sensors Article

Strain Modal Analysis of Small and Light Pipes Using Distributed Fibre Bragg Grating Sensors Jun Huang 1,3, *, Zude Zhou 1 , Lin Zhang 2 , Juntao Chen 1 , Chunqian Ji 3 and Duc Truong Pham 3 1 2 3

*

School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan 430070, China; [email protected] (Z.Z.); [email protected] (J.C.) School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China; [email protected] School of Engineering, University of Birmingham, Birmingham B15 2TT, UK; [email protected] (C.J.); [email protected] (D.T.P.) Correspondence: [email protected]; Tel.: +44-781-8909-889

Academic Editors: Christophe Caucheteur and Tuan Guo Received: 29 July 2016; Accepted: 13 September 2016; Published: 25 September 2016

Abstract: Vibration fatigue failure is a critical problem of hydraulic pipes under severe working conditions. Strain modal testing of small and light pipes is a good option for dynamic characteristic evaluation, structural health monitoring and damage identification. Unique features such as small size, light weight, and high multiplexing capability enable Fibre Bragg Grating (FBG) sensors to measure structural dynamic responses where sensor size and placement are critical. In this paper, experimental strain modal analysis of pipes using distributed FBG sensors ispresented. Strain modal analysis and parameter identification methods are introduced. Experimental strain modal testing and finite element analysis for a cantilever pipe have been carried out. The analysis results indicate that the natural frequencies and strain mode shapes of the tested pipe acquired by FBG sensors are in good agreement with the results obtained by a reference accelerometer and simulation outputs. The strain modal parameters of a hydraulic pipe were obtained by the proposed strain modal testing method. FBG sensors have been shown to be useful in the experimental strain modal analysis of small and light pipes in mechanical, aeronautic and aerospace applications. Keywords: small and light pipe; strain modal analysis; Fibre Bragg Grating; strain mode shape

1. Introduction Vibration fatigue of hydraulic pipes subjected to dynamic forces is one of the major failure factors in the mechanical, automotive, and aeronautic industries [1]. Therefore, experimental modal analysis or testing of pipes is essential for understanding and evaluating their dynamic characteristics in design and operation. Electrical strain gauges and accelerometers are commonly used for strain and displacement modal analysis of pipes to consider dynamic properties and fatigue strength [2]. However, these conventional approaches face significant challenges where the size, weight, amount and location of sensing elements are issues, especially for the small and light pipes in the aeronautic and aerospace fields [3,4]. For example, strong electromagnetic interference may affect the results of strain gauges and accelerometers for multipoint and accurate response measurements. Accelerometers installed on the tested pipes would bring a mass-loading effect which would affect the dynamic characteristics of the pipes [5,6]. Unlike conventional electrical techniques, Fibre Bragg Grating (FBG) sensors adopt light as a sensing signal, which gives FBG sensors various attractive characteristics, such as small size, light weight, immunity to electromagnetic interference, and large multiplexing capability [7–9]. In recent years, FBG sensors have been investigated and applied for dynamic strain measurement and modal Sensors 2016, 16, 1583; doi:10.3390/s16101583

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analysis [10–13]. For example, distributed FBG sensors were used for strain modal analysis and damage identification of a plate structure [14], and embedded fibre grating sensors were utilised in experimental modal analysis on an aircraft wing model [15]. FBG optical sensors were chosen for monitoring the dynamic response of a composite antenna sub-reflector [16]. Strain modal analysis of a helicopter main rotor blade using FBG sensors was reported [17] and modal analysis of a helicopter blade model using a long FBG interrogated by optical frequency domain reflectometry was demonstrated [18]. However, the strain modal measurement of hydraulic pipes has not been studied widely. On the basis of the previous research of the dynamic strain measurement of hydraulic systems [19], FBG sensors are used for experimental strain modal analysis of small and light pipes in this paper. The theoretical backgrounds including the matrix of strain frequency response function and the modal parameter identification method are described. An experimental strain modal testing and analysis case of a cantilever pipe using distributed FBG sensors is carried out. As a reference, the natural frequencies and strain mode shapes are calculated with finite element simulation. The analysis results and modal assurance criteria (MAC) values indicate that the natural frequencies and strain mode shapes of the tested pipe acquired by FBG sensors are in good agreement with the results obtained by the reference accelerometer and simulation outputs. The strain modal parameters of a hydraulic pipe were obtained by the proposed strain modal testing method using FBG sensors. Distributed FBG sensors could be widely used in the experimental strain modal analysis of small and light pipes in mechanical, aeronautic and aerospace applications. 2. Theoretical Background Although the strain mode shape and displacement mode shape are both intrinsic dynamic characteristics of a structure and they correspond to each other, the strain mode shape can capture local structural defects (such as cracks) more effectively [20]. By investigating the stress-strain distribution, experimental strain modal analysis is important for vibration fatigue analysis and damage identification [21]. Their light weight and excellent multiplexing capability enable FBG sensors to capture not only the local key parts of the structure but also the overall dynamic response information in experimental strain modal analysis [22]. 2.1. Strain Modal Analysis The frequency response function (FRF) is defined as the ratio of the output response of a structure due to an excitation force, which can be obtained by transforming the measured time data (input load and output response) from the time domain to the frequency domain using the fast Fourier transform (FFT) algorithm. The strain frequency response function (SFRF) is a fundamental measurement that isolates the inherent dynamic properties of a mechanical structure. The excitation force and structural strain responses have to be measured simultaneously to define the SFRFs. Modal parameters (frequency, damping, and strain mode shape) can be obtained from a set of SFRF measurements. The matrix of the SFRF [Hε ] is given by [20]:  m

[ H ε ] No Ni =

  Y ∑ r  r =1

ε ϕ ψ1r 1r ε ϕ ψ2r 1r ··· ψεNo r ϕ1r

ε ϕ ψ1r 2r ε ϕ ψ2r 2r ··· ψεNo r ϕ2r

··· ··· ···

 −1  Yr = kr − ω 2 mr + jωcr

ε ϕ ψ1r Ni r ε ϕ ψ2r Ni r ··· ψεNo r ϕ Ni r

    

(1)

(2)

where Ni represents the number of inputs and No represents the number of output measurements; m is the total number of modes considered; ψirε is the normalised value of the r-th strain mode at point i, and φlr is the normalised value of the r-th displacement mode at point l; kr , mr , and cr are the r-th modal stiffness, modal mass and modal damping, respectively; ω is the frequency of excitation. The

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columns of the matrix correspond to the strain responses due to the excitation points along the rows of the matrix. The elements of the SFRF matrix can be expressed as: m

Hilε =

ψirε ϕlr 2 r =1 k r − ω mr + jωcr

∑

(3)

The element of the SFRF matrix is the ratio between the Fourier transform of the strain response at point i induced by an excitation force applied at point l. From Equations (2) and (3), it can be found that any column of the SFRF matrix contains the information about the strain mode ψrε for all the modes, any element of the SFRF matrix contains the information about kr , mr , and cr for all the modes. Therefore, it is only necessary to excite one selected point and acquire the strain responses at all the measurement points, which enable the modal parameters (frequency, damping, strain mode shape) to be obtained after data processing. 2.2. Strain Modal Parameter Identification After the SFRFs are acquired by measurement and data processing of strain responses, the next step is the modal parameter identification, which is the process of estimating a parametric model (poles and mode shapes) [16]. There are several methods proposed for modal parameter identification such as the complex exponential algorithm (CEA), least squares complex exponential (LSCE), polyreference frequency domain (PFD), and eigensystem realisation algorithm (ERA). The SFRF can be described by the residues (in the numerator) and the poles (in the denominator) for each of the modes of the system with m DOFs (degrees of freedom). m

Hilε

∑

(s) =

r =1



∗ Rilr Rilr − 2j (s − sr ) 2j (s − sr∗ )

Rilr = p



ψirε ϕlr

1 − ζ r2 ωr mr q sr = −ζ r ωr + j 1 − ζ r2 ωr

(4)

(5) (6)

where Rilr and sr are the r-th modal residue and pole, respectively; ω r and ζ r are the r-th undamped natural frequency and damping ratio, respectively. The method of rational fraction orthogonal polynomial (RFOP) is used to estimate the poles of the system in this paper. The poles of the system can be obtained by power polynomial solutions and stabilisation diagrams. The stabilisation criterion about frequency, damping and mode shape can be utilised to acquire the number of the poles. When the poles are estimated, the method of least-squares frequency-domain (LSFD) was used to estimate the strain mode shape vector from the measured SFRFs. The estimation model of the strain frequency response function is given by: N

ε

[ H (ω )] No Ni =

∑

r =1



 [ Rr ] [ Rr∗ ] [ L] − − 2 + [U ] ∗ 2j ( jω − sr ) 2j ( jω − sr ) ω

(7)

where [L] is the lower residual matrix, and [U] is the upper residual matrix. When the poles of system sr are applied to Equation (7), the strain frequency response function can be transferred to the linear equation related to residues, upper residuals and lower residuals, which could be solved by the least square method.

the least square method. As any column of the SFRF matrix contains all the information about the strain modes, according to Equation (4) the residue vector [Rr] of one column can be expressed as: T  1r   R1r     Sensors 2016, 16, 1583 4 of 11 R  lr 2r   2 r   (8)  Rr        1   r2 r mr   the strain modes, according As any column of the SFRF matrixcontains  all the information about  No r as: column can be expressed No r   Rone to Equation (4) the residue vector [Rr ] of

 T strain mode shape  vector  can be obtained. According to Equation (8), thenormalised ε ψ1r R1r    ψε   2r   R2r  ϕlr 3. Experiment and Comparison with Finite Element Analysis (FEA)    (8) [ Rr ] =  .  = p . Simulation  2 . 1 − ζ r ωr m r   ..   .  To investigate the strain modal analysis of pipes using distributed FBG sensors, a strain modal ψεNo r R No r testing and analysis case was carried out. The measured modal parameters of natural frequencies and According strain mode were with strain the results the reference to shapes Equation (8), compared the normalised modeof shape vector canaccelerometer be obtained. and finite element analysis (FEA). 3. Experiment and Comparison with Finite Element Analysis (FEA) Simulation 3.1. Experimental Setup To investigate the strain modal analysis of pipes using distributed FBG sensors, a strain modal The schematic diagram the experimental strain modal setupofisnatural depicted in Figure and 1. It testing and analysis case was of carried out. The measured modal testing parameters frequencies mainly contains a small-diameter pipe, accelerometer, impact hammer,and FBGfinite sensors, data strain mode shapes were comparedcantilever with the results of the reference accelerometer element acquisition unit, FBG interrogator, laptop, and optical fibre. The impact hammer (2302-10, Endevco, analysis (FEA). San Juan Capistrano, California, CA, USA) with a load cell was used to excite the pipe at the exciting 3.1. Experimental point (as shown Setup in Figure 1). An accelerometer (4508B, B & K, Næ rum, Denmark) with mass of 4.8 g was The bonded on surfaceofof pipe as a strain reference measurement theinacceleration schematicthe diagram thethe experimental modal testing setuptois detect depicted Figure 1. It response of the tested pipe along the X axis direction, as shown inimpact Figurehammer, 1. ElevenFBG FBGsensors, sensors were mainly contains a small-diameter cantilever pipe, accelerometer, data installed on the surface of the tested pipe (as shown in Figure 1) by epoxy adhesive to measure strain acquisition unit, FBG interrogator, laptop, and optical fibre. The impact hammer (2302-10, Endevco, responses at differentCA, points. dataaacquisition andused signal analysis 7700,point B & (as K,) San Juan Capistrano, USA)Awith load cell was to excite thesystem pipe at(PULSE the exciting was used to collect the experimental data of the accelerometer and impact A shown in Figure 1). and An process accelerometer (4508B, B & K, Nærum, Denmark) with mass ofhammer. 4.8 g was dynamic FBG interrogator with a maximum sample rate of 4 kHz, a minimum wavelength bonded on the surface of the pipe as a reference measurement to detect the acceleration response of resolution of 1 along pm, and channels was to detect the wavelength shifts of FBG sensors the tested pipe the four X axis direction, asemployed shown in Figure 1. Eleven FBG sensors were installed on in real time. The dynamic FBG interrogator is based on the tunable Fabry-Perot (F-P) filter the surface of the tested pipe (as shown in Figure 1) by epoxy adhesive to measure strain responses technology. at different points. A data acquisition and signal analysis system (PULSE 7700, B & K,) was used to diameter, wall thickness and of the tested pipe are 13 mm, 1.5 mm,FBG and collectThe andinner process the experimental data of thelength accelerometer and impact hammer. A dynamic 985 mm, respectively. One end of the pipe was fixed and the other one is free as shown in Figure 1. interrogator with a maximum sample rate of 4 kHz, a minimum wavelength resolution of 1 pm, and The material of the tested pipe is 304 stainless steel and its density, Young’s modulus, and Poisson’s four channels was employed to detect the wavelength shifts of FBG sensors in real time. The dynamic 3, 200 GPa, and 0.31, respectively. ratiointerrogator are 7930 kg/m FBG is based on the tunable Fabry-Perot (F-P) filter technology.

Figure Figure1. 1. Schematic Schematic diagram diagram of of experimental experimentalsetup. setup.

The inner diameter, wall thickness and length of the tested pipe are 13 mm, 1.5 mm, and 985 mm, respectively. One end of the pipe was fixed and the other one is free as shown in Figure 1. The material of the tested pipe is 304 stainless steel and its density, Young’s modulus, and Poisson’s ratio are 7930 kg/m3 , 200 GPa, and 0.31, respectively.

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The FBG sensors were glued on the surface of the tested pipe and spaced in intervals of 90 mm along the length (Z direction) to detect the strain responses under impacting load induced by the hammer. The initial wavelengths locations sensors are listed 1. Theses The FBG sensors were glued and on the surfaceofofthe theFBG tested pipe and spacedininTable intervals of 90 FBG mm sensors were multiplexed in two opticalthe fibre cables connected to two channelsload (Ch1: FBG1, by FBG2, along the length (Z direction) to detect strain responses under impacting induced the FBG3, FBG4, FBG6; Ch2:and FBG7, FBG8,of FBG9, FBG10 and FBG11) of in theTable FBG 1. interrogator. hammer. TheFBG5 initialand wavelengths locations the FBG sensors are listed Theses FBG sensors were multiplexed in two optical fibre cables connected to two channels (Ch1: FBG1, FBG2, The locations of FBGFBG9, (Fibre FBG10 Bragg Grating) sensors. FBG3, FBG4, FBG5 andTable FBG6;1.Ch2: FBG7, FBG8, and FBG11) of the FBG interrogator. Sensor Number Wavelength (nm) Distance from Fixed End (mm) Table 1. The locations of FBG (Fibre Bragg Grating) sensors. FBG1 1295.388 35 FBG2 1290.417 125 Sensor Number Wavelength (nm) Distance from Fixed End (mm) FBG3 1288.557 215 FBG1 1295.388 35 FBG4 1309.855 305125 FBG2 1290.417 FBG5 1304.282 395215 FBG3 1288.557 FBG4 1309.855 FBG6 1298.596 485305 FBG5 1304.282 FBG7 1309.965 575395 FBG6 1298.596 485 FBG8 1288.398 665 FBG7 1309.965 575 FBG9 1290.245 755665 FBG8 1288.398 FBG10 1304.615 845755 FBG9 1290.245 FBG10 1304.615 FBG11 1298.676 935845 FBG11

1298.676

935

3.2. Finite Element Analysis 3.2. Finite Element Analysis To validate experimental results, numerical modal analysis of the tested pipe under To validate experimental numerical modal analysis of the the software tested pipe under to investigation investigation was performed results, with finite element analysis using ANSYS determine was performed withand finite element analysis using the software ANSYS to determine natural frequencies natural frequencies normalised strain mode shapes. The finite element geometry model with the and normalised strain mode shapes. The finite element geometry model with the mesh of the is mesh of the pipe is developed, as shown in Figure 2. The left end of the pipe is fixed and thepipe other developed, ascoordinate shown in Figure 2. is The left end of the pipe is fixed theAother one is free.onAthe coordinate one is free. A system created at the fixed end of theand pipe. solution path surface system is created at the fixed end of the pipe. A solution path on the surface of the pipe along the of the pipe along the length (Z direction) is chosen to acquire the strain responses in different strain length (Z direction) is chosen to acquire the strain responses in different strain mode shapes. mode shapes.

Figure 2. Finite element geometry model.

4. 4. Results Resultsand andDiscussion Discussion The dynamic acceleration The impacting impacting force force and and dynamic acceleration response response were were measured measured simultaneously simultaneously as as the the input The time time domain domain signals signals associated associated input excitation excitation and and output output response response of of the the acceleration acceleration FRF. FRF. The with with the the impact impact hammer hammer and and related related acceleration acceleration response response acquired acquired by by the the accelerator accelerator are are shown shown in in Figure 3a,c, respectively. The frequency domain of the impacting force is obtained by using Figure 3a,c, respectively. The frequency domain of the impacting force is obtained by using the the FFT FFT algorithm, algorithm, as as shown shown in in Figure Figure 3b. 3b. Figure Figure 3d 3d shows shows the the estimated estimated acceleration-over-force acceleration-over-force FRF FRF and and the the identified first four natural frequencies which occur at 14.8 Hz, 94.7 Hz, 258.0 Hz and 500.0 Hz with identified first four natural frequencies which occur at 14.8 Hz, 94.7 Hz, 258.0 Hz and 500.0 Hz with a afrequency frequencyresolution resolution Hz. ofof 0.10.1 Hz.

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15

10

15 10

(a) (a)

(b) (b)

10-1

10

10-2

Force Force (N) (N)

5

Force Force (N) (N)

-1

5 0 0 -5

10-2

10-3 10-3

-5 10-4

-10 0.0 -10 0.0

1.0

1.5

2.0

0.5

1.0(s) Time

1.5

2.0

10-4

0

100

200

400

500

600

0

100

200 300 (Hz)400 Frequency

500

600

103

(c) (c) 2 2 Amplitude ((m/s Amplitude ((m/s )/N))/N)

150 100

300

Frequency (Hz)

Time (s)

150

2 2 Acceleration Acceleration ) ) (m/s(m/s

0.5

100 50 50 0 0 -50 -50 -100 -100 0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0(s) Time

1.5

2.0

(d) (d) 92.7

103 102 102 101 101 100

500.0 500.0

258.0 258.0

92.7

14.8 14.8

100 10-1 10-1 10-2 10-2 10-3 10-3

0

100

200

400

500

600

0

100

200 300 (Hz)400 Frequency

300

500

600

Time (s) Frequency (Hz) Figure 3. Signals of impacting forceand andacceleration. acceleration. (a) force; (b) (b) frequency spectrum of Figure 3. Signals of impacting force (a)impacting impacting force; frequency spectrum of

Figure 3. Signals of impacting force and(c) acceleration. (a)(d) impacting force; (b) frequency spectrum of impacting force; acceleration; acceleration FRF. impacting force; (c) acceleration; (d) acceleration FRF.

impacting force; (c) acceleration; (d) acceleration FRF.

The strain responses of the tested pipe under the impacting load induced by the hammer are

The strain thethe tested pipe under the impacting load induced by the The strain of tested pipe undersignal the impacting load induced by (FBG1) the hammer are are captured byresponses the responses 11 FBG of sensors. The time domain of one of the FBG sensors ishammer shown captured by the 11 sensors. Thetime time domain signal of the FBG sensors (FBG1) is shown captured by the sensors. The domain signal ofone one the sensors (FBG1) isan shown in Figure 4a.11 AsFBG theFBG response does not decay to zero before theofof end ofFBG the sampling window, in Figure 4a. As the response does not decay to zero before the end of the sampling window, an exponential window is used to reduce the leakage effects in the response spectrum. Using the FFT in Figure 4a. As the response does not decay zero before the end of the sampling window, an exponential window is used to reduce the leakage in the algorithm, the strain SFRF obtained bythe FBG1 sensoreffects is shown in response Figure 4b.spectrum. The otherUsing SFRFs of FFT the exponential window is used to reduce leakage effects inthe the response spectrum. Using the FFT algorithm, the strain SFRF obtained by FBG1 sensor is shown in Figure 4b. The other SFRFs of the FBG FBG sensors could also be obtained in the same way. Therefore, one column of the SFRF matrix algorithm, the strain SFRF obtained by FBG1 sensor is shown in Figure 4b. The other SFRFs of the FBG sensors couldforalso obtained the sameidentification way. Therefore, one column could be acquired the be strain modalin parameter of the tested pipe.of the SFRF matrix sensors could also be obtained in the same way. Therefore, one column of the SFRF matrix could be could be acquired for the strain modal parameter identification of the tested pipe. acquired for the strain modal parameter identification of the tested pipe.

(a) (a)

60 60

3

10 102

Amplitude ( Amplitude ( ) )

30

Strain Strain ()()

(b) (b)

103

30 0 0 -30 -30 -60 -60

102 101 101 100 100 10-1 10-1 10-2 10-2

0

2

4

6

8

10

0

100

200

400

500

600

0

2

4 Time (s) 6

8

10

0

100

200Frequency 300 (Hz)400

500

600

Time (s)

300

Frequency (Hz)

Figure 4. Signal of FBG sensor (FBG1). (a) strian response; (b) strian frequency response function.

Figure 4. Signal of FBG sensor (FBG1). response;(b) (b)strian strian frequency response function. Figure 4. Signal of FBG sensor (FBG1).(a) (a)strian strian response; frequency response function. The RFOP method mentioned above was applied to estimate the modal parameters of the pipe method mentioned above was applied to estimate the modal parameters from The a setRFOP of SFRFs by FBG sensors. According to Equations poles of the pipe The RFOP methodmeasured mentioned above was applied to estimate the(4)–(6), modalthe parameters of the pipe from a set of SFRFs measured by FBG sensors. According to Equations (4)–(6), the poles of the pipe

from a set of SFRFs measured by FBG sensors. According to Equations (4)–(6), the poles of the pipe could be estimated. The stabilisation diagram is represented in Figure 5. As the exponential window adds artificial damping to all of the modes of the pipe in a known manner, the estimated damping

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ratios after curve fitting could be obtained by subtracting the artificial damping. The estimated first four modal parameters (natural frequencies and damping ratios) of the tested pipe from a set of SFRFs obtained by FBG sensors can be found in Table 2.

Figure 5. Stabilisation diagram. Table 2. The estimated natural frequencies and damping ratios of the pipe. Mode Number

Natural Frequency (Hz)

Damping Ratio (%)

1 2 3 4

14.8 92.8 259.0 507.3

0.471 0.204 0.195 0.221

The comparisons of natural frequencies from experimental measurements (FBG sensors and accelerometer) and numerical analysis (FEA simulation) are depicted in Figure 6. The experimental results demonstrate a good agreement between the natural frequencies provided by FBG sensors and 2016, 16, 1583 8 of 12 FEA Sensors simulation calculation results. 600

Natural frequency [Hz]

500

FBG sensors FEA simulation accelerometer

400 300 200 100 0 1

2

3

4

Mode number Figure 6. Comparisonofofexperimental experimental and and analysis Figure 6. Comparison analysisnatural naturalfrequencies. frequencies.

-0.5

FEA simulation FBG measured

1.0

n mode shape 2

n mode shape 1

0.0

0.5 0.0 -0.5

FEA simulation FBG measured

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600 Sensors 2016, 16, 1583

FBG sensors FEA simulation accelerometer

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Natural frequency [Hz]

500

It is observed that unlike the natural frequencies obtained by FBG sensors, the third and fourth 400 mode natural frequencies captured by the accelerometer are, respectively, 0.35% and 0.93% smaller than the results calculated by FEA simulation. Similarly, the accelerometer-measured third and fourth 300 mode natural frequencies are 0.39% and 1.44% smaller than the results obtained by FBG sensors, respectively. It is mainly caused by the mass of the accelerometer placed on the pipe. The mass added 200 to the pipe by the accelerometer has a negative effect on the modal testing, which causes the measured natural frequencies to be smaller than the correct values. After the poles of100 the tested pipe are estimated, the LSFD method mentioned above was used to estimate the strain mode shape vector according to Equations (7) and (8). All the strain mode shapes here are normalised with0 respect to the strain amplitude of the point from which the distance is 35 mm to the fixed end in the length direction. of the 1 The comparisons 2 3 normalised 4 strain mode shapes for the first four modes in the length direction (Z direction) obtained by FEA simulation calculation and FBG Mode number sensor measurements are shown in Figure 7. It is observed that the normalised strain mode shapes of Figure Comparison of experimental and analysis natural frequencies. these two approaches are6.in good agreement.

Strain mode shape 2

Strain mode shape 1

1.0

FEA simulation FBG measured

0.0

-0.5

(a)

-1.0 0

200

400

600

800

FEA simulation FBG measured

0.5 0.0 -0.5

(b)

-1.0

0

1000

200

Strain mode shape 3

Strain mode shape 4

FEA simulation FBG measured

1.0 0.5 0.0 -0.5

(c)

-1.0 0

200

400

600

Length (mm)

400

600

800

1000

Length (mm)

Length (mm)

800

1000

FEA simulation FBG measured

1

0

-1

(d) 0

200

400

600

800

1000

Length (mm)

Figure 7. Comparison Comparison of normalised strain FEA simulation simulation and Figure 7. of normalised strain mode mode shapes shapes between between the the results results of of FEA and experiment, the strain mode shapes along the Z direction of the pipe: (a) mode 1; (b) mode experiment, the strain mode shapes along the Z direction of the pipe: (a) mode 1; (b) mode 2; (c) mode 2; 3; (c) (d) mode mode 3; 4. (d) mode 4.

Modal assurance criteria (MAC) are usually employed to determine the quality of the measured Modal assurance criteria (MAC) are usually employed to determine the quality of the measured mode shapes, which ranges between 0 and 1 [23]. If the value of the MAC is close to 0, then there is mode shapes, which ranges between 0 and 1 [23]. If the value of the MAC is close to 0, then there is little correlation between the two modes. However, if the value of the MAC approaches 1.0, it means little correlation between the two modes. However, if the value of the MAC approaches 1.0, it means that the two modes are highly correlated and almost identical. The correlation of the strain mode that the two modes are highly correlated and almost identical. The correlation of the strain mode shapes between experimental testing by FBG sensors and FEA calculation (shown in Figure 7) was shapes between experimental testing by FBG sensors and FEA calculation (shown in Figure 7) was calculated using the MAC and the results are shown in Figure 8. The diagonal MAC values are close calculated using the MAC and the results are shown in Figure 8. The diagonal MAC values are close to to 1.0, and the non-diagonal values are nearly close to zero. It is indicated that there is a high 1.0, and the non-diagonal values are nearly close to zero. It is indicated that there is a high correlation correlation between the two strain mode shapes (acquired by experiments using FBG sensors and between the two strain mode shapes (acquired by experiments using FBG sensors and FEA simulation), FEA simulation), and the experimental modal testing results by the FBG sensors are accurate and and the experimental modal testing results by the FBG sensors are accurate and reliable. reliable.

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Figure 8. MAC correlation strainmode mode shapes shapes between of of experimental testing and and Figure 8. MAC correlation ofofstrain betweenthe theresults results experimental testing FEA calculation. Figure 8. MAC correlation of strain mode shapes between the results of experimental testing and FEA calculation. FEA calculation.

5. Application 5. Application 5. Application The proposed strain modal analysis method using FBG sensors was used to measure the strain The proposed strain modal analysis method using FBGsensors sensors was used toThe measure the modal parameter ofstrain hydraulic pipe in the hydraulic system, as shown in used Figure tested pipestrain The proposed modal analysis method using FBG was to 9. measure the strain modal parameter of hydraulic hydraulic pipe innine the hydraulic system, Figure The pipe was filled with oil and FBG sensors were as adopted toinin detect its9.9. dynamical modal parameter of hydraulic pipe in the hydraulic system, asshown shown Figure The tested testedstrain pipe was filledresponses withfilled hydraulic oil andofnine FBGnine sensors were adopted to detect its dynamical strainof responses on the hydraulic surface the under impacting load induced by to a hammer. interval the was with oil pipe and FBG sensors were adopted detect itsThe dynamical strain FBG sensors isthe 100 mm and installation positions of load the sensors shown in of Figure 9.ofThe responses onthe surface of the pipe under impacting induced by aare hammer. The interval the on the surface of pipe under impacting load induced byFBG a hammer. The interval the FBG sensors hydraulic platform includes a motor, pump, hydraulic oil tank, hydraulic valves, tested hydraulic FBG sensors is 100 mm and installation positions of the FBG sensors are shown in Figure 9. The is 100 mm and installation positions of the FBG sensors are shown in Figure 9. The hydraulic platform pipe, fixtures. The tested hydraulic pipe was fixedvalves, byoil two fixtures at twovalves, ends oftested the pipe. hydraulic platform includes a motor, pump, hydraulic tank, hydraulic hydraulic includes aand motor, pump, hydraulic oil tank, hydraulic tested hydraulic pipe, and fixtures. The pipe, and fixtures. The tested hydraulic pipe was fixed by two fixtures at two ends of the pipe.

tested hydraulic pipe was fixed by two fixtures at two ends of the pipe.

Figure 9. Hydraulic platform and tested pipe with FBG sensors. Figure 9. Hydraulicplatform platform and and tested tested pipe sensors. Figure 9. Hydraulic pipewith withFBG FBG sensors.

A set of SFRFs could be obtained from the data acquired by the FBG sensors, and the first four natural frequencies and damping ratios of the tested hydraulic pipe are estimated and shown in Table 3. The first four normalised strain mode shapes along the length direction of the tested pipe are shown

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A set of SFRFs could be obtained from the data acquired by the FBG sensors, and the first four Sensors 2016, 16, 1583 10 of in 11 natural frequencies and damping ratios of the tested hydraulic pipe are estimated and shown Table 3. The first four normalised strain mode shapes along the length direction of the tested pipe are shown10. in The Figure 10. distributions The strain distributions the under tested different pipe under different could be in Figure strain of the testedofpipe modes couldmodes be found in the found in the strain mode shapes. strain mode shapes. Table ratios of of the the tested tested pipe. pipe. Table 3. 3. The The natural natural frequencies frequencies and and damping damping ratios

Mode Number Mode Number 1 21 2 3 3 44

Damping Ratio (%) Damping Ratio (%) 0.665 0.665 0.305 0.305 0.145 0.145 0.195 0.195

Mode 1 Mode 2 Mode 3 Mode 4

1.0

0.5

Strain mode shape

Natural Frequency (Hz) Natural Frequency (Hz) 74.6 74.6 209.8 209.8 410.4 410.4 670.6 670.6

0.0

-0.5

-1.0 0

100

200

300

400

500

600

700

800

900

1000

Length (mm)

Figure 10. Normalised Normalised strain strain mode mode shapes shapes along along the length direction of the tested pipe. Figure

6. Conclusions Conclusions As strain measurements measurementsare aredirectly directlyrelated relatedtoto stress, fatigue and failure, strain modal analysis stress, fatigue and failure, strain modal analysis has has a great potential in evaluating the dynamic performance of and small andpipes. light With pipes.some Withunique some a great potential in evaluating the dynamic performance of small light unique advantages, FBG sensors for the experimental strain modal of analysis of small advantages, FBG sensors could becould used be for used the experimental strain modal analysis small and light and pipes distributed to detect distributed dynamic strain responses, theweight size and sensing pipeslight to detect dynamic strain responses, where thewhere size and ofweight sensingofelements elements are restricted. In this the experimental strain modal of light smallpipes and light are restricted. In this paper, thepaper, experimental strain modal analysis ofanalysis small and usingpipes FBG using FBG sensors is investigated. Strain modal analysis of a cantilever small-diameter pipe using sensors is investigated. Strain modal analysis of a cantilever small-diameter pipe using FBG sensors FBG sensors has been carried frequencies out. The Natural frequencies andarestrain mode shapes are has been carried out. The Natural and strain mode shapes experimentally acquired experimentally acquired and of compared with the resultsand of finite a reference and finite and compared with the results a reference accelerometer elementaccelerometer simulation analysis. The element simulation analysis. good agreement the results the MAC can values indicate good agreement between theThe results and the MACbetween values indicate thatand FBG sensors be used for that FBG sensors strain can bemodal used for the experimental modal The analysis of small and lightparameters pipes. The the experimental analysis of small andstrain light pipes. first four strain modal first four strain modal parameters ofby thethe hydraulic pipe were obtained by method. the proposed strain modal of the hydraulic pipe were obtained proposed strain modal testing testing method. Acknowledgments: This work was supported by the National Natural Science Foundation of China (51475343), the National High Technology Research and Development Program of China (863 Program) (Grant No. Acknowledgments: This work was supported by the National Natural Science Foundation of China (51475343), 2015AA042101), and the Fundamental Research Funds for the Central Universities (Grant No. 2015III003). the National High Technology Research and Development Program of China (863 Program) (Grant Author Contributions: the experiments and wrote paper;Universities J.C. performed the No. experiments and No. 2015AA042101), andJ.H. thedesigned Fundamental Research Funds for thethe Central (Grant 2015III003). analyzed the data; Z.Z., L.Z., Q.J. and D.T.P. reviewed and revised the paper. Author Contributions: J.H. designed the experiments and wrote the paper; J.C. performed the experiments and Conflicts of Interest: The authors declare no conflict of interest. analyzed the data; Z.Z., L.Z., Q.J. and D.T.P. reviewed and revised the paper.

Conflicts of Interest: The authors declare no conflict of interest. References 1.

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