+ All Categories
Home > Documents > Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical...

Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical...

Date post: 29-Mar-2020
Category:
Upload: others
View: 10 times
Download: 0 times
Share this document with a friend
91
analogies mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements analogies potential and flow quantities Impedances Measurement of acoustical impedances distributed elements coupling interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back Acoustics II: electrical-mechanical- acoustical analogies Kurt Heutschi 2013-01-18
Transcript
Page 1: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Acoustics II:electrical-mechanical-

acousticalanalogies

Kurt Heutschi2013-01-18

Page 2: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Motivation

I microphones, loudspeakers consist of mechanical,acoustical and electrical subsystems

I due to their interaction an analysis has to considerall subsystems in integral way

I the fundamental differential equations have identicalform in all systems

I introduction of analogies and conversion ofmechanical and acoustical systems into electricalones

I excellent tools available for analysis of electricalnetworks

Page 3: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

mechanical systems

Page 4: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Describing quantities

At a specific point of a mechanical system, the twoquantities of interest are:

I velocity u = dxdt

I force F

From u, further quantities are derived to describe themovement:

I displacement x =∫udt

I acceleration a = dudt

Page 5: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

mechanical elements

Page 6: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Mechanical massideal mass:

I incompressible → each point of the mass hasidentical velocity

I physical description: Fres = m · a (Newton)

ports 1 and 2:

u1 = u2 = u

F1 − F2 = mdu

dt

Page 7: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Springideal spring:

I stiffness s independent of excursion x

I physical description: F = s · x (Hook’s law)

I no force difference along the spring

F1 = F2 = F

F = s

∫u1 − u2dt

Page 8: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Frictionideal friction:

I proportionality between friction force and velocity

I physical description: F = R · uI no force difference along the friction element

F1 = F2 = F

F = R(u1 − u2)

Page 9: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Links

ideal link:

I connecting element for mechanical building blocks

I no mass and incompressible

Page 10: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Lever

ideal lever:

I frictionless and massless

I mechanical transformer

F1l1 − F2l2 = 0

u1l2 + u2l1 = 0

Page 11: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Mechanical sources

Mechanical sources:

I force sourceI example: conductor in a magnetic field → force∼ B × I but independent of velocity

I velocity sourceI example: motor with a flywheel → velocity

independent of load (force)

Page 12: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

resonance system: spring pendulum

I given: external force: F (t) = F̂ sin(ωt)

I unknown: movement of massI excursion x(t)I velocity u(t) = dx/dtI acceleration a(t) = d2x/dt2

Page 13: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

resonance system: spring pendulum:

solution?

equilibrium of forces:

Facceleration + Ffriction + Fspring = F

differential equation for x in complex writing forharmonic excitation:

md2x

dt2+ R

dx

dt+ sx = F̂ e jωt

Page 14: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

resonance system: spring pendulum:

solution?

general solution of the differential equation:

x(t) =F̂ e jωt

(jω)2m + jωR + s=

F̂ e jωt

jω(jωm + R + s

)

Page 15: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

resonance system: spring pendulum:

solution?

withZm = jωm + R +

s

the quantities become:

x(t) =F̂ e jωt

jωZm

u(t) =F̂ e jωt

Zm

a(t) =jωF̂ e jωt

Zm

Page 16: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

resonance system: spring pendulum:

solution?

amplitude responses:

Page 17: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

resonance system: spring pendulum

consequences for microphones:

I majority of mics use a membrane (mechanicalspring-mass resonance system)

I condition for flat amplitude response:I operation below resonance if electrical output

proportional to membrane excursionI operation at resonance if electrical output

proportional to membrane velocity

Page 18: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

acoustical systems

Page 19: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

describing quantities

fundamental acoustical quantities:

I sound pressure p

I sound particle velocity v or volume flow Q =∫SvdS

movie: rohr-open5-4

Page 20: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Acoustical elements

Page 21: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Acoustical mass

Acoustical mass:

I accelerated but not compressed air

I realization: tube of length l , diameter d whereλ� l and λ� d

∆F = ρAldv

dtNewton

whereρ: density of air

Page 22: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Acoustical compliance

Acoustical compliance:

I compressed but not accelerated air

I realization: cavity V with opening area A (largestdimension l � λ)

Page 23: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Acoustical compliance

I experiment: virtual piston presses on opening A withforce F

I piston sinks in by ∆l = Fs

( s: stiffness of thecompliance )

with the assumption of adiabatic behavior (notemperature exchange):

s = c2ρA2

V

wherec : speed of sound

Page 24: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Acoustical compliance

I position of the opening is irrelevant

Page 25: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Acoustical resistance

Acoustical resistance:

I element introduces loss (conversion of sound energyinto heat)

I realization: porous material, small tube

resistance of a small tube:

∆p = v8lη

r 2

where∆p: sound pressure difference on both sidesv : sound particle velocityl : length of the tuber : radius of the tube (r � l)η: dynamic viscosity of air: 1.82× 10−5Nsm−2

Page 26: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Acoustical resistance

I acoustical resistance of a tube is alwaysaccompanied by an acoustical mass

I mass behavior can be neglected for small diametersand low frequencies

Page 27: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

analogies

Page 28: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Analog quantities

potential quantity flow quantityelectrical system: voltage currentmechanical system: velocity u force Facoustical system: sound pressure p volume flow Q

→ dual analogy would be possible as well

Page 29: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Analog quantities

Amplitude scaling:

I conversion of acoustical and mechanical quantitiesinto electrical ones requires amplitude and unitconversion factors

I arbitrary amplitude conversion factors are allowed

I here: = 1

Page 30: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Analog quantities

mechanical → electrical:

U = G1u with G1 = 1Vs

m

I =1

G2F with G2 = 1

N

A

Page 31: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Analog quantities

acoustical → electrical:

U = G3p with G3 = 1Vm2

N

I =1

G4Q with G4 = 1

m3

As

Page 32: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Impedances: general

Conversion of mechanical and acoustical elements intoanalog electrical ones is performed using impedancedefinition:

impedance =potentialQuantity

flowQuantity

Page 33: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Impedances: mechanical mass

impedance Z of the mechanical mass:

Z =u

F=

u

m dudt

with u = u0ejωt follows du

dt= u0jωe

jωt and finally

Z =1

jωm

I mechanical mass , electrical capacitor

I inertia of the mass is understood relative toreference system at rest → capacitors always atground potential

Page 34: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Impedances: mechanical spring

impedance Z of the mechanical spring:

Z =u

F=

u

s∫udt

with u = u0ejωt follows

∫udt = u0

1jωe jωt and finally

Z = jω1

s

I mechanical spring , electrical inductance

Page 35: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Impedances: mechanical friction

impedance Z of a mechanical friction element:

Z =u

F=

u

Ru=

1

R

I mechanical friction element , electrical resistance

Page 36: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Impedances: acoustical mass

impedance Z of the acoustical mass:

Z =p

Q=

∆FA

Av=ρAl dv

dt

AAv

with v = v0ejωt follows dv

dt= v0jωe

jωt and finally

Z = jωρl

A

I acoustical mass , electrical inductance

Page 37: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Impedance: acoustical compliance

impedance Z of the acoustical compliance:

Z =p

Q=

∆FA

Av=

c2ρA2

V∆l

AAv=

c2ρA2

V

∫vdt

AAv

with v = v0ejωt follows

∫vdt = v0

1jωe jωt and finally

Z =c2ρ

jωV

I acoustical compliance , electrical capacitor

Page 38: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Impedance: acoustical resistance

impedance Z of the acoustical resistance:

Z =p

Q=

8lη

πr 4

wherel : length of the tuber : radius of the tube (r � l)η: dynamic viscosity in air = 1.82 ×10−5 Nsm−2

I acoustical resistance , electrical resistance

Page 39: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Impedances: overview

electrical mechanical acoustical

Z = R resistance resistance resistance

Z = 1jωC

capacitor mass compliance

Z = jωL inductance spring mass

Page 40: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Impedances: overview

electrical mechanical acoustical

Z = R R R = 1Rm

G1G2 R = RmG3G4

Z = 1jωC

C C = m 1G1G2

C = Vρc2

1G3G4

Z = jωL L L = 1sG1G2 L = ρl

AG3G4

Page 41: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Examples of acoustical andmechanical networks

Page 42: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Example: acoustical resonance circuit

combination of acoustical mass and compliance:

Page 43: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Example: acoustical resonance circuit

combination of acoustical mass and compliance:

→ parallel resonance circuit

Page 44: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Example: acoustical resonance circuit

combination of acoustical mass and compliance:

Page 45: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Example: acoustical resonance circuit

combination of acoustical mass and compliance:

→ series resonance circuit

Page 46: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Example: acoustical muffler

acoustical muffler:

I no damping for low frequency air flow

I high damping for high frequency sound

I → acoustical low pass filter

Page 47: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Example: acoustical transmission line

description of tube-like structures (cross sect. area = A)but length > λ:

I subdivision of the tube into small sections of lengthl (l � λ)

I representation of each section by mass andcompliance properties

I equivalent network: L,C , L T-section withI L = ρl

2AI C = Al

ρc2

Page 48: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Example: acoustical transmission line

numerical example:

I hard terminated tube of length 24 cm

I cross sectional area A = 10−4 m2

I discretization: l = 0.04 m

Page 49: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Example: acoustical transmission line

Spice simulation of the voltage at the hard termination:

theoretically expected resonance frequencies:1λ4

= 0.24m → 354 Hz, 3λ4

= 0.24m → 1063 Hz

5λ4

= 0.24m → 1771 Hz, 7λ4

= 0.24m → 2479 Hz

Page 50: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Example: mechanical resonance circuit

combinations of mechanical mass and spring:

Page 51: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Example: mechanical resonance circuit

combinations of mechanical mass and spring:

Page 52: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Example: mechanical resonance circuit

I sinusoidal force excitation

I analysis of the mechanical spring pendulum withhelp of an equivalent electrical circuit.

Page 53: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Example: mechanical spring pendulum

I x = FF/s

I u = FR/R

I a = FM/m

Page 54: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Example: mechanical vibration damper

I strategy in case of sinusoidal vibration →implementation of an additional resonance system

Page 55: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Example: mechanical vibration damper

I additional series resonance circuit

I at resonance: impedance → 0

I short-circuit for the excitation force

Page 56: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Measurement of acousticalimpedances

Page 57: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Measurement of acoustical impedances

I investigation of unknown acoustical impedancesI e.g. quality control of wind instrumentsI e.g. audiometric tests → input impedance of the

middle ear

I measurement methodsI combination of sound pressure and sound particle

sensor, e.g. µFlownI two pressure sensors and known acoustical

resistance

Page 58: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Measurement of acoustical impedances

method with two pressure sensors and known acousticalresistance

evaluation strategies:

I loudspeaker signal controlled for constant differencep1 − p2 → p2 ∼ Z

I calculation from measured spectra

Page 59: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

distributed acousticalelements

Page 60: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

distributed acoustical elements

I simulation of acoustical behavior of long tubes:I series of short sections, each represented by

lumped elementsI or introduction of a distributed acoustical element→ four pole

Page 61: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

distributed acoustical elements

I assumption: 2 plane waves traveling in oppositedirections

p̌(x) = Ae−jkx + Be jkx

v̌(x) =A

ρce−jkx − B

ρce jkx

Page 62: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

distributed acoustical elements

I definition:I p̌(x = 0) ≡ p̌1 and p̌(x = d) ≡ p̌2

I v̌(x = 0) ≡ v̌1 and v̌(x = d) ≡ v̌2

Page 63: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

distributed acoustical elements

insert above definition:

I p̌(x = 0) = p̌1 and p̌(x = d) = p̌2

I v̌(x = 0) = v̌1 and v̌(x = d) = v̌2

in:

p̌(x) = Ae−jkx + Be jkx

v̌(x) =A

ρce−jkx − B

ρce jkx

and resolve for p̌2 and v̌2

Page 64: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

distributed acoustical elements

yields the four pole equations:

p̌2 =p̌1

2e−jkd +

p̌1

2e jkd +

v̌1ρc

2e−jkd − v̌1ρc

2e jkd

= p̌1 cosh(jkd)− v̌1ρc sinh(jkd)

v̌2 =1

ρc

p̌1

2e−jkd − 1

ρc

p̌1

2e jkd +

v̌1

2e−jkd +

v̌1

2e jkd

= −p̌11

ρcsinh(jkd) + v̌1 cosh(jkd)

Page 65: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

distributed acoustical elements

representation of the above four-pole relations by T-typecircuit:

with:

Z1 = jρc1− cos(kd)

sin(kd)

Z2 =−jρc

sin(kd)

Page 66: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

distributed acoustical elements

impedance at the entrance of tube of length d with hardtermination:

p̌1

v̌1= Z1 + Z2

p̌1

v̌1=

jρc − jρc cos(kd)− jρc

sin(kd)= −jρc cot(kd)

Page 67: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

coupling of mechanical,acoustical and electrical

systems

Page 68: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

interfaces

Page 69: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

interfaces

”linking” of different subsystems:

I transformation of different physical quantities byhelp of interfaces:

I type 1: conversion of potential quantity intopotential quantity and flow quantity into flowquantity

I type 2: conversion of potential quantity into flowquantity and vice versa

Page 70: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Interfaces: transformer

interface type 1: transformer

relations for the transformer:

U2 = nU1 (1)

I2 =1

nI1 (2)

Page 71: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

interface: gyrator

interface type 2: gyrator

relations for the gyrator:

U2 = I1m (3)

I2 = U11

m(4)

Page 72: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Dual conversion

Page 73: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Dual conversion

motivation: elimination of gyrators

I selection of an arbitrary conversion factor r

I dual conversion of a suitable region of the network(cut in half all gyrators)

I gyrators → transformers

I series arrangement → parallel arrangement

I parallel arrangement → series arrangement

I L → C , C → L, R → 1/R

Page 74: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Dual conversion

Page 75: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Dual conversion

Page 76: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

application:thin absorber

in front of a hard wall

Page 77: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Thin absorber in front of a hard wall

discussion of the behavior of a thin porous layer in frontof a hard wall by an equivalent network for perpendicularincidence

→ equivalent electrical network?

Page 78: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Thin absorber in front of a hard wall

Page 79: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Thin absorber in front of a hard wall

I absorption: dissipated power according to iR · uI max. absorption for

I ZS → 0I RS = Z0

I m→∞ and f →∞

Page 80: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

application:measurement of the

impedance at the earETH-Diss. Alfred

Stirnemann

Page 81: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Impedance measurement for the diagnosis

of middle ear diseases

idea:

I estimation of middle-ear transfer characteristicsfrom measured impedance at the entrance of theear canal

I detection of possible diseases in the middle-ear

I e.g. Otosclerosis

Page 82: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Measurement of the impedance at the ear

- principal structure

Page 83: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Model of the middle ear - equivalent

network

Page 84: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Model of the middle ear - equivalent

network

complete equivalent network

Page 85: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Model of the middle ear - equivalent

network

dual conversion

Page 86: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Model of the middle ear - equivalent

network

network after dual conversion and neglect of irrelevantelements

vTR : velocity of ear drumvST : velocity of inner ear window

Page 87: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Diagnosis

diagnosis:

I measurement of the input impedance

I estimation of the element values

I comparison with limits → indication for possibledisease

Page 88: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Experimentsexperiments - model fit:

Page 89: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Experiments

experiments - comparison of impedances:

Page 90: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

Experiments

experiments - velocity of inner ear window:

Page 91: Acoustics II: electrical-mechanical- acoustical analogiesisistaff/courses/ak2/... · mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance

analogies

mechanicalsystems

describing quantities

mechanical elements

Mechanical sources

Mechanical resonance -spring pendulum

acoustical systems

describing quantities

Acoustical elements

analogies

potential and flowquantities

Impedances

Measurement of acousticalimpedances

distributed elements

coupling

interfaces

Dual conversion

thin absorbers

measurement ofthe impedance atthe ear

Model of the middle ear

Diagnosis

Experiments

back

eth-acoustics-2


Recommended