Nonlinear vibrations emerging from damaged structures are suitable indicators for detecting defects. When a crack arises, its behavior could be approximated like a bilinear stiffness. According to this scheme, typical nonlinear phenomena as the presence of superharmonics in the dynamic response and the variation of the oscillation frequency in time emerge. These physical consequences give the opportunity to study damage detection procedures with relevant improvements with respect to the typical strategies based on linear vibrations, such as high sensitivity to small damages, no need for an accurate comparison model, and behavior not influenced by environmental conditions. This paper presents a methodology, which aims at finding suitable nonlinear phenomena for the damage detection of three contact-type damages in a panel representing a typical aeronautical structural component. At first, structural simulations are executed using MSC Nastran models and reduced dynamic models in MATLAB in order to highlight relevant nonlinear behaviors. Then, proper experimental tests are developed in order to look for the nonlinear phenomena identified: presence of superharmonics in the dynamic response and nonlinear behavior of the lower frequency of vibration, computed using the CWT (continuous wavelet transform). The proposed approach exhibits the possibility to detect and localize contact-type damages present in a realistic assembled structure.

In the actual aeronautical field, the damage detection methodologies are achieving a significant importance due to their great influence on both safety and cost/time optimization. Traditional maintenance schemes based on statistical predictions of times between controls are not optimal for cost savings. Moreover, maintenance implies two kinds of costs for operators [

The traditional approaches to damage detection are represented by the NDT (nondestructive testing) techniques. They are off-line and localized inspections, developed since the early to mid of 1960s, mainly related to visual inspections, ultrasonic inspections, eddy currents, acoustic emissions, radiography, thermography, and shearography [

The typical SHM methods are the linear vibrational schemes [

Nonlinear vibrational methods in SHM context are emerging in order to overcome the previous limitations. The basic idea around them is that a damage causes a nonlinear behaviour in the structure where it grows. For this reason, it is sufficient to detect the nonlinearities in order to state elements on the defect presence. According to [

In order to justify the phenomena emerging from nonlinearities suitable for the damage detection procedure here discussed, a short theoretical background on nonlinear systems is mandatory.

A nonlinear dynamical system owns particular characteristics compared to those of the linear processes. Firstly, the dynamical response to an input excitation depends both on the frequency of the excitation and its amplitude. For a linear system, instead, the only dependence is that from the excitation frequency. Then, nonlinear systems exhibit sensitivity to initial conditions. In the case of linear time-invariant systems, the impulse response is the linear combination of specific shapes, called eigenvectors, vibrating at specific eigenfrequencies. These invariant sets do not depend on initial conditions. Moreover, given a certain group of harmonics in the input signal, the output of a nonlinear system may present subharmonics or superharmonics. Finally, due to the high sensitivity to initial conditions, nonlinear systems can exhibit a chaotic behavior also with deterministic nonlinear models.

The mathematician Volterra proposed an extension of the conventional input-output relation used to describe dynamical systems in the time domain [

The terms

The terms

The Volterra series can be used in order to justify the superharmonic presence in the output of nonlinear systems. Indeed, in the case of a periodic excitation composed by the harmonic

The response exhibits multiple integers of the harmonic

Another relevant aspect described by the Volterra series is the absence of homogeneity. While each single convolution of the series is homogeneous, the response is the combination of homogeneous convolutions with different numbers of input functions. Consequently, differences in the amplitude of the input cause a nonhomogeneous system response.

In order to highlight the two effects underlined at the end of the previous paragraph for nonlinear systems and relate them to a bilinear spring mechanism representing a damage, it is here proposed a simple 1 degree of freedom (dof) example coming from the work of Neild [

1 dof system with bilinear spring.

The equations of motion for the two different situations are here proposed:

The solutions of the equations for the two cases can be easily found once the initial conditions are defined. Considering the motion evolving from an initial position

Through the analytic calculation of [

Normalized frequency with different amplitudes.

Normalized superharmonics with different amplitudes.

The results clearly underline the consequences of the bilinear spring in the system response. Figure

The panel analyzed in the current work reproduces a typical aeronautical structure. It is the one used in [

Aluminum panel used for the analyses.

The three damaged scenarios are represented by the absence of connections between one stiffener and the base plate (two lines of screws are used to connect each stiffener to the base plate). They are called D2, D3, and D5. Table

Damaged scenario characteristics.

Case | Description |
---|---|

D2 | Separation, 80 mm long; ending stiffener position |

D3 | Separation, 120 mm long; central stiffener position |

D5 | Separation, 40 mm long; ending stiffener position |

Figures

Detail of D2 damage.

Detail of D3 damage.

Detail of D5 damage.

Table

First 20 eigenfrequencies for the undamaged panel and for the three damages with their % variations.

Mode | D2, |
D2, % | D3, |
D3, % | D5, |
D5, % | |
---|---|---|---|---|---|---|---|

1 | 7.95 | 7.79 | −2.01 | 7.95 | 0.00 | 7.83 | −1.51 |

2 | 11.38 | 11.37 | −0.09 | 11.38 | 0.00 | 11.38 | 0.00 |

3 | 23.67 | 23.44 | −0.97 | 23.67 | 0.00 | 23.51 | −0.68 |

4 | 27.90 | 26.67 | −4.41 | 27.90 | 0.00 | 27.60 | −1.08 |

5 | 47.72 | 43.15 | −9.58 | 47.72 | 0.00 | 46.56 | −2.43 |

6 | 49.82 | 48.93 | −1.79 | 49.82 | 0.00 | 49.11 | −1.43 |

7 | 72.67 | 71.01 | −2.28 | 72.58 | −0.12 | 71.47 | −1.65 |

8 | 83.43 | 73.63 | −11.75 | 83.43 | 0.00 | 80.68 | −3.30 |

9 | 87.70 | 85.73 | −2.25 | 87.57 | −0.15 | 86.44 | −1.44 |

— | — | 93.42 | — | — | — | — | — |

10 | 99.22 | 99.41 | 0.19 | 98.89 | −0.33 | 98.20 | −1.03 |

11 | 101.11 | 99.88 | −1.22 | 101.10 | −0.01 | 99.76 | −1.34 |

12 | 101.22 | 109.68 | 8.36 | 100.97 | −0.25 | 100.20 | −1.01 |

13 | 114.46 | 114.63 | 0.15 | 114.45 | −0.01 | 112.51 | −1.70 |

14 | 117.42 | 117.48 | 0.05 | 115.53 | −1.61 | 116.92 | −0.43 |

15 | 118.05 | 117.86 | −0.16 | 117.70 | −0.30 | 117.86 | −0.16 |

16 | 121.78 | 121.75 | −0.02 | 121.77 | −0.01 | 121.75 | −0.02 |

17 | 121.82 | 126.57 | 3.90 | 121.65 | −0.14 | 119.86 | −1.61 |

18 | 128.90 | 131.10 | 1.71 | 128.88 | −0.02 | 128.15 | −0.58 |

19 | 137.98 | 137.16 | −0.59 | 137.68 | −0.22 | 137.36 | −0.45 |

20 | 139.43 | 139.27 | −0.11 | 139.34 | −0.06 | 139.33 | −0.07 |

Figures

Mode 1.

Mode 2.

Mode 3.

Mode 4.

Mode 5.

Mode 6.

Figures

Undamaged panel–D2 damaged panel MAC.

Undamaged panel–D3 damaged panel MAC.

Undamaged panel–D5 damaged panel MAC.

Each MAC matrix has been obtained by comparing the transversal (out of plane) displacements of a set of 120 nodes equally spaced along the lateral and longitudinal directions of the panel.

Figure

In order to highlight the presence of nonlinear phenomena, preliminary numerical tests have been run both using complete FEM (finite element method) models with the software MSC Nastran and reduced models with the software MATLAB. The panel has been analyzed as a free-free structure.

The original undamaged panel model has been developed in Nastran by using 12800 CQUAD4 elements (12000 for the base plate mesh, 800 for the transversal supports) and 560 CBEAM elements for the stiffeners. Each damaged scenario has been obtained from the original model by using a certain amount of GAP elements, connecting pairs of nodes in the vertical direction: they are nonlinear tools used to implement a bilinear stiffness. In particular, the penalty approach has been used in order to simulate the contact mechanism in each damaged area: when there is a contact between the two nodes connected by the gap, a stiffness of 10^{9} N/m acts in order to avoid interpositions. The cards used to implement these gaps are CGAP and PGAP for the property definition [

For each scenario, a limited number of gap elements have been used only for the central pairs of nodes involved in order to reduce the time required for the analyses with the SOL 129 (“nonlinear transient solution”): damages D2 and D3 present three gaps and damage D5 presents two gaps.

The campaign of FEM analyses with complete models has regarded the following tests:

Double sine excitation tests: a 5 Hz vertical (out of plane) sinusoidal excitation and a 15 Hz one have been located on the opposite sides of the damaged stiffener, with amplitudes of 50 N and duration of 10 s. Acceleration stories in the excited nodes and in the other three nodes along the damaged stiffener have been windowed with the Hanning window and finally analyzed by using the FFT (all the elaborations have been executed in MATLAB). Figures

The double sine excitation at the indicated frequencies has been chosen in order to look for possible linear combinations, sum, or difference of the excitation frequencies in the outputs. The use of a difference between the two frequencies matching a structural eigenfrequency determines a relevant response, as indicated in the literature [

In the case of the damaged scenario D2, a 30 Hz response in the results has been obtained. Figure

Tests for superharmonics: for each damaged scenario, two different numerical tests have been executed applying a 50 N amplitude vertical sinusoidal force next to each damaged area. It has been decided to use the same frequencies of the previous tests, in order to highlight possible similarities in the results. Thus, in the first case, the frequency is 5 Hz and in the second case, it is 15 Hz for a duration of 12 s. In the case of the 15 Hz excitation, scenario D2 has exhibited the superharmonic 30 Hz highlighted in the tests done before (Figure

Decay tests: for each damaged scenario, an impulsive vertical load has been applied next to each damaged area, represented as a triangular excitation of duration 0.01 s and amplitude 20 N. A damping factor of 1% has been set for a frequency about 7 Hz, which is close to the first eigenfrequency of the undamaged plate. Each simulation with SOL 129 has lasted 5 s, with a time step suitable as to follow the very rapid load story (an initial step of 0.001 s has been used). Accelerations in time have been computed in each excitation node and in a node located in the damaged area. Then, they have been processed by using the continuous wavelet transform (CWT) based on the Morlet wavelets [

Excitation and measuring nodes for D2 damage along the second stiffener.

Excitation and measuring nodes for D3 damage along the second stiffener.

Excitation and measuring nodes for D5 damage along the second stiffener.

D2, 30 Hz normalized responses, double excitation tests.

D3, 20 Hz normalized responses, double excitation tests.

D2, 25 Hz normalized responses, double excitation tests.

D3, 25 Hz normalized responses, double excitation tests.

D5, 25 Hz normalized responses, double excitation tests.

D2, direct response FFT, 15 Hz excitation test.

D3, decay test CWT, acceleration in the excitation node.

D3, decay test CWT, acceleration in the damaged area.

The analyses described before have the limitation of long times required, and this does not give the opportunity to model the entire damaged areas with the gaps. For this reason, further analyses have been executed by using reduced-order models, following the Craig-Bampton approach [

Sine excitation tests: a vertical sinusoidal excitation (amplitude = 50 N) at different frequencies (5, 10, 15, 20, 25, and 30 Hz) has been placed in a node located next to each damaged area. The direct acceleration responses, obtained by using the forward finite difference method on the velocities, have been windowed with the Hanning window and then elaborated through the FFT. The first simulations done on undamped models have shown unclear responses in those cases exhibiting nonlinear behaviors: indeed, the absence of damping causes a limited dissipation of those contributions at frequencies higher than the half of the sampling frequency (time step has been taken equal to 0.005 s). For this reason, tests have been repeated on damped models: a damping matrix proportional to the stiffness matrix has been used, with the proportional constant taken as to give a damping of 1% for a frequency of 7 Hz. The function used to solve the system of nonlinear ordinary differential equation is “ode15s,” a stiff solver. In this case, tests have been done for the damaged scenarios D2 and D5, as D3 has not exhibited superharmonics in the previous simulations. Tables

Figures

A theoretical interpretation of this aspect comes from several works in which it is stated that, in the case of crack breathing into beam structures, the higher first superharmonic behavior is excited when the excitation frequency matches one-half of the eigenfrequency [

Sine excitation tests with multiple acceleration measurements: once the previous tests have revealed the presence of superharmonics, the 20 Hz excitation simulations have been repeated by computing the accelerations in 10 nodes along the damaged stiffeners, at a distance one to each other of about 106.6 mm (see Figure

Sine excitation tests repeated with a different force position. The multiple acceleration measurement tests have been executed exciting in node 12, which is located on the opposite side of the damages D2 and D5 (see Figures ^{2}, while the 20 Hz one is 3.995 m/s^{2}).

Decay tests: as done for the tests with complete models, an impulsive vertical load has been applied next to each damaged area, with the same characteristics of the one used before. A damping matrix proportional to the stiffness one has been introduced in the model. 4 s acceleration signals with a resolution of 0.005 s have been recorded in the excitation node and in a node close to each damaged area. Each signal has been transformed into its analytic version and processed by using the CWT with Morlet wavelets. Results have highlighted relevant fluctuations of the main vibration frequency around an eigenfrequency after a transition time, for the accelerations in the damaged area nodes (Figures

As observed for the D3 results with the complete models, the fluctuations in this case are around the second eigenfrequency (11 Hz). On the other hand, they are around the first eigenfrequency (8 Hz) for D2 and D5 cases. This fact is due to the relation between the position of the excitation node and the mode shapes: the first mode, with a frequency about 8 Hz, is a torsion (see Figure

D2, superharmonic contributions, sine excitation tests.

1st superharmonic | 2nd superharmonic | 3rd superharmonic | |
---|---|---|---|

5 Hz | Yes | No | Yes |

10 Hz | Yes | No | Yes |

15 Hz | Yes | Yes | Yes |

20 Hz | Yes | Yes | Yes |

25 Hz | Yes | Yes | No |

D5, superharmonic contributions, sine excitation tests.

1st superharmonic | 2nd superharmonic | 3rd superharmonic | |
---|---|---|---|

20 Hz | Yes | No | Yes |

25 Hz | Yes | No | No |

D2, direct response FFT, 20 Hz sine excitation.

D2, direct response FFT, 25 Hz sine excitation.

D5, direct response FFT, 20 Hz sine excitation.

D5, direct response FFT, 25 Hz sine excitation.

Positions of the measuring nodes for experimental superharmonic tests.

D2, 20 Hz excitation, normalized 1st superharmonic.

D5, 20 Hz excitation, normalized 1st superharmonic.

D2, 20 Hz excitation in node 12, normalized 1st superharmonic.

D2, decay test CWT, acceleration in the damaged node.

D3, decay test CWT, acceleration in the damaged node.

D5, decay test CWT, acceleration in the damaged node.

The numerical tests described have been executed with the aim of finding suitable nonlinear phenomena for an experimental damage detection technique. From the results discussed, it has been decided to focus on two phenomena highlighted by the use of the reduced-order simulations: the presence of superharmonics with sine excitation tests and the fluctuations of the main vibration frequency for decay tests. Both these aspects are theoretically justified by the analyses described in Section

In order to exhibit a comparison between the numerical results obtained in the cases of complete and reduced models, Figures

D2, 20 Hz excitation 1st superharmonic, full model.

D2, 20 Hz excitation 1st superharmonic, reduced model.

D2, node 93 acceleration, FFT, full model.

D2, node 93 acceleration, FFT, reduced model.

The comparisons shown must be completed highlighting that full models consider a limited area for the gap mechanism implementation in each damaged place. Moreover, reduced model simulations have been executed introducing a damping has written in Section

It is important to underline that the numerical simulations have been based on model assumptions. In particular, the intuitive idea of the local bilinear stiffness behavior of each contact has been followed without an energetic formalization of the contact mechanism. Moreover, the contact has been modeled only in each damaged area, so where it has been supposed that the nonlinear phenomena show their main effects. Finally, each link between the stiffeners and the baseplate has been realized merging the common nodes.

The detail of time integration parameters and model setup for the numerical analyses is present in [

The experimental tests have been conducted with the panel suspended by the use of an elastic cable, which supports the structure at the two extremities. Figure

Detail of the hanging system.

All the accelerometers are located on the side of the baseplate without the stiffeners.

Tests for superharmonics have been conducted by forcing the reference structure and the damaged scenarios in two different positions. The first one corresponds to that of node 12 and the second one is the place of node 93. They are located on the opposite sides of the damaged stiffeners, as shown in Figures

Detail of the rod for panel-shaker connection.

Load levels used for the sine excitation tests.

Load level | Voltage range (V) | Load range (N) |
---|---|---|

1 | 0.05 | 12.4 |

2 | 0.1 | 23.6 |

3 | 0.15 | 34.1 |

4 | 0.2 | 44.3 |

5 | 0.25 | 54 |

6 | 0.3 | 62.7 |

For each excitation, ten accelerations have been recorded along the second stiffener in those points corresponding to the measuring nodes of the numerical simulations. Figure

20 Hz sinusoidal excitation in node 12, with the six load levels

20 Hz sinusoidal excitation in node 93, with the six load levels

25 Hz sinusoidal excitation in node 12, with the six load levels

25 Hz sinusoidal excitation in node 93, with the six load levels

The choice for the excitation frequencies result from the numerical analyses.

By analyzing the direct responses, the presence of superharmonics has been found also for the reference condition. Moreover, several superharmonics have been detected for each damaged scenario analyzed [

Number of superharmonic peaks (

Case | ||||
---|---|---|---|---|

Reference | 7 | 160 | 2 | 60 |

D2 | 2 | 80 | 24 | 980 |

D3 | 2 | 60 | 26 | 820 |

D5 | 19 | 700 | 9 | 820 |

The analyses of the direct responses for the 25 Hz excitation have revealed the presence of superharmonics with more limited contributions. For this reason, it has been decided to focus on the 20 Hz excitation cases. As done for the numerical tests, the first superharmonic contributions (40 Hz) in all the measuring nodes have been compared. In particular, for all the cases, these contributions and their normalized values (obtained dividing the data by the highest value in each test) have been reported into figures representing the first superharmonics along the measuring nodes for all the load levels.

Figures

40 Hz FFT magnitudes, 20 Hz sine excitation force applied to node 12, reference structure.

40 Hz FFT magnitudes, 20 Hz sine excitation force applied to node 93, reference structure.

Figures

40 Hz FFT magnitudes, 20 Hz sine excitation force applied to node 12, D2 scenario.

40 Hz FFT magnitudes, 20 Hz sine excitation force applied to node 93, D2 scenario.

Figures

40 Hz FFT magnitudes, 20 Hz sine excitation force applied to node 12, D3 scenario.

40 Hz FFT magnitudes, 20 Hz sine excitation force applied to node 93, D3 scenario.

Figures

40 Hz FFT magnitudes, 20 Hz sine excitation force applied to node 12, D5 scenario.

40 Hz FFT magnitudes, 20 Hz sine excitation force applied to node 93, D5 scenario.

The analyses of the first superharmonics have revealed the possibility to localize the end damages (D2 and D5) in a qualitative way by comparing the responses to the different load levels and exciting at the opposite sides. Only for those cases of end damages, growing trend oriented towards a predominant direction has been observed. Moreover, for the case of excitation in node 12 in the D2 scenario, the nonhomogeneity explained by the Volterra series has resulted in the increasing of the amount of the contributions of those nodes located in the damaged area with the augmentation of the load level. Therefore, the load level acts like a trigger able to activate the nonlinear contact mechanism. Figure

D2, 20 Hz sine excitation force in node 12, ratios between load levels.

Similar conclusions for the qualitative localization are possible analyzing other superharmonics. In particular, [

Decay tests have been executed by using the PCB instrumented hammer shown in Figure

Instrumented hammer used for decay tests.

For each damaged scenario, two impulsive excitations have been applied: the first one located next to the damaged region and the second one located at one end of the damaged stiffener. For each test, three accelerations have been recorded: one in point

Positions of the measuring nodes for decay tests, damage D2.

Positions of the measuring nodes for decay tests, damage D3.

Positions of the measuring nodes for decay tests, damage D5.

The CWT has been firstly applied considering a frequency range between 5 Hz and 30 Hz. This allows the elimination of the low-frequency contribution related to the elastic cable used to suspend the structure. By the data processing, the main frequency content in time has been obtained. Two contributions have been observed in the acceleration stories: one at 8 Hz, so close to the first eigenfrequency of the undamaged structure, and one at 24 Hz [

D2, maximum frequency, excitation near the damage.

D2, maximum frequency, excitation far from the damage.

Fluctuations in the contribution are evident. The oscillations for point

Undamaged case, excitation near the damage.

Undamaged case, excitation far from the damage.

In this case, the oscillation ranges are the same for all the points. The visible abrupt ends are due to the absence of further more acceleration time data after the final time indicated: the test recordings last 14-15 s.

The elaborations done for the D3 case are the same as those for the D2 one, with the exception that the range of frequencies used for the second analyses goes from 5 Hz to 13 Hz. In the test done exciting near the damage, the acceleration analysis for point

D3, maximum frequency, point

D3, maximum frequency, excitation far from the damage.

As visible from Figure

The elaborations done for the D5 case are the same as those for the previous cases. The frequency ranges for the results reported go from 5 Hz to 10 Hz. Figures

D5, maximum frequency, excitation near the damage.

D5, maximum frequency, excitation far from the damage.

Again, the trend of the amplitudes increasing moving from node 12 to point

The analyses from decay tests have underlined some relevant points. At first, the main frequency fluctuation expected from numerical tests has been found for all the measurement points. As understood by the superharmonic tests, this element is the consequence of the nonlinear behavior of the structure itself. Then, it clearly emerges that the oscillations found for the lower dominant frequencies have higher amplitudes when related to the damaged area. Thus, comparing these amplitudes among different acceleration elaborations, it is possible to detect and localize the damages. Finally, the nonlinear phenomenon observed, justified by the variation of the local stiffness due to the contact, has shown some remarkable elements. Among them, all the fluctuations identified by the CWT elaborations have exhibited a 2 Hz oscillation behavior, with higher contributions at the top and the bottom of each one of them. Also, the local effect of damages has resulted stronger than the global nonlinearity of the structure. In this context, the CWT has resulted a powerful tool for the identification of a nonlinear and nonstationary phenomenon.

In order to define a proper quantitative index for the localization, Tables

D2, excitation near the damage, localization indexes.

Index | Node 12 | Point |
Point |
---|---|---|---|

7.93 | 7.86 | 7.78 | |

8.74 | 8.82 | 8.82 | |

0.81 | 0.96 | 1.04 |

D2, excitation far from the damage, localization indexes.

Index | Node 12 | Point |
Point |
---|---|---|---|

7.87 | 7.86 | 7.79 | |

8.75 | 8.81 | 8.93 | |

0.88 | 0.95 | 1.14 |

D3, excitation far from the damage, localization indexes.

Index | Point |
Point |
---|---|---|

9.66 | 9.38 | |

11.67 | 11.67 | |

2.01 | 2.29 |

D5, excitation near the damage, localization indexes.

Index | Node 12 | Point |
Point |
---|---|---|---|

7.93 | 7.87 | 7.83 | |

8.62 | 8.62 | 8.95 | |

0.69 | 0.75 | 1.12 |

D5, excitation far from the damage, localization indexes.

Index | Node 12 | Point |
Point |
---|---|---|---|

7.93 | 7.87 | 7.60 | |

8.75 | 8.75 | 9.60 | |

0.82 | 0.88 | 2.00 |

In order to obtain the value reported, proper time intervals have been fixed for the data available in order to include only the dominant frequency oscillations. For all the cases, the frequency range increases from node 12 to point

The paper has presented an extension of the nonlinear vibrational methods in a structure reproducing an aeronautical panel, in which three different contact-type damages have been taken into account.

The results presented based on the search for two nonlinear behaviours (superharmonics, variation of the lower frequency of vibration in time) have highlighted the following points in relation to the damage detection procedure:

The nonlinear phenomena here investigated have also emerged with the undamaged structure. Thus, the detection itself by finding these characteristics has not resulted a possibility.

The localization of the contact damages has been possible through the comparisons of the responses measured. In particular, the variations of the lower frequency of vibration in time have resulted usefulness to detect all the damages analyzed, without requiring the amount of elaboration done for the end defect detection using the superharmonics. Indeed, the amplitude of these variations has resulted sensitivity to the damage location. Moreover, the CWT used to analyze the decay tests data has been identified as a powerful tool for underlining such a nonlinear and nonstationary behaviour.

The numerical procedure executed in order to highlight the presence of suitable nonlinear phenomena has been able to underline effects also found in the experimental tests. Therefore, the contact mechanism model implemented using the local penalty approach has resulted validity for understanding possible nonlinear evidences but needs to be furtherly elaborated in order to ensure true predictions of the analyzed structure. Despite the model assumptions reported in Section

Finally, in order to better explore the nonlinear vibrational methods for applicative cases, further analyses are mandatory: extension of the methods used (step and sine for superharmonics, CWT used for the monitoring of superharmonics in time, and decay tests with more damages of different extensions), new conditions (composites structures, different boundary conditions), and different kinds of damages.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Special thanks are due to Mauro Terraneo from Vicoter for his support during the experimental activity and to Dr. Kamal Rezvani for his previous activity on the design, manufacturing, and testing of the reference panel used in this work.