Displacement Response Reconstruction of Slender Flexible Structures Based on Cubic Spline Fitting Method

How to reconstruct a dynamic displacement of slender flexible structures is the key technology to develop smart structures and structural health monitoring (SHM), which are beneficial for controlling the structural vibration and protecting the structural safety. In this paper, the displacement reconstruction method based on cubic spline fitting is put forward to reconstruct the dynamic displacement of slender flexible structures without the knowledge of mode-shapes and applied loading. The obtained strains and displacements are compared with the results calculated by ABAQUS to check the method’s validity. It can be found that the proposed method can accurately identify the strains and displacement of slender flexible structures undergoing linear vibrations, nonlinear vibrations, and parametric vibrations. Under the concentrated force, the strains of slender flexible structures will change suddenly along the axial direction. With locally densified measurement points, the present reconstruction method still works well for the strain concentration problem.


Introduction
Slender flexible structures are widely used in various engineering fields. Generally, they have small structural damping and low natural frequencies, which makes them sensitive to the influence of internal and/or external disturbances. For example, the ocean current may cause vortex-induced vibration (VIV) of risers and free-spanning pipelines in offshore engineering (Chaplin et al., 2005). The airflow may give rise to wind-induced vibration of chimneys and stay cables in civil engineering (Le and Caracoglia, 2016;Jing et al., 2017). The flexible attachment of aircraft in aerospace engineering may vibrate easily, which is induced by solar wind, space dust and its own motion (Omidi et al., 2015). Also, flexible manipulators in mechanical engineering can be affected by their elastic deformation and vibrate frequently (Lochan et al., 2016). Mechanical vibrations of structures may lead to excessive wear of bearings, loosening of fasteners, structural and mechanical failures, lower production efficiency and tremendous structural safety risks. The development of smart structures and structural health monitoring (SHM) is in the great interest of controlling the structural vibration and protecting the structural safety. Real-time reconstruction of structural deformation using measured strain data is the key technology for actuation and control of smart structures.
It is a great challenge for directly measuring the dynamic displacement distribution of the slender flexible structures due to the limitations of measuring methods and devices. Setting discrete strain gages at local positions is widely used to monitor the dynamic motion. The displacement distribution of the flexible structure needs to be reconstructed using the discrete measuring strains. In recent years, many investigators have developed several methods for structural displacement reconstruction. Tessler and Spangler (2005) employed an inverse finite element method (iFEM) to reconstruct the three-dimensional deformations in the plate and flat shell structures using discrete strain data. Subsequently, the iFEM methodology was improved by Gherlone et al. (2012) to reconstruct the structural displacement of truss, beam and frame structures, without the knowledge of material properties, applied loading, or damping. Ko et al. (2007) developed displacement theories to calculate the deformed shapes for a variety of large aspect ratio (the ratio of structural length to the dimension of the structural section) flexible structures, such as aircraft wings, wind turbine blades and bridges. The displacement theories are based on the classical beam equations and the deformed shapes of structures are determined by integrating discretely measured strains. Modal analysis approach is also widely used in the displacement reconstruction. Foss and Haugse (1995) applied strain and displacement mode shapes to conduct a transformation from measured strains to displacements of a flat cantilevered plate. Li and Ulsoy (1999) used the modal analysis approach for the high-precision measurement of a beam's oscillatory displacement, which is beneficial to improve the accuracy of the line boring machining.
The slender flexible structures, such as risers, stay cables and transition lines, have extremely large aspect ratios and multi-order natural frequencies. Hence, vibrations of such structures exhibit multi-mode (Chaplin et al., 2005;Song et al., 2011). Modal analysis approach is effective for the displacement reconstruction of this type of slender flexible structures. Assuming that the deviation of the long flexible structures from the vertical is little and the dynamic motion is approximately linear, the deformation of the structure is expressed as a sum of eigenfunctions. The mode weight of each eigenfunction can be acquired using discrete measuring strains, and then the dynamic displacements can be determined. Trim et al. (2005) used the modal analysis approach to reconstruct the VIV response of riser model with an aspect ratio of 1400. Similar displacement reconstruction technology was also employed to identify the VIV displacement of the slender flexible cylinders (Lie and Kaasen, 2006;Gao et al., 2014Gao et al., , 2016Gao et al., , 2017Han et al., 2017;Xu et al., 2018aXu et al., , 2018bXu et al., , 2018c. However, the application of modal analysis approach needs to obey some assumptions and constraints. For example, the slender flexible structures should deviate little from the vertical, and the dynamic motions of the structures should be approximately linear. In many engineering fields, large deformation and strong nonlinear behaviors usually exist in the dynamic response of slender flexible structures. Besides, the basic thought of the modal analysis approach is to transform the time-space varying displacement into the superposition of multiple modes. Thus, the mode shapes of the structure must be obtained before applying the modal analysis approach. However, the mode shapes of the slender flexible structure with complicated boundary cannot be presented by simple shape functions and they need to be identified by the finite element method. While the natural characteristics of a strong nonlinear system vary from the structural response and it is difficult to acquire the mode shapes.
Nowadays, modal analysis approach is widely used to reconstruct the oscillation displacement of slender flexible structures. There are many constraints and limitations for the application of modal analysis approach. For example, the mode shapes of structure should be known and the modal analysis approach is not available for strong nonlinear systems. Hence, the main aim of this paper is to propose a robust approach for displacement reconstruction of slender flexible structures, namely the cubic spline fitting method. The cubic spline fitting method can be employed to identify the dynamic response of slender flexible structures without knowing the structural mode shapes and applied loading. Besides, the cubic spline fitting method is available to the response reconstruction of slender flexible structures with complicated boundaries and strong nonlinear factors. According to the cubic spline fitting method, the strain distribution function of the slender flexible structures can be firstly obtained using discrete strain measurements based on the cubic spline fitting. Then, the displacement function of the structures can be acquired by integrating the strain function double times. The effectiveness of the cubic spline fitting method is checked by the finite element models and experimental data from model tests.

Theoretical background
Cubic spline fitting is widely used for interpolation of numeric data to obtain a smooth continuous function in many engineering fields (Cao et al., 2017;Li et al., 2018). For cubic spline, each piece is a third-degree polynomial function. The resulting spline will be continuous and its first and second derivatives are still continuous. The strain function of flexible structures is identified firstly using cubic spline fitting method.
The boundary points and the measurement points are seen as the interpolation nodes, which divide the strain function into several sections. The strain function can be obtained using the strains at the interpolation nodes. The firstorder partial derivative of the strain function to z at the node can be written as (Sauer, 2012): where z is the axial coordination, t donates the time, n is the node number. If p i is known, we can obtain the cubic Hermite spline polynomial in the interval [z i , z i+1 ] shown as: where , the cubic Hermite spline polynomial satisfies the following conditions: ε test (t, z i ) is the strain at the node, and p i can be calculated by using the continuity of the second-order partial derivative of the strain function at nodes. The second-order partial derivative of the strain function in the interval [z i , z i+1 ] can be expressed as: The second-order partial derivative of the strain function in the interval [z i-1 , z i ] can be written as: Hence, Eq. (8) can be simplified as: Eq. (9) contains (n-1) linear equations, but there are (n+1) unknowns (p 0 , p 1 , …, p n ). To determine these unknowns, we need additional boundary conditions. There are four additional boundary conditions in total for cubic spline fitting (Cao et al., 2017).
If the first-order partial derivatives of the strain function at the boundary points are known, boundary condition I can be used.
If the second-order partial derivatives of the strain function at the boundary points are known, boundary condition II is useful.
Boundary condition III: This boundary condition is used for periodic problems.
Boundary condition IV is the not-a-knot boundary condition. If the first three boundary conditions are unknown, we can use the boundary condition IV as a compromise.
According to Eq. (9) and the additional boundary condition, p i is obtained. Hence, the segmented strain function can be reconstructed and expressed as: The relationship between structural strain and displacement can be expressed as (Trim et al., 2005): where R is the distance between the strain gauge and the neutral layer of the structure; is the displacement function; is the second-order partial derivative to z and can be calculated by integrating two times. In the interval , is expressed as: where C i and D i are the integration coefficients, which can be calculated from the first and second derivative continuous condition of displacement at measuring points and the boundary conditions of displacement. According to the first and second partial derivative continuous condition of displacement at measuring points, we can obtain Eq.
Eq. (19) includes (2n-2) linear equations, but the number of unknowns (C i and D i ) is 2n. Hence, two equations are still needed to obtain these unknowns. Taking the boundary conditions of displacement shown as Eq. (20) or Eq. (21) into account, the integration coefficients are determined.
3 Method validation It is not easy to measure the displacement distribution of slender flexible structures directly via experiment. Hence, commercial finite element software (ABAQUS) was employed to verify the reconstruction method based on cubic spline fitting. For the sake of simplification, the dynamic responses of a flexible cylinder undergoing various kinds of loads were simulated by ABAQUS. Then, the strain and displacement functions of the flexible cylinder were reconstructed using discrete strains from strain field obtained by ABAQUS. The validity of the reconstruction method based on cubic spline fitting was checked by comparing the difference between the reconstructed displacement and the calculated results from ABAQUS.
The calculation model was a two-dimensional flexible cylinder with an aspect ratio of 1500, as shown in Fig. 1. The length of the flexible cylinder was 30 m. The interface of the flexible cylinder was a pipe with an outer diameter of 20 mm and a wall thickness of 2 mm. The material density was 7850 kg/m 3 . The Young's modulus was 210 GPa and the Poisson's ratio was 0.3. The 3-node quadratic beam element B22 was employed to disperse the flexible cylinder. The flexible cylinder was divided into 120 units along the axial direction. The flexible cylinder was simply supported at both ends and the axial tension was set as 2000 N. 29 nodes with equal spacing were selected as measurement points (G1-G29). The essential parameters of the flexible cylinder model are presented in Table 1. The dynamic responses of the flexible cylinder undergoing different types of loads were solved by ABAQUS. The structural damping was set as 0, and the explicit solver was employed to calculate the dynamic responses of the flexible cylinder.
Before conducting the dynamic calculation, the natural frequencies of the flexible cylinder were obtained, as listed in Table 2. The theoretical values of the natural frequencies calculated from Eq. (22) are also presented as a comparison.
where j is the mode order. The first ten natural frequencies calculated from ABAQUS are identical to the theoretical values within the permissible range of error, which indicates the validity of the dynamic results of ABAQUS.
3.1 Displacement reconstruction of linear dynamic response Three cases (Cases 1, 2 and 3) were conducted to verify the displacement reconstruction of the flexible cylinder whose dynamic response is linear. In Case 1, the flexible cylinder model experienced the uniformly distributed load in the y-direction, as shown in Eq. (23).
The load Q 1 had five frequency components and the dominant frequency is f 8 , which made sure that multi-order mode vibrations of the flexible cylinder can be excited. Fig. 2 shows the root-mean-square (RMS) of the microstrain of the flexible cylinder in Case 1. The grey ball in Fig. 2a is the RMS of strains calculated directly from ABAQUS, which can be seen as the real strains of the flexible cylinder. The blue line donates the RMS of the reconstructed strains based on cubic spline fitting. To compare the method based on cubic spline fitting and another reconstruction approach, the result of modal analysis approach is also presented by the red dashed line. The modes involved in the calculation are from the 1st to the 30th. The RMS of the microstrains of the flexible cylinder undergoing load Q 1 has eight peaks versus z/L. The RMS of the reconstructed microstrains based on cubic spline fitting and modal analysis approach shows very little discrepancies from the real results. Fig. 2b shows the errors of RMS of the strains between the real values and the reconstructed values. The errors of the reconstructed method based on cubic spline fitting vary from the axial coordinate. The maximum value of the errors  is 2.80%, which is observed at the ends of the flexible cylinder. However, at the most axial position, the errors are smaller than 1%. The RMS of strain reconstructed by modal analysis approach has much smaller errors than the cubic spline fitting method. The maximum error of the modal analysis approach is 0.35%, which is also found at the ends of the flexible cylinder. Fig. 3 shows the RMS of dimensionless displacement of the flexible cylinder in Case 1. The RMS of dimensionless displacement also has 8 peaks versus the axial position and the maximum RMS of the displacement is 1.0D. The RMS of the reconstructed displacements based on the cubic spline fitting and modal analysis approach both agree well with the displacement obtained from ABAQUS. The maximum error of the cubic spline method is 0.24%, while it is 0.39% for the modal analysis method. Obviously, the errors of the modal analysis approach are much larger than those of the cubic spline method at most axial positions. It is interesting that the errors of the strains are opposite, as shown in Fig. 2b. When the modal analysis approach is performed to reconstruct the dynamic response, the mean error of the displacements is 0.14% and the mean error of the strain is 0.03%. It seems that the small error of the strains is enlarged in the process of transformation from strain to displacement. The amplitudes of the strain mode-shapes are proportional to the mode number squared, which means that the low modes are very sensitive to the error of strains (Lie and Kaasen, 2006). Hence, the amplification error of displacements is mainly due to the fact that the errors of strains at low modes are enlarged.
In Case 2, the unevenly distributed load Q 2 acted on the flexible cylinder in y-direction.
Load Q 2 caused the 6th-10th vibration mode of the flexible cylinder to be excited and the 8th-order mode dominated the oscillations. Fig. 4 shows the RMS of microstrain of the flexible cylinder in Case 2. Under the unevenly distributed load Q 2 , the strains of the flexible cylinder can be accurately identified based on cubic spline fitting. The maximum error of the strains is 1.50% and the mean error is 0.23%. However, the error of the modal analysis method is much smaller with a mean value of 0.02%. Fig. 5 presents the RMS of dimensionless displacement of the flexible cylinder in Case 2. The comparison between the reconstructed dimensionless displacements based on cubic spline fitting and the results calculated from ABAQUS is satisfactory. The mean error of the cubic spline fitting method is 0.23%, and the mean error of modal analysis approach is 0.31%.  In Case 3, a concentrated force F 1 was applied to the flexible cylinder at the midpoint in y-direction.
Under the concentrated force F 1 , the stress concentration issue was found at the midpoint of the flexible cylinder. The validity of the reconstruction method based on cubic spline fitting was checked when the strain changes suddenly along the axial direction. Fig. 6 shows the RMS of microstrain of the flexible cylinder in Case 3. Due to the concentrated force, there is a very thin peak near the midpoint of the flexible cylinder. The RMS of the reconstructed strains based on cubic spline fitting and modal analysis approach both show significant deviation from the values of ABAQUS. The error of the cubic spline fitting method reaches 37.94% near the midpoint of the flexible cylinder. The results of the modal analysis approach also have much larger errors and the maximum error is 43.48% near the midpoint. RMS of dimensionless displacement of the flexible cylinder in Case 3 is presented in Fig. 7. The RMSs of reconstructed displacement based on the two methods are distinct from the ABAQUS results. The maximum error of the reconstructed displacements can be 75.29%, which indicates the reconstruction methods based on cubic spline fitting and modal analysis approach failed. The measurement points near the midpoint of the flexible cylinder are too few to catch the sudden variation of the strains, which causes the failure of the reconstruction. Hence, to catch the strain in-formation comprehensively, more measurement points are needed near the midpoint of the flexible cylinder. Three measurement points are evenly added between G14 and G15 and between G15 and G16, respectively. With the six additional measurement points, the reconstructed strains based on cubic spline fitting agree well with the strains from ABAQUS. The mean error of strains decreases remarkably to 0.26%. The reconstructed displacements are extremely consistent with the displacements obtained by ABAQUS. The mean error of the displacements based on cubic spline fitting reduces to 0.13%. The modal analysis approach with six additional measurement points still does not work well. The maximum error of the strains still reaches 29.71% near the midpoint of the flexible cylinder and the errors of the displacements are also at a high level. It can be seen that the cubic spline fitting method takes advantages over the modal analysis approach when the strain suddenly changes along the axial direction.

Displacement reconstruction of parametric vibration
The parametric vibrations of the slender flexible structures are the common issues in many engineering fields. For example, the heave of the floating oil platform in waves may cause the axial tension of marine risers to vary periodically, which induces the parametric vibration of the marine risers. Case 4 was put forward to check the accuracy of the displacement reconstruction of a flexible cylinder undergoing parametric vibrations. In Case 4, a periodic changed axi-  HAN Qing-hua et al. China Ocean Eng., 2019, Vol. 33, No. 2, P. 226-236 231 al tension T v was applied to the flexible cylinder in z-direction. The load in y-direction is Q 2 .
(26) Fig. 8 shows the RMS of microstrain of the flexible cylinder in Case 4. The RMS of the reconstructed strains based on cubic spline fitting is in fair agreement with the results of ABAQUS. The maximum error of the strains is 3.58% and the mean error is 0.54%. In the present study, the flexible cylinder model was simply supported at both ends and boundary condition II with P 0 =P n =0 was used to reconstruct the strain function. As aforementioned, boundary condition IV can be useful if the first three boundary conditions (I, II and III) are unknown. Hence, the boundary condition IV was employed for reconstructing the strains to check the validity. It can be seen from Fig. 8 that the reconstructed strains with not-a-knot boundary show small deviation from those with boundary condition II, except for the ends of the flexible cylinder. Although the error of the strain is large at four points near the ends of the flexible cylinder, the error is identical to that with boundary condition II at almost all axial positions. Fig. 9 shows the RMS of dimensionless displacement of the flexible cylinder in Case 4. The reconstructed displacement with boundary condition II and IV are closely similar to the displacement obtained from ABAQUS. The mean error of the displacement with boundary condition II is 0.2%. The error of the displacement with boundary condition IV can reach 0.89% near the ends and the mean value is 0.23%. Fig. 10 shows time histories and spectra of the dimensionless displacement at the midpoint in Case 4. The reconstructed time histories of the displacement based on boundary condition II and IV are consistent with the results obtained by ABAQUS. As shown in Fig. 10b, the spectra of displacements have more than one peak, which indicates the vibrations of the flexible cylinder have multiple frequencies.
The spectra of the reconstructed displacements show little deviation from those of the result from ABAQUS. The results of Case 4 indicate that the displacement reconstruction method based on cubic spline fitting works well for the flexible cylinder undergoing parametric vibrations. Besides, if boundary condition I, II and III are unknown, boundary condition IV works well and can be used as a compromise.
3.3 Displacement reconstruction of nonlinear dynamic response In engineering practice, the vibrations of slender flexible structures are usually affected by non-negligible nonlinear factors. In Case 5, 11 nonlinear springs were evenly applied to the flexible model to verify the displacement reconstruction regarding the nonlinear dynamic problem, as shown in Fig. 11. The relation between restoring force and spring deformation is expressed as Eq. (27). The load in ydirection is Q 2 .   RMS of dimensionless displacement of the flexible cylinder in Case 5 is shown in Fig. 13. The RMS of the reconstructed displacements is consistent with the results calculated by ABAQUS. The mean error of cubic spline fitting with boundary condition II is 0.20%, while the mean error of cubic spline fitting with boundary condition IV is 0.23%. Fig. 14 presents time histories and spectra of the dimensionless displacement at the midpoint in Case 5. The time histories of displacements show strong harmonic behaviours due to the nonlinear factor. The reconstructed displacements agree well with those calculated from ABAQUS.    Besides, the spectra of the reconstructed displacements are similar to the results of ABAQUS. It can be concluded that the dynamic displacement of the flexible cylinder undergoing nonlinear vibrations can be exactly reconstructed by the method base on cubic spline fitting.

Method application
To further verify the validity of the displacement reconstruction method based on cubic spline fitting, this displacement reconstruction approach was applied to the response reconstruction of the flexible cylinder undergoing vortex-induced vibration. Many researchers have conducted model tests to investigate the vortex-induced vibration of the flexible cylinder. Due to the fact that the original experimental data obtained by other researchers is difficult to be acquired, the experimental data from our own model tests was used to check the applicability of the cubic spline fitting method. We had carried out model tests to investigate the vortex-induced vibrations of a yawed flexible cylinder at different yaw angles (a), and some experimental data was published by Xu et al. (2018c).
In the model tests, the flexible cylinder model was a coaxial composite tube with a length of 5.6 m. The inner part was made of a copper pipe with an outer diameter of 8.0 mm and a wall thickness of 1.0 mm. The outer part was made of a silicone tube with an outer diameter of 16 mm and a wall thickness of 4.0 mm. The mass per unit length was 0.3821 kg/m. The bending stiffness of the model was totally supplied by the inner copper pipe, with a value of   17.45 Nm 2 . The flexible cylinder model was simply supported at both ends. The axial tension of 300 N was applied on one end of the model. Seven measurement points were evenly set on the axial direction of the model. Two pairs of strain gages were attached to the outer surface of the copper pipe at each measurement point to acquire the vibration strains in cross-flow and in-line directions. The flexible cylinder model was submerged in water and suffered uniform flow. The velocity of uniform flow varied from 0.05 m/s to 1.00 m/s with an interval of 0.05 m/s.
The VIV of the displacement responses of the flexible cylinder model were reconstructed by the modal analysis approach of Xu et al. (2018c). In the current research, the displacement responses of the flexible cylinder model during the tests were also reconstructed based on the cubic spline fitting method to do a further validation. The reconstructed displacement results of the flexible cylinder model at certain condition (the yaw angle a is 0°, and the uniform flow velocity U is 0.50 m/s) are presented as an example. Fig. 15 shows the dimensionless displacement of the flexible cylinder model undergoing vortex-induced vibration.
The RMS values of the dimensionless displacement reconstructed by the cubic spline fitting method and the modal analysis approach agree quite well with each other. The errors between the two reconstruction methods are below 0.5%. For further comparison, time histories and spectra of the displacement of the flexible cylinder model at z=0.25L are presented in Fig. 16. It is observed that the time histories and spectra of the reconstructed displacement based on the two reconstruction methods are also close to each other. It can be concluded that the displacement reconstruction method based on cubic spline fitting is effective to the practical engineering issues.

Conclusions
The displacement reconstruction method based on cubic spline fitting was proposed to identify the dynamic deformation of slender flexible structures. The strains and displacements of the flexible cylinder undergoing linear vibrations, nonlinear vibrations and parametric vibrations were reconstructed using the discrete strains calculated by ABAQUS. The validity of the reconstruction method was checked by comparing the reconstructed results and the results acquired from ABAQUS. Some conclusions are drawn as follows.
(1) The reconstruction method in this paper is effective for obtaining the dynamic deformation of slender flexible structures without the knowledge of mode-shapes and applied loading. The errors of the reconstructed strains and displacements are acceptable.
(2) If the strain changes suddenly along the axial direction of the slender flexible structures, more measurement points near the break point are needed to catch the strain information comprehensively. With locally densified measurement points, the strains and displacements of slender  HAN Qing-hua et al. China Ocean Eng., 2019, Vol. 33, No. 2, P. 226-236 235 flexible structures under concentrated force can be reconstructed accurately based on cubic spline fitting.
(3) Additional boundary conditions are acquired to reconstruct the strain function based on cubic spline fitting. If the boundary conditions I, II and III are unknown, boundary condition IV (not-a-knot condition) can be used as a compromise. The errors of the reconstruction method with boundary condition IV are still small at the most axial position of the slender flexible structures.