Experimental observationsof a mechanical cochlear model R. S. Chadwick

and D. Adler

Facultyof Mechanical Engineering, TechnionIsraelInstituteof Technology, ttaifa, Israel (Received10 December1973;revised29 April 1975) A two chamber cochlear model was built with the basilar membrane modeled by a thin elasticbeam. For

various inputfrequencies thesteady-state envelopes of thebeamoscillations weremeasured usingseveral combinations of elasticbeamsand viscousfluids.The frequencydependence of the thickness of the periodic

boundary layeradjacent to thebeamwasalsodetermined. The experimental resultsarein general

agreement withtheoretical predictions, andtheabilityof viscous damping to altertheshape of the envelopes is confirmed. An opticalleversuitable for measuring relatively smalldisplacements is also described.

SubjectClassification:65.26.65.3 5.

INTRODUCTION

Mechanical cochlear models have been used by several authors as an aid in understanding the mechanics of the

cochlea(e.g., yon B•kgsy, 1960; Tonndorf, 1959; Lesser and Berkley, 1972; and Soroka, 1974). In these models, some attempt was made to scale dimensions and physical properties in order to obtain dynamic similarity with the human cochlea. It seems, however, that no systematic attempt was made to compare the experimental results of the models with theory. A strict comparison was not possible because the existing theo-

layer has a retarding effect on the flow which modifies the longitudinal pressure distribution in the eoehlear canals. This point was recognized by previous authors (e.g., Zwisloeki, 1965), but an even stronger effect was overlooked. Namely, the normal velocity gradient in the Stokes layer produces a longitudinal surface trac-

tion r = (vwp•'/2)•/•'u whichacts on the basilar membrane. Since at a given distance along the coehlear duet the fluid velocity above and below the membrane is rr ra-

diansout of phase(this point is provedin the theory), these surface tractions produce a distributed moment

along.the beam. This momententers the differential

ries were not complete in the s,ensethat the effects of

equation for the displacement of the beam in the form

viscous damping were not determined by known parameters of the problem. The theory of Inselberg and Chad-

y(w, v)Vt, which is consistentwith the classical assumption that the dampingis proportional to the normal velocity of the beam. In this ease, however, the coefficient y is frequency dependent, and increases with the

wick (1975) and Chadwick et al. (1975) overcomes this difficulty, and for this reason a mechanical model has been built which complies as closely as possible to the conditions inherent in this theory. I.

DESCRIPTION

It is clear

OF THEORY

In the mathematical model, the cochlea is represented of Corti are not included) filled with a viscous incom-

pressible fluid and separated by a thin uniform EulerBernoulli beam (the basilar membrane). The system is driven at one end by the piston-like movement of the stapes. The solution of the resulting initial and boundary value problem is obtained in closed form in terms

of the five functions, u•(x, t), u2(x, t), p•(x, t), p•.(x, t), and r/(x, t), which, respectively, denote the average longitudinal fluid velocity in the upper and lower chamber, the pressure in the upper and lower chamber, and the transverse displacement of the beam. The two independent variables are the distance from the stapes x and time t. The independent parameters of the problem are given in Table I. A significant difference between this and previous interaction

of eoehlear

between

mechanics

the viscous

is the treatment

fluid

of the

and the basilar

membrane. Due to the oseillatory nature of the flow, a Stokes layer is formed on the walls of the eoehlear duet and the basilar membrane. This unsteady boundary

layer has a thickness6=0(v/w)x/•', andthereforebecomes thinner as the frequency is increased.

706

that the thickness

of the basilar

membrane

must be nonzero in this analysis (if not, the moment would be identically zero). Previous analysis, which

by two rectangular chambers (the scala media and organ

theories

square root of the frequency.

This

J. Acoust. Soc.Am.,Vol.58, No.3, September 1975

did not specifythe thickness, necessarilymissedthis importanteffect andwere forced to compensatefor it by specifyingarbitrary dampingfunctions. This theory avoids this difficulty since the damping is completely

specifiedby the knownparameters of the problem. The equations of motion for the model are

hux= nt ,

(2)

PUt = - Px - p•'u ,

(3)

where

u=

1

,

1

P = •(Pl -- P•.),

o••' 12pb/Ed•'

y =6p(2uw) •/•'/Ehd•' ,

fiL-24lEd3 ,

r= (2uw/h•.)1/gß

The beam is assumed to be hinged at both ends, resulting in the boundary conditions •

r/(O, t)=r/(L,t)=0,

(4)

t)= n=(z,, t)=0. Copyright ¸ 1975by theAcoustical Society of America

706

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707

Chadwick and Adler: Mechanical cochlear model

TABLE I.

Independent parameter

of mathematical

707

transparent,

model.

except the left end, which is covered by an

elastic membrane (M). A beam (BM) 63 cm long and Symbol

Physical Quantity

close fitting on the sides divides the box into two cham-

L

beam length height of each chamber beam thickness fluid density beam density kinematic viscosity of fluid Young's modulus of beam frequency of input signal amplitude of pressure difference between upper and lower chambers at stapes end (x=0)

bers (F), and is fixed to the side walls with low-friction bearings (B), thus simulating the zero displace-

h d p

E

Po

ment, zero moment, and conditions of the theory as

close as possible. -A gap (G) is left at the right between the end of the beam

helicotrema,

and the end wall

to stimulate

the

which was assumed to be represented

in

the theory by th• boundary condition Pt (L, t)= p•.(L, t). The system is driven on the left by an oscillating pis-

ton (P), the action of which is transmitted by water through a rigid pipe, and produces a displacement of the

elastic membrane. The pistonhas a variable stroke The pressure equilization by the helicotrema yields

at x = L

t) - 0 .

and its frequency could vary between 0.5 and 10 Hz. ' Another degree of freedom is provided by an air column

(5)

The whole system is driven by

p(O, t) = Posinwt .

(6)

The solution of Eqs. 1-6 for the steady-state beam displacement envelope, r/•.sv, is given by

•/•.sv =

d,.

+

dan --

-

,

(AC) of controllable active height by moving the plug (PL), which has the effect of modifying the output impedance of the driving system. This proved to be very useful in making fine adjustments of the amplitude of vibration

The entire assembly sits on two eccentric cams (C)

(7)

where

that are used for calibration purposes by raising the box a known amount. The need for this is explained in Sec. III.

The dimensions of the apparatus were determined by convenience only. No attempt was made to obtain dynamic similarity with the human cochlea, since, as

•.=(- •A.+ C.) •+• B., n

(8)

'

+

Cn 4W4 =nL4 • ,

mentioned before, the prime purpose of the present experiment is to verify the mathematical model. III.

D,=-2P 0/n•.

Details of the development of the mathematic• model and its solution can be found in Inselberg and Chadwick

(1975) and Chadwick et al. (1975).

METHODS

THE

MECHANICAL

The mathematical

a calibrated

decided

MODEL

marked

of the vibrations.

neces-

divisions.

A

It is estimated

that the

a mechanical

model

The second method is based on an optical lever principle which is potentially very useful when direct micro-

which PL

taken to allow variation of the main parameters listed in Table I such as beam thickness, densities of beam

//////////////////////////////////

and liquid, fluid viscosity, Young's modulusof the beam, frequency of the input signal, and pressure difference amplitude. Using this mechanical model, the effects of these main parameters as predicted by the theory could and the relevance

0.1-mm

sufficiently accurate to compare the experimental results with theory. Using this method the steady-state envelopes were determined by measuring the displacements point by point along the length of the beam.

will represent the mathematical model including all its geometrical and mechanical simplifications. Care was

be verified

with

error in making these measurementswas ñ 5%, which is

model described previously

to construct

scale

stroboscopic light source was used to allow visual ob-

sarily includes a number of assumptions and simplifications to allow formulation and solution of the problem. Its agreement with physiological cochlear mechanics cannot be investigated directly since the necessary experiments which would yield the required quantitative irfformation are not possible in a living ear. It was therefore

OF MEASUREMENT

Two methods were used to measure the steady state beam envelopes. The first was simply visual observation of the vibrating beam with an objective lens having

servation

II.

of the beam.

of the mathematical

M

model

corffirmed. FIG.

The model consists of a fluid-filled box having the inside dimensions' 68 cm long, 20 cm high, and 8 cm

wide (Fig. 1). All of the walls of the box are rigid and

1.

Schematic

of mechanical

model.

The letters

denote

the following: C, ham; F, fluid; W, water; B, bearing; beam; M, elastic membrane; AC, air column; PL, plug; P, piston; and G, gap.

J. Acoust.Soc. Am., Vol. 58, No. 3, September1975

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708

Chadwick and Adler: Mechanical cochlear model

708

Ix01> w). As a practical application, suppose it is desired to measure a deflection y = 0.05 mm at a magnification of FLUID

100 with an accuracy of 5%, using a mirror 1 mm wide.

MOTION

Then fromEq.

•

•.•

FIG. 2.

and fromEq.

9, l ~1 m.

Often it is difficult or impractical to determine x0 by measurement. In such a case, it is necessary to calibrate the system by moving the mirror a known amount and recording the resulting deflection of the light on the screen. This was the reason for using the cams shown in Fig. 1.

SCREEN

BEAM

11, x0 ~lcm,

MOTION

Optical lever system for measuring vertical displace-

ments.

scopic viewing is inconvenient due to focusing restrictions. The beam is first painted dull black and then small front surface mirrors of good quality are glued to the surface. A converging beam of light is focused near a mirror and the deflection is projected on a screen

The unsteady boundary-layer thickness was measured by observing the motion of fine carbon particles suspended in glycerine-water solution having a viscosity of 1.3 poise at 24 øC. The thickness of the layer could be determined visually by observing (through the objective

lens) the phase difference between the particles near the beam surface and the particles far from the surface. The slower moving particles near the surface respond to changes in the pressure gradient more rapidly than the particles far from the surface which have more inertia. ,

(Fig. 2). In the first approximation the mirror does not

The

rotate, but remains horizontal as it moves up and down. The magnification is caused by the mirror intercepting rays of different angles as it moves through the light

the carbon particles move in phase with the bulk of the fluid was used as a measure of the boundary-layer thick-

minimum

vertical

distance

from

the beam

at which

ness.

beam.

IV.

From geometrical considerations one can show that the relationship between vertical deflections of the mirror

Ay and th• corresponding vertical deflections on the screen A y is given by

r= where

+ t/Xo)

l is the distance

(9) between

the mirror

Altogether, 12 experiments were carried out in which the steady-state beam envelopes were measured. Two beams (stainless steel and rubber) were tested in water and glucose. In each combination, measurements were

taken for 1, 5, and 9 Hz. thickness

and the

screen, and x0 is the horizontal component of the vector drawn between the mirror

EXPERIMENT

and the potential focus

measurements

In addition, boundary-layer were

taken

beam vibrating in a glycerine-water frequencies.

for

a stainless

steel

mixture for six

(Fig. 2). In the first series of experiments, Because any real mirror will have a finite width, say

2w, it produces a spread of the signal on the screen o r which is given by

2wy(xo +l)

(10)

The ratio AY/•rr is the signal-to-noiseratio wherethe is due to the finite

width

of the mirror.

From

Eqs. 9 and 10,

AY/r•r= (Ay/2y)(x•- w•')/Xo w.

•

But ymin=Aymax/2 , where Ayma , is the maximumvertical movement

light beam.

of the mirror

such that it doesn't

(• Y/o'It)max = x'•0 -- •x0 ß w

(11)

From Eq. 0 it follows that the magnification increases as the light is focused closer to the mirror, while from Eq. 11 it follows that the signal-to-noise ratio improves

as the light is focused farther from the mirror (for

what

conditions

it was decided to see

the mechanical

model

behaved

linearly. Linearity implies that the normalized (with respect to the maximum amplitude) steady-state envelopes should be identical irrespective of the amplitude of the input signal. This was found to be the case over the frequency range 1-10 Hz, provided the maximum beam amplibade did not exceed approximately 3 min. Greater amplitudes resulted in a nonlinearity primarily due to small jumping motions of the beam in the bearings, which obviously must be avoided.

leave the

In that ease,

water.

Since the theory is a linear, under

where y is the vertical component of the vector drawn between the mirror and the potentiai focus. "noise"

was used, and the eochlear model was filled with distilled

O'y=X•O_ /•2 ,

a stainless-steel

beam, 2.45 mm thick, havingE = 2.0 x 10•' dyn/cm•',

R was found that this beam was too stiff to show a

measurable change in the envelope even when a fluid with high viscosity was used. A nondimensional group of parameters which determines the relative effects of the viscous damping force to the bending force is given by

h •l•') k-2 r oo Sl•'L•p l•tl=

J. Acoust. Soc. Am., Vol. 58, No. 3, September 1975

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Chadwick andA•ller: Mechanical cochlear model

709

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