# Engineering Acoustics/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 advantages in
power density, compactness, flexibility, 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. In spite of these advantages, there are also some
disadvantages. One of the main drawbacks with the hydraulic fluid power
systems is the vibration and noise generated by them. The health and
safety issues relating to noise, vibration and harshness (NVH) have been
recognized for many years and legislation is now placing clear demands
on manufacturers to reduce noise levels \[1\]. Hence, a lot of attention
has been paid on reducing the noise of hydraulic fluid power systems
both from the industrial and academic researchers. A good understanding
of the noise generation, transmission and propagation is very important
in order to improve the NVH performance of hydraulic fluid power
systems.

## Sound in fluids

The speed of sound in fluids can be determined using the following
relation.

$c = \sqrt {\frac{K}{\rho}}$ where K - fluid bulk modulus, $\rho$- 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 $c = \sqrt {\frac{2e9}{889}}= 1499.9 m/s$

## 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 displacement 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 classified under two
categories:

\(i\) fluidborne noise and

\(ii\) Structureborne noise

==

```{=html}
<center>
```
Fluidborne Noise (FBN)

```{=html}
</center>
```
==

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 separated from each other by an
angular pitch equal to $\phi = \frac {360}{n}$ 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 kinematic flow ripple. The amplitude of the
kinematic ripple can be theoretically 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 superimposed 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.

==

```{=html}
<center>
```
Structureborne Noise (SBN)

```{=html}
</center>
```
==

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.

```{=html}
<center>
```
![](Pump_noise.png "Pump_noise.png")

```{=html}
</center>
```
```{=html}
<center>
```
**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.**

```{=html}
</center>
```
## 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:

```{=html}
<center>
```
$\frac {}{} P = Ae^{-k x} + Be^{-k x}$ \...\...\...(1)

```{=html}
</center>
```
```{=html}
<center>
```
$Q = \frac {1}{Z_{0}}(Ae^{-k x} - Be^{-k x})$\.....(2)

```{=html}
</center>
```
where $\frac {}{} A$ and $\frac {}{} B$ are frequency dependent complex
coefficients which are directly proportional to pump (source) flow
ripple, but also functions of the source impedance $\frac {}{} Z_{s}$,
characteristic impedance of the pipe $\frac {}{} Z_{0}$ and the
termination impedance $\frac {}{} Z_{t}$. These impedances ,usually vary
as the system operating pressure and flow rate changes, can be
determined experimentally.

```{=html}
<center>
```
**Fig.2 Schematic of a pump connected to a hydraulic line**

```{=html}
</center>
```
For complex systems with several system components, the pressure and
flow ripples are estimated using the transformation matrix approach. For
this, the system components can be treated as lumped impedances (a
throttle valve or accumulator), or distributed impedances (flexible hose
or silencer). Various 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 transferred 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 transferred 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 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 availabale 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 hydraulic 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. The
amplitude of fluid pulsations can be reduced, at the source, with the
use of an hydraulic attenuator(5).

\(ii\) **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.

## Hydraulic System noise

```{=html}
<center>
```
![](Noise.png "Noise.png")

```{=html}
</center>
```
```{=html}
<center>
```
**Fig.3 Domain of hydraulic system noise generation and transmission
(Figure recreated from \[1\])**

```{=html}
</center>
```
## 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.

5\. *Hydraulic Power System Analysis*, A. Akers, M. Gassman, & R. Smith,
Taylor & Francis, New York, 2006,

6\. *Experimental studies of the vibro-acoustic characteristics of an
axial piston pump under run-up and steady-state operating conditions*,
Shaogan Ye et al., 2018, Measurement, 133.

7\. *Sound quality evaluation and prediction for the emitted noise of
axial piston pumps*, Junhui Zhang, Shiqi Xia, Shaogan Ye et al., 2018,
Applied Acoustics 145:27-40.

Back to main page


# Engineering Acoustics/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 95 dB (A) - CA. V.C. 27151
.Without a muffler the typical car exhaust noise would exceed 110 dB.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

![](ProRacer.jpg "ProRacer.jpg")

## 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

A low-pass filter is a circuit that passes low frequency signals but
stops the high frequency signals. Once the low pass filter is set by the
user at a specific cutoff frequency, all frequencies lower than that
will be passed through the filter, while higher frequencies will be
attenuated in amplitude. This circuit is made up of passive components
(resistor, capacitors and inductors) capable of accomplishing this
objective.
![](inductive_law_pass_filter.jpg "inductive_law_pass_filter.jpg")

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.

### Description of the muffler to cancel 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

![](Open-twister.gif "Open-twister.gif")

### 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 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-of-sight through the baffles. Frequently the
materials line on the channels. Attenuation 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 also 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.


# Engineering Acoustics/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.

```{=html}
<center>
```
***Figure 1***

```{=html}
</center>
```
# 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. \[1\] \[2\]\[3\]

```{=html}
<center>
```
***Figure 2***

```{=html}
</center>
```
V: cavity volume

$\rho$: ambient density

c: speed of sound

S: cross-section area of orifice

K: stiffness

$M_a$: acoustic mass

$C_a$: 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:

```{=html}
<center>
```
$K = \left(\frac{\rho c^2}{V}\right)S^2$

```{=html}
</center>
```
The equation that describes the Helmholtz resonator is the following:

```{=html}
<center>
```
$S \hat{P}_e =\frac{\hat{q}_e}{j\omega S}(-\omega ^2 M + j\omega R + K)$

```{=html}
</center>
```
$\hat{P}_e$: 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:

```{=html}
<center>
```
$M_a = \frac{L \rho}{S}$

```{=html}
</center>
```
```{=html}
<center>
```
$C_a = \frac{\pi V}{\rho c^2}$

```{=html}
</center>
```
```{=html}
<center>
```
$L ' = \ L + \ 1.7 \ re$

```{=html}
</center>
```
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)

```{=html}
<center>
```
$\omega_r^2 = \frac{1}{M_a C_a}$

```{=html}
</center>
```
```{=html}
<center>
```
$f_r = c 2 \pi \sqrt{\frac{S}{L' V}}$

```{=html}
</center>
```
The sharpness of the resonance peak is measured by the quality factor Q
of the Helmholtz resonator as follows:

```{=html}
<center>
```
$Q = 2 \pi \sqrt{V \left(\frac{L'} {S}\right)^3}$

```{=html}
</center>
```
$f_r$: resonance frequency in Hz

$\omega_r$: 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.

## 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.

```{=html}
<center>
```
***Figure 3***

```{=html}
</center>
```
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 $\delta$ 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
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).

```{=html}
<center>
```
![](theplot.gif "theplot.gif")

```{=html}
</center>
```
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 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.

```{=html}
<center>
```
![](curve.gif "curve.gif")

```{=html}
</center>
```
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.

# Useful Websites

This link: 1 takes you to the website of EXA
Corporation, a developer of PowerFlow for Computational Fluid Dynamics
(CFD) analysis.

This link:
2 is a
small news article about the current use of(CFD) software to model
sunroof buffeting.

This link:
3
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.

------------------------------------------------------------------------

Back to main page


# Engineering Acoustics/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 inconveniences.

!Muffler type \"Cherry
bomb\"\|right{width="150"}

## The absorber muffler

The muffler is composed of a tube covered by sound absorbing material.
The tube is perforated so that some part of the sound wave goes through
the perforation to the absorbing material. 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 bent
metal sheet.

The advantages of this method are a low back pressure and the relatively
simple design. The inconvenient aspect of this method is a low sound
damping compared to the other techniques, especially at low frequency.

Mufflers using the absorption technique are usually installed on sports
vehicles to increase the performances of the engine because of their low
back pressure. A trick to improve their
muffling ability consist of lining up several \"straight\" mufflers.

## The reflector muffler

Principle: Sound wave reflection is used to create a maximum amount of
destructive interferences

!Destructive
interference\|right{width="400"}

### 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. !Wave
reflection\|right{width="250"}

### Definition of the reflection

The sound is a travelling 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 to the frontier). The figure 1 shows two medium A and B
and the 3 waves: incident transmitted and reflected.

### Example

If plane 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

Mufflers using the reflection technique are most commonly used because
they damp the noise much better than the absorber muffler. However, they
often create higher back pressure which can
lower the performance of the engine at higher rpm\'s. While some engines
develope maximum horsepower at lower rpm\'s (say, under 2800 rpm), most
do not and would thus yield a greater net horsepower (at the higher
rpm\'s)with no muffler at all.
!Schema{width="400"}

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 high-end mufflers use the reflection principle together with a
cavity(shown in red below) known as a Helmholtz
Resonator
to further dampen the noise.

![](Muffler_resonator.png "Muffler_resonator.png"){width="300"}

## 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

!Muffler working with waves
reflections{width="500"}

P stands for Pressure \[Pa\] and U stand for volumetric flowrate\[m3/s\]

$\begin{bmatrix} P1 \\ U1 \end{bmatrix}$=$\begin{bmatrix}  T1 \end{bmatrix}
\begin{bmatrix} P2 \\ U2 \end{bmatrix}$ and
$\begin{bmatrix} P2 \\ U2 \end{bmatrix}$=$\begin{bmatrix}  T2 \end{bmatrix}
\begin{bmatrix} P3 \\ U3 \end{bmatrix}$ and
$\begin{bmatrix} P3 \\ U3 \end{bmatrix}$=$\begin{bmatrix}  T3 \end{bmatrix}
\begin{bmatrix} P4 \\ U4 \end{bmatrix}$

So, finally: $\begin{bmatrix} P1 \\ U1 \end{bmatrix}$=
$\begin{bmatrix} T1 \end{bmatrix}
\begin{bmatrix} T2 \end{bmatrix}
\begin{bmatrix} T3 \end{bmatrix}
\begin{bmatrix} P4 \\ U4 \end{bmatrix}$

with

$\begin{bmatrix} T_i \end{bmatrix}$=$\begin{bmatrix}  cos (k L_i) & j sin (k L_i) \frac{\rho c}{S_i}  \\ j sin (k L_i) \frac{\rho c}{S_i} & cos (k L_i) \end{bmatrix}$

Si stands for the cross section area

k is the angular
velocity

$\ \rho$ is the medium density

c is the speed of sound of the medium

### Results

!Schema{width="400"}

https://commons.wikimedia.org/wiki/File:Transmission_loss.png#Source_code
Matlab
code
of the 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 3000 Hz. 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.

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 Helmholtz Resonator is
$\begin{bmatrix} 1 & 0 \\ \frac{1}{Z} & 1 \end{bmatrix}$ with
$\ Z = j \rho ( \frac{\omega L_i}{S_i} - \frac{c^2}{\omega V})$

## Links

More informations about the Transfer Matrice Method :
www.scielo.br/pdf/jbsmse/v27n2/25381.pdf

General informations 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>

Back to Main page


# Engineering Acoustics/Sound Absorbing Structures and Materials

## Introduction

Noise can be defined as unwanted sound. There are many cases and
applications that reducing noise level is of great importance. Loss of
hearing is only one of the effects of continuous exposure to excessive
noise levels. Noise can interfere with sleep and speech, and cause
discomfort and other non-auditory effects. Moreover, high level noise
and vibration lead to structural failures as well as reduction in life
span in many industrial equipments. As an example in control valves, the
vibration caused by flow instability occasionally defects the feedback
to the control system and resulting in extreme oscillations. The
importance of noise issue could be well understood by looking at
regulations that have been passed by governments to restrict noise
production in society. Industrial machinery, air/surface transportation
and construction activities are assumed to be main contributors in noise
production or so called \"noise pollution\".

## Noise Control Mechanisms

-   **Active Noise control**

Modifying and canceling sound field by electro-acoustical approaches is
called active noise control. There are two methods for active control.
First by utilizing the actuators as an acoustic source to produce
completely out of phase signals to eliminate the disturbances. second
method is to use flexible and vibro-elastic materials to radiate a sound
field interfering with the disturbances and minimize the overall
intensity. The latter method is called active structural acoustic
control (ASAC).

-   **Passive Noise Control**

Passive noise control refers to those methods that aim to suppress the
sound by modifying the environment close to the source. Since no input
power is required in such methods, Passive noise control is often
cheaper than active control, however the performance is limited to mid
and high frequencies. active control works well for low frequencies
hence, the combination of two methods may be utilized for broadband
noise reduction.

\[\[Image:Absorbing_Mechanisms.jpg\|center\|thumb\|400px\|

```{=html}
<center>
```
Figure 1: Noise Control Mechanisms

```{=html}
</center>
```
\]\]

## Sound Absorption

Sound waves striking an arbitrary surface are either reflected,
transmitted or absorbed; the amount of energy going into reflection,
transmission or absorption depends on acoustic properties of the
surface. The reflected sound may be almost completely redirected by
large flat surfaces or scattered by a diffused surface. When a
considerable amount of the reflected sound is spatially and temporally
scattered, this status is called a diffuse reflection, and the surface
involved is often termed a diffuser. The absorbed sound may either be
transmitted or dissipated. A simple schematic of surface-wave
interactions is shown in figure 2.
\[\[Image:Sound_treatment.jpg\|center\|thumb\|500px\|

```{=html}
<center>
```
Figure 2: surface-sound interaction- absorption (left), reflection
(middle) and diffusing (right)

```{=html}
</center>
```
\]\] Sound energy is dissipated by simultaneous actions of viscous and
thermal mechanisms. Sound absorbers are used to dissipate sound energy
and to minimize its reflection.[^1] The absorption coefficient $\alpha$
is a common quantity used for measuring the sound absorption of a
material and is known to be the function of the frequency of the
incident wave. It is defined as the ratio of energy absorbed by a
material to the energy incident upon its surface.

### Sound Absorbing Coefficient

The absorbing coefficient can be mathematically presented as follows:

$\alpha=1-\frac{I_R}{I_I}$

*where α, $I_R$, and $I_I$ are the sound absorption coefficient,
one-sided intensity of the reflected sound and the one-sided intensity
of the incident sound, respectively.*

from the above equation, it can be observed that the absorption
coefficient of materials varies from 0 to 1. there are several standard
methods to measure sound absorption coefficient. In one of the common
approaches,a plane wave impedance tube that is equipped with two
microphones is utilized.The experimental setup and dimensions are
according to ASTM E1050/ISO 10534-2.^2The method is done by
evaluating the transfer function, $\hat{h}(f)$ between two microphones
spaced ***s***apart, and a distance***l*** from sample to get absorption
coefficient using the following equations:

$\hat{h}=\frac{\hat{p_1}}{\hat{p_2}}$

$\hat{r}=\frac{\hat{h}-e^{-jks}}{e^{jks}+\hat{h}}$

$\alpha=1-|\hat{r}|^2$

*Where, $\hat{p_1},\hat{p_2}$ are complex pressure amplitude measured by
Mic. 1 and Mic. 2 respectively. **k** is the wave number, **s** is the
microphones spacing and **$\alpha$** is the absorption coefficent.*

According to the standard technique,[^3] frequency is limited by
microphone spacing as well as tube diameter. It is also recommended that
**$0.05\frac{c}{s} < f < 0.45\frac{c}{s}$** to guarantee the plane wave
propagation. The coefficient of commercial absorbing materials is
specified in terms of a noise reduction coefficient (NRC) which refers
to the average of absorption coefficients at **250 Hz, 500 Hz, 1,000 Hz,
and 2,000 Hz**. Average values of some acoustic insulating materials
that are used in buildings are tabulated in table 1. Based on their
construction and material structure, sound absorbers are categorized as
**non-porous** and **porous** absorbers.

\[\[Image:Imp_tube.jpg\|center\|thumb\|400px\|

```{=html}
<center>
```
Figure 3: Two-microphone method to obtain sound absorption coefficient

```{=html}
</center>
```
\]\]

  Material                        Sound absorption coefficient
  ------------------------------- ------------------------------
  6 mm cork sheet                 0.1-0.2
  6 mm porous rubber sheet        0.1-0.2
  12 mm fiberboard on battens     0.3-0.4
  50 mm slag wool or glass silk   0.8-0.9
  Hardwood                        0.3
  100 mm mineral wool             0.65

  : **Table 1 - Sound absorbing coefficient of common absorbents** [^4]

### Non-Porous Absorbers ( Absorbing Resonators )

There are Two types of non-porous absorbers that are common in
industrial applications. **Panel (membrane) resonators** and **Helmholtz
resonators**. Panel absorbers are light, thin and non-porous sheets or
membranes that are tuned to absorb sound waves over a specific frequency
range. The structural resistance of the panel to fast shaping leads to
sound absorption. Panel absorbers are defined by their geometry and
structural vibration properties. Helmholtz Resonators or cavity
absorbers are perforated structures containing very small pores; one
example is the acoustic liners that are used inside the aircraft engine
frame to suppress the noise emission from the compression and combustion
stages. Similar structures are applied in fans and ventilators used in
ventilation and air-conditioning systems. The size of the opening, the
length of the neck, and the volume of the cavity govern the resonant
frequency of the resonator and hence the absorption performance.

### Porous Absorbers

Porous sound absorbers correspond to materials where sound propagation
takes place in a network of interconnected pores such that viscous and
thermal interaction cause acoustic energy to be dissipated and converted
to heat. Absorptive treatment such as mineral wool, glass fiber or
high-porosity foams reduces reflection sound. Porous absorbers are in
fact thermal materials and usually not effective sound barriers. The
need for significant thickness compared to operating sound wavelength
makes porous absorbers dramatically inefficient and impractical at low
frequencies.

\[\[Image:Typic_absorption.jpg\|center\|thumb\|500px\|

```{=html}
<center>
```
Figure 2: Typical variation of sound absorbing coefficient for different
absorbers

```{=html}
</center>
```
\]\]

## Physical Characteristic Properties of Porous Absorbers

The propagation of sound in a porous material is a phenomenon that
governed by physical characteristics of a porous medium, namely porosity
($\phi$) , tortuosity (*q*), flow resistivity ($\sigma$), viscous
characteristic length ($\Lambda$) and thermal characteristic
length($\Lambda'$).

-   **Porosity**

Defined as the ratio of interconnected void volume (air volume in open
pores) and the total volume. Most commercial absorbers have high
porosity (greater than 0.95). The higher porosity the easier interaction
between solid-fluid phases which leads to more sound attenuation.

$\phi = \frac{V_0}{V_T}$

$V_0$ = volume of the void space.

$V_T$ = total volume of the porous material.

-   **Tortuosity** [^5]

the physical characteristics corresponds to the "non-straightness" of
the pore network inside the porous material. It shows how well the
porous material prevents direct flow through the porous medium. The more
complex the path, the more time a wave is in contact with the absorbent
and hence the more energy dissipation and more absorbing capability. If
the Porous absorber is not conductive, one method to measure it is to
saturate the absorbent with an electrically conducting fluid and measure
the electrical resistivity of the saturated sample, $R_s$, and compare
to the resistivity of the fluid itself, $R_f$ then the tortuosity can be
expressed as follows:

$q = \phi\frac{R_s}{R_f}$

-   **Flow resistivity**

The pressure drop required to drive a unit flow through the material can
be related to the viscous losses of the propagating sound waves inside
the porous absorber and denoted as flow resistivity . For a wide range
of porous materials, the flow resistivity is the major factor for sound
absorption. The unit of flow resistance is **$N.s/m^4$ or $Rayls/m$**
and defined as the ratio of static pressure drop **$\Delta P$** to a
volume flow **(*U*)** for a small sample thickness **(*d*)**.

$\sigma = \frac{\Delta P}{Ud}$

-   **Characteristic lengths** [^6]

Two more important microstructural properties are the characteristic
viscous length $\Lambda$ and characteristic thermal length $\Lambda$'
that contribute viscous and thermal dissipation. The former is related
to smaller pores and the latter is related to the larger pores of porous
aggregate. The thermal length $\Lambda '$ is the twice ratio of volume
to surface area in connected pores. This is geometric and can be
measured directly. The viscous length, $\Lambda$, is nearly the same,
but each integral is weighted by the square of the fluid velocity
***v*** inside the pores and hence, cannot be measured directly.

$\Lambda ' = 2\frac{\int dV}{\int dS}$

$\Lambda = 2\frac{\int v^2_{fluid}dV}{\int v^2_{fluid} dS}$

# Acoustic modeling of Porous Absorbents

## Wave equation in rigid porous absorbents

the plane wave equation derived from the linearized equations of
conservation of mass and momentum should be modified to account for the
effects of porosity, tortuosity and flow resistance. The modified wave
equation[^7] that governs the sound propagation in compressible-gas
filled in rigid porous materials is given by:

$\frac{\partial^2 P}{\partial x^2} -(\frac{q\rho_0}{k_{eff} })*\frac{\partial^2 P}{\partial t^2}-(\frac{\sigma\phi}{k_{eff}})*\frac{\partial p}{\partial t} = 0$

where, p = sound pressure within the pores of material

$\rho_0$ = density of compressible gas

$k_{eff}$ = effective bulk modulus of the gas

*q* = tortuosity

$\phi$ = porosity

$\sigma$ = flow resistivity

The acoustical behavior of a absorptive porous layer can also be
investigated from its basic acoustic quantities: the complex wave number
and characteristic impedance. These quantities are obtained as a part of
solution of the modified plane wave equation and can be used to
determine the absorption coefficient and surface impedance. The most
practical and common values for the complex wave number and the surface
impedance are based on semi-empirical methods and correlated using the
regression analysis. One important correlation is suggested by Delany
and Bazely [^8]

$k' = \alpha+j\beta = \frac{\omega}{c} [1+0.0978(\frac{\rho_0 f}{\sigma})^{-0.700}-j0.189(\frac{\rho_0 f}{\sigma})^{-0.595}]$

$z' = R+jX =  \rho_0 c[1+0.0571(\frac{\rho_0 f}{\sigma})^{-0.754}-j0.087(\frac{\rho_0 f}{\sigma})^{-0.732}]$

Where, *f* = frequency *σ* = flow resistance

## Effective Density

By assuming rigid-frame pore network in the absorbent, the solid phase
would be completely motionless and the frame bulk modulus is
considerably greater than that of compressible gas, hence it can be
modeled as an effective fluid using the wave equation for a fluid with
complex effective fluid density and complex effective bulk modulus. In
this situation the dynamic density accounts for the viscous losses and
the dynamic bulk modulus for the thermal losses. The effective density
relation as a function of dynamic tortuosity was proposed by Johnson *et
al.* [^9]

$\rho_{eff}=q(1+\frac{\sigma\phi}{j\omega\rho_0 q} G(\omega))\rho_0$

$G(\omega)=\sqrt{1+j\frac{4q^2\mu\rho_0\omega}{\sigma^2\Lambda^2\phi^2}}$

where,

*μ* = gas viscosity

*ω* = 2*πf*

## Effective Bulk Module

Another factor that affects the sound propagation in the absorbent is
the thermal interaction in material due to the heat exchange between the
acoustic wave front traveling in the compressible fluid and the solid
phase. Champoux and Allard [^10]., have introduced a function
$G'(\omega)$ to evaluate effective bulk module for the gas. as it is
observed in the following formula this would be the function of thermal
characteristic length ($\Lambda '$).

$k_{eff}=\frac{\gamma p_0}{\gamma-(\gamma-1)(1-j\frac{8\mu}{\Lambda '^2 Pr^2\omega\rho_0} G'(\omega))\rho_0}$

$G'(\omega)=\sqrt{1+j\frac{\Lambda '^2 Pr^2\omega\rho_0}{16\mu}}$

where,

*γ* = gas specific heat ratio (for air \~ 1.4)

*Pr* = fluid Prantdl number

## References

[^1]: Cox, T. J. and P. D\'antonio, Acoustic Absorbers and Diffusers,
    SponPress,(2004)

[^2]: ASTM E1050 - 08 Standard Test Method for Impedance and Absorption
    of Acoustical Materials Using A Tube, Two Microphones and A Digital
    Frequency Analysis System

[^3]:

[^4]: Link common absorbing
    materials,
    absorbing coefficients.

[^5]:

[^6]:

[^7]: Fahy, F., Foundations of Engineering Acoustics, Academic Press
    London, (2001).

[^8]: Delany, M.E., and Bazley, E.N., \"Acoustical properties of fibrous
    absorbent materials\" Applied Acoustics, vol. 3, 1970, pp. 105-116.

[^9]: Johnson, D.L., Koplik, J., and Dashen, R.,\"Theory of dynamic
    permeability and tortuosity in fluid-saturated porous media,\"
    Journal of Fluid Mechanics, vol. 176, 1987, pp. 379-402

[^10]: Champoux, Y., and Allard, J.F., \"Dynamic tortuosity and bulk
    modulus in air saturated porous media,\" Journal of Applied Physics,
    vol. 70, no. 4, 1991, pp. 1975-1979.


# Engineering Acoustics/Noise from cooling fans

## Proposal

As electric/electronic devices get smaller and functional, the noise of
cooling device becomes important. This 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

Inside a desktop computer, there may be three (or more) fans. Usually
there is a fan on the heat sink of the CPU, in the rear of the power
supply unit, on the case ventilation hole, and maybe on the graphics
card, plus one on the motherboard chipset if it is a 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
were first introduced, there was no need to have a fan on the CPU at
all, but most modern CPUs cannot function even for several seconds
without a cooling fan, and some CPU\'s (such as Intel\'s
Prescott core) have extreme cooling requirements,
which often causes more and more noise. The type of fan used in a
desktop computer is almost always an axial
fan "wikilink"), while centrifugal fans are commonly
used in laptop computers. Several fan types are shown here (pdf
format). Different
fan types have different characteristics of noise generation and
performance. The axial flow fan is the main type 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.

```{=html}
<center>
```
![](Noisespectrum.gif "Noisespectrum.gif")

```{=html}
</center>
```
(\*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 rotor)

### 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
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 non-uniform 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 "wikilink") 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.

```{=html}
<center>
```
![](Stall.gif "Stall.gif")

```{=html}
</center>
```
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
seriese of phenomenon causes unstedy 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 spread 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 steady 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\'.
(some images and
animations.)
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 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 500**mm** fan can change the
noise power by 50**dB** (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.  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.  Restricting certain aerodynamic phenomenon for noise reduction 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 camparison of various
bearings used in fans.

## Links to Interesting Sites about Fan Noise

Some practical issue of PC noise are presented at the following sites:

-   Cooling Fan Noise Comparison - Sleeve Bearing vs. Ball Bearing (pdf
    format)
-   Brief explanation of fan noise origins and noise reduction
    suggestions
-   Effect of sweep angle
    comparison
-   Comparisons of noise from various 80mm
    fans
-   Noise reduction of a specific desktop
    case
-   Noise reduction of another specific desktop
    case
-   Informal study for noise from CPU cooling
    fan
-   Informal study for noise from PC case
    fans
-   Active fan speed optimizators for minimum noise from desktop
    computers
-   Some general fan noise reduction
    technics
-   Various applications and training in - Brüel &
    Kjær

## 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
Gerauschmessverfahren fur Ventilatoren. Forschungsbericht FLT 3/1/31/87,
Forschungsvereinigung fur Luft- und Trocknungstechnik e. V.,
Frankfurt/Main, Germany

back to **Engineering Acoustics**


# Engineering Acoustics/Noise from turbine blades

## Introduction

Sound prediction in fluid flows is hard to predict because of the
non-linearity of governing equations. The sound production occurs at
high Reynolds number, where the nonlinear inertial terms are much higher
than the viscous terms. The sound production is a very small portion of
energy in fluid flow, especially in open space for subsonic flows.
Aeroacoustics provide approximations of such flows, and the difference
between the actual flow and the reference flow is identified as the
source of sound. The sound field is obtained through the Green\'s
function, in which the Green\'s function is the linear response of the
fluid flow to an impulsive sound source, expressed in delta function of
space and time. The Green\'s function is as below:

```{=html}
<center>
```
$\frac{1}{c_0^2}\frac{d^2G}{dt^2} =\delta(x-y)\delta(t-\tau)$

```{=html}
</center>
```
Aeroacoustics is a field of study that focuses on sound from fluid flow,
and is often used to predict sound in turbine flows.

```{=html}
<center>
```
![](Blade11.png "Blade11.png"){width="188" height="188"}

```{=html}
</center>
```
## Blade displacement noise(monopole)

Blade displacement noise is a monopole source of sound, and can be
severe for turbomachinery and helicopter blades. The simplest model of a
monopole is a radially expanding sphere. In an infinite homogenous
medium, a pulsating sphere will produce a spherical wave as below:

```{=html}
<center>
```
$p(r,t) = (A/r)e^{j(wt-kr)}$

```{=html}
</center>
```
where A is determined by an approximate boundary condition. In a sphere
of average radius *a*, vibrating radially in a complex speed
$U_0{exp(jwt)}$ The specific acoustic impedance for the spherical wave
is

```{=html}
<center>
```
$z(a) = \rho_0{c}{cos\theta_a}e^{j\theta_a}$

```{=html}
</center>
```
Where $cot{\theta_a} = ka$. The pressure of the surface is then

```{=html}
<center>
```
$p(a,t) = {\rho_0}{cU_0}cos\theta_a}{e^{i(wt-ka+\theta_a)}$

```{=html}
</center>
```
Then A becomes

```{=html}
<center>
```
$A = {\rho_0}{cU_0}cos\theta_a}{e^{i(ka+\theta_a)}$

```{=html}
</center>
```
So, the pressure at any distance r\>a is

```{=html}
<center>
```
$p(r,t) = {\rho_0}{cU_0}cos\theta_a}{e^{i(wt-k(r-a)+\theta_a)}$

```{=html}
</center>
```
## Tonal noise(Dipole)

Tonal noise at blade passing frequency(BPF) is an example of a dipole
source. While volume displacement is a monopole source, fluctuating
pressures is a dipole source, and unsteady Reynolds stress or transport
of momentum would be a quadrupole source. The fluctuating blade
pressures (dipoles) are always an important source of sound for rotating
machinery. Steady rotating forces and unsteady rotating forces would
classify as dipole blade forces, and examples of these are uniform
stationary inflow and non-uniform stationary inflow, non-uniform
unstationary inflow, vortex shedding and secondary flows. If two
monopole sources of equal strength but opposite direction are close
enough, it resembles a dipole. A rigid sphere whose center is
oscillating back and forth is another example of a dipole. The net force
exerted on the fluid by the sphere, in accordance to Newton's third law,
is the surface integral of $p(a,e,t)e_r$. Symmetry requires that this
force have only a *z* component, so the force is

```{=html}
<center>
```
$F(t) = F_z(t)e_z = e_z{a^2}\int_{0}^{2\pi}\int_{0}^{\pi}p(a,\theta,t)\cos{\theta}\sin{\theta}{d\theta}dz$

```{=html}
</center>
```
## Noise from wind turbine blade(Flutter)

Flutter has been a problem traditionally related to compressor and fan
blades. Over the years fan blades has decreased in blade and disc
thickness and increased in aspect ratio, in efforts to increase it lift
coefficient. This leads to the decrease in blade stiffness of the bladed
disc assembly and its natural frequencies, and as a result can lead to
flutter motion. Flutter boundaries are very sensitive to mode-shapes,
and reduced frequencies play a secondary role. Flutter would bring
pressure fluctuations and would be a source of dipole sound.

```{=html}
<center>
```
![](Wind_Turbine_blade.png "Wind_Turbine_blade.png"){width="500"
height="500"}

```{=html}
</center>
```
## Noise from gas turbines

In a gas turbine, there are three main sources of noise such as intake,
exhaust, and casing. Intake noise is created by the interaction of the
axial air compressor rotor and stator, and is a function of blade
number, tip speed, and pressure increase. Intake noise is less than
exhaust noise in overall, but its high frequency content sounds are much
larger than that of the exhaust noise. Exhaust noise has higher
amplitude and has lower frequency due to combustion process. Typically,
the inlet and exhaust sound power levels range from 120 dB to over 155
dB. Casing noise is generated through high speed misaligned mechanical
components in the turbine housing radiating to the outer casing. In
principle, gas turbine noise come from aerodynamic sources. High
aerodynamic turbulence are combustion are present in the operation of
gas turbine. Combustion would be a monopole source of sound along with
rotating shock waves. Dipole sources of sound would mainly be from
fluctuating forces on blades and guide vanes, and free jets would be a
quadrupole noise.

```{=html}
<center>
```
![](Gas_turbine_internal_cooling_model_.png "Gas_turbine_internal_cooling_model_.png"){width="500"
height="500"}

```{=html}
</center>
```
## External References

1.  Pierce, A. D., & Beyer, R. T. (1990). Acoustics: An Introduction to
    Its Physical Principles and Applications. 1989 Edition. 2.
    Kinsler, L. E., Frey, A. R., Coppens, H. B., Sanders, J. V., &
    Saunders, H. (1983). Fundamentals of acoustics.
2.  Kinsler, L. E., Frey, A. R., Coppens, H. B., Sanders, J. V., &
    Saunders, H. (1983). Fundamentals of acoustics.
3.  <http://www.sandia.gov/>
4.  <http://www.sonobex.com/gas-turbines/>


# Engineering Acoustics/International Space Station Acoustics Challenges

The International Space Station
(ISS) is a
research facility laboratory made up of several different modules in a
Low Earth Orbit. This facility represents a union of several different
space station projects from various different nations. The long-term
goals of ISS is to develop the technology necessary for human-based
space and planetary exploration and colonization (including life support
systems, safety precautions, environmental monitoring in space), new
ways to treat diseases, more efficient methods of producing materials,
more accurate measurements than would be possible to achieve on Earth,
and a more complete understanding of the Universe.

In order to achieve these objectives, human space flight centers train
highly qualified astronauts in order to best perform different tasks on
board the ISS. Astronauts spend months at a training facility and in a
variety of conditions which simulate the environment of space before
going to ISS. A part of astronauts are cross trained to perform a number
of tasks. A pilot, for example, might also be trained to carry out
scientific experiments, or to work on equipment repairs. Astronauts
schedule is usually very dense on ground, and this becomes even more
critical on board the ISS. Therefore, care needs to be taken in order to
provide astronauts a safe and habitable working environment once they
are on orbit.

One of the major problems on board the ISS is the presence of white
noise. Basically, each module of ISS have equipment such as fans, pumps,
compressors, avionics, and other hardware or systems that serve ISS
functionality and astronauts' life support needs. These equipment
present a significant acoustics challenge to the ISS because of
difficulties with controlling the noise produced by various elements
provided by international partners. The excessive noise levels from
machinery or equipment on ISS has shown to affect crews' hearing,
habitability, safety, productivity, annoyance, and sleep interference.
Crew performance concerns include inability to effectively communicate
and understand what is being said or what is going on around them (e.g.
intelligibility,
speech interference, inability to hear alarms or other important
auditory cues such as an equipment item malfunctioning, inability to
concentrate, strain in vocal cords).

## Normal Hearing Response and Hearing Threshold

The sensitivity of hearing mechanism is highly dependent upon the
frequency content of the received sound. Human ears can detect sound
between frequencies of 20 to 20000 Hz. Intensity of sound is measured in
decibels (dB). The decibel is a logarithmic unit used to describe a
ratio (10 log10 (p2/p1)2). The ratio may be power, sound pressure,
voltage or intensity or several other things. Sound power level (in dB)
of typical sources could be viewed
here.
There are several different units to quantize the perceived sound by
human ears. We will introduce some of them. The phon is a unit that is
related to dB by the psycho-physically measured frequency response of
the ear. At 1 kHz, readings in phons and dB are, by definition, the
same. For all other frequencies, the phon scale is determined by the
results of experiments in which healthy volunteers are asked to adjust
the loudness of a signal at a given frequency until they judge its
loudness is equal to that of a 1 kHz signal
exercise. To convert from
dB to phons, a graph of such
results is needed. Such a graph depends on sound level: it becomes
flatter at high sound levels. Hearing threshold defines the level at
which the ear barely perceives the sound. The threshold is
frequency-dependent and corresponds to the 0-phon curve. Damage to the
hearing mechanism has the effect of raising the hearing threshold, which
indicates that higher sound levels are required in order to be heard.
The degree of shift in the hearing threshold is used as an index of the
amount of hearing impairment incurred; criteria for hearing damage are
usually based on shifts in this threshold.

## dBA

The human ear does not respond equally to all frequencies: we are much
more sensitive to sounds in the frequency range about 1 kHz to 4 kHz
than to very low or high frequency sounds. For this reason, sound meters
are usually fitted with a filter whose response to frequency is a bit
like that of the human ear. If the sound pressure level is given in
units of dB(A) or dBA, the \"A weighting filter\" is used. dBA roughly
corresponds to the inverse of the 40 dB (at 1 kHz) equal-loudness curve.
Sound pressure level on the dBA scale is easy to measure
link and is therefore widely
used. It is still different from loudness, however, because the filter
does not respond in quite the same way as the ear. dBA different
sources

## Noise Exposures

![](Table_1_Permissible_noise_exposures.jpg "Table_1_Permissible_noise_exposures.jpg"){width="50"}

Exposure to excessive and prolonged noise is one major cause of hearing
disorders worldwide. Hearing damage result in an increase in hearing
threshold. Table 1 shows the duration of exposure to higher sound
intensities, set by Occupational Safety and Health Act (OSHA), which
will result in no more damage to hearing than that produced by 8 h at 90
dBA. In addition people must not be exposed to steady sound levels above
115 dBA, regardless of the duration.

When the daily noise exposure is composed of two or more periods of
noise exposure of difference levels, their combined effect should be
considered, rather than the individual effect of each. When people are
exposed to different sound levels during the day, the mixed dose (D)
must be calculated by using the following formula:

$D=C1/T1+C2/T2+...+Cn/Tn$

where Cn is the total exposure time at a given noise level and Tn is the
total exposure time permitted at that level. If the sum of the fractions
equals or exceeds 1, then the mixed exposure is considered to exceed the
limit value.
source

## Intelligibility

The listeners' speech comprehension is diminished by the ambient noise
and the distortion of the system. One could ensure that a message is
clear and intelligible in all situations by measuring the
"intelligibility" of the system. Speech intelligibility is the degree to
which speech can be understood by a listener in a noisy environment. For
satisfactory communication the average speech level should exceed that
of the noise by 6 dB, but lower S/N ratios can be acceptable (Moore,
1997). The most common method to rate the speech interference effect of
noise is called the Preferred Speech Interference Level (PSIL) shown in
the figure bellow.

![](PSIL.jpg "PSIL.jpg")

## Noise detection and protection on board the ISS

In order to determine the total exposure to noise during a given period
astronauts wear an audio dosimeter. The astronauts of the ISS are
exposed to an average noise level of 72dBA for the entire duration of
their stay on the ISS, which can last up to six months. One of the
medical flight rules sets noise exposure limits based on a 24-hour
exposure level. If the 24-hour noise exposure levels measured by the
audio dosimeters exceed 65 dBA, then the crewmembers are directed to
wear approved hearing protection devices. Design specifications of 60
dBA for work areas and 50 dBA for sleep areas have been agreed upon as
"safety limits" for ISS operations. The use of hearing protection
devices (HPDs) is suggested if noise exposure levels exceed 60 dBA, or
if the crewmembers are exposed to high intermittent noise periods (e.g.
use of exercise devices such as the treadmill, airlock repress, or other
short term high noises). The ISS specifications take into account the
impact of noise on crew hearing (both temporary and permanent threshold
shifts), as well as habitability and performance (disrupted
communication, irritability, impaired sleep, etc.). Use of HPDs during
sleep provides additional lowering of the noise input to the inner ear
and aids recovery from acoustic trauma sustained during the day.
Recovery is more robust in quieter environments such as in an adequately
quieted crew sleep station or with the use of hearing protection during
high noise exposure periods. Although hearing protection headsets are
available, astronauts do not use them all the time, as they are
uncomfortable to wear continuously and make communication with other
crewmembers difficult. Since hearing needs to be tested in a quiet
environment, researchers\' efforts to record in-flight changes in
hearing have not been successful because of the continuous noise on the
ISS. Some of the astronauts\' reports could be found in the following
website: SpaceRef

## Further Research

-   Acoustical Testing Laboratory
-   Leuven Measurement Systems
-   Test on
    Humans

## References


# Engineering Acoustics/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 $\Omega$) and a V
blades stator, in a unique Rotor/Stator configuration. The source
frequencies are multiples of $B \Omega$, that is to say $mB \Omega$. For
the moment we don't have access to the source levels $F_{m}$. The noise
frequencies are also $mB \Omega$, not depending on the number of blades
of the stator. Nevertheless, this number V has a predominant role in the
noise levels ($P_{m}$) and directivity, as it will be discussed later.

*Example*

*For an airplane air-conditioning ventilator, reasonable data are :*

*$B=13$ and $\Omega = 12000$ 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 must focus on the
interaction between those levels and the noise levels.

The transfer function ${{F_m } \over {P_m }}$ contains the following
part :

```{=html}
<center>
```
$\sum\limits_{s =  - \infty }^{s =  + \infty } {e^{ - {{i(mB - sV)\pi } \over 2}} J_{mB - sV} } (mBM)$

```{=html}
</center>
```
Where m is the Mach number and $J_{mB - sV}$ the Bessel function of
mB-sV order. In order to 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 ${{F_m } \over {P_m }}$ 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
$F_{m}$. This will lead us to choose to minimize the Bessel functions
for the most significative 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 explicited 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 $\theta = 0$)

```{=html}
<center>
```
![](Acoustics_1313.JPG "Acoustics_1313.JPG")

```{=html}
</center>
```
For B=13 and V=19, the sound level is very low on the axis but high
perpendicularly to the axis (for $\theta = Pi/2$)

```{=html}
<center>
```
![](Acoustics_1319.jpg "Acoustics_1319.jpg")

```{=html}
</center>
```
## External references


# Engineering Acoustics/Noise control with self-tuning Helmholtz resonators

## Introduction

Many engineering systems create unwanted acoustic noise. Noise may be
reduced using engineering noise control methods. One noise control
method popular in mufflers is the Helmholtz resonator, see
here. It is comprised
of a cavity connected to the system of interest through one or several
short narrow tubes. The classical examples are in automobile exhaust
systems. By adding a tuned Helmholtz resonator, sound is reflected back
to the source.

Helmholtz resonators have been exploited to enhance or attenuate sound
fields at least since ancient Greek times where they were used in
ancient amphitheaters to reduce reverberation. Since this time,
Helmholtz resonators have found widespread use in reverberant spaces
such as churches and as mufflers in ducts and pipes. The Helmholtz
resonator effect underlies the phenomena of sunroof buffeting seen
here.
One advantage of the Helmholtz resonator is its simplicity. However, the
frequency range over which Helmholtz resonators are effective is
relatively narrow. Consequently these devices need to be precisely tuned
to the noise source to achieve significant attenuation.

## Noise and vibration control

There are four general categories for noise and vibration control:[^1]

1.  **Active systems:** load or unload the unwanted noise by using
    actuators such as loudspeakers
    1
    and Acoustics/Active Control
2.  **Passive systems:** achieve sound attenuation by using
    2

    :   2.1. reactive devices such as Helmholtz resonators and expansion
        chambers.
    :   2.2. resistive materials such as acoustic linings and porous
        membranes
3.  **Hybrid systems:** use both active and passive elements to achieve
    sound reduction
    3
4.  **Adaptive-passive systems:** use passive devices whose parameters
    can be varied in order to achieve optimal noise attenuation over a
    band of operating frequencies.

## Lumped element model of the Helmholtz resonator

The Helmholtz resonator is an acoustic filter element. If dimensions of
the Helmholtz resonator are smaller than the acoustic wavelength, then
dynamic behavior of the Helmholtz resonator can be modelled as a lumped
system see 4.
It is effectively a mass on a spring and can be treated so
mathematically. The large volume of air is the spring and the air in the
neck is the oscillating mass. Damping appears in the form of radiation
losses at the neck ends, and viscous losses due to friction of the
oscillating air in the neck. Figure 1 shows this analogy between
Helmholtz resonator and a vibration absorber.

\[\[Image: HR and vibration absorber sys.JPG\|frame\|center\|

```{=html}
<center>
```
Figure 1. Helmholtz resonator and vibration absorber

```{=html}
</center>
```
\]\]

## Parameters definition

```{=html}
<table align=center border=2>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Parameter

```{=html}
</th>
```
```{=html}
<th>
```
definition

```{=html}
</th>
```
```{=html}
<th>
```
Parameter

```{=html}
</th>
```
```{=html}
<th>
```
definition

```{=html}
</th>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
$M_a$

```{=html}
</td>
```
```{=html}
<td>
```
Acoustic mass of the resonator

```{=html}
</td>
```
```{=html}
<td>
```
$C_a$

```{=html}
</td>
```
```{=html}
<td>
```
Acoustic compliance

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
$\rho$

```{=html}
</td>
```
```{=html}
<td>
```
Density of the fluid

```{=html}
</td>
```
```{=html}
<td>
```
P

```{=html}
</td>
```
```{=html}
<td>
```
pressure at the neck entrance

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
ω

```{=html}
</td>
```
```{=html}
<td>
```
Excitation frequency

```{=html}
</td>
```
```{=html}
<td>
```
S

```{=html}
</td>
```
```{=html}
<td>
```
Cross section area of the neck

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
a

```{=html}
</td>
```
```{=html}
<td>
```
Radius of the neck

```{=html}
</td>
```
```{=html}
<td>
```
L

```{=html}
</td>
```
```{=html}
<td>
```
Actual neck length

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
y

```{=html}
</td>
```
```{=html}
<td>
```
Displacement in the direction pointing inward the neck

```{=html}
</td>
```
```{=html}
<td>
```
V

```{=html}
</td>
```
```{=html}
<td>
```
Cavity volume of the Helmholtz resonator

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
F

```{=html}
</td>
```
```{=html}
<td>
```
Force applied in the y direction at the resonator neck entrance

```{=html}
</td>
```
```{=html}
<td>
```
L~eff~

```{=html}
</td>
```
```{=html}
<td>
```
Effective neck length (the mass inside the neck and the mass near the
neck edges)

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
## Theoretical analysis of Helmholtz resonators

For a neck which is flanged at both ends, L~eff~ is approximately:

$L_\text{eff} = \ L + \ 1.7 \ a$

The acoustic mass of a Helmholtz resonator is given by

$M_a = L_\text{eff}\ \rho\ S$

The stiffness of the resonator is defined as the reciprocal of the
compliance, and it is defined as

$k_r = \frac{dF}{dy}$

where

$F = P\ S$

For adiabatic system with air as an ideal gas, the thermodynamic process
equation for the resonator is

$PV^{\gamma} = \text{constant} \,$

Differentiating this equation gives

$V^\gamma dP + P \gamma V^{\gamma -1} dV = 0 \,$

The change in the cavity volume is

$dV = -Sdy\$

substituting these into differential equation, it can be re-casted as

$V^\gamma \frac{dF}{S} - P \gamma V^{\gamma -1} S dy = 0 \Rightarrow \frac{dF}{dy} = \frac{P \gamma S^2}{V} = k_r$

or considering $P = \rho R\ T$ and $c=\sqrt{\gamma R T}$, resonator
stiffness is then:

$k_r =\frac{\rho c^2\ S^2}{V}$

where c is the speed of sound, and $\rho$ is the density of the medium.

Two source of damping in the Helmholtz resonator can be considered:
sound radiation from the neck and viscous losses in the neck, which in
many cases can be neglected compared to radiation losses.

1\. **Sound radiation from the neck:** Sound radiation resistance is a
function of the outside neck geometry. For a flanged pipe, the radiation
resistance is approximately[^2]

$R_r =\frac{\rho c k^2\ S^2}{2\pi}$

where k is the wave number, $k =\frac{w}{c}$

2\. **Viscous losses in the neck:** The mechanical resistance due to
viscous losses can be considered as[^3]

$R_v =2R_s S\frac{ \ (L+a)}{\rho c a}$

where R~s~ for a sufficiently large neck diameter is

$R_s =0.00083\sqrt{\frac{w}{2\pi}}$, where ω is the excitation
frequency.

The mechanical impedance of the mechanical system is defined as the
ratio of the driving force and the velocity of the system at the driving
point.The mechanical impedance of a driven mass-spring--damper system is

$\hat{Z}_m=\frac{\hat{F}}{\hat{u}}= R_m+j(\omega m - \frac{k_r}{w})$

according to the analogy between Helmholtz resonator and
mass-spring--damper system (vibration absorber), the mechanical
impedance of a Helmholtz resonator is obtained by replacing mass and
damping from Helmholtz resonator system in above equation:

$\hat{Z}_{mres}= (R_v+\frac{\rho c k^2\ S^2}{2\pi})+j(\omega \rho L_{eff} S - \frac{\rho c^2\ S^2}{wV})$

The natural frequency of a Helmholtz resonator,w~0~, is the frequency
for which the reactance is zero:

$w_0 =c\sqrt{\frac{S}{L_\text{eff}V}}$,

and the acoustic impedance of the Helmholtz resonator is

$\hat{Z}_{res}= \frac{\hat{Z}_{mres}}{\ S^2}$

Resonance occurs when the natural frequency of the resonator is equal to
the excitation frequency. Helmholtz resonators are typically used to
attenuate sound pressure when the system is originally at resonance.A
simple open-ended duct system with a side branch Helmholtz resonator and
the analogous electrical circuit of the system is shown below. For an
undamped resonator, the impedance at resonance is zero, and therefore
according to electrical analogy in Fig.2 the Helmholtz resonator becomes
a short circuit. There is no current flowing in the elements in the
right. On the other hand, the undamped Helmholtz resonator at resonance
causes all reflection of acoustic waves back to the source, while in
damped resonator some current will flow through the branch to the right
of the Helmholtz resonator and reduce the magnitude of attenuation.

\[\[Image: HR analogy.JPG\|frame\|center\|300px\|

```{=html}
<center>
```
Figure 2. open duct system with a side branch Helmholtz resonator with
electrical circuit analogy near junction point A

```{=html}
</center>
```
\]\]

**1- Effect of Resonator Volume on sound attenuation**

Figure 3 shows the frequency response of the above duct system without
Helmholtz resonator, and with two different volume Helmholtz resonators
with the same natural frequency. The excitation frequency axis is
normalized with respect to the fundamental frequency of the straight
pipe system, which was also chosen as the natural frequency of the
resonator. The maximum attenuation of sound pressure for duct systems
with side branch Helmholtz resonators occurs when the natural frequency
of the resonator is equal to the excitation frequency. By comparing two
curves with different colors, blue and gray, it can be seen that to
increase the effective bandwidth of attenuation of a Helmholtz
resonator, the device should be made as large as possible. It should be
mention that in order to minimize the effects of standing waves within
the device, the dimensions do not exceed a quarter wavelength of the
resonator natural frequency .

\[\[Image: volume var HR.png\|frame\|center\|300px\|

```{=html}
<center>
```
Figure 3. Effect of Resonator Volume on sound attenuation

```{=html}
</center>
```
\]\]

**2- Effect of Resonator Damping on sound attenuation**

The effect of Helmholtz resonator damping(Resulting from radiation
resistance and viscous losses in the neck) on the frequency response of
the duct system is shown in Figure 5. The lightly damped Helmholtz
resonator is not robust with respect to changes in the excitation
frequency, since the sound pressure in the duct system can be amplified
if the noise frequency shifts to the vicinity of either of the two
system resonances. To increase the robustness of Helmholtz resonator
with respect to changes in the excitation frequency, damping is useful
to add to the resonators to decrease the magnitude of the resonant
peaks. Such increase in robustness decreases performance, since the
maximum attenuation is significantly less for heavily damped Helmholtz
resonators. The motivation for creating a tunable Helmholtz resonator
stems from this trade off between robustness and performance. A tunable
Helmholtz resonator, capable of adjusting its natural frequency to match
the excitation frequency, would be able to guarantee the high
performance of a lightly damped Helmholtz resonator and track changes in
frequency.

\[\[<File:Damping> HR.png\|frame\|center\|300px\|

```{=html}
<center>
```
Figure 4. Effect of Resonator Damping on sound attenuation

```{=html}
</center>
```
\]\]

## Adaptive Helmholtz resonator

The tunable Helmholtz resonator is a variable volume resonator, which
allows the natural frequency to be adjusted.As shown in Figure 5, a
variable volume Helmholtz resonator can be achieved by rotating an
internal radial wall inside the resonator cavity with respect to an
internal fixed wall. The movable wall is fixed to the bottom end plate
which is attached to a DC motor to provides the motion to change the
volume.

\[\[Image: movable HR4.JPG\|frame\|center\|100px\|

```{=html}
<center>
```
Figure 5. Variable volume Helmholtz resonator

```{=html}
</center>
```
\]\]

To determine the sound pressure and volume velocity at any position
along the duct such as the microphone position in Figure 2, we should
first determine the pressure and velocity at the speaker.

\[\[Image: speaker analo.JPG\|frame\|center\|100px\|

```{=html}
<center>
```
Figure 5. Equivalent circuit for duct system in figure 2

```{=html}
</center>
```
\]\]

\[\[Image: equivalent circuit.JPG\|frame\|center\|100px\|

```{=html}
<center>
```
Figure 6. Simplified equivalent circuit

```{=html}
</center>
```
\]\]

The acoustic impedance for the system termination, which is an unflanged
open pipe is approximately

$Z_1 =\frac{\rho c}{S_p}(\frac{1}{4})(ka)^2+j 0.6 ka$

where S~p~ is the cross sectional area of the pipe, and a~p~ is the
radius of the pipe. The impedance at point 2 is

$\frac{Z_2}{\frac{\rho c} {S_p}}=\frac{\frac{Z_1}{\frac{\rho c} {S_p}}+j tan(kL_1)}{1+j\frac{Z_1}{\frac{\rho c} {S_p}}tan(kL_1)}$

where L~1~ is the length of the pipe separating termination form the
resonator. The resonator acoustic impedance is the same as what is shown
above. The acoustic impedance at point 3 is given by

$Z_3 =\frac{Z_2 Z_{res}}{Z_2+ Z_{res}}$

The impedance at point 4 can be determined by

$\frac{Z_4}{\frac{\rho c} {S_p}}=\frac{\frac{Z_3}{\frac{\rho c} {S_p}}+j tan(kL_2)}{1+j\frac{Z_3}{\frac{\rho c} {S_p}}tan(kL_2)}$

finally the impedance at the speaker is given by

$\frac{Z_{sys}}{\frac{\rho c} {S_{enc}}}=\frac{\frac{Z_4}{\frac{\rho c} {S_{enc}}}+j tan(kL_{enc})}{1+j\frac{Z_4}{\frac{\rho c} {S_{enc}}}tan(kL_{enc})}$

where S~enc~ is the cross section area of the speaker enclosure, and
L~enc~ is the length of the enclosure aperture from the speaker.

From figure 6 with the system impedance (Z\_{sys}), the pressure and
velocity at the speaker can be determined. considering transfer
matrices, the pressure and velocity at any location in the duct system
may be computed from the pressure and velocity at the speaker.The first
transfer matrix may be used to relate the pressure and velocity at the
point downstream in a straight pipe to the pressure and velocity at the
origin of the pipe.

$\begin{bmatrix} P_d(l) \\ U_d(l) \end{bmatrix}$=$\begin{bmatrix}  cos (k L) & -j sin (k L) \frac{\rho c}{S_p}  \\ -j \frac{sin (k L)}{\frac{\rho c}{S_p}} & cos (k L) \end{bmatrix}
\begin{bmatrix} P_d(0) \\ U_d(0) \end{bmatrix}$

The second matrix relates the pressure and velocity immediately
downstream of the sidebranch, to the pressure and velocity immediately
before the side branch.

$\begin{bmatrix} P_d(3) \\ U_d(3) \end{bmatrix}$=$\begin{bmatrix}  1 & 0  \\  -\frac {1}{Z_{res}} & 1 \end{bmatrix}
\begin{bmatrix} P_d(2) \\ U_d(2) \end{bmatrix}$

Correct combination of theses transfer matrices may be used to determine
the pressure occurring in the system at the location of the microphone
in figure 2.

## Appendix A: Matlab Code for straight pipe with Helmholtz resonator

``` {.matlab .numberLines startFrom="1"}
%This Matlab code is used for calculating pressure at the place of
%microphone for the pipe without Helmholtz resonator
clear all
clc
V0=1;
for f=10:1:200;
   omega=2*pi*f;
   freq(f-9)=f;
   c=343;                                     % speed of sound
   rho=1.21;                                  % density of the medium (air)
   ap=0.0254;                                 % radius of the pipe
   Sp=pi*(ap^2);                              % cross sectional area of the pipe
   k=omega/c;                                 % wave numbe
   Lenc=0.10;                                 %distance separating the enclosure aperture and the speaker face
   L1=0.34+(0.6*ap);                          % length of the pipe separating the termination from the resonator
   L2=0.62+(0.85*ap);                         % length of the pipe between the resonator and speaker enclosure
   Lx=0.254;                                  % length of the pipe from HR to micropjone
   Lm=Lx+L2;                                  %distance from speaker to microphon
   Ld=L1+L2;
   Rm=1;                                      % loudspeaker coil resistance
   Bl=7.5;
   Cm=1/2000;                                 %Complience of the speaker
   OmegaN=345;                                %Natural frequency of the speaker(55HZ)
   a=0.0425;                                  %effective radius of the diaphragm
   Senc=pi*(a^2);                             % cross sectional area of the speaker enclosure  (Transformer ratio)
   Mm=0.01;                                   %Air load mass on both side of the driver
   Gamma=1.4;                                 %Specific heat ratio for air
   P0=10^5;
   b=0.38;
   Pref=20e-6;                                %Reference pressure for sound pressure level

   %calculate the system impedance
   Z1=(rho*c*Sp)*(((1/4)*((k*ap)^2)+i*(0.6*k*ap))/(Sp^2));      %open ended unflanged pipe
   % impedance at point 4( after speaker)
   Z4=(rho*c/Sp)*(((Z1/(rho*c/Sp))+i*tan(k*Ld))/(1+i*(Z1/(rho*c/Sp))*tan(k*Ld)));
   Zsys=(rho*c/Senc)*(((Z4/(rho*c/Senc))+i*tan(k*(Lenc)))/(1+i*(Z4/(rho*c/Senc))*tan(k*(Lenc))));
   %calculating the impedance of the loud speaker
   Induct=(Mm/(Senc^2));
   Resist=((Bl^2)/((Senc^2)*(Rm)));
   Drivimp=Resist+(i*omega*Induct)+(1/(i*omega*Cm));           % impedance moadel for speaker
   Impedance=Zsys+ Drivimp;
   Voltage=V0*Bl/((Rm)*Senc);
   Velocity=Voltage/Impedance;
   Pressure=Velocity*Zsys;
   Pressure=Pressure*sqrt(2);
   Velocity=Velocity*sqrt(2);
   TR1=[Pressure;Velocity];
   TR15=[cos(k*(Lenc)) -i*(rho*c/Senc)*sin(k*(Lenc)); -i*(sin(k*(Lenc)))/(rho*c/Senc) cos(k*(Lenc))];
   TR3=[cos(k*Lm) -i*(rho*c/Sp)*sin(k*Lm); -i*(sin(k*Lm))/(rho*c/Sp) cos(k*Lm)];
   Tf=TR3*TR15*TR1;
   Pmike=Tf(1)/(sqrt(2));                                % pressure at microphone
   Vmike=Tf(2);                                          % velocity at microphone
   MagP(f-9)=20*log10(Pmike/Pref);
end;
nondim=freq/(OmegaN/2/pi);
plot(nondim,MagP,'k-.');
title('frequency response of system with Helmholtz Rsonator');
xlabel('Normalized Excitation Frequency (Hz)');
ylabel('Sound pressure level (dB)');
```

## References

[^1]: Robert J. Bernhard, Henry R. Hall, and James D. Jones.
    Adaptive-passive noise control. Proceeding of Inter-Noise 92, pages
    427-430, 1992.

[^2]: L. Kinsler, A. Frey, A. Coppens, and J. Sandesr. Fundamentals of
    Acoustics. John Wiley and Sons, New York, NY, Third edition, 1982.

[^3]: On the theory and design of acoustic resonators. The journal of
    the Acoustical society of America, 25(6):1037-1061, 1953.