PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Modal Analysis for damage detection in structures by non-contact
measurements with a commercial microphone array
Olaf BÖLKE 1; Jan HEIMANN1; Joaquin GARCIA2
12

gfai tech GmbH, Berlin, Germany

Universidad Nacional de Tres de Febrero (UNTREF), Buenos Aires, Argentina

ABSTRACT
When monitoring critical structures, fatigue fracture, deformations, holes and much more are cases of failure
which must be detected at an early stage. Changes of the modal parameters (eigenfrequencies, attenuationratios, and mode shapes) of the structure give information about the extent and the location of thedeterioration. Conventional measurement methods (i.e. acceleration sensors, laser vibrometers, etc.) forvibration analysis have the disadvantage that they can either "detune" the vibration modes due to their own
weight and/or require a long measurement time due to punctual measurements. In contrast, the use of asuitable microphone array allows the high-resolution acquisition of the entire surface vibration covered by
the array. Thus, the modal parameters of interest are determined by measuring the pressure fluctuations in
the near field of the structure. A commercial acoustic camera with 120 microphones (Fibonacci120, gfai techGmbH) is used for this purpose. On the basis of artificially generated failure cases (load fracture,inhomogeneities, etc.) on application-oriented, large-area structures, a method for the detection of failurecases using a microphone array is demonstrated.
Keywords: Microphone Arrays, SONAH, Modal Analysis

1. INTRODUCTION
The dynamic response of a structure is an important aspect of understanding the structure’s
behavior. Strong vibrations can generate not only unwanted noises but also structural wear and thus
material weakness. The risk of rupture in structures under load conditions leads to necessary
understand the structure’s dynamic response when a vibration is present. This can be achieved bymonitoring the natural frequencies or magnitudes of frequency response functions to detect faults andmechanical failures [1]. Typically, experimental modal analysis (EMA) is carried out to obtain themodal parameters such as mode shapes, eigenfrequencies, and damping ratios. This consist in applying
a defined force (e.g by a shaker or an impact hammer) and record the structural vibration responseusing different transducers such as accelerometers and laser Doppler vibrometers (LDV). However,the laser scanning of several points can be time-consuming when large surfaces are under study. On
the other hand, attached external sensors on the structure can "detune" its natural response due to theadded mass. Thus, recent studies proved that non-contact measurements can be used, especially forthe analysis of very light and fragile structures [1].
This paper is focused on detecting failure cases on large-area structures. The EMA is used todetermine the eigenmodes by measuring the sound pressure with a contactless microphone array.
Artificial inhomogeneities (additional mass and notches) are applied to a plate surface to test thedetection of typical structural failure. In current studies, such as [2], qualitative investigations of thedetermined modes have already been carried out, whereby the used algorithms usually provide
unsatisfying results. In this study the measured pressure fluctuations are assigned to the surface
vibrations using the near-field holography method SONAH („Statistically Optimal Near-field
Acoustical Holography“) [3]. Furthermore, the analyses of the data collected by a commercial
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boelke@gfaitech.de 2 joaquin.garcia1900@hotmail.com

microphone array (Fibonacci120 AC Pro, gfai tech GmbH) are carried out with the software„NoiseImage“. The Modal Assurance Criterium (MAC) is used to evaluate the change in modal
shape[4]. The MAC values are usually interpreted as a statistical indicator for the comparability oftwo modal forms.

2. THEORY
2.1 Experimental Modal Analysis
The response of a structure to a punctual force can be described by modal superposition [5].
Considering a plate with mode shapes ߶௥ (‫ݔ‬, ‫ )ݕ‬for the eigenfrequencies ߱௥ and modal damping
ratios ߦ௥ , then the transverse displacement ߟ of the surface is given by:ஶߟ(‫ݔ‬, ‫ݕ‬, ‫ = )ݐ‬෍௥ୀଵ‫ܨ‬෨ ߶௥ (‫ݔ‬, ‫߶ )ݕ‬௥ (‫ݔ‬௙ , ‫ݕ‬௙ ) ௜ఠ௧݁ ,‫ܯ‬௥ ߱௥ଶ െ ߱ ଶ + 2݅ߦ௥ ߱௥ ߱(1)for a harmonic force ‫ܨ‬෨ at position (‫ݔ‬௙ , ‫ݕ‬௙ ), generalized mass ‫ܯ‬௥ and angular frequency ߱ for allmodes ‫ݎ‬.
Experimental modal analysis (EMA) is used to find the modal properties (߶௥ , ߱௥ , ߦ௥ ) in the model
given in (1). This is done by fitting frequency response functions (FRFs), which are determined fromthe measured force excitation and the measured system response. The modal parameters are estimatedusing the Polyreference Least Squares Complex Frequency (POLY-LSCF) algorithm [6]. ThePOLYLSCF algorithm uses (weighted) least square approaches of multiple-input-multiple-output
frequency response functions. The stability diagrams determined during the analysis allow an easy
analysis even of complex systems with highly damped modes and/or large modal overlap. Further
information on mathematical implementation and validation using experimental and numerical models
can be found in the work of Peeter et. all [7] and Phillips et. all [6].
2.2 Modal Assurance Criterion
The Modal Assurance Criterion (MAC) is defined as the normalized scalar product of two modalvectors ‫ݒ‬௜ , ‫ݒ‬௝ :ଶ‫ܥܣܯ‬௜௝ =ห‫ݒ‬௜ு ‫ݒ‬௝ ห൫‫ݒ‬௜ு ‫ݒ‬௜ ൯൫ ‫ݒ‬௝ு ‫ݒ‬௝ ൯,(2)where (ή)ு is the complex conjugate transpose (Hermitian) of a matrix.
The MAC is a statistical indicator that can be used as a measure of the quantitative comparabilityof two modal forms. A MAC value of 0 indicates that the modes are not consistent and a value of 1
represents a fully consistent mode shape [4]. Therefore, the MAC matrix (cross correlation of allmodes in a given range) obtained by EMA and/or FEA consists of zeros, except in the diagonal wherethe values are close to 1. This least squares-based form of linear regression analysis yields an indicatorthat is most sensitive to the largest difference between comparative values and results in modal
assurance criterion that is insensitive to small changes or small magnitudes. Under certain conditions,such as a stationary, linear system state, the MAC can also be taken as a criterion for the orthogonalityof the compared modes. However, the MAC values must be interpreted according to the measurement
situation: On the one hand an insufficient local resolution of two orthogonal mode shapes can lead tohigh MAC values, while on the other hand lower values can be achieved e.g. by noisy measurement
signals [4].
2.3 SONAH
Acoustic near-field holography describes a method of estimating the sound field on the source
surface by measuring the acoustic quantities (usually sound pressure or sound velocity) at a small
distance from the source surface. The requirements on the microphone array when using traditional
nearfield acoustical holography (NAH) are very high. A uniform microphone arrangement is required,
where the array covers the entire sound source. In addition, the minimum microphone distance mustbe less than half the wavelength at the highest frequency of interest. However, SONAH does not havethese restrictions: It can not only work with irregular arrays, it also gives good results when the arrayis smaller than the source [3]. The latter is due to the fact that, unlike traditional NAH, the strong143

spatial window effects do not occur [8].
The publication of Puhle et. al [9] describes, among other things, the mathematical implementationof two advanced NAH methods such as SONAH and HELS (Helmholtz Equation Least-Squares). Inaddition, Puhle et. al. also qualitative validates the experimentally determined mode shapes fromacoustic measurements using SONAH and HELS with LDV-measurements. This work continues theinvestigations by concentrating not only on the amplitudes of the calculated surface oscillations, butalso on the complex amplitude including the spatially distributed phase. Thus, the MAC is used toquantify the quality of certain mode shapes using FEM, LDV and SONAH.

3. MEASUREMENT SETUP
The vibration response of a stainless-steel plate (600 mm x 600 mm x 4mm) to a force excitation
was investigated. The plate was supported by four rubber bands in a metal frame, which were guided
through corresponding holes in the corners of the plate (diameter 7 mm). Thus, free boundary
conditions could be realized for the relevant frequency range (100 to 1000 Hz). Using a Shaker-Stinger
setup (PCB SmartShaker with an integrated power amplifier, model K2007E01), the plate was excitedperpendicular to the surface, whereby the force was measured using a force sensor (PCB type 208C02),
see Fig. 1 (left). White noise with frequencies from 100 to 1000 Hz was used as an excitation signal.
The plate response was recorded by measuring the sound pressure oscillation with a commerciallyavailable microphone array (acoustic camera "Fibonacci120", gfai tech GmbH). In order to improve
the signal-to-noise ratio, the measurements were performed in an anechoic chamber. The microphonearray used is a multifunctional array and is well suited for both SONAH and conventional
beamforming. Preliminary investigations have shown that the best results can be obtained with amicrophone distance of approx. 14 mm to the measuring surface, see Fig. 1 (right). The pressureoscillations were recorded using a data acquisition system (mcdRec 721B, gfai tech GmbH) with asampling rate of 48 kHz and averaged over 16 seconds. The particle velocity on the surface of theplate was determined using the nearfield hologram module of the Noise Image software (gfai techGmbH), which is based on the SONAH algorithm.

Figure 1 – Setup for determination of the plate mode shapes with a commercial microphone array(Fibonacci120 AC Pro, gfai tech GmbH) in an anechoic chamber. Left: rubber mounted plate withmicrophone array. Right: Alignment of shaker, plate, and parallel microphone array.
Three measurements were carried out to investigate the effects of a geometric change on the modeshapes: 1) plate without changes, 2) plate with simulated fatigue fracture (ca. 2 mm deep and 30 cm
long notch, horizontal on the lower third of the plate) and 3) plate with additional mass (300 mm x 25mm x 5 mm brass beam, horizontal on the lower third of the plate).
In a preliminary study [10], the mode shapes were simulated and compared with the results ofmeasurements using laser Doppler vibrometry (LDV) and SONAH to verify the measurement
methodology. Figures 2 and 3 show representative mode shapes determined by FEM (left column),
LDV (middle column) and SONAH (right column). The normalized amplitude (upper row) and thecorresponding phase (lower row) of the plate are presented. Both the mode shapes based on LDV andSONAH correspond qualitatively well to the values estimated by FEM. It is noticeable, however,144

Figure 2: Amplitude (top) and phase (bottom) ofmode No 8; FEM (left), LDV 170 Hz (middle),SONAH 172 Hz (right) [10]Figure 3: Amplitude (top) and phase (bottom) ofmode No 14; FEM (left), LDV 324 Hz (middle),SONAH 324 Hz (right) [10]that the vibration amplitudes in the outer edge of the plate deviate from the LDV results or thecalculated values, respectively. It can be assumed that this is related to the microphone density
decreasing with the distance to the center of the plate. However, it should be noted that the phase of
the vibration mode can be determined with less noise by SONAH than with the comparable LDV,
whose quality depends strongly on the surface quality.

4. RESULTS
The three different panel designs are compared to assess the suitability of the microphone arrayfor error detection based on the mode shapes determined. Due to the minor changes in the geometry,
only slight shifts of the resonance frequency (<30Hz) in the transfer function between force anddetermined sound pressure fluctuations can be determined. However, it was found that some modes
are no longer excited by the fix driving point position due to the asymmetric change in geometry. In
the following, therefore, only the mode shapes that could be determined for all three plate
configurations are compared.
Figures 4 to 11 show the amplitude (upper row) and phase (lower row) of the mode shapes basedon SONAH. The columns correspond to the three different plate configurations (from left to right): 1)
plate without changes, 2) plate with a notch, and 3) plate with an additional mass. The determined
Auto-MAC values of the presented mode shaped showed exclusively high values in the diagonal,
which can be interpreted as a strong linear independence from the determined modes. Where thechanges in the oscillation mode due to the simulated fatigue break (case 2) in the frequency range
investigated here (100 to 600 Hz) shows only minor changes in the mode shape (see e.g. Fig. 6), theadditional mass (case 3) shows a more significant change compared to the reference case (case 1).
Due to the additional mass in the lower part of the plate at case 3, a stiffening occurs, which causes a
significantly changed mode shape beginning from the 2nd mode shown here (Figure 5, right). Due tothe horizontal orientation, a higher influence on modes that would have an oscillation node in theregion of mass can be seen here (compare, for example, Figures 10 and 11).

Figure 4: Amplitude (top) and phase (bottom) of
the plate mode No 1; Left: 163 Hz (Case 1);
Middle: 158 (Case 2); Hz; Right: 163 Hz (Case 3)

Figure 5: Amplitude (top) and phase (bottom) of
the plate mode No 2; Left: 165 Hz (Case 1);
Middle: 164 (Case 2); Hz; Right: 173 Hz (Case 3)

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Figure 6: Amplitude (top) and phase (bottom) of
the plate mode No 3; Left: 186 Hz (Case 1);
Middle: 184 (Case 2); Hz; Right: 196 Hz (Case 3)

Figure 7: Amplitude (top) and phase (bottom) of
the plate mode No 4; Left: 206 Hz (Case 1);
Middle: 204 (Case 2); Hz; Right: 237 Hz (Case3)

Figure 8: Amplitude (top) and phase (bottom) of
the plate mode No 5; Left: 280 Hz (Case 1);
Middle: 277 (Case 2); Hz; Right: 283 Hz (Case 3)

Figure 9: Amplitude (top) and phase (bottom) of
the plate mode No 6; Left: 313 Hz (Case 1);
Middle: 306 (Case 2); Hz; Right: 314 Hz (Case 3)

Figure 10: Amplitude (top) and phase (bottom) of
the plate mode No 7; Left: 324 Hz (Case 1);
Middle: 322 (Case 2); Hz; Right: 329 Hz (Case 3)

Figure 11: Amplitude (top) and phase (bottom) of
the plate mode No 8; Left: 350 Hz (Case 1);
Middle: 346 (Case 2); Hz; Right: 433 Hz (Case 3)

Figure 10: Amplitude (top) and phase (bottom) of
the plate mode No 9; Left: 523 Hz (Case 1);
Middle: 520 (Case 2); Hz; Right: 517 Hz (Case 3)

Figure 11: Amplitude (top) and phase (bottom) of
the plate mode No 10; Left: 528 Hz (Case 1);
Middle: 527 (Case 2); Hz; Right: 532 Hz (Case 3)

Figure 12: MAC-Values by Eq. (2)between plate (case 1) and plate withnotch (case 2)

Figure 13: MAC-Values by Eq. (2)between plate (case 1) and plate withmass (case 3)

By means of the MAC values according to Eq. (2), the different mode shapes can be compared
qualitatively. Figures 12 and 13 show the MAC values for the cases 2 and 3, respectively, with respect
to the plate without modification, case 1 (see Figs. 4 to 11). As expected, the largest values are shownon the diagonal. Only the modes 1 and 5 show MAC-values above 50%. It can be assumed that this isdue to the low microphone density near the edges of the plate, so that the vibration field can no longer
be resolved high enough to clearly separate the mode shapes.
In accordance with the previous observations, the MAC values in Fig. 12 show much lower values
than in Fig. 13, which highlights the much stronger influence of the additional mass on the modeshapes compared to the notch. However, it can be shown that for the first three modes determined inthis investigation, slight deviations of the MAC values can also be observed for the plate with thenotch.

5. CONCLUSIONSIt has been shown that with a microphone array using SONAH the eigenmodes of a vibrating plate
can be quantitatively well estimated. The preliminary investigations [10] showed a good agreement
between the simulated and LDV vibration modes, both in amplitude and phase distribution. When
comparing the test cases examined here, the influence of an additional mass on the test structure andthe influence (within certain limits) of a notch from the certain mode shapes could be determined bycontactless microphone array measurements.
The changes in the mode shapes indicate, among other things, a stiffening in the area in which theadditional mass is located. Therefore, comparisons of the specific mode shapes allow conclusions tobe drawn about the position and type of geometry changes. With an appropriate learned algorithm,
e.g. via pattern recognition, a control algorithm based on a microphone array could be created, which
can distinguish between different error or failure cases. The advantage of the microphone arraymeasurement technology over the usual LDV-based measurement methods lies in the simplified andless time-consuming measurement method, which measures the entire surface of the array. Future
works are focused on further investigations to fully clarify the limitations of the array design used inthis investigation.

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