analogies

mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems Acoustics II: describing quantities Acoustical elements analogies electrical-mechanical- potential and flow quantities Impedances Measurement of acoustical impedances acoustical distributed elements coupling interfaces Dual conversion analogies thin absorbers measurement of the impedance at the ear Kurt Heutschi Model of the middle ear Diagnosis 2013-01-18 Experiments back analogies Motivation mechanical systems describing quantities mechanical elements I microphones, loudspeakers consist of mechanical, Mechanical sources Mechanical resonance - spring pendulum acoustical and electrical subsystems acoustical systems I due to their interaction an analysis has to consider describing quantities Acoustical elements all subsystems in integral way analogies potential and flow I the fundamental differential equations have identical quantities Impedances form in all systems Measurement of acoustical impedances distributed elements I introduction of analogies and conversion of coupling mechanical and acoustical systems into electrical interfaces Dual conversion ones thin absorbers I excellent tools available for analysis of electrical measurement of the impedance at networks the ear Model of the middle ear Diagnosis Experiments back analogies

mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements analogies mechanical systems 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 analogies Describing quantities mechanical systems describing quantities mechanical elements Mechanical sources At a specific point of a mechanical system, the two Mechanical resonance - spring pendulum quantities of interest are: acoustical systems describing quantities dx I velocity u = Acoustical elements dt analogies I force F potential and flow quantities Impedances From u, further quantities are derived to describe the Measurement of acoustical impedances distributed elements movement: coupling R interfaces I displacement x = udt Dual conversion du thin absorbers I acceleration a = dt measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies

mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements analogies mechanical elements 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 analogies Mechanical mass mechanical ideal mass: systems describing quantities I incompressible → each point of the mass has mechanical elements Mechanical sources identical velocity Mechanical resonance - spring pendulum I physical description: F = m · a (Newton) acoustical systems res describing quantities Acoustical elements analogies potential and flow quantities Impedances Measurement of acoustical impedances distributed elements coupling interfaces Dual conversion ports 1 and 2: thin absorbers measurement of the impedance at u1 = u2 = u the ear du Model of the middle ear Diagnosis F1 − F2 = m Experiments dt back analogies Spring ideal spring: mechanical systems describing quantities I stiffness s independent of excursion x mechanical elements Mechanical sources I physical description: F = s · x (Hook’s law) Mechanical resonance - spring pendulum acoustical systems I no force difference along the spring 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 F1 = F2 = F the ear Z Model of the middle ear Diagnosis F = s u1 − u2dt Experiments back analogies Friction mechanical ideal friction: systems describing quantities mechanical elements I proportionality between friction force and velocity Mechanical sources Mechanical resonance - I physical description: F = R · u spring pendulum acoustical systems I no force difference along the friction element 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 F = F = F the ear 1 2 Model of the middle ear Diagnosis F = R(u1 − u2) Experiments back analogies Links mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements ideal link: analogies potential and flow I connecting element for mechanical building blocks quantities Impedances I no mass and incompressible 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 analogies Lever mechanical systems describing quantities ideal lever: mechanical elements Mechanical sources Mechanical resonance - I frictionless and massless spring pendulum acoustical systems I mechanical transformer describing quantities Acoustical elements analogies potential and flow quantities Impedances Measurement of acoustical impedances distributed elements coupling interfaces Dual conversion thin absorbers F1l1 − F2l2 = 0 measurement of the impedance at u1l2 + u2l1 = 0 the ear Model of the middle ear Diagnosis Experiments back analogies Mechanical sources mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum Mechanical sources: acoustical systems describing quantities I force source Acoustical elements analogies I example: conductor in a magnetic field → force potential and flow quantities ∼ B × I but independent of velocity Impedances Measurement of acoustical I velocity source impedances distributed elements I example: motor with a flywheel → velocity coupling interfaces independent of load (force) Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies resonance system: spring pendulum 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 ˆ coupling I given: external force: F (t) = F sin(ωt) interfaces Dual conversion I unknown: movement of mass thin absorbers I excursion x(t) measurement of I velocity u(t) = dx/dt the impedance at 2 2 the ear I acceleration a(t) = d x/dt Model of the middle ear Diagnosis Experiments back analogies resonance system: spring pendulum: mechanical solution? systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum equilibrium of forces: acoustical systems describing quantities Acoustical elements Facceleration + Ffriction + Fspring = F analogies potential and flow quantities Impedances Measurement of acoustical impedances distributed elements differential equation for x in complex writing for coupling harmonic excitation: interfaces Dual conversion 2 thin absorbers d x dx m + R + sx = Fˆ ejωt measurement of 2 the impedance at dt dt the ear Model of the middle ear Diagnosis Experiments back analogies resonance system: spring pendulum: mechanical solution? systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements general solution of the differential equation: analogies potential and flow quantities Impedances ˆ jωt ˆ jωt Measurement of acoustical F e F e impedances x(t) = = 2   distributed elements (jω) m + jωR + s jω jωm + R + s coupling jω interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies resonance system: spring pendulum: mechanical solution? systems describing quantities mechanical elements Mechanical sources with Mechanical resonance - s spring pendulum Zm = jωm + R + acoustical systems jω describing quantities Acoustical elements the quantities become: analogies potential and flow quantities Impedances ˆ jωt Measurement of acoustical F e impedances x(t) = distributed elements jωZm coupling jωt interfaces Fˆ e Dual conversion u(t) = thin absorbers Zm jωt measurement of jωFˆ e the impedance at a(t) = the ear Z Model of the middle ear m Diagnosis Experiments back analogies resonance system: spring pendulum: mechanical solution? systems describing quantities mechanical elements Mechanical sources Mechanical resonance - amplitude responses: 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 analogies resonance system: spring pendulum mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum consequences for microphones: acoustical systems describing quantities I majority of mics use a membrane (mechanical Acoustical elements spring-mass resonance system) analogies potential and flow quantities I condition for flat amplitude response: Impedances Measurement of acoustical I operation below resonance if electrical output impedances distributed elements proportional to membrane excursion coupling I operation at resonance if electrical output interfaces Dual conversion proportional to membrane velocity thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies

mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements analogies acoustical systems 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 analogies describing quantities mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems fundamental acoustical quantities: describing quantities Acoustical elements I sound pressure p analogies R potential and flow I sound particle velocity v or volume flow Q = vdS quantities S Impedances Measurement of acoustical impedances distributed elements coupling movie: rohr-open5-4 interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies

mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements analogies Acoustical elements 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 analogies Acoustical mass mechanical systems Acoustical mass: describing quantities mechanical elements I accelerated but not compressed air Mechanical sources Mechanical resonance - spring pendulum I realization: tube of length l, diameter d where acoustical systems λ  l and λ  d describing quantities Acoustical elements analogies potential and flow quantities Impedances Measurement of acoustical impedances distributed elements coupling interfaces Dual conversion dv thin absorbers ∆F = ρAl Newton measurement of dt the impedance at the ear where Model of the middle ear Diagnosis ρ: density of air Experiments back analogies Acoustical compliance mechanical systems describing quantities mechanical elements Mechanical sources Acoustical compliance: Mechanical resonance - spring pendulum I compressed but not accelerated air acoustical systems describing quantities I realization: cavity V with opening area A (largest Acoustical elements analogies dimension l  λ) 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 analogies Acoustical compliance mechanical systems describing quantities mechanical elements Mechanical sources I experiment: virtual piston presses on opening A with Mechanical resonance - spring pendulum force F acoustical systems F I piston sinks in by ∆l = ( s: stiffness of the describing quantities s Acoustical elements compliance ) analogies potential and flow quantities with the assumption of adiabatic behavior (no Impedances Measurement of acoustical temperature exchange): impedances distributed elements 2 coupling 2 A interfaces s = c ρ Dual conversion V thin absorbers where measurement of the impedance at c: speed of sound the ear Model of the middle ear Diagnosis Experiments back analogies Acoustical compliance mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum I position of the opening is irrelevant 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 analogies Acoustical resistance mechanical systems Acoustical resistance: describing quantities mechanical elements I element introduces loss (conversion of sound energy Mechanical sources Mechanical resonance - spring pendulum into heat) acoustical systems I realization: porous material, small tube describing quantities Acoustical elements analogies resistance of a small tube: potential and flow quantities Impedances 8lη Measurement of acoustical ∆p = v impedances r 2 distributed elements coupling where interfaces Dual conversion ∆p: sound pressure difference on both sides thin absorbers v: sound particle velocity measurement of the impedance at l: length of the tube the ear r: radius of the tube (r  l) Model of the middle ear −5 −2 Diagnosis η: dynamic viscosity of air: 1.82 × 10 Nsm Experiments back analogies Acoustical resistance mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements I acoustical resistance of a tube is always analogies accompanied by an acoustical mass potential and flow quantities Impedances I mass behavior can be neglected for small diameters Measurement of acoustical impedances and low frequencies distributed elements coupling interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies

mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements analogies 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 analogies Analog quantities mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements potential quantity flow quantity analogies potential and flow electrical system: voltage current quantities Impedances mechanical system: velocity u force F Measurement of acoustical impedances acoustical system: sound pressure p volume flow Q distributed elements coupling interfaces → dual analogy would be possible as well Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Analog quantities mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum Amplitude scaling: acoustical systems describing quantities Acoustical elements I conversion of acoustical and mechanical quantities analogies into electrical ones requires amplitude and unit potential and flow quantities Impedances conversion factors Measurement of acoustical impedances I arbitrary amplitude conversion factors are allowed distributed elements coupling I here: = 1 interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Analog quantities mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum mechanical → electrical: acoustical systems describing quantities Acoustical elements analogies Vs potential and flow quantities U = G1u with G1 = 1 Impedances m Measurement of acoustical impedances 1 N distributed elements I = F with G = 1 G 2 A coupling 2 interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Analog quantities mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical → electrical: acoustical systems describing quantities Acoustical elements analogies Vm2 potential and flow quantities U = G3p with G3 = 1 Impedances N Measurement of acoustical 3 impedances 1 m distributed elements I = Q with G4 = 1 coupling G4 As interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Impedances: general mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems Conversion of mechanical and acoustical elements into describing quantities Acoustical elements analog electrical ones is performed using impedance analogies definition: potential and flow quantities Impedances Measurement of acoustical potentialQuantity impedances impedance = distributed elements flowQuantity coupling interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Impedances: mechanical mass mechanical systems describing quantities impedance Z of the mechanical mass: mechanical elements Mechanical sources Mechanical resonance - u u spring pendulum Z = = F du acoustical systems m dt describing quantities Acoustical elements jωt du jωt with u = u0e follows dt = u0jωe and finally analogies potential and flow quantities 1 Impedances Z = Measurement of acoustical impedances jωm distributed elements coupling interfaces I mechanical mass , electrical capacitor Dual conversion thin absorbers I inertia of the mass is understood relative to measurement of reference system at rest → capacitors always at the impedance at the ear ground potential Model of the middle ear Diagnosis Experiments back analogies Impedances: mechanical spring mechanical systems describing quantities mechanical elements Mechanical sources impedance Z of the mechanical spring: Mechanical resonance - spring pendulum acoustical systems u u describing quantities Z = = R Acoustical elements F s udt analogies jωt R 1 jωt potential and flow with u = u e follows udt = u e and finally quantities 0 0 jω Impedances Measurement of acoustical impedances 1 distributed elements Z = jω coupling s interfaces Dual conversion I mechanical spring electrical inductance thin absorbers , measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Impedances: mechanical friction mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems impedance Z of a mechanical friction element: describing quantities Acoustical elements analogies u u 1 potential and flow Z = = = quantities F Ru R Impedances Measurement of acoustical impedances distributed elements I mechanical friction element , electrical resistance coupling interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Impedances: acoustical mass mechanical systems describing quantities mechanical elements Mechanical sources impedance Z of the acoustical mass: Mechanical resonance - spring pendulum acoustical systems p ∆F ρAl dv describing quantities Z = = A = dt Acoustical elements Q Av AAv analogies potential and flow jωt dv jωt quantities with v = v0e follows dt = v0jωe and finally Impedances Measurement of acoustical impedances ρl distributed elements Z = jω coupling A interfaces Dual conversion thin absorbers I acoustical mass , electrical inductance measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Impedance: acoustical compliance mechanical systems describing quantities mechanical elements Mechanical sources impedance Z of the acoustical compliance: Mechanical resonance - spring pendulum 2 2 acoustical systems ∆F 2 A 2 A R p A c ρ V ∆l c ρ V vdt describing quantities Z = = = = Acoustical elements Q Av AAv AAv analogies potential and flow jωt R 1 jωt quantities with v = v0e follows vdt = v0 jω e and finally Impedances Measurement of acoustical impedances c2ρ distributed elements Z = coupling jωV interfaces Dual conversion thin absorbers I acoustical compliance , electrical capacitor measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Impedance: acoustical resistance mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - impedance Z of the acoustical resistance: spring pendulum acoustical systems p 8lη describing quantities Z = = Acoustical elements Q πr 4 analogies potential and flow quantities where Impedances Measurement of acoustical l: length of the tube impedances distributed elements r: radius of the tube (r  l) −5 −2 coupling η: dynamic viscosity in air = 1.82 ×10 Nsm interfaces Dual conversion I acoustical resistance electrical resistance thin absorbers , measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Impedances: overview mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems electrical mechanical acoustical describing quantities Acoustical elements analogies Z = R resistance resistance resistance potential and flow quantities 1 Impedances Z = capacitor mass compliance Measurement of acoustical jωC impedances distributed elements Z = jωL inductance spring mass coupling interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Impedances: overview mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems electrical mechanical acoustical describing quantities Acoustical elements 1 analogies Z = RRR = G G R = R G G Rm 1 2 m 3 4 potential and flow quantities 1 1 V 1 Impedances Z = CC = m C = 2 Measurement of acoustical jωC G1G2 ρc G3G4 impedances distributed elements Z = jωLLL = 1 G G L = ρl G G coupling s 1 2 A 3 4 interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies

mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements Examples of acoustical and analogies potential and flow quantities Impedances mechanical networks 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 analogies Example: acoustical resonance circuit mechanical systems describing quantities combination of acoustical mass and compliance: 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 analogies Example: acoustical resonance circuit mechanical systems describing quantities combination of acoustical mass and compliance: 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 → parallel resonance circuit Model of the middle ear Diagnosis Experiments back analogies Example: acoustical resonance circuit mechanical systems describing quantities combination of acoustical mass and compliance: 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 analogies Example: acoustical resonance circuit mechanical systems describing quantities combination of acoustical mass and compliance: 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 → series resonance circuit Model of the middle ear Diagnosis Experiments back analogies Example: acoustical muffler mechanical systems describing quantities mechanical elements Mechanical sources acoustical muffler: Mechanical resonance - spring pendulum I no damping for low frequency air flow acoustical systems describing quantities Acoustical elements I high damping for high frequency sound analogies I → acoustical low pass filter 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 analogies Example: acoustical transmission line mechanical description of tube-like structures (cross sect. area = A) systems describing quantities but length > λ: mechanical elements Mechanical sources Mechanical resonance - I subdivision of the tube into small sections of length spring pendulum acoustical systems l (l  λ) describing quantities Acoustical elements I representation of each section by mass and analogies compliance properties potential and flow quantities Impedances I equivalent network: L, C, L T-section with Measurement of acoustical impedances ρl I L = 2A distributed elements Al I C = coupling ρc2 interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Example: acoustical transmission line mechanical systems describing quantities mechanical elements Mechanical sources numerical example: Mechanical resonance - spring pendulum acoustical systems I hard terminated tube of length 24 cm describing quantities −4 2 Acoustical elements I cross sectional area A = 10 m analogies potential and flow I discretization: l = 0.04 m 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 analogies Example: acoustical transmission line mechanical systems Spice simulation of the voltage at the hard termination: 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 theoretically expected resonance frequencies: 1λ 3λ measurement of = 0.24m → 354 Hz, = 0.24m → 1063 Hz the impedance at 4 4 the ear 5λ 7λ Model of the middle ear 4 = 0.24m → 1771 Hz, 4 = 0.24m → 2479 Hz Diagnosis Experiments back analogies Example: mechanical resonance circuit mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum combinations of mechanical mass and spring: 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 analogies Example: mechanical resonance circuit mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum combinations of mechanical mass and spring: 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 analogies Example: mechanical resonance circuit 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 I sinusoidal force excitation measurement of the impedance at the ear I analysis of the mechanical spring pendulum with Model of the middle ear Diagnosis help of an equivalent electrical circuit. Experiments back analogies Example: mechanical spring pendulum 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 I x = FF /s thin absorbers measurement of I u = FR /R the impedance at the ear I a = FM /m Model of the middle ear Diagnosis Experiments back analogies Example: mechanical vibration damper mechanical systems I strategy in case of sinusoidal vibration → describing quantities mechanical elements implementation of an additional resonance system 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 analogies Example: mechanical vibration damper 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 I additional series resonance circuit thin absorbers I at resonance: impedance → 0 measurement of the impedance at the ear I short-circuit for the excitation force Model of the middle ear Diagnosis Experiments back analogies

mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements Measurement of acoustical analogies potential and flow quantities Impedances 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 analogies Measurement of acoustical impedances mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum I investigation of unknown acoustical impedances acoustical systems I e.g. quality control of wind instruments describing quantities Acoustical elements I e.g. audiometric tests → input impedance of the analogies middle ear potential and flow quantities Impedances I measurement methods Measurement of acoustical impedances I combination of sound pressure and sound particle distributed elements sensor, e.g. µFlown coupling interfaces I two pressure sensors and known acoustical Dual conversion resistance thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Measurement of acoustical impedances mechanical method with two pressure sensors and known acoustical systems describing quantities resistance 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 evaluation strategies: measurement of the impedance at I loudspeaker signal controlled for constant difference the ear Model of the middle ear p1 − p2 → p2 ∼ Z Diagnosis Experiments I calculation from measured spectra back analogies

mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements distributed acoustical analogies potential and flow quantities Impedances elements 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 analogies distributed acoustical elements mechanical systems describing quantities mechanical elements Mechanical sources I simulation of acoustical behavior of long tubes: Mechanical resonance - spring pendulum I series of short sections, each represented by acoustical systems lumped elements describing quantities Acoustical elements I or introduction of a distributed acoustical element analogies → four pole 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 analogies distributed acoustical elements 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 I assumption: 2 plane waves traveling in opposite coupling directions interfaces Dual conversion thin absorbers −jkx jkx measurement of pˇ(x) = Ae + Be the impedance at the ear Model of the middle ear A −jkx B jkx Diagnosis vˇ(x) = e − e Experiments ρc ρc back analogies distributed acoustical elements 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 I definition: thin absorbers I pˇ(x = 0) ≡ pˇ1 andp ˇ(x = d) ≡ pˇ2 measurement of the impedance at I vˇ(x = 0) ≡ vˇ1 andv ˇ(x = d) ≡ vˇ2 the ear Model of the middle ear Diagnosis Experiments back analogies distributed acoustical elements mechanical systems describing quantities mechanical elements insert above definition: Mechanical sources Mechanical resonance - spring pendulum I pˇ(x = 0) =p ˇ1 andp ˇ(x = d) =p ˇ2 acoustical systems describing quantities I vˇ(x = 0) =v ˇ1 andv ˇ(x = d) =v ˇ2 Acoustical elements analogies in: potential and flow quantities Impedances Measurement of acoustical −jkx jkx impedances pˇ(x) = Ae + Be distributed elements coupling interfaces A −jkx B jkx Dual conversion vˇ(x) = e − e thin absorbers ρc ρc measurement of and resolve forp ˇ andv ˇ the impedance at 2 2 the ear Model of the middle ear Diagnosis Experiments back analogies distributed acoustical elements mechanical systems describing quantities yields the four pole equations: mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems pˇ1 −jkd pˇ1 jkd vˇ1ρc −jkd vˇ1ρc jkd describing quantities pˇ2 = e + e + e − e Acoustical elements 2 2 2 2 analogies =p ˇ1 cosh(jkd) − vˇ1ρc sinh(jkd) potential and flow quantities Impedances Measurement of acoustical impedances distributed elements 1 pˇ 1 pˇ vˇ vˇ coupling 1 −jkd 1 jkd 1 −jkd 1 jkd vˇ2 = e − e + e + e interfaces ρc 2 ρc 2 2 2 Dual conversion thin absorbers 1 = −pˇ1 sinh(jkd) +v ˇ1 cosh(jkd) measurement of ρc the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies distributed acoustical elements mechanical representation of the above four-pole relations by T-type systems describing quantities circuit: 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 with: coupling interfaces Dual conversion 1 − cos(kd) Z = jρc thin absorbers 1 sin(kd) measurement of the impedance at the ear −jρc Model of the middle ear Z2 = Diagnosis sin(kd) Experiments back analogies distributed acoustical elements mechanical systems impedance at the entrance of tube of length d with hard describing quantities mechanical elements termination: 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 pˇ1 Dual conversion = Z1 + Z2 thin absorbers vˇ1 measurement of the impedance at pˇ1 jρc − jρc cos(kd) − jρc the ear = = −jρc cot(kd) Model of the middle ear vˇ1 sin(kd) Diagnosis Experiments back analogies

mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems coupling of mechanical, describing quantities Acoustical elements analogies potential and flow acoustical and electrical quantities Impedances Measurement of acoustical impedances systems distributed elements coupling interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies

mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements analogies interfaces 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 analogies interfaces mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum ”linking”of different subsystems: acoustical systems describing quantities I transformation of different physical quantities by Acoustical elements help of interfaces: analogies potential and flow I type 1: conversion of potential quantity into quantities Impedances potential quantity and flow quantity into flow Measurement of acoustical impedances quantity distributed elements coupling I type 2: conversion of potential quantity into flow interfaces quantity and vice versa Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Interfaces: transformer mechanical systems interface type 1: transformer 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 relations for the transformer: distributed elements coupling interfaces Dual conversion thin absorbers U2 = nU1 (1) measurement of 1 the impedance at I2 = I1 (2) the ear n Model of the middle ear Diagnosis Experiments back analogies interface: gyrator mechanical systems interface type 2: gyrator 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 relations for the gyrator: distributed elements coupling interfaces Dual conversion U = I m (3) thin absorbers 2 1 measurement of 1 the impedance at I2 = U1 (4) the ear m Model of the middle ear Diagnosis Experiments back analogies

mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements analogies Dual conversion 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 analogies Dual conversion mechanical systems describing quantities mechanical elements Mechanical sources motivation: elimination of gyrators Mechanical resonance - spring pendulum acoustical systems I selection of an arbitrary conversion factor r describing quantities Acoustical elements I dual conversion of a suitable region of the network analogies (cut in half all gyrators) potential and flow quantities Impedances I gyrators → transformers Measurement of acoustical impedances distributed elements I series arrangement → parallel arrangement coupling interfaces I parallel arrangement → series arrangement Dual conversion I L → C, C → L, R → 1/R thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Dual conversion 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 analogies Dual conversion 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 analogies

mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities application: Acoustical elements analogies potential and flow thin absorber quantities Impedances Measurement of acoustical impedances distributed elements in front of a hard wall coupling interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Thin absorber in front of a hard wall mechanical systems discussion of the behavior of a thin porous layer in front describing quantities mechanical elements of a hard wall by an equivalent network for perpendicular Mechanical sources Mechanical resonance - incidence 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 → equivalent electrical network? Experiments back analogies Thin absorber in front of a hard wall 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 analogies Thin absorber in front of a hard wall mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities I absorption: dissipated power according to iR · u Acoustical elements analogies I max. absorption for potential and flow quantities I ZS → 0 Impedances Measurement of acoustical I RS = Z0 impedances distributed elements I m → ∞ and f → ∞ coupling interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies

mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - application: spring pendulum acoustical systems describing quantities measurement of the Acoustical elements analogies potential and flow quantities impedance at the ear Impedances Measurement of acoustical impedances distributed elements ETH-Diss. Alfred coupling interfaces Dual conversion Stirnemann thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Impedance measurement for the diagnosis mechanical of middle ear diseases systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities idea: Acoustical elements analogies I estimation of middle-ear transfer characteristics potential and flow quantities from measured impedance at the entrance of the Impedances Measurement of acoustical impedances ear canal distributed elements I detection of possible diseases in the middle-ear coupling interfaces Dual conversion I e.g. Otosclerosis thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Measurement of the impedance at the ear mechanical - principal structure 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 analogies Model of the middle ear - equivalent mechanical network 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 analogies Model of the middle ear - equivalent mechanical network systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum complete equivalent network 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 analogies Model of the middle ear - equivalent mechanical network systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum dual conversion 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 analogies Model of the middle ear - equivalent mechanical network systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum network after dual conversion and neglect of irrelevant acoustical systems describing quantities elements Acoustical elements analogies potential and flow quantities Impedances Measurement of acoustical impedances distributed elements coupling interfaces Dual conversion thin absorbers vTR : velocity of ear drum measurement of v : velocity of inner ear window the impedance at ST the ear Model of the middle ear Diagnosis Experiments back analogies Diagnosis mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems diagnosis: describing quantities Acoustical elements I measurement of the input impedance analogies potential and flow quantities I estimation of the element values Impedances Measurement of acoustical impedances I comparison with limits → indication for possible distributed elements disease coupling interfaces Dual conversion thin absorbers measurement of the impedance at the ear Model of the middle ear Diagnosis Experiments back analogies Experiments mechanical experiments - model fit: 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 analogies Experiments mechanical experiments - comparison of impedances: 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 analogies Experiments mechanical experiments - velocity of inner ear window: 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 analogies

mechanical systems describing quantities mechanical elements Mechanical sources Mechanical resonance - spring pendulum acoustical systems describing quantities Acoustical elements eth-acoustics-2 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