Damage Identification Based on the Nodal Line Active Modulation Method

A novel vibration-based method (VBM) for damage identification, nodal line active modulation (NLAM), is proposed in this study. Unlike traditional VBMs, the NLAM can identify damage by taking advantage of the nodal line effect, which is that when a defect is located on the nodal lines, the dynamic responses of the structure remain nearly unchanged. The basic idea of NLAM is introduced through a simply supported beam. In numerical study, we have discussed the detection steps of the method in detail by detecting a cleft detection in an aluminum plate. In experimental research, piezoelectric transducers work as both actuators and sensors for an added mass damaged detection in carbon fiber reinforced plastic laminates, while a scanning laser Doppler vibrometer is employed for measuring the full-field dynamic response. The damage is visualized from the highlighted nodal lines of the modulated operational deflection shapes. The present investigation provides a new idea for VBM damage detection, which will have practical applications for structural health monitoring.


Introduction
Vibration-based methods (VBMs) for damage detection and structural health monitoring (SHM) have been extensively researched and are widely used in various engineering fields [1][2][3].With well-established excitation and measurement technology, traditional VBMs are suitable for the detection of large-scale structures due to their high efficiency in global structure detection.In recent years, with the application of new materials and structures, VBMs have also been further developed with advanced signal processing technology [4,5].Many VBMs have been used for damage detection in lightweight materials, such as carbon fiber reinforced plastics (CFRP) [6,7], honeycomb structures [8,9], and lattice truss core sandwich structures [10,11].Due to the low frequency adjusting the exciting harmonic force.The rest of this paper is arranged as follows.Section 2 introduces the NLAM method with its main processing procedure, and conducting to detect cleft defect in an aluminum plate damage numerically.The method is validated by experimental test in Sect.3. Finally, the paper is summarized and concluded in Sect. 4.

Nodal Line Active Modulation (NLAM) Method
For traditional VBMs, it is no chance to detect damage when it is located on nodal lines due to the undetectable variations in modal frequencies and the modal shapes.However, from a different perspective, the no significant change in modal charactersimplicate that the defect could be located exactly on nodal lines.This insight provides us with a totally different detection approach for damage identification.If we can manipulate the excitation signals to make the vibration shapes of the damaged structure to be the same as the intact ones, it could be logically asserted that the defect is on the nodal lines of these shapes.Since the modal shapes are the inherent properties of a structure that cannot be modulated conveniently, operational deflection shapes (ODSs) are adopted in this method, which are the linear combinations of the modal shapes, and the excitation is adjustable by controlling the magnitude and phase of the excitation signals of each actuator.Furthermore, in order to accurately locate the damage from several nodal lines, multiple ODS nodal lines excited at different frequencies are considered comprehensively.The following sections describe the fundamental principle and the procedure of the proposed NLAM method.

Fundamental Principle of the NLAM Method
For describing the proposed NLAM method clearly, we consider forced vibration of a simply supported beam as shown in Fig. 1.When two harmonic loads F 1 sin(ωt) and F 2 sin(ωt), with a certain same frequency ω, are applied at different positions to excite the beam separately, two forced vibration shapes, also known as ODSs, can be obtained as the dash lines in Fig. 1.Now examine the vibration amplitudes of an arbitrary point C in the structure.Since the amplitudes the above two loads at point C are v 1 and v 2 respectively t, we can easily determine the coefficient α 1 and α 2 to make α 1 v 1 +α 2 v 2 0.Then, by applying α 1 F 1 sin(ωt) and α 2 F 2 sin(ωt) to excite the beam simultaneously, the vibration amplitude at point C would be compensated to zero, as shown in Fig. 1.In other words, if the damage located at point C, the specific excitation by combining α 1 F 1 sin(ωt) and α 2 F 2 sin(ωt) is applied on the beam, the ODSs of the intact and damaged beam should be the same.Based on this idea, the procedure of the NLAM method can be undertaken as shown in Fig. 2. First, two ODSs of intact structure are measured corresponding to F 1 sin(ωt) and F 2 sin(ωt).Second, two ODSs of the damaged structure are also measured corresponding to the same excitations.Since the damage location is unknown, the third step is to determine the appropriate coefficients α 1 and α 2 , such that the linearly superposed ODSs of both the intact and damaged structure are the same.Then, the damage can be located at the nodal point of the ODSs excited by the modulated excitations of α 1 F 1 sin(ωt) and α 2 F 2 sin(ωt).
In the damage detection procedure, another key issue is the required numbers of actuators and sensors.According to Figs. 1 and 2, it is clear that at least two actuators are required for one single damage.It is because the method requires compensating the vibration magnitude at the damage location.The number of sensors is not strict limited, but it needs to ensure the vibration shapes that can verify the response amplitudes between damages and health structures.
However, in real structure inspections, due to the influence of damping, the vibration shapes contain amplitudes and phases, which makes the detection more complicated.Another major disadvantage of measuring the ODS of the whole structure is very time consuming, especially for the damaged structure, as it is quite important to locate damage instantly after it occurs.Thus, we improve the NLAM method and describe the real detection procedure in detail for a plate in the following sections.

Detail Procedure of the NLAM Method
We numerically analyzed an aluminum plate with cleft damage.The size of the plate was 300 × 300 × 1 mm.Nine piezoelectric transducers (PZTs) with 14 mm diameter and 1 mm thickness were attached to the back of the plate, as shown in Fig. 3.A small cleft was built in the numerical model as the defect.The center of the cleft was located at (75 mm, 50 mm), and the center of the plate was the origin point.The peripheral PZTs were evenly arranged on a circle with a diameter of 120 mm.It is worth noting that in order to excite more modal shape components, the peripheral PZTs were rotated 15 degrees counterclockwise in directions other than on the symmetric axes of the square plate, which are where the nodal lines of most modal shapes are located.
Additionally, the proportional damping factors are applied to the material in the simulation, which led to complex vibrations that are reproducible in experiments.Although it is difficult to determine the values of these parameters exactly as they are in a real material, especially over a wide frequency range, this does not affect the procedure and results of the NLAM.It is important to modulate the phase of the excitation signal as well as the magnitude to obtain the appropriate ODSs.Therefore, complex frequency analysis is adopted in this study.
The core concept of the NLAM algorithm is to find a suitable combined excitation mode, including the magnitude and phase, of each actuator, which can excite both intact and damaged structures to the same ODSs.The proposed method is implemented through following steps: coordinate on the plate (i.e. the positions of the measuring points), the third subscript f corresponds to the frequency, and the last subscript a represents the number of actuating PZT.(ii) The response voltages of the non-excited PZTs as sensors (eight sensors total) are also recorded in another complex matrix V I s f a , where the superscript I represents the intact condition, the first subscript s corresponds to the PZTs that are working as sensors, and the last two subscripts have the same meaning as in S xy f a .The reason that the voltages of sensors are measured is that it is quite time consuming if the ODSs of both the intact and damaged structures are measured for comparison, especially when detecting the damaged structure, which is not suitable for real-time SHM.A simple and convenient process is adopted in this study.Instead of comparing the whole structure ODSs between intact and damaged structures, we only analyze several response voltage signals, including the magnitudes and phases, measured by the PZTs.By modulating the excitation signals of several chosen PZTs, the measured voltages of other PZTs can remain the same in both the intact and damaged structure.The details and calculations of the exitation modulation and voltage comparison are described in step (iv).We can then assume that the ODSs of the damaged structure remain the same as the intact one.Both the numerical and experimental results are shown in the following sections to validate this assumption.(iii) When a defect appears in the structure, such as the cleft shown in Fig. 3, by repeating step (ii), we can obtain another set of voltage responses from the PZT sensors V D s f a , where the superscript D refers to damaged structures.ODSs do not need to be measured in the damaged condition, which may greatly improve the efficiency of detection and monitoring.(iv) In order to obtain the ODSs that are not affected by the defect, which indicate the damage location on the nodal lines, three PZTs working as actuators exciting simultaneously with certain amplitudes and phases are selected for detection, while the remaining PZTs work as sensors for the voltage comparison.For example, by applying certain amplitudes and phases to PZTs number 1, 2 and 3 (working as actuators), the voltages detected by the other six PZTs (working as sensors) of the intact and damaged structures could remain the same in both amplitude and phase.We then assume that the entire ODSs of the structure also remain unchanged, as discussed in step (ii).Equation (1) shows the calculation process. 3 The coefficients k f a are complex numbers that represent the amplitudes and phases applied to each actuator denoted by subscript a with different frequencies f .It is worth noting that, instead of real excitation and measurement for the ODSs, only a numerical calculation of Eq. ( 1) is required for this step, which is quite efficient for on-line monitoring.Due to the calculated and actual measurement errors, including that the damage has a certain size that cannot be located exactly on the nodal lines, Eq. ( 1) cannot strictly be satisfied.Therefore, the values of k f a in Eq. ( 1) are calculated by a least square, meaning that the resulting k f a can make the expression 9 its minimum value.The operator • gives the module a complex value.Since we can choose any three of nine PZTs as actuators for the above procedure, there are 84 combinations in total.For each combination, we can calculate a set of coefficients k n f a (n 1, 2, ..., 84).(v) Based on the actuated coefficients k n f a obtained in step (iv), the ODSs, the nodal lines of which show the location of the damage, can be constructed by Eq. (2).
Because the PZTs, as additional masses attached to the plate, will reduce the amplitude of the vibration at local areasand thus cause a misjudgment of the nodal lines, we calculated the root mean square (RMS) values of all the S xy f a values at each point, as shown in Eq. ( 3), which indicates the vibration energy distribution of the structure.
where n f and n a are the numbers of actuating frequencies and actuators, respectively, which are both nine in this example.We modify the ODSs to M O DS n xy f as defined in Eq. ( 4), which reduces the impact of the structural mass distribution on the ODSs, and can improve the accuracy of NLAM for complex structures.A threshold T H (0 < T H < 1) is then chosen (in this study, T H 0.1) to identify and highlight the nodal lines using Eq.(6).
By superimposing these nodal line patterns at each frequency by equation ( 7), we can obtain more precise nodal line images at each frequency Obviously, the defects located on these highlighted nodal lines at all frequencies have values that are close to one.Thus, the damage position can be imaged by multiplying all these nodal line patterns for each frequency.The normalized damage index is defined in equation (8).

Numerical Results
Figure 4 shows the nodal lines obtained by Eq. ( 6) in step (vi).The red dots represent the center of the cleft damage.
It is quite clear that the damage is located exactly on the highlighted nodal lines at all frequencies, as predicted by the proposed method.To verify the assumption presented in step (ii), we also measured the ODSs and calculated the highlighted nodal lines of the damaged structure (see Fig. 4b), although it is not required for the NLAM.It can be clearly seen that the highlighted nodal lines of the intact and damaged structures are almost exactly the same at each frequency, which indicates that the existence of the cleft does not change the ODSs at the modulated excitation conditions (i.e., the cleft is located on these nodal lines as expected).
The damage imaging obtained from Eq. ( 6) is shown in Fig. 5. From the local damage imaging, we find that the peak

Experimental Validation
In this section, experimental work is conducted to validate the proposed NLAM technique.As the method is a vibration deflection shapes analysis, it is applicable to any material structures, even without material parameters.In our experiment, we applied this technique for damage detection in a CFRP composite plate.The specimen with attached actuator/sensor PZTs and the experimental setup are shown in Fig. 6.The size of the plate is 400 mm × 400 mm × 1.6 mm, and it contains 8 layers ([0/45/90/-45] s ).The manufacturer did not provide the specific material parameters of the laminate.Nine PZT wafers with dimeson of 15 mm diameters and 1 mm thicknesses were attached to the plate.The peripheral PZTs were arranged on a circle with diameter of 100 mm and a 15-degree counterclockwise rotation.Two steel magnetic cylinders with 8 mm diameters, 9 mm height and 3.4 g mass(for a total mass of 6.8 g) were magnetically attached to the plate from both sides and acted as the damage, which is more convenient and repeatable than making a cleft.Based on the analysis process of NLAM, we determined that the method is applicable to any type of damage, as well as to Because the stiffness and thickness of the CFPR are larger than that of the aluminum plate in the simulation, in order to locate the damage more accurately and reduce the acoustic noise during the experiment, we decided to excite the material at ultrasonic frequencies (21 kHz to 29 kHz with an interval of 1 kHz) for detection.Generally, when the damage size does not exceed approximately a quarter of the wavelength of the standing wave, the frequency detection is higher and the damage location is found more accurately.The primary reason for this is that the NLAM method requires that the damage should not be oversized so that it straddles two standing wave peaks, which may lead to inaccuracy in the nodal line detection.Furthermore, in order to more efficiently obtain the ODSs responses of the structure at each frequency, we superimposed the nine frequency harmonic signals to one excitation signal, so that the structural responses to each of the nine frequencies can be obtained in one measurement.The time domain of one period and the amplitude of the frequency domain are shown in Fig. 7.We found that the synthesized signal can well meet the excitation requirements of this method.
A scanning laser Doppler vibrometer (SLDV, Polytec PSV-500) was employed to generate the excitation signal and measure the out of plane velocity of the CFPR.The measuring area was 160 × 160 mm with an interval of 2 mm; in total, 6561 points were measured.To enhance laser reflection, a layer of reflective film was pasted to the front surface of the specimen.A high voltage amplifier (Aigtek ATA-2021H) was used to amplify the signal and output to actuate the PZTs.The voltage signals of the sensor PZTs were obtained and recorded by an oscilloscope (Tektronix MSO44).
Following the steps in the procedure section, we first measured the ODSs of the intact structure actuated by each PZT wafer separately.Meanwhile, the oscilloscope recorded the voltage signals of the other sensing PZT.Using a Fourier transform, the real and imaginary parts of the response signals at each frequency were obtained.Then, after applying the adhesive, we measured and calculated the additional mass damage magnetically at (16,23), as well as the real and imaginary parts of the voltage signals of only the sensing PZTs.Using Eq. ( 8), the damage imaging of the CFRP was obtained and is shown in Fig. 8, in which the circled outlined by a red dotted line shows the position of the actual magnetic mass.
From Fig. 8, we can see that the peak of the damage index was located at (14, 24), which was within the damage area and proves the effectiveness of the proposed NLAM method.

Conclusions
A novel vibration-based damage detection method, NLAM, was proposed and validated in this study.Unlike to traditional vibration identification methods, in which it is difficult to detect damage when it located on the nodal lines because the dynamic responses of structures have not been affected, NLAM takes advantage of this phenomenon.By modulating and controlling the excitation from several actuators, including the magnitudes and phases, we can make the ODSs of both the intact and damaged structures be the same, which indicates that the damage is located exactly on the nodal lines.In order to improve the detection efficiency, we analyze the signals of several measuring points (by PZTs) when comparing the ODSs.Considering the ODS results actuated by modulated excitation signals at multiple frequencies, we can obtain a damage imaging map that precisely shows the location of the defect.From the procedure steps of the NLAM, it is clear that aside from the large amount of measurements required to obtain the base signals of the intact structure (including the ODSs excited by each actuator separately and the voltage signals detected by the sensors), it is an efficient method that requires only the voltage signals from sensors to calculate the final damage imaging results.Therefore, NLAM is quite suitable for online SHM.Another advantage of this proposed method is that since NLAM is a vibration-based method, it is suitable for the detection of any type of damage in any kind of material structures.Both numerical and experimental results validated the effectiveness and the advantages of NLAM.This proposed method provides a new analytical approach for SHM.

Fig. 1
Fig. 1 Schematic diagram of node modulation in a simply supported beam

Fig. 2 Fig. 3
Fig. 2 Flow diagram of NLAM procedure In order to highlight the nodal lines, which can locate the damage more accurately, we first square the modified ODSs and normalize them to the maximum value with Eq. (5).

Fig. 4 Fig. 6
Fig. 4 The highlighted nodal lines of a the intact structure (the red dots represent the center of the cleft damage); and b the structure with cleft

Fig. 7
Fig. 7 Time domain (one period) and frequency domain of the synthesized excitation signal

Fig. 8
Fig. 8 Damage imaging of CFRP (the circled outlined with a red dotted line shows the position of the actual magnetic mass)