Moulton  Lectures

 

On

Electro-Acoustics

 

 

 

Presented by:

Dave L Moulton and Vanessa Berthe

 

 

 

 

Location:  Thales Acoustics Harrow UK

Date:  20-November-2002

 

Content

 

This lecture is more of a presentation of the mathematical model with some technical comments and explanations thrown in by myself.

 

The mechanical mathematical model developed in lecture 6 has been successfully put onto a spreadsheet by Vanessa Berthe and structured so that the affect of individual performance parameters can be seen on a dynamic graph.

 

The model has been compared alongside real attenuation performance curves for Thales Acoustics passive hearing protectors.  In particular these headsets have included the Slimgard II and Crewgard.  Others headsets have also been used for model comparison, such as the commercially available Peltor hearing protector purchased from RS.

 

 

On with the Lecture/Presentation

 

Below are the generalized representations of the mechanical model for a simplistic hearing protector and the corresponding transfer function.

 

 

Figure 7.1

 

 

Figure 7.2

 

 

The coefficients for the transfer function in terms of the generalized model parameters are shown below.  I will leave it to the class to prove these expressions for themselves, it’s a long rigorous process.

 

N₃ = C_FC_C(M_L + M_S) (R_C + R_F)…………………………………………..[7.1]

 

N₂ = C_FC_C[ R_F R_C + (R_C + R_F) R_L ] + (C_C + C_F) (M_L + M_S)  ………….…[7.2]

 

N₁ = R_FC_F + R_CC_C + (C_C + C_F) R_L ………………………………...……..[7.3]

 

D₅ = C_V M_LM_SC_FC_C (R_C + R_F) …………………………………………..[7.4]

 

D₄ = C_V[M_L(R_F R_C C_FC_C + M_S(C_C+ C_F)+ M_S R_LC_FC_C(R_C+R_F)]………….[7.5]

 

D₃ = C_V[ R_L R_FR_CC_F C_C +M_L(R_FC_F+R_CC_C)+ M_SR_L(C_C+C_F)+ C_FC_C(M_L+MS)(R_C+R_F)] ………………………………………..….[7.6]

 

D₂ = C_V[ (M_L+R_L(R_FC_F+ R_CC_C )]+ C_FC_C[R_F( R_C +R_F)+R_LR_C]+

+ (M_L + M_S) (C_C + C_F)……………………………………………..[7.7]

 

D₁ = C_VR_L + R_FC_F + R_CC_C +(C_C + C_F) R_L …………………………….....[7.8]

 

Vanessa’s model uses these expressions and applies predicted and measured parameter values to the individual components.

 

The individual parameters have been calculated and measured in the lab on actual earcups, earcushions and shell volumes.

 

The cushion compliance and Resistance has been measured using Force versus Deflection, and Displacement versus Time curves, obtained through rigorous physical measurements. A more detailed examination of these measurement methods will be covered in Lecture 8.

 

The Graphical Results and Simulations of Vanessa’s Excel model are shown in the sections below:

 

Graphical Results and Simulations

 

Note:  The modeled curve is the black one.  The actual attenuation measurements are shown as the data points with error bars.  The error bars represent the maximum and minimum attenuation levels over a range of repeated test measurements.

 

1.     The simulated performance of a Passive Jetgard Earshell with Cushions.

 

 

 

 

 

 

             

 

 

 

 

End of Lecture

 

 

 

 

 

 

Figure 7.3

 

2.     The simulated performance of a Passive Slimgard II Earshell with   Cushions.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 7.4

 

The Slimgard II  results show the difference between  the empty earshell  and when it is fitted with standard components such as foam and an Active Noise Reduction (ANR) Module.  In this case the ANR module is only in the earshell to take up space and is not in its operational mode.

 

In theory the simulation for the empty earcup should be most representative of the mathematical model, however the above results show a degree of inaccuracy in the model.  This could be down to the difficulty in providing accurate values for the model parameters.  Also the real attenuation results tend to vary significantly between real subjects and this could also account for the discrepancy.

3.     The simulated performance of a Commercially Available Hearing Protector produced by Peltor.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 7.5

 

 

These results show that the model is very close to the real measurement. It also shows that in this case the introduction of foam into the earshell has negligible affect on the attenuation performance.

 

 

 

4.    The simulated performance of a Commercially Available Hearing Protector with an un-modeled cushion leak.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 7.6

 

 

The above results show how badly the model performs when the headset earcushion has a built in leak path which has not been added to the mathematical model.

 

The Basic Model

 

The basic model simulated in Excel provides us with the ability to see the affect of the performance parameters on the overall attenuation response. The model allows us to have a reference curve (all parameters are at fixed values) and then to have a working curve which changes as the individual parameters are varied.  The comparison of the working curve and the reference curve give us a good visual indication of the performance direction due to parametric changes.

 

Some examples are shown below:

 

Changes in Earshell Mass.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 7.7

Changes in Cushion Resistance (System still heavily damped)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 7.8

 

The above curve shows how reducing the cushion Resistance greatly affects the attenuation at the low frequency end of the spectrum.  The combination of the earshell Mass and the cushion parameters produce a second order system, very similar to that described in lecture 5.  In lecture 5 we looked at the solutions to the second order differential equation described by a Mass, Spring and Damper.  As a reminder the defining function is shown below:

 

 

Where this function (deterministic equation) is satisfied by the condition:

 

In this case the Resistive Forces within the Cushion are still greater than the Stiffness forces within the Cushion. The system does not oscillate or have any overshoot.

 

If we were to change the balance such that the Stiffness forces become greater than the resistive forces then the inequality would become complex.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 7.9

 

 

 

This condition is satisfied when:

 

 

 

Increasing the size of the Leak Path Between the Cushion and the Skin

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 7.10

 

Increasing the Earshell Volume

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 7.11

 

 

Increasing the Earshell Vector area A_S

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 7.12

 

 

 

End of Lecture