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Devanjith FonsekaSummer 2016 Xerox Fellowship Report

ANALYSIS OF MICRO-CHAMBER FOR COCHLEA MEASUREMENT

Mechanical Engineering Department, University of Rochester

Abstract

The main purpose of this project was to develop theoretical models that simulate the pressure and velocity responses for a membrane attached to the micro-chamber. Two types of models were developed to simulate the micro-chamber: a one-dimensional acoustic model and a lumped element model, valid at low frequencies, when the wavelength is greater than the micro-chamber dimensions. Both models gave similar results in the low frequency limit and the overall shape of the experimental data was consistent with that expected from these models. A simulation of the experimental procedure to find the compliance as a function of frequency was performed to verify the assumption that the pressure at the membrane for a membrane of unknown stiffness is the same as that of calibrated membrane stiffness. This assumption was found to hold only for a low frequency range and for membranes stiffer than the calibrated one.

Introduction

The micro-chamber was specifically built to measure the compliance of a membrane attached to a slit in one of its chambers. It is composed of a network of tubes connected to a source, either a speaker or a piezoelectric transducer at one end, and branches out to tubes containing the membrane, pressure release port and the feeder tubes as seen in figure 1.

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Figure 1: CAD drawing of the micro-chamber.

The source is placed at the oval window and vibrates the oval window upon activation. This vibration generates fluid velocity and pressure waves which travel along the lengths of the tubes. As a result, there is a pressure being exerted at the membrane which causes it to deflect back and forth with a velocity which is dependent on its stiffness. A laser vibrometer, focused on the membrane is used to measure the velocity of the membrane.

There are two ways to model the micro-chamber. One method is a one-dimensional acoustical model of the micro-chamber which uses the solutions to the pressure and velocity wave equations to generate solutions determined by the boundary conditions of the chamber. This approach is very straightforward but offers little insight to the effect each component within the chamber has on the response at the membrane. This method can also be more computationally demanding depending on the complexity of the micro-chamber. The alternative approach is the lumped element method which is an approximation only suitable for applications where the dimensions of the tubes and the frequencies are small enough that they can be considered as a lumped element as opposed to a continuum. The range of frequencies for which this assumption is valid is no more than a few kilohertz, because the dimensions of the tubes are on the order of millimeters. The advantage of using this method is that each tube, cavity and membrane has a specific acoustic impedance, inertance and compliance which gives intuition as to which components influence the response.

The theoretical models are based on a one-dimensional equivalent schematic of the micro-chamber as shown in figure 2. This is possible due to gravity being neglected and the pressure being a function of absolute length relative to the source. The experimental set-up allows for further simplifications to be made. The feeder tubes are closed whilst the micro-chamber is in operation, which implies that the velocity at the tube ends should be zero. Since the fluid is incompressible and due to the conservation of volume flowrate, this also implies that the fluid velocity entering the tubes at the junction should be also zero allowing one to neglect the feeder tubes. However, the inclusion of the tubes makes them resemble a Helmholtz resonator (band pass filter). The simplified model is shown in figure 3.

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Figure 2: One-dimensional representation of micro-chamber. Tube 1 is from the oval window to the junction, which connects to the other tubes. Tubes 4 and 5, and, 6 and 7 are of equal dimensions; they form the feeder tubes. Tubes 2 and 3 contain the membrane and the pressure release port respectively.

Figure 3: Simplified schematic of the micro-chamber.

Acoustic Model

I. Governing equations of this model

The one dimensional pressure wave equations is

.c2o∂x2

∂ p2− ∂t2

∂ p2= 0

Assuming the solution to this wave equation varies time harmonically,

,(x)ep = p iωt

the solution to the pressure wave equation for the spatial dimension, , isx

.(x) e epi = Ai−jk xi + Bi

jk xi

The pressure wave is related to the velocity via the impedance, byZ,

,up+,i = Z i +,i

,up−,i = − Z i −,i

where subscripts + and – denote forward and backward travelling waves.

Using the above definition, the velocity wave equation can be written in terms of the pressure wave equation and impedance as

,(x)ui = Zi

A ei−jk xi

− Zi

B eijk xi

where is the wavenumber given byki

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.jki = cω + αi

The subscript denotes the tubes in the micro-chamber, the angular frequency, the speed of soundi ω c in water and the absorption coefficient which accounts for damping in the model. There are severalα ways to include damping in the micro-chamber. The options include viscous damping, thermo-viscous damping and boundary layer absorption damping; the most significant effect is shown by boundary layer damping. This is a result of the dimensions of the tubes being small enough that both the thermal and viscous boundary layer is of comparable thickness with the diameter. The expression for isα

,(1 ) α = 1R√ ωμ

2ρc2 + Prγ−1

where is the radius of the tube, is the density, is the ratio of the specific heats ( and Pr isR ρ γ /C ) Cp v the Prandtl number.

To get the solution, the coefficients and need to be determined by the use of boundary conditions.Ai Bi

II. Boundary Conditions

The boundary conditions applicable to the chamber are: conservation of mass flowrate, continuity of pressure and termination boundary conditions.

1. PZT response boundary condition ,(l )us = u1 1

where is the velocity of the source. One assumption made in the calculation was that the sourceuS displacement is independent of frequency.

2. Conservation of mass at the junction connecting tube 1 with tubes 2,3,4 and 5

,u (l ) u (l ) u (l ) u (l1) u (l1)s1 1 1 = s2 2 1 + s3 3 1 + s4 4 + s5 5 where is the surface area of the tube.s

3. Continuity of pressure at the junction connecting tube 1 with tubes 2, 3, 4 and 5.

,(l ) (l )p1 1 = p2 1

,(l ) (l )p1 1 = p3 1

,(l ) (l )p1 1 = p4 1

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.(l ) (l )p1 1 = p5 1

4. Termination at membrane

,u (l +l )2 1 2

p (l +l )2 1 2 = ω−i Kmem

Where is the stiffness of the membrane measured in .Kmem N

m3

5. Zero pressure condition at pressure release port (exposure to atmospheric pressure)

.(l )p3 1 + l3 = 0

6. Continuity between tubes 4 and 6, and tubes 5 and 7

,u (l ) u (l )s4 4 1 + l4 = s6 6 1 + l4

,u (l ) u (l )s5 5 1 + l5 = s7 7 1 + l7

,(l ) (l )p4 1 + l4 = p6 1 + l6

.(l ) (l )p5 1 + l5 = p7 1 + l5

7. Rigid termination at the end of tubes 6 and 7.

,(l )u6 1 + l4 + l6 = 0

.(l )u7 1 + l5 + l7 = 0

For the simplified model, only boundary conditions involving i = 1, 2 and 3 are applicable.

The above expressions were inputted into Mathematica so that the coefficients could be determined.

III. Results

The velocity and pressure transfer function plots as a function of frequency are shown in figures 4 through 7 for both the simplified and complete model. As per the experimental group, the compliance of the membranes measured during a hydrostatic experiment was approximately 5nm/Pa which corresponds to a stiffness of . Additional membrane stiffnesses which were an order of magnitude above andx10 N/m2 8 3 below the measured stiffness were plotted to understand the effects of the membrane stiffness on the response. This isolates the membrane resonances from the tube resonances.

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The pressure and velocity transfer functions show that the overall trend of the curves is similar for both the complete and simplified chamber. Membranes of lower stiffness undergo resonance at lower frequencies as expected and vice versa. It is also seen that the pressure and velocity transfer functions have different resonant frequencies for a specific membrane stiffness. One interesting observation is that the inclusion of the feeder tubes causes the resonances to shift slightly to the left and there is a dip at around 5.2 kHz in both the pressure and velocity plots. The tube resonances beyond the domain at 35 kHz are shifted to 25 kHz, as seen in figure 6 and 7, in the complete model but they are beyond the domain of interest for experiments.

Figure 4: Velocity transfer function for the simplified model. Resonant frequency for the membrane of stiffness is 0.74 kHz.x10 N/m2 8 3

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Figure 5: Pressure transfer function for the simplified model. Resonant frequency for the membrane of stiffness is 1.12 kHz.x10 N/m 2 8 3

Figure 6: Velocity transfer function for the entire model. Resonant frequency for the membrane of stiffness is 0.71 kHz.x10 N/m2 8 3

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Figure 7: Pressure transfer function for the complete model. Resonant frequency for the membrane of stiffness is 1.09 kHz.x10 N/m 2 8 3

Lumped Element Model

I. Introduction to this model

As mentioned previously, the lump element method can be used only for instances when the product of the wavelength and the length of the tube, is much less than 1. In this method, the acousticalkl, impedances can be represented as electrical impedances with the pressure and volume velocity ( ), beingq equivalent to voltage and current, respectively. Note that the volume velocity is akin to current and not velocity because the volume velocity is a conserved quantity even with varying cross-section.

Each component in the tube can be modelled by either an inductor, capacitor or a resistor having a specific acoustical impedance ( ). A tube is similar to an inductor because the pressure causes the mass/qp of fluid within the tube to accelerate. Therefore the derived expression for the impedance of a tube is

,Z in = siωρl

where and is the length and cross-sectional area of the tube respectively.l s

Using a similar approach, the impedance of a cavity or a membrane behaves like that of a capacitor because the pressure results in the compression of either the fluid in the cavity or the membrane. This makes them resemble a spring with a specific compliance. Therefore these impedances are

,Zcap,mem = Kmemiωsmem

,Zcap,cavity = Viωρc2

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where is the stiffness of the membrane, is the area of the membrane and is the volume ofKmem smem V the cavity.

Resistors in acoustical systems represent dissipative terms such as viscosity and radiation of energy to the atmosphere at the pressure release port. There is no definitive expression for the resistance to account for boundary layer damping and thermal effects could be neglected because the fluid is in thermal equilibrium. However, viscous effects can be considered by using Pouseille’s flow equation and solving for . q

p

The viscous damping impedance is

,Zvisc = 8lμ πR4

and the radiation impedance is

.Zrad = 2Πcρ ω0

2

Assigning the impedances to the respective parts of the micro-chamber yields the circuit in figure 8. Every tube within the chamber has an inductor and resistor combination, and cavities only consist of a capacitor. The radiation term is only included in tube 3 because it is the only tube which is exposed to the atmosphere. The volume velocity and the pressure can be found by calculating the current and pressure through the capacitor C2 respectively.

Length corrections were added to tubes 2, 3, 4 and 5 because the pressure causes perturbations in the ends of adjacent tubes. Therefore the lengths of tubes are increased by a correction, , ofl∆

,l .85R∆ = 0

where R is once again the radius of the tube.

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Figure 8: Lumped element circuit for the micro-chamber.

II. Results

The velocity and pressure transfer functions were plotted in figures 9 through 13 to make a comparison with the acoustical model. As in the acoustical model, inclusion of the two feeder tubes has the same effect in the lumped element model; the peaks are shifted slightly to the left and there is a dip at 5.2 kHz as a result. Note that the peaks are sharper than that of the acoustical model; this is due to viscous damping not being as significant as boundary layer damping

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Figure 9: Velocity transfer function for the lumped element model. Resonant frequency for the membrane of stiffness is 0.68 kHz.x10 N/m2 8 3

Figure 10: Velocity transfer function for the complete model. Resonant frequency for the membrane of stiffness is 0.69 KHz.x10 N/m2 8 3

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Figure 11: Pressure transfer function for the simplified lumped element model. Resonant frequency for the membrane of stiffness is 1.02 kHz.x10 N/m2 8 3

Figure 12: Pressure transfer function for the complete lumped element model. Resonant frequency for the membrane of stiffness is 1.00 KHz.x10 N/m2 8 3

The versatility of the lumped element models is that an expression for the resonant frequency can be derived from the response transfer function. By setting the denominator equal to zero in both the velocity and pressure transfer functions, and solving for frequency yields

, fmem,vel = 12Π√ s2

Kmem

ρ( + )l3s3

l2s2

for the velocity transfer function and

, fmem,pres = 12Π√ ( + )s ρ2

Kmem l1s1

l2s2

( + + )l l2 1s s2 1

l l2 3s s2 3

l l3 1s s3 1

for the pressure transfer function.

Discussion of the two models

To compare both models quantitatively, both models were plotted on the same axis in figure 14. The comparison shows that the resonant frequency is approximately the same, with the deviation of the maximas being less than 4%, for all models irrespective of whether they are simplified or complete renditions of the micro-chamber. However in the high frequency domain (frequencies greater than 20 kHz) the lumped element models converge whereas the acoustical model diverges to form a peak at

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around 35 kHz (tube resonance). This is a result of the lumped element model being an approximation only valid in the low frequency limit. The dip also occurs at the same frequency in both models.

The dip is attributed to be a result of the feeder tubes since they are not present in the simplified models. The feeder tubes bear resemblance to a Helmholtz resonator since it is a tube which is connected to a cavity. Helmholtz resonators are band-pass filter which restrict flow when excited at its resonant frequency. The resonant frequency of a Helmholtz resonator is given as

. fHelmholtz = c2Π√ s4

l s l4 6 6

When the resonant frequency of a Helmholtz resonator is computed, it coincides with the frequency at which the dip occurs since the pressure drops to a minimum at this point (transmission coefficient becomes zero when a Helmholtz resonator resonates).

The effects of the micro-chamber dimensions on the response can be found by looking at the expressions of the natural frequencies since all the peaks within the frequency range can be defined by those expressions. The membrane resonances are dependent on the dimensions of tubes 1, 2 and 3 whereas micro-chamber resonances depend on the dimensions of tubes 4, 5, 6 and 7.

Figure 14: Plot of the velocity transfer function of the lumped element model from the simplified and complete chamber, and the response from the simplified one-dimensional model.

Experimental Results

The theoretically expected results from the models are compared with the empirical results to validate the accuracy of the models. Velocity data measured at the slit for a membrane of stiffness of isx10 N/m2 8 3 shown in figure 15.

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The shape of the response is similar to the shape of the response from the theoretical models of the complete chamber although the peaks occur at different frequencies. By the use of , a stiffness off mem approximately 1/8 of which was used is required to generate the desired peak at 0.225 kHz. However, the experimental group did report highly inconsistent data in the frequency domain till 0.300 kHz so there exists a possibility that the collection of peaks at 0.7 kHz is a result of the membrane. Several assumptions were made by the experimental group to compute the pressure so it yielded the same shape as the velocity response.

Figure 15: Velocity of the membrane for a membrane compliance of Pa. There are resonantx105 −9 /m peaks at 0.225 kHz and 4.5 kHz. Low frequency data is unreliable and can be neglected.

Simulation of experimental procedure to yield compliance

To measure the compliance of a membrane, the experimental group makes the assumption that the pressure measured at the slit for a membrane of specific stiffness is the same as the pressure at the membrane of an unknown compliance. This assumption will obviously breakdown in a limit such as when the stiffness is zero because the pressure at the slit should be zero. By simulating the way they measure the compliance from the data, the bounds to which this assumption works for both the stiffness and frequency range can be determined.

The procedure to calculate the compliance involves dividing the displacement as a result of an unknown stiffness by the calibrated pressure at the slit for a membrane of known stiffness. This provides the compliance as a function of frequency as shown in figure 16. The simulation revealed that the compliance is flat only for the curve for which both the pressure and velocity were a result of the same membrane stiffness. For membranes which had stiffnesses greater than which was calibrated, the correct compliance can be determined in the low frequency domain as long as the frequency is less than the calibrated membrane’s resonance frequency. This is shown in figure 16 with the flat region of the compliance curves

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being on the horizontal lines which depicts the expected compliance. Membranes of lower stiffnesses to that of the calibrated membrane do not show this flat compliance unless the difference in stiffness is very small. Note that at higher frequencies, the compliance converges to that of the calibrated membrane.

Figure 16: This plot simulates the measured compliance for different stiffnesses using a calibrated stiffness of . The horizontal lines show the expected compliance values for the respectivex102 8 /mN 3 membrane compliances which were plotted.

Conclusion and Discussion

A one-dimensional acoustical and a lumped element model were used to simulate the velocity and pressure response of a membrane mounted in a micro-chamber. Both models have a simplified representation of the chamber with the closed feeder tubes being neglected. The lumped element model and the acoustical model yielded similar responses in the low frequency domain but the lumped element model breaks down at higher frequencies (frequencies above 20 kHz). It could also be inferred that compliant membranes undergo resonance at lower frequencies than stiffer membranes. The frequency at which the membrane resonances occur for the pressure and velocity transfer functions are predicted by

and , respectively. Another interesting phenomenon is the existence of the dip at aroundfmem,vel fmem,pres 5.2 kHz which is indicated in the complete micro-chamber model of both the acoustical and lumped element model, but not in their simplified models. This is a result of the feeder tubes resembling a Helmholtz resonator since 5.2 kHz corresponds to its natural frequency, . Comparison of thef Helmholtz

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predicted results with the empirical data show that the overall shapes of the plot agree but the resonances appear at frequencies which are to be expected by a much more compliant membrane. However the low frequency data (0-0.3 kHz) was not coherent so there is uncertainty in determining the membrane resonance experimentally.

The experimental procedure to calculate tissue compliance was simulated using the assumption that the pressure at the membrane is the same as that of a calibration membrane of known stiffness. This method predicts the correct compliance value for frequencies below the calibration membrane resonance frequency, as long as the compliance of the tissue is less than the calibration membrane compliance.

The model would have been more accurate if the response of the transducer displacement was taken into consideration. It was assumed that the displacement is independent of frequency but this assumption is questionable due to the additional loading of the chamber.

Acknowledgements

I would like to thank Professor Gracewski for all the help she has provided during the course of this research project. This project would not have been possible if not for her guidance and teachings over the course of the year. I would also like to thank the experimental group of Nam lab for answering questions and for giving data from the micro-chamber. Last but not least, I thank Professor Blackstock for helping me understand concepts in acoustics.

References

Blackstock, David T (2000) fundamentals of physical acoustics . Austin, Texas: John Wiley and Sons, Inc

Appendix

Table 1: Dimensions of the parameters of the micro-chamber

Parameter Dimension l1 0.0095 m

l2 0.00042 m

l3 0.014 m

,l4 l5 0.026 m

,l6 l7 0.023 m

s1 .85×10 m 7 −7 2

s2 ×10 m 5 −7 2

s3 .85×10 m 7 −7 2

,s4 s5 .01×10 m 2 −6 2

,s6 s7 .94×10 m 5 −6 2

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,R1 R3 0.0005 m

,R4 R5 0.0008 m

,R6 R7 0.001375 m d 0.001 m w 0.0005 m

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