The paper presents methods and experimental results of the semi-active control of noise transmission in a curved glass shell with attached piezoelectric macro fiber composite (MFC) actuators. The semi-active noise control is achieved via active elasticity control of piezoelectric actuators by connecting them to an active electric shunt circuit that has a negative effective capacitance. Using this approach, it is possible to suppress the vibration of the glass shell in the normal direction with respect to its surface and to increase the acoustic transmission loss of the piezoelectric MFC-glass composite structure. The effect of the MFC actuators connected to the negative capacitance shunt circuit on the surface distribution of the normal vibration amplitude is studied using frequency-shifted digital holography (FSDH). The principle of the used FSDH method is described in the paper. The frequency dependence of the acoustic transmission loss through the piezoelectric MFC-glass composite structure is estimated using measurements of the specific acoustic impedance of the curved glass shell. The specific acoustic impedance is measured using two microphones and a laser Doppler vibrometer (LDV). The results from the LDV measurements are compared with the FSDH data. The results of the experiments show that using this approach, the acoustic transmission loss in a glass shell can be increased by 36 dB in the frequency range around 247 Hz and by 25 dB in the frequency range around 258 Hz. The experiments indicate that FSDH measurements provide an efficient tool that can be used for fast and accurate measurements of the acoustic transmission loss in large planar structures.

## I. INTRODUCTION

It is an everyday acoustic experience that glass windows represent a substantial path for noise entering the interior of buildings. From the point of view of noise perception by humans, a noise in the frequency range from 2 kHz to 5 kHz is a frequent source of inner ear impairment due to the highest sensitivity of the human ear to this range.^{1} There exist several passive methods for the suppression of noise transmission through glass windows, such as laminated glass technology and double glazing.^{2} A big disadvantage of passive methods is the fact that their implementation goes hand in hand with a considerable increase in weight of the laminated glass structure, which is frequently unacceptable in applications where the total weight of the system is a consideration. A well-known tool for eliminating noise transmission is active noise control (ANC), which is based on the destructive interference of disturbing noise waves with secondary ones. Such an approach has recently been demonstrated in the work by Zhu *et al.*,^{3} where a sound transmission reduction by 10 dB in a broadband frequency range of up to 600 Hz was achieved. ANC can also be used to improve the sound insulation of double-glazed windows. A comprehensive overview that gives an insight into the physics of active noise control of double-glazed windows using several systems of loudspeakers and microphones inside the cavity has been presented by Jakob and Möser. They achieved nearly 8 dB and 5 dB reductions in the total sound pressure level using feedforward^{4} and adaptive feedback controllers,^{5} respectively. Unfortunately, the implementation of ANC methods in the frequency range from 2 kHz to 5 kHz is an extremely challenging task, which is very difficult to accomplish in real-life systems. Therefore, the search for alternative methods of glass window noise suppression functional in the considered frequency range has become an important mission.

A very promising approach that merges the advantages of passive and active methods is the active piezoelectric shunt damping (APSD). The APSD method is based on the change of the vibrational response of a mechanical structure using piezoelectric actuators shunted by electronic circuits which have a negative capacitance (NC). This principle was originally presented by Date *et al.*^{6} Several successful implementations of the APSD method have been achieved since that time, e.g. noise shielding and absorption using thin piezoelectric polymer membranes,^{7–9} vibration suppression of cantilever beams and plates.^{10–12} In order to improve the performance of APSD devices in varying operational conditions, several adaptive APSD systems have been developed.^{13–15} A rather general description of the vibrational response of large planar structures with attached piezoelectric actuators has been developed using an introduction of Piezo-Electro-Mechanical structural members by Alessandroni *et al.*^{16,17} and Porfiri *et al.*^{18} Theoretical analysis of the suppression of noise transmission through glass plates is presented in the work by Nováková *et al.*^{19}

The experiment analyzed the possibility of increasing the acoustic transmission loss of sound transmitted through planar or curved glass shells using attached piezoelectric macro fiber composite (MFC) actuators shunted by NC circuits. The analysis of this kind of noise isolation device (NID) was focused on theoretical calculations, which indicate key features that control the sound transmission through the NID. In addition, a detailed analysis of the particular arrangement of MFC actuators on a glass shell was performed using a finite element method (FEM) model. The FEM simulations showed that the essential difficulty that prevents an easy implementation of the APSD methods to large planar NIDs is the presence of a large number of vibration modes that are difficult to suppress by a limited number of actuators.

Another issue associated with the development of large planar NIDs is the availability of compact, accurate, and fast methods of the acoustic transmission loss (TL) measurement, i.e. the measurement of the physical quantity that expresses the noise shielding efficiency of a device. There exist two frequently used conventional techniques to determine TL. The first technique is the laboratory measurement of sound insulation of building elements,^{20} which is based on a determination of average sound pressure levels in two reverberation rooms. The second technique is a very precise four-microphone transfer function method in the acoustic tube.^{21,22} Unfortunately, neither of these techniques is suitable for the development of large planar NIDs. The former technique requires voluminous reverberation rooms and is rather time consuming, which makes it difficult to apply to measurements of NIDs with a low operational stability. The latter technique is limited to measurements of TL through devices of a few centimeters’ dimension.

The aforementioned issues have motivated the work, which is presented in this article. First, we constructed a NID, which consists of a curved glass plate with attached NC-shunted piezoelectric MFC actuators, according to the earlier design.^{19} The noise shielding effect is achieved by using the control of the elastic properties of the piezoelectric actuator, which actively tunes the specific acoustic impedance of the NID. Second, a method for fast measurement of acoustic TL through large planar NIDs was developed. The method is based on the measurement of specific acoustic impedance using two microphones and a laser Doppler vibrometer (LDV).^{23,24} Advantages of such an approach are (i) its simplicity, (ii) the speed of data acquisition, and (iii) the possibility to use the acquired data for direct comparison with numerical results of the FEM model and its calibration. The disadvantage of this method is its lower accuracy in the high frequency range due to the presence of several vibrational nodes and antinodes on the surface of the NID, which is not detectable using a single-point LDV measurement. In order to eliminate this drawback, we have developed frequency-shifted digital holography (FSDH), which allows fast and precise measurements of vibration amplitudes over the whole inspected area of the NID with high lateral resolution.

A limiting factor for the application of FSDH is the maximum measured value of the vibration amplitude attainable without a risk of ambiguity. The value is approximately 80 nm for the experimental system with the frequency of the doubled Nd: YAG line^{25} and it has been used for a visualization of piezoelectric transformer vibrations.^{26} Considering the NIDs at moderate incident sound pressure levels, we estimated that their vibration amplitudes would be in a range from dozens to thousands of nanometers. This was the reason why we put an effort into the modification of the FSDH method with the aim of increasing its dynamic range. This effort resulted in the development of a tool useful for measurements of the surface distribution of the normal vibration amplitude, which can be employed for the refinement of acoustic TL measurements in the high frequency range.

## II. ACTIVE PIEZOELECTRIC SHUNT DAMPING AND ACOUSTIC TRANSMISSION LOSS

Figure 1 shows a scheme of the considered NID. Figure 1(a) shows the side view of the NID, which consists of the curved glass shell, which is fixed along the edges in a rigid frame. Figure 1(b) shows the front view of the NID and the indicated placement of the MFC actuators. The principle of sound transmission through the curved glass plate can be explained as follows: The incident sound wave of an acoustic pressure *p _{i}* hits the glass shell, which produces vibrations with an amplitude

*W*

_{0}in the middle point. An interaction of the incident sound wave with the glass shell results in a situation where a part of the incident wave is reflected and a part is transmitted, producing acoustic pressures

*p*and

_{r}*p*, respectively. The physical quantity which expresses the sound shielding efficiency of the NID is called acoustic transmission loss (TL). TL is defined as a ratio of the acoustic powers of the incident and transmitted acoustic waves, respectively and it is usually expressed in decibels:

_{t} It is a straightforward task^{27} to express the above formula in terms of the specific acoustic impedance *Z _{w}* of the NID:

where *Z _{a}* =

*ϱ*

_{0}

*c*is the specific acoustic impedance of air,

*ϱ*

_{0}is the air density and

*c*is the sound velocity in the air. The specific acoustic impedance of the NID is defined as

where *v* and Δ*p* = (*p _{i}* +

*p*) −

_{r}*p*are the normal component of the vibration velocity and the acoustic pressure difference at the opposite sides of the NID, respectively. If the incident sound wave has a harmonic time dependence of an angular frequency

_{t}*ω*, Eq. (3) can be rewritten as:

where Δ*P*_{0} is the amplitude of the acoustic pressure difference at the opposite sides of the glass shell and $j= \u2212 1 $. It is clear that when the vibration amplitude *W*_{0} is reduced the value of the transmission loss TL is reduced as well.

The above-described logic immediately leads to the noise shielding principle, in which the amplitude of glass shell vibrations is reduced due to the action of attached MFC actuators. A convenient way of explaining the effect of MFC actuators on the glass shell vibrations is through the consideration of the elastic properties of the whole composite structure of the NID. It is a known fact that in a curved shell, which is fixed in a rigid frame, the vibration displacement in the normal direction is linearly coupled with the in-plane strain.^{19} Therefore, an increase in the in-plane elastic compliance (i.e. the Young’s modulus) of one part of the composite yields a decrease in the vibration amplitude *W*_{0} in the normal direction. Such an increase in the Young’s modulus of the attached MFC actuators is achieved by means of the active elasticity control (AEC) method. The AEC method implements the idea by Date *et al.*^{6} that the effective Young’s modulus of a piezoelectric actuator *Y _{Eff}* can be controlled by the value of the connected shunt capacitance

*C*:

_{NC} where *Y _{S}* is the Young’s modulus of piezoelectric MFC actuator material,

*k*is the electromechanical coupling factor, and

*α*=

*C*

_{NC}/

*C*is the ratio of the shunt circuit capacitance

_{S}*C*

_{NC}over the static capacitance

*C*of the piezoelectric element. It follows from Eq. (5) that the effective Young’s modulus reaches infinity when

_{S} where $ C S \u2032 $ and tan*δ _{S}* are the real part and the loss tangent of the MFC actuator capacitance, respectively.

In order to control the elasticity of the MFC actuator, it follows from Eq. (6) that the connected shunt circuit should have a negative effective capacitance. Such a situation is achieved by using an electrical circuit with an operational amplifier shown in Fig. 2. The implemented NC is based on the negative impedance inverter, whose conventional implementation is analyzed in detail in Date *et al.*^{6} In our modified implementation, the resistor *R*_{3} was introduced in order to suppress the saturation of the operational amplifier due to its nonzero input voltage offset. The effective capacitance of the circuit shown in Fig. 2 can be expressed as:

The optimal adjustment of the variable resistors *R*_{0} and *R*_{1} at given frequency *ω*_{0} can be calculated by combining real and imaginary parts of Eqs. (6) and (7):

## III. VIBRATION MEASUREMENTS BY MEANS OF DIGITAL HOLOGRAPHY

### A. Digital holography principles

A digital hologram is an array of numbers, which represent a digitally sampled (using CCD or CMOS camera) state of the optical field generated by the superposition of two optical waves: first, the wave reflected from an object and second, the reference optical wave. The implementation of digital holography consists of two steps: (i) the recording of a digital hologram and (ii) the digital reconstruction of the numerical model of the optical wave reflected from an object. Since the interaction of optical waves with a hologram is completely described by the diffraction theory, the optical wave reflected from an object can be reconstructed numerically from a digital hologram in the form of an array of complex numbers representing the amplitude and phase of the optical field at a given position and time.

When a diffusely reflecting object located at a distance *d* from a digital camera with a pixel size Δ*ξ* × Δ*η* is considered, the complex field *U* representing the diffracted wave in the image plane can be reconstructed from the recorded hologram *h*(Δ*ξ*, Δ*η*) by a multiplication with the reference wave *r*(Δ*ξ*, Δ*η*). (A planar reference wave is defined as *r*(Δ*ξ*, Δ*η*) = 1.) The complex field *U* of the reconstructed image is computed using the Sommerfeld formula, which describes the optical field of a diffracted wave in the distance *d* from the hologram. When the stored hologram consists of *N* × *M* discrete values and the pixel spacing in the reconstructed image is Δ*x* × Δ*y*, the Sommerfeld integral can be solved e.g. by the Fresnel approximation, which can be expressed in a discrete finite form as follows:

where λ is the wavelength of the light, and *n* and *m* are integers going from 1 to *N* and *M*, respectively. A very efficient way of computing the complex field *U* given by Eq. (10) is the FFT-algorithm.

For the purpose of vibration measurements using digital holography it is convenient to express the intensity *I* and phase *φ* of the complex field *U*. If we denote *C* and *S* the real and imaginary parts of the complex field *U*, respectively, i.e. *U* = *C* + *jS*, the intensity *I* and phase *φ* are equal to:

It should be noted that the pixel distances in the reconstructed image Δ*x* and Δ*y*, are different from those of the hologram matrix, Δ*ξ* and Δ*η*: Δ*x* = λ*d*/(*N*Δ*ξ*) and Δ*y* = λ*d*/(*M*Δ*η*).

### B. Vibration measurement principles

The experimentally easiest and most frequently used holographic method for a vibration analysis is the time-average holographic interferometry.^{28} Let us consider a diffusely reflecting surface of an object, which harmonically oscillates around its reference (e.g. equilibrium) position. The displacement of any point *p* on the surface at the given time *t* is denoted by the function *d*(*p*, *t*) = *d*(*p*)sin(*ωt*) where *ω* is the angular frequency and *d*(*p*) is the amplitude of vibration. Such an oscillating surface is holographically recorded using an exposure time considerably longer than the period of the oscillation *T* ≈ 2*π*/*ω*. The reconstructed intensity field of the optical wave reflected from the object given by Eq. (11) at the position *p* is proportional to the square of the zero-order Bessel function of the first kind *J*_{0}:

where Ω(*p*) = *d*(*p*) *e*(*p*) denotes the phase with the sensitivity vector *e*(*p*) defined by the particular arrangement of the holographic experiment.

The key principle in the visualization of vibration modes using digital holography is the computation of the phase Ω(*p*) from the reconstructed intensity *I*(*p*). The points with maximal measured intensity, i.e. *J*_{0}(0) = 1, are identified as the brightest fringes and correspond to the nodal lines of the vibration modes. At the same time, there exist points where the intensity of the reconstructed field becomes minimal, i.e. $I ( p ) = J 0 2 \Omega ( p ) =0$. Such points are identified as dark fringes. At these points, the value of Ω(*p*) equals the well-known arguments of the zeros of the zero-order Bessel function of the first kind. The centers of the dark fringes can be located and they form contour lines (isolines) with the same value of phase. Using this procedure, it is possible to interpolate the profile of the vibrating surface. This procedure works well for vibrations with large amplitudes with a slope. In this situation, the dark fringes (known phase values) are dense enough and the interpolation error can be neglected. The advantage of this approach lies in its good resistance against the noise in the reconstructed intensity field.

However, in a case where the density of the dark fringes in the reconstructed intensity field is too low (small amplitudes) or too high (large vibration amplitudes), it is not possible to use time-average holography due to its low sensitivity to small vibration amplitudes, as well as due to the limiting Nyquist criterion for large vibration amplitudes. One way of overcoming this difficulty is to shift the frequency of the reference wave and to employ the principle of frequency-shifted holography. In frequency-shifted holography,^{29,30} the frequency of the reference wave is modulated by an integer multiple *nω* of the object vibration frequency *ω*, i.e. *f _{ref}*(

*t*) =

*e*

^{jnωt}. If the object wave

*U*is not modulated and the reference wave

_{P}*U*is modulated by

_{R}*f*, the resulting intensity in the hologram plane at time

_{ref}*t*is proportional to $I ( t ) = U P + f ref ( t ) U R 2 $. Using the solution of the interference holographic equation and the properties of the Bessel functions we obtain:

The above equation represents a generalization of Eq. (13), where no modulation is applied and, therefore, *n* = 0. The benefits of the frequency modulation can be clarified by the decomposition of the Bessel function into the power series:

for |Ω| < ∞, and searching its limiting value. For very small amplitudes (Ω → 0) the variation of intensity field proportional to *J _{n}* is equal to:

It is evident that the value *n* = 1 should be used for very small amplitude measurements in order to get the maximum measurement sensitivity. Unfortunately, it is not possible to use interpolating methods, since there are few or no dark fringes in the intensity field. Therefore, the phase must be evaluated at every single pixel.

The easiest approach is a direct inversion of the first monotone interval of the Bessel function *J*_{1}(Ω)^{25} or an inversion of *J*_{0} Bessel function in different intervals,^{31} but there exists a risk of ambiguity for amplitudes greater than *d _{M}*(

*p*) = 1.84/

*e*(

*p*). A more robust method of a single pixel phase evaluation is the creation of an overdetermined system of holograms captured for several values of

*n*≥ 1. The least squares (LS) method can then be applied to find the value Ω, when the residual:

is a minimum. The slashed function $ J n \u2032 ( \Omega ) $ is a model Bessel function and *J _{n}* is the measured value. Although this method is slower compared to the interpolating method, it can be used outside the first monotonous interval.

Obviously, the key prerequisite of an accurate single pixel phase computation is a low noise level in a reconstructed intensity field. In order to suppress speckle noise, it is convenient to capture a hologram of a static object in its equilibrium position, in addition to the hologram of the oscillating object. In this case, the complex fields are computed using Eq. (10).^{32} The real and imaginary parts of the reconstructed complex field of the vibrating object are proportional to:

where Ω_{s} and Ω_{v} are temporally constant (static) and temporally variable (vibrating) spatial phases, respectively. The real and imaginary parts of the reconstructed complex field of the static object in the equilibrium position are equal to:

where Ω_{d} is a slowly varying phase drift. In order to replace the high-frequency phase (speckle) noise Ω_{s} by the low-frequency phase drift, one can calculate the new orthogonal components:

using the approximate expressions:

A low-pass filter is subsequently applied to the new orthogonal components. The low-pass filter can be implemented by the averaging filter with three adjustable parameters: the height and width of the average window and the number of iterations. Moreover, if the “salt and pepper” noise, which is usually produced by the electronic devices, impairs the real and imaginary parts, we recommend using a median filter before the averaging filter is applied.

The real and imaginary parts *C _{N}* and

*S*can be used to compute the so called “wrapped phase” field Δ

_{N}*φ*that takes into account the information about both the phase drift field Ω

_{d}and the sign of the Bessel function:

where arctan(*x*, *y*) gives the arc tangent of *y*/*x*, taking into account in which quadrant the point (*x*, *y*) is located. In a general form, the wrapped phase corresponds to:

The phase field can also be used for the detection of zero-crossing points.^{33} The filtered intensity field is recalculated using Eq. (11) employing the low-pass filtered components *S _{N}* and

*C*and used for LS phase calculation. Although the noise is greatly reduced, there is still some noise remaining in the intensity field. A shadowing effect in the intensity field can be observed in Fig. 3, where the wires connected to the actuators shade the measured region. Below, these pixels are called “outliers”. Since the values of intensity are the main input in the algorithm for phase evaluation, the calculated phase field is also corrupted by the outliers and the remaining noise. In order to reduce the outliers and the remaining noise, we propose the following procedure.

_{N}Based on the assumption that the amplitude distribution is a differentiable function, we have removed the outliers and interpolated the phase field in order to get continuous and smooth intensity data and to remove the remaining noise. Single-pixel reconstruction methods help to increase the dynamic range of the measurement in the case of small vibration amplitude measurement. When compared to common interpolating methods it does not bring any benefit to large vibration amplitudes. Therefore, three different techniques for the phase field computation depending on the magnitude and slope of vibration amplitudes can be used:

(i) In the case of large vibration amplitudes (approximately 1400 nm in the measurement performed) where the density of dark fringes is high, the conventional *J*_{0} interpolating technique is used. The high dark fringes density makes the method nearly insensitive to the choice of the interpolating function. When the density of the dark fringes is too high (particular fringes are not distinguishable anymore), a higher order modulation *n* in Eq. (14) must be used.

(ii) Middle scale vibration amplitudes (in the range of approximately from 400 nm to 1400 nm) can be reconstructed using the aforementioned interpolating methods as well, but the phase distribution strongly depends on the choice of the interpolating function thanks to the low density of dark fringes. We therefore propose to detect dark fringes in intensity fields, which were reconstructed from holograms captured with different modulations, and to combine them in order to increase the density of the dark fringes. In this way the phase is reconstructed more accurately and is less affected by the choice of the interpolating function. It should also be noted that except the dark fringes, where the intensity is minimal, the maxima of the Bessel function can also be detected. However, their accurate location is more affected by the presence of noise.

(iii) In the case of very small vibration amplitudes, even the combination of dark fringes from different modulations is not dense enough and therefore, a single-pixel phase reconstruction has to be used.

## IV. EXPERIMENT SETUP

### A. Acoustic measurement setup

As the first step of the experiment, the NID system was constructed. The curved shape of the glass plate was manufactured as follows: First, the glass shaping mold with the profile *z _{max}*sin(

*πx*/

*a*)sin(

*πy*/

*b*) where

*a*= 0.42 m,

*b*= 0.3 m, and

*z*= 5 mm was carved out of an iron block. The 4-mm thick flat glass measuring 0.445 m × 0.318 m was put on top of the iron mold and thermally treated in the furnace to achieve its lying down to the mold and forming a curved glass plate of the above-specified profile. On top of the curved glass shell the MFC actuators were attached using epoxy glue. The negative capacitance circuit was connected in parallel to all MFC actuators.

_{max}Since this approach to suppressing noise transmission through the NID is based on the control of the elastic properties of the piezoelectric MFC actuators, which actively tune the specific acoustic impedance of the NID, we have developed a simple and fast method for the TL measurement, which is based on the measurement of the specific acoustic impedance. The method (originally introduced in Refs. 23 and 24) employs the values of sound pressure difference at the opposite sides of the glass plate and the values of the normal velocity of the glass plate. Figure 4 shows the experimental setup for the approximative measurements of the acoustic transmission loss. The thermally shaped glass shell was clamped in an iron frame of the inner dimensions of 0.42 × 0.3 m. This structure forms the lid of the soundproof box with a loudspeaker UNI-PEX P-500 that generates the incident sound wave. The microphone IN, inside the box, and the microphone OUT, outside the soundproof box, measure the difference between the acoustic pressures Δ*p*. The microphones are located approximately 1 cm above and below the middle point of the glass plate. A laser Doppler vibrometer measures the amplitude of the vibration velocity *V* of the selected (usually the middle) point of the glass plate. The specific acoustic impedance *Z _{w}* is approximated by the ratio Δ

*p*/

*V*and the value of the acoustic TL is then computed using Eq. (2).

By doing this, we have achieved a simple and fast method which has an acceptable accuracy in the low frequency range. However, the simplicity of this method causes several issues that affect its precision and that need to be carefully checked:

First, due to the presence of the modal structure of the sound field within the box the interior microphone may not measure the local pressure that is applied to the curved glass shell. After performing several numerical simulations of the acoustic box using the FEM software COMSOL it emerged that the sound pressure distribution is quite uniform under the glass plate surface up to 700 Hz. In the higher frequency range, there is a modal structure of the sound pressure in the acoustic box. However, our FEM simulations and further estimates indicate that it has a smaller effect on the macroscopic value of the TL than the resonant vibration modes of the glass plate.

Second, due to the directivity of the glass shell the exterior microphone may not be accurately measuring the total sound pressure that is being transmitted through the panel. Again, the COMSOL simulations indicated that this is not a serious issue since the acoustic wave outside the box travels more or less in the free space and the sound pressure distribution on the surface of the curved glass remains uniform up to 1 kHz.

Third, the vibration of the center does not capture the overall vibration to be used in Eq. (3) for the specific acoustic impedance. Due to the presence of several vibrational nodes and antinodes on the surface of the NID, which is not detectable using a single-point LDV measurement, the accuracy of the acoustic TL measurement setup may be considerably lower in the high frequency range. On the other hand, the values of TL determined using this procedure can be directly compared with numerical data from the FEM model in COMSOL and computed using the same procedure, which was used in the experiment. In addition, this drawback can be completely eliminated by employing the FSDH method instead of the single point LDV measurement.

### B. Digital holography setup

Figure 5 shows the digital holography setup which is based on the Mach-Zehnder type of holographic interferometer. The laser beam has a wavelength of 532 nm and a power of 100 mW. Behind the mechanical shutter, the beam is split in two by the polarizing beam splitter equipped with half-wavelength retardation plates. Half-wavelength retardation plates help to set the intensities in both beams as well as the polarization of each beam. The first beam acts as a reference wave and it could be further attenuated if necessary by a set of gray filters placed in the filter wheels. Each beam is frequency-shifted by an acousto-optic frequency modulator - Bragg cell with a fundamental frequency of 40 MHz. The beams are spatially filtered and the reference beam is collimated. The object beam illuminates the measured window and the light scattered from its surface interferes with the reference wave. The negative lens reduces the imaging angle and the measured window having size of 425 × 300 mm can therefore be measured. The setup is designed as an in-line scheme which makes the sensitivity vector:

The camera has a resolution of 2048 × 2048 pixels of the pixel size 1.75 × 1.75 *μ*m. The camera is connected to the computer via a USB interface enabling a frame rate of 6.5 FPS. A sequence of 6 frames was captured. Frequencies of Bragg cells are chosen to fulfill the equation Ω_{B2} = Ω_{B1} + *ω* + *ω _{C}*/

*M*, where

*ω*is the frequency of the object vibrations,

*ω*is the camera frame frequency and

_{C}*M*is the number of frames used in heterodyne detection. Thus a sequence of

*M*phase shifted digital holograms is recorded and processed by the formula

to get a filtered digital hologram *h*. The filtered hologram free of undesirable diffraction terms then enters the Eq. (10) for further processing.

## V. RESULTS

At first, the acoustic transmission loss through the curved glass shell was measured in the frequency range from 200 Hz to 1 kHz. Subsequently, a frequency of 286 Hz was identified with a minimum value of the acoustic transmission loss, which corresponds to the first dominant vibration mode of the curved glass shell. Finally, two values 247 Hz and 258 Hz of a frequency *ω*_{0} were chosen and the NC was adjusted according to Eqs. (8) and (9). The values of *C*_{0} = 4.7 *μ*F and *R*_{2} = 10 kΩ were fixed in the NC shown in Fig. 2. At the frequency of 247 Hz, the particular adjustment of the NC was following: *R*_{3} = 570 Ω, *R*_{0} = 5745 Ω, and *R*_{1} = 78.95 Ω. At the frequency of 258 Hz, the adjustable resistors in the NC were set to: *R*_{3} = 572 Ω, *R*_{0} = 5617 Ω, and *R*_{1} = 69.71 Ω.

Then, the NC was connected to the MFC actuators attached to the curved glass shell and the frequency dependence of the acoustic TL was measured again with two given particular adjustments of the NC specified above. The frequency dependencies of the acoustic transmission loss and curved glass shell vibration amplitude normalized to the value of sound pressure difference at the opposite sides of the curved glass shell measured in the frequency range from 230 Hz to 400 Hz are presented in Figs. 6(a) and 6(b), respectively. The curves with circle markers refer to situations when the NC is off. Two curves marked by triangles and squares refer to two adjustments of the NC that yield maxims in the acoustic TL at frequencies 247 Hz and 258 Hz, respectively. LDV measurements indicated increases in the TL by 36.6 dB and 25.0 dB due to the action of the NC at the frequencies 247.0 Hz and 257.9 Hz, respectively. Curves with empty and filled markers correspond to experimental data acquired by LDV and FSDH methods, respectively. There can be seen a qualitative agreement with acoustic transmission loss and normalized vibration amplitude data measured using LDV and FSDH techniques in Figs. 6(a) and 6(b).

The profiles of the vibration modes of the curved glass shell were measured using digital holography in the frequency range from 250 Hz to 265 Hz in the two situation, when the NC was disconnected and when the NC was connected and adjusted at the frequency 258 Hz. Figure 7 shows the profile of the vibration mode at the frequency of 258 Hz when the connected NC circuit was off 7(a) and on 7(b). At this frequency, the amplitudes of the vibration of the middle point were 715.2 nm and 87.8 nm in situations when the NC circuit was off and on. That corresponds to an increase in the TL by approximately 18 dB due to the effect of the NC circuit. Such a result is broadly consistent with the acoustic measurements. The greater values obtained using LDV measurements are caused by a much finer resolution in the frequency scale of the LVD measurement.

The measured distribution of the vibration amplitude on the surface of the curved glass shell was used to compute the specific acoustic impedance and TL of the NID. The computed frequency dependence of TL is presented by filed markers in Fig. 6(a). The the obtained data indicate that the FSDH method can be applied to experimental testing of adaptive vibration suppression systems.

## VI. CONCLUSIONS

In order to carry our this project, a noise isolation device (NID) consisting of a curved glass shell fixed in a rigid frame was constructed. Piezoelectric MFC actuators were glued to the surface of the shell and subsequently connected to an active shunt circuit with negative capacitance.

Two methods of measuring acoustic transmission loss (TL) were developed and tested. First, a simple and fast method is based on the measurement of specific acoustic impedance using two microphones and a laser Doppler vibrometer (LDV). The method is applicable to measuring TL in the frequency range from 50 to 700 Hz with an accuracy comparable to the conventional methods of TL measurement. In the higher frequency range, the accuracy of the method may be lower. However, the results of the developed measurement method of TL can be directly and without any limitations compared with the predictions of the FEM model disregarding the frequency range. Frequency dependencies of the TL were measured using a purpose-built acoustic setup. The approximate increases of 36 dB and 25 dB in the acoustic transmission loss were achieved due to the action of the NC shunt circuit in the narrow frequency ranges around 247 Hz and 258 Hz.

The second method of measuring acoustic transmission loss (TL) is based on the measurement of the vibration modes of the curved glass shell with the attached piezoelectric MFC actuators were measured using frequency-shifted digital holography (FSDH). Using this approach, more detailed data on the noise transmission through the curved glass shell can be obtained. As an example, the displacement distributions over the surface of the curved glass shell at the frequency range from 250 Hz to 265 Hz without and with the action of the connected NC circuit were measured using FSDH. The profiles of the vibration amplitudes of curved glass shell and corresponding values of TL were computed. We have demonstrated that the TL values measured by the means of the FSDH are consistent with the TL values measured using the LDV method. That suggests that FSDH can provide a fast and accurate method for measuring the acoustic TL of large planar structures. The particular implementation of this method is a subject of future research.

These results have successfully demonstrated the potential of the APSD method as an efficient and simple way to control noise transmission through curved glass shells. The greatest drawback of the particular implementation of the APSD method in the constructed NID is its narrow frequency range of the efficient suppression of noise transmission. However, several methods^{15,34,35} have been developed which allow the broadening of the frequency range through making simple changes in the arrangements of the NC shunt circuit.

## ACKNOWLEDGMENT

The authors would like to express their sincere gratitude to Martin Černík, Pavel Márton and Jakub Nečásek of the Institute of Mechatronics and Computer Engineering at the Technical University of Liberec (TUL). This work would not have be possible without the help and support from Václav Kopecký, the Dean of the Faculty of Mechatronics, Informatics and Interdisciplinary Studies at the TUL. The authors thank Iva Holasová and Julie Volfová for reading the manuscript and for the English corrections.

This work was supported by Czech Science Foundation Project No.: GACR 13-10365S and co-financed by the Ministry of Education, Youth and Sports of the Czech Republic within the Project No. NPU LO1206. The work of P. P. and R. D. was also supported by the Ministry of Education of the Czech Republic within the SGS project no. 78001/115 at the Technical University of Liberec.

## REFERENCES

*10140-2 acoustics — laboratory measurement of sound insulation of building elements — part 2: Measurement of airborne sound insulation*(2010).