México Unido contra la Delincuencia (MUCD)

The theory for equivalent circuit modeling is found in section 5.3.1.1. The equivalent circuit for the absorber system described in this section is of a simpler form than that described in chapters five and six. It consists of only a mechanical and acoustical section such that the analogue circuit that describes its motion can be given as:

is the resistive losses in the cabinet, M is the

determined by eighing the plate, ZAF is equal to ρ0c/SD if the absorber is within the impedance tube or ion 5.19 if in free field. RAD, can be estimated from a loudspeaker

in p 2 AD M R_AD CAD AB R AB C AF Z

Figure 4.13 Equivalent circuit for a membrane absorber

CABis the compliance of the cabinet, RAB AD

mass of the plate, RADis the resistive losses due to the mounting of the plate, CADis the compliance of the surround and ZAFis the radiation impedance which is dependent on whether the absorber is in a tube or in free field conditions (see section 5.3.3). All of the above quantities are in acoustic units.

4.4.1.1 Finding Parameters

Before the above circuit can be modeled and the surface impedance found it is necessary to determine some of the constants in the system. MADis easily

w

is given by equat

model, RABis determined from the Delany and Bazley model of porous media [10] and

CAB = V/ρ0c2. CAD is the hardest to determine and requires a simple experimental procedure as described in section 4.4.1.1.1.

4.4.1.1.1 Determining CAD

The compliance of the surround is given by its displacement when given a force F.

ssuming Hook’s law the force applied to the spring (surround) should be proportional stiffness of the surround and from that to determine its compliance the following test set up was used:

Figure 4.14 Experimental setup for measuring compliance of surround

he surround, the isplacement was then measured to a hundredth of a millimeter by the dial gauge which

ia

nd hence the compliance could be found. A

to its displacement. In order to find the

Added mass Dial Gauge

Clamped Support

Surround

Incremental changes in the mass applied to the plate displaced t d

was held with a clamp stand to keep it from moving. Masses were positioned carefully such that equal force was applied to the entire surround. The plate used was rigid hardboard so it was only the compl nce of the surround being measured rather than that of the plate itself. A graph as shown in Figure 4.15 was plotted of force applied versus displacement from which the stiffness a

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10-4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Deflection of Surround, m F or c e A ppl ied, N

round (Force Applied versus Deflection of Surround) Finding Compliance of Sur

Measured Data Points Least Squarres Best Fit Line

nd

(4.2)

easurement is given as 1.26% which is significantly less than the 10% error often quoted as the error in the compliance as measured by MLSSA in the added mass method of small-signal parameters [26] (see section 5.3.3).

Having determined the parameters governing the pistonic case the equivalent circuit can be solved easily. The equivalent circuit for the clamped case is the same but deriving the compliance and mass are difficult and as such an equivalent piston of the plate can be considered.

4.4.1.1.2 Finding the Equivalent Piston of a Clamped Plate

Figure 4.15 Results from experiment to determine compliance of surrou

A regression analysis was performed on the measured data and the corresponding best fit line plotted. The gradient of the best fit line gives the stiffness of the surround, the reciprocal of which gives the compliance. The compliance of the surround was found to be: -1 μmN 2.59 205.49± = AB C

Assuming a clamped plate with a Poisson ratio, μ thickness, h and Young’s modulus, E

it is possible to determine the equivalent mass and compliance of the plate as an equivalent piston. Rossi [27] presents such theory. By calculating the kinetic and potential energy of the plate as it is deflected by the standard parabolic deflection and integrating over its area it is possible to find an equivalent mass and compliance as if the plate were a pistonic resonator. Rossi gives the equivalent mass as one fifth of the mass of the clamped plate and the compliance as:

(

)

These values can be put into the equivalent circuit model such that the pistonic case can be compared with the clamped case.

.4.1.2 Results

lerometer tests so as to eliminate the uncertainties in the boundary conditions of the ounting rings:

4

The equivalent circuit was implemented in MATLAB using the code in Appendix G plotting the equivalent circuit for the pistonic case and the clamped condition yields

Figure 4.16. The scenario that was modeled was the same as with the initial acce

0 50 100 150 200 250 300 350 400 0 0.2 0.4 0.6 0.8 1

Equivalent Circuit Model of Membrane Absorber With and Without Surround Mounting

o n C o e ffi c ie n t With Surround Without Surround 55 Hz 95 Hz Frequency, Hz A b s o rp ti

Figure 4.16 Absorption curve from equivalent circuit model of membrane with and without surround

ll absorption of the system as it shows large inaccuracies in prediction f the absorption bandwidth. These inaccuracies result because a plate is a complex element, the equivalent piston model calculated an equivalent compliance and mass assuming the plate was at its fundamental resonance so outside of that range it is unsurprising that the model is inaccurate.