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Damage Detection Using Measurement Response Data of Beam Structure Subject to a Moving Mass

Abstract

The moving load problem is divided into two typical types: moving force and moving mass. The moving mass problem includes the time-varying mass effect. This work considers damage detection of the beam structure subject to a moving load including the inertia effect based on the only measurement data from strain gages and accelerometers without any baseline data. This experiment compares the feasibility of damage detection methods depending on the measurement sensors of strain gages and accelerometers. The measurement data are transformed to the proper orthogonal modes (POMs) in the time domain and the frequency domain, respectively. The magnitude of the moving mass and its velocity are also evaluated as test variables in this experiment. It is shown in the beam tests that the measured strain data can be more explicitly utilized in detecting damage than the acceleration data, and the mass magnitude and its velocity affect the feasibility of damage detection.

Keywords:
damage detection; moving mass; proper orthogonal mode; inertia effect; strain gage; accelerometer

1 NTRODUCTION

The dynamic response of the beam structure traversed by moving systems has been studied during the last several years. The moving load problem is divided into two typical types, namely moving force and moving mass, depending on the inertia interaction of the moving mass and its supporting member. Neglecting the inertia interaction of the moving load and its supporting structure, the external load acts as the moving force. The moving force problem assumes the magnitude of the moving force is constant because the inertia effect is neglected. Considering the inertia interaction of the mass and beam leads to a moving load in the form of moving mass.

A time-varying (non-stationary) system due to the inertia effect leads to the change of the modal parameters at each time instant. The velocity of the moving mass and the ratio of the moving mass to that of the supporting structure affect the dynamic responses of the supporting member.

There have been many attempts to analytically solve moving load problems such as moving mass and moving sprung mass problems. (Pesterev et al., 2003Pesterev, A.V., Bergman, L.A., Tan, C.A., Tsao, T.C., Yang, B., (2003). On asymptotics of the solution of the moving oscillator problem, Journal of Sound and Vibration 260(3): 519-536.) investigated the asymptotic behavior of the solution for the moving oscillator problem. They discussed that in case of infinite spring stiffness, the moving oscillator problem leads to the moving mass problem. For small spring stiffness, it is equivalent to the moving force problem. (Azam et al., 2013Azam, S.E., Mofid, M., Khoraskani, R.A., (2013). Dynamic response of Timoshenko beam under moving mass, Scientia Iranica A 20(1): 50-56.) evaluated the dynamic responses of a Timoshenko beam subjected to a moving mass and a moving sprung mass using Hamilton's principle. (Mao and Lu, 2013Mao, L. and Lu, Y. (2013). Critical speed and resonance criteria of railway bridge response to moving trains, Journal of Bridge Engineering-ASCE 18(2): 131-141.) developed a resonance severity indicator for an assessment of the resonance effect involving moving loads and moving masses. (Dehestani et al., 2009Dehestani, M., Mofid, M., Vafai, A., (2009). Investigation of critical influential speed for moving mass problems on beams, Applied Mathematical Modelling 33: 3885-3895.) presented an analytical-numerical method to determine the dynamic response of beams carrying a moving mass with various boundary conditions and investigated critical influential speeds in the moving mass problems. (Yavari et al., 2002Yavari, A., Nouri, M., Mofid, M., (2002). Discrete element analysis of dynamic response of Timoshenko beams under moving mass, Advances in Engineering Software 33(3): 143-153.) investigated the effects of beam thickness and moving mass velocity on the dynamic response of Timoshenko beams under moving mass using the discrete element technique. (Garibaldi et al., 2011Garibaldi, L., Fasana, A., Marchesiello, S., Bellino, A., (2011). Time variant identification of damaged structures excited by moving loads, Proceedings of the 8th International Conference on Structural Dynamics, EURODYN 2011, Leuven, Belgium.) established the connection among the natural frequency, signal amplitude and nonlinearity to eliminate the nonlinear contribution. The method can estimate the carriage characteristics such as mass and velocity.

The derived dynamic responses of the beam structure subject to a moving mass have been applied to its damage detection. (Bilello and Bergman, 2004Bilello, C. and Bergman, L.A., (2004). Vibration of damaged beams under a moving mass: theory and experimental validation, Journal of Sound and Vibration 274: 567-582.) provided an analytical tool for the analysis of a damaged beam under a moving mass. (Bellino et al., 2009Bellino, A., Garibaldi, L., Marchesiello, S., (2009). Time-varying output-only identification of a cracked beam, Key Engineering Materials 413-414: 643-650.) presented a time-varying identification method to detect a crack in a beam with a moving mass. Considering that natural frequencies of real structures are affected by environmental conditions such as the mass and the velocity of the train, (Bellino et al., 2010Bellino, A., Garibaldi, L., Fasana, A., Marchesiello, S., (2010). Damage detection in time-varying systems by means of the PCA method, Proceedings of ISMA: 783-794.) the relationship between the natural frequencies and the load characteristics. They proposed a damage identification method to detect the presence of damage in a railway bridge crossed by a train using the principal component analysis. (Wahab and De Roeck, 1999Wahab M.M.A. and De Roeck, G., (1999). Damage detection in bridges using modal curvatures: applications to a real damage scenario, Journal of Sound and Vibration 336: 217-235.) investigated the application of the change in modal curvatures to detect damage in a prestressed concrete bridge. (Zhu and Hao, 2007Zhu, X.Q. and Hao, H., (2007). Damage detection of bridge beam structures under moving loads, IMAC-XXV: A Conference & Exposition on Structural Dynamics, Florida, USA.) developed a method to detect the damage in a bridge beam structure under load environments modeled by a group of vehicle-bridge interaction forces moving at a prescribed velocity. (Pala and Reis, 2013Pala, Y. and M. Reis, M., (2013). Dynamic response of a cracked beam under a moving mass load, Journal of Engineering Mechanics 139(9): 1229-1238.) studied the effects of inertial, centripetal, and Coriolis forces on the dynamic response of a simply supported beam with a single crack under moving mass load. Analyzing the time response of a cracked beam carrying moving mass and combining continuous and discrete wavelet transforms, (Gokdag, 2011Gokdag, H., (2011). Wavelet-based damage detection method for a beam-type structure carrying moving mass, Structural Engineering and Mechanics 38(1): 81-97.) proposed a damage detection method. This research mentions that strain data outperform displacement data at the same point in revealing the damage signature. (Cavadas et al., 2013Cavadas, F., Smith, L.F.C., Figueiras, J., (2013). Damage detection using data-driven methods applied to moving-load responses, Mechanical Systems and Signal Processing 39: 409-425.) presented a damage detection approach for using moving-load responses as time series based on two data-driven methods of moving principal component analysis and robust regression analysis. Using the generalized multi-symplectic formulations, Hu et al. (2013) considered the dynamic behavior of continuous beams under moving load. (Nikkhoo et al., 2014Nikkhoo A., Hassanabadi M.E., Azam S.E., Amiri J.V., (2014), Vibration of a thin rectangular plate subjected to series of moving inertial loads, Mechanics Research Communications 55: 105-113.) studied the vibration of a rectangular plate due to multiple traveling masses.

This work investigates damage detection of a beam structure subject to a moving load including the inertia effect based on only measurement data from strain gages and accelerometers without any baseline data. The feasibility of damage detection by using accelerometers and strain gages is compared. The measurement data are transformed to proper orthogonal mode (POM) to reduce the inertia and noise effects and to investigate the damage present. The magnitude of the moving mass, its velocity and the types of measurement sensors are utilized as the test variables in this experiment. It is shown in the beam tests that the measured strain data can be more explicitly utilized in detecting damage than the acceleration data, and the mass magnitude and its velocity affect the applicability of damage detection.

2 YNAMIC RESPONSE OF A BEAM STRUCTURE SUBJECT TO A MOVING MASS

The moving mass analysis should be considered to predict the dynamic responses of a beam structure subject to a moving mass. The dynamic response of the beam in Figure 1 is affected by the traveling velocity of the moving mass and its magnitude. The moving mass problem assumes that the stiffness of the moving system is infinite and neglects the possibility that the mass may separate from the beam. Based on energy methods, the equation of motion for an Euler-Bernoulli beam of length L travelled by a mass M with velocity v(t) pressed by

where y(x,t) is the vertical deflection of the beam; m and I are the mass per a unit length and the area moment of inertia, respectively; (t) = vt is the instantaneous position of the mass along the beam; δ(x) is the Dirac delta function and g is the acceleration of gravity. It is shown in Eq. (1) that the dynamic responses of the beam are affected by the velocity of the moving mass and the ratio of the moving mass to that of the supporting structure. Taking them as the test variables, this study performs the experiments using the measurement sensors of strain gages and accelerometers to evaluate the feasibility of damage detection.

Figure 1
A beam structure subject to a moving mass.

3 EXPERIMENTAL STUDY

The applicability of damage detection on a simply supported beam is examined depending on the test variables and measurement sensors. A steel beam in Figure 2 is 1200mm in length, and its gross cross-section is 100mm x 11mm. The damage is located at 350mm between nodes 3 and 4 from the left end, and its cross-section is 110mm x 8mm. The response data in the time domain are sequentially measured by 11 strain gages and accelerometers installed in advance at the bottom of the beam without the action of any external force. The damage is traced by the measurement data only and without any baseline data. Both experiments using strain gages and accelerometers are conducted using DYTRAN model 3055B1 uniaxial accelerometers and TML strain gages of type FLA-5-11, respectively. The data in the time domain are collected by a DEWETRON model DEWE-43.

Figure 2
A simply supported beam structure subject to a moving mass.

A battery-operated toy car functions as the moving mass by using a sum of the self-weight of the car and the weight of steel magnets. It is shown in Figure 3 that the stiffness of the toy car is very high. The moving velocity and the time required from an end support to the other end support decreases with an increase in the moving mass. Accelerating gradually from rest, the moving mass does not travel at a constant velocity, and the system is considered a time-varying system. This study utilized the average velocity expressed by the distance covered with respect to the time taken. The mass magnitude is not inversely proportional to its velocity. A response dataset is collected by the moving mass in a rightward direction and another dataset in a leftward direction.

Figure 3
A battery-operated toy car and magnet mass.

The total weight of the steel beam as the supporting member is 13kg. The moving masses, including the weight of steel magnets and the self-weight of the car, are 2.3kg, 2.5kg and 2.74kg.

Experimental data are collected from two different types of measurement sensors of strain gages and accelerometers, and the feasibility of the measurement sensors for damage detection are compared. For the first test, the damage detection of the steel beam is performed using the measurement data from the strain gages. The strain gages are bonded in the interval of 100mm on the bottom of the beam section. Two datasets in the rightward and leftward directions according to the movement directions of the car are simultaneously collected by 11 strain gages as shown in Figure 2. The moving mass and its average velocity obtained in the experiment are displayed in Table 1. Figure 4 represents the strain responses in the time domain as each mass of 2.3kg, 2.5kg and 2.74kg moves an end support to the other end support. In the plots, R and L denote the rightward and leftward directions, respectively. It is observed that the plots do not exhibit the information on the damage because of the external noise included in the measured data. Most noise comes from the vibration of the motor in the car. To minimize the noise effect included in each strain dataset, the POM corresponding to the first proper orthogonal value (POV) is extracted from three strain datasets within the time that the moving mass arrives at the same position as the strain gage. The POMs of strains at all nodes are displayed in Figure 5. The plots show that it is not easy to explicitly find the damage location based on the strain curves regardless of the inertia effect. Considering that the variation in the slope of the strain curves is sensed, the slope curves of the strains are described in Figure 5. The slope indicates the variation of the strain along the nodes. It is found that the damage is located at the nodes to exhibit the abrupt variation of strains as shown in three different cases. The plots also show that the damage can be more explicitly detected as the moving mass increases or the average moving velocity decreases.

Figure 4
Strain response in the time domain: (a) 2.3-R, (b) 2.3-L. (c) 2.5-R, (d) 2.5-L, (e) 2.74-R, (f) 2.74-L.

Figure 5
Strain POM and its slope along the beam member: (a) 2.3-R, (b) 2.3-R, (c) 2.3-L , (d) 2.3-L, (e) 2.5-R, (f) 2.5-R, (g) 2.5-L, (h) 2.5-L, (i) 2.74-R, (j) 2.74-R, (k) 2.74-L, (l) 2.74-L

Table 1
Test variables of mass magnitude and its velocity.

Another beam test for damage detection was performed by the measurement of accelerometers as shown in Figure 6. The acceleration responses in the time domain are measured at 11 nodal positions and are represented in Figure 7. It is shown that the acceleration responses, including external noise in the time domain, do not provide any information on the damage. The acceleration response in the time domain is transformed to the FFT in the frequency domain. Figure 8 represents the magnitude of the responses in the frequency domain. Neglecting a slight difference in the first resonance frequency due to the inertia effect, three FFT datasets in the neighborhood of the first resonance frequency are selected because it is not easy to collect the exact data at the first resonance frequency moment. In addition, the datasets are transformed to the POM. Figure 9 represents the POM curves. It is observed that they do not have consistent shapes depending on the mass magnitude so that the damage cannot be recognized at the same location. The measurement data from accelerometers are more sensitive to the external noise and the mass magnitude than from the strain gages. The beam structure is characterized by the flexural behavior such as the curvature. Based on the flexural characteristics, the abrupt variation of the curvature should be located near the damage region.

Figure 6
A simply supported beam subject to a moving mass (accelerometer sensors).

Figure 7
Acceleration responses in the time domain: 2.3-R, (b) 2.3-L, (c) 2.5-R, (d) 2.5-L, (e) 2.74-R, (f) 2.74-L.

Figure 8
FFT transform: (a) 2.3-R, (b) 2.3-L, (c) 2.5-R, (d) 2.5-L, (e) 2.74-R, (f) 2.74-L.

Figure 9
POM and its curvature: (a) 2.3-R, (b) 2.3-R, (c) 2.3-L , (d) 2.3-L, (e) 2.5-R, (f) 2.5-R, (g) 2.5-L, (h) 2.5-L, (i) 2.74-R, (j) 2.74-R, (k) 2.74-L, (l) 2.74-L.

This study utilizes the POM curvature to evaluate the damage of the beam. The POM curvature corresponding to the l-th (l = 1,2,...,n) POM at each location k on the structure can be numerically obtained using a central difference approximation:

where Ckl is the second derivatives of the kth flexural dimension, l denotes the corresponding POV, and h is the distance between two successive nodes. Figure 9 represents the POM curvature curves. The plots on the POM and its curvature corresponding to the mass magnitudes of 2.3kg, and 2.5kg in Figures 9(a) through (d) and (e) through (h) do not provide information on the damage, respectively, but the plots of 2.74kg in Figures 9(i) through (l) do. In the case of the mass of 2.74kg, the curvature curve taken from the mass and the beam moving in the leftward direction provide more accurate information than the rightward direction. This indicates that it is not easy to detect damage during the acceleration process from rest. From the experiments, the measurement from strain gages provides more conservative damage information than the accelerometers. In addition, the accelerometers are sensitively affected by the mass magnitude of the car unlike the strain gages.

The applicability of the damage detection method at beam structure considering subject to a moving mass is investigated by a numerical experiment. The test beam in Figure 2 is also utilized as the analytical model. A beam with a length of 1.2 m was modeled using 12 beam elements. The beam has an elastic modulus of 2.0 x 105 Mpa and a unit mass of 7,860 kg/m3 The beam's gross cross section is 100mm x 10mm and its damage section is established as the reduction of 30% flexural rigidity at element 4. The damping matrix is assumed as a Rayleigh damping to be expressed by the stiffness matrix and a proportional constant of 0.0001. The mass of 2.3kg is assumed to travel with a velocity of 337.8mm/sec This work neglects the external noise and the analysis is carried out following the procedure in Figure 10.

Figure 10
Flow chart for finite element analysis.

The dynamic responses of the beam are obtained by solving the second order differential equation. The displacement responses at all nodes are displayed in Figure 11(a). It is shown that the dynamic transverse responses in the time domain do not provide the explicit information on the damage. Summing the absolute values of all displacements in the time span [t0 tf] = [0 3.5524 sec] where t0 and tf represent the beginning and final time to be integrated, respectively, the responses are normalized as follows:

Figure 11
Numerical results: (a) displacement responses, (b) normalized responses of summed-absolute displacements, (c) flexural curvature, (d) slope of curvature, (e) POM corresponding to the first POV, (f) slope of the POM.

Where denotes the normalized displacement of summed-absolute displacements at node i and l is the number of time step to be integrated. And |•| is the absolute and |•| is the norm. Figure 11(b) and 11(c) represent the normalized displacement and its curvature. The curvature is estimated by a central difference method. Both plots also do not indicate explicit damage. Considering the curvature is proportional to the flexural strain, Figure 11(d) displays the slope of the curvature related to the flexural strain. It is observed that the slope between two adjacent nodes 4 and 5 in the neighborhood of the damage is abruptly changed. It is expected that the damage can be detected by analyzing the displacements of the beam subject to a moving mass.

The 300 displacement response data sets between 1.2569 sec. and 1.4197 sec. are extracted to transform to the POM. Figure 11(e) and 11(f) display the POM corresponding to the first POV. It is observed that the POM curvature is abruptly changed between two nodes 4 and 5. From this numerical experiment, the validity of the proposed method to detect damage based on the flexural responses of beam structure due to the moving mass is verified.

4 CONCLUSIONS

This work investigated the feasibility of damage detection of a damaged beam structure subject to a moving mass depending on the mass magnitude, its velocity and the measurement sensors. The experimental results represent that the damage detection using strain gages provides more accurate information on the damage than accelerometers. It is observed that the damage detection from strain gages can be taken more explicitly as the mass magnitude increases and its velocity decreases. The measurement data from accelerometers are more sensitive to the external noise and the mass magnitude than strain gages. The experimental results show that it is not easy to detect the damage when using accelerometers during the acceleration process from rest.

Acknowledgements

This research was supported by the Chung-Ang University research grant in 2014. This research is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2057431).

References

  • Azam, S.E., Mofid, M., Khoraskani, R.A., (2013). Dynamic response of Timoshenko beam under moving mass, Scientia Iranica A 20(1): 50-56.
  • Abu Husain N., Ouyang H., (2011), Detection of damage in welded structure using experimental modal data, Journal of Physics: Conference Series 305, doi:10.1088/1742-6596/305/1/012120.
    » https://doi.org/10.1088/1742-6596/305/1/012120
  • Bellino, A., Garibaldi, L., Marchesiello, S., (2009). Time-varying output-only identification of a cracked beam, Key Engineering Materials 413-414: 643-650.
  • Bellino, A., Garibaldi, L., Fasana, A., Marchesiello, S., (2010). Damage detection in time-varying systems by means of the PCA method, Proceedings of ISMA: 783-794.
  • Bilello, C. and Bergman, L.A., (2004). Vibration of damaged beams under a moving mass: theory and experimental validation, Journal of Sound and Vibration 274: 567-582.
  • Cavadas, F., Smith, L.F.C., Figueiras, J., (2013). Damage detection using data-driven methods applied to moving-load responses, Mechanical Systems and Signal Processing 39: 409-425.
  • Dehestani, M., Mofid, M., Vafai, A., (2009). Investigation of critical influential speed for moving mass problems on beams, Applied Mathematical Modelling 33: 3885-3895.
  • Garibaldi, L., Fasana, A., Marchesiello, S., Bellino, A., (2011). Time variant identification of damaged structures excited by moving loads, Proceedings of the 8th International Conference on Structural Dynamics, EURODYN 2011, Leuven, Belgium.
  • Gokdag, H., (2011). Wavelet-based damage detection method for a beam-type structure carrying moving mass, Structural Engineering and Mechanics 38(1): 81-97.
  • Mao, L. and Lu, Y. (2013). Critical speed and resonance criteria of railway bridge response to moving trains, Journal of Bridge Engineering-ASCE 18(2): 131-141.
  • Nikkhoo A., Hassanabadi M.E., Azam S.E., Amiri J.V., (2014), Vibration of a thin rectangular plate subjected to series of moving inertial loads, Mechanics Research Communications 55: 105-113.
  • Pesterev, A.V., Bergman, L.A., Tan, C.A., Tsao, T.C., Yang, B., (2003). On asymptotics of the solution of the moving oscillator problem, Journal of Sound and Vibration 260(3): 519-536.
  • Pala, Y. and M. Reis, M., (2013). Dynamic response of a cracked beam under a moving mass load, Journal of Engineering Mechanics 139(9): 1229-1238.
  • Wahab M.M.A. and De Roeck, G., (1999). Damage detection in bridges using modal curvatures: applications to a real damage scenario, Journal of Sound and Vibration 336: 217-235.
  • Yavari, A., Nouri, M., Mofid, M., (2002). Discrete element analysis of dynamic response of Timoshenko beams under moving mass, Advances in Engineering Software 33(3): 143-153.
  • Zhu, X.Q. and Hao, H., (2007). Damage detection of bridge beam structures under moving loads, IMAC-XXV: A Conference & Exposition on Structural Dynamics, Florida, USA.

Appendix

Assume that measurement data are expressed by a finite number of points in space and frequency. Expressing a set of FRF response data as Û, suppose that m linear snapshots of the response Ûi of size n are obtained by response measurements written as

where the superscript (j) represents the FRF data at the j-th frequency Ωj, n denotes the number of measurement positions of the structure and m is the total number of frequency observations.

The POD of this discrete field involves solving the eigenvalue problem. Let m × m matrix be defined as

Solving the eigenvalue problem of Eq. (A2) at the core of the POD method, Eq. (A2) satisfies

where the eigenvalues are arranged as follows:

where the eigenvalues λk are the POVs and each eigenvector φk of the extreme value problem is associated with a POV λk. φ1 represents the eigenvector corresponding to the largest eigenvalue λ1.

The POMs may be used as a basis for the decomposition of U. The POM associated with the greatest POV is the optimal vector. If the eigenvalues are normalized to unit magnitude, they represent the relative energy captured by the corresponding POM. The eigenvalue reflects the relative kinetic energy associated with the corresponding mode. The energy is defined as the sum of the POVs. The POMs are written as

The POMs are arranged as follows:

Publication Dates

  • Publication in this collection
    Dec 2015

History

  • Received
    07 Mar 2015
  • Reviewed
    06 May 2015
  • Accepted
    22 June 2015
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