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Electrical-Mechanical-Acoustical Analogies

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Electrical-Mechanical-Acoustical Analogies 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 ^ coupling I given: external force: F (t) = F sin(!t) 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 = F^ ej!t 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 ^ j!t ^ j!t Measurement of acoustical F e F e impedances x(t) = = 2 distributed elements (j!) m + j!R + s j! j!m + R + s coupling j! 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 = j!m + R + acoustical systems j! describing quantities Acoustical elements the quantities become: analogies potential and flow quantities Impedances ^ j!t Measurement of acoustical F e impedances x(t) = distributed elements j!Zm coupling j!t interfaces F^ e Dual conversion u(t) = thin absorbers Zm j!t measurement of j!F^ 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
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