Photogrammetry-based structural damage detection by tracking a visible laser line

Abstract

Research works on photogrammetry have received tremendous attention in the past few decades. One advantage of photogrammetry is that it can measure displacement and deformation of a structure in a fully non-contact, full-field manner. As a non-destructive evaluation method, photogrammetry can be used to detect structural damage by identifying local anomalies in measured deformation of a structure. Numerous methods have been proposed to measure deformations by tracking exterior features of structures, assuming that the features can be consistently identified and tracked on sequences of digital images captured by cameras. Such feature-tracking methods can fail if the features do not exist on captured images. One feasible solution to the potential failure is to artificially add exterior features to structures. However, painting and mounting such features can introduce unwanted permanent surficial modifications, mass loads, and stiffness changes to structures. In this article, a photogrammetry-based structural damage detection method is developed, where a visible laser line is projected to a surface of a structure, serving as an exterior feature to be tracked; the projected laser line is massless and its existence is temporary. A laser-line-tracking technique is proposed to track the projected laser line on captured digital images. Modal parameters of a target line corresponding to the projected laser line can be estimated by conducting experimental modal analysis. By identifying anomalies in curvature mode shapes of the target line and mapping the anomalies to the projected laser line, structural damage can be detected with identified positions and sizes. An experimental investigation of the damage detection method was conducted on a damaged beam. Modal parameters of a target line corresponding to a projected laser line were estimated, which compared well with those from a finite element model of the damaged beam. Experimental damage detection results were validated by numerical ones from the finite element model.

Keywords

Structural damage detection photogrammetry vision-based method laser-line-tracking technique experimental modal analysis

Introduction

Vibration-based damage detection is one application of structural dynamics. An assumption of the application is that changes in structural properties of a structure, such as mass and stiffness, due to occurrence of damage can cause changes in its modal parameters, including natural frequencies, modal damping ratios and mode shapes. Inversely, one can detect the damage by identifying and processing the changes in modal parameters. In practice, workability of a vibration-based damage detection method largely depends on accuracy of modal parameter estimation. Recent advances of camera and computer technologies have benefited the usage of photogrammetry for structural dynamics and its application to vibration-based damage detection. By tracking movements of exterior features of a structure that appear on digital images captured by cameras, its full-field displacement and deformation can be measured in a fully non-contact manner, and its modal parameters can be estimated for damage detection.

In the field of experimental structural dynamics, digital image correlation (DIC) techniques have been used to measure vibration and estimate modal parameters;^1–6 they have also been used for structural damage detection. Ghorbani et al.⁷ generated detailed cracked maps of a full-scale masonry wall based on its surface deformation fields measured using a three-dimensional DIC technique. Seguel and Meruane⁸ identified debonding in an aluminum honeycomb sandwich panel based on its full-field vibration shapes measured using a three-dimensional DIC system. Before applying a DIC technique, a surface of a structure is usually prepared by painting a high-contrast speckle pattern so that its digital images captured at different instants can be compared to track deformation of the surface. When certain locations of a structure instead of a surface are of interest, one can attach visible targets to the structure and displacement of the locations can be obtained by tracking the attached targets.^9–11 Lecompte et al.¹² developed a point-tracking technique to measure static deformation of a cracked concrete beam and detect its cracks, where infrared light-emitting diodes (LEDs) were attached to the beam and served as targets. Compared with results from a DIC system, the density of displacement measured using the point-tracking technique was low as it depended on the number of attached LEDs. Wahbeh et al.¹³ mounted LEDs to a structure and measured its two-dimensional vibration. The LEDs that served as targets were tracked by locating their centers on each captured digital image. Rucka and Wilde¹⁴ attached circular points to a lateral surface of a cracked beam that underwent static deformation, which was measured by tracking the attached points on captured digital images. Wavelet transforms of the measured deformation were used to detect the crack of the beam. Feng and Feng¹¹ tracked vibration of multiple black dots attached to a lateral surface of a beam and the tracked vibration compared well with those measured using laser displacement sensors and accelerometers.

The DIC and target-tracking techniques require visible exterior features to be painted on or mounted to a structure, which can introduce unwanted permanent surficial modifications, mass loads and stiffness changes. When such features do not exist or are difficult to add, one can apply target-less techniques, such as edge detection and motion magnification techniques,^15–17 to track displacement and deformation of a structure. Poudel et al.¹⁸ extracted mode shapes of a cracked beam from captured digital images using an edge detection technique, and wavelet transforms of differences between the extracted mode shapes and referenced ones were used to detect the cracks. Pakrashi et al.¹⁹ applied an edge detection technique called Sobel technique to convert digital images to black-and-white binary ones, and a pattern recognition process was used to extract the first mode shape of a cracked beam; the cracks could be identified using the extracted mode shape. Javh et al.^20,21 developed a gradient-based optical-flow modal analysis technique using digital cameras, where black-and-white lines were drawn on surfaces of structures, such as beams and cymbals. Chen et al.²² proposed a motion magnification technique to measure modal parameters of structures. The technique was then applied to evaluate natural frequencies of an outdoor antenna tower and an outdoor bridge by Chen et al.²³ and Chen et al.,²⁴ respectively. Yang et al.¹⁷ proposed a target-less blind identification technique for full-field output-only modal analysis based on a motion magnification technique; damage in beam components of a frame structure was detected based on mode shapes estimated by the proposed technique. Guo and Zhu²⁵ proposed a real-time target-less efficient technique for vibration measurement, where Lucas–Kanade tracking algorithm and a modified inverse compositional algorithm were used. However, the proposed technique could be undermined due to changes in shading, lighting and background conditions.

In this work, a photogrammetry-based structural damage detection method is developed; it consists of three techniques: a laser-line-tracking technique, an experimental modal analysis technique for a target line and a damage identification technique. In the laser-line-tracking technique, a digital camera captures a series of digital images of a visible laser line projected to a surface of a structure and a target line corresponding to the projected laser line is tracked. Projecting the laser line to a structure does not cause any structural modifications, since a laser line is massless and its existence is temporary. Modal parameters of the target line are estimated using the experimental modal analysis technique, and positions and sizes of structural damage can be identified based on estimated mode shapes of the target line. The developed structural damage detection method was experimentally investigated on a damaged cantilever beam. Modal parameters of a target line estimated by the experimental modal analysis technique were compared with those from a finite element model of the beam. Experimental damage identification results were validated by those from the finite element model.

The remaining part of this article is outlined as follows. The laser-line-tracking technique is described in section “Laser-line-tracking technique,” based on which the experimental modal analysis technique is described in section “Experimental modal analysis of a target line.” The damage detection technique is described in section “Structural damage detection technique.” Experimental investigations of the laser-line-tracking, experimental modal analysis and damage identification techniques are presented in section “Experimental investigation.” Conclusions of this work are presented in section “Conclusion.”

Methodology

Laser-line-tracking technique

A three-dimensional schematic of the laser-line-tracking technique is shown in Figure 1(a). In this technique, a digital camera is facing a surface of a beam with a tilt angle $θ$ and a field-of-view (FOV) plane is formed on a portion of the beam surface. A line laser is projecting a straight visible laser line to the beam surface along the length of the beam. The portion of the projected laser line that lies on the FOV plane is termed as the FOV laser line. The line laser is placed at a position with an orientation so that the plane, which is defined by the line laser and FOV laser line, is perpendicular to the beam surface. Based on a pinhole perspective imaging model of a camera,²⁶ an image plane can be defined in front of the lens of the camera. A target line corresponding to the FOV laser line is formed on the image plane. It is the target line along with its background mapped to the image plane that appears on digital images captured by the camera. As shown in Figure 1(b), when the beam undergoes transverse deformation, the target line corresponding to the FOV laser line will undergo in-plane deformation on the image plane, that is, on a captured digital image.

Figure 1.

(a) Three-dimensional schematic of the laser-line-tracking technique, (b) a view of the schematic from the lens in (a) with the beam undergoing transverse deformation, (c) a cross-sectional view of the schematic in (a), and (d) a geometric description of the cross-sectional view in (c).

In the three-dimensional schematic shown in Figure 1(a), a Cartesian coordinate system $O - uvw$ is defined, under which the undeformed surface of the beam is parallel with a plane defined by the $v$ and $w$ axes; the $u$ axis is perpendicular to the flat surface and aligns with the transverse direction of the beam. A cross-sectional view of the three-dimensional schematic is shown in Figure 1(c), where a point on the FOV laser line and that on the corresponding target line are denoted by $P$ and $P_{i}$ , respectively. When the beam deforms transversely, $P$ moves to point $P'$ and $P_{i}$ moves to point $P_{i}'$ on the image plane. The kinematic relation between the displacement of $P$ in the transverse direction and that of $P_{i}$ on the image plane can be derived based on a geometric description of the cross-sectional view, which is shown in Figure 1(d). The lens center of the camera is denoted by point $C$ . The transverse displacement of $P$ , that is, the length of $PP'$ , is denoted by $d$ and the displacement of $P_{i}$ on the image plane, that is, the length of $P_{i} P_{i}'$ , is denoted by $d_{i}$ . A projective ray, that is, a straight line, starting from $C$ to $P'$ intersects the $u$ axis at point $P_{a}'$ and the length of $P_{i} P_{a}'$ is denoted by $d'$ . Lengths of $O P_{i}$ and $OC$ are denoted by $a$ and $b$ , respectively. Note that $a$ is a parameter related to the stand-off distance of the camera from the beam. Since $PP' ∥ P_{i} P'_{a}$ , $Δ P_{i} CP'_{a}$ and $Δ PCP'$ are similar triangles, and one has

\frac{d}{| CP |} = \frac{d'}{| C P_{i} |}

(1)

where |·| denotes the length of a straight line, and rearranging equation (1) yields

d' = \frac{| C P_{i} |}{| CP |} d

(2)

Let $α = ∠ P'_{i} P_{i} P'_{a}$ and a straight line passing through $P_{i}$ and $P'_{i}$ on the image plane can be expressed by a slope–intercept form

v = \tan α (u - a)

(3)

A straight line passing through $C$ and $P'$ can be expressed by a two-point form using $u$ and $v$ coordinates of $C$ and $P'_{a}$ , which are $(0, b)$ and $(a + d', 0)$ , respectively, and the two-point form can be expressed by

\frac{v - 0}{u - (a + d')} = \frac{b - 0}{0 - (a + d')}

(4)

and rearranging equation (4) yields

v = - \frac{b}{a + d'} u + b

(5)

Coordinates of $P_{i}'$ , denoted by $(u_{{P'}_{i}}, v_{{P'}_{i}})$ , can be obtained by solving a linear equation set consisting of equations (3) and (5), and they can be expressed by

{\begin{matrix} u_{{P'}_{i}} = \frac{bd'}{\tan α (a + d') + b} + a \\ v_{{P'}_{i}} = \tan α [\frac{bd'}{\tan α (a + d') + b}] \end{matrix}

(6)

The length of $P_{i} P_{i}'$ , that is, $d_{i}$ , can be calculated by

\begin{matrix} d_{i} = \sqrt{{(u_{P'_{i}} - a)}^{2} + v_{{P'}_{i}}^{2}} \\ = \frac{bd'}{\sin α (a + d') + b \cos α} \end{matrix}

(7)

When the beam deforms transversely with a small amplitude and the stand-off distance of the camera is much larger than the deformation amplitude of the beam, one has $d' << a$ and $d' << b$ . Hence, one has

\frac{1}{\sin α (a + d') + b \cos α} \approx \frac{1}{a \sin α + b \cos α}

(8)

and $d_{i}$ in equation (7) can be approximated by

d_{i} = \frac{bd'}{a \sin α + b \cos α}

(9)

Substituting equation (2) into equation (9) yields

d_{i} = \frac{b | C P_{i} |}{(a \sin α + b \cos α) | CP |} d

(10)

Since $a$ , $b$ , $| CP |$ , $| C P_{i} |$ , and $α$ are constants in the setup, the kinematic relation between $d_{i}$ and $d$ can be considered linear and expressed by equation (10). By tracking the target line on captured digital images and quantifying its deformations, one can quantify transverse deformations of the beam along the FOV laser line with the ratio $b | C P_{i} | / ((a \sin α + b \cos α) | CP |)$ in equation (10).

An intensity profile of a target line corresponding to an FOV laser line resembles a one-dimensional Gaussian distribution²⁷

\tilde{I} (y) = A e^{- \frac{{(y - {\tilde{y}}_{0})}^{2}}{σ^{2}}}

(11)

where $y$ denotes the $y$ coordinate of a two-dimensional Cartesian coordinate system on an image plane, $A$ is the amplitude of the distribution, ${\tilde{y}}_{0}$ is the $y$ coordinate of the center of the intensity profile, and $σ$ determines the width of the intensity profile; $\tilde{I}$ attains its maximum value when $y = {\tilde{y}}_{0}$ . Note that the $x$ and $y$ axes of the coordinate system are in the horizontal and vertical directions of the image plane, respectively. When the centers are tracked along $x$ axis of the image plane, a gray-gravity rule²⁸ can be used, where the y coordinate of the center of an intensity profile is calculated by

{\bar{y}}_{0} (x) = \frac{Ξ [I (x, y) \times y]}{Ξ y}

(12)

where $I (x, y)$ denotes discrete intensity values at $(x, y)$ and $Ξ$ denotes summation over all pixels of the intensity profile at $x$ along $y$ axis. Assume that $I$ keeps resembling one-dimensional Gaussian distributions when the beam deforms transversely. A laser-line-tracking technique is proposed that an FOV laser line be tracked by tracking centers of intensity profiles of its corresponding target line on captured images.

Experimental modal analysis of a target line

As shown in section “Laser-line-tracking technique,” when the beam in Figure 1(a) undergoes transverse vibration, the target line will undergo in-plane vibration on digital images captured by the camera, and one can track the corresponding vibrating target line using the gray-gravity rule. Assuming that the beam is a linear structure and its transverse vibration has a small amplitude, the ratio between $d_{i}$ and $d$ is constant and the time-domain response of the tracked target line can be represented by a linear combination of mode shape vectors of the target line

{\bar{y}}_{0} (x, t) = \sum_{j = 1}^{J} ϕ_{j} (x) q_{j} (t)

(13)

where $J$ is the number of modes of the target line, $ϕ_{j}$ is the jth mode shape vector of the target line with $ϕ_{j} (x)$ being its entry at $x$ , and $q_{j}$ is the jth modal coordinate with $q_{j} (t)$ being its value at time $t$ . When the captured digital images have $p \times q$ pixels with $p$ and $q$ denoting numbers of pixels along $x$ and $y$ axes, respectively, $ϕ_{j}$ is a q-dimensional vector if the target line spans the length of the image plane. Assuming that the vibration of the beam is solely caused by excitation $f_{s}$ applied to a point on the beam, a generalized frequency response function between ${\bar{y}}_{0}$ at $x$ and $f_{s}$ can be expressed by

H (x, ω) = \frac{{\bar{Y}}_{0} (x, ω)}{F_{s} (ω)}

(14)

where ${\bar{Y}}_{0}$ and $F_{s}$ are Fourier transforms of ${\bar{y}}_{0}$ and $f_{s}$ , respectively, and $ω$ denotes a circular frequency. Modal parameters of the target line can be estimated by analyzing $H$ with an experimental modal analysis algorithm. More importantly, since the relation between deformations of the FOV laser line and target line can be considered linear with a constant ratio, the estimated modal parameters of the target line can be considered equal to those of the beam along the FOV laser line.

Structural damage detection technique

A curvature mode shape is the second-order derivative of a mode shape of a structure and damage can introduce local anomalies to curvature mode shapes.²⁹ On digital images captured by the camera in the setup of the laser-line-tracking technique, the local anomalies in curvature mode shape of the structure can be mapped to corresponding curvature mode shapes of the target line. Inversely, identifying the local anomalies in the curvature mode shapes of the target line can assist in detecting the damage by mapping the local anomalies to the FOV laser line.

A non-model-based damage detection technique by Xu et al.³⁰ is used here to identify local anomalies in curvature mode shapes of a target line, by mapping which locations and sizes of damage in a structure along a corresponding FOV laser line can be identified. An overview of the damage detection technique is provided here, and more details of the technique can be found in Xu et al.³⁰ An entry of a curvature mode shape vector of a target line at $x$ can be calculated using a central difference scheme

ϕ_{j}^{″} (x) = \frac{ϕ_{j} (x + h) - 2 ϕ_{j} (x) + ϕ_{j} (x - h)}{h^{2}}

(15)

where a prime denotes spatial derivative and $h$ is the step size for calculating ${ϕ ″}_{j}$ . It is shown by Xu et al.³⁰ that a mode shape from a polynomial that fits a mode shape of a damaged structure can well approximate that of a corresponding undamaged structure, provided that the undamaged structure is geometrically smooth and made of materials that have no mass and stiffness discontinuities. By comparing a curvature mode shape vector of a target line with that from a polynomial that fits the corresponding mode shape of the target line, a curvature damage index (CDI) is defined and its value at $x$ can be expressed by

δ_{j} (x) = {[ϕ_{j}^{″} (x) - ϕ_{j}^{p ″} (x)]}^{2}

(16)

where $ϕ_{j}^{p}$ is a mode shape vector from a polynomial that fits $ϕ_{j}$ ; $ϕ_{j}^{p} (x)$ denotes the entry of $ϕ_{j}^{p}$ at $x$ . Locations and sizes of the damage can be identified by mapping neighborhoods with high CDI values to the FOV laser line. A polynomial fit can be expressed by

ϕ^{p} (x) = \sum_{n = 0}^{n = k} a_{n} x^{n}

(17)

where $k$ is the order of the polynomial fit and $a_{n}$ are coefficients of the polynomial fit, which can be determined by solving a linear equation set

[\begin{matrix} 1 & x_{1} & x_{1}^{2} & \dots & x_{1}^{k} \\ 1 & x_{2} & x_{2}^{2} & \dots & x_{2}^{k} \\ 1 & x_{3} & x_{3}^{2} & \dots & x_{3}^{k} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 1 & x_{m} & x_{m}^{2} & \dots & x_{m}^{k} \end{matrix}] [\begin{matrix} a_{0} \\ a_{1} \\ a_{2} \\ ⋮ \\ a_{k} \end{matrix}] = [\begin{matrix} ϕ_{j} (x_{1}) \\ ϕ_{j} (x_{2}) \\ ϕ_{j} (x_{3}) \\ ⋮ \\ ϕ_{j} (x_{m}) \end{matrix}]

(18)

where $m$ is the number of measurement points of a mode shape to be fit. A proper order of the polynomial fit is two plus the least $k$ value, with which a fitting index

fit (k) = \frac{RMS (ϕ)}{RMS (ϕ - ϕ^{p}) + RMS (ϕ)} \times 100 %

(19)

is larger than $90 %$ , where $RMS (\cdot)$ denotes the root-mean-square of a vector. When multiple mode shapes are available, one can identify the damage in neighborhoods with consistently high CDI values corresponding to different mode shapes. An auxiliary CDI is defined to assist identification of the neighborhoods, which can be expressed by

δ_{a} (x) = Γ {\hat{δ}}_{j} (x)

(20)

where $Γ$ denotes summation over all available mode shapes and ${\hat{δ}}_{j}$ is a normalized CDI associated with $δ_{j}$ in equation (16) with the maximum value being $1$ .

Experimental investigation

Experimental setup

Experimental investigations of the laser-line-tracking, experimental modal analysis, and damage detection techniques were conducted on a damaged cantilever beam. Dimensions of the beam and experimental setup of the investigations are shown in Figures 2(a) and (b), respectively. One end of the beam was clamped by a bench vice to simulate a fixed boundary. The damage was in the form of a machined area with thickness reduced by 20% on one side of the beam, as shown in Figure 2(c). A visible laser line was projected to the intact surface of the beam using a class-2 line laser, Bosch GLL 50; the projected laser line is shown in Figure 2(d). The intact surface of the beam had a uniform background, without which the intensity profile of the target line associated with the projected laser line would not resemble a one-dimensional Gaussian distribution. A solid straight line was drawn on the intact surface along its length to assist in positioning the beam and line laser so that the projected laser line could be aligned with the length of the beam. Note that one could use a level to position the beam and line laser, with which the solid line would not need to be drawn on the beam. The plane that was defined by the line laser and projected laser line was perpendicular to the intact surface. A digital camera, Photron AX200, was used to capture digital images of a portion of the projected laser line on the beam. The camera was placed in front of the beam with $θ = 45 °$ , and its lens was facing the projected laser line. Note that $θ$ is critical to the developed damage detection method. If $θ$ is too small, for example, $θ = 5 °$ , deformations of a target line will become so small that estimated mode shapes will have low signal-to-noise ratios and cannot be used for damage identification. If $θ$ is too large, for example, $θ = 85 °$ , a target line will become so narrow that the center of the one-dimensional Gaussian distribution associated with the target line cannot be accurately identified; estimated mode shapes will also have low signal-to-noise ratios and cannot be used for damage identification. Hence, $θ$ was chosen to be $45 °$ here. A series of digital images with $64 \times 1024 pixels$ were captured with a frame rate of $500 fps$ by the camera. The distance between the camera and beam was the one with which a target line corresponding to the projected laser line was centered on a captured digital image. With such a distance, dynamic ranges of vibration measurements by the camera could be maximized. An impact hammer, PCB 086C05, was used to excite the beam on $P$ in the form of single impacts, and the location of $P$ is shown in Figure 2(a). When the camera captured images, the illumination environment was a dark indoor condition without any external light sources. Hence, only the target line corresponding to the FOV laser line, which was the portion of the project laser line, existed on captured digital images. Geometrical features, such as the width and edges, of the beam and the drawn solid line did not appear on the captured digital images. The position and length of the FOV laser line in the experimental setup are shown in Figure 2(a). If the illumination environment of the experimental had been noisy, a high-power line laser would become necessary so that a target line in captured images could outstand and its intensity profile could resemble a one-dimensional Gaussian distribution.

Figure 2.

(a) Dimensions of the damaged aluminum cantilever beam, (b) the setup of the beam for experimental investigations, (c) damage in the form of a machined area on a side of the beam, and (d) the projected laser line along the length of the beam.

Laser-line-tracking results

A captured digital image of the target line is shown in Figure 3(a) and its measured intensity profile along $y$ axis at $x = 512 pixels$ is shown in Figure 3(b). The profile consisted of $64$ discrete intensity values at 64 pixels along $y$ axis. By fitting the one-dimensional Gaussian distribution in equation (11) to the measured intensity profile, one could obtain a fit intensity profile with ${\tilde{y}}_{0} = 32.70 pixels$ , as shown in Figure 3(b). Comparison between the measured and fit intensity profiles validated that an intensity profile of a target line corresponding to an FOV laser line resembled a one-dimensional Gaussian distribution. Using the gray-gravity rule in the laser-line-tracking technique in equation (12), one could track the center of the measured intensity profile at ${\bar{y}}_{0} = 32.82 pixels$ , which compared well with that of the fit intensity profile.

Figure 3.

(a) Target line on a captured digital image; (b) measured and fit intensity profiles of the image in (a) at $x = 512 pixels$ , and centers of the measured intensity profile from the fit intensity profile (fit center) and from the gray-gravity rule in the laser-line-tracking technique (gray-gravity center); and (c) centers of the target line using the laser-line-tracking technique (gray-gravity center) and from fit intensity profiles (fit center).

A total of $1024$ centers of measured intensity profiles of the target line at $1024 pixels$ along $x$ axis on the digital image were tracked using the gray-gravity rule. Resulting ${\bar{y}}_{0}$ were compared with ${\tilde{y}}_{0}$ from fit intensity profiles associated with the measured intensity profiles, as shown in Figure 3(c). Computation times for tracking one center using the gray-gravity rule and that from a fit intensity profile were $7.27 \times 10^{- 6}$ and $0.02 s$ , respectively. The root-mean-square of differences between the 1024 $y_{0}$ from the laser-line-tracking technique and the 1024 ${\tilde{y}}_{0}$ from the fit intensity profiles was $0.1404 pixel$ . Comparison between the computation times and the root-mean-square indicated that the use of the laser-line-tracking technique had a higher efficiency than but almost the same accuracy as the use of fit intensity profiles in tracking centers of a target line corresponding to an FOV laser line.

Experimental modal analysis results

Experimental modal analysis was conducted on the target line with single impacts to $P$ on the target line. Single impacts $f_{s}$ and ${\bar{y}}_{0}$ of eight sampling periods were measured and the duration of each sampling period was 32 s. Sampling frequencies of $f_{s}$ and ${\bar{y}}_{0}$ had the same value as the frame rate of the camera, that is, 500 Hz. Time histories of $f_{s}$ and ${\bar{y}}_{0}$ at $x = 512 pixels$ in the first sampling period are shown in Figures 4(a) and (b), respectively. The single impact $f_{s}$ had non-zero values after its occurrence and ${\bar{y}}_{0}$ at $x = 512 pixels$ did not decay to zero within the sampling period. A rectangular window was applied to $f_{s}$ in each sampling period so that its values before and after occurrence of an impact were zero; a exponential window was applied to ${\bar{y}}_{0}$ in each sampling period so that its amplitudes at all $x$ coordinates became almost zero at the end of a sampling period. Windowed $f_{s}$ and ${\bar{y}}_{0}$ in the first sampling period are shown in Figure 4(c) and (d), respectively.

Figure 4.

Measured (a) $f_{s}$ and (b) ${\bar{y}}_{0}$ at $x = 512 pixels$ in the first sampling period, and windowed (c) $f_{s}$ and (d) ${\bar{y}}_{0}$ at $x = 512 pixels$ in the first sampling period.

FRFs calculated from $H_{1}$ formulation³¹ using windowed $f_{s}$ and ${\bar{y}}_{0}$ at $x = 512 pixels$ in the first one, two, four, and eight sampling periods are shown in Figure 5(a). Noise floors of the FRFs were lowered by increasing the number of sampling periods. After eight averages, the noise floors were still relatively high, and they hindered identification of higher order modes of the target line. The noise floors could become lower and some higher order modes could be identified, if a digital camera with a higher resolution had been used. Coherence functions associated with the FRFs in Figure 5(a) using windowed $f_{s}$ and ${\bar{y}}_{0}$ at $x = 512 pixels$ in the first two, four, and eight sampling periods are shown in Figure 5(b). Relatively high coherence function values could be found in neighborhoods of frequencies where four magnitude peaks existed in the FRF calculated using measured $f_{s}$ and ${\bar{y}}_{0}$ in the eight sampling periods, indicating the four frequencies could be natural frequencies of the target line and the natural frequencies could be accurately estimated when only the FRFs at $x = 512 pixels$ were available. FRFs at $x = 800 pixels$ in the first one, two, four, and eight sampling periods are shown in Figure 5(c). Similar to the FRFs in Figure 5(a), noise floors of the frequency response functions were lowered by increasing the number of sampling periods. In coherence functions associated with the FRFs in Figure 5(c), relatively high coherence function values could be found in neighborhoods of frequencies where four magnitude peaks existed in the FRFs calculated using measured $f_{s}$ and ${\bar{y}}_{0}$ in the eight sampling periods shown in Figure 5(c), while relatively low coherence function values could be found in the neighborhood of frequencies where an additional magnitude peak of the FRF existed. It was indicated that the five frequencies could be natural frequencies of the target line, which could be accurately estimated; however, the mode shape corresponding to the additional mode might not be accurately estimated.

Figure 5.

(a) Magnitude of $H (512, ω)$ using $f_{s}$ and ${\bar{y}}_{0} (512, t)$ in the first one, two, four, and eight sampling periods; (b) coherence functions associated with $H (512, ω)$ obtained using $f_{s}$ and ${\bar{y}}_{0} (512, t)$ in the first two, four, and eight sampling periods; (c) the magnitude of $H (800, ω)$ with $f_{s}$ and ${\bar{y}}_{0} (800, t)$ in the first one, two, four, and eight sampling periods; (d) coherence functions associated with $H (800, ω)$ obtained using $f_{s}$ and ${\bar{y}}_{0} (800, t)$ in the first two, four, and eight sampling periods; and (e) the composite FRF associated $1024$ measured FRFs of the target line.

In order to better identify modes of the target line, a composite FRF, that is, the sum of magnitudes of available FRFs, was calculated and shown in Figure 5(e), where six peaks could be identified. A total of $1024$ FRFs along the length of the target line were analyzed using an experimental modal analysis algorithm, PolyMax in LMS Test.Lab v17, to estimate modal parameters of the target line, including natural frequencies and mode shapes. The estimated natural frequencies are listed in Table 1. A finite element model of the beam was constructed with the elastic modulus and Poisson’s ratio of the beam being $68.5 GPa$ and $0.3$ , respectively; it consists of 22,125 eight-node shell (S8R) elements. The first six natural frequencies of the beam from the finite element model are listed in Table 1 to compare with those from the experimental modal analysis of the target line. The first five natural frequencies of the target line compared well with those from the finite element model. Note that the sixth measured natural frequency of the target line was $241.40 Hz$ and had a difference of $- 6.3 %$ . The reason was that the sampling frequency of ${\bar{y}}_{0}$ was $500 Hz$ , the associated Nyquist frequency was $250 Hz$ , and the actual sixth natural frequency of the target line was greater than $250 Hz$ . By unfolding the measured sixth natural frequency of the target line with respect to Nyquist frequency of $250 Hz$ , one could obtain an unfolded natural frequency of $258.6 Hz$ , which had a difference of $0.4 %$ , compared with the sixth natural frequency from the finite element model.

Table 1.

Comparison between natural frequencies of the target line from the experimental modal analysis (experimental) and those from a finite element model of the beam (numerical).

Mode	Experimental (Hz)	Numerical (Hz)	Difference (%)
1	3.10	3.08	0.6
2	18.71	18.86	−0.8
3	54.86	54.40	0.8
4	103.96	103.96	0.0
5	177.50	176.01	0.8
6	241.40	257.52	−6.3

Mode shapes from the finite element model of the beam along the FOV laser line are shown in Figure 6, and those of the target line were mapped to the beam along the FOV laser line and shown in Figure 6. A modal assurance criterion (MAC) matrix between the mode shapes from the finite element model and those of the target line is listed in Table 2. Diagonal entries of the MAC matrix were all over $94 %$ , indicating that the mode shapes of the same modes compared well with each other. Some off-diagonal entries of the MAC matrix had relatively large values since the FOV laser line was on a portion of the beam and some mode shapes of the beam along the FOV laser line were similar. Since the resolution of the digital camera was not high enough, peaks corresponding to the sixth mode in the magnitudes of the $1024$ individual FRFs were masked under their noise floors. As a result, the coherence function values in the neighborhood of the sixth natural frequency of the target line were low and the sixth mode shape of the target line had a low signal-to-noise ratio, as shown in Figure 6(f), although a peak corresponding to the sixth mode could be identified in the composite FRF in Figure 5(e). Use of the sixth mode shape for damage detection of the beam would be excluded hereafter, though the diagonal entry of the MAC matrix associated with the sixth mode shape of the target line and that on the FOV laser line from the finite element model was larger than $94 %$ .

Figure 6.

Mode shapes from the finite element model of the beam (numerical), mapped mode shapes of the target line (experimental), and mapped mode shapes from polynomials that fit the mode shapes of the target line (polynomial: experimental) associated with the (a) first, (b) second, (c) third, (d) fourth, (e) fifth, and (f) sixth modes.

Table 2.

Entries of the MAC matrix in percent between the first six mode shapes of the target line from the experimental modal analysis and those from the finite element model along the FOV laser line.

Mode	1	2	3	4	5	6
1	99.89	76.48	10.57	17.24	11.37	2.29
2	73.52	99.97	1.60	33.90	0.09	0.07
3	10.10	2.54	99.80	0.20	73.92	0.01
4	19.16	37.16	0.97	99.88	0.64	55.41
5	11.63	0.39	78.21	0.01	99.27	0.20
6	2.87	0.00	0.58	58.18	0.23	94.92

MAC: modal assurance criterion; FOV: field-of-view.

Structural damage identification results

The first five mode shapes of the target line were fit by polynomials with proper orders determined based on $fit$ in equation (19), and the proper orders are listed in Table 3. Mode shapes from the polynomial fits were mapped to the beam along the FOV laser line and shown in Figure 6. The mapped mode shapes from the polynomial fits compared well with those of the target line and also well with those from the finite element model of the beam along the FOV laser line. A numerical smoothing technique, which was local regression that used weighted linear least squares and a linear polynomial model, was applied to reduce measurement noise in each mode shape of the target line before calculating its associated curvature mode shape. Curvature mode shapes of the target line were calculated and mapped to the beam along the FOV laser line as shown in Figure 7. Anomalies could be observed in the mapped curvature mode shapes by comparing them with those from the polynomial fits. CDIs associated with the first five mapped curvature mode shapes of the target line are shown in Figure 8, and their associated auxiliary CDI is shown in Figure 9. The position and size of the damage could be identified in the neighborhood with high auxiliary CDI values.

Table 3.

Proper orders of polynomials that fit measured mode shapes of the target line (experimental) and those from the finite element model along the FOV laser line (numerical).

Mode	1	2	3	4	5
Experimental	5	5	5	6	7
Numerical	5	5	5	6	7

FOV: field-of-view.

Figure 7.

Curvature mode shapes from the finite element model of the beam (numerical), mapped curvature mode shapes of the target line (experimental), and mapped curvature mode shapes from polynomials that fit the mode shapes of the target line (polynomial: exp.) and those from the finite element model (polynomial: num.) associated with the (a) first, (b) second, (c) third, (d) fourth, and (e) fifth modes. The locations of the damage ends are indicated by gray lines.

Figure 8.

CDIs associated with the (a) first, (b) second, (c) third, (d) fourth, and (e) fifth modes from the finite element model of the beam (numerical) and those of the target line (experimental). The locations of the damage ends are indicated by gray lines.

Figure 9.

Auxiliary CDIs associated with CDIs from the finite element model of the beam (numerical) and experimental modal analysis of the target line (experimental). The locations of the damage ends are indicated by gray lines.

The first five mode shapes of the beam along the FOV laser line from the finite element model were then fit by polynomials with proper orders determined based on $fit$ in equation (19), and the proper orders are listed in Table 3. Curvature mode shapes from the polynomial fits are shown in Figure 7. Similar to the experimental results of the mapped curvature mode shapes of the target line, anomalies can be observed in the curvature mode shapes from the finite element model of the beam by comparing them with those from the polynomial fits. CDIs associated with the first five mode shapes from the finite element model of the beam along the FOV laser line are shown in Figure 9, and their associated auxiliary CDI is shown in Figure 8. The position and size of the damage could also be identified in the neighborhood with high auxiliary CDI values. More importantly, the auxiliary CDI associated with the mapped mode shapes of the target line and mode shapes of the beam from the finite element model along FOV laser line compared well with each other, especially in the neighborhood of the damage.

Conclusion

A photogrammetry-based structural damage detection method is developed and experimental investigations of the method were conducted on a damaged beam. In the developed method, a digital camera is used to measure vibration of a structure by tracking a projected laser line. Structural damage can be detected by identifying anomalies in mapped curvature mode shapes of a target line that corresponds to the projected laser line. It was experimentally shown that a measured intensity profile of a target line on a digital image, which corresponded to a projected laser line, resembled a one-dimensional Gaussian distribution and centers of measured intensity profiles of the target line could be efficiently and accurately tracked using the gray-gravity rule in a laser-line-tracking technique. An experimental modal analysis technique is proposed to estimate modal parameters of a target line on a series of digital images captured by a camera. It was experimentally shown that estimated natural frequencies and mapped mode shapes of the target line compared well with those from a finite element model of the damaged beam. Anomalies in curvature mode shapes of the target line could be observed by comparing them with those from polynomials that fit the mode shapes of the target line. Using CDIs that quantify the anomalies in mapped curvature mode shapes, the damage of the beam was accurately detected in neighborhoods with consistently high CDI values corresponding to available mode shapes. Experimental damage detection results compared well with numerical ones from the finite element model. The developed damage detection method has two limitations in practice. One is that it requires a uniform background where a projected laser line exists, and the other is that it requires a high-power line laser in cases whose illumination environments are noisy.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The author is grateful for the faculty startup support from the Department of Mechanical and Materials Engineering at the University of Cincinnati.

ORCID iD

Yongfeng Xu

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