Expansion of real-time operating data from limited measurements to obtain full-field displacement data has been performed for structures in air. This approach has shown great success, and its main advantage is that the applied forces do not need to be identified. However, there are applications where structures may be immersed in water and the full-field real-time response may be needed for design and structural health assessments. This paper presents the results of a structure submersed in water to identify full-field response using only a handful of measured data. The same approach is used to extract the full-field displacements, and the results are compared to the actual full-field measured response. The advantage of this approach is that the force does not need to be identified and, most importantly, the damping and fluid–structure interaction does not need to be identified in order to perform the expansion. The results show excellent agreement with the measured data.

## Introduction

Dynamic strain response of the structure is of great concern for health monitoring of the structures under operating loads [13]. The inspection/testing/monitoring of the structure is a concern for its safety and potential damage to the environment [46]. One approach of getting the response is by identifying the actual loads and structural conditions, as well as building an appropriate model. This is the forward approach of predicting the dynamic response. However, the input force and model parameters are approximate and much effort is expended to make them better represent the actual case [79]. The actual characteristics are not always well known or understood, and therefore, the estimated response and dynamic strain or stress is not well known [10,11]. These misunderstood loads and boundary condition can cause a displacement or strain distribution that is different from those used for response simulation and thus gives an inaccurate depiction of how the structure behaves and performs.

Rather than estimating the input force, an alternative approach to identify the full-field displacement or strain obtained from limited sets of measurements is used. This approach utilizes the response at limited sets of measurement locations together with orthogonal expansion functions to identify the full-field displacement, which is then used in conjunction with the finite element back substitution process to obtain full-field dynamic strain. The great advantage of this process is the actual load and complicated boundary condition (fluid–structure coupling interaction) is not necessarily known. There are some approaches that focus on predicting the full-field response from limited sets of measurement. Avitabile and Pingle [12,13] used noncontact measurement technique, with digital image correlation and dynamic photogrammetry, as well as full-field techniques to measure the surface response. Both the full-field displacement and strain of an analytical box-beam model and a base-upright structure was determined in that work. Baqersad et al. [1417] used a three-dimensional point tracking technique in conjunction with a modal expansion method to obtain the full-field dynamic displacements and strain of rotating and nonrotating wind turbine blades from the limited set of optical measurements. Chierichetti and Ruzzene [18] employed an iterative technique based on modal decomposition that maps the displacement field of the vibrating structure by using experimental data in conjunction with the numerical model of the test structure.

Previous work using these expansion techniques has shown great success with structures in air. However, to the author's knowledge, there is no study that has been done using this expansion technique applied for structures underwater (Sedlar et al. [19] built the reduced finite element model (FEM) using the same technique for the beam submerged in water and used the added mass to describe the fluid–structure interaction). The purpose of this work is to study this expansion technique for an underwater application. The fluid–structure coupling and interaction is more difficult than the analysis in an air medium. Fluid–structure coupling interaction is another approach to address this problem that shows promise but has other difficulties and assumptions that need to be determined [2022]. Instead, this paper explores using only the response of a handful of measured points with an expansion procedure utilizing the modal shapes of a finite element model to address this problem and obtain full-field dynamic displacement as well as full-field dynamic strain.

This work only involves the vibrating structure with relatively small amplitude. The availability of this response expansion technique is only discussed when the structure is vibrating in the linear regime. The effectiveness of this method addressing the large amplitude vibration response is beyond the scope of this work and is not addressed in this paper.

## Overview of Response Approach and Expansion Process

Generally, the finite element modeling process involves the discretization and assembly of the mass and stiffness matrices followed by the application of boundary conditions and loads which is then solved to obtain dynamic displacement and dynamic strain [2224]. In the case of a large wind turbine blade, consideration needs to be given to the forces that result from the turbine blade/air interaction due to the large size of the very flexible wind turbine blade. The determination of these forces would involve some fluid–structure coupling interactions. While an in-water application will have different coupling effects, such as water has mass and force loading effect on the underwater structure such that it can change the frequency and damping of the structure, these would need to be determined from a fluid–structure coupling analysis. The general overview of the finite element modeling process is shown schematically in Fig. 1 for a large wind turbine blade in air (as an example of one of many possible applications). The finite element model needs to be accurately identify the fluid–structure interaction effects in order to get a proper solution of dynamic response.

For the approach considered in this work using the expansion methodology employed, the difference is that the actual application of the loads and boundary conditions and the solution of the system set of equations are not specifically performed. Rather, the sparse set of displacements, measured from an actual operating event, is used with a set of orthogonal expansion functions to obtain the full-field displacement solution. This displacement solution is then used with the normal recovery of the stress–strain solution in the finite element modeling process. This procedure is shown schematically in Fig. 2 where the limited set of measurement degrees-of-freedom (DOF) are used with expansion processes to obtain the full-field displacement for the system. With this approach, the finite element modeling solution process is intercepted (as seen in the hatched out region for loads and boundary conditions and the solution for displacement) and replaced with the expansion of the limited set of measured degrees-of-freedom; these limited set of measured degrees-of-freedom can be obtained from an in-air application or an in-water application (as studied in this paper).

The actual expansion process has been presented in Refs. [12] and [25]. The basic theoretical approach is summarized below and utilizes concepts from model reduction and model expansion as the underlying methodology for the expansion approach used for this work. This is followed by a description of the test article and cases studied for this paper.

## Theoretical Background

The underlying procedure utilizes model reduction and model expansion of real-time operating data. This creates a mapping between the limited set of measured degrees-of-freedom and the full-field space of the finite element model. Several model reduction methods such as Guyan, improved reduced system, and system equivalent reduction–expansion process (SEREP) have commonly been used for expansion of data at limited points [2628]. For this work, SEREP [29] is employed to expand measured displacements of these measured points (reduced model) to provide a full-field displacement solution (full model). Using the SEREP process, the expansion matrix can be determined from the eigensolution of the full space finite element model.

The SEREP technique has been presented in earlier work cited in the references; only summarizing equations are list below. The relationship between the full set of DOF and a reduced set of degree-of-freedom is given as
${Xn}={XaXd}=[T]{Xa}$
(1)

where subscript n denotes the full model, a denotes the reduced DOF, and d denotes the omitted DOF. In this study, ${Xa}$ is the operating displacement response at the test points and ${Xn}$ is the full-field expanded response.

The transformation matrix $[T]$ can project the full mass and stiffness matrices to a smaller size. The reduced matrices can be written as
$[Ma]=[T]T[Mn][T]$
(2)

$[Ka]=[T]T[Kn][T]$
(3)
The SEREP procedure produces reduced matrices for mass and stiffness that yield the exact frequencies and mode shapes as those obtained from the eigensolution of the full size matrix. The SEREP transformation matrix $[T]$, containing the mapping information between the full model and reduced model, can be formulated as
$[T]=[Un][Ua]g$
(4)

where $[Un]$ is the modal vector matrix containing m columns, i.e., m modes. $[Ua]g$ is the generalized inverse matrix of $[Ua]$ which is a rows submatrix of $[Un]$. The a rows are selected to correspond with the node number of the test points in the full model.

Equation (1) can be used to expand the measured operating data from several measured points to the full field and can be written as
$[ERTOn]=[T][RTOa]$
(5)

where RTO indicates real-time operating data and ERTO indicates expanded real-time operating data with the a and n subscripts as defined previously.

The procedure is shown schematically in Fig. 3 and basically can be decomposed into three steps. (The schematic in Fig. 3 is specifically related to the particular structure used in this study.) For this study a known force is applied to show the process and validate the technique; in a real application, the force would be unknown and unmeasured and is not needed for the process—only the measured response at limited points is required.

1. (1)

An arbitrary force pulse is generated to excite a specific frequency range of interest and response measured.

2. (2)

The expansion matrix is used to expand measured response to the full field of the finite element model.

3. (3)

The full-field displacement is applied to the finite element model to obtain full-field strain at all time steps.

## Structure and Model Description

To present the method, a simple fin-like structure is involved in this study, as shown in Fig. 4(a). The structure is assembled from three aluminum substructures—a rectangular plate ($8×2×0.25$ in or $203.2×50.8×6.35$ mm) and two identical blocks ($4×2×3$ in or $101.6×50.8×76.2$ mm). The two blocks are held together with the plate by bolts. The structure has four soft flexible supports attached at the corners of the base to allow for a free–free condition.

The FEM of this structure, shown in Fig. 4(b), is created using FEMAP. The three substructures are modeled as one entity and meshed by cubic solid elements. The boundary condition is modeled as linear soft spring with one end attached to the block and the other fixed to ground and simulates the free–free condition due to the softness of the support. The eigensolution of the model is performed in Nastran to get the mode shapes. The first six modes are essentially “rigid body modes” due to the soft flexible boundary condition; the rest of the modes are flexible body modes. The frequency and flexible shapes of the model are shown in Figs. 4(c) and 4(d), respectively. The first three flexible modes are first bending mode, first torsional mode, and second bending mode.

## Test Cases Studied

The structure is shown in several different configurations as seen in Fig. 5. Figure 5(a) shows the structure mounted in an upright orientation, in air, on an optical table, resting on its soft mount arrangement; the structure is seen in the middle portion of Fig. 5(a) with a shaker mounted as shown in the lower left and the three lasers used to measure the response are shown in the upper right. Figure 5(b) shows the structure mounted in an inverted orientation supported by wooden dowel rods suspending the structure in an empty fish tank (on the left) and filled with water (on the right). Figure 5(c) shows the structure with the strain gage mounted at the base of the cantilevered fin.

Before any dynamic response studies were performed, experimental modal tests were performed both in air and in water to obtain preliminary modal information; the test configuration used is shown in Fig. 5(b). The results are presented in Table 1 and are included to show the differences between the in-air and in-water modal results for reference.

Three separate test configurations were studied as part of this work to validate the expansion process in air and submersed in water. These cases were as follows:

Case 1: The structure mounted upright in air on the optical table to validate earlier work performed in Refs. [12], [15], and [25] (these results are needed to substantiate the results in water and earlier work done by others on different structures)

Case 2: The structure mounted in an inverted orientation suspended by dowel rods into an empty fish tank (these results are needed to confirm the measurement setup was not substantially different than case 1)

Case 3: The structure mounted in an inverted orientation suspended by dowel rods into a fish tank filled with water

In all cases, the structure was excited by a force pulse to excite the lower flexible modes of the structure. The shaker was attached to the base (on the large block) with an impedance head to measure force and acceleration; the shaker was offset from the centerline to cause bending and torsion modes to be excited. The force pulse used is shown in Fig. 6. The upper-left plot shows the pulse in time domain and the upper-right plot shows its frequency spectrum. Plot in the bottom gives an expanded view of the pulse in time domain. This is the actual measured force pulse used to excite the structure.

The force used to excite the structure was intentionally designed to only excite the three lower order flexible modes for the demonstration case shown here. As such, only three measured degrees-of-freedom are necessary to form the SEREP transformation matrix. The three points were selected to form a full ranked inversion matrix for the SEREP process and be sufficient to describe the three modes excited by the pulse. These three points are shown in Fig. 7 on the physical structure as well as on the finite element structure with corresponding node points identified. (It is important to state that for this study only three modes were necessary. If a wider frequency range with more modes were activated, then additional measurements and modes would be needed. This technique is not restricted but for the study here, only three modes were necessary to demonstrate the technique.)

The response was measured with three separate lasers simultaneously to capture the velocity time response. This response was transformed to the frequency domain, converted to a displacement spectrum and inverse transformed to obtain a displacement time response; some filtering of the rigid body modes that were not important to the dynamic strain was also performed. Also attached on the structure was an ICP dynamic strain gage that was used for validation of the expanded dynamic strain results.

For the testing of the structure in the fish tank (with and without water), the structure was supported (hung) by wooden dowel rods to support the structure from the top of the tank. Both configurations are shown in Fig. 8.

With these preliminary statements for the measurements made, excitation force selected, position of the measurement points determined, and the specific test cases studied are now addressed.

Case 1: Upright structure on optical table

The results of this case were performed to reconfirm the results of studies performed using earlier structures. This was to assure that the results behaved as expected from earlier work. These results are essentially the same as those results in case 2 and are not presented here to avoid redundancy and for brevity. Case 2 is set up and oriented exactly the same as case 3 and it is best to show and compare those results.

Case 2: Structure in fish tank with no water

The structure is placed in an inverted orientation and supported with dowel rods in a fish tank, but with no water in the tank. The wooden dowel rods provide a very soft support which simulates a free–free boundary condition when compared to the flexible modes of the structure. The test excitation and measurement locations are all the same as described previously. The lasers are positioned such that they measure the structure's response through the plexiglass side panel of the fish tank and are shown.

Using the methodology of the expansion process described, the structure's response was measured due to the force pulse. The laser velocity at the three measured locations was processed to obtain the displacement response. The force was also used to compute the response for the existing finite element model. The results of the measured displacement and the calculated finite element model response are shown and overlaid in Fig. 9 for validation purposes. The responses are organized as follows.

The left column corresponds to the top laser head point which is halfway down along the length of the fin, the middle column corresponds to one of the top corners of the structure whereas the right column corresponds to the other corner of the structure. Now the top row of the figure contains the full time response and the middle row contains an expanded view of that time response to clearly see the accuracy of the results; the bottom row contains the frequency spectrum of the response for reference.

The red curve (light curve in print) is related to the measured displacement by laser vibrometer and the blue curve (dark curve in print) is related to the calculated result from the finite element model. The FE model is solved in modal space. The measured frequency, measured damping, and mode shapes of FE model is used to form the modal space equation. Then the modal solution is converted into physical space to obtain the simulated full-field displacement.

Because the response has both rigid body and flexible motion, the response is dominated by the rigid body modes but these do not have significant contribution to the dynamic strain of the flexible modes. Therefore, a high-pass filter (Butterworth high-pass filter, cutoff frequency 300 Hz for the data in air and 200 Hz for the data in water) was used to filter the low frequency displacement and to concentrate on the flexible dynamic behavior. While this does have a slight effect on the early part of the transient, the filter was applied to all signals to provide a proper comparison.

Overall, the results in Fig. 9 are considered to be very good.

The measured displacement was then expanded to the full space of the finite element model and used in abaqus to obtain full-field strain for the structure. Because only the flexible modes are of interest in the dynamic response, the transformation matrix only utilized the flexible body modes of the system. The expanded displacement is then imported into abaqus, and the full-field strain of the structure can be viewed.

To verify the correctness of the expansion technique, the strain gage mounted on the structure was compared to the abaqus results. The comparison between the measured strain and the expanded strain from measured displacement of three points is shown in Fig. 10; the red curve is related to the measured strain and the black curve (dark curve in print) is related to the model prediction using the expanded displacements. Note that the expanded strain is not located near any of the measurement points. There is a slight effect of the high-pass filter that is seen at the beginning of the transient response; however, the comparison is very good overall.

Case 3: Structure in fish tank with water

The methodology is proven to be accurate when the structure is in air. The same process is performed with the structure in water using the same steps as case 2. The major difference is the fluid–structure interaction due to the water that will have a direct effect on the response of the structure. In addition, the laser response is now obtained with the lasers shooting through the plexiglass and water with the structure submerged. The structure is shown in Fig. 8(b) submersed in water. The displacement response of the structure is shown in Fig. 11; the same notation in case 2 for laser heads applies here. The upper row shows the measured displacement in time domain and the bottom row shows displacement spectrum. However, there is no comparison to the finite element model in this case because no model exists for comparison. All the same parameters and processing were used for this case.

After the displacement in water is measured, the full-field displacement of the structure can be predicted using the same method with the data measured without water. It should be noted that in this process, the transformation matrix which maps the three measured points to the full-field displacement is performed using the transformation matrix which contains the mode shapes of the finite element model; this does not include any effects of the fluid–structure interaction that might exist.

The expanded displacement of the full field is imported into abaqus, and the full-field strain can be viewed. The measured strain from the ICP dynamic strain gage is compared to the abaqus predicted strain from the expanded displacement data and is shown in Fig. 12. Again, there is a slight phasing due to the high-pass filter effects. Overall this is a very good comparison.

The response of the structure is very different in air and in water as seen in Fig. 13. At first glance the strain in water is higher than the strain in air but then more rapidly decays to zero. This is likely due to the effects of the water further loading the structure and increasing the curvature thereby increasing the strain. Figure 14 shows the curvature difference in air (left) and in water (right) and clearly shows the increased curvature and strain.

### Observations.

The expansion of limited sets of measured data to obtain full-field displacement and full-field strain has been performed successfully in prior studies [12,13]. The full-field displacement as well as the full-field strain can be identified for transient operating conditions. These earlier studies have involved linear structures in normal operating conditions such as large wind turbine blades [14]. The expansion has been shown to obtain very accurate results even with blades in a rotating condition [1517]. This expansion has also been shown to be very good for certain nonlinear applications as well [30].

However, until this study, there has not been any effort to understand if this methodology would work in an underwater environment. The cases presented here clearly show promise for this technique to work equally well without knowledge of the applied force, with imprecise boundary conditions, and now without any knowledge of the forces resulting from fluid–structure interaction.

## Conclusion

The expansion of limited sets of measured data to obtain full-field displacement and full-field strain has been performed successfully for a structure in a submersed water condition. The method of predicting the full-field dynamic strain from expanding measured displacement response of several points in water is shown to be a successful technique. The advantage of this technique is that there is no need to know the input force on the structure or the fluid–structure interaction; only the response of several measured points and the mode shapes extracted from the finite element model are needed to predict the response of the model at every point.

Using this expansion methodology, there is no need to measure or know the applied force and there is no need to understand the fluid–structure interaction (and resulting forces). The measured response at limited points used in conjunction with the orthogonal expansion obtained from the finite element model is sufficient to form the prediction. Results shown in this work confirm that this technique can provide very useful results without knowledge of the fluid–structure interaction effects or the actual applied forces on the structure. The predicted strain response in water using limited measurement points produced very good results; these results provide full-field dynamic transient strain from limited measurements in water.

## Acknowledgment

Some of the work presented herein was partially funded by NSF Civil, Mechanical and Manufacturing Innovation (CMMI) project entitled “Collaborative Research: Enabling Noncontact Structural Dynamic Identification with Focused Ultrasound Radiation Force” and by Department of the Air Force AFRL/RWK “Nonstationary System State Identification using Complex Polynomial Representations.” The authors are grateful for the support obtained. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the particular funding agency.

## Funding Data

• Air Force Research Laboratory (Grant No. FA8651-16-2-0006).

• Division of Civil, Mechanical and Manufacturing Innovation (Grant No. 1266019).

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