Transient Strain Measurements of a Suspended Cable Under Impact Loading Using FBG Sensors

1998 IEEE SENSORS JOURNAL, VOL. 9, NO. 12, DECEMBER 2009

Transient Strain Measurements of a SuspendedCable Under Impact Loadings Using

Fiber Bragg Grating SensorsChien-Ching Ma and Cheng-Wei Wang

Abstract—The theoretical and experimental studies of fiberBragg grating (FBG) sensors is investigated. In the theoreticalstudies, the coupled-mode theory with transfer matrix methodis used to simulate the dynamic responses of the FBG sensorssubjected to a sinusoidal strain wave. The numerical resultsindicate that the increase of frequency and amplitude of strainwave result in signal distortion, hence the signal distortion limitis established to analyze the FBG dynamic sensing range. In theexperimental studies, the FBG sensors as well as the strain gageare used to measure simultaneously the transient responses ofdynamic strain for a suspended cable under impact loadings. Thefrequency spectra of the measuring results for transient responsesof dynamic strain are analyzed and the natural frequencies ofthe cable are determined. The natural frequencies obtainedfrom experimental measurements have excellent agreement fordifferent sensors and different impact positions. In addition, theexperimental results also have good correspondence with the the-oretical analysis and the finite-element method (FEM). Accordingto the available experimental results, it is proved that the FBGsensor has excellent ability of transient strain measurement forstructures with curved surface and large curvature and has betterperformance than the traditional strain gage.

Index Terms—Coupled-mode theory, dynamic strain, fiberBragg grating (FBG), impact loading, natural frequency, sus-pended cable, transient response.

I. INTRODUCTION

T HE fiber Bragg grating (FBG) sensors have attracted moreand more attention in recent years due to their low weight,

small size, high sensitivity, and immunity to electromagneticfields [1]–[3]. In addition, it requires no electrical power at thesensing point. They can be used to measure temperature, strain,displacement, pressure, etc. For strain sensing, both the changeof the grating period due to physical elongation of the sensorand the change in effective index due to photoelastic effectwith shift the Bragg wavelength. In strain sensor applications,FBGs has been successfully applied to health monitoring ofcivil engineering structures such as bridges, dams and tunnels[4], [5]. Traditionally, for the purpose of position tracking

Manuscript received November 18, 2008; revised July 23, 2009; acceptedAugust 24, 2009. Current version published November 04, 2009. This work wassupported in part by National Science Council (Republic of China) under GrantNSC96-2221-E002-210-MY3. The associate editor coordinating the review ofthis paper and approving it for publication was Prof. Istvan Barsony.

The authors are with the Department of Mechanical Engineering, NationalTaiwan University, Taipei, Taiwan 106, China (e-mail: [email protected];[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSEN.2009.2031327

control, linear variable differential transformer (LVDT) sensorshave been widely used for displacement measurement. Overa wide range, the LVDT sensor has good characteristics forresolution in the order of micrometers and there is no electricalcontact between the moving specimen and the coils of thesensor [6]–[8]. However, compared with the FBG sensor, theLVDT is relatively large in physical size and heavy in weight.Besides, there are difficulties for the measurement in the orderof submicrometers due to lower sensitivity of the LVDT. In anattempt to overcome these disadvantages of the LVDT sensor,Chunag and Ma [9] presented a rather simple but effectivemethod to set up an FBG sensing system and successfullyused it for the application of tracking control of a multilayerpiezoelectric actuator.

In order to understand the optical properties of fiber grat-ings, Erdogen [10] introduced the coupled-mode theory to de-scribe the principle and spectral characteristics for uniform andnonuniform gratings. In addition, as a strain sensor, the spectralanalysis of a FBG subjected to strain fields is important. Tai [11]simulated the maximum reflectivity of a FBG corresponding tovarious nonuniform static strain conditions. Sikora [12] mod-eled the behavior of an apodized FBG under impulsive strainof rectangular shape. Minardo et al. [13] numerically investi-gated the response of FBG to the longitudinal ultrasonic wavesin terms of shape and wavelength changes. Ling et al. [14] pre-sented a simulation method for evaluating dynamic strain dis-tribution along a uniform FBG using its reflection spectrum.

Since the FBG sensors are based on the wavelength encodingof the measurand signal information, the demodulators whichconvert the wavelength shifts into electrical signals should beincluded in sensing systems for easy-reading and real-timemonitoring. Therefore, many demodulators have been devel-oped such as the Fabry–Perot tunable filter system [15], [16],the imbalanced Mach–Zehnder interferometer [17], [18], thelong-period fiber grating [19]–[21], the chirped fiber grating[22], the FBG [23], [24], and the tilted FBG [25]. Amongthese methods, the FBG filter has the smallest specificationof full-width at half maximum (FWHM) compared with othertypes of fiber gratings, an FBG filter-based wavelength-opticalintensity demodulation technique has the highest sensitivity.The FBG filter-based demodulation technique [10], [23], [24] isused in this study to obtain higher sensitivity. Furthermore, theFBG filter is placed at the output of the broadband light sourceto enhance the signal-to-noise ratio (SNR) of the demodulationsystem.

To investigate the ability of dynamic strain measurement,the FBG sensors are used in this study to measure the transient

1530-437X/$26.00 © 2009 IEEE

MA AND WANG: TRANSIENT STRAIN MEASUREMENTS OF A SUSPENDED CABLE UNDER IMPACT LOADINGS USING FBG SENSORS 1999

responses of a suspended cable under impact loadings. Cablesare very efficient structural elements and hence have a widerange of practical applications in civil engineering and inelectrical industry, including cable-stayed bridges, suspensionbridges, guyed towers, and cable-supported roofs. A history ofthe derivations and solutions of the equations for the static anddynamic response of cables can be found in [26].

Besides theoretical investigations of cables, experimental ver-ifications of the solutions for the natural frequencies have alsobeen carried out in the literature. Lee and Perkins [27] inves-tigated experimentally the near resonant responses of the sus-pended elastic cables driven by harmonic excitation. Cheng andPerkins [28] compared theoretical and experimental forced re-sponses and natural frequencies of a sagged horizontal cablesupporting an array of discrete masses. Lin and Perkins [29]presented a theoretical model that describes the frequency re-sponses of arbitrarily complex and sagged cable/mass systemsand compared to those obtained experimentally using modaltesting techniques.

Since cables are crucial elements for overall structural safetyof a cable-stayed bridge, the monitoring of cable tension forceis very important and necessary. Russell and Lardner [30] de-termined the tension at the base of an inclined cable by usingthe modern cable theory [31] that considers the sag-extensibilitywithout bending stiffness. Zui et al. [32] proposed the practicalformulas to estimate cable tension force by the vibration methodthat takes account of both sag-extensibility and bending stiff-ness. Kim et al. [33], [34] introduced a comparative study ofthe existing vibration-based tension estimation techniques forcable supported bridges.

The purpose of this study is to apply the FBG sensors to themeasurement of transient responses of a suspended cable underimpact loadings, to analyze the frequency spectra of experi-mental results, and to compare the measured natural frequencieswith the theoretical predictions and the finite element method.With the accurate measurement of natural frequencies, the FBGsensors may have practical applications to the tension determi-nation and damage detection of cables and hence to the healthmonitoring of cable structures. According to the available exper-imental results, it is proved in this study that the FBG sensor hasexcellent ability of transient strain measurement for structureswith curved surface and large curvature and has better perfor-mance than the traditional strain gage.

II. DYNAMIC SENSING PRINCIPLE AND SIMULATION OF FBG

A. Theoretical Background

The FBG is defined as a small periodical perturbation to theeffective index of refraction of the optical fiber core de-scribed by [10]

(1)

where is the “dc” index change spatially averaged overa grating period, is the fringe visibility of the index change,

is the nominal period, and describes grating chirp. Bycoupled-mode theory, the first-order differential equations de-scribing mode propagation through the grating in -direction are

(2a)

(2b)

where and are the amplitudes of the forward- andbackward-propagating modes, respectively. isthe “ac” coupling coefficient. is the general “dc” self-couplingcoefficient defined as

(3)

where for the uniform grating, is theBragg wavelength which would be changed when the grating issubjected to various strain condition.

The length of the uniform grating is assumed to be , so thelimit of the grating is defined as . To obtain thereflectivity of the uniform Bragg grating subjected to nonuni-form strain, the grating is assumed to be divided into smallsections and with uniform coupling properties. The number ofsections cannot be arbitrarily large, since the coupled-modetheory approximations are not valid when a uniform grating sec-tion is only a few grating periods long. Hence, the is con-strained as

(4)

By defining and as the field amplitudes after traversingthe section , the propagation through this uniform section canbe described by a transfer matrix defined such that

(5)

where (6) is shown at the bottom of the page in which isthe length of each section and . As a result, thepropagation through the entire grating can be described by

(7)

where . By assuming thatno backward-going wave exists for [i.e., ], thereflectivity of the entire grating is then calculated by

(8)

For dynamic strain measurement, an optical filter which pos-sesses a linear wavelength dependent transmittance is necessaryto interpret the time-varied reflected spectrum of FBG sensor

(6)


Fig. 1. Schematic of the uniform FBG under sinusoidal strain wave.

into the electrical signal. The linear filter is assumed inthe form of

where is the filtering slope and is equal to zero at .When the intensity of the light incident on the FBG sensor is

, the output light power can be expressed as

(9)

B. Simulation Procedure and Results

Fig. 1 shows a uniform FBG which is subjected to a sinu-soidal strain wave with velocity , amplitude , wavelength

, frequency , and period . The time response of outputlight power is simulated for a period.

The grating’s parameters for the simulation are chosen to besimilar to parameters of grating used in this study. The gratinglength is , the Bragg wavelength is ,the effective refractive index is , the “dc” indexchange is , and the fringe visibility is .The parameters of the demodulation system are assumed to beas follows: Optical wavelength range is

, the incident light spectrum is , and thelinear filter spectrum is .

The simulation results of output light power responses fortwo cases are shown in Fig. 2. In the first case [Fig. 2(a)], thesimulation is performed for a fixed strain amplitude

and for three different nondimentional frequencyof strain ( , , and

). The response for is similar to sinusoidalfunction and agrees with sinusoidal strain wave. However, it isnoted that the response for higher frequency of strain shows ob-vious difference from sinusoidal function by observing the re-sults for and . In the second case[Fig. 2(b)], the simulation is performed for a fixed frequencyof strain and for three different strain ampli-tude ( , 100 , and 1000 ). Theresponse for is similar to sinusoidal func-tion and agrees with sinusoidal strain wave. We also find thatthe response for higher strain amplitude is different from sinu-soidal function for 100 and 1000 . Therefore, theincrease of frequency and amplitude of strain result in gettingworse of the signal distortion.

Fig. 2. The time responses of output light power for: (a) different frequency ofstrain and (b) different amplitude of strain.

Fig. 3. The signal accuracy versus frequency and amplitude of strain.

In order to establish a precise signal distortion limit of outputlight power response, the signal accuracy is defined as

(10)

According to the definition indicated in (10), the simulationresult of the signal accuracy versus frequency and amplitude ofstrain is shown in Fig. 3. The signal accuracy declines graduallywith the increase of frequency and amplitude of strain. Here,the signal distortion limit is defined when the signal accuracy is90% (i.e., the solid line in Fig. 3). The lower left-hand side ofthe signal distortion limit is the region suitable for measurementapplication and is named “good region” in which the output lightpower response is assumed to be not distorted. On the contrary,the upper right-hand side of the signal distortion limit is named“bad region” in which the response is assumed to be distorted.The concept of signal distortion limit is established to discussthe dynamic sensing range of FBG in Section III. The result ofthis analysis provides a useful tool for the practical measuringapplication of the dynamic strain.


Fig. 4. The definition diagram of cable.

III. TRANSIENT STRAIN MEASUREMENT OF SUSPENDED CABLE

A. Theoretical Determination of Natural Frequencies

The simplest method to determine the natural frequencies ofcable is the flat taut string theory

(11)

where , , , and denote the th natural frequency, lengthof cable, cable tension, and mass of cable per unit length, respec-tively. However, this simple formula neglects bending stiffnessof the cable and is only valid for a flat long slender cable.

The cable theory considering bending stiffness is used in thisstudy. As shown in Fig. 4, an elastic cable suspended betweentwo fixed supports at the same level is considered. The followingassumptions are made in the analysis.

1) The sag-to-span ratio is sufficiently small .2) The cable vibrates only within the - plane and its motion

in the -direction is negligibly small.3) The derivative cable tension caused by vibration is negli-

gibly small.The equation of motion in the -direction becomes [28]

(12)

where is flexural rigidity of cable, and is deflectionin the -direction.

By means of variable separation of

(13)

Equation (12) can be transformed into the following form

(14)

(15)

where is the circular natural frequency.The general solution of (14) is

(16)

in which

(17a)

(17b)

where and .

Fig. 5. Experimental setup of: (a) the suspended cable with the FBG sensorsand (b) the wavelength demodulation system.

By assuming that the cable is clamped at both ends, the fol-lowing characteristic equation is obtained:

(18)

The natural frequencies can be determined by numerical solvingthe roots of the characteristic equation.

B. Experimental Setup

As shown in Fig. 5(a), a single-strand steel cable with length, diameter , mass per unit length

, Young’s modulus , and tensionforce is clamped at both ends on the tensile tester.The tension force in the cable is controlled by moving the leftend of the cable horizontally and measured simultaneously bythe load cell.

There are two kinds of FBG sensor on the cable, includingsurface mounted FBG sensor and two-point mounted FBGsensor. The surface mounted FBG sensor is attached to thecable surface with strain gage cement on the full grating zone.The two-point mounted FBG sensor, different from traditionalmounting method, is glued to the cable surface with epoxyat both end of grating zone. The grating must be prestrainedin the measurement range for the two-point mounted FBGsensor. The Bragg wavelengths of the two FBG sensors before


Fig. 6. The transient responses of the cable under impact at position D.

experimental measurement are both 1559.3 nm, and the gratinglengths are both 10 mm. A resistive foil strain gage with gagelength 1 mm is also used to compare the results from the FBGsensors. The positions of the surface mounted FBG sensor, thetwo-point mounted FBG sensor, and the strain gage are situated40, 60, and 80 mm from the right end of the cable, respectively.

The cable is impacted transversely by a steel rod at the posi-tion D (middle of the cable) and the position E (1/3 length of thecable from the left end). The transient responses induced by theimpact loadings are measured by three sensors simultaneously.

The wavelength demodulation system of the FBG sensors isshown in Fig. 5(b). A broadband source with working wave-length ranging from 1525 to 1570 nm and output power up to20 dBm is used. In order to perform the simultaneous mea-surement of the two FBG sensors, a 1 2 coupler is used todivide the light beam from broadband source into two paths.The FBG behind the 1 2 coupler is used as filter to shapethe light source spectrum. The isolator and the circulator canprevent the reflected light of FBG filter and FBG sensor fromcoming back to the light source. The output light power fromthe circulator is transformed by the photodiode into the elec-trical power which is received by the oscilloscope. If the Braggwavelength of the FBG sensor lies in the positive linear sectionof the shaped source spectrum, the output power will increaseas the FBG sensor is elongated and will decrease as the FBGsensor is compressed.

C. Experimental Results and Discussion

Fig. 6 shows the transient responses of dynamic strain forthe cable under impact at position D measured with the twoFBG sensors and the strain gage simultaneously. The verticalaxis represents the voltage value recorded by the oscilloscope.The time period of the complicated transient responses is 0.1 sand the number of recorded samples is 2.5 . The peak-to-peak values of voltage are indicated in the figure. The mountingpositions of the three sensors are different due to the limitedspace of the cable, hence the measured signals for the transientstrain responses of the three sensors are different.

Fig. 7. The frequency spectra of the cable under impact at position D mea-sured with: (a) the surface mounted FBG sensor; (b) the two-point mountedFBG sensor; and (c) the strain gage.

The frequency spectrum of the cable can be constructed fromtaking the fast Fourier transform (FFT) from the time-domaintransient responses measured by the three sensors. Fig. 7 showsthe frequency spectra of the measured responses presented inFig. 6. The indicated peaks in the spectra correspond to the nat-ural frequencies of the cable. Due to the fact that the impact po-sition D is at the middle of the cable, the induced cable modesare mainly symmetric and the power distribution observed inFig. 7 is concentrated on the symmetric modes.


Fig. 8. The time-varied frequency spectra of the cable under impact at positionD measured by the strain gage.

The short-time Fourier transform (STFT) with Hammingwindow function is also taken in order to understand thefrequency variation with time of the experimental signals.Fig. 8 shows the time-varied frequency spectra of the measuredresponses presented in Fig. 6 for the strain gage. The poweris higher and the duration is longer for symmetric modescompared with antisymmetric modes.

It is noted from the dynamic strain measurement in Fig. 6 thatthe result obtained by the strain gage has relatively large noisebefore the measurement which is not observed by two FBG sen-sors. Although the nature frequencies, which correspond to thepeaks in the frequency spectra presented in Fig. 7, are almost thesame, we can see that relatively larger noise also appears in thespectrum for strain gage. The reason is that the steel cable usedin this study has curved surface and 1 mm in diameter. Hence, itis not easy to mount the strain gage in the steel cable with largecurvature and result in the noise appears at the transient responsemeasurement and frequency spectrum. This disadvantage is notobserved in the dynamic strain measurements by two FBG sen-sors. Hence, the FBG sensors are more suitable and convenientthan the strain gage to measure the dynamic strain for structureswith curved surface and large curvature.

According to the frequency spectra from FFT [Fig. 7(c)] andthe time-varied frequency spectra from STFT (Fig. 8) measuredby the strain gage, the natural frequencies and the maximumstrain amplitudes of the induced cable modes by impacting atposition D can be obtained. The relationship between the in-duced cable modes and the signal distortion limit (Fig. 3) dis-cussed in the previous section is shown in Fig. 9 with strain wavevelocity . It is worthy to note that the inducedcable modes by impact at position D are all located in the “goodregion,” which means that the experimental responses measuredby two FBG sensors (Fig. 6) are not distorted.

The measured natural frequencies of cable under impact atposition D using the FBG sensors and the strain gage (Table I)have excellent agreement with one another even though the re-sponses obtained from the three sensors are quite different. Thecomparison of the experimental natural frequencies (average ofthe three sensors) with theoretical and numerical results is listedin Table II. The natural frequencies from the string theory are

Fig. 9. The relationship between the signal distortion limit and the inducedcable modes by impacting at position D measured with the strain gage.

TABLE ITHE MEASURED NATURAL FREQUENCIES OF THE

CABLE UNDER IMPACT AT POSITION D

much lower than the experimental results due to the neglectof bending stiffness. The results from the cable theory consid-ering bending stiffness and the experiment have excellent agree-ment with error less than 1% for most modes. The finite-elementmethod (FEM) with package software ABAQUS is also used toanalyze numerically the natural frequencies of the cable. The re-sults from FEM also have excellent agreement with the resultsfrom the cable theory and experiment. The blank space “--” inTables I and II indicates that this resonant frequency is not ob-served from the experimental measurement.

Fig. 10 shows the transient responses of dynamic strain for thecable under impact at position E measured with the two FBGsensors and the strain gage simultaneously. The responses ofimpact at position E are very complicated and similar to the re-sponses of impact at position D in time domain. We can see fromFig. 10 that relatively larger noise is observed for the transientresponse of dynamic strain measured by strain gage. Fig. 11shows the frequency spectra from FFT of the transient responsesin Fig. 10 for the three sensors. Due to the fact that the impactposition E is asymmetric, the excited cable modes include thesymmetric and antisymmetric modes. Therefore, there are more


TABLE IITHE COMPARISON OF THE EXPERIMENTAL, THEORETICAL, AND NUMERICAL

NATURAL FREQUENCIES OF THE CABLE UNDER IMPACT AT POSITION D

Fig. 10. The transient responses of the cable under impact at position E.

cable modes observed in the frequency spectra of impact at posi-tion E (Fig. 11) than the spectra of impact at position D (Fig. 7).Fig. 12 shows the time-varied frequency spectra from STFT ofthe measured responses in Fig. 10 for the strain gage. Differentfrom the results of impact at position D in Fig. 8, both the sym-metric and the antisymmetric modes have long duration time forthe results of impact at position E in Fig. 12.

With the information of frequency spectra from FFT[Fig. 11(c)] and the time-varied frequency spectra from STFT[Fig. 12] measured by the strain gage, the natural frequenciesand the maximum strain amplitudes of the induced cable modesby impacting at position E can be obtained. The relationshipbetween the induced cable modes and the signal distortion limitis shown in Fig. 13. The induced cable modes by impacting atposition E are all located in the “good region,” which meansthe experimental responses of the two FBG sensors (Fig. 10)are also not distorted.

The measured natural frequencies of cable under impact atposition E using the different sensors (Table III) have excellentagreement with one another. The results of different impact po-sitions D and E (Tables I and III), also have excellent agree-ment with each other. The comparison of the experimental nat-

Fig. 11. The frequency spectra of the cable under impact at position E mea-sured with: (a) the surface mounted FBG sensor; (b) the two-point mountedFBG sensor; and (c) the strain gage.

ural frequencies (average of the three sensors) with theoreticaland numerical results is listed in Table IV. Similar to the resultsof impact at position D (Table II), the natural frequencies fromthe string theory are much lower than the experimental resultswhile the cable theory and the FEM have excellent agreementwith experiment.

In order to understand the relationship between the stringtheory and the cable theory, the variations of natural frequen-cies with respect to the ratio of length and diameter from the two


Fig. 12. The time-varied frequency spectra of the cable under impact at positionE measured by the strain gage.

Fig. 13. The relationship between the signal distortion limit and the inducedcable modes by impacting at position E measured with the strain gage.

TABLE IIITHE MEASURED NATURAL FREQUENCIES OF THE

CABLE UNDER IMPACT AT POSITION E

theories are shown in Fig. 14(a). With the increase of length-todiameter ratio, the natural frequencies from the two theoriesget close gradually, which means the string theory is suitable

TABLE IVTHE COMPARISON OF THE EXPERIMENTAL, THEORETICAL, AND NUMERICAL

NATURAL FREQUENCIES OF THE CABLE UNDER IMPACT AT POSITION E

Fig. 14. The variation of: (a) natural frequency and (b) natural frequencyratio with respect to length/diameter of the cable according to string and cabletheories.

to model very slender cable. Fig. 14(b) shows the natural fre-quency ratio of the string and cable theory. If the cable mod-eling error by the string theory less than 10% (i.e., the naturalfrequency ratio of the string and cable theories more than 0.9) isneeded, the length-to-diameter ratio of the cable must be more


than 1000. However, the length-to-diameter ratio of the cableused in our experiment is only 280, hence the string theory isnot suitable to predict the natural frequencies of this cable.

IV. CONCLUSION

The dynamic response of a FBG subjected to a sinusoidalstrain wave is investigated. The FBG under nonuniform dy-namic strain is considered to be piecewise-uniform, hence thereflectivity calculations for the FBG are carried out by using thecoupled-mode theory with transfer matrix method. A linear filterused to transform the variation of optical spectrum into lightpower signal is also considered. The simulation results indicatethat the increase of frequency and amplitude of strain result inthe distortion of output light power signal. The concept of thesignal distortion limit is established to understand the dynamicsensing range of FBG.

In addition to the dynamic sensing principle and simulationof FBG, the transient strain measurements of a suspended cableunder impact loadings using the FBG sensors is performed inthis study. Two kinds of FBG sensors, the surface mounted FBGsensor and the two-point mounted FBG sensor, are used as wellas the strain gage to measure the dynamic strain simultaneously.The natural frequencies of the cable are obtained from analyzingthe frequency spectrum of the measured transient response. Fordifferent sensors and different impact positions, the measurednatural frequencies all have excellent agreement. The experi-mental results also agree well with the cable theory and theFEM.

The traditional strain gage was usually used to measurestrains for flat surfaces. It is noted in this study that thereis noise appeared in the measurement of dynamic strain forstructures with curved surface and large curvature. Accordingto the experimental results, it is proved that the FBG sensorshave excellent ability to measure the transient strain responseand natural frequencies for steel cables with small diametersand large curvatures.

REFERENCES

[1] A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C.G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,”J. Lightw. Technol., vol. 15, pp. 1442–1463, Aug. 1997.

[2] K. O. Hill and G. Meltz, “Fiber Bragg gratings technology fundamen-tals and overview,” J. Lightw. Technol., vol. 15, pp. 1263–1276, Aug.1997.

[3] Y. J. Rao, “In-fibre Bragg grating sensors,” Measure. Sci. Technol., vol.8, pp. 355–375, Apr. 1997.

[4] M. Nellen, “Optical fiber Bragg grating for structural monitoring incivil engineering,” in Proc. 16th Congr. IABSE, Lucerne, Switzerland,2003, vol. 24, pp. 86–89.

[5] H. N. Li, D. S. Li, and G. B. Song, “Recent applications of fiber opticsensors to health monitoring in civil engineering,” Eng. Structures, vol.26, pp. 1647–1657, Sep. 2004.

[6] D.-J. Choi, C.-T. Rim, S. Kim, and Y. K. Kwak, “High sensitivity in-ductive sensing system for position measurement,” in Proc. 17th IEEEInstrum. Meas. Technol. Conf., 2000, vol. 2, pp. 595–599.

[7] S. C. Saxena and S. B. L. Seksena, “A self-compensated smart LVDTtransducer,” IEEE Trans. Instrum. Meas., vol. 38, pp. 748–753, 1989.

[8] S. T. Wu, S. C. Mo, and B. S. Wu, “An LVDT-based self-actuating dis-placement transducer,” Sens. Actuators A., vol. 141, no. 2, pp. 558–564,2008.

[9] K.-C. Chuang and C.-C. Ma, “Tracking control of a multilayerpiezoelectric actuator using a fiber Bragg grating displacement sensorsystem,” IEEE Trans. Ultrasonics, Ferroelectrics, and FrequencyControl, 2009.

[10] T. Erdogan, “Fiber grating spectra,” J. Lightw. Technol., vol. 15, pp.1277–1294, Aug. 1997.

[11] H. Tai, “Simple numerical simulation of strain measurement,” Proc.SPIE, vol. 4772, pp. 13–24, 2002.

[12] A. Sikora, “Impulse strain processing using uniform fibre Bragggrating. Numerical simulation,” Proc. SPIE, vol. 6159, pp.61592C–61592C, 2006.

[13] A. Minardo, A. Cusano, R. Bernini, L. Zeni, and M. Giordano, “Re-sponse of fiber Bragg gratings to longitudinal ultrasonic waves,” IEEETrans. Ultrasonics, Ferroelectrics, and Frequency Control, vol. 52, pp.304–312, Feb. 2005.

[14] H. Y. Ling, K. T. Lau, W. Jin, and K. C. Chan, “Characterization of dy-namic strain measurement using reflection spectrum from a fiber Bragggrating,” Opt. Commun., vol. 270, no. 1, pp. 25–30, Feb. 2007.

[15] A. D. Kersey, T. A. Berkoff, and W. W. Morey, “Multiplexed fiberBragg grating strain-sensor system with a fiber Fabry-Perot wavelengthfilter,” Opt. Lett., vol. 18, no. 16, pp. 1370–1372, Aug. 1993.

[16] Y. L. Lo, “In-fiber Bragg grating sensors using interferometric in-terrogations for passive quadrature signal processing,” IEEE Photon.Technol. Lett., vol. 10, pp. 1003–1005, Jul. 1998.

[17] A. D. Kersey, T. A. Berkoff, and W. W. Morey, “High-Resolutionfiber-grating based strain sensor with interferometric wavelength-shiftdetection,” Electron. Lett., vol. 28, no. 3, pp. 236–238, Jan. 1992.

[18] M. H. Song, S. Z. Yin, and P. B. Ruffin, “Fiber Bragg grating strainsensor demodulation with quadrature sampling of a Mach-Zehnder in-terferometer,” Appl. Opt., vol. 39, no. 7, pp. 1106–1111, Mar. 2000.

[19] R. W. Fallon, L. Zhang, L. A. Everall, J. A. R. Williams, and I. Bennion,“All-Fibre optical sensing system: Bragg grating sensor interrogated bya long-period grating,” Measure. Sci. Technol., vol. 9, pp. 1969–1973,Dec. 1998.

[20] H. J. Patrick, G. M. Williams, A. D. Kersey, J. R. Pedrazzani, and A.M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber gratingsensor for strain/temperature discrimination,” IEEE Photon. Technol.Lett., vol. 8, pp. 1223–1225, Sep. 1996.

[21] V. Bhatia, D. Campbell, R. O. Claus, and A. M. Vengsarkar, “Simulta-neous strain and temperature measurement with long-period gratings,”Opt. Lett., vol. 22, no. 9, pp. 648–650, May 1997.

[22] S. Kim, S. Kim, J. Kwon, and B. Lee, “Fiber Bragg grating strainsensor demodulator using a chirped fiber grating,” IEEE Photon.Technol. Lett., vol. 13, pp. 839–841, Aug. 2001.

[23] I. C. Song, S. K. Lee, S. H. Jeong, and B. H. Lee, “Absolute strainmeasurements made with fiber Bragg grating sensors,” Appl. Opt., vol.43, no. 6, pp. 1337–1341, Feb. 2004.

[24] J. Mora, J. L. Cruz, M. V. Andres, and R. Duchowicz, “Simple high-resolution wavelength monitor based on a fiber Bragg grating,” Appl.Opt., vol. 43, no. 4, pp. 744–749, Feb. 2004.

[25] S. C. Kang, S. Y. Kim, S. B. Lee, S. W. Kwon, S. S. Choi, and B.Lee, “Temperature-independent strain sensor system using a tilted fiberBragg grating demodulator,” IEEE Photon. Technol. Lett., vol. 10, pp.1461–1463, Oct. 1998.

[26] H. M. Irvine, Cable Structures. Cambridge, MA: MIT Press, 1981.[27] C. L. Lee and N. C. Perkins, “Experimental investigation of isolated

and simultaneous internal resonances in suspended cables,” J. Vibra-tion and Acoustics, vol. 117, pp. 385–391, Oct. 1995.

[28] S. P. Cheng and N. C. Perkins, “Theoretical and experimental anal-ysis of the forced response of sagged cable/mass suspensions,” J. Appl.Mech., vol. 61, pp. 944–948, Dec. 1994.

[29] H. P. Lin and N. C. Perkins, “Free vibration of complex cable/masssystems: Theory and experiment,” Journal of Sound and Vibration, vol.179, no. 1, pp. 131–149, Jan. 1995.

[30] J. C. Russell and T. J. Lardner, “Experimental determination of fre-quencies and tension for elastic cables,” J. Eng. Mech., vol. 124, no.10, pp. 1067–1072, Oct. 1998.

[31] M. S. Triantafyllou and L. Grinfogel, “Natural frequencies and modesof inclined cables,” J. Structural Eng., vol. 112, no. 1, pp. 139–148,Jan. 1986.

[32] H. Zui, T. Shinke, and Y. Namita, “Practical formulas for estimationof cable tension by vibration method,” J. Structure Eng., vol. 122, no.6, pp. 651–656, Jun. 1996.

[33] B. H. Kim and T. Park, “Estimation of cable tension force using the fre-quency-based system identification method,” J. Sound and Vibration,vol. 304, pp. 660–676, Jul. 2007.

[34] B. H. Kim, T. Park, H. Shin, and T. Y. Yoon, “A comparative studyof the tension estimation methods for cable supported bridges,” SteelStructures, vol. 7, pp. 77–84, Mar. 2007.


Chien-Ching Ma was born in Taipei, Taiwan, in1956. He received the B.S. degree in agricultureengineering from the National Taiwan University,Taipei, Taiwan, in 1978, and the M.S. and Ph.D.degrees in mechanical engineering from BrownUniversity, Providence, RI, in 1982 and 1984,respectively.

From 1984 to 1985, he worked as a Postdoctoralin the Engineering Division of Brown University. In1985, he joined the faculty of the Department of Me-chanical Engineering, National Taiwan University, as

an Associate Professor. He was promoted to Full Professor in 1989. He is cur-rently a Distinguished Professor of the Department of Mechanical Engineering,National Taiwan University. His research interests are in the fields of wave prop-agation in solids, fracture mechanics, fiber Bragg grating sensors, solid me-chanics, piezoelectric material, and vibration analysis.

Prof. Ma has received the Distinguished Research Award from the NationalScience Council (NSC) of Taiwan three times. He has been elected as Fellow ofthe American Society of Mechanical Engineers.

Cheng-Wei Wang received the B.S. and M.S. de-grees in mechanical engineering from the NationalTaiwan University, Taipei, Taiwan, in 2006 and 2008,respectively.

His research interests are in the field of vibrationanalysis and fiber Bragg grating sensors.

Documents

Transient Strain Measurements of a Suspended Cable Under Impact Loading Using FBG Sensors