Research on Deformation Reconstruction Based on Structural Curvature of CFRP Propeller With Fiber Bragg Grating Sensor Network

The deformation and reconstruction of the composite propeller under the static load in the laboratory is studied so as to provide the basic research for the deformation and reconstruction of the underwater deformed propeller. The fiber Bragg grating (FBG) sensor is proposed to be used for strain monitoring and deformation reconstruction of the carbon fiber reinforced polymer (CFRP) propeller, and a reconstruction algorithm of structural curvature deformation of the CFRP propeller based on strain information is presented. The reconstruction algorithm is verified by using variable-thickness CFRP laminates in the finite element software. The results show that the relative error of the reconstruction algorithm is within 8%. Then, an experimental system of strain monitoring and deformation reconstruction for the CFRP propeller based on the FBG sensor network is built. The propeller blade is loaded in the form of the cantilever beam, and the blade deformation is reconstructed by the strain measured by the FBG sensor network. Compared with the blade deformation measured by three coordinate scanners, the reconstruction relative error is within 15%.


Introduction
Over the past few decades, polymer composites have been gaining attention as potential replacement materials for marine propellers due to their favourable benefits such as high specific strength, corrosion resistance, good vibration-reducing performance, superb designability, fatigue, and their potential for passive adaptation. For the propellers on marine vehicles, the working conditions are very complicated, and their deformations are difficult to monitor. However, the deformation of the composite blade plays an important role in improving the performance of the propeller. Lee and Lin [1] used a genetic algorithm to obtain the smallest pitch stacking sequence first, and then the propeller is displaced in the opposite direction to form a predeformed propeller. The predeformed propeller finally meets the demands of optimization and outperforms the traditional metal propeller. Blasques et al. [2] developed thrust by cutting laminates to control the deformed shape of blades. Chen et al. [3] introduced a novel morphing composite propeller blade with the improved efficiency and less acoustic emission. Maung et al. [4] presented a curved fiber path method to demonstrate the improvements possible for the optimization of an adaptive propeller by using the bend-twist coupling. The influence of the deformation of the composite blade on the torque and efficiency of the propeller is shown in these literatures, so the research on the deformation and reconstruction of the blade is of great significance. The fiber Bragg grating (FBG) sensor [5,6] has the advantages of small size, light weight, immunity to electromagnetic interference, and ease to realize distributed measurement, and it also has many advantages in monitoring composite materials. Cusano et al. [7] demonstrated that the FBG could be used to measure static and dynamic strains. In the FBG monitoring of the composite propeller, in 2001, Zetterlind et al. [8] first investigated the feasibility of embedding FBG sensors in composite propeller blades for strain measurement. In 2013, Herath et al. [9] measured the strain of the propeller by pasting the FBG sensor array on the surface of the composite blade. The above researches verified the feasibility and applicability by using the FBG sensor technology for online monitoring of the propeller. The FBG sensor cannot directly measure the deformation of the structure, which can only measure the strain and temperature of the structure, hence it is necessary to convert the strain to deformation through the deformation reconstruction method.
Previous scholars have done a lot of researches on structural deformation reconstruction methods based on strain information and successively put forward the Ko displacement theory, modal superposition method, inverse finite element method, and structural curvature algorithm. The Ko displacement theory is applicable to large thin plate flexible structures, and its calculation accuracy is closely related to the number of sensors [10,11]. The mode superposition method is to calculate the displacement-strain conversion matrix through the structural strain mode matrix and the corresponding displacement mode matrix. It has strict requirements for the sensor placement and the description of the material properties of the structure [12,13]. The inverse finite element method is based on the elastic theory of the medium thick plate. Its basic idea is to solve the transfer function between the strain field and the displacement field of the structure based on the least square variational equation, so as to solve the inverse problem of the transformation of the strain into displacement [14][15][16]. The structural curvature algorithm is to convert the strain information of the structure to the surface curvature information, and then the structural curve reconstruction theory based on the curvature information can realize the deformation reconstruction of the structure. The structural curvature algorithm is not limited by the structural materials and can be applied to the deformation reconstruction of the composite material propeller, which can simplify the processing of the anisotropic material properties of composite materials. In this paper, the structural curvature method is used to reconstruct the deformation of the carbon fiber reinforced polymer (CFRP) propeller.
In terms of the structural curvature algorithm research, Roesthuis [17] of the Netherlands and others established a flexible nickel-titanium needle, which was inserted into the soft tissue movement model and obtained the structural strain information through 12 FBG sensors arrays arranged on the needle. According to the strain and its relationship with the curvature information, the three-dimensional shape of the needle was reconstructed, and the results showed that the Guoping DING et  Glaser et al. [19] converted the strain information of the structure measured by the strain gauge into the curvature information and reconstructed the structure shape in real time based on the curvature information. The results showed the difference between the measurement and the reconstruction by the curvature algorithm fluctuated very little. Yi et al. [20] installed the FBG sensor on the cylindrical memory alloy group in a staggered orthogonal arrangement. The relationship coefficient between the curvature and FBG wavelength shift was obtained by the calibration method, and the curvature was calculated accordingly. The curvature information of the discrete points was used to reconstruct the spatial shape of the sensor by the spatial curve fitting method. These documents showed that the deformation of the structure could be reconstructed based on the relationship between the strain and curvature information.
This paper will explore whether the FBG technology and structural curvature method can reconfigurate a CFRP propeller shape. Firstly, a linear discretization method for variable-thickness CFRP laminates is proposed. The finite element method (FEM) simulation is used to verify the influence of different degrees of dispersion on the deformation reconstruction of variable-thickness CFRP laminates. Then, the discretization method of the CFRP propeller is proposed by variable-thickness CFRP laminates and the feasibility of linear discretization applied to the deformation reconstruction of the CFRP propeller is verified by simulation. Finally, a propeller strain-deformation reconstruction experimental system based on the FBG sensor network is constructed.

Obtaining curvature information
In the post-processing of the simulation software, the strain values at various places of each laminate can be obtained. How to convert the strain information to the curvature information is crucial. According to the knowledge of material mechanics, the transformation relationship between the strain information and the curvature information corresponding to the structural micro-element can be obtained based on the pure bending theory of the flexible beam. As shown in Fig. 1(a), a segment of the structural element with a length of L and a thickness of h is cut off from the flexible beam with the constant cross-section, and it is assumed that it satisfies the ideal deformation conditions. In Fig.  1(b), M is the structural element segment subjected to a pair of applied bending moments, 1 2 O O is the neutral layer of the structure, and ρ is the curvature radius of the neutral layer 1 (1) From (1), we can get where L is the length of the element, L Δ is the change of the length of the structural element, θ is the central angle corresponding to the arc when the element is deformed, k is the curvature of the element, and ε is the strain. It can be seen from (2) that as long as the thickness of the element h and the strain value of the deformed element structure ε are known, the curvature information k of the corresponding measurement point can be obtained. For the experimental structure model, the thickness of the element structure is constant, and the magnitude of the curvature k at a certain point has a linear relationship with the magnitude of strain ε , so the corresponding curvature information can be obtained as long as the strain information of the structure is known.

Curve reconstruction based on curvature information
After obtaining the multi-point curvature information of the structure, the position of the deformation curve is obtained by recursion, so as to realize the deformation reconstruction of the structure. The recursive process is to establish the moving coordinate system a b c − − , changing with the curve on the space curve, and assume that  of the curve. The schematic diagram of the spatial curve reconstruction is shown in Fig. 2. σ is as follows: 10 1 0 0 0 (2) Rotate (1) σ around the axis 1 c by the 1 α degree to obtain a new coordinate system (3) Continue to rotate the angle 1 θ of ( ) σ , which is determined by (8) and (9):

Deformation reconstruction of variablethickness CFRP laminates
The finite element analysis model of variablethickness CFRP laminates is established in the simulation software Abaqus. The length of variable-thickness CFRP laminates is 240 mm, the width is 80 mm, and the thickness gradually decreases from 2 mm to 0.4 mm. The left end face of the model is set as a fixed constraint, and a concentrated force of 100 N is applied along the negative direction of the Z axis at the center of the right end face, and the attribute of the divided mesh is C3D8R cell.
Based on the above theoretical knowledge, variable-thickness CFRP laminates are discretized firstly. The discretization process is shown in Fig. 3. The variable-thickness CFRP laminates with uniform cross-section changes are divided into similar variable-thickness CFRP laminates. When the number of partitioning n is large enough, the maximum thickness 1 t and the minimum thickness 2 t of variable-thickness CFRP laminates will be close enough, and each region is approximately equivalent to constant-thickness CFRP laminates with the cross-section thickness of 1 2 ( )/2 t t + . Variable-thickness CFRP laminates are discretized, which is named according to the different degrees of dispersion. When the differences of the thickness element are 0.4 mm and 0.2 mm, they are named as the discrete 1 and the discrete 2, respectively. On the variable-thickness CFRP laminates with the length of 240 mm, 5 nodes are selected at equal intervals, as shown in Fig. 4, and the strain and displacement values of the five nodes are extracted from the finite element analysis. The strain information is converted into the curvature information, which is processed continuously and then input into the reconstruction algorithm to obtain the reconstructed displacement values, which is shown in Table 1.
As can be seen from Table 1, under the concentrated load, the reconstruction error of the discrete 2 is smaller than that of the discrete 1, but the reconstruction deflection error of both methods are within 8%.  In conclusion, the feasibility of the strain information based reconstruction algorithm is verified, and the deformation of variable-thickness CFRP laminates can be reconstructed more accurately within a small error range. The smaller the thickness difference of discrete elements is, the higher the accuracy of reconstruction will be.

Deformation reconstruction of the CFRP propeller model
The CFRP propeller studied in this paper is composed of three parts: metal hub, metal embedded parts, and CFRP blade. The three-dimensional geometric model of the CFRP propeller is shown in Fig. 5. In Fig. 6, the metal embedded parts and CFRP blade are curing into a whole by molding. The design parameters of the CFRP propeller are shown in Table 2. The CFRP propeller selected in this paper is symmetrically and evenly laid with 11 layers on one side, and the thickness of each layer is 0.   When the reconstruction method is applied to the CFRP propeller, it is also necessary to carry out linear discretization of the propeller according to the geometric shape, thickness, and other characteristics of the propeller. The discretization idea of the CFRP propeller is shown in Fig. 7. First of all, the CFRP propeller can be divided into several discrete reconstruction regions. Then, the geometry with the irregular surface shape is approximately equivalent to the regular constant section geometry. Finally, the deformation of the CFRP propeller structure is reconstructed based on the reconstruction algorithm. In the finite element software, the hub end of the CFRP propeller is set as the fixed constraint, and a concentrated force of 50 N is applied at the blade tip, and the attribute of the divided mesh is C3D8R cell.
According to the finite element analysis results of the CFRP propeller under the concentrated force and the variation trend of blade strain gradient, a curve on the blade is selected, as shown in Fig. 8. Six nodes with an average distribution on the curve are selected to verify the feasibility of the propeller reconstruction algorithm based on the strain information. The nodes are named successively starting from the blade root. In the verification of the deformation reconstruction algorithm for variable-thickness CFRP laminates, it is concluded that when the thickness difference of the discrete elements is 0.4 mm, the reconstruction error can be controlled within 8%. In the finite element software, the elements where these six points are located are divided finely to guarantee the maximum thickness difference of the discrete elements, less than 0.4 mm. The strain information of each element along the X and Y directions is extracted and the equivalent thickness of the element is calculated. The strain information of the elements where the 6 points are located is converted into curvature information and then processed continuously. Finally, the reconstruction deformation of the CFRP propeller is obtained based on the reconstruction algorithm. Table 3 shows the reconstruction deformation error  It can be seen from the above results that the reconstruction error of each node gradually increases as the distance from the blade root increases. This is because the reconstruction algorithm is based on the coordinates of the first point to calculate the coordinates of the next point, which causes the accumulation of errors. However, the reconstruction relative error of each node is within 10%, which indicates that the reconstruction algorithm can reconstruct the deformation of the CFRP propeller.

Experimental system construction
The FBG calibration experiment precedes the propeller loading experiment. As shown in Fig. 9, a strain gauge is pasted next to the FBG sensor, and the strain gauge data are used as a reference value for calibration. The FBG used in this project is produced by Shenzhen Changge Photoelectric Co., LTD. The bandwidth is less than 0.5 nm, the side-mode suppression ratio is greater than 12 dB, the reflectivity is greater than 80%, the grating length is 5 mm, and the type of optical fiber is Acrylate SWF-28e.
According to the principle of the deformation reconstruction algorithm of the CFRP propeller based on the strain information, the more measured points on the reconstruction curve are, the better the reconstruction effect is. However, considering the size of the blades and other constraints, three lateral representative curves are selected, including 11 measuring points. In the spatial curve reconstruction theory based on the curvature information, it is necessary to obtain the orthogonal curvature information in two orthogonal directions of each measuring point. Therefore, two mutually perpendicular FBG sensors are arranged on the selected 11 measuring points. The FBG sensor pasted along the leading edge and trailing edge directions is defined as the X direction, and the other direction is defined as the Y According to the established FBG sensor network, the FBG sensor is pasted on the corresponding measuring points on the CFRP propeller. The CFRP propeller with the FBG sensor pasted is shown in Fig. 12.
The cantilevered load condition is used here because it resembles the typical loading state of a propeller, and it provides a good measure for assessing structural responses such as the bend-twist character. According to the existing conditions of the laboratory, the following experimental system is designed in this paper. The schematic diagram of the system is shown in Fig. 13. The experimental system is composed of the CFRP propeller, universal beam, FBG sensing system, loading device, three-dimensional coordinate scanning system, etc. In this experimental system, the propeller is fixed on the universal beam with bolts and nuts, and the universal beam is fixed on the test bench with G clamps to form a stable support. A heavy object is suspended at the blade tip of the propeller to make the propeller deform. The FBG sensor network is arranged on the propeller. The wavelength signal of the FBG sensor is obtained by the FBG wavelength demodulator (in this experiment, the wavelength demodulator is a static demodulator, and the sampling frequency is 1 HZ), so as to obtain the strain value, which is input into the CFRP propeller reconstruction algorithm to obtain the reconstructed value. Meanwhile, the propeller blades before and after loading are scanned by three coordinate scanners, respectively, and the measured deformation of the propeller is obtained. Finally, the actual deformation of the propeller is compared with the reconstruction deformation to verify the feasibility and accuracy of the deformation reconstruction method in this paper. According to the system block diagram shown in Fig. 13, the experimental system is built, as shown in Fig. 14.
Before formally loading the CFRP propeller blade, the reference model of the propeller blade before deformation is obtained by scanning with a three coordinate scanners, and the signals of 22 FBG sensors are normal in the relevant software. The 2 kg weight is hung at the tip of the CFRP propeller blade and the FBG sensor network collects strain data for 15 seconds after the system stabilized. At the same time, the CFRP propeller blade is scanned in all directions by three coordinate scanners, and the deformed solid 3-dimensional model is generated in the corresponding computer software. In the corresponding software, the deformed CFRP propeller blade can be viewed. After the data collection of the 2 kg weight loading are completed, 3 kg and 5 kg weights are used for loading, and the experimental process of loading is the same as that of 2 kg.

Experimental results and analysis
At the end of the experiment, the wavelength shifts of the 22 FBG sensors at 11 measuring points on the CFRP propeller under different loads can be obtained. The obtained strain information of measuring points is input into the deformation reconstruction algorithm of the CFRP propeller to reconstruct the deformation, then the reconstructed deformation is compared with the measured deformation obtained from scanning, and the error is calculated. The results are shown in Tables 4, 5, and 6.   In order to observe the reconstruction effect under different loads more visually, the measured and reconstructed displacements of the CFRP propeller under different concentrated loads are shown in Fig. 15. Comparison of relative errors between measured displacements and reconstructed displacements of CFRP propellers under different concentrated loads is shown in Fig. 16. From the above results, they can be seen: (1) The reconstruction error of each node gradually increases as the distance from the blade root increases. This is because the reconstruction algorithm is based on the coordinates of the first point to calculate the coordinates of the next point, which causes the accumulation of errors. However, the reconstructed displacements of each measuring point on the CFRP propeller blade under different concentrated loads are in good agreement with the measured displacements, and the maximum relative errors is within 15%.
(2) In this experiment, three deformation curves of the CFRP propeller surface are reconstructed. It can be found that the data of five measuring points on the first curve are compared with the data of three measuring points on the second and third curves under the loads of 2 kg, 3 kg, and 5 kg weights. It can be seen from Fig. 16 that the reconstruction error of the first curve is the smallest on the whole, indicating that the more data of strain measuring points are, the higher accuracy of the deformation reconstruction algorithm of the CFRP propeller based on the strain information is.

Conclusions
This paper studies a CFRP propeller deformation reconstruction algorithm based on the curvature information and verifies the feasibility of the algorithm through simulation and experiments. Then, the FBG sensor technology is applied to the deformation detection of the CFRP propeller. According to the surface characteristics of the CFRP propeller, the FBG sensor network is constructed to realize the reconstruction of the deformation curve on the propeller.
(1) In the finite element software, variablethickness CFRP laminates are verified with the accuracy of the deformation curve reconstruction based on the simulated strain information. Moreover, the influence of different dispersion degrees on the reconstruction effect is explored. When the thickness difference of discrete elements is 0.4 mm, the deformation reconstruction relative error is within 8%.
(2) A representative curve is selected on the blade, and 6 nodes on the curve are uniformly selected for simulation verification. The results show that the relative error is within 10%, and the feasibility of the deformation reconstruction method on the propeller curve based on the structural curvature method is verified.
(3) The propeller strain-deformation reconstruction experimental system based on the FBG sensing network is constructed, and the concentrated force load is applied at the tip of the CFRP propeller and reconstructs the blade deformation through the strain information obtained by the FBG sensor network. Compared with the actual deformation of the blade measured by the three coordinate scanners, the reconstruction relative error is within 15%.

Discussion
Although the current study is limited only to the cantilever beam analysis, it is expected to be used as a reference, and it can be extended towards experiments on manufacturing hydrofoils in the future. Considering the costs, we only use a single FBG sensor instead of deploying multiple FBG sensors on one fiber. If the FBG sensor network is deployed on a propeller operating underwater, the sensor network lead wire needs to be drawn from the hub end of the propeller and connected to the demodulation equipment through the optical fiber slip ring. Figure 12 shows the sensor network constructed under the assumption that the oar is not rotating. In this project, when the blade is loaded with 5 kg weight, the reconstructed deformation at the maximum deformation position of the blade is 0.23 mm, the measured deformation is 0.258 mm, and the reconstructed relative error is within 15%. The propeller in this paper is similar to that in [21]. Figures 10 -12 in [21] show that the deformation error of 0.038 mm has little influence on the performance of the propeller.