functions representing the solutions to the wave equation previously discussed,
i.e.
with
and
are dependent upon the boundary and initial conditions. If it is assumed that the wave is
propogating through a string, the initial conditions are related to the specific disturbance in the
string at t=0. These specific disturbances are determined by location and type of contact and can
be anything from simple oscillations to violent impulses. The effects of boundary conditions are
less subtle.
The most simple boundary conditions are the Fixed Support and Free End. In practice, the Free
End boundary condition is rarely encountered since it is assumed there are no transverse forces
holding the string (e.g. the string is simply floating).
For a Fixed Support:
The overall displacement of the waves travelling in the string, at the support, must be zero.
Denoting x=0 at the support, This requires:
Therefore, the total transverse displacement at x=0 is zero.
57
The sequence of wave reflection for incident, reflected and combined waves are illustrated
below. Please note that the wave is traveling to the left (negative x direction) at the beginning.
The reflected wave is ,of course, traveling to the right (positive x direction).
t=0
58
t=t1
59
t=t2
60
t=t3
For a Free Support:
Unlike the Fixed Support boundary condition, the transverse displacment at the support does not
need to be zero, but must require the sum of transverse forces to cancel. If it is assumed that the
angle of displacement is small,
and so,
But of course, the tension in the string, or T, will not be zero and this requires the slope at x=0 to
be zero:
i.e.
61
Again for free boundary, the sequence of wave reflection for incident, reflected and combined
waves are illustrated below:
t=0
62
t=t1
63
t=t2
64
t=t3
Other Boundary Conditions:
There are many other types of boundary conditions that do not fall into our simplified categories.
As one would expect though, it isn't difficult to relate the characteristics of numerous "complex"
systems to the basic boundary conditions. Typical or realistic boundary conditions include massloaded, resistance-loaded, damped loaded, and impedance-loaded strings. For further
information, see Kinsler, Fundamentals of Acoustics, pp 54-58.
Here is a website with nice movies of wave reflection at different BC's: Wave Reflection
http://www.ap.stmarys.ca/demos/content/osc_and_waves/wave_reflection/wave_reflection.html
Wave Properties
To begin with, a few definitions of useful variables will be discussed. These include; the wave
number, phase speed, and wavelength characteristics of wave travelling through a string.
The speed that a wave propogates through a string is given in terms of the phase speed, typicaly
in m/s, given by:
65
where
is the density per unit length of the string.
The wavenumber is used to reduce the transverse displacement equation to a simpler form and
for simple harmonic motion, is multiplied by the lateral position. It is given by:
where
Lastely, the wavelength is defined as:
and is defined as the distance between two points, usually peaks, of a periodic waveform.
These "wave properties" are of practical importance when calculating the solution of the wave
equation for a number of different cases. As will be seen later, the wave number is used
extensively to describe wave phenomenon graphically and quantitatively.
For
further
information:
http://scienceworld.wolfram.com/physics/Wavenumber.html
Wave
Properties
Forced Vibrations
1.forced vibrations of infinite string suppose there is a string very long , at x=o there is force
exerted on it.
F(t)=Fcos(wt)=Real{Fexp(jwt)}
use the boundary condition at x=o,
neglect the reflect wave
it is easy to get the wave form
66
where w is the angular velocity, k is the wave number.
according to the impedance definition
it represent the characteristic impedance of the string. obviously, it is purely resistive, which is
like the restance in the mechanical system.
The dissipated power
Note: along the string, all the variable propagate at same speed.
link title a useful link to show the time-space property of the wave.
67
Some interesting animation of the wave at different boundary conditions.
1.hard boundary( which is like a fixed end)
2.soft boundary ( which is like a free end)
68
3.from low density to high density string
4.from high density to low density string
69
Part 3: Applications
Room Acoustics and Concert Halls
Introduction
From performing on many different rooms and stages all over the United States, I thought it
would be nice to have a better understanding and source about the room acoustics. This
Wikibook page is intended to help to the user with basic to technical questions/answers about
room acoustics. Main topics that will be covered are: what really makes a room sound good or
bad, alive or dead. This will lead into absorption and transmission coefficients, decay of sound in
the room, and reverberation. Different use of materials in rooms will be mentioned also. There is
no intention of taking work from another. This page is a switchboard source to help the user find
information about room acoustics.
Sound Fields
Two types of sound fields are involved in room acoustics: Direct Sound and Reverberant Sound.
Direct Sound
The component of the sound field in a room that involves only a direct path between the source
and the receiver, before any reflections off walls and other surfaces.
Reverberant Sound
The component of the sound field in a room that involves the direct path and the path after it
reflects off of walls or any other surfaces. How the waves deflect off of the mediums all depends
on the absorption and transmission coefficients.
Good example pictures are shown at Crutchfield Advisor
http://akamaipix.crutchfield.com/ca/reviews/20040120/roomacoustics1a.gif, a Physics Site from
MTSU http://physics.mtsu.edu/~wmr/reverb1f1.gif, and Voiceteacher.com
http://www.voiceteacher.com/art/bounce.gif
Room Coefficients
In a perfect world, if there is a sound shot right at a wall, the sound should come right back. But
because sounds hit different materials types of walls, the sound does not have perfect reflection.
From 1, these are explained as follows:
Absorption & Transmission Coefficients
The best way to explain how sound reacts to different mediums is through acoustical energy.
When sound impacts on a wall, acoustical energy will be reflected, absorbed, or transmitted
70
through the wall.
Absorption Coefficient:
Transmission Coefficient:
If all of the acoustic energy hits the wall and goes through the wall, the alpha would equal 1
because none of the energy had zero reflection but all absorption. This would be an example of a
dead or soft wall because it takes in everything and doesn't reflect anything back. Rooms that are
like this are called Anechoic Rooms which looks like this from Axiomaudio
http://www.axiomaudio.com/archives/22chamber.jpg.
If all of the acoustic energy hits the wall and all reflects back, the alpha would equal 0. This
would be an example of a live or hard wall because the sound bounces right back and does not
go through the wall. Rooms that are like this are called Reverberant Rooms like this McIntosh
http://www.roger-russell.com/revrm3.jpg room. Look how the walls have nothing attached to
them. More room for the sound waves to bounce off the walls.
71
Room Averaged Sound Absorption Coefficient
Not all rooms have the same walls on all sides. The room averaged sound absorption coefficient
can be used to have different types of materials and areas of walls averaged together.
RASAC:
Absorption Coefficients for Specific Materials
Basic sound absorption Coefficients are shown here at Acoustical Surfaces.
Brick, unglazed, painted alpha ~ .01 - .03 -> Sound reflects back
An open door alpha equals 1 -> Sound goes through
Units are in Sabins.
Sound Decay and Reverberation Time
In a large reverberant room, a sound can still propagate after the sound source has been turned
off. This time when the sound intensity level has decay 60 dB is called the reverberation time of
the room.
Great Reverberation Source
72
Great Halls in the World
Foellinger Great Hall
Japan
Budapest
Carnegie Hall in New York
Carnegie Hall
Pick Staiger at Northwestern U
Concert Hall Acoustics
References
[1] Lord, Gatley, Evensen; Noise Control for Engineers, Krieger Publishing, 435 pgs
Created by Kevin Baldwin
73
Bass Reflex Enclosure Design
Introduction
Bass-reflex enclosures improve the low-frequency response of loudspeaker systems. Bass-reflex
enclosures are also called "vented-box design" or "ported-cabinet design". A bass-reflex
enclosure includes a vent or port between the cabinet and the ambient environment. This type of
design, as one may observe by looking at contemporary loudspeaker products, is still widely
used today. Although the construction of bass-reflex enclosures is fairly simple, their design is
not simple, and requires proper tuning. This reference focuses on the technical details of bassreflex design. General loudspeaker information can be found here.
Effects of the Port on the Enclosure Response
Before discussing the bass-reflex enclosure, it is important to be familiar with the simpler sealed
enclosure system performance. As the name suggests, the sealed enclosure system attaches the
loudspeaker to a sealed enclosure (except for a small air leak included to equalize the ambient
pressure inside). Ideally, the enclosure would act as an acoustical compiance element, as the air
inside the enclosure is compressed and rarified. Often, however, an acoustic material is added
inside the box to reduce standing waves, dissipate heat, and other reasons. This adds a resistive
element to the acoustical lumped-element model. A non-ideal model of the effect of the
enclosure actually adds an acoustical mass element to complete a series lumped-element circuit
given in Figure 1. For more on sealed enclosure design, see the Sealed Box Subwoofer Design
page.
74
Figure 1. Sealed enclosure acoustic circuit.
In the case of a bass-reflex enclosure, a port is added to the construction. Typically, the port is
cylindrical and is flanged on the end pointing outside the enclosure. In a bass-reflex enclosure,
the amount of acoustic material used is usually much less than in the sealed enclosure case, often
none at all. This allows air to flow freely through the port. Instead, the larger losses come from
the air leakage in the enclosure. With this setup, a lumped-element acoustical circuit has the
following form.
Figure 2. Bass-reflex enclosure acoustic circuit.
In this figure, ZRAD represents the radiation impedance of the outside environment on the
loudspeaker diaphragm. The loading on the rear of the diaphragm has changed when compared
to the sealed enclosure case. If one visualizes the movement of air within the enclosure, some of
the air is compressed and rarified by the compliance of the enclosure, some leaks out of the
enclosure, and some flows out of the port. This explains the parallel combination of MAP, CAB,
and RAL. A truly realistic model would incorporate a radiation impedance of the port in series
with MAP, but for now it is ignored. Finally, MAB, the acoustical mass of the enclosure, is
included as discussed in the sealed enclosure case. The formulas which calculate the enclosure
parameters are listed in Appendix B.
It is important to note the parallel combination of MAP and CAB. This forms a Helmholtz
75
resonator (click here for more information). Physically, the port functions as the "neck" of the
resonator and the enclosure functions as the "cavity." In this case, the resonator is driven from
the piston directly on the cavity instead of the typical Helmholtz case where it is driven at the
"neck." However, the same resonant behavior still occurs at the enclosure resonance frequency,
fB. At this frequency, the impedance seen by the loudspeaker diaphragm is large (see Figure 3
below). Thus, the load on the loudspeaker reduces the velocity flowing through its mechanical
parameters, causing an anti-resonance condition where the displacement of the diaphragm is a
minimum. Instead, the majority of the volume velocity is actually emitted by the port itself
instead of the loudspeaker. When this impedance is reflected to the electrical circuit, it is
proportional to 1 / Z, thus a minimum in the impedance seen by the voice coil is small. Figure 3
shows a plot of the impedance seen at the terminals of the loudspeaker. In this example, fB was
found to be about 40 Hz, which corresponds to the null in the voice-coil impedance.
Figure 3. Impedances seen by the loudspeaker diaphragm and voice coil.
Quantitative Analysis of Port on Enclosure
The performance of the loudspeaker is first measured by its velocity response, which can be
found directly from the equivalent circuit of the system. As the goal of most loudspeaker designs
is to improve the bass response (leaving high-frequency production to a tweeter), low frequency
approximations will be made as much as possible to simplify the analysis. First, the inductance
of the voice coil, LE, can be ignored as long as
. In a typical loudspeaker, LE is
of the order of 1 mH, while RE is typically 8?c), thus an upper frequency limit is approximately 1
kHz for this approximation, which is certainly high enough for the frequency range of interest.
Another approximation involves the radiation impedance, ZRAD. It can be shown [1] that this
value is given by the following equation (in acoustical ohms):
Where J1(x) and H1(x) are types of Bessel functions. For small values of ka,
76
an
d
Hence, the low-frequency impedance on the loudspeaker is represented with an acoustic mass
MA1 [1]. For a simple analysis, RE, MMD, CMS, and RMS (the transducer parameters, or ThieleSmall parameters) are converted to their acoustical equivalents. All conversions for all
parameters are given in Appendix A. Then, the series masses, MAD, MA1, and MAB, are lumped
together to create MAC. This new circuit is shown below.
Figure 4. Low-Frequency Equivalent Acoustic Circuit
Unlike sealed enclosure analysis, there are multiple sources of volume velocity that radiate to the
outside environment. Hence, the diaphragm volume velocity, UD, is not analyzed but rather U0 =
UD + UP + UL. This essentially draws a "bubble" around the enclosure and treats the system as a
source with volume velocity U0. This "lumped" approach will only be valid for low frequencies,
but previous approximations have already limited the analysis to such frequencies anyway. It can
be seen from the circuit that the volume velocity flowing into the enclosure, UB = ? U0,
compresses the air inside the enclosure. Thus, the circuit model of Figure 3 is valid and the
relationship relating input voltage, VIN to U0 may be computed.
In order to make the equations easier to understand, several parameters are combined to form
other parameter names. First, ωB and ωS, the enclosure and loudspeaker resonance frequencies,
respectively, are:
Based on the nature of the derivation, it is convenient to define the parameters ω0 and h, the
Helmholtz tuning ratio:
A parameter known as the compliance ratio or volume ratio, α, is given by:
77
Other parameters are combined to form what are known as quality factors:
This notation allows for a simpler expression for the resulting transfer function [1]:
where
Development of Low-Frequency Pressure Response
It can be shown [2] that for ka < 1 / 2, a loudspeaker behaves as a spherical source. Here, a
represents the radius of the loudspeaker. For a 15" diameter loudspeaker in air, this low
frequency limit is about 150 Hz. For smaller loudspeakers, this limit increases. This limit
dominates the limit which ignores LE, and is consistent with the limit that models ZRAD by MA1.
Within this limit, the loudspeaker emits a volume velocity U0, as determined in the previous
section. For a simple spherical source with volume velocity U0, the far-field pressure is given by
[1]:
It is possible to simply let r = 1 for this analysis without loss of generality because distance is
only a function of the surroundings, not the loudspeaker. Also, because the transfer function
magnitude is of primary interest, the exponential term, which has a unity magnitude, is omitted.
Hence, the pressure response of the system is given by [1]:
78
Where H(s) = sG(s). In the following sections, design methods will focus on | H(s) | 2 rather than
H(s), which is given by:
This also implicitly ignores the constants in front of | H(s) | since they simply scale the response
and do not affect the shape of the frequency response curve.
Alignments
A popular way to determine the ideal parameters has been through the use of alignments. The
concept of alignments is based upon filter theory. Filter development is a method of selecting the
poles (and possibly zeros) of a transfer function to meet a particular design criterion. The criteria
are the desired properties of a magnitude-squared transfer function, which in this case is | H(s) | 2.
From any of the design criteria, the poles (and possibly zeros) of | H(s) | 2 are found, which can
then be used to calculate the numerator and denominator. This is the "optimal" transfer function,
which has coefficients that are matched to the parameters of | H(s) | 2 to compute the appropriate
values that will yield a design that meets the criteria.
There are many different types of filter designs, each which have trade-offs associated with
them. However, this design is limited because of the structure of | H(s) | 2. In particular, it has the
structure of a fourth-order high-pass filter with all zeros at s = 0. Therefore, only those filter
design methods which produce a low-pass filter with only poles will be acceptable methods to
use. From the traditional set of algorithms, only Butterworth and Chebyshev low-pass filters
have only poles. In addition, another type of filter called a quasi-Butterworth filter can also be
used, which has similar properties to a Butterworth filter. These three algorithms are fairly
simple, thus they are the most popular. When these low-pass filters are converted to high-pass
filters, the
transformation produces s8 in the numerator.
More details regarding filter theory and these relationships can be found in numerous resources,
including [5].
Butterworth Alignment
The Butterworth algorithm is designed to have a maximally flat pass band. Since the slope of a
function corresponds to its derivatives, a flat function will have derivatives equal to zero. Since
as flat of a pass band as possible is optimal, the ideal function will have as many derivatives
equal to zero as possible at s = 0. Of course, if all derivatives were equal to zero, then the
function would be a constant, which performs no filtering.
79
Often, it is better to examine what is called the loss function. Loss is the reciprocal of gain, thus
The loss function can be used to achieve the desired properties, then the desired gain function is
recovered from the loss function.
Now, applying the desired Butterworth property of maximal pass-band flatness, the loss function
is simply a polynomial with derivatives equal to zero at s = 0. At the same time, the original
polynomial must be of degree eight (yielding a fourth-order function). However, derivatives one
through seven can be equal to zero if [3]
With the high-pass transformation
,
It is convenient to define ?c) = ω / ω3dB, since
or -3 dB. This
defintion allows the matching of coefficients for the | H(s) | 2 describing the loudspeaker
response when ω3dB = ω0. From this matching, the following design equations are obtained [1]:
Quasi-Butterworth Alignment
The quasi-Butterworth alignments do not have as well-defined of an algorithm when compared
to the Butterworth alignment. The name "quasi-Butterworth" comes from the fact that the
transfer functions for these responses appear similar to the Butterworth ones, with (in general)
the addition of terms in the denominator. This will be illustrated below. While there are many
types of quasi-Butterworth alignments, the simplest and most popular is the 3rd order alignment
(QB3). The comparison of the QB3 magnitude-squared response against the 4th order
Butterworth is shown below.
Notice that the case B = 0 is the Butterworth alignment. The reason that this QB alignment is
80
called 3rd order is due to the fact that as B increases, the slope approaches 3 dec/dec instead of 4
dec/dec, as in 4th order Butterworth. This phenomenon can be seen in Figure 5.
Figure 5: 3rd-Order Quasi-Butterworth Response for
Equating the system response | H(s) | 2 with | HQB3(s) | 2, the equations guiding the design can be
found [1]:
Chebyshev Alignment
The Chebyshev algorithm is an alternative to the Butterworth algorithm. For the Chebyshev
response, the maximally-flat passband restriction is abandoned. Now, a ripple, or fluctuation is
allowed in the pass band. This allows a steeper transition or roll-off to occur. In this type of
application, the low-frequency response of the loudspeaker can be extended beyond what can be
achieved by Butterworth-type filters. An example plot of a Chebyshev high-pass response with
0.5 dB of ripple against a Butterworth high-pass response for the same ω3dB is shown below.
81
Figure 6: Chebyshev vs. Butterworth High-Pass Response.
The Chebyshev response is defined by [4]:
Cn(?c)) is called the Chebyshev polynomial and is defined by [4]:
cos[ncos ? 1(?c))]
| ?c) | < 1
cosh[ncosh ? 1(?c))]
| ?c) | > 1
Fortunately, Chebyshev polynomials satisfy a simple recursion formula [4]:
C0(x) = 1
C1(x) = x
Cn(x) = 2xCn ? 1 ? Cn ? 2
For more information on Chebyshev polynomials, see the Wolfram Mathworld: Chebyshev
Polynomials page.
When applying the high-pass transformation to the 4th order form of
response has the form [1]:
82
, the desired
The parameter ε determines the ripple. In particular, the magnitude of the ripple is 10log[1 + ε2]
dB and can be chosen by the designer, similar to B in the quasi-Butterworth case. Using the
recursion formula for Cn(x),
Applying this equation to | H(j?c)) | 2 [1],
Thus, the design equations become [1]:
Choosing the Correct Alignment
With all the equations that have already been presented, the question naturally arises, "Which
one should I choose?" Notice that the coefficients a1, a2, and a3 are not simply related to the
parameters of the system response. Certain combinations of parameters may indeed invalidate
one or more of the alignments because they cannot realize the necessary coefficients. With this in
mind, general guidelines have been developed to guide the selection of the appropriate
alignment. This is very useful if one is designing an enclosure to suit a particular transducer that
cannot be changed.
The general guideline for the Butterworth alignment focuses on QL and QTS. Since the three
coefficients a1, a2, and a3 are a function of QL, QTS, h, and α, fixing one of these parameters
yields three equations that uniquely determine the other three. In the case where a particular
transducer is already given, QTS is essentially fixed. If the desired parameters of the enclosure are
83
already known, then QL is a better starting point.
In the case that the rigid requirements of the Butterworth alignment cannot be satisfied, the
quasi-Butterworth alignment is often applied when QTS is not large enough.. The addition of
another parameter, B, allows more flexibility in the design.
For QTS values that are too large for the Butterworth alignment, the Chebyshev alignment is
typically chosen. However, the steep transition of the Chebyshev alignment may also be utilized
to attempt to extend the bass response of the loudspeaker in the case where the transducer
properties can be changed.
In addition to these three popular alignments, research continues in the area of developing new
algorithms that can manipulate the low-frequency response of the bass-reflex enclosure. For
example, a 5th order quasi-Butterworth alignment has been developed [6]. Another example [7]
applies root-locus techniques to achieve results. In the modern age of high-powered computing,
other researchers have focused their efforts in creating computerized optimization algorithms
that can be modified to achieve a flatter response with sharp roll-off or introduce quasi-ripples
which provide a boost in sub-bass frequencies [8].
References
[1] Leach, W. Marshall, Jr. Introduction to Electroacoustics and Audio Amplifier Design. 2nd ed.
Kendall/Hunt, Dubuque, IA. 2001.
[2] Beranek, L. L. Acoustics. 2nd ed. Acoustical Society of America, Woodbridge, NY. 1993.
[3] DeCarlo, Raymond A. "The Butterworth Approximation." Notes from ECE 445. Purdue
University. 2004.
[4] DeCarlo, Raymond A. "The Chebyshev Approximation." Notes from ECE 445. Purdue
University. 2004.
[5] VanValkenburg, M. E. Analog Filter Design. Holt, Rinehart and Winston, Inc. Chicago, IL.
1982.
[6] Kreutz, Joseph and Panzer, Joerg. "Derivation of the Quasi-Butterworth 5 Alignments."
Journal of the Audio Engineering Society. Vol. 42, No. 5, May 1994.
[7] Rutt, Thomas E. "Root-Locus Technique for Vented-Box Loudspeaker Design." Journal of
the Audio Engineering Society. Vol. 33, No. 9, September 1985.
[8] Simeonov, Lubomir B. and Shopova-Simeonova, Elena. "Passive-Radiator Loudspeaker
System Design Software Including Optimization Algorithm." Journal of the Audio Engineering
Society. Vol. 47, No. 4, April 1999.
84
Appendix A: Equivalent Circuit Parameters
Name
Electrical Equivalent
Voice-Coil
Resistance
RE
Driver
(Speaker)
Mass
See CMEC
Mechanical Equivalent
Acoustical Equivalent
MMD
Driver
(Speaker)
LCES = (Bl)2CMS
Suspension
Compliance
CMS
Driver
(Speaker)
Suspension
Resistance
RMS
Enclosure
Compliance
CAB
Enclosure
Air-Leak
Losses
RAL
Acoustic
Mass of Port
MAP
Enclosure
Mass Load
See CMEC
See MMC
MAB
LowFrequency
Radiation
Mass Load
See CMEC
See MMC
MA1
Combination
Mass Load
85
Appendix B: Enclosure Parameter Formulas
Figure 7: Important dimensions of bass-reflex enclosure.
Based on these dimensions [1],
VB = hwd (inside enclosure volume)
SB = wh (inside area of the side the speaker is
mounted on)
86
cair = specific heat of air at constant volume
cfill = specific heat of filling at constant volume
(Vfilling)
ρ0 = mean density of air (about 1.3 kg/m3)
ρfill = density of filling
γ = ratio of specific heats for air (1.4)
c0 = speed of sound in air (about 344 m/s)
ρeff = effective density of enclosure. If little or no filling (acceptable assumption in a bass-reflex
system but not for sealed enclosures),
87
New Acoustic Filter For Ultrasonics Media
Introduction
Acoustic filters are used in many devices such as mufflers, noise control materials (absorptive
and reactive), and loudspeaker systems to name a few. Although the waves in simple (singlemedium) acoustic filters usually travel in gases such as air and carbon-monoxide (in the case of
automobile mufflers) or in materials such as fiberglass, polyvinylidene fluoride (PVDF) film, or
polyethylene (Saran Wrap), there are also filters that couple two or three distinct media together
to achieve a desired acoustic response. General information about basic acoustic filter design can
be perused at the following wikibook page [Acoustic Filter Design & Implementation]. The
focus of this article will be on acoustic filters that use multilayer air/polymer film-coupled media
as its acoustic medium for sound waves to propagate through; concluding with an example of
how these filters can be used to detect and extrapolate audio frequency information in highfrequency "carrier" waves that carry an audio signal. However, before getting into these specific
type of acoustic filters, we need to briefly discuss how sound waves interact with the
medium(media) in which it travels and how these factors can play a role when designing acoustic
filters.
Changes in Media Properties Due to Sound Wave
Characteristics
As with any system being designed, the filter response characteristics of an acoustic filter are
tailored based on the frequency spectrum of the input signal and the desired output. The input
signal may be infrasonic (frequencies below human hearing), sonic (frequencies within human
hearing range), or ultrasonic (frequencies above human hearing range). In addition to the
frequency content of the input signal, the density, and, thus, the characteristic impedance of the
medium (media) being used in the acoustic filter must also be taken into account. In general, the
characteristic impedance
for a particular medium is expressed as...
where
= (equilibrium) density of medium
= speed of sound in medium
The characteristic impedance is important because this value simultaneously gives an idea of
how fast or slow particles will travel as well as how much mass is "weighting down" the
particles in the medium (per unit area or volume) when they are excited by a sound source. The
speed in which sound travels in the medium needs to be taken into consideration because this
factor can ultimately affect the time response of the filter (i.e. the output of the filter may not
radiate or attentuate sound fast or slow enough if not designed properly). The intensity
of a
88
sound wave is expressed as...
.
is interpreted as the (time-averaged) rate of energy transmission of a sound wave through a
unit area normal to the direction of propagation, and this parameter is also an important factor in
acoustic filter design because the characteristic properties of the given medium can change
relative to intensity of the sound wave traveling through it. In other words, the reaction of the
particles (atoms or molecules) that make up the medium will respond differently when the
intensity of the sound wave is very high or very small relative to the size of the control area (i.e.
dimensions of the filter, in this case). Other properties such as the elasticity and mean
propagation velocity (of a sound wave) can change in the acoustic medium as well, but focusing
on frequency, impedance, and/or intensity in the design process usually takes care of these other
parameters because most of them will inevitably be dependent on the aforementioned properties
of the medium.
Why Coupled Acoustic Media in Acoustic Filters?
In acoustic transducers, media coupling is employed in acoustic transducers to either increase or
decrease the impedance of the transducer, and, thus, control the intensity and speed of the signal
acting on the transducer while converting the incident wave, or initial excitation sound wave,
from one form of energy to another (e.g. converting acoustic energy to electrical energy).
Specifically, the impedance of the transducer is augmented by inserting a solid structure (not
necessarily rigid) between the transducer and the initial propagation medium (e.g. air). The
reflective properties of the inserted medium is exploited to either increase or decrease the
intensity and propagation speed of the incident sound wave. It is the ability to alter, and to some
extent, control, the impedance of a propagation medium by (periodically) inserting (a) solid
structure(s) such as thin, flexible films in the original medium (air) and its ability to
concomitantly alter the frequency response of the original medium that makes use of multilayer
media in acoustic filters attractive. The reflection factor and transmission factor
respectively, between two media, expressed as...
and
and
,
,
are the tangible values that tell how much of the incident wave is being reflected from and
transmitted through the junction where the media meet. Note that
is the (total) input
impedance seen by the incident sound wave upon just entering an air-solid acoustic media layer.
89
In the case of multiple air-columns as shown in Fig. 2,
is the aggregate impedance of each
air-column layer seen by the incident wave at the input. Below in Fig. 1, a simple illustration
explains what happens when an incident sound wave propagating in medium (1) and comes in
contact with medium (2) at the junction of the both media (x=0), where the sound waves are
represented by vectors.
Fig. 1
Illustration of How Incident, Reflected, and Transmitted Are Related
As mentioned above, an example of three such successive air-solid acoustic media layers is
shown in Fig. 2 and the electroacoustic equivalent circuit for Fig. 2 is shown in Fig. 3 where
= (density of solid material)(thickness of solid material) = unit-area (or volume)
characteristic acoustic impedance of medium, and
mass,
wavenumber. Note that in the case of a multilayer, coupled acoustic medium in an acoustic filter,
the impedance of each air-solid section is calculated by using the following general purpose
impedance ratio equation (also referred to as transfer matrices)...
where
is the (known) impedance at the edge of the solid of an air-solid layer (on the right)
and
is the (unknown) impedance at the edge of the air column of an air-solid layer.
90
Fig 2: Polymer Films In Acoustic
LPF w/ Related Impedances
Fig 3: Electroacoustic
Equivalent Circuit of Coupled Acoustic Media Layers
Effects of High-Intensity, Ultrasonic Waves in Acoustic
Media in Audio Frequency Spectrum
When an ultrasonic wave is used as a carrier to transmit audio frequencies, three audio effects
are associated with extrapolating the audio frequency information from the carrier wave: (a)
beating effects, (b) parametric array effects, and (c) radiation pressure.
Beating occurs when two ultrasonic waves with distinct frequencies
and
propagate in the
same direction, resulting in amplitude variations which consequently make the audio signal
information go in and out of phase, or "beat", at a frequency of
.
Parametric array effects occur when the intensity of an ultrasonic wave is so high in a particular
medium that the high displacements of particles (atoms) per wave cycle changes properties of
that medium so that it influences parameters like elasticity, density, propagation velocity, etc. in
a non-linear fashion. The results of parametric array effects on modulated, high-intensity,
ultrasonic waves in a particular medium (or coupled media) is the generation and propagation of
audio frequency waves (not necessarily present in the original audio information) that are
generated in a manner similar to the nonlinear process of amplitude demodulation commonly
inherent in diode circuits (when diodes are forward biased).
91
Another audio effect that arises from high-intensity ultrasonic beams of sound is a static (DC)
pressure called radiation pressure. Radiation pressure is similar to parametric array effects in that
amplitude variations in the signal give rise to audible frequencies via amplitude demodulation.
However, unlike parametric array effects, radiation pressure fluctuations that generate audible
signals from amplitude demodulation can occur due to any low-frequency modulation and not
just from pressure fluctuations occurring at the modulation frequency
or beating frequency
.
An Application of Coupled Media in Acoustic Filters
Transmission
Factor
vs.
Frequency
92
For
Fig 4: Test Setup For
Acoustic
Filter
Fig 5: Test Setup For Radiation
Pressure Factor vs. Frequency For Acoustic Filter
Figs. 1 - 3 were all from a research paper entitled New Type of Acoustics Filter Using Periodic
Polymer Layers for Measuring Audio Signal Components Excited by Amplitude-Modulated
High_Intensity Ultrasonic Wavessubmitted to the Audio Engineering Society (AES) by Minoru
Todo, Primary Innovator at Measurement Specialties, Inc., in the October 2005 edition of the
AES Journal. Figs. 4 and 5 below, also from this paper, are illustrations of test setups referred to
in this paper. Specifically, Fig. 4 is a test setup used to measure the transmission (of an incident
ultrasonic sound wave) through the acoustic filter described by Figs. 1 and 2. Fig. 5 is a block
diagram of the test setup used for measuring radiation pressure, one of the audio effects
mentioned in the previous section. It turns out that out of all of the audio effects mentioned in the
previous section that are caused by high-intensity ultrasonic waves propagating in a medium,
sound waves produced from radiated pressure are the hardest to detect when microphones and
preamplifiers are used in the detection/receiver system. Although nonlinear noise artifacts occur
due to overloading of the preamplifier present in the detection/receiver system, the bulk of the
nonlinear noise comes from the inherent nonlinear noise properties of microphones. This is true
because all microphones, even specialized measurement microphones designed for audio
spectrum measurements that have sensitivity well beyond the threshold of hearing, have
nonlinearities artifacts that (periodically) increase in magnitude with respect to increase at
ultrasonic frequencies. These nonlinearities essentially mask the radiation pressure generated
because the magnitude of these nonlinearities are orders of magnitude greater than the radiation
pressure. The acoustic (low-pass) filter referred to in this paper was designed in order to filter out
the "detrimental" ultrasonic wave that was inducing high nonlinear noise artifacts in the
measurement microphones. The high-intensity, ultrasonic wave was producing radiation pressure
(which is audible) within the initial acoustic medium (i.e. air). By filtering out the ultrasonic
wave, the measurement microphone would only detect the audible radiation pressure that the
ultrasonic wave was producing in air. Acoustic filters like these could possibly be used to
93
detect/receive any high-intensity, ultrasonic signal that may carry audio information which may
need to be extrapolated with an acceptable level of fidelity.
References
[1] Minoru Todo, "New Type of Acoustic Filter Using Periodic Polymer Layers for Measuring
Audio Signal Components Excited by Amplitude-Modulated High-Intensity Ultrasonic Waves,"
Journal of Audio Engineering Society, Vol. 53, pp. 930-41 (2005 October)
[2] Fundamentals of Acoustics; Kinsler et al, John Wiley & Sons, 2000
[3] ME 513 Course Notes, Dr. Luc Mongeau, Purdue University
[4] http://www.ieee-uffc.org/archive/uffc/trans/Toc/abs/02/t0270972.htm
Created by Valdez L. Gant
94
Noise in Hydraulic Systems
Noise in Hydraulic Systems
Hydraulic systems are the most preferred source of power transmission in most of the industrial
and mobile equipments due to their power denstiy, compactness, flexiblity, fast response and
efficiency. The field hydraulics and pneumatics is also known as 'Fluid Power Technology'.
Fluid power systems have a wide range of applications which include industrial, off-road
vehicles, automotive system and aircrafts. But, one of the main problems with the hydraulic
systems is the noise generated by them. The health and safety issues relating to noise have been
recognized for many years and legislation is now placing clear demands on manufacturers to
reduce noise levels [1]. Hence, noise reduction in hydraulic systems demands lot of attention
from the industrial as well as academic researchers. It needs a good understanding of how the
noise is generated and propagated in a hydraulic system in order to reduce it.
Sound in fluids
The speed of sound in fluids can be determined using the following relation.
where K - fluid bulk modulus, ρ- fluid density, c - velocity of sound
Typical value of bulk modulus range from 2e9 to 2.5e9 N/m2. For a particular oil, with a density
of 889 kg/m3,
speed of sound
Source of Noise
The main source of noise in hydraulic systems is the pump which supplies the flow. Most of the
pumps used are positive displacement pumps. Of the positive dispalcement pumps, axial piston
swash plate type is mostly preferred due to their controllability and efficiency.
The noise generation in an axial piston pump can be classifeid under two categories (i)
fluidborne nose and
(ii) Structureborne noise
Fluidborne Noise (FBN)
95
Among the positive displacement pumps, highest levels of FBN are generated by axial piston
pumps and lowest levels by screw pumps and in between these lie the external gear pump and
vane pump [1]. The discussion in this page is mainly focused on axial piston swash plate type
pumps. An axial piston pump has a fixed number of displacement chambers arranged in a
circular pattern seperated from each other by an angular pitch equal to
where n is the
number of displacement chambers. As each chamber discharges a specific volume of fluid, the
discharge at the pump outlet is sum of all the discharge from the individual chambers. The
discontinuity in flow between adjacent chambers results in a kinemtic flow ripple. The amplitude
of the kinematic ripple can be theoretical determined given the size of the pump and the number
of displament chambers. The kinematic ripple is the main cause of the fluidborne noise. The
kinematic ripples is a theoretical value. The actual flow ripple at the pump outlet is much larger
than the theoretical value because the kinematic ripple is combined with a compressibility
component which is due to the fluid compressibility. These ripples (also referred as flow
pulsations) generated at the pump are transmitted through the pipe or flexible hose connected to
the pump and travel to all parts of the hydraulic circuit.
The pump is considered an ideal flow source. The pressure in the system will be decided by
resistance to the flow or otherwise known as system load. The flow pulsations result in pressure
pulsations. The pressure pulsations are supreimposed on the mean system pressure. Both the
flow and pressure pulsations easily travel to all part of the circuit and affect the performance of
the components like control valve and actuators in the system and make the component vibrate,
sometimes even resonate. This vibration of system components adds to the noise generated by
the flow pulsations. The transmission of FBN in the circuit is discussed under transmission
below.
A typical axial piston pump with 9 pistons running at 1000 rpm can produce a sound pressure
level of more than 70 dBs.
Structure borne Noise (SBN)
In swash plate type pumps, the main sources of the structureborne noise are the fluctuating forces
and moments of the swas plate. These fluctuating forces arise as a result of the varying pressure
inside the displacement chamber. As the displacing elements move from suction stroke to
discharge stroke, the pressure varies accordingly from few bars to few hundred bars. This
pressure changes are reflected on the displacement elements (in this case, pistons) as forces and
these force are exerted on the swash plate causing the swash plate to vibrate. This vibration of
the swash plate is the main cause of structureborne noise. There are other components in the
system which also vibrate and lead to structureborne noise, but the swash is the major
contributor.
96
Fig. 1
shows an exploded view of axial piston pump. Also the flow pulsations and the oscillating
forces on the swash plate, which cause FBN and SBN respectively are shown for one
revolution of the pump.
Transmission
FBN
The transmission of FBN is a complex phenomenon. Over the past few decades, considerable
amount of research had gone into mathematical modeling of pressure and flow transient in the
circuit. This involves the solution of wave equations, with piping treated as a distributed
parameter system known as a transmission line [1] & [3].
Lets consider a simple pump-pipe-loading valve circuit as shown in Fig. 2. The pressure and
flow ripple at ay location in the pipe can be described by the relations:
.........(1)
.....(2)
where
and
are frequency dependent complex coefficients which are directly proportional
to pump (source) flow ripple, but also functions of the source impedance
, characteristic
97
impedance of the pipe
and the termination impedance
. These impedances ,usually vary as
the system operating pressure and flow rate changes, can be determined experimentally.
Fig.2
Schematic
of
a
pump
connected to a hydraulic line
Fig.3 Impedance representation of
pump-pipe-valve system
For complex systems with several system compenents, the pressure and flow ripples are
estimated using the tranformation matrix approach. For this, the system compenents can be
treated as lumped impedances (a throttle valve or accumulator), or distrubuted impedances
(flexible hose or silencer). Variuos software packages are available today to predict the pressure
pulsations.
SBN
The transmission of SBN follows the classic source-path-noise model. The vibrations of the
swash plate, the main cause of SBN, is transfered to the pump casing which encloses all the
rotating group in the pump including displacement chambers (also known as cylinder block),
pistons and the swash plate. The pump case, apart from vibrating itself, transfers the vibration
down to the mount on which the pump is mounted. The mount then passes the vibrations down to
the main mounted structure or the vehicle. Thus the SBN is transfered from the swash plate to
the main strucuture or vehicle via pumpcasing and mount.
Some of the machine structures, along the path of transmission, are good at transmitting this
vribational energy and they even resonate and reinforce it. By converting only a fraction of 1%
of the pump structureborne noise into sound, a member in the transmission path could radiate
more ABN than the pump itself [4].
Airborne noise (ABN)
Both FBN and SBN , impart high fatigue loads on the system components and make them
vibrate. All of these vibrations are radiated as airborne noise and can be heard by a human
98
operator. Also, the flow and pressure pulsations make the system components such as a control
valve to resonate. This vibration of the particular component again radiates airborne noise.
Noise reduction
The reduction of the noise radiated from the hydraulic system can be approached in two ways.
(i) Reduction at Source - which is the reduction of noise at the pump. A large amount of open
literature are availbale on the reduction techniques with some techniques focusing on reducing
FBN at source and others focusing on SBN. Reduction in FBN and SBN at the source has a large
influence on the ABN that is radiated. Even though, a lot of progress had been made in reducing
the FBN and SBN separately, the problem of noise in hydarulic systems is not fully solved and
lot need to be done. The reason is that the FBN and SBN are interlated, in a sense that, if one
tried to reduce the FBN at the pump, it tends to affect the SBN characteristics. Currently, one of
the main researches in noise reduction in pumps, is a systematic approach in understanding the
coupling between FBN and SBN and targeting them simultaneously instead of treating them as
two separte sources. Such an unified approach, demands not only well trained researchers but
also sophisticated computer based mathematical model of the pump which can accurately output
the necessary results for optimization of pump design.
(ii) Reduction at Component level - which focuses on the reduction of noise from individual
component like hose, control valve, pump mounts and fixtures. This can be accomplished by a
suitable design modification of the component so that it radiates least amount of noise.
Optimization using computer based models can be one of the ways.
99
Hydraulic System noise
Fig.4
Domain of hydraulic system noise generation and transmission (Figure recreated from [1])
References
1. Designing Quieter Hydraulic Systems - Some Recent Developements and Contributions, Kevin
Edge, 1999, Fluid Power: Forth JHPS International Symposium.
2. Fundamentals of Acoustics L.E. Kinsler, A.R. Frey, A.B.Coppens, J.V. Sanders. Fourth
Edition. John Wiley & Sons Inc.
3. Reduction of Axial Piston Pump Pressure Ripple A.M. Harrison. PhD thesis, University of
Bath. 1997
4. Noise Control of Hydraulic Machinery Stan Skaistis, 1988. MARCEL DEKKER , INC.
100
Basic Acoustics of the Marimba
Introduction
One of my favorite instruments is the marimba. Like a xylophone, a marimba has octaves of
wooden bars that are struck with mallets to produce tones. Unlike the harsh sound of a
xylophone, a marimba produces a deep, rich tone. Marimbas are not uncommon and are played
in most high school bands. Now, while all the trumpet and flute and clarinet players are busy
tuning up their instruments, the marimba player is back in the percussion section with her feet up
just relaxing. This is a bit surprising, however, since the marimba is a melodic instrument that
needs to be in tune to sound good. So what gives? Why is the marimba never tuned? How would
you even go about tuning a marimba? To answer these questions, the acoustics behind (or
within) a marimba must be understood.
Components of Sound
What gives the marimba its unique sound? It can be boiled down to two components: the bars
and the resonators. Typically, the bars are made of rosewood (or some synthetic version of
wood). They are cut to size depending on what note is desired, then the tuning is refined by
shaving wood from the underside of the bar.
101
Example
***Rosewood bar, middle C, 1 cm thick***
The equation that relates the length of the bar with the desired
frequency comes from the theory of modeling a bar that is free at
both ends.
This theory yields the following equation:
***
***
where t is the thickness of the bar, c is the speed of sound in the bar,
and f is the frequency of the note.
***For rosewood, c = 5217 m/s.
For middle C, f=262 Hz.***
Therefore, to make a middle C key for a rosewood marimba, cut the bar to be:
***
***
The resonators are made from metal (usually aluminum) and their lengths also differ depending
on the desired note. It is important to know that each resonator is open at the top but closed by a
stopper at the bottom end.
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Example
***Aluminum resonator, middle C***
The equation that relates the length of the resonator with the
desired frequency comes from modeling the resonator as a pipe
that is driven at one end and closed at the other end.
A "driven"
pipe is one that has a source of excitation (in this case, the
vibrating key) at one end.
***
This model yields the following:
***
where c is the speed of sound in air and f is the frequency of the note.
***For air, c = 343 m/s.
For middle C, f = 262 Hz.***
Therefore, to make a resonator for the middle C key, the resonator length
should be:
***
***
Resonator Shape
The shape of the resonator is an important factor in determining the quality of sound that can be
produced. The ideal shape is a sphere. This is modeled by the Helmholtz resonator. (For more
see Helmholtz Resonator page) However, mounting big, round, beach ball-like resonators under
the keys is typically impractical. The worst choices for resonators are square or oval tubes. These
103
shapes amplify the non-harmonic pitches sometimes referred to as "junk pitches". The round
tube is typically chosen because it does the best job (aside from the sphere) at amplifying the
desired harmonic and not much else.
As mentioned in the second example above, the resonator on a marimba can be modeled by a
closed pipe. This model can be used to predict what type of sound (full and rich vs dull) the
marimba will produce. As shown in the following figure, each pipe is a "quarter wave resonator"
that amplifies the sound waves produced by of the bar. This means that in order to produce a full,
rich sound, the length of the resonator must exactly match one-quarter of the wavelength. If the
length is off, the marimba will produce a dull or off-key sound for that note.
Why would the marimba need tuning?
In the theoretical world where it is always 72 degrees with low humidity, a marimba would not
need tuning. But, since weather can be a factor (especially for the marching band) marimbas do
not always perform the same way. Hot and cold weather can wreak havoc on all kinds of
percussion instruments, and the marimba is no exception. On hot days, the marimba tends to be
sharp and for cold days it tends to be flat. This is the exact opposite of what happens to string
instruments. Why? The tone of a string instrument depends mainly on the tension in the string,
which decreases as the string expands with heat. The decrease in tension leads to a flat note.
Marimbas on the other hand produce sound by moving air through the resonators. The speed at
which this air is moved is the speed of sound, which varies proportionately with temperature! So,
as the temperature increases, so does the speed of sound. From the equation given in example 2
from above, you can see that an increase in the speed of sound (c) means a longer pipe is needed
to resonate the same note. If the length of the resonator is not increased, the note will sound
sharp. Now, the heat can also cause the wooden bars to expand, but the effect of this expansion is
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insignificant compared to the effect of the change in the speed of sound.
Tuning Myths
It is a common myth among percussionists that the marimba can be tuned by simply moving the
resonators up or down (while the bars remain in the same position.) The thought behind this is
that by moving the resonators down, for example, you are in effect lengthening them. While this
may sound like sound reasoning, it actually does not hold true in practice. Judging by how the
marimba is constructed (cutting bars and resonators to specific lengths), it seems that there are
really two options to consider when looking to tune a marimba: shave some wood off the
underside of the bars, or change the length of the resonator. For obvious reasons, shaving wood
off the keys every time the weather changes is not a practical solution. Therefore, the only option
left is to change the length of the resonator. As mentioned above, each resonator is plugged by a
stopper at the bottom end. So, by simply shoving the stopper farther up the pipe, you can shorten
the resonator and sharpen the note. Conversely, pushing the stopper down the pipe can flatten the
note. Most marimbas do not come with tunable resonators, so this process can be a little
challenging. (Broomsticks and hammers are common tools of the trade.)
Example
***Middle C Resonator lengthened by 1 cm***
For ideal conditions, the length of the middle C (262 Hz) resonator should be
32.7 cm as shown in example 2.
Therefore, the change in frequency for this
resonator due to a change in length is given by:
***
***
If the length is increased by 1 cm, the change in frequency will be:
***
***
105
The acoustics behind the tuning a marimba go back to the design that each resonator is to be ?/4
of the total wavelength of the desired note. When marimbas get out of tune, this length is no
longer exactly equal to ?/4 the wavelength due to the lengthening or shortening of the resonator
as described above. Because the length has changed, resonance is no longer achieved, and the
tone can become muffled or off-key.
Conclusions
Some marimba builders are now changing their designs to include tunable resonators. Since any
leak in the end-seal will cause major loss of volume and richness of the tone, this is proving to be
a very difficult task. At least now, though, armed with the acoustic background of their
instruments, percussionists everywhere will now have something to do when the conductor says,
"tune up!"
Links and Referneces
1. http://www.gppercussion.com/html/resonators.html
2. http://www.mostlymarimba.com/
3. http://www.outback.chi.il.us/~bonnysu/craftymusicteachers/bassmarimba/index.html
106
How an Acoustic Guitar works
Introduction
sound vibrations that contribute to sound production. First of all, there are the strings. Any string
that is under tension will vibrate at a certain frequency. The weight and length of the string, the
tension in the string, and the compliance of the string determine the frequency at which it
vibrates. The guitar controls the length and tension of six differently weighted strings to cover a
very wide range of frequencies. Second, there is the body of the guitar. The guitar body is
connected directly to one end of each of the strings. The body receives the vibrations of the
strings and transmits them to the air around the body. It is the body?tm)s large surface area that
allows it to "push" a lot more air than a string. Finally, there is the air inside the body. This is
very important for the lower frequencies of the guitar. The air mass just inside the sound hole
oscillates, compressing and decompressing the compliant air inside the body. In practice this
concept is called a Helmholtz resonator. Without this, it would difficult to produce the wonderful
timbre of the guitar.
The Strings
The strings of the guitar vary in linear density, length, and tension. This gives the guitar a wide
range of attainable frequencies. The larger the linear density is, the slower the string vibrates.
The same goes for the length; the longer the string is the slower it vibrates. This causes a low
frequency. Inversely, if the strings are less dense and/or shorter they create a higher frequency.
The resonance frequencies of the strings can be calculated by
107
The string length, L, in the equation is what changes when a player presses on a string at a
certain fret. This will shorten the string which in turn increases the frequency it produces when
plucked. The spacing of these frets is important. The length from the nut to bridge determines
how much space goes between each fret. If the length is 25 inches, then the position of the first
fret should be located (25/17.817) inches from the nut. Then the second fret should be located
(25-(25/17.817))/17.817 inches from the first fret. This results in the equation
When a string is plucked, a disturbance is formed and travels in both directions away from point
where the string was plucked. These "waves" travel at a speed that is related to the tension and
linear density and can be calculated by
The waves travel until they reach the boundaries on each end where they are reflected back. The
link below displays how the waves propagate in a string.
Plucked String @ www.phys.unsw.edu
The strings themselves do not produce very much sound because they are so thin. They can't
"push" the air that surrounds them very effectively. This is why they are connected to the top
plate of the guitar body. They need to transfer the frequencies they are producing to a large
surface area which can create more intense pressure disturbances.
The Body
The body of the guitar transfers the vibrations of the bridge to the air that surrounds it. The top
plate contributes to most of the pressure disturbances, because the player dampens the back plate
and the sides are relatively stiff. This is why it is important to make the top plate out of a light
springy wood, like spruce. The more the top plate can vibrate, the louder the sound it produces
will be. It is also important to keep the top plate flat, so a series of braces are located on the
inside to strengthen it. Without these braces the top plate would bend and crack under the large
stress created by the tension in the strings. This would also affect the magnitude of the sound
being transmitted. The warped plate would not be able to "push" air very efficiently. A good
experiment to try, in order to see how important this part of the guitar is in the amplification
108
process, is as follows:
1. Start with an ordinary rubber band, a large bowl, adhesive tape, and plastic wrap.
2. Stretch the rubber band and pluck it a few times to get a good sense for how loud it is.
3. Stretch the plastic wrap over the bowl to form a sort of drum.
4. Tape down one end of the rubber band to the plastic wrap.
5. Stretch the rubber band and pluck it a few times.
6. The sound should be much louder than before.
The Air
The final part of the guitar is the air inside the body. This is very important for the lower range of
the instrument. The air just inside the soundhole oscillates compressing and expanding the air
inside the body. This is just like blowing across the top of a bottle and listening to the tone it
produces. This forms what is called a Helmholtz resonator. For more information on Helmholtz
resonators go to Helmholtz Resonance. This link also shows the correlation to acoustic guitars in
great detail. The acoustic guitar makers often tune these resonators to have a resonance
frequency between F#2 and A2 (92.5 to 110.0 Hz). Having such a low resonance frequency is
what aids the amplification of the lower frequency strings. To demonstrate the importance of the
air in the cavity, simply play an open A on the guitar (the second string). Now, as the string is
vibrating, place a peice of cardboard over the soundhole. The sound level is reduced
dramatically. This is because you've stopped the vibration of the air mass just inside the
soundhole, causing only the top plate to vibrate. Although the top plate still vibrates a transmitts
sound, it isn't as effective at transmitting lower frequency waves, thus the need for the Helmholtz
resonator.
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Specific application-automobile muffler
General information about Automobile muffler
Introduction
A muffler is a part of the exhaust system on an automobile that plays a vital role. It needs to have
modes that are located away from the frequencies that the engine operates at, whether the engine
be idling or running at the maximum amount of revolutions per second.A muffler that affects an
automobile in a negative way is one that causes noise or discomfort while the car engine is
running.Inside a muffler, you'll find a deceptively simple set of tubes with some holes in them.
These tubes and chambers are actually as finely tuned as a musical instrument. They are
designed to reflect the sound waves produced by the engine in such a way that they partially
cancel themselves out.( cited from www.howstuffworks.com )
It is very important to have it on the automobile. The legal limit for exhaust noise in the state of
California is 95dB (A) - CA. V.C. 27151 .Without a muffler the typical car exhaust noise would
exceed 110dB.A conventional car muffler is capable of limiting noise to about 90 dB. The
active-noise canceling muffler enables cancellation of exhaust noise to a wide range of
frequencies.
The Configuration of A automobile muffler
110
How Does automobile muffler function?
General Concept
The simple and main part of designing the automobile muffler is to use the low-pass filter. It
typically makes use of the change of the cross section area which can be made as a chamber to
filter or reduce the sound wave which the engine produced.
Low Pass Filter
the formula to be used:
Human ear sound reaction feature
When these pressure pulses reach your ear, the eardrum vibrates back and forth. Your brain
interprets this motion as sound. Two main characteristics of the wave determine how we
perceive the sound:
1.sound wave frequency. 2.air wave pressure amplitude.
It turns out that it is possible to add two or more sound waves together and get less sound.
111
Discription of the muffler to cancle the noise
The key thing about sound waves is that the result at your ear is the sum of all the sound waves
hitting your ear at that time. If you are listening to a band, even though you may hear several
distinct sources of sound, the pressure waves hitting your ear drum all add together, so your ear
drum only feels one pressure at any given moment. Now comes the cool part: It is possible to
produce a sound wave that is exactly the opposite of another wave. This is the basis for those
noise-canceling headphones you may have seen. Take a look at the figure below. The wave on
top and the second wave are both pure tones. If the two waves are in phase, they add up to a
wave with the same frequency but twice the amplitude. This is called constructive interference.
But, if they are exactly out of phase, they add up to zero. This is called destructive interference.
At the time when the first wave is at its maximum pressure, the second wave is at its minimum.
If both of these waves hit your ear drum at the same time, you would not hear anything because
the two waves always add up to zero.
Benefits of an Active Noise-Canceling Muffler
1.By using an active muffler the exhaust noise can be easily tuned, amplified, or nearly
eliminated.
2.The backpressure of a conventional muffler can be essentially eliminated, thus increasing
engine performance and efficiency.
3.By increasing engine efficiency and performance, less fuel will be used and the emissions will
be reduced.
Absorptive muffler
Lined ducts
It can be regarded as simplest form of absorptive muffler. Attach absorptive material to the bare
walls of the duct.( in car that is the exhaustion tube) The attenuation performance improves with
the thickness of absorptive material.
The attenuation curves like a skewed bell. Increase the thickness of the wall will get the lower
maximum attenuation frequency. For higher frequency though, thinner absorbent layers are
effective, but the large gap allows noise to pass directly along. Thin layers and narrow passages
are therefore more effective at high frequencies. For good absorption over the widest frequency
112
range, thick absorbent layers and narrow passages are best.
Parallel and block-line-of-sight baffles
Divide the duct into several channels or turn the flow channels so that there is no direct line-ofsight through the baffles. Frequently the materials line on the channels. Attnuation improves with
the thickness of absorptive material and length of the baffle. Lined bends can be used to provide
a greater attenuation and attenuate best at high frequency. Comparatively, at low frequency
attenuation can be increased by adding thicker lining.
Plenum chambers
They are relatively large volume chambers, usually fabricated from sheet metal, which
interconnect two ducts. The interior of the chamber is lined with absorbing material to attenuate
noise in the duct. Protective facing material may aslo be necessary if the temperature and
velocity conditions of the gas stream are too severe.
The performance of a plenum chamber can be improved by: 1.increase the thickness of the
absorbing lining 2.blocking the direct line of sight from the chamber inlet to the outlet. 3.increase
the cross-sectional area of the chamber.
References And Other Links
http://www.howstuffworks.com - howstuffworks
http://www.thecarforum.com/ - car forum
http://widget.ecn.purdue.edu/~me413/Index.html - acoustic noise control course
113
Bessel Functions and the Kettledrum
What is a kettledrum
A kettledrum is a percussion instrument with a circular drumhead mounted on a "kettle-like"
enclosure. When one strikes the drumhead with a mallet, it vibrates which produces its sound.
The pitch of this sound is determined by the tension of the drumhead, which is precisely tuned
before playing. The sound of the kettledrum (called the Timpani in classical music) is present in
many forms of music from many difference places of the world. It most famous role (no pun
intended) to those acquainted with classic movies was as the "drum" in the theme for 2001:A
Space Odyssey
The math behind the kettledrum: the brief version
When one looks at how a kettledrum produces sound, one should look no farther than the
drumhead. The vibration of this circular membrane (and the air in the drum enclosure) is what
produces the sound in this instrument. The mathematics behind this vibrating drum are relatively
simple. If one looks at a small element of the drum head, it looks exactly like the situation for the
114
vibrating string (see:). The only difference is that there are two dimensions where there are
forces on the element, the two dimensions that are planar to the drum. As this is the same
situation, we have the same equation, except with another spatial term in the other planar
dimension. This allows us to model the drumhead using a helmholtz equation. The next step
(solved in detail below) is to assume that the displacement of the drumhead (in polar
coordinates) is a product of two separate functions for theta and r. This allows us to turn the PDE
into two ODES which are readily solved and applied to the situation of the kettledrum head. For
more info, see below.
The math behind the kettledrum: the derivation
So starting with the trusty general Helmholtz equation:
Where k is the wave number, the frequency of the forced oscillations divided by the speed of
sound in the membrane.
Since we are dealing with a circular object, it make sense to work in polar coordinates (in terms
of radius and angle) instead of rectangular coordinates. For polar coordinates the Laplacian term
of the helmholtz relation (
) becomes
Now lets assume that:Ψ(r,θ) = R(r)Θ(θ)
This assumption follows the method of separation of variables. (see Reference 3 for more info)
Substituting this result back into our trusty Helmholtz equation gives the following:
r2 / R(d2R / dr2 + 1 / rdR / dr) + k2r2 = ? 1 / Θd2Θ / dθ2
Since we separated the variables of the solution into two one-dimensional functions, the partial
derivatives become ordinary derivatives. Both sides of this result must equal the same constant.
For simplicity, i will use λ as this constant. This results in the following two equations:
d2Θ / dθ2 = ? λ2Θ
d2R / dr2 + 1 / rdR / dr + (k2 ? λ2 / r2)R = 0
The first of these equations readily seen as the standard second order ordinary differential
equation which has a harmonic solution of sines and cosines with the frequency based on λ. The
second equation is what is known as Bessel's Equation. The solution to this equation is
cryptically called Bessel functions of order λ of the first and second kind. These functions, while
sounding very intimidating, are simply oscillatory functions of the radius times the wave number
that are unbounded at when kr (for the function of the second kind) approaches zero and
diminish as kr get larger. (For more information on what these functions look like see References
1,2, and 3)
115
Now that we have the general solution to this equation, we can now model a infinite radius
kettledrum head. However, since i have yet to see an infinate kettle drum, we need to constrain
this solution of a vibrating membrane to a finite radius. We can do this by applying what we
know about our circular membrane: along the edges of the kettledrum, the drum head is attached
to the drum. This means that there can be no displacement of the membrane at the termination at
the radius of the kettle drum. This boundary condiction can be mathematically discribed as the
following:
R(a) = 0
Where a is the arbirary radius of the kettledrum. In addition to this boundary condition, the
displacement of the drum head at the center must be finite. This second boundary condition
removes the bessel function of the second kind from the solution. This reduces the R part of our
solution to:
R(r) = AJλ(kr)
Where Jλ is a bessel function of the first kind of order λ. Apply our other boundary condition at
the radius of the drum requires that the wave number k must have discrete values, (jmn / a) which
can be looked up. Combining all of these gives us our solution to how a drumhead behaves
(which is the real part of the following):
The math behind the kettledrum:the entire drum
The above derivation is just for the drum head. An actual kettledrum has one side of this circular
membrane surrounded by an enclosed cavity. This means that air is compressed in the cavity
when the membrane is vibrating, adding more complications to the solution. In mathematical
terms, this makes the partial differential equation non-homogeneous or in simpler terms, the right
side of the Helmholtz equation does not equal zero. This result requires significantly more
derivation, and will not be done here. If the reader cares to know more, these results are
discussed in the two books under references 6 and 7.
Sites of interest
As one can see from the derivation above, the kettledrum is very interesting mathmatically.
However, it also has a rich historical music tradition in various places of the world. As this
page's emphasis is on math, there are few links provided below that reference this rich history.
A discussion of persian kettledrums: Kettle drums of Iran and other countries
http://www.drumdojo.com/world/persia/kettledrums.htm
A discussion of kettledrums in classical music: Kettle drum Lit.
http://cctr.umkc.edu/user/mgarlitos/timp.html
116
A massive resource for kettledrum history, construction and technique" Vienna Symphonic
Library http://www.vsl.co.at/en-us/70/3196/3198/5675.vsl
Wikibooks sister cite, references under Timpani: Wikipedia reference
References
1.Eric W. Weisstein. "Bessel Function of the First Kind." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
2.Eric W. Weisstein. "Bessel Function of the Second Kind." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html
3.Eric W. Weisstein. "Bessel Function." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/BesselFunction.html
4.Eric W. Weisstein et al. "Separation of Variables." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/SeparationofVariables.html
5.Eric W. Weisstein. "Bessel Differential Equation." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/BesselDifferentialEquation.html
6. Kinsler and Frey, "Fudamentals of Acoustics", fourth edition, Wiley & Sons
7. Haberman, "Applied Partial Differential Equations", fourth edition, Prentice Hall Press
117
Filter Design and Implementation
Introduction
Acoustic filters, or mufflers, are used in a number of applications requiring the suppression or
attenuation of sound. Although the idea might not be familiar to many people, acoustic mufflers
make everyday life much more pleasant. Many common appliances, such as refrigerators and air
conditioners, use acoustic mufflers to produce a minimal working noise. The application of
acoustic mufflers is mostly directed to machine components or areas where there is a large
amount of radiated sound such as high pressure exhaust pipes, gas turbines, and rotary pumps.
Although there are a number of applications for acoustic mufflers, there are really only two main
types which are used. These are absorptive and reactive mufflers. Absorptive mufflers
incorporate sound absorbing materials to attenuate the radiated energy in gas flow. Reactive
mufflers use a series of complex passages to maximize sound attenuation while meeting set
specifications, such as pressure drop, volume flow, etc. Many of the more complex mufflers
today incorporate both methods to optimize sound attenuation and provide realistic
specifications.
In order to fully understand how acoustic filters attenuate radiated sound, it is first necessary to
briefly cover some basic background topics. For more information on wave theory and other
material necessary to study acoustic filters please refer to the references below.
Basic Wave Theory
Although not fundamentally difficult to understand, there are a number of alternate techniques
used to analyze wave motion which could seem overwhelming to a novice at first. Therefore,
only 1-D wave motion will be analyzed to keep most of the mathematics as simple as possible.
This analysis is valid, with not much error, for the majority of pipes and enclosures encountered
in practice.
Plane-Wave Pressure Distribution in Pipes
The most important equation used is the wave equation in 1-D form (See [1],[2],
http://mathworld.wolfram.com/WaveEquation1-Dimensional.html,
http://en.wikibooks.org/wiki/Acoustic:Transverse_vibrations_of_strings#Characterization_of_th
e_mechanical_system for information).
Therefore, it is reasonable to suggest, if plane waves are propagating, that the pressure
distribution in a pipe is given by:
where Pi and Pr are incident and reflected wave amplitudes respectively. Also note that bold
notation is used to indicate the possiblily of complex terms. The first term represents a wave
118
travelling in the +x direction and the second term, -x direction.
Since acoustic filters or mufflers typically attenuate the radiated sound power as much as
possible, it is logical to assume that if we can find a way to maximize the ratio between reflected
and incident wave amplitude then we will effectively attenuated the radiated noise at certain
frequencies. This ratio is called the reflection coefficient and is given by:
It is important to point out that wave reflection only occurs when the impedance of a pipe
changes. It is possible to match the end impedance of a pipe with the characteristic impedance of
a pipe to get no wave reflection. For more information see [1] or [2].
Although the reflection coefficient isn't very useful in its current form since we want a relation
describing sound power, a more useful form can be derived by recognizing that the power
intensity coefficient is simply the magnitude of reflection coefficient square [1]:
As one would expect, the power reflection coefficient must be less than or equal to one.
Therefore, it is useful to define the transmission coefficient as:
which is the amount of power transmitted. This relation comes directly from conservation of
energy. When talking about the performance of mufflers, typically the power transmission
coefficient is specified.
Basic Filter Design
For simple filters, a long wavelength approximation can be made to make the analysis of the
system easier. When this assumption is valid (e.g. low frequencies) the components of the
system behave as lumped acoustical elements. Equations relating the various properties are
easily
derived
under
these
circumstances,
see
http://en.wikibooks.org/wiki/Acoustic:Acoustics_of_pipes%2C_enclosures%2C_and_cavities_at
_low_frequency for further information.
The following derivations assume long wavelength. Practical applications for most conditions
are given later.
119
Low-Pass Filter
Tpi for Low-Pass Filter
These are devices that attenuate the radiated sound power at higher frequencies. This means the
power transmission coefficient is approximently 1 across the band pass at low frequencies(see
figure to right).
This is equivalent to an expansion in a pipe, with the volume of gas located in the expansion
having an acoustic compliance (see figure to right). Continuity of acoustic impedance at the
junction, see [1], gives a power transmission coefficient of:
where k is the wavenumber, L & S1 are length and area of expansion respectively, and S is the
area
of
the
pipe.
(see
Java
Applet
at:
http://www.ndted.org/EducationResources/CommunityCollege/Ultrasonics/Physics/acousticimpedance.htm and
http://en.wikibooks.org/wiki/Acoustic:Boundary_Conditions_and_Forced_Vibrations#Wave_Pro
perties)
The cut-off frequency is given by:
120
High-Pass Filter
Tpi for High-Pass Filter
These are devices that attenuate the radiated sound power at lower frequencies. Like before, this
means the power transmission coefficient is approximently 1 across the band pass at high
frequencies (see figure to right).
This is equivalent to a short side brach (see figure to right) with a radius and length much smaller
than the wavelength (lumped element assumption). This side branch acts like an acoustic mass
and applies a different acoustic impedance to the system than the low-pass filter. Again using
continuity of acoustic impedance at the junction yields a power transmission coefficient of the
form [1]:
where a and L are the area and effective length of the small tube, and S is the area of the pipe.
The cut-off frequency is given by:
121
Band-Stop Filter
Tpi for Band-Stop Filter
These are devices that attenuate the radiated sound power over a certain frequency range (see
figure to right). Like before, the power transmission coefficient is approximently 1 in the band
pass region.
Since the band-stop filter is essentially a cross between a low and high pass filter, one might
expect to create one by using a combination of both techniques. This is true in that the
combination of a lumped acoustic mass and compliance gives a band-stop filter. This can be
realized as a helmholtz resonator (see [Helmholtz Resonator] or figure to right). Again, since the
impedance of the helmholtz resonator can be easily determined, continuity of acoustic
impedance at the junction can give the power transmission coefficient as [1]:
where Sb is the area of the neck, L is the effective length of the neck, V is the volume of the
helmholtz resonator, and S is the area of the pipe. It is interesting to note that the power
transmission coefficient is zero when the frequency is that of the resonance frequency of the
helmholtz. This can be explained by the fact that at resonance the volume velocity in the neck is
large with a phase such that all the incident wave is reflected back to the source [1].
The zero power transmission coefficient location is given by:
This frequency value has powerful implications. If a system has the majority of noise at one
frequency component, the system can be "tuned" using the above equation, with a helmholtz
resonator, to perfectly attenuate any transmitted power (see examples below).
122
Helmholtz
Resonator as a Muffler, f = 60 Hz
Helmholtz
Resonator as a Muffler, f = fc
Design
If the long wavelength assumption is valid, typically a combination of methods described above
are used to design a filter. A specific design procedure is outlined for a helmholtz resonator, and
other
basic
filters
follow
a
similar
procedure
(see
[1
http://www.silex.com/pdfs/Exhaust%20Silencers.pdf]).
Two main metrics need to be identified when designing a helmholtz resonator [3]:
(1) - Resonance frequency desired:
where
.
(2) - Transmission loss:
based on TL level. This constant is found from a
TL graph (see [HR http://mecheng.osu.edu/~selamet/docs/2003_JASA_113(4)_19751985_helmholtz_ext_neck.pdf] pp. 6).
This will result in two equations with two unknowns which can be solved for the unknown
dimensions of the helmholtz resonator. It is important to note that flow velocities degrade the
amount of transmission loss at resonance and tend to move the resonance location upwards [3].
In many situations, the long wavelength approximation is not valid and alternative methods must
be examined. These are much more mathematically rigorous and require a complete
understanding acoustics involved. Although the mathematics involved are not shown, common
filters used are given in the section that follows.
Actual Filter Design
As explained previously, there are two main types of filters used in practice: absorptive and
reactive. The benefits and drawback of each will be briefly expained, along with their relative
applications
(see
[Absorptive
Mufflers
http://en.wikibooks.org/wiki/Acoustic:specific_applicationautomobile_muffler#Absorptive_muffler].
123
Absorptive
These are mufflers which incorporate sound absorbing materials to transform acoustic energy
into heat. Unlike reactive mufflers which use destructive interferance to minimize radiated sound
power, absorptive mufflers are typically straight through pipes lined with multiple layers of
absorptive materials to reduce radiated sound power. The most important property of absorptive
mufflers is the attenuation constant. Higher attenuation constants lead to more energy dissipation
and lower radiated sound power.
Advantages of Absorptive Mufflers [3]:
(1) - High amount of absorption at larger frequencies.
(2) - Good for applications involving broadband (constant across the spectrum) and narrowband
(see [1]) noise.
(3) - Reduced amount of back pressure compared to reactive mufflers.
Disadvantages of Absorptive Mufflers [3]:
(1) - Poor performance at low frequencies.
(2) - Material can degrade under certain circumstances (high heat, etc).
Examples
Absorptive Muffler
There are a number of applications for absorptive mufflers. The most well known application is
in racecars, where engine performance is desired. Absorptive mufflers don't create a large
amount of back pressure (as in reactive mufflers) to attenuate the sound, which leads to higher
muffler performance. It should be noted however, that the radiate sound is much higher. Other
applications include plenum chambers (large chambers lined with absorptive materials, see
picture below), lined ducts, and ventilation systems.
124
Reactive
Reactive mufflers use a number of complex passages (or lumped elements) to reduce the amount
of acoustic energy transmitted. This is acomplished by a change in impedance at the
intersections, which gives rise to reflected waves (and effectively reduces the amount of
transmitted acoustic energy). Since the amount of energy transmitted is minimized, the reflected
energy back to the source is quite high. This can actually degrade the performance of engines
and other sources. Opposite to absorptive mufflers, which dissipate the acoustic energy, reactive
mufflers keep the energy contained within the system. See [Reactive Mufflers
http://en.wikibooks.org/wiki/Acoustic:Car_Mufflers#The_reflector_muffler Reactive Mufflers]
for more information.
Advantages of Reactive Mufflers [3]:
(1) - High performance at low frequencies.
(2) - Typically give high insertion loss, IL, for stationary tones.
(3) - Useful in harsh conditions.
Disadvantages of Reactive Mufflers [3]:
(1) - Poor performance at high frequencies.
(2) - Not desirable characteristics for broadband noise.
Examples
Reflective Muffler
Reactive mufflers are the most widely used mufflers in combustion engines[1]. Reactive
mufflers are very efficient in low frequency applications (especially since simple lumped
element analysis can be applied). Other application areas include: harsh environments (high
temperature/velocity engines, turbines, etc), specific frequency attenuation (using a helmholtz
like device, a specific frequency can be toned to give total attenuation of radiated sound power),
and a need for low radiated sound power (car mufflers, air conditioners, etc).
Performance
There are 3 main metrics used to describe the performance of mufflers; Noise Reduction,
Insertion Loss, and Transmission Loss. Typically when designing a muffler, 1 or 2 of these
125
metrics is given as a desired value.
Noise Reduction (NR)
Defined as the difference between sound pressure levels on the source and receiver side. It is
essentially the amount of sound power reduced between the location of the source and
termination of the muffler system (it doesn't have to be the termination, but it is the most
common location) [3].
where Lp1 and Lp2 is sound pressure levels at source and receiver respectively. Although NR is
easy to measure, pressure typically varies at source side due to standing waves [3].
Insertion Loss (IL)
Defined as difference of sound pressure level at the receiver with and without sound attenuating
barriers. This can be realized, in a car muffler, as the difference in radiated sound power with just
a straight pipe to that with an expansion chamber located in the pipe. Since the expansion
chamber will attenuate some of the radiate sound power, the pressure at the receiver with sound
attenuating barriers will be less. Therefore, a higher insertion loss is desired [3].
where Lp,without and Lp,with are pressure levels at receiver without and with a muffler system
respectively. Main problem with measuring IL is that the barrier or sound attenuating system
needs to be removed without changing the source [3].
Transmission Loss (TL)
Defined as the difference between the sound power level of the incident wave to the muffler
system and the transmitted sound power. For further information see [Transmission Loss
http://freespace.virgin.net/mark.davidson3/TL/TL.html] [3].
with
where It and Ii are the transmitted and incident wave power respectively. From this expression, it
is obvious the problem with measure TL is decomposing the sound field into incident and
transmitted waves which can be difficult to do for complex systems (analytically).
126
Examples
(1) - For a plenum chamber (see figure below):
in dB
where α is average absorption coefficient.
Plenum
Chamber
Transmission Loss vs.
Theta
(2) - For an expansion (see figure below):
where
Expansion
in
NR, IL, & TL for
Infinite Pipe
Expansion
(3) - For a helmholtz resonator (see figure below):
127
in dB
Helmholtz
Resonator
TL for Helmholtz
Resonator
Links
[1] - Muffler/silencer applications and descriptions of performance criteria [Exhaust Silencers]
http://www.silex.com/pdfs/Exhaust%20Silencers.pdf
[2]
Engineering
Acoustics,
http://widget.ecn.purdue.edu/~me513/
Purdue
University
-
[ME
[3] - Sound Propagation [Animations] http://widget.ecn.purdue.edu/~me513/animate.html
[4] - Exhaust Muffler [Design] http://myfwc.com/boating/airboat/Section3.pdf
[5] - Project Proposal &
References
[1] - Fundamentals of Acoustics; Kinsler et al, John Wiley & Sons, 2000
[2] - Acoustics; Pierce, Acoustical Society of America, 1989
[3] - ME 413 Noise Control, Dr. Mongeau, Purdue University
128
513].
Flow-induced oscillations of a Helmholtz
resonator and applications
Introduction
The importance of flow excited acoustic resonance lies in the large number of applications in
which it occurs. Sound production in organ pipes, compressors, transonic wind tunnels, and open
sunroofs are only a few examples of the many applications in which flow excited resonance of
Helmholtz resonators can be found.[4] An instability of the fluid motion coupled with an
acoustic resonance of the cavity produce large pressure fluctuations that are felt as increased
sound pressure levels. Passengers of road vehicles with open sunroofs often experience
discomfort, fatigue, and dizziness from self-sustained oscillations inside the car cabin. This
phenomenon is caused by the coupling of acoustic and hydrodynamic flow inside a cavity which
creates strong pressure oscillations in the passenger compartment in the 10 to 50 Hz frequency
range. Some effects experienced by vehicles with open sunroofs when buffeting include:
dizziness, temporary hearing reduction, discomfort, driver fatigue, and in extreme cases nausea.
The importance of reducing interior noise levels inside the car cabin relies primarily in reducing
driver fatigue and improving sound transmission from entertainment and communication
devices. This Wikibook page aims to theoretically and graphically explain the mechanisms
involved in the flow-excited acoustic resonance of Helmholtz resonators. The interaction
between fluid motion and acoustic resonance will be explained to provide a thorough explanation
of the behavior of self-oscillatory Helmholtz resonator systems. As an application example, a
description of the mechanisms involved in sunroof buffeting phenomena will be developed at the
end of the page.
Feedback loop analysis
As mentioned before, the self-sustained oscillations of a Helmholtz resonator in many cases is a
continuous interaction of hydrodynamic and acoustic mechanisms. In the frequency domain, the
flow excitation and the acoustic behavior can be represented as transfer functions. The flow can
be decomposed into two volume velocities.
qr: flow associated with acoustic response of cavity
qo: flow associated with excitation
Figure 1 shows the feedback loop of these two volume velocities.
129
Figure
1
Acoustical characteristics of the resonator
Lumped parameter model
The lumped parameter model of a Helmholtz resonator consists of a rigid-walled volume open to
the environment through a small opening at one end. The dimensions of the resonator in this
model are much less than the acoustic wavelength, in this way allowing us to model the system
as a lumped system.
where re is the equivalent radius of the orifice.
Figure 2 shows a sketch of a Helmholtz resonator on the left, the mechanical analog on the
middle section, and the electric-circuit analog on the right hand side. As shown in the Helmholtz
resonator drawing, the air mass flowing through an inflow of volume velocity includes the mass
inside the neck (Mo) and an end-correction mass (Mend). Viscous losses at the edges of the neck
length are included as well as the radiation resistance of the tube. The electric-circuit analog
shows the resonator modeled as a forced harmonic oscillator. [1] [2][3]
130
Figure
2
V: cavity volume
ρ: ambient density
c: speed of sound
S: cross-section area of orifice
K: stiffness
Ma: acoustic mass
Ca: acoustic compliance
The equivalent stiffness K is related to the potential energy of the flow compressed inside the
cavity. For a rigid wall cavity it is approximately:
The equation that describes the Helmholtz resonator is the following:
131
: excitation pressure
M: total mass (mass inside neck Mo plus end correction, Mend)
R: total resistance (radiation loss plus viscous loss)
From the electrical-circuit we know the following:
The main cavity resonance parameters are resonance frequency and quality factor which can be
estimated using the parameters explained above (assuming free field radiation, no viscous losses
and leaks, and negligible wall compliance effects)
The sharpness of the resonance peak is measured by the quality factor Q of the Helmholtz
resonator as follows:
fr: resonance frequency in Hz
ωr: resonance frequency in radians
L: length of neck
L': corrected length of neck
From the equations above, the following can be deduced:
-The greater the volume of the resonator, the lower the resonance frequencies.
-If the length of the neck is increased, the resonance frequency decreases.
132
Production of self-sustained oscillations
The acoustic field interacts with the unstable hydrodynamic flow above the open section of the
cavity, where the grazing flow is continuous. The flow in this section separates from the wall at a
point where the acoustic and hydrodynamic flows are strongly coupled. [5]
The separation of the boundary layer at the leading edge of the cavity (front part of opening from
incoming flow) produces strong vortices in the main stream. As observed in Figure 3, a shear
layer crosses the cavity orifice and vortices start to form due to instabilities in the layer at the
leading edge.
Figure 3
From Figure 3, L is the length of the inner cavity region, d denotes the diameter or length of the
cavity length, D represents the height of the cavity, and δ describes the gradient length in the
grazing velocity profile (boundary layer thickness).
The velocity in this region is characterized to be unsteady and the perturbations in this region
will lead to self-sustained oscillations inside the cavity. Vortices will continually form in the
opening region due to the instability of the shear layer at the leading edge of the opening.
Applications to Sunroof Buffeting
How are vortices formed during buffeting?
In order to understand the generation and convection of vortices from the shear layer along the
sunroof opening, the animation below has been developed. At a certain range of flow velocities,
self-sustained oscillations inside the open cavity (sunroof) will be predominant. During this
period of time, vortices are shed at the trailing edge of the opening and continue to be convected
along the length of the cavity opening as pressure inside the cabin decreases and increases. Flow
133
visualization experimentation is one method that helps obtain a qualitative understanding of
vortex formation and conduction.
The animation below, shows in the middle, a side view of a car cabin with the sunroof open. As
the air starts to flow at a certain mean velocity Uo, air mass will enter and leave the cabin as the
pressure decreases and increases again. At the right hand side of the animation, a legend shows a
range of colors to determine the pressure magnitude inside the car cabin. At the top of the
animation, a plot of circulation and acoustic cavity pressure versus time for one period of
oscillation is shown. The symbol x moving along the acoustic cavity pressure plot is
synchronized with pressure fluctuations inside the car cabin and with the legend on the right. For
example, whenever the x symbol is located at the point where t=0 (when the acoustic cavity
pressure is minimum) the color of the car cabin will match that of the minimum pressure in the
legend (blue).
The perturbations in the shear layer propagate with a velocity of the order of 1/2Uo which is half
the mean inflow velocity. [5] After the pressure inside the cavity reaches a minimum (blue color)
the air mass position in the neck of the cavity reaches its maximum outward position. At this
point, a vortex is shed at the leading edge of the sunroof opening (front part of sunroof in the
direction of inflow velocity). As the pressure inside the cavity increases (progressively to red
color) and the air mass at the cavity entrance is moved inwards, the vortex is displaced into the
neck of the cavity. The maximum downward displacement of the vortex is achieved when the
pressure inside the cabin is also maximum and the air mass in the neck of the Helmholtz
resonator (sunroof opening) reaches its maximum downward displacement. For the rest of the
remaining half cycle, the pressure cavity falls and the air below the neck of the resonator is
moved upwards. The vortex continues displacing towards the downstream edge of the sunroof
where it is convected upwards and outside the neck of the resonator. At this point the air below
134
the neck reaches its maximum upwards displacement.[4] And the process starts once again.
How to identify buffeting
Flow induced tests performed over a range of flow velocities are helpful to determine the change
in sound pressure levels (SPL) inside the car cabin as inflow velocity is increased. The following
animation shows typical auto spectra results from a car cabin with the sunroof open at various
inflow velocities. At the top right hand corner of the animation, it is possible to see the inflow
velocity and resonance frequency corresponding to the plot shown at that instant of time.
It is observed in the animation that the SPL increases gradually with increasing inflow velocity.
Initially, the levels are below 80 dB and no major peaks are observed. As velocity is increased,
the SPL increases throughout the frequency range until a definite peak is observed around a 100
Hz and 120 dB of amplitude. This is the resonance frequency of the cavity at which buffeting
occurs. As it is observed in the animation, as velocity is further increased, the peak decreases and
disappears. In this way, sound pressure level plots versus frequency are helpful in determining
increased sound pressure levels inside the car cabin to find ways to minimize them. Some of the
methods used to minimize the increased SPL levels achieved by buffeting include: notched
deflectors, mass injection, and spoilers.
135
Useful Websites
This link: http://www.exa.com/ takes you to the website of EXA Corporation, a developer of
PowerFlow for Computational Fluid Dynamics (CFD) analysis.
This link: http://www.cd-adapco.com/press_room/dynamics/20/saab.html
article about the current use of(CFD) software to model sunroof buffeting.
is a small news
This link: http://www.cd-adapco.com/products/brochures/industry_applications/autoapps.pdf is
a small industry brochure that shows the current use of CFD for sunroof buffeting.
References
[1] Acoustics: An introduction to its Physical Principles and Applications ; Pierce, Allan D.,
Acoustical Society of America, 1989.
[2] Prediction and Control of the Interior Pressure Fluctuations in a Flow-excited Helmholtz
resonator ; Mongeau, Luc, and Hyungseok Kook., Ray W. Herrick Laboratories, Purdue
University, 1997.
[3]Influence of leakage on the flow-induced response of vehicles with open sunroofs ; Mongeau,
Luc, and Jin-Seok Hong., Ray W. Herrick Laboratories, Purdue University.
[4]Fluid dynamics of a flow excited resonance, part I: Experiment ; P.A. Nelson, Halliwell and
Doak.; 1991.
[5]An Introduction to Acoustics ; Rienstra, S.W., A. Hirschberg., Report IWDE 99-02,
Eindhoven University of Technology, 1999.
Wiki page created by Paloma Y. Mejia
Questions and/or comments? Send e-mail to [email protected]
136
Acoustics in Violins
Acoustics of the Violin
For detail anatomy of violin, please refer to Atelierla Bussiere.
How Does A Violin Make Sound?
General Concept
When a violinist bows a string, which can produce vibrations with abundant harmonics. The
vibrations of the strings are structurally transmitted to the bridge and the body of the instrument
through the bridge. The bridge transmits the vibrational energy produced by the strings to the
body through its feet, further triggering the vibration of body. The vibration of the body
determines sound radiation and sound quality, along with the resonance of the cavity.
137
String
The vibration pattern of the strings can be easily be observed. To the naked eye, the string
appears to move back and forth in a parabolic shape (see figure), which resembles the first mode
of free vibration of a stretched string. The vibration of strings was first investigated by Hermann
Von Helmholtz, the famous mathematician and physicist in 19th century. A surprising scenario
was discovered that the string actually moves in an inverse "V" shape rather than parabolas (see
figure). What we see is just an envelope of the motion of the string. To honor his findings, the
motion of bowed strings had been called "Helmholtz motion."
Bridge
The primary role of the bridge is to transform the motion of vibrating strings into periodic
driving forces by its feet to the top plate of the violin body. The configuration of the bridge can
be referred to the figure. The bridge stands on the belly between f holes, which have two primary
functions. One is to connect the air inside the body with outside air, and the other one is to make
the belly between f holes move more easily than other parts of the body. The fundamental
frequency of a violin bridge was found to be around 3000 Hz when it is on a rigid support, and it
is an effective energy-transmitting medium to transmit the energy from the string to body at
frequencies from 1 KHz to 4KHz, which is in the range of keen sensitivity of human hearing. In
138
order to darken the sound of violin, the player attaches a mute on the bridge. The mute is actually
an additional mass which reduces the fundamental frequency of the bridge. As a result, the sound
at higher frequencies is diminished since the force transferred to the body has been decreased.
On the other hand, the fundamental frequency of the bridge can be raised by attaching an
additional stiffness in the form of tiny wedges, and the sound at higher frequencies will be
amplified accordingly.
The sound post connects the flexible belly to the much stiffer back plate. The sound post can
prevent the collapse of the belly due to high tension force in the string, and, at the same time,
couples the vibration of the plate. The bass bar under the belly extends beyond the f holes and
transmits the force of the bridge to a larger area of the belly. As can be seen in the figure, the
motion of the treble foot is restricted by the sound post, while, conversely, the foot over bass bar
can move up and down more easily. As a result, the bridge tends to move up and down, pivoting
about the treble foot. The forces appearing at the two feet remain equal and opposite up to 1
KHz. At higher frequencies, the forces become uneven. The force on the soundpost foot
predominates at some frequencies, while it is the bass bar foot at some.
Body
The body includes top plate, back plate, the sides, and the air inside, all of which serve to
transmit the vibration of the bridge into the vibration of air surrounding the violin. For this
reason, the violin needs a relatively large surface area to push enough amount of air back and
forth. Thus, the top and back plates play important roles in the mechanism. Violin makers have
traditionally pay much attention on the vibration of the top and back plates of the violin by
listening to the tap tones, or, recently, by observing the vibration mode shapes of the body plates.
The vibration modes of an assembled violin are, however, much more complicated.
The vibration modes of top and back plates can be easily observed in a similar technique first
performed by Ernest Florens Friedrich Chaldni (1756 ? 1827), who is often respectfully referred
"the father of acoustics." First, the fine sand is uniformly sprinkled on the plate. Then, the plate
can be resonated, either by a powerful sound wave tuned to the desired frequencies, by being
bowed by a violin bow, or by being excited mechanically or electromechanically at desired
frequencies. Consequently, the sand disperses randomly due to the vibration of plate. Some of
the sand falls outside the region of plate, while some of the sand is collected by the nodal
regions, which have relatively small movement, of the plate. Hence, the mode shapes of the plate
139
can be visualized in this manner, which can be refered to the figures in the reference site, Violin
Acoustics. The first seven modes of the top and back plates of violin are presented, with nodal
lines depicted by using black sands.
The air inside the body is also important, especially in the range of lower frequencies. It is like
the air inside a bottle when you blow into the neck, or, as known as Helmholtz resonance, which
has its own modes of vibration. The air inside the body can communicate with air outside
through the f holes, and the outside air serves as medium carrying waves from the violin.
see www.violinbridges.co.uk for more articles on bridges and accoustics.
Sound Radiation
A complete description of sound radiation of a violin should include the information about
radiation intensity as functions both of frequency and location. The sound radiation can be
measured by a microphone connected to a pressure level meter which is rotatably supported on a
stand arm around the violin, while the violin is fastened at the neck by a clip. The force is
introduced into the violin by using a miniature impact hammer at the upper edge of the bridge in
the direction of bowing. The detail can be referred to Martin Schleske, master studio for
violinmaking . The radiation intensity of different frequencies at different locations can be
represented by directional characteristics, or acoustic maps. The directional characteristics of a
violin can be shown in the figure in the website of Martin Schleske, where the radial distance
from the center point represents the absolute value of the sound level (re 1Pa/N) in dB, and the
angular coordinate of the full circle indicates the measurement point around the instrument.
According to the directional characteristics of violins, the principal radiation directions for the
violin in the horizontal plane can be established. For more detail about the principal radiation
direction for violins at different frequencies, please refer to reference (Meyer 1972).
References And Other Links
•
Violin Acoustics http://www.phys.unsw.edu.au/music/violin/
•
Paul Galluzzo's Homepage http://www2.eng.cam.ac.uk/~pmg26/home_frame.html
•
Martin Schleske, master studio for violinmaking
http://www.schleske.de/index.php?lang=en&http://www.schleske.de/06geigenbauer/en_a
kustik3schall3messmeth.shtml
•
Atelierla Bussiere http://www.atelierlabussiere.com/
•
Fletcher, N. H., and Rossing, T. D., The physics of musical instrument, Springer-Verlag,
1991
•
Meyer, J., "Directivity of bowed stringed instruments and its effect on orchestral sound in
concert halls", J. Acoustic. Soc. Am., 51, 1972, pp. 1994-2009
140
Moving Coil Loudspeaker
Moving Coil Transducer
The purpose of the acoustic transducer is to convert electrical energy into acoustic energy. Many
variations of acoustic transducers exist, although the most common is the moving coil-permanent
magnet transducer. The classic loudspeaker is of the moving coil-permanent magnet type.
The classic electrodynamic loudspeaker driver can be divided into three key components:
1) The Magnet Motor Drive System
2) The Loudspeaker Cone System
3) The Loudspeaker Suspension
141
Figure 1 Cut-away of
a moving coil-permanent magnet loudspeaker
The Magnet Motor Drive System
The main purpose of the Magnet Motor Drive System is to establish a symmetrical magnetic
field in which the voice coil will operate. The Magnet Motor Drive System is comprised of a
front focusing plate, permanent magnet, back plate, and a pole piece. In figure 2, the assembled
drive system is illustrated. In most cases, the back plate and the pole piece are built into one
piece called the yoke. The yoke and the front focusing plate are normally made of a very soft
cast iron. Iron is a material that is used in conjunction with magnetic structures because the iron
is easily saturated when exposed to a magnetic field. Notice in figure 2, that an air gap was
intentionally left between the front focusing plate and the yoke. The magnetic field is coupled
through the air gap. The magnetic field strength (B) of the air gap is typically optimized for
uniformity across the gap. [1]
142
Figure 2 Permanent
Magnet Structure
When a coil of wire with a current flowing is place inside the permanent magnetic field, a force
is produced. B is the magnetic field strength, l is the length of the coil, and I is the current
flowing through the coil.
F = Bli
143
Figure
3
Voice
Coil
Mounted in Permanent Magnetic Structure
The coil is excited with the AC signal that is intended for sound reproduction, when the changing
magnetic field of the coil interacts with the permanent magnetic field then the coil moves back
and forth in order to reproduce the input signal. The coil of a loudspeaker is known as the voice
coil.
Figure 4 Photograph - Voice Coil
The Loudspeaker Cone System
On a typical loudspeaker, the cone serves the purpose of creating a larger radiating area allowing
144
more air to be moved when excited by the voice coil. The cone serves a piston that is excited by
the voice coil. The cone then displaces air creating a sound wave. In an ideal environment, the
cone should be infinitely rigid and have zero mass, but in reality neither is true. Cone materials
vary from carbon fiber, paper, bamboo, and just about any other material that can be shaped into
a stiff conical shape. The loudspeaker cone is a very critical part of the loudspeaker. Since the
cone is not infinitely rigid, it tends to have different types of resonance modes form at different
frequencies, which in turn alters and colors the reproduction of the sound waves. The shape of
the cone directly influences the directivity and frequency response of the loudspeaker. When the
cone is attached to the voice coil, a large gap above the voice coil is left exposed. This could be a
problem if foreign particles make their way into the air gap of the voice coil and the permanent
magnet structure. The solution to this problem is to place what is known as a dust cap on the
cone to cover the air gap. Below a figure of the cone and dust cap are shown.
Figure 6 Cone and
Dust Cap attached to Voice Coil
The Loudspeaker Suspension
Most moving coil loudspeakers have a two piece suspension system, also known as a flexure
system. The combination of the two flexures allows the voice coil to maintain linear travel as the
voice coil is energized and provide a restoring force for the voice coil system. The two piece
145
system consists of large flexible membrane surrounding the outside edge of the cone, called the
surround, and an additional flexure connected directly to the voice coil, called the spider. The
surround has another purpose and that is to seal the loudspeaker when mounted in an enclosure.
Commonly, the surround is made of a variety of different materials, such as, folded paper, cloth,
rubber, and foam. Construction of the spider consists of different woven cloth or synthetic
materials that are compressed to form a flexible membrane. The following two figures illustrate
where the suspension components are physically at on the loudspeaker and how they function as
the loudspeaker operates.
Loudspeaker
Suspension
146
Figure
7
System
Figure
8
Moving
Loudspeaker
Modeling the Loudspeaker as a Lumped System
Before implementing a loudspeaker into a specific application, a series of parameters
characterizing the loudspeaker must be extracted. The equivalent circuit of the loudspeaker is
key when developing enclosures. The circuit models all aspects of the loudspeaker through an
equivalent electrical, mechanical, and acoustical circuit. Figure 9 shows how the three equivalent
circuits are connected. The electrical circuit is comprised of the DC resistance of the voice coil,
Re, the imaginary part of the voice coil inductance, Le, and the real part of the voice coil
inductance, Revc. The mechanical system has electrical components that model different
physical parameters of the loudspeaker. In the mechanical circuit, Mm, is the electrical
capacitance due to the moving mass, Cm, is the electrical inductance due to the compliance of
the moving mass, and Rm, is the electrical resistance due to the suspension system. In the
147
acoustical equivalent circuit, Ma models the air mass and Ra models the radiation impedance[2].
This equivalent circuit allows insight into what parameters change the characteristics of the
loudspeaker. Figure 10 shows the electrical input impedance as a function of frequency
developed using the equivalent circuit of the loudspeaker.
9
Loudspeaker
Analogous
Figure
Circuit
Figure
10 Electrical Input Impedance
References
[1] The Loudspeaker Design Cookbook 5th Edition; Dickason, Vance., Audio Amateur Press,
1997. [2] Beranek, L. L. Acoustics. 2nd ed. Acoustical Society of America, Woodbridge, NY.
1993.
Links
This page was created by Joe Land.
148
Attenuation of Sound Waves
Introduction
When sound travels through a medium, its intensity diminishes with distance. This weakening in
the energy of the wave results from two basic causes, scattering and absorption. The combined
effect of scattering and absorption is called attenuation. For small distances or short times the
effects of attenuation in sound waves can usually be ignored. Yet, for practical reasons it should
be considered. So far in our discussions, sound has only been dissipated by the spreading of the
wave, such as when we consider spherical and cylindrical waves. However this dissipation of
sound in these cases is due to geometric effects associated with energy being spread over an
increasing area and not actually to any loss of total energy.
Types of Attenuation
As mentioned above, attenuation is caused by both absorption and scattering. Absorption is
generally caused by the media. This can be due to energy loss by both viscosity and heat
conduction. Attenuation due to adsorption is important when the volume of the material is large.
Scattering, the second cause of attenuation, is important when the volume is small or in cases of
thin ducts and porous materials.
Viscosity and Heat conduction
Whenever there is a relative motion between particles in a media, such as in wave propogation,
energy loss occurs. This is due to stress from viscous forces between particles of the medium.
The energy lost is converted to heat. Because of this, the intensity of a sound wave decreases
more rapidly than the inverse square of distance. Viscosity in gases is dependant upon
temperature for the most part. Thus as you increase the temperature you increase the viscous
forces.
Boundary Layer Losses
A special type of adsorption occurs when a sound wave travels over a boundary, such as a fluid
flowing over a solid surface. In such a situation, the fluid in immediate contact with the surface
must be at rest. Subsiquent layers of fluid will have a velocity that increases as the distance from
the solid surface increases such as in the figure below.
149
The velocity gradient causes an internal stress associated with viscosity, that leads to a loss of
momentum. This loss of momentum leads to a decrease in the amplitude of a wave close to the
surface. The region over with the velocity of the fluid decreases from its nominal velocity to that
of zero is called the acoustic boundary layer. The thickness of the acoustic boundary layer do to
viscosity can be expressed as
Where
is the shear viscosity number. Ideal fluids would not have a boundary layer thickness
since
.
Relaxation
Attenuation can also occur by a process called relaxation. One of the basic assumptions prior to
this discussion on attenuation was that when a pressure or density of a fluid or media depended
only on the instantaneous values of density and temperature and not on the rate of change in
these variables. However, whenever a change occurs, equilibrium is upset and the media adjusts
until a new local equilibrium is achieved. This does not occur instantaneously, and pressure and
density will vary in the media. The time it takes to achieve this new equilibrium is called the
relaxation time, math> \theta \,</math> . As a consequence the speed of sound will increase from
an initial value to that of a maximum as frequancy increases. Again the losses associated with
relaxation are due to mechanical energy being transformed into heat.
150
Modeling of losses
The following is done for a plane wave. Losses can be introduced by the addition of a complex
expression for the wave number
which when substituded into the time-solution yeilds
with a new term of
which resulted from the use of a complex wave number. is known as
the absorption coefficient with units of nepers per unit distance (the neper is dB to base e) and
is related to the phase speed. The absorption coefficient is frequency dependant and is
generally proportional to the square of sound frequency. However, its relationship does vary
when considering the different absorption mechanisms as shown below.
The velocity of the particles can be expressed as
The impedance for this travelling wave would be given by
From this we can see that the rate of decrease in intensity of an attenuated wave is
References
Wood, A. A Textbook of Sound. London: Bell, 1957.
151
Blackstone, David. Fundamentals of Physical Acoustics. New York: Wiley, 2000.
Attenuation considerations in Ultrasound http://www.ndted.org/EducationResources/CommunityCollege/Ultrasonics/Physics/attenuation.htm
152
Car Mufflers
Introduction
A car muffler is a component of the exhaust system of a car. The exhaust system has mainly 3
functions:
1) Getting the hot and noxious gas from the engine away from the vehicle
2) Reduce exhaust emission
3) Attenuating the noise output from the engine
The last specified function is the function of the car muffler. It is necessary because the gas
coming from the combustion in the pistons of the engine would generate an extremely loud noise
if it were sent directly in the ambient surrounding through the exhaust valves. There are mainly 2
techniques used to dampen the noise: the absorption and the reflection. Each technique has its
advantages and inconvenient.
The absorber muffler
The muffler is composed of a tube covered by an sound absorbing stuff. The tube is perforated so
that some part of the sound wave goes through the perforation to the absorbing stuff. The
absorbing material is usually made of fiberglass or steel wool. The dampening material is
protected from the surrounding by a supplementary coat made of a bend metal sheet.
The advantages of this method are a low back pressure a relatively simple design. The
inconvenient of this method is a low sound damping compared to the other techniques, especially
at low frequency.
The mufflers using the absorption technique are usually sports vehicle because they increase the
performances of the engine because of their low back pressure . A trick to improve there
muffling ability consist of lining up several "straight" mufflers.
153
The reflector muffler
Principle: Sound wave reflection is used to create a maximum amount of destructive
interferences
Definition of destructive interferences
Let's consider the noise a person would hear when a car drives past. This sound would physically
correspond to the pressure variation of the air which would make his ear-drum vibrate. The curve
A1 of the graph 1 could represent this sound. The pressure amplitude is a function of the time at
a certain fixed place. If another sound wave A2 is produced at the same time, the pressure of the
two waves will add. If the amplitude of A1 is exactly the opposite of the amplitude A2, then the
sum will be zero, which corresponds physically to the atmospheric pressure. The listener would
thus hear nothing although there are two radiating sound sources. A2 is called the destructive
interference.
Definition of the reflection
The sound is a traveling wave i.e. its position changes in function of the time. As long as the
wave travels in the same medium, there is no change of speed and amplitude. When the wave
reaches a frontier between two mediums which have different impedances, the speed, and the
pressure amplitude change (and so does the angle if the wave does not propagate perpendicularly
154
to the frontier). The figure 1 shows two medium A and B and the 3 waves: incident transmitted
and reflected.
Example
If plan sound waves are propagating across a tube and the section of the tube changes at a point
x, the impedance of the tube will change. A part of the incident waves will so be transmitted in
the part of the tube with the new section value and the other part of the incident waves will be
reflected.
Animation http://en.wikibooks.org/wiki/Engineering_Acoustics/Car_Mufflers:Animation
The muffler using the reflection technique are the most commonly used because they damp the
noise much better than the absorber muffler. There have nevertheless a higher back pressure
which lower the performances of the engine. The actual best way to take the best power of the
engine would simply be not to use any muffler.
The upper right image represents a Car Muffler typical architecture. It is composed of 3 tubes.
There are 3 areas separated by plates, the part of the tubes located in the middle area are
perforated. Small quantity of pressure "escapes" from the tubes through the perforation and
cancel one another.
Some muffler using the reflection pricipe also incorporate some cavities which dampen the
noise. These cavities are called in accoutics Helmotz Resonators. This feature is usually only
available for up market class mufflers.
155
Back pressure
Car engines are 4 stroke cycle engines. Out of these 4 strokes, only one produces the power, this
is when the explosion occurs and pushes the pistons back. The other 3 strokes are necessary evil
that don't produce energy. They on the contrary consume energy. During the exhaust stroke, the
remaining gas from the explosion is expelled from the cylinder. The higher the pressure behind
the exhaust valves (i.e. back pressure), and the higher effort necessary to expel the gas out of the
cylinder. So, a low back pressure is preferable in order to have a higher engine horsepower.
Muffler Modeling by Transfer Matrix Method
This method is easy to use on computer to obtain theoretical values for the transmission loss of a
muffler. The transmission loss gives a value in dB that correspond to the ability of the muffler to
dampen the noise.
Example
P stands for Pressure [Pa] and U stand for volume velocity [m3/s]
=
So, finaly:
and
=
and
=
156
=
with
=
Si stands for the cross section area
k is the angular velocity
is the medium density
c is the speed of sound of the medium
Results
Matlab
code
of
the
http://upload.wikimedia.org/wikibooks/en/6/69/Matlab_code.jpg
graph
above.
Comments
The higher the value of the transmission loss and the better the muffler.
The transmission loss depends on the frequency. The sound frequency of a car engine is
approximately between 50 and 3000Hz. At resonance frequencies, the transmission loss is zero.
These frequencies correspond to the lower peaks on the graph.
The transmission loss is independent of the applied pressure or velocity at the input.
157
The temperature (about 600 Fahrenheit) has an impact on the air properties : the speed of sound
is higher and the mass density is lower.
The elementary transfer matrice depends on the element which is modelled. For instance the
transfer matrice of a Helmotz Resonator is
with
The transmission loss and the insertion loss are different terms. The transmission loss is 10 times
the logarithm of the ratio output/input. The insertion loss is 10 times the logarithm of the ratio of
the radiated sound power with and without muffler.
Links
More information about the Transfer Matrice Method :
www.scielo.br/pdf/jbsmse/v27n2/25381.pdf
General information about filters: Filter Design & Implementation
General information about car mufflers: http://auto.howstuffworks.com/muffler.htm
Example of car exhaust manufacturer
http://www.performancepeddler.com/manufacturer.asp?CatName=Magnaflow
158
Noise from cooling fans
Proposal
As electric/electronic devices get smaller and functional, the noise of cooling device becomes
important. My page will explain the origins of noise generation from small axial cooling fans
used in electronic goods like desktop/laptop computers. The source of fan noises includes
aerodynamic noise as well as operating sound of the fan itself. This page will be focused on the
aerodynamic noise generation mechanisms.
Introduction
If one opens his desktop computer, he may find three (or more) fans. Each fan is on the heat sink
of the CPU, in the back panel of the power supply unit, on the case ventilation hole, and maybe
on the graphic card, plus on the motherboard chipset if it is very recent one. The noise from a
computer that annoys people is mostly due to cooling fans if the hard drive(s) is fairly quiet.
When Intel Pentium processors first introduced, there was no need to have a fan on CPU at all,
but CPUs of these days cannot function even for several seconds without a cooling fan, and still
require more and more amount of blow, which causes more and more noise. The type of fans
used in a desktop computer is most likely axial fans, and centrifugal blowers are used in laptop
computers.
Several
fan
types
are
shown
here
(pdf
format)
http://www.etrinet.com/tech/pdf/aerodynamics.pdf. Different fan types have different
characteristics of noise generation and performance. The axial flow fan is mainly considered in
this page.
Noise Generation Mechanisms
The figure below shows a typical noise spectrum of a 120 mm diameter electronic device
cooling fan. One microphone is used at the point 1 m far from the upstream side of the fan. The
fan has 7 blades, 4 struts for motor mounting and operates at 13V. Certain amount of load is
applied. The blue plot is background noise of anechoic chamber, and the green one is sound
loudness spectrum when the fan is running.
159
(*BPF = Blade Passing Frequency) Each noise elements shown in this figure is caused by one or
more of following generation mechanisms.
Blade Thickness Noise - Monopole (But very weak)
Blade thickness noise is generated by volume displacement of fluid. Fan blades has its thickness
and volume. As the rotor rotates, the volume of each blade displaces fluid volume, then they
consequently fluctuate pressure of near field, and noise is generated. This noise is tonal at the
running frequency and generally very weak for cooling fans, because their RPM is relatively
low. Therefore, thickness of fan blades hardly affects to electronic cooling fan noise.
(This kind of noise can become severe for high speed turbomachines like helicopter rotors.)
Tonal Noise by Aerodynamic Forces - Dipole
Uniform Inlet Flow (Negligible)
The sound generation due to uniform and steady aerodynamic force has very similar
characteristic as the blade thickness noise. It is very weak for low speed fans, and depends on fan
RPM. Since at least of ideal steady blade forces are necessary for a fan to do its duty, even in an
160
ideal condition, this kind of noise is impossible to be avoided. It is known that this noise can be
reduced by increasing the number of blades.
Non-uniform Inlet Flow
Non-uniform (still steady) inlet flow causes non-uniform aerodynamic forces on blades as their
angular positions change. This generates noise at blade passing frequency and its harmonics. It is
one of the major noise sources of electronic cooling fans.
Rotor-Casing interaction
If the fan blades are very close to a structure which is not symmetric, unsteady interaction forces
to blades are generated. Then the fan experiences a similar running condition as lying in nonuniform flow field.
Impulsive Noise (Negligible)
This noise is caused by the interaction between a blade and blade-tip-vortex of the preceding
blade, and not severe for cooling fans.
Rotating Stall
Click here to read the definition and an aerodynamic description of stall.
The noise due to stall is a complex phenomenon that occurs at low flow rates. For some reason,
if flow is locally disturbed, it can cause stall on one of the blades. As a result, the upstream
passage on this blade is partially blocked. Therefore, the mean flow is diverted away from this
passage. This causes increasing of the angle of attack on the closest blade at the upstream side of
the originally stalled blade, the flow is again stalled there. On the other hand, the other side of
the first blade is un-stalled because of reduction of flow angle.
161
repeatedly, the stall cell turns around the blades at about 30~50% of the running frequency, and
the direction is opposite to the blades. This series of phenomenon causes unsteady blade forces,
and consequently generates noise and vibrations.
Non-uniform Rotor Geometry
Asymmetry of rotor causes noise at the rotating frequency and its harmonics (not blade passing
frequency obviously), even when the inlet flow is uniform and steady.
Unsteady Flow Field
Unsteady flow causes random forces on the blades. It spreads the discrete spectrum noises and
makes them continuous. In case of low-frequency variation, the spreaded continuous spectral
noise is around rotating frequency, and narrowband noise is generated. The stochastic velocity
fluctuations of inlet flow generates broadband noise spectrum. The generation of random noise
components is covered by the following sections.
Random Noise by Unsteady Aerodynamic Forces
Turbulent Boundary Layer
Even in the stady and uniform inlet flow, there exist random force fluctuations on the blades.
That is from turbulent blade boundary layer. Some noise is generated for this reason, but
dominant noise is produced by the boundary layer passing the blade trailing edge. The blade
trailing edges scatter the non-propagating near-field pressure into a propagatable sound field.
Incident Turbulent
Velocity fluctuations of the intake flow with a stochastic time history generate random forces on
blades, and a broadband spectrum noise.
Vortex Shedding
For some reason, a vortex can separate from a blade. Then the circulating flow around the blade
starts to be changed. This causes non-uniform forces on blades, and noises. A classical example
for this phenomenon is 'Karman vortex street'
http://www.galleryoffluidmechanics.com/vortex/karman.htm. (some images and animations
http://www2.icfd.co.jp/menu1/karmanvortex/karman.html.) Vortex shedding mechanism can
occur in a laminar boundary layer of low speed fan and also in a turbulent boundary layer of high
frequency fan.
Flow Separation
Flow separation causes stall explained above. This phenomenon can cause random noise, which
spreads all the discrete spectrum noises, and turns the noise into broadband.
Tip Vortex
Since cooling fans are ducted axial flow machines, the annular gap between the blade tips and
162
the casing is important parameter for noise generation. While rotating, there is another flow
through the annular gap due to pressure difference between upstream and downstream of fan.
Because of this flow, tip vortex is generated through the gap, and broadband noise increases as
the annular gap gets bigger.
Installation Effects
Once a fan is installed, even though the fan is well designed acoustically, unexpected noise
problem can come up. It is called as installation effects, and two types are applicable to cooling
fans.
Effect of Inlet Flow Conditions
A structure that affects the inlet flow of a fan causes installation effects. For example Hoppe &
Neise [3] showed that with and without a bellmouth nozzle at the inlet flange of 500mm fan can
change the noise power by 50dB (This application is for much larger and noisier fan though).
Acoustic Loading Effect
This effect is shown on duct system applications. Some high performance graphic cards apply
duct system for direct exhaustion.
The sound power generated by a fan is not only a function of its impeller speed and operating
condition, but also depends on the acoustic impedances of the duct systems connected to its inlet
and outlet. Therefore, fan and duct system should be matched not only for aerodynamic noise
reasons but also because of acoustic considerations.
Closing Comment
Noise reduction of cooling fans has some restrictions:
1. 1. Active noise control is not economically effective. 80mm cooling fans are only 5~10
US dollars. It is only applicable for high-end electronic products.
2. 2. Restricting certain aerodynamic phenomenon for noise reducion can cause serious
performance reduction of the fan. Increasing RPM of the fan is of course much more
dominant factor for noise.
Different stories of fan noise are introduced at some of the linked sites below like active RPM
control or noise comparison of various bearings used in fans.
Links to Interesting Sites about Fan Noise
•
Cooling Fan Noise Comparison - Sleeve Bearing vs. Ball Bearing (pdf format)
http://www.silentpcreview.com/files/ball_vs_sleeve_bearing.pdf
•
Brief explanation of fan noise origins and noise reduction suggestions
163
http://www.jmcproducts.com/cooling_info/noise.shtml
•
Effect of sweep angle comparison http://www.ansys.com/industries/tm-fan-noise.htm
•
Comparisons of noise from various 80mm fans http://www.directron.com/noise.html
•
Noise reduction of a specific desktop case
http://www.xlr8yourmac.com/G4ZONE/G4_fan_noise.html
•
Noise reduction of another specific desktop case
http://www.xlr8yourmac.com/systems/quicksilver_noise/quieting_quicksilver_noise.html
•
Informal study for noise from CPU cooling fan http://www.cpemma.co.uk/ Informal
study for noise from PC case fans
•
Informal study for noise from PC case fans
http://www.tomshardware.com/2004/06/15/fighting_fan_noise_pollution/index.html
•
Active fan speed optimizators for minimum noise from desktop computers
•
Some general fan noise reduction technics
http://www.diracdelta.co.uk/science/source/f/a/fan%20noise/source.html
•
Various applications and training in - Br?/4el & Kjær http://www.bkhome.com/
References
[1] Neise, W., and Michel, U., "Aerodynamic Noise of Turbomachines"
[2] Anderson, J., "Fundamentals of Aerodynamics", 3rd edition, 2001, McGrawHill
[3] Hoppe, G., and Neise, W., "Vergleich verschiedener Gerauschmessnerfahren fur
Ventilatoren. Forschungsbericht FLT 3/1/31/87, Forschungsvereinigung fur Luft- und
Trocknungstechnik e. V., Frankfurt/Main, Germany
164
Human Vocal Fold
Physiology of Vocal Fold
Human vocal fold is a set of lip-like tissues located inside the larynx, and is the source of sound
for a human and many animals.
The Larynx is located at the top of trachea. It is mainly composed of cartilages and muscles, and
the largest cartilage, thyroid, is well known as the "Adam's Apple."
The organ has two main functions; to act as the last protector of the airway, and to act as a sound
source for voice. This page focuses on the latter function.
Links on Physiology: Discover The Larynx
http://sprojects.mmi.mcgill.ca/larynx/notes/n_frames.htm
Voice Production
Although the science behind sound production for a vocal fold is complex, it can be thought of as
similar to a brass player's lips, or a whistle made out of grass. Basically, vocal folds (or lips or a
pair of grass) make a constriction to the airflow, and as the air is forced through the narrow
opening, the vocal folds oscillate. This causes a periodical change in the air pressure, which is
perceived as sound.
Vocal Folds Video http://www.entusa.com/normal_larynx.htm
When the airflow is introduced to the vocal folds, it forces open the two vocal folds which are
nearly closed initially. Due to the stiffness of the folds, they will then try to close the opening
again. And now the airflow will try to force the folds open etc... This creates an oscillation of the
vocal folds, which in turn, as I stated above, creates sound. However, this is a damped
oscillation, meaning it will eventually achieve an equilibrium position and stop oscillating. So
how are we able to "sustain" sound?
As it will be shown later, the answer seems to be in the changing shape of vocal folds. In the
opening and the closing stages of the osillation, the vocal folds have different shapes. This
affects the pressure in the opening, and creates the extra pressure needed to push the vocal folds
open and sustain oscillation. This part is explained in more detail in the "Model" section.
This flow-induced oscillation, as with many fluid mechanics problems, is not an easy problem to
model. Numorous attempts to model the oscillation of vocal folds have been made, ranging from
a single mass-spring-damper system to finite element models. In this page I would like to use my
single-mass model to explain the basic physics behind the oscillation of a vocal fold.
Information on vocal fold models:
http://www.ncvs.org/ncvs/tutorials/voiceprod/tutorial/model.html
165
Model
Figure 1: Schematics
The most simple way of simulating the motion of vocal folds is to use a single mass-springdamper system as shown above. The mass represents one vocal fold, and the second vocal fold is
assumed to be symmetry about the axis of symmetry. Position 3 respresents a location
immediately past the exit (end of the mass), and position 2 represents the glottis (the region
between the two vocal folds).
The Pressure Force
The major driving force behind the oscillation of vocal folds is the pressure in the glottis. The
Bernoulli's equation from fluid mechanics states that:
-----EQN 1
Neglecting potential difference and applying EQN 1 to positions 2 and 3 of Figure 1,
-----EQN 2
Note that the pressure and the velocity at position 3 cannot change. This makes the right hand
166
side of EQN 2 constant. Observation of EQN 2 reveals that in order to have oscillating pressure
at 2, we must have oscillation velocity at 2. The flow velocity inside the glottis can be studied
through the theories of the orifice flow.
The constriction of airflow at the vocal folds is much like an orifice flow with one major
difference: with vocal folds, the orifice profile is continuously changing. The orifice profile for
the vocal folds can open or close, as well as change the shape of the opening. In Figure 1, the
profile is converging, but in another stage of oscillation it takes a diverging shape.
The orifice flow is described by Blevins as:
-----EQN 3
Where the constant C is the orifice coefficient, governed by the shape and the opening size of the
orifice. This number is determined experimentally, and it changes throughout the different stages
of oscillation.
Solving equations 2 and 3, the pressure force throughout the glottal region can be determined.
The Collision Force
As the video of the vocal folds shows, vocal fods can completely close during oscillation. When
this happens, the Bernoulli equation fails. Instead, the collision force becomes the dominating
force. For this analysis, Hertz collision model was applied.
FH = kHdelta3 / 2(1 + bHdelta') -----EQN 4
where
Here delta is the penetration distance of the vocal fold past the line of symmetry.
Simulation of the Model
The pressure and the collision forces were inserted into the equation of motion, and the result
was simulated.
167
Figure
2:
Area
Opening
and
Volumetric Flow Rate
Figure 2 shows that an oscillating volumetric flow rate was achieved by passing a constant
airflow through the vocal folds. When simulating the oscillation, it was found that the collision
force limits the amplitude of oscillation rather than drive the oscillation. Which tells us that the
pressure force is what allows the sustained oscillation to occur.
The Acoustic Output
This model showed that the changing profile of glottal opening causes an oscillating volumetric
flow rate through the vocal folds. This will in turn cause an oscillating pressure past the vocal
folds. This method of producing sound is unusual, because in most other means of sound
production, air is compressed periodically by a solid such as a speaker cone.
Past the vocal folds, the produced sound enters the vocal tract. Basically this is the cavity in the
mouth as well as the nasal cavity. These cavities act as acoustic filters, modifying the character
of the sound. These are the characters that define the unique voice each person produces.
168
Related Links
FEA Model http://biorobotics.harvard.edu/pubs/gunter-jasa-pub.pdf
Two Mass Model http://www.mat.unb.br/~lucero/JSV05.pdf
References
[1] Fundamentals of Acoustics; Kinsler et al, John Wiley & Sons, 2000
[2] Acoustics: An introduction to its Physical Principles and Applications ; Pierce, Allan D.,
Acoustical Society of America, 1989.
[3] Blevins, R.D. (1984). Applied Fluid Dynamics Handbook. Van Nostrand Reinhold Co. 8182.
[4] Horacek, J., Sidlof, P., Svec, J.G. Self-Oscillations of Human Vocal Folds. Institute of
Thermomechanics, Academy of Sciences of the Czech Republic
[5] Lucero, J., Cataldo, E., Leta, F.R., Nicolato, L. (2005). Synthesis of voiced sounds using lowdimensional models of the vocal cords and time-varying subglottal pressure. Mechanics
Research Communications.
[6] Titze, I.R. (1988). The physics of small-amplitude oscillation of the vocal folds. J. Acoust.
Soc. Am. 83, 1536-1552
Created by Shohei Shibata
169
Microphone Design and Operation
Introduction
Microphones are devices which convert pressure fluctuations into electrical signals. There are
two main methods of accomplishing this task that are used in the mainstream entertainment
industry. They are known as dynamic microphones and condenser microphones. Piezoelectric
crystals can also be used as microphones but are not commonly used in the entertainment
industry.
For
further
information
on
piezoelectric
transducers
see:
http://en.wikibooks.org/wiki/Acoustic:Piezoelectric_Transducers Dynamic microphones.
This type of microphone coverts pressure fluctuations into electrical current. These microphones
work by means of the principal known as Faraday?tm)s Law. The principal states that when an
electrical conductor is moved through a magnetic field, an electrical current is induced within the
conductor. The magnetic field within the microphone is created using permanent magnets and
the conductor is produced in two common arrangements.
Figure 1: Sectional View of Moving-Coil Dynamic Microphone
The first conductor arrangement is made of a coil of wire. The wire is typically copper and is
attached to a circular membrane or piston usually made from lightweight plastic or occasionally
aluminum. The impinging pressure fluctuation on the piston causes it to move in the magnetic
field and thus creates the desired electrical current. Figure 1 provides a sectional view of a
moving-coil microphone.
170
Figure 2: Dynamic Ribbon Microphone
The second conductor arrangement is a ribbon of metallic foil suspended between magnets. The
metallic ribbon is what moves in response to a pressure fluctuation and in the same manner, an
electrical current is produced. Figure 2 provides a sectional view of a ribbon microphone. In both
configurations, dynamic microphones follow the same principals as acoustical transducers. For
further information about transducers go to:
http://en.wikibooks.org/wiki/Acoustic:Acoustic_Transducers_-_The_Loudspeaker
Condenser Microphones
This type of microphone converts pressure fluctuations into electrical potentials through the use
of changing an electrical capacitor. This is why condenser microphones are also known as
capacitor microphones. An electrical capacitor is created when two charged electrical conductors
are placed at a finite distance from each other. The basic relation that describes capacitors is:
Q=C*V
where Q is the electrical charge of the capacitor?tm)s conductors, C is the capacitance, and V is
the electric potential between the capacitor?tm)s conductors. If the electrical charge of the
conductors is held at a constant value, then the voltage between the conductors will be inversely
proportional to the capacitance. Also, the capacitance is inversely proportional to the distance
between the conductors. Condenser microphones utilize these two concepts.
171
Figure 3: Sectional View of Condenser Microphone
The capacitor in a condenser microphone is made of two parts: the diaphragm and the backplate.
Figure 3 shows a section view of a condenser microphone. The diaphragm is what moves due to
impinging pressure fluctuations and the backplate is held in a stationary position. When the
diaphragm moves closer to the backplate, the capacitance increases and therefore a change in
electric potential is produced. The diaphragm is typically made of metallic coated Mylar. The
assembly that houses both the backplate and the diaphragm is commonly referred to as a capsule.
To keep the diaphragm and backplate at a constant charge, an electric potential must be
presented to the capsule. There are various ways of performing this operation. The first of which
is by simply using a battery to supply the needed DC potential to the capsule. A simplified
schematic of this technique is displayed in figure 4. The resistor across the leads of the capsule is
very high, in the range of 10 mega ohms, to keep the charge on the capsule close to constant.
Figure 4: Internal Battery Powered Condenser Microphone
Another technique of providing a constant charge on the capacitor is to supply a DC electric
potential through the microphone cable that carries the microphones output signal. Standard
microphone cable is known as XLR cable and is terminated by three pin connectors. Pin one
connects to the shield around the cable. The microphone signal is transmitted between pins two
and three. Figure 5 displays the layout of dynamic microphone attached to a mixing console via
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XLR cable.
Figure 5: Dynamic Microphone Connection to Mixing Console via XLR Cable
The first and most popular method of providing a DC potential through a microphone cable is to
supply +48 V to both of the microphone output leads, pins 2 and 3, and use the shield of the
cable, pin 1, as the ground to the circuit. Because pins 2 and 3 see the same potential, any
fluctuation of the microphone powering potential will not affect the microphone signal seen by
the attached audio equipment. This configuration can be seen in figure 6. The +48 V will be
stepped down at the microphone using a transformer and provide the potential to the backplate
and diaphragm in a similar fashion as the battery solution.
Figure 6: Condenser Microphone Powering Techniques
The second method of running the potential through the cable is to supply 12 V between pins 2
and 3. This method is referred to as T-powering. The main problem with T-powering is that
potential fluctuation in the powering of the capsule will be transmitted into an audio signal
because the audio equipment analyzing the microphone signal will not see a difference between a
potential change across pins 2 and 3 due to a pressure fluctuation and one due to the power
source electric potential fluctuation.
Finally, the diaphragm and backplate can be manufactured from a material that maintains a fixed
charge. These microphones are termed electrets. In early electret designs, the charge on the
material tended to become unstable over time. Recent advances in science and manufacturing
have allowed this problem to be eliminated in present designs.
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Conclusion
Two branches of microphones exist in the entertainment industry. Dynamic microphones are
found in the moving-coil and ribbon configurations. The movement of the conductor in dynamic
microphones induces an electric current which is then transformed into the reproduction of
sound. Condenser microphones utilize the properties of capacitors. Creating the charge on the
capsule of condenser microphones can be accomplished by battery, phantom powering, Tpowering, and by using fixed charge materials in manufacturing.
References
-Sound Recording Handbook. Woram, John M. 1989.
-Handbook of Recording Engineering Fourth Edition. Eargle, John. 2003.
Microphone Manufacturers Links
http://www.akgusa.com/ AKG
http://www.audio-technica.com/cms/site/c35da94027e94819/index.html Audio Technica
http://www.audixusa.com/ Audix
http://www.bkhome.com/bk_home.asp Bruel & Kjaer
http://www.neumannusa.com/mat_dev/FLift/open.asp Neumann
http://www.rode.com.au/ Rode
http://www.shure.com/ Shure
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Piezoelectric Transducers
Introduction
Piezoelectricity from the greek word "piezo" means pressure electricity. Certain crystalline
substances generate electric charges under mechanical stress and conversely experience a
mechanical strain in the presence of an electric field. The piezoelectric effect describes a
situation where the transducing material senses input mechanical vibrations and produces a
charge at the frequency of the vibration. An AC voltage causes the piezoelectric material to
vibrate in an oscillatory fashion at the same frequency as the input current.
Quartz is the best known single crystal material with piezoelectric properties. Strong
piezoelectric effects can be induced in materials with an ABO3, Perovskite crystalline structure.
'A' denotes a large divalent metal ion such as lead and 'B' denotes a smaller tetravalent ion such
as titanium or zirconium.
For any crystal to exhibit the piezoelectric effect, its structure must have no center of symmetry.
Either a tensile or compressive stress applied to the crystal alters the separation between positive
and negative charge sights in the cell causing a net polarization at the surface of the crystal. The
polarization varies directly with the applied stress and is direction dependent so that compressive
and tensile stresses will result in electric fields of opposite voltages.
Vibrations & Displacements
Piezoelectric ceramics have non-centrosymmetric unit cells below the Curie temperature and
centrosymmetric unit cells above the Curie temperature. Non-centrosymmetric structures provide
a net electric dipole moment. The dipoles are randomly oriented until a strong DC electric field
is applied causing permanent polarization and thus piezoelectric properties.
A polarized ceramic may be subjected to stress causing the crystal lattice to distort changing the
total dipole moment of the ceramic. The change in dipole moment due to an applied stress causes
a net electric field which varies linearly with stress.
Dynamic Performance
The dynamic performance of a piezoelectric material relates to how it behaves under alternating
stresses near the mechanical resonance. The parallel combination of C2 with L1, C1, and R1 in
the equivalent circuit below control the transducers reactance which is a function of frequency.
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Equivalent Electric Circuit
Frequency Response
The graph below shows the impedance of a piezoelectric transducer as a function of frequency.
The minimum value at fn corresponds to the resonance while the maximum value at fm
corresponds to anti-resonance.
Resonant Devices
Non resonant devices may be modeled by a capacitor representing the capacitance of the
piezoelectric with an impedance modeling the mechanically vibrating system as a shunt in the
circuit. The impedance may be modeled as a capacitor in the non resonant case which allows the
circuit to reduce to a single capacitor replacing the parallel combination.
For resonant devices the impedance becomes a resistance or static capacitance at resonance. This
is an undesirable effect. In mechanically driven systems this effect acts as a load on the
transducer and decreases the electrical output. In electrically driven systems this effect shunts the
driver requiring a larger input current. The adverse effect of the static capacitance experienced at
resonant operation may be counteracted by using a shunt or series inductor resonating with the
static capacitance at the operating frequency.
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Applications
Mechanical Measurement
Because of the dielectric leakage current of piezoelectrics they are poorly suited for applications
where force or pressure have a slow rate of change. They are, however, very well suited for
highly dynamic measurements that might be needed in blast gauges and accelerometers.
Ultrasonic
High intensity ultrasound applications utilize half wavelength transducers with resonant
frequencies between 18 kHz and 45 kHz. Large blocks of transducer material is needed to
generate high intensities which is makes manufacturing difficult and is economically impractical.
Also, since half wavelength transducers have the highest stress amplitude in the center the end
sections act as inert masses. The end sections are often replaced with metal plates possessing a
much higher mechanical quality factor giving the composite transducer a higher mechanical
quality factor than a single-piece transducer.
The overall electro-acoustic efficiency is:
Qm0 = unloaded mechanical quality factor
QE = electric quality factor
QL = quality factor due to the acoustic load alone
The second term on the right hand side is the dielectric loss and the third term is the mechanical
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loss.
Efficiency is maximized when:
then:
The maximum ultrasonic efficiency is described by:
Applications of ultrasonic transducers include:
Welding of plactics
Atomization of liquids
Ultrasonic drilling
Ultrasonic cleaning
Ultrasound
Non destructive testing
etc.
More Information and Source of Information
MorganElectroCeramics http://morganelectroceramics.com
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Microphone Technique
General Technique
1. A microphone should be used whose frequency response will suit the frequency range of
the voice or instrument being recorded.
2. Vary microphone positions and distances until you achieve the monitored sound that you
desire.
3. In the case of poor room acoustics, place the microphone very close to the loudest part of
the instrument being recorded or isolate the instrument.
4. Personal taste is the most important component of microphone technique. Whatever
sounds right to you, is right.
Working Distance
Close Miking
When miking at a distance of 1 inch to about 3 feet from the sound source, it is considered close
miking. This technique generally provides a tight, present sound quality and does an effective job
of isolating the signal and excluding other sounds in the acoustic environment.
Leakage
Leakage occurs when the signal is not properly isolated and the microphone picks up another
nearby instrument. This can make the mixdown process difficult if there are multiple voices on
one track. Use the following methods to prevent leakage:
•
Place the microphones closer to the instruments.
•
Move the instruments farther apart.
•
Put some sort of acoustic barrier between the instruments.
•
Use directional microphones.
3 to 1 Rule
The 3:1 distance rule is a general rule of thumb for close miking. To prevent phase anomalies
and leakage, the instruments should be placed at least three times as far as the distance between
the instrument and the microphone.
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Distant Miking
Distant miking refers to the placement of microphones at a distance of 3 feet or more from the
sound source. This technique allows the full range and balance of the instrument to develop and
it captures the room sound. This tends to add a live, open feeling to the recorded sound, but
careful consideration needs to be given to the acoustic environment.
Accent Miking
Accent miking is a technique used for solo passages when miking an ensemble. A soloist needs
to stand out from an ensemble, but placing a microphone to close will sound unnaturally present
compared the distant miking technique used with the rest of the ensemble. Therefore, the
microphone should be placed just close enough the soloist that the signal can be mixed
effectively without sounding completely excluded from the ensemble.
Ambient Miking
Ambient miking is placing the microphones at such a distance that the room sound is more
prominent than the direct signal. This technique is used to capture audience sound or the natural
reverberation of a room or concert hall.
Stereo and Surround Technique
Stereo
Stereo miking is simply using two microphones to obtain a stereo left-right image of the sound.
A simple method is the use of a spaced pair, which is placing two identical microphones several
feet apart and using the difference in time and amplitude to create the image. Great care should
be taken in the method as phase anomalies can occur due to the signal delay. This risk of phase
anomaly can be reduced by using the X/Y method, where the two microphones are placed with
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the grills as close together as possible without touching. There should be an angle of 90 to 135
degrees between the mics. This technique uses only amplitude, not time, to create the image, so
the chance of phase discrepancies is unlikely.
Surround
To take advantage of 5.1 sound or some other surround setup, microphones may be placed to
capture the surround sound of a room. This technique essentially stems from stereo technique
with the addition of more microphones. Because every acoustic environment is different, it is
difficult to define a general rule for surround miking, so placement becomes dependent on
experimentation. Careful attention must be paid to the distance between microphones and
potential phase anomalies.
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Placement for Varying Instruments
Amplifiers
When miking an amplifier, such as for electric guitars, the mic should be placed 2 to 12 inches
from the speaker. Exact placement becomes more critical at a distance of less than 4 inches. A
brighter sound is achieved when the mic faces directly into the center of the speaker cone and a
more mellow sound is produced when placed slightly off-center. Placing off-center also reduces
amplifier noise.
Brass Instruments
High sound-pressure levels are produced by brass instruments due to the directional
characteristics of mid to mid-high frequencies. Therefore, for brass instruments such as trumpets,
trombones, and tubas, microphones should face slightly off of the bell's center at a distance of
one foot or more to prevent overloading from windblasts.
Guitars
Technique for acoustic guitars is dependent on the desired sound. Placing a microphone close to
the sound hole will achieve the highest output possible, but the sound may be bottom-heavy
because of how the sound hole resonates at low frequencies. Placing the mic slightly off-center at
6 to 12 inches from the hole will provide a more balanced pickup. Placing the mic closer to the
bridge with the same working distance will ensure that the full range of the instrument is
captured.
Pianos
Ideally, microphones would be placed 4 to 6 feet from the piano to allow the full range of the
instrument to develop before it is captured. This isn't always possible due to room noise, so the
next best option is to place the microphone just inside the open lid. This applies to both grand
and upright pianos.
Percussion
One overhead microphone can be used for a drum set, although two are preferable. If possible,
each component of the drum set should be miked individually at a distance of 1 to 2 inches as if
they were their own instrument. This also applies to other drums such as congas and bongos. For
large, tuned instruments such as xylophones, multiple mics can be used as long as they are
spaced according to the 3:1 rule.
Voice
Standard technique is to put the microphone directly in front of the vocalist's mouth, although
placing slightly off-center can alleviate harsh consonant sounds (such as "p") and prevent
overloading due to excessive dynamic range.
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Woodwinds
A general rule for woodwinds is to place the microphone around the middle of the instrument at
a distance of 6 inches to 2 feet. The microphone should be tilted slightly towards the bell or
sound hole, but not directly in front of it.
Sound Propagation
It is important to understand how sound propagates due to the nature of the acoustic environment
so that microphone technique can be adjusted accordingly. There are four basic ways that this
occurs:
Reflection
Sound waves are reflected by surfaces if the object is as large as the wavelength of the sound. It
is the cause of echo (simple delay), reverberation (many reflections cause the sound to continue
after the source has stopped), and standing waves (the distance between two parallel walls is
such that the original and reflected waves in phase reinforce one another).
Absorption
Sound waves are absorbed by materials rather than reflected. This can have both positive and
negative effects depending on whether you desire to reduce reverberation or retain a live sound.
Diffraction
Objects that may be between sound sources and microphones must be considered due to
diffraction. Sound will be stopped by obstacles that are larger than its wavelength. Therefore,
higher frequencies will be blocked more easily that lower frequencies.
Refraction
Sound waves bend as they pass through mediums with varying density. Wind or temperature
changes can cause sound to seem like it is literally moving in a different direction than expected.
Sources
•
Huber, Dave Miles, and Robert E. Runstein. Modern Recording Techniques. Sixth
Edition. Burlington: Elsevier, Inc., 2005.
•
Shure, Inc. (2003). Shure Product Literature. Retrieved November 28, 2005, from
http://www.shure.com/scripts/literature/literature.aspx.
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Sealed Box Subwoofer Design
Introduction
A sealed or closed box baffle is the most basic but often the cleanest sounding subwoofer box
design. The subwoofer box in its most simple form, serves to isolate the back of the speaker from
the front, much like the theoretical infinite baffle. The sealed box provides simple construction
and controlled response for most subwoofer applications. The slow low end roll-off provides a
clean transition into the extreme frequency range. Unlike ported boxes, the cone excursion is
reduced below the resonant frequency of the box and driver due to the added stiffness provided
by the sealed box baffle.
Closed baffle boxes are typically constructed of a very rigid material such as MDF (medium
density fiber board) or plywood .75 to 1 inch thick. Depending on the size of the box and
material used, internal bracing may be necessary to maintain a rigid box. A rigid box is important
to design in order to prevent unwanted box resonance.
As with any acoustics application, the box must be matched to the loudspeaker driver for
maximum performance. The following will outline the procedure to tune the box or maximize
the output of the subwoofer box and driver combination.
Closed Baffle Circuit
The sealed box encloser for subwoofers can be modeled as a lumped element system if the
dimensions of the box are significantly shorter than the shortest wavelength reproduced by the
subwoofer. Most subwoofer applications are crossed over around 80 to 100 Hz. A 100 Hz wave
in air has a wavelength of about 11 feet. Subwoofers typically have all dimensions much shorter
than this wavelength, thus the lumped element system analysis is accurate. Using this analysis,
the following circuit represents a subwoofer enclosure system.
where all of the following parameters are in the mechanical mobility analog
Ve - voltage supply
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Re - electrical resistance
Mm - driver mass
Cm - driver compliance
Rm - resistance
RAr - rear cone radiation resistance into the air
XAf - front cone radiation reactance into the air
RBr - rear cone radiation resistance into the box
XBr - rear cone radiation reactance into the box
Driver Parameters
In order to tune a sealed box to a driver, the driver parameters must be known. Some of the
parameters are provided by the manufacturer, some are found experimentally, and some are
found from general tables. For ease of calculations, all parameters will be represented in the SI
units meter/kilogram/second. The parameters that must be known to determine the size of the
box are as follows:
f0 - driver free-air resonance
CMS - mechanical compliance of the driver
SD - effective area of the driver
Resonance of the Driver
The resonance of the driver is either provided by the manufacturer or must be found
experimentally. It is a good idea to measure the resonance frequency even if it is provided by the
manufacturer to account for inconsistent manufacturing processes.
The following diagram shows the setup for finding resonance:
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Where voltage V1 is held constant and the variable frequency source is vaied until V2 is a
maximum. The frequency where V2 is a maximum is the resonance frequency for the driver.
Mechanical Compliance
By definition compliance is the inverse of stiffness or what is commonly referred to as the spring
constant. The compliance of a driver can be found by measuring the displacement of the cone
when known masses are place on the cone when the driver is facing up. The compliance would
then be the displacement of the cone in meters divided by the added weight in newtons.
Effective Area of the Driver
The physical diameter of the driver does not lead to the effective area of the driver. The effective
diameter can be found using the following diagram:
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From this diameter, the area is found from the basic area of a circle equation.
Acoustic Compliance
From the known mechanical compliance of the cone, the acoustic compliance can be found from
the following equation:
CAS = CMSSD2
From the driver acoustic compliance, the box acoustic compliance is found. This is where the
final application of the subwoofer is considered. The acoustic compliance of the box will
determine the percent shift upwards of the resonant frequency. If a large shift is desire for high
SPL applications, then a large ratio of driver to box acoustic compliance would be required. If a
more flattened response is desire for high fidelity applications, then a lower ratio of driver to box
acoustic compliance would be required. Specifically, the ratios can be found in the following
figure using line (b) as reference.
CAS = CAB*r
r - driver to box acoustic compliance ratio
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Sealed Box Design
Volume of Box
The volume of the sealed box can now be found from the box acoustic compliance. The
following equation is used to calculate the box volume
VB= CAB&gamma
Box Dimensions
Fom the calculated box volume, the dimensions of the box can then be designed. There is no set
formula for finding the dimensions of the box, but there are general guidelines to be followed. If
the driver was mounted in the center of a square face, the waves generated by the cone would
reach the edges of the box at the same time, thus when combined would create a strong diffracted
wave in the listening space. In order to best prevent this, the driver should be either be mounted
offset of a square face, or the face should be rectangular.
The face of the box which the driver is set in should not be a square.
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Acoustic Guitars
I plan to discuss the workings of an acoustic guitar, and how the topics that we have studied
apply. This will largely be vibrations of strings and vibrations of cavities.
Introduction
The acoustic guitar is one of the most well known musical instruments. Although precise dates
are not known, the acoustic guitar is generally thought to have originated sometime during the
Renaissance in the form of a lute, a smaller fretless form of what is known today. After evolving
over the course of about 500 years, the guitar today consists of a few major components: the
strings and neck, the bridge, soundboard, head, and internal cavity.
Strings, Neck, and Head
The strings are what actually create vibration on the guitar. On a standard acoustic, there are six
strings, each with a different constant linear density. Strings run along the length of the neck, and
are wound around adjustable tuning pegs located on the head. These tuning pegs can be turned to
adjust the tension in the string. This allows a modification of the wave speed, governed by the
equation
c2=T/ρ
where c is the wave speed [m/s] as a function of tension [N], T, and rho is is the linear density
[kg/m^3]. The string is assumed to fixed at the head (x=0) and mass loaded at the bridge (x=L).
To determine the vibrating frequency of an open string, a general harmonic solution (GHS) is
assumed, y(x,t) = Aexp(j(wt ? kx)) + Bexp(j(wt ? kx))
To solve for coefficients A and B, boundary conditions at x=0 and x=L are evaluated. At x=0,
string velocity (dy/dx) must be zero at all times because that end is assumed to be fixed.
Applying this knowledge to the GHS produces
y(x,t) = ? 2jAsin(kx) * exp(jwt)
Alternatively, at the bridge (a.k.a the mass load at x=L), the bridge and soundboard (along with
any other piece that may vibrate) is assumed to be a lumped element of mass m. The overall goal
with this boundary condition is to determine the velocity of the mass. From Newton's second law
(F=ma), the only force involved is the tension force in the string. The y-component of this force
divided by mass m equals the acceleration. Knowing that acceleration equals velocity times jw
(a=jwu),
Image:H:\pu.data\Desktop\string tension.bmp
u(L,t) = ? T / (j * w * m) * (dy / dx)
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evaluated at x=L. Combining the two boundary equations and simplifying, a final equation can
be obtained
cot(kL) = (m / ms)kL
where k is the wavenumber (w/c), L is the string length, m is the lumped mass of the guitar body,
ms is the total mass of the string (linear density times length), w is the frequency, and c is the
wave speed. If the ratio of m/ms is large (which in a guitar's case, it is), these frequencies are
designated by kL=n*pi. Simplified, the fundamental frequency can be given by
f = sqrt(T / rho) / 2L
Therefore to adjust the resonance frequency of the string, either change the tension (turn the
tuning knob), change the linear density (play a different string), or adjust the length (use the
fretboard).
To determine the location of the frets, musical notes must be considered. In the musical world, it
is common practice to use a tempered scale. In this scale, an A note is set at 440 Hz. To get the
next note in the scale, multiply that frequency by the 12th root of 2 (approximately 1.059), and
an A-sharp will be produced. Multiply by the same factor for the next note, and so on. With this
in mind, to increase f by a factor of 1.059, a corresponding factor should be applied to L. That
factor is 1/17.817, with L in inches. For example, consider an open A string, vibrating at 440 Hz.
For a 26 inch string, the position of the first fret is (26/17.817=1.459) inches from the head. The
second fret will be ((26-1.459)/17.817) inches from the first, and so on.
Bridge
The bridge is the connection point between the strings and the soundboard. The vibration of the
string moves the assumed mass load of the bridge, which vibrates the soundboard, described
next.
Soundboard
The soundboard increases the surface area of vibration, increasing the initial intensity of the note,
and is assisted by the internal cavity.
Internal Cavity
The internal cavity acts as a Helmholtz resonator, and helps to amplify the sound. As the sound
board vibrates, the sound wave is able to resonate inside.
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Basic Room Acoustic Treatments
Room Acoustic Treatments for "Dummies"
Introduction
Many people use one or two rooms in their living space as "theatrical" rooms where theater or
music room activities commence. It is a common misconseption that adding speakers to the room
will enhance the quality of the room acoustics. There are other simple things that can be done to
increase the acoustics of the room to produce sound that is similar to "theater" sound. This site
will take you through some simple background knowledge on acoustics and then explain some
solutions that will help improve sound quality in a room.
Room Sound Combinations
The sound you hear in a room is a combination of direct sound and indirect sound. Direct sound
will come directly from your speakers while the other sound you hear is reflected off of various
objects in the room.
The Direct sound is comming right out of the TV to the listener, as you can see with the heavy
black arrow. All of the other sound is reflected off surfaces before they reach the listener.
Good and Bad Reflected Sound
Have you ever listened to speakers outside? You might have noticed that the sound is thin and
dull. This occurs because when sound is reflected, it is fuller and louder than it would if it were
in an open space. So when sound is reflected, it can add a fullness, or spaciousness. The bad part
of reflected sound occurs when the reflections amplify some notes, while cancelling out others,
making the sound distorted. It can also affect tonal quality and create an echo-like effect. There
191
are three types of reflected sound, pure reflection, absorption, and diffusion. Each relectoin type
is important in creating a "theater" type acoustic room.
Reflected Sound
Reflected sound waves, good and bad, effect the sound you hear, where it comes from, and the
quality of the sound when it gets to you. The bad news when it comes to reflected sound is
standing waves.more on standing waves These waves are created when sound is reflected back
and forth between any two parallel surfaces in your room, ceiling and floor or wall to wall.
Standing waves can distort noises 300Hz and down. These noises include the lower mid
frequency and bass ranges. Standing waves tend to collect near the walls and in corners of a
room, these collecting standing waves are called room resonance modes.
Finding your room resonance modes
First, specify room dimensions (length, width, and height). Then follow this example:
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193
Working with room resonance modes to increase sound quality
1. There are some room dimensions that produce the largest amount of standing waves.
a. Cube
b. Room with 2 out of the three dimensions equal
c. Rooms with dimensions that are multiples of each other
2. Move the chair or sofa away from the walls or corners to reduce standing wave effects
Absorbed
The sound that humans hear is actually a form of acoustic energy. Different materials absorb
different amounts of this energy at different frequencies. When considering room acoustics, there
should be a good mix of high frequency absorbing materials and low frequency absorbing
materials. A table including information on how different common household absorb sound can
be found on the website:
http://www.crutchfieldadvisor.com/learningcenter/home/speakers_roomacoustics.html?page=2#
materials_table
Diffused Sound
Using devices that diffuse sound is a fairly new way of increasing acoustic performance in a
room. It is a means to create sound that appears to be "live". They can replace echo-like
reflections without absorbing too much sound.
Some ways of determining where diffusive items should be placed were found on
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http://www.crutchfieldadvisor.com/ShpU9sw2hgbG/learningcenter/home/speakers_roomacoustics.html?page=4:
1.) If you have carpet or drapes already in your room, use diffusion to control side wall
reflections.
2.) A bookcase filled with odd-sized books makes an effective diffusor.
3.) Use absorptive material on room surfaces between your listening position and your front
speakers, and treat the back wall with diffusive material to re-distribute the reflections.
How to Find Overall Trouble Spots In a Room
Every surface in a room does not have to be treated in order to have good room acoustics. Here is
a simple method of finding trouble spots in a room.
1.) Grab a friend to hold a mirror along the wall near a certian speaker at speaker height.
2.) The listener sits in a spot of normal viewing.
3.) The friend then moves slowly toward the listening position (stay along the wall).
4.) Mark each spot on the wall where the listener can see any of the room speakers in the mirror.
5.) Congradulations! These are the trouble spots in the room that need an absorptive material in
place. Dont forget that diffusive material can also be placed in those positions.
References Sound
http://www.ecoustics.com/Home/Accessories/Acoustic_Room_Treatments/Acoustic_Room_Tre
atment_Articles/
http://www.audioholics.com/techtips/roomacoustics/roomacoustictreatments.php
http://www.diynetwork.com/diy/hi_family_room/article/0,2037,DIY_13912_3471072,00.html
http://www.crutchfieldadvisor.com/ShpU9sw2hgbG/learningcenter/home/speakers_roomacoustics.html?page=1
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Boundary Conditions and Wave Properties
Boundary Conditions
The functions representing the solutions to the wave equation previously discussed,
i.e.
with
and
are dependent upon the boundary and initial conditions. If it is assumed that the wave is
propogating through a string, the initial conditions are related to the specific disturbance in the
string at t=0. These specific disturbances are determined by location and type of contact and can
be anything from simple oscillations to violent impulses. The effects of boundary conditions are
less subtle.
The most simple boundary conditions are the Fixed Support and Free End. In practice, the Free
End boundary condition is rarely encountered since it is assumed there are no transverse forces
holding the string (e.g. the string is simply floating).
- For a Fixed Support:
The overall displacement of the waves travelling in the string, at the support, must be zero.
Denoting x=0 at the support, This requires:
Therefore, the total transverse displacement at x=0 is zero.
- For a Free Support:
Unlike the Fixed Support boundary condition, the transverse displacment at the support does not
need to be zero, but must require the sum of transverse forces to cancel. If it is assumed that the
angle of displacement is small,
and so,
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But of course, the tension in the string, or T, will not be zero and this requires the slope at x=0 to
be zero:
i.e.
- Other Boundary Conditions:
There are many other types of boundary conditions that do not fall into our simplified categories.
As one would expect though, it isn't difficult to relate the characteristics of numerous "complex"
systems to the basic boundary conditions. Typical or realistic boundary conditions include massloaded, resistance-loaded, damped loaded, and impedance-loaded strings. For further
information, see Kinsler, Fundamentals of Acoustics, pp 54-58.
Wave Properties
To begin with, a few definitions of useful variables will be discussed. These include; the wave
number, phase speed, and wavelength characteristics of wave travelling through a string.
The speed that a wave propagates through a string is given in terms of the phase speed, typicaly
in m/s, given by:
where
is the density per unit length of the string.
The wavenumber is used to reduce the transverse displacement equation to a simpler form and
for simple harmonic motion, is multiplied by the lateral position. It is given by:
where
Lastly, the wavelength is defined as:
and is defined as the distance between two points, usually peaks, of a periodic waveform.
These "wave properties" are of practical importance when calculating the solution of the wave
equation for a number of different cases. As will be seen later, the wave number is used
extensively to describe wave phenomenon graphically and quantitatively.
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For further information: Wave Properties at
http://scienceworld.wolfram.com/physics/Wavenumber.html
Edited by: Mychal Spencer
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Rotor Stator Interactions
An important issue for the aeronautical industry is the reduction of aircraft noise. The
characteristics of the turbomachinery noise are to be studied. The rotor/stator interaction is a
predominant part of the noise emission. We will present an introduction to these interaction
theory, whose applications are numerous. For example, the conception of air-conditioning
ventilators requires a full understanding of this interaction.
Noise emission of a Rotor-Stator mechanism
A Rotor wake induces on the downstream Stator blades a fluctuating vane loading, which is
directly linked to the noise emission.
We consider a B blades Rotor (at a rotation speed of c) and a V blades stator, in a unique
Rotor/Stator configuration. The source frequencies are multiples of Bc), that is to say mBc. For
the moment we don't have access to the source levels Fm. The noise frequencies are also mBc,
not depending on the number of blades of the stator. Nevertheless, this number V has a
predominant role in the noise levels (Pm) and directivity, as it will be discussed later.
Example
For an airplane air-conditioning ventilator, reasonable data are :
B = 13 and c = 12000 rnd/min
The blade passing frequency is 2600 Hz, so we only have to include the first two multiples (2600
Hz and 5200 Hz), because of the human ear high-sensibility limit. We have to study the
frequencies m=1 and m=2.
Optimization of the number of blades
As the source levels can't be easily modified, we have to focuse on the interaction between those
levels and the noise levels.
The transfer function
contains the following part :
Where m is the Mach number and JmB
? sV
the Bessel function of mB-sV order. In order to
199
minimize the influence of the transfer function, the goal is to reduce the value of this Bessel
function. To do so, the argument must be smaller than the order of the Bessel function.
Back to the example :
For m=1, with a Mach number M=0.3, the argument of the Bessel function is about 4. We have
to avoid having mB-sV inferior than 4. If V=10, we have 13-1x10=3, so there will be a noisy
mode. If V=19, the minimum of mB-sV is 6, and the noise emission will be limited.
Remark :
The case that is to be strictly avoided is when mB-sV can be nul, which causes the order of the
Bessel function to be 0. As a consequence, we have to take care having B and V prime numbers.
Determination of source levels
The minimization of the transfer function
is a great step in the process of reducing the noise
emission. Nevertheless, to be highly efficient, we also have to predict the source levels Fm. This
will lead us to choose to minimize the Bessel functions for the most significant values of m. For
example, if the source level for m=1 is very higher than for m=2, we will not consider the Bessel
functions of order 2B-sV. The determination of the source levels is given by the Sears theory,
which will not be explained here.
Directivity
All this study was made for a privilegiate direction : the axis of the Rotor/Stator. All the results
are acceptable when the noise reduction is ought to be in this direction. In the case where the
noise to reduce is perpendicular to the axis, the results are very different, as those figures shown :
For B=13 and V=13, which is the worst case, we see that the sound level is very high on the axis
(for θ = 0)
Image:1313bis.jpg
For B=13 and V=19, the sound level is very low on the axis but high perpendicularly to the axis
(for θ = Pi / 2)
Image:1319bis.jpg
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External references
Prediction of rotor wake-stator interaction noise by P. Sijtsma and J.B.H.M. Schulten
http://www.nlr.nl/documents/publications/2003/2003-124-tp.pdf
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License
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