Engineering Acoustics/Electro-acoustic analogies

From Wikibooks, open books for an open world
< Engineering Acoustics
Jump to: navigation, search

EA1.jpg

Authors · Print · License

Edit this template

Part 1: Lumped Acoustical Systems1.11.21.31.41.51.61.71.81.91.101.11

Part 2: One-Dimensional Wave Motion2.12.22.3

Part 3: Applications3.13.23.33.43.53.63.73.83.93.103.113.123.133.143.153.163.173.183.193.203.213.223.233.24

Electro-acoustical Analogies[edit]

Acoustical Mass[edit]

Consider a rigid tube-piston system as following figure.

Acousticalmass.gif

Piston is moving back and forth sinusoidally with frequency of f. Assuming f << \frac{c}{l\ or\ \sqrt{S}} (where c is sound velocity c=\sqrt{\gamma R T_0}), volume of fluid in tube is,

\Pi_v=S\ l


Then mass (mechanical mass) of fluid in tube is given as,

M_M= \Pi_v \rho_0 = \rho_0 S\ l


For sinusoidal motion of piston, fluid move as rigid body at same velocity as piston. Namely, every point in tube moves with the same velocity.

Applying the Newton's second law to the following free body diagram,

FBD.gif


SP'=(\rho_0Sl)\frac{du}{dt}
\hat{P}=\rho_0l(j\omega)\hat{u}=j\omega(\frac{\rho_0l}{S})\hat{U}

Where, plug flow assumption is used.

"Plug flow" assumption:
Frequently in acoustics, the velocity distribution along the normal surface of
fluid flow is assumed uniform. Under this assumption, the acoustic volume velocity U is
simply product of velocity and entire surface. U=Su

Acoustical Impedance[edit]

Recalling mechanical impedance,

\hat{Z}_M=\frac{\hat{F}}{\hat{u}}=j\omega(\rho_0Sl)

acoustical impedance (often termed an acoustic ohm) is defined as,

\hat{Z}_A=\frac{\hat{P}}{\hat{U}}=\frac{Z_M}{S^2}=j\omega(\frac{\rho_0l}{S})\quad  \left[\frac{N s}{m^5}\right]

where, acoustical mass is defined.

M_A=\frac{\rho_0l}{S}

Acoustical Mobility[edit]

Acoustical mobility is defined as,

 \hat{\xi}_A=\frac{1}{\hat{Z}_A}=\frac{\hat{U}}{\hat{P}}

Impedance Analog vs. Mobility Analog[edit]

Imp-mov.gif

Acoustical Resistance[edit]

Acoustical resistance models loss due to viscous effects (friction) and flow resistance (represented by a screen).

File:Ra analogs.png rA is the reciprocal of RA and is referred to as responsiveness.

Acoustical Generators[edit]

The acoustical generator components are pressure, P and volume velocity, U, which are analogus to force, F and velocity, u of electro-mechanical analogy respectively. Namely, for impedance analog, pressure is analogus to voltage and volume velocity is analogus to current, and vice versa for mobility analog. These are arranged in the following table.


E-a analogy.gif


Impedance and Mobility analogs for acoustical generators of constant pressure and constant volume velocity are as follows:

File:Acoustic gen.png

Acoustical Compliance[edit]

Consider a piston in an enclosure.

File:Enclosed Piston.png

When the piston moves, it displaces the fluid inside the enclosure. Acoustic compliance is the measurement of how "easy" it is to displace the fluid.

Here the volume of the enclosure should be assumed to be small enough that the fluid pressure remains uniform.

Assume no heat exchange 1.adiabatic 2.gas compressed uniformly , p prime in cavity everywhere the same.

from thermo equitation File:Equ1.jpg it is easy to get the relation between disturbing pressure and displacement of the piston File:Equ3.gif where U is volume rate, P is pressure according to the definition of the impendance and mobility, we can getFile:Equ4.gif

Mobility Analog VS Impedance Analog

File:Comp.gif

Examples of Electro-Acoustical Analogies[edit]

Example 1: Helmholtz Resonator

Example1holm.JPG

Assumptions - (1) Completely sealed cavity with no leaks. (2) Cavity acts like a rigid body inducing no vibrations.

Solution:

- Impedance Analog -
File:Example2holm1sol.JPG

Example 2: Combination of Side-Branch Cavities

File:Exam2prob.JPG

Solution:

- Impedance Analog -
File:Exam2sol.JPG

Back to Main page