Structural health monitoring of cylindrical bodies under impulsive hydrodynamic loading by distributed FBG strain measurements

A sketch of the employed experimental setup is depicted in figure 1. It allows to perform free fall impacts of bodies on a quiescent water free surface from a height up to 2.5 m. The water is contained in a tank made up of stainless steel and glass, which is internally 1.85 m large, 1.50 m long and 1 m deep. The tank is filled with water up to a height of 0.6 m in order to avoid water overflow during the experiments. The circular cylinder impacting the water is connected to a sledge of aluminum that runs along two vertical rails rigidly connected to the roof of the laboratory and to the water tank. The rails are about 3 m long and are positioned at a mutual distance of 65 cm. The system is designed to release the cylinder 15 cm above the water surface, which is a sufficient distance to prevent contact between the sledge and the water jet generated by the cylinder water entry. The mass of the structure impacting the water, including the cylinder, the sledge and the sensors, is 3.1 kg.

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The cylinder position along the z axis during free fall and water entry is measured through a linear potentiometer membrane (Spectrasymbol thinpot) embedded in one of the two rails. The velocity of the cylinder along its run is calculated by numerical derivation of the recorded time history of the position through a central differencing scheme.

Digital images are acquired by means of a high speed $\text{Phanto}{{\text{m}}^{\circledR}}$ Miro 110 camera recording through a transparent side of the tank. Such a camera is monochromatic and features a CMOS sensor of $1280\times 800$ pixels. The acquisition speed ranges from 1600 fps at full resolution to $400\,000$ fps at reduced resolution. Here we employ a speed of 2000 fps with a maximum definition of $1152\times 720$ pixels. Triggering is performed by means of a Markteck Optoelectronics MTRS4720D photodiode with a response time of 1 μs. The analog signals are synchronously acquired by a National Instrument NI USB-6009 at a sample rate of 10 kHz.

The cylindrical specimen has a diameter of 240 mm, a length of 150 mm and a thickness of 1.8 mm. The cylinder is made up of three layers of plain weave fabric E-glass, with a nominal weight of $480\pm 20$ g m⁻³ and a fiber volume fraction of 60%. The nominal flexural modulus, the Poisson ratio and the mass density are 33 GPa, 0.28 and 2540 kg m⁻³, respectively.

The strain at selected locations is measured through fiber Bragg gratings. The measuring principle of such sensing devices is based on a periodic perturbation of the refractive index in the core of an optical fiber extending over a certain length. This part of the fiber is called Bragg grating, which has the property to reflect a specific wavelength, called the Bragg wavelength ${{\lambda}_{\text{B}}}$. This wavelength is related to the period of the grating ${{\Lambda}_{\text{B}}}$ from the simple relationship ${{\lambda}_{\text{B}}}=2{{n}_{r}}{{\Lambda}_{\text{B}}}$, where n_r is the average effective refractive index. When the signal traveling in the fiber arrives in the area of the Bragg grating, it can be transmitted, reflected back or exit from the fiber depending on its wavelength. If the portion of the fiber with the grating is glued or inserted into the structure, the structural deformation will result in a deformation of the grating, which in turn changes the reflected wavelength. By sending a broadband signal, we measure the incremental strain $\delta \epsilon$ by measuring the shift of the peak wavelength $\delta {{\lambda}_{\text{B}}}$ in reflected signal as follows:

$$\begin{eqnarray}&&\delta {{\lambda}_{\text{B}}}={{\lambda}_{\text{B}}}\delta \epsilon \left(1-{{p}_{\epsilon}}\right)\end{eqnarray} \tag{ 1 }$$

where ${{p}_{\epsilon}}$ is the photoelastic coefficient of the fiber equal to 0.22. Actually, also a temperature variation would produce a shift in the peak wavelength. However, being the slamming event duration very short (in the order of 50 ms), temperature compensation is not needed in this particular case.

Here, we employ one optical fiber with five gratings of 3 mm in cascade with different Bragg wavelengths. Such sensors are glued on the internal surface of the cylinder at its mid span through a cyanoacrylate adhesive and protected by a thin layer of silicone. Measurements have been performed after 48 h of curing. The measuring points with Bragg wavelengths ${{\lambda}_{\text{B}}}$ of 557.2, 1542.9, 1535.0, 1528.7 and 1526.8 nm are located at 95.5°, 112.2°, 128.9°, 171.8°, and 205.3° with respect to the point where the cylinder is connected to the sledge, as depicted in figure 1. The high speed image of the falling cylinder impacting the water in the right panel of figure 1 illustrates the high deformation of the cylinder during one of the impact events.

Strains are synchronously measured by an interrogation system, which integrates a laser source with an average optical power output of 3 mW and a wavelength bandwidth of 80 nm. The repeatability is  ±  3 pm and the strain resolution is about 1 pm. The sampling frequency is 4 kHz.

We here propose a modal decomposition method for the reconstruction of the deformation of the cylinder under dynamic impulsive loading using distributed FBG strain measurements. The analytical procedure is an enhanced version of the one developed in previous work by some of the authors and fully described in [27, 28].

We assume that a finite number of mode shapes allows to reconstruct the deformation of the body with sufficient accuracy [75] and we compute the elastic response of the structure in terms of modal coordinates. Displacements and strains can be expressed as $\boldsymbol{d}(t)=\boldsymbol{\varPhi}\boldsymbol{\mu}(t)$ and $\boldsymbol{\epsilon}(t)=\boldsymbol{\varPsi}\boldsymbol{\mu}(t)$, respectively, where $\boldsymbol{d}(t)$ and $\boldsymbol{\epsilon}(t)$ are vectors containing displacements d_n(t) and strains ${{\epsilon}_{n}}(t)$ as functions of time t, n denoting the measurement location. In the previous expressions, $\boldsymbol{\varPhi}$ and $\boldsymbol{\varPsi}$ are $N\times M$ matrices (N  =  number of measurement points and M  =  number of mode shapes, $M\leqslant N$) collecting the normalized modal displacement ${{\Phi}_{n,m}}$ and modal strain components ${{\Psi}_{n,m}}$ at the measuring locations, respectively, m denoting the modal order, while $\boldsymbol{\mu}(t)$ is a vector that contains the time-varying modal coordinates ${{\mu}_{m}}(t)$. At each time instant, the strain values ${{\epsilon}_{n}}(t)$ are read from the FBG sensors and the modal coordinates $\boldsymbol{\mu}(t)$ are computed as

$$\begin{eqnarray}&&\boldsymbol{\mu}(t)={{\left(\boldsymbol{\varPsi}^{{T}}\boldsymbol{\varPsi}\right)}^{-1}}\boldsymbol{\varPsi}^{{T}}\boldsymbol{\epsilon}(t)\end{eqnarray} \tag{ 2 }$$

The matrices $\boldsymbol{\varPhi}$ and $\boldsymbol{\varPsi}$ are characteristics of the investigated structure and could be obtained analytically, numerically or experimentally, once before monitoring and without requiring any further updating. This represents a great advantage of this methodology if compared to FEM methods, in terms of real time calculations at low computational cost.

If the mode shapes are known in every point of the structure, it is possible to reconstruct the overall deformation by simply substituting matrix $\boldsymbol{\varPhi}$ with a matrix $\boldsymbol{\varphi}$ collecting the normalized modal displacements at the points of interest. In the application example, for instance, matrix $\boldsymbol{\varphi}$ contains modal components ${{\varphi}_{\vartheta,m}}$, where ϑ is the polar coordinate and m denotes the order of the mode. Displacements $\boldsymbol{w}_{{\vartheta}}(t)$ and strains $\boldsymbol{\varepsilon}_{{\vartheta}}(z,t)$ at any angular location are:

$$\begin{eqnarray}&&\boldsymbol{w}_{{\vartheta}}(t)=\boldsymbol{\varphi}{{\left(\boldsymbol{\varPsi}^{{T}}\boldsymbol{\varPsi}\right)}^{-1}}\boldsymbol{\varPsi}^{{T}}\boldsymbol{\epsilon}(t)\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\boldsymbol{\varepsilon}_{{\vartheta}}(t)=\boldsymbol{\psi}{{\left(\boldsymbol{\varPsi}^{{T}}\boldsymbol{\varPsi}\right)}^{-1}}\boldsymbol{\varPsi}^{{T}}\boldsymbol{\epsilon}(t)\end{eqnarray} \tag{ 4 }$$

In the problem at hand, the normalized radial displacement and circumferential strain for every mode shape can be written by enforcing the theory of thin walled cylinders as follows:

$$\begin{eqnarray}&&\boldsymbol{\varphi}_{{\vartheta,m}}=\cos \left(m\vartheta \right)\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\boldsymbol{\psi}_{{\vartheta,m}}(z)=\frac{\left(1-{{m}^{2}}\right)z}{{{R}^{2}}}\cos \left(m\vartheta \right)\end{eqnarray} \tag{ 6 }$$

where z is the distance from the neutral surface of the structure along the thickness of the shell. Equation (6) derives from the general formulation of circumferential strain for thin walled cylinders:

$$\begin{eqnarray}&&\boldsymbol{\varepsilon}(z)=\boldsymbol{\varepsilon}_{{0}}-z\chi \end{eqnarray} \tag{ 7 }$$

It should be noted that, in the strain formulation, the elongation of the middle surface $\boldsymbol{\varepsilon}_{{0}}$ has been eliminated, under the assumption that, in case of loading conditions that are not axial symmetrical with respect to the revolution axis of the cylinder, membrane stresses can be neglected. In the formulation proposed in [27] the change of curvature, χ, was reduced to the second derivative of the radial displacement w with respect to circumferential coordinate ϑ. In all experimental and numerical results of the present paper, the following expression is used, which is more accurate in case of inextensible deformation:

$$\begin{eqnarray}&&\chi =\frac{1}{{{R}^{2}}}\left(\frac{\partial v}{\partial \vartheta}+\frac{{{\partial}^{2}}w}{\partial {{\vartheta}^{2}}}\right)=\frac{1}{{{R}^{2}}}\left(w+\frac{{{\partial}^{2}}w}{\partial {{\vartheta}^{2}}}\right)\end{eqnarray} \tag{ 8 }$$

where R is the radius of the cylinder, v is the tangential displacement and $\frac{\partial v}{\partial \vartheta}=w$ because the membrane stresses have been neglected.

Experiments have been carried out in order to validate the reconstruction technique and to benchmark its performance against its previous version. To this end, free falling experiments are conducted using six different drop heights, from 25 cm to 75 cm, with steps of 25 cm.

Table 1 presents the average strain reconstruction errors of the present method, compared to the previous ones in [27], for a drop height of 75 cm. Each error component is computed as

$$\begin{eqnarray}&&\text{Er}{{\text{r}}_{n}}\left(\% \right)=\sqrt{\frac{1}{K-1}\underset{k=1}{\overset{K}{\sum}}\frac{{{\left({{{\hat{\epsilon}}}_{n}}(k)-{{\epsilon}_{n}}(k)\right)}^{2}}}{{{\left(\text{ma}{{\text{x}}_{k}}\left({{\epsilon}_{n}}(k)\right)-\text{mi}{{\text{n}}_{k}}\left({{\epsilon}_{n}}(k)\right)\right)}^{2}}}}\end{eqnarray} \tag{ 9 }$$

where we denote with ${{\hat{\epsilon}}_{n}}(k)$ and ${{\epsilon}_{n}}(k)$ reconstructed and measured strains at the kth time sample at sensor n, respectively, K  =  300 being the length of strain signals. The reconstruction error, in equation (9), is computed by using four FBG sensors for reconstruction purposes and sensor n for comparison. Thus, the error quantifies the difference between a local FBG measurement and the reconstructed strain using the remaining FBG sensors. The average reconstruction errors are all of the order of $10 \%$, which could be considered a good result, given that only 4 sensors are employed, all located in a limited portion (about a quarter) of the circle. In fact, as it will be shown in section 5.1, the reconstruction error depends upon the number and spatial layout of the FBG sensors. With regard to the results summarized in table 1, it should also be noted that the reconstruction technique is more efficient for small strains and displacements, where the modal approach is more accurate. Indeed, limiting the sampling to the very first part of the time history (e.g. the first 4 ms), or to the very last part at the end of the free decay, where the strains are small, the relative error decreases to less than $5 \%$.

Table 1. Average strain reconstruction errors (expressed as a percentage), calculated through equation (9), at different sensors' locations, for a drop height of 75 cm (the errors are computed as average values among three test repetitions).

	Sensor used for comparison
	$\text{Er}{{\text{r}}_{1}}$	$\text{Er}{{\text{r}}_{2}}$	$\text{Er}{{\text{r}}_{3}}$	$\text{Er}{{\text{r}}_{4}}$	$\text{Er}{{\text{r}}_{5}}$
Present study	13.5	7.6	11.9	10.5	9.8
Method in Panciroli et al [27]	15.2	9.2	15.2	10.9	11.1

In terms of deformed shape, the proposed method yields the results of figure 2, where, for different drop heights and at different time instants, the deformed shape of the cylinder, measured through the high speed camera, is compared to that reconstructed by the present technique and to that reconstructed with the method reported in [27]. Again, the accuracy of the presented method, further enhanced with respect to the previous one [27, 28], is apparent from the presented results.

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We comment that, in the presented example, the strains are larger than usually expected in real applications. This, however, is not a limitation in generality considering that (i) the reconstruction technique is based on a modal assumption which works better for smaller strains/displacements, as noted above, and because (ii) FBG sensors are quite effective in measuring very small strains, as demonstrated in several applications, not limited to naval and aerospace structures, but also including civil structures where strains are typically very small [29–40].

In the first part of the parametric study, spatial variations of ${{I}_{\epsilon}}$ and I_d with and without damage are investigated. The considered analysis cases are depicted in figure 4. In particular, for each set of reference sensors, an undamaged and a damaged case are considered, where, as described above, damage is simulated as a localized delamination taking place at one half of the thickness. Damage is assumed to have an angular extension $\Delta ={{30}^{\circ}}$ and to be located at $\gamma ={{180}^{\circ}}$. In sets 1 and 2 six reference sensors with a mutual angular spacing $\delta ={{60}^{\circ}}$ are considered, with $\alpha ={{10}^{\circ}}$ for set 1 and $\alpha ={{25}^{\circ}}$ for set 2. In this way, reference sensors are equally spaced along the circumference of the cylinder cross-section in both sets, one reference sensor falls into the damaged region in set 1, while no reference sensor falls into the damaged region in set 2. In sets 3 and 4 six reference sensors with a mutual angular spacing $\delta ={{15}^{\circ}}$ are considered, with $\alpha ={{180}^{\circ}}$ for set 3 and $\alpha ={{220}^{\circ}}$ for set 4. In this way, reference sensors are equally spaced in one quarter of the circumference, one reference sensor falls into the damaged region in set 3, while no reference sensor falls into the damaged region in set 4.

Figures 5 and 6 show ${{I}_{\epsilon}}$ versus β for all considered cases. First of all, it should be noted that the reconstruction error for the undamaged cylinder depends upon the number and spatial layout of the FBG sensors. In the presented numerical example, only 6 of such sensors are used, thus allowing to achieve a fairly accurate reconstruction of strains, with a maximum deviation of about $10 \%$ at a few locations far from the reference sensors, when these are uniformly distributed along the circle. On the contrary, when reference sensors are deployed only in one quarter of the circle, strain reconstruction errors can occur far from the strain sensors, while strain reconstruction is accurate elsewhere. A major consequence of this circumstance is that the layout of reference sensors results crucial for damage detection, as this is feasible only when no significant strain reconstruction error occurs in the undamaged case. With regard to the results obtained in the damaged cases, it is noted that, when reference sensors are uniformly distributed along the circle, regardless of the presence of a reference sensor onto the damaged region, damage systematically introduces large and measurable deviations between reconstructed and measured strains, resulting in maximum values of ${{I}_{\epsilon}}$ up to about 200. Considering that ${{I}_{\epsilon}}=0$ when the control sensor coincides with one reference sensor, the locally maximum error tends to be at intermediate locations between two reference sensors. Furthermore, larger values of ${{I}_{\epsilon}}$ are observed around the damaged area, which might indicate the feasibility of damage localization. With reference to the comparison between damaged and undamaged cases, we observe that, depending on the spatial distribution of the sensors, the errors observed in the damaged cases are generally much larger and observed in larger portions of the structure (compare figures 5(b) and (d)).

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Figures 7 and 8 show I_d versus β for the same cases considered in figures 5 and 6. Similarly to the case of using strains, when reference sensors are deployed only along one quarter of the circle and, in particular, in set 3, some displacement reconstruction errors occur far from the reference sensors, which practically impedes damage detection. On the contrary, displacement reconstruction is accurate everywhere, with ${{I}_{\text{d}}}\cong 0~\forall \beta$, for reference sensors uniformly distributed along the circle. In such cases and, in particular, in sets 1 and 2, damage introduces a measurable deviation between reconstructed and measured displacements but no clear localization of the error is evidenced.

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In order to get more insight into the presented results and to better appreciate the deviations in predicted and measured strains and displacements, figures 9–12 show time histories of ${{\epsilon}_{C}}\left(\beta,t\right)$ and ${{d}_{C}}\left(\beta,t\right)$ for different values of β and for all considered analysis cases. From these figures, the accuracy in strain and displacement reconstruction achieved in the undamaged case when reference sensors are uniformly deployed along the circle (set 1 and 2) is apparent, as time series referring to measured and predicted quantities are almost perfectly overlapped. On the contrary, the time series referring to the undamaged case with reference sensors' set 3 and 4 confirm the existence of the already mentioned prediction errors far from the reference sensors, highlighting that such errors mostly occur in the initial loading phase of the structural response. The same figures highlight the significant deviations between reconstructed and measured strains and displacements in the damaged cases.

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With the purpose of investigating the sensitivity of the reconstruction error with respect to damage severity, ${{I}_{\epsilon}}$ and I_d have been computed by increasing damage extension, $\Delta$, for different values of β. In light of the previous results, only the case with reference sensors uniformly deployed along the whole circle is considered using, in particular, reference set number 2. The results, illustrated in figure 13, show that, for relatively small damage, both ${{I}_{\epsilon}}$ and I_d exhibit a clear and monotonic increase with increasing damage severity. This allows a robust detection of damage existence and also permits to track its evolution either using deviation in strain or in displacement as damage sensitive features. The same results also show that, when damage approaches a control sensor, ${{I}_{\epsilon}}$ undergoes an abrupt change in slope, which can potentially enable damage localization. Other changes in slope are observed in both ${{I}_{\epsilon}}$ and I_d when one or more reference sensors fall into the damaged region. In particular, the slope of I_d changes only when one reference sensor falls into the damaged region. In a dense deployment of reference sensors, this information can be useful for evaluating damage extension.

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