- Email: [email protected]
Estimation of mode shapes of beam-like structures by a moving lumped mass
Engineering Structures 180 (2019) 654–668
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Estimation of mode shapes of beam-like structures by a moving lumped mass Yao Zhang, Haisheng Zhao , Seng Tjhen Lie ⁎
T
School of Civil & Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
ARTICLE INFO
ABSTRACT
Keywords: Mode shape Beam like structures Moving lumped mass Non-stationary IF
This paper presents a method to estimate mode shapes of beam like structures by using the acceleration of a moving lumped mass. In fact, the coupled frequencies of the vehicle-beam interaction system are time varying and the non-stationary instantaneous frequencies (IFs) contain information of mode shapes. The theoretical analysis in this paper shows that the mode shapes can be re-constructed by using the IFs; therefore, extracting mode shapes becomes a problem of IF estimation. A modified time-frequency analysis method based on weighted polynomial chirplet transform is developed to estimate the non-stationary IFs. Moreover, a new sampling algorithm based on accumulative measured energy is proposed to reconstruct the mode shapes, in which more data are sampled at the area with higher measured energy, making it more robust to noise. The proposed method is more convenient since only a lumped mass with a single accelerometer is required, and it is more practical because external exciter is not required, and the surface roughness can be the source of excitation. Numerical simulations and laboratory scale experiments have been carried out, which show that the proposed method performs well in extracting mode shapes, even if the travelling speed is high.
1. Introduction
characteristic matrix which is the multiplication of inverse mass matrix and stiffness matrix, and mode shapes are corresponding eigenvectors. Because eigenvectors are more vulnerable and sensitive to noise of the characteristic matrix than eigenvalues, low level noise may bring high level errors on eigenvectors. Therefore, extracting mode shapes with high accuracy is usually more difficult than natural frequencies. In conventional modal analysis, estimation of mode shapes usually has higher error than natural frequencies [16,17]. In spectrum analysis, the measurement noise can easily contaminate the magnitude. Since mode shapes are also very important in damage detection and FE model validation and considering the indirect method has been successfully employed in practice, some researchers attempted to use indirect approach to identify mode shapes, which is a tougher task in practice than extracting natural frequencies [18]. Zhang et al. [19] first attempted to extract the mode shape squares of beam like structures by using the acceleration of a passing vehicle equipped with tapping device. Because it requires exciter installed on the vehicle, it is somehow inconvenient in practice. Yang et al. [20] proposed a theoretical method to construct the mode shapes from the vertical acceleration of a passing vehicle, and they found the road roughness inversely affected the proposed method. To overcome this inverse effect, Malekjafarian and Obrien [21] evaluated the mode shapes by applying Short Time Frequency Domain Decomposition to dynamic responses of two following quarter-cars travelling over a bridge. Later, they [22] used multiple laser
Modal analysis is important in civil and mechanical engineering because modal parameters can be used to validate finite element (FE) models [1] and monitor the health condition of structures [2]. In conventional approaches, they are measured through common modal analysis techniques like Frequency Response Function method, Fourier spectral element method, and output-only decomposition method in which a lot of sensors are installed on the structures [3–5]. These approaches are still frequently used in practice; however, they are usually costly and time consuming. Recently, indirect measurement of modal parameters by using dynamic responses of a moving vehicle has been investigated [6] because it only requires a single sensor installed on the vehicle which is more convenient in practice. Yang et al. [7] first proposed this concept to extract the natural frequencies of bridges, and they experimentally validated the feasibility of this method on real bridges by using both travelling vehicle [8] and hand-drawn cart [9]. Several studies have also been conducted to identify bridge frequencies by using dynamic responses of a passing vehicle [10–13]. Moreover, Gonzalez et al. [14] and Keenahan et al. [15] tried to identify the damping by using this indirect method and they carried out numerical verifications. Identification of natural frequencies and mode shapes is a typical eigenvalue problem. Natural frequencies are eigenvalues of a
⁎
Corresponding author. E-mail address: [email protected] (H. Zhao).
https://doi.org/10.1016/j.engstruct.2018.11.074 Received 25 June 2018; Received in revised form 28 November 2018; Accepted 28 November 2018 0141-0296/ © 2018 Elsevier Ltd. All rights reserved.
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
measurements from a passing vehicle to evaluate the mode shapes, and a novel approach is also proposed to estimate the mode shapes by using a travelling truck trailer system equipped with an external excitation [23]. Kong et al. [24] used two trailers and subtracted the response of one trailer by the response of the other with a time shift. Qi and Au [25] identified the bridge mode shapes based on the dynamic response of a passing vehicle under impact excitation. However, they only presented numerical studies and no experimental validation was carried out. In this study, a new mode shape extraction method for beam like structure is proposed by using a moving lumped mass. The IFs of the mass-beam interaction system change with the location of the lumped mass and therefore they contain the information of mode shapes. Then, extracting mode shapes becomes a problem of evaluating IFs. A modified time-frequency analysis method based on weighted polynomial chirplet transform is proposed to evaluate the non-stationary IFs and a new sampling algorithm based on accumulative measured energy is proposed to reconstruct the mode shapes, in which more points are sampled at the area with higher measured energy. The proposed method is more convenient since only a lumped mass with a single accelerometer is required, and it is more practical because external exciter is not required, and the surface roughness can be the source of excitation. Numerical simulations and laboratory scale experiments have been carried out, indicating that the proposed method performs well in extracting lower order mode shapes, even if the travelling speed is high.
ml EI n4 4 q¨ + q = 2 n 2 l3 n
bn
mv
d2qv = dt 2
+ EI
mv g
= f (t ) (x f (t )
xc )
n
n (x ) qn (t ) n
¨i i (x c ) q
2 n (x c )
2 bn qn
q¨n +
ml
r¨ (x c )
g
(5)
i n
=
2m v ml
n (x c )
of
¨i i (x c ) q
the
beam,
+ r¨ (xc ) + g
i n
=
bn 2mv n2 (x c )
(7)
ml
It can be found from Eq. (7) that the natural frequencies of the interaction system are functions of the mode shapes; therefore, extracting mode shapes becomes a typical problem of identifying the IFs of the interaction system n (x )
=
n sign(x )
2 bn 2 n
1
(8)
where n is a constant coefficient to normalize the nth mode shape, sign(x ) is a function to determine the sign of the value based on prior knowledge. Two issues should be addressed: one is that although the simply supported boundary condition is used here, the method is applicable for other types of boundary condition which is verified in the following experimental study; the other is that the method is only effective when the weight of lumped mass cannot be ignored compared to the beam weight, otherwise the IFs are the time independent natural frequencies of the beam. 2.2. IF estimation based on weighted polynomial chirplet transform Short Time Fourier Transform (STFT) and Continuous Wavelet Transform (CWT) are usually used to produce time-frequency distributions for signals in civil engineering; however, neither STFT nor CWT can achieve a good frequency resolution due to the restriction of the Heisenberg-Gabor inequality. Chirplet Transform (CT) initially designed for the analysis of chirp-like signals with linear IF law has been improved by using the chirplet kernel with a polynomial nonlinear IF law [26]; however, it performs under expectation when the noise level is high. Therefore, a modified algorithm based on weighted polynomial chirplet transform to estimate the IF is proposed in this study. First of all, consider a nonlinear chirp signal, s (t ) , and its analytical signal can be easily obtained by Hilbert Transform:
(1) (2)
(3)
Using the modal superposition method, the vertical displacement of the beam can be written as
ub (x , t ) =
2m v
1+
where qv (t ) and ub (x , t ) are the vertical displacements of the lumped mass and the beam, respectively; f (t ) is the interaction force and (·) is the Dirac delta function. They are in permanent contact; therefore, the following equation can be obtained as
qv = ub (x c , t ) + r (xc )
n (x c )
The nth natural frequency of the interaction system, including the lumped mass and the beam, can be approximated as
Fig. 1 shows a typical simplified model of a lumped mass travelling over a simply supported Euler beam. The beam has length l , bending stiffness EI , mass m per unit length, and surface roughness profile r (x ) ; the lumped mass is travelling at a constant speed, v , and x c is the coordinate of the contact point. For simplicity, damping is temporarily ignored herein. The governing equations of the interaction system are given as
m
mv
(6)
2.1. Interaction of a lumped mass passing through a beam
4u b x4
2 ¨n n (x c ) q
By considering that the nth frequency = n2 2l 2 (EI /m )1/2 , Eq. (5) can be simplified as
1+
2. Theory
2u b t2
mv
(9)
z (t ) = s (t ) + jH (s (t )) The analytical signal z (t ) can be further written as:
(
(4)
z (t ) = a (t )exp j 2
where n (x ) = sin n x / l is the nth mode shape and qn (t ) is the corresponding modal coordinate. Substituting Eqs. (2)–(4) into Eq. (1), multiplying m (x ) on both siders and integrating over the beam, one can obtain
t 0
f ( )d +
)
(10)
where a (t ) and f ( ) are the instantaneous amplitude and frequency, is the initial phase. The polynomial chirplet transform (PCT) of z (t ) is defined as [26]: Fig. 1. Simplified model of a lumped mass travelling through a beam.
655
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
PCT(t 0, =
,
+
1,
,
n;
) n+1
z (t )exp
j
tk 1 k 1
k 1t k=2
=
+
(
a (t )exp j 2
t 0
f ( )d +
t 0k
1
w (t
(t )
) w (t
t 0 )exp( j t )dt t 0 )exp( j t )dt (11)
where
w (t
t0) =
1 2
n+ 1
(t ) =
k 1t k=2
exp
1 t 2
tk 1 k 1
t0k
2
(12)
1
(13)
Fig. 3. Illustration of accumulative energy-based sampling algorithm.
It should be noted that w shown in Eq. (12) is Gaussian function, which is frequently used in time-frequency analysis as a window function. When the IF of the de-chirplet operator is equal to the real IF of the analytical signal z (t ) , that is, d (t )/dt = f (t ) , the PZT could produce a high-quality time-frequency distribution for z (t ) and the result can have good concentration. Fig. 2 shows the algorithm to evaluate the IF trajectory: in the initial step, a predefined threshold, , and the polynomial order, n should be set and the polynomial coefficients are assumed as zero; for the ith step, the ith PCT is calculated and the ith IF can be obtained by picking the peaks in PCT, and then the polynomial coefficients can be calculated by weighted least square method; once the difference between the evaluated IFs in two constitutive steps is less than the predetermined threshold, the final estimation of IF trajectory can be determined. It should be noted that when calculating the coefficients in the ith step, t 0 and te indicate the start time and end time of the travelling, respectively, while f0 and fe are the natural frequencies of the interaction system for the lumped mass standing at the start-point and the endpoint, respectively. This is because when the lumped mass is just standing on the beam, the frequency of the interaction system can be identified accurately.
2.3. Accumulative energy-based sampling algorithm Usually, the measurement points are sampled uniformly; however, some points may have more significant measurement noise, especially for those near the nodal points. Therefore, an accumulative energybased sampling algorithm is proposed. Considering that the amplitude in PCT at the resonance frequency usually represents the vibration energy, the accumulative energy at a certain time, tc, is defined as tc
AE (tc ) =
0
Amp (t )dt
(14)
The basic idea of this sampling algorithm is that more data can be sampled at the time interval with higher accumulative measured energy and Signal-to-Noise Ratio (SNR) and vice vasa, which can be seen in Fig. 3; therefore, the extracted mode shapes are more robust to noise. If the total accumulative energy is AE and N points are required, the data will be sampled at time t1, t2, …, tN, which can be determined by:
AE (ti ) =
(i N
1) AE , i = 1, 1
,N
Fig. 2. Flowchart of weighted polynomial chirplet transform based IF estimation algorithm. 656
(15)
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
3. Numerical simulations
Before extracting the mode shapes by the proposed method, the signal is filtered by a band-pass filter so that only one single frequency component is contained. In numerical simulations, three bands, 1–10 Hz, 20–30 Hz, 40–60 Hz are used for the first three modes. In this study, the value of σ is set as 0.3, which is identical to that in Ref. [26]. It should be noted that when σ is in the range of 0.2–0.5, it shows little influence on the identified IFs and re-constructed mode shapes. The polynomial order, n, is also important: it should be neither too small nor too large, otherwise, underfitting or overfitting can be observed. It can be determined as follows: the number of half-sine pulses that are contained in one specific mode shape should be evaluated by prior knowledge, and n can then be calculated by multiplying this number by 4 because quadric function can usually fit a half-sine pulse well. Therefore, n can be obtained as 4, 8,12 for the first three mode shapes. Moreover, is set as 0.001 herein. Fig. 5 shows the initial PCTi=1 of the acceleration at t0 = 1 s and t0 = 5 s where polynomial coefficients are zero, and the locations of peaks (IFi=1) can be obtained by searching the maximum values which are also defined as Ampi=1(t0). For example, for the first mode, IFi=1(t0 = 1) = 5.86 Hz, Ampi=1(t0 = 1) = PCTi=1(t0 = 1, ω = 5.86 × 2π, α = 0) = 0.66, IFi=1(t0 = 5) = 5.37 Hz, Ampi=1 (t0 = 5) = PCTi=1(t0 = 5, ω = 5.37 × 2π, α = 0) = 0.21. Fig. 6 shows the extracted mode shapes by the proposed method, in which the solid line represents the analytical solution and the circle indicates the extracted mode shapes sampled by using the sampling algorithm based on accumulative energy. For each vibrational mode, 200 points are sampled. It is observed that the first three mode shapes are identified with good accuracy where fewer points are sampled near the nodal points.
3.1. FE model FE model is constructed by using general purpose commercial software ABAQUS in this study. The simply supported Euler beam has length of 20 m, cross-sectional area of 1.8 m2, moment of inertia of 0.3375 m4, density of 2400 kg/m3, and elastic modulus of 30 GPa. Therefore, the first three natural frequencies of the beam are 6.012, 24.048 and 54.108 Hz. The beam is modeled by 2-node plane beam elements in length of 0.5 m. The lumped mass is 10,000 kg and it passes over the beam at 2 m/s. The time step of the analysis is 0.001 s. It is noteworthy that the parameters of the FE model are selected to simulate a heavy road roller with high stiffness travelling over a short-span bridge, so that the proposed method is possible to be used in full scale structures. 3.2. Extraction of mode shapes Fig. 4 shows the time history of vertical acceleration of the lumped mass and its corresponding time frequency representation by using STFT in which “Hanning” window with length of 2048 is used. It can be seen clearly that the frequency resolution is quite low by using STFT due to limited window length and therefore it is almost impossible to extract accurate IF with high resolution.
3.3. Effect of measurement noise Because the measurement noise due to either surface roughness or equipment makes the identification more difficult, it is necessary to examine the resistance of the proposed method to noise. Three levels of Gaussian noise, 5%, 10%, and 20% are considered herein which are added directly to the acceleration response as follows:
Sn (t ) = S (t ) + nl random(t )max[S (t )]
(16)
where S (t ) and Sn (t ) are original and polluted signals, respectively, nl is the noise level and random(t ) is a function to generate random number between −1 and 1. Fig. 7 shows the original acceleration signal and polluted signals with different noise levels. Then the extracted mode shapes by applying the proposed method to the contaminated signals are shown in Figs. 8–10. The proposed method performs quite well when the noise level is up to 10% and almost no deviation can be found between the extracted mode shapes based on accelerations with and without noise. When the noise increases to 20%, the first two mode shapes can be identified well; however, the extracted third mode shape is less accurate near the nodal points. It is noteworthy that the number of points with less accuracy are comparatively fewer than others with higher accuracy due to the proposed accumulative energy-based sampling algorithm. 3.4. Effect of weight of lumped mass It is found in Eq. (7) that the IFs change more significantly if the lumped mass is heavier and they may be time independent constants if the weight of lumped mass can be neglectable; therefore, it is necessary to investigate the effect of weight of lumped mass on the proposed method. Six more lumped masses are used herein for parametric study: 200 kg, 1000 kg, 2000 kg, 5000 kg, 15,000 kg and 20,000 kg. By following the same procedure, the first three mode shapes can be extracted for each lumped mass. Table 1 lists the MAC values between the extracted mode shape and theoretical ones. Generally, the proposed method performs best when the lumped mass of 10,000 kg (mv/ ml = 11.6%) is used and it performs quite well if the mass is in the range of 5000 kg (mv/ml = 5.8%) to 15,000 kg (mv/ml = 17.4%); however, it fails if the mass is less than 1000 kg (mv/ml = 1.16%).
Fig. 4. (a) Vertical acceleration response of the traveling lumped mass and (b) its corresponding time frequency representation (STFT). 657
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 5. PCT at t0 = 1 s and t0 = 5 s (a) the first mode, (b) the second mode, (c) the third mode.
Fig. 6. Normalized mode shapes, (a) the first mode, (b) the second mode, (c) the third mode.
When the mass is only 200 kg, the IFs are almost constant; therefore, the mode shapes cannot be identified, and the MAC values are not available. When mv increases to 1000 kg, the first two mode shapes can be extracted but the MAC values are relatively low, and the third mode shape cannot be identified because higher vibrational modes cannot be excited. All the first three mode shapes can only be identified when mv increases to 2000 kg; however, the MAC values are still not high. When the lumped mass of 5000 kg is used, the identified first three mode shapes shown in Fig. 11 are quite accurate and the MAC values are all over 0.997. The first three mode shapes match best with theoretical ones when mv is 10,000 kg and the MAC values are over 0.998. The proposed method performs quite well in extracting the first mode shape when mv increases to 15,000 kg and 20,000 kg (Fig. 12), while the reconstructed higher mode shapes show slight deviations concentrated at the nodal points. This may be because the proposed method assumes linear vibration and the mode shapes of the beam do not change due to the travelling lumped mass, which may not be true when the lumped mass is too heavy.
Fig. 7. Original acceleration signal and polluted signals with different noise levels (5–5.5 s). 658
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 8. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode by using the acceleration with 5% noise.
3.5. Effect of travelling speed
frequencies are 3.87, 15.27 and 34.40 Hz. First, a lumped mass of 5000 kg is used to extract the first three mode shapes, which are shown in Fig. 15. The re-constructed mode shapes match quite well with the theoretical ones. Then a test vehicle which is modeled as a single degree of freedom mass-spring system is adopted to estimate the first three mode shapes. It has mass of 5000 kg and stiffness of 5 × 106 N/m. Fig. 16 shows the estimated first two mode shapes: the first one can be re-constructed quite well but the second one show slight deviations near the end nodes. However, it should be admitted that the third mode shape cannot be identified by applying the proposed method to the acceleration of the test vehicle, because the vehicle’s frequency is much lower than the bridge’s third frequency and it performs like a low-pass filter so that the higher frequency component is filtered. This is also the reason that a lumped mass which has very high stiffness is recommended for the proposed method, for example, a heavy road roller with high stiffness. However, considering that a general test vehicle is more valuable in practice, how to extract higher order mode shapes by using the dynamic responses of the test vehicle should be further investigated in the future.
The travelling speed is also quite important for the proposed method because it saves more time if the travelling speed is higher. In the case of baseline, the velocity of 2 m/s is considered, while velocities of 1 m/s and 4 m/s are investigated herein. The estimated first three mode shapes for these two cases are shown in Figs. 13 and 14, respectively. As expected, when the travelling speed is lower, the proposed method performs better; while it is also observed that when the lumped mass passes over the beam at 4 m/s, the identified mode shapes are almost as accurate as those extracted by the acceleration of lumped mass with velocity of 2 m/s. This is an advantage of the proposed method that it still performs well when the lumped mass travels over the whole beam by using 5 s. However, it is admitted that when the travelling speed increases to 10 m/s, the proposed method can hardly identify the higher mode shapes due to very limited data. 3.6. Example of a bridge inspected by a lumped mass and a test vehicle A bridge with lower frequencies is adopted for further demonstration. The parameters used are identical to those used in Ref. [20]: the bridge has length of 30 m, moment of inertia of 0.175 m4, mass density of 1000 kg/m, and elastic modulus of 27.5 GPa; the first three natural
3.7. Effect of accuracy of f0 and fe Two frequencies, f0 and fe, are quite important in the proposed
Fig. 9. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode by using the acceleration with 10% noise. 659
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 10. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode by using the acceleration with 20% noise.
acceleration of the lumped mass to estimate the IFs, the re-constructed first three mode shapes shown in Fig. 18 match the theoretical ones well. However, when it is applied on the acceleration with 10% noise, only the extracted first mode shape matches the theoretical one well and large deviations concentrated at the nodal points can be found in the second and third mode shapes. Similar conclusion can be obtained when HHT is used (Fig. 19). In fact, HHT performs quite well if the signal is noise-free and the estimated IFs and the re-constructed mode shapes are accurate; however, it cannot estimate the IFs accurately if the signal is subject to 10% noise. Therefore, specific filtering technique is required before PCT or HHT can be used to extract the IFs and reconstruct mode shapes.
Table 1 MAC of the first three modes for different lumped masses. mv (kg)
200
1000
2000
5000
10,000
15,000
20,000
Mode 1 Mode 2 Mode 3
– – –
0.9914 0.9963 –
0.9916 0.9980 0.9928
0.9998 0.9991 0.9972
0.9999 0.9993 0.9981
0.9998 0.9972 0.9943
0.9998 0.9968 0.9934
method, and therefore, the effect of accuracy of f0 and fe is investigated herein. For each mode, the two frequencies are assumed to have 5% errors, that is, f0 = fe = 5.71 Hz for the first mode, f0 = fe = 22.85 Hz for the second mode and f0 = fe = 51.40 Hz for the third mode. Fig. 17 shows the extracted first three mode shapes when f0 and fe have 5% errors. The proposed method still performs well but large deviations can be observed near the end nodes, because the frequencies, f0 and fe have quite large errors. Fortunately, f0 and fe can usually be measured accurately since they are measured when the lumped mass stops on the beam.
3.9. Effect of boundary condition A fixed-fixed beam is investigated herein to verify the availability of the proposed method for different boundary conditions. The beam is identical to that used in Section 3.2 and the first three frequencies increase to 13.63, 37.57, and 73.65 Hz due to the fixed-fixed boundary condition. Fig. 20 shows the extracted first three mode shapes and they match very well with those obtained from numerical simulations.
3.8. Comparison of WPCT, PCT and HHT Two more IF estimation algorithms, PCT and HHT are used herein for better demonstration. When PCT is applied on the noise-free
Fig. 11. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode by using the 5000 kg lumped mass. 660
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 12. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode by using the 20,000 kg lumped mass.
Fig. 13. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode when the lumped mass passes through the beam by 1 m/s.
Fig. 14. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode when the lumped mass passes through the beam by 4 m/s.
661
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 15. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode when the lumped mass passes through the bridge.
Fig. 16. Estimated mode shapes (a) the first mode, (b) the second mode when the test vehicle passes through the bridge.
3.10. Example of two-span continuous beam
differentiate these two modes, indicating the limitation of the method in distinguishing closed modes. Therefore, how to improve the performance of the proposed method on multi-span beams and closed modes should be further investigate in the future.
Since the multi-span continuous bridge is widely used in practice, it is necessary to investigate whether the proposed method is still applicable for this case. A simply supported two-span continuous beam having length of 40 m is adopted herein, and other properties are identical to those in Section 3.2. The first four frequencies are 6.01, 9.39, 24.05 and 30.43 Hz, respectively. It is noteworthy that the third frequency (24.05 Hz) is nearly 80% of the fourth frequency (30.43 Hz), which can be approximated as closed modes. Fig. 21 shows the identified first four mode shapes, coinciding with the FEM simulations well; nevertheless, it is admitted that slight deviations can be observed near the nodal points. Moreover, the proposed method has also been applied to three-span continuous beam, and the corresponding first five frequencies are 6.01, 7.67, 11.28, 24.05, and 27.27 Hz, respectively. The fourth frequency is 88% of the fifth frequency, which are two closer modes than the previous case. However, the proposed method fails to reconstruct the fourth and fifth mode shapes since it is difficult to
4. Experimental study Fig. 22 shows the experimental setup: a thin steel beam (2.4 kg) with dimension of 120 cm × 4.4 cm × 0.6 cm clamped at two ends, a linear motion unit pulling a lumped mass to travel at the top surface with a constant speed, and an accelerometer attached on the lumped mass. The vertical acceleration was recorded at sampling frequency of 1000 Hz. Four cases including two travelling speeds and two lumped masses were investigated in this study. For Case 1 and Case 2, the lumped mass is 0.5 kg and the travelling speeds are 5 cm/s and 10 cm/s, respectively; while for Case 3 and Case 4, the lumped mass is 1 kg and the travelling speeds are 5 cm/s and 10 cm/s. The first three frequencies of the beam are 24.5, 59.6 and 116.1 Hz, respectively. The first three 662
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 17. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode when f0 and fe have 5% errors.
Fig. 18. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode when PCT is applied.
Fig. 19. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode when HHT is applied.
663
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 20. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode when the beam is fixed at both ends.
Fig. 21. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode, (d) the fourth mode of a two-span continuous beam.
Fig. 22. Experimental setup.
664
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 23. Time history of vertical accelerations (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4.
mode shapes were reconstructed by the proposed method and 200 points were sampled for each mode. For better comparison, the mode shapes were also identified by the conventional method based on Frequency Response Functions to perform as baseline (25 points per mode). Fig. 23 shows the measured vertical accelerations for these four cases with different mass weights and travelling speeds. Generally, the higher the speed, the larger the amplitude of acceleration. The vibration is mainly caused by the sliding friction due to uneven surface roughness; and a few sudden peaks can be found because of the impacts between the lumped mass and the beam. Fig. 24 shows the time-frequency analysis of the acceleration signals in which the “Hanning” window with length of 2048 is used. The first three time-varying IFs of the interaction system can be observed in all four cases. Each impact between the lumped mass and the beam shown in the time history introduces an obvious horizontal “line” in the timefrequency representation, because the delta function in time domain is uniform in frequency domain. The IF trajectories are wider if the travelling speed is higher, making the IF extraction more difficult, although higher traveling speed is more expected in practice. Moreover, it is observed that for each mode, the corresponding IF trajectory becomes blur and undetectable near the nodal points. Three band-pass filters with bands of 10–30 Hz, 30–70 Hz, 70–120 Hz are used for the first three modes. Figs. 25–28 show the extracted first three mode shapes for the four cases. In comparison with the mode shapes constructed by the conventional method, those extracted by the proposed method show a high level of accuracy for all four cases and more importantly, they consist of more sampled points. In particular, it is observed that the heavier lumped mass is preferable because the IFs may change more significantly which benefits the proposed method. However, it should be noted that the lumped mass
cannot be too heavy because the assumption of the proposed method may be invalid. It is also found that even the travelling speed increased to 10 cm/s and the lumped mass passed through the beam by only 10 s (Case 2 and Case 3), the accuracy of identified mode shapes is still high. 5. Conclusion A novel method to extract the mode shapes of beam like structure by using a moving lumped mass is proposed in this study. Through the analytical study, extracting mode shapes becomes a typical problem of time varying IF estimation of the dynamic response of the travelling lumped mass. A modified time-frequency analysis method which is more accurate than conventional time-frequency analysis method like STFT and CWT is proposed to evaluate the time varying Ifs, and a measured energy-based sampling algorithm is proposed to reconstruct the mode shapes, which samples points with higher SNR; therefore, it is more robust to noise. Numerical simulation has been conducted, and it is observed from the numerical results that the proposed method performs quite well in reconstructing the mode shapes. Parametric study has also been carried out: even if 20% noise is added to the original signal, the proposed method still performs well especially for low order mode shapes; the lumped mass should be selected carefully because the variation of IFs cannot be detected if the lumped mass is too light and the proposed method may fail if the lumped mass is too heavy; although lower travelling speed is preferable, higher travelling speed which is more practical for this method can also give satisfactory results. Laboratory experimental study is carried out to validate the proposed method, even if the lumped mass is as light as 20% of the beam mass and the total travel time is as short as 10 s. The proposed method is more convenient because only a lumped 665
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 24. Time frequency analysis (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4.
Fig. 25. Estimated mode shapes for Case 1 (a) the first mode, (b) the second mode, (c) the third mode.
666
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 26. Estimated mode shapes for Case 2 (a) the first mode, (b) the second mode, (c) the third mode.
Fig. 27. Estimated mode shapes for Case 3 (a) the first mode, (b) the second mode, (c) the third mode.
Fig. 28. Estimated mode shapes for Case 4 (a) the first mode, (b) the second mode, (c) the third mode.
667
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
mass with a single accelerometer is required, and it is more practical because external exciter is not required with the roughness being the source of excitation. Moreover, it is more robust to noise because only points with higher SNR are sampled.
[11]
Acknowledgement
[12]
The authors would like to thank the School of Civil and Environmental Engineering at Nanyang Technological University, Singapore for kindly supporting this research topic.
[13]
[14]
Appendix A. Supplementary material
[15]
Supplementary data to this article can be found online at https:// doi.org/10.1016/j.engstruct.2018.11.074.
[16]
References
[17] [19]
[1] Altunisik AC, Bayraktar A, Ozdemir H. Seismic safety assessment of eynel highway steel bridge using ambient vibration measurement. Smart Struct Syst 2012;10(2):131–54. [2] Fan W, Qiao PZ. Vibration based damage identification methods: a review and comparative study. Struct Health Monit 2011;10:83–111. [3] Khiem N, Toan L. A novel method for crack detection in beam like structures by measurements of natural frequencies. J Sound Vib 2014;333:4084–103. [4] Dilena M, Limongelli M, Morassi A. Damage localization in bridges via the FRF interpolation method. Mech Syst Sig Process 2015;52:162–80. [5] Caldwell RA, Feeny BF. Output-only modal identification of a nonuniform beam by using decomposition methods. J Vib Acoust 2014;136(4):041010. [6] Yang YB, Yang JP. State-of-the-art review on modal identification and damage detection of bridges by moving test vehicles. Int J Struct Stab Dyn 2018;18(2):1850025. [7] Yang YB, Lin CW, Yau JD. Extracting bridge frequencies from the dynamic response of a passing vehicle. J Sound Vib 2004;272:471–93. [8] Lin CW, Yang YB. Use of a passing vehicle to scan the fundamental bridge frequencies: an experimental verification. Eng Struct 2005;27:1865–78. [9] Yang YB, Chen WF, Yu HW, Chan CS. Experimental study of a hand-drawn cart for measuring the bridge frequencies. Eng Struct 2013;57:222–31. [10] Oshima Y, Kobayashi Y, Yamaguchi T, Sugiura K. Eigenfrequency estimation for
[20] [21] [22] [23] [24] [25] [26]
668
bridges using the response of a passing vehicle with excitation system. Proceedings of the 4th International Conference on Bridge Maintenance, Seoul, Republic of Korea. 2008. Kim CW, Isemoto R, Toshinami T, Kawatani M, McGetrick PJ, O’Brien EJ. Experimental investigation of drive-by bridge inspection. Proceedings of the 5th International Conference on Structural Health Monitoring of Intelligent Infrastructure, Cancun, Mexico. 2011. Siringoringo DM, Fujino Y. Estimating bridge fundamental frequency from vibration response of instrumented passing vehicle: analytical and experimental study. Adv Struct Eng 2012;15(3):417–33. Nagayama T, Reksowardojo A, Su D, Mizutani T, Zhang C. Bridge natural frequency estimation by extracting the common vibration component from the responses of two vehicles. 6th international conference on advances in experimental structural engineering, Urbana-Champaign, USA. 2015. Gonzalez A, Obrien EJ, McGetrick PJ. Identification of damping in a bridge using a moving instrumented vehicle. J Sound Vib 2012;331:4115–31. Keenahan J, OBrien EJ, McGetrick PJ, González A. The use of a dynamic trucktrailer drive-by system to monitor bridge damping. Struct Health Monit 2014;13:143–57. Ewins DJ. Modal testing: theory, practice and application. 2nd ed England: Research Studies Press Ltd; 2000. Wenzel H, Pichler P. Ambient vibration monitoring. England: John Wiley & Sons Ltd.; 2005. Zhang Y, Wang LQ, Xiang ZH. Damage detection by mode shape squares extracted from a passing vehicle. J Sound Vib 2012;331:291–307. Yang YB, Li YC, Chang KC. Constructing the mode shapes of a bridge from a passing vehicle: a theoretical study. Smart Struct Syst 2014;13:797–819. Malekjafarian A, Obrien EJ. Identification of bridge mode shapes using short time frequency domain decomposition of the responses measured in a passing vehicle. Eng Struct 2014;81:386–97. Obrien EJ, Malekjafarian A. A mode shape-based damage detection approach using laser measurement from a vehicle crossing a simply supported bridge. Struct Control Health Monit 2016;23:1273–86. Malekjafarian A, Obrien EJ. On the use if a passing vehicle for the estimation of bridge mode shapes. J Sound Vib 2017;397:77–91. Kong X, Cai CS, Kong B. Numerically extracting bridge modal properties from dynamics responses of moving vehicles. ASCE J Eng Mech 2016;142(6):04016025. Qi ZQ, Au FTK. Identifying mode shapes of girder bridges using dynamic responses extracted from a moving vehicle under impact excitation. Int J Struct Stab Dyn 2017;17(8):1750081. Peng ZK, Meng G, Chu FL, Lang ZQ, Zhang WM, Yang Y. Polynomial chirplet transform with application to instantaneous frequency estimation. IEEE Trans Instrum Meas 2011;60(9):3222–9.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Estimation of mode shapes of beam-like structures by a moving lumped mass Yao Zhang, Haisheng Zhao , Seng Tjhen Lie ⁎
T
School of Civil & Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
ARTICLE INFO
ABSTRACT
Keywords: Mode shape Beam like structures Moving lumped mass Non-stationary IF
This paper presents a method to estimate mode shapes of beam like structures by using the acceleration of a moving lumped mass. In fact, the coupled frequencies of the vehicle-beam interaction system are time varying and the non-stationary instantaneous frequencies (IFs) contain information of mode shapes. The theoretical analysis in this paper shows that the mode shapes can be re-constructed by using the IFs; therefore, extracting mode shapes becomes a problem of IF estimation. A modified time-frequency analysis method based on weighted polynomial chirplet transform is developed to estimate the non-stationary IFs. Moreover, a new sampling algorithm based on accumulative measured energy is proposed to reconstruct the mode shapes, in which more data are sampled at the area with higher measured energy, making it more robust to noise. The proposed method is more convenient since only a lumped mass with a single accelerometer is required, and it is more practical because external exciter is not required, and the surface roughness can be the source of excitation. Numerical simulations and laboratory scale experiments have been carried out, which show that the proposed method performs well in extracting mode shapes, even if the travelling speed is high.
1. Introduction
characteristic matrix which is the multiplication of inverse mass matrix and stiffness matrix, and mode shapes are corresponding eigenvectors. Because eigenvectors are more vulnerable and sensitive to noise of the characteristic matrix than eigenvalues, low level noise may bring high level errors on eigenvectors. Therefore, extracting mode shapes with high accuracy is usually more difficult than natural frequencies. In conventional modal analysis, estimation of mode shapes usually has higher error than natural frequencies [16,17]. In spectrum analysis, the measurement noise can easily contaminate the magnitude. Since mode shapes are also very important in damage detection and FE model validation and considering the indirect method has been successfully employed in practice, some researchers attempted to use indirect approach to identify mode shapes, which is a tougher task in practice than extracting natural frequencies [18]. Zhang et al. [19] first attempted to extract the mode shape squares of beam like structures by using the acceleration of a passing vehicle equipped with tapping device. Because it requires exciter installed on the vehicle, it is somehow inconvenient in practice. Yang et al. [20] proposed a theoretical method to construct the mode shapes from the vertical acceleration of a passing vehicle, and they found the road roughness inversely affected the proposed method. To overcome this inverse effect, Malekjafarian and Obrien [21] evaluated the mode shapes by applying Short Time Frequency Domain Decomposition to dynamic responses of two following quarter-cars travelling over a bridge. Later, they [22] used multiple laser
Modal analysis is important in civil and mechanical engineering because modal parameters can be used to validate finite element (FE) models [1] and monitor the health condition of structures [2]. In conventional approaches, they are measured through common modal analysis techniques like Frequency Response Function method, Fourier spectral element method, and output-only decomposition method in which a lot of sensors are installed on the structures [3–5]. These approaches are still frequently used in practice; however, they are usually costly and time consuming. Recently, indirect measurement of modal parameters by using dynamic responses of a moving vehicle has been investigated [6] because it only requires a single sensor installed on the vehicle which is more convenient in practice. Yang et al. [7] first proposed this concept to extract the natural frequencies of bridges, and they experimentally validated the feasibility of this method on real bridges by using both travelling vehicle [8] and hand-drawn cart [9]. Several studies have also been conducted to identify bridge frequencies by using dynamic responses of a passing vehicle [10–13]. Moreover, Gonzalez et al. [14] and Keenahan et al. [15] tried to identify the damping by using this indirect method and they carried out numerical verifications. Identification of natural frequencies and mode shapes is a typical eigenvalue problem. Natural frequencies are eigenvalues of a
⁎
Corresponding author. E-mail address: [email protected] (H. Zhao).
https://doi.org/10.1016/j.engstruct.2018.11.074 Received 25 June 2018; Received in revised form 28 November 2018; Accepted 28 November 2018 0141-0296/ © 2018 Elsevier Ltd. All rights reserved.
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
measurements from a passing vehicle to evaluate the mode shapes, and a novel approach is also proposed to estimate the mode shapes by using a travelling truck trailer system equipped with an external excitation [23]. Kong et al. [24] used two trailers and subtracted the response of one trailer by the response of the other with a time shift. Qi and Au [25] identified the bridge mode shapes based on the dynamic response of a passing vehicle under impact excitation. However, they only presented numerical studies and no experimental validation was carried out. In this study, a new mode shape extraction method for beam like structure is proposed by using a moving lumped mass. The IFs of the mass-beam interaction system change with the location of the lumped mass and therefore they contain the information of mode shapes. Then, extracting mode shapes becomes a problem of evaluating IFs. A modified time-frequency analysis method based on weighted polynomial chirplet transform is proposed to evaluate the non-stationary IFs and a new sampling algorithm based on accumulative measured energy is proposed to reconstruct the mode shapes, in which more points are sampled at the area with higher measured energy. The proposed method is more convenient since only a lumped mass with a single accelerometer is required, and it is more practical because external exciter is not required, and the surface roughness can be the source of excitation. Numerical simulations and laboratory scale experiments have been carried out, indicating that the proposed method performs well in extracting lower order mode shapes, even if the travelling speed is high.
ml EI n4 4 q¨ + q = 2 n 2 l3 n
bn
mv
d2qv = dt 2
+ EI
mv g
= f (t ) (x f (t )
xc )
n
n (x ) qn (t ) n
¨i i (x c ) q
2 n (x c )
2 bn qn
q¨n +
ml
r¨ (x c )
g
(5)
i n
=
2m v ml
n (x c )
of
¨i i (x c ) q
the
beam,
+ r¨ (xc ) + g
i n
=
bn 2mv n2 (x c )
(7)
ml
It can be found from Eq. (7) that the natural frequencies of the interaction system are functions of the mode shapes; therefore, extracting mode shapes becomes a typical problem of identifying the IFs of the interaction system n (x )
=
n sign(x )
2 bn 2 n
1
(8)
where n is a constant coefficient to normalize the nth mode shape, sign(x ) is a function to determine the sign of the value based on prior knowledge. Two issues should be addressed: one is that although the simply supported boundary condition is used here, the method is applicable for other types of boundary condition which is verified in the following experimental study; the other is that the method is only effective when the weight of lumped mass cannot be ignored compared to the beam weight, otherwise the IFs are the time independent natural frequencies of the beam. 2.2. IF estimation based on weighted polynomial chirplet transform Short Time Fourier Transform (STFT) and Continuous Wavelet Transform (CWT) are usually used to produce time-frequency distributions for signals in civil engineering; however, neither STFT nor CWT can achieve a good frequency resolution due to the restriction of the Heisenberg-Gabor inequality. Chirplet Transform (CT) initially designed for the analysis of chirp-like signals with linear IF law has been improved by using the chirplet kernel with a polynomial nonlinear IF law [26]; however, it performs under expectation when the noise level is high. Therefore, a modified algorithm based on weighted polynomial chirplet transform to estimate the IF is proposed in this study. First of all, consider a nonlinear chirp signal, s (t ) , and its analytical signal can be easily obtained by Hilbert Transform:
(1) (2)
(3)
Using the modal superposition method, the vertical displacement of the beam can be written as
ub (x , t ) =
2m v
1+
where qv (t ) and ub (x , t ) are the vertical displacements of the lumped mass and the beam, respectively; f (t ) is the interaction force and (·) is the Dirac delta function. They are in permanent contact; therefore, the following equation can be obtained as
qv = ub (x c , t ) + r (xc )
n (x c )
The nth natural frequency of the interaction system, including the lumped mass and the beam, can be approximated as
Fig. 1 shows a typical simplified model of a lumped mass travelling over a simply supported Euler beam. The beam has length l , bending stiffness EI , mass m per unit length, and surface roughness profile r (x ) ; the lumped mass is travelling at a constant speed, v , and x c is the coordinate of the contact point. For simplicity, damping is temporarily ignored herein. The governing equations of the interaction system are given as
m
mv
(6)
2.1. Interaction of a lumped mass passing through a beam
4u b x4
2 ¨n n (x c ) q
By considering that the nth frequency = n2 2l 2 (EI /m )1/2 , Eq. (5) can be simplified as
1+
2. Theory
2u b t2
mv
(9)
z (t ) = s (t ) + jH (s (t )) The analytical signal z (t ) can be further written as:
(
(4)
z (t ) = a (t )exp j 2
where n (x ) = sin n x / l is the nth mode shape and qn (t ) is the corresponding modal coordinate. Substituting Eqs. (2)–(4) into Eq. (1), multiplying m (x ) on both siders and integrating over the beam, one can obtain
t 0
f ( )d +
)
(10)
where a (t ) and f ( ) are the instantaneous amplitude and frequency, is the initial phase. The polynomial chirplet transform (PCT) of z (t ) is defined as [26]: Fig. 1. Simplified model of a lumped mass travelling through a beam.
655
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
PCT(t 0, =
,
+
1,
,
n;
) n+1
z (t )exp
j
tk 1 k 1
k 1t k=2
=
+
(
a (t )exp j 2
t 0
f ( )d +
t 0k
1
w (t
(t )
) w (t
t 0 )exp( j t )dt t 0 )exp( j t )dt (11)
where
w (t
t0) =
1 2
n+ 1
(t ) =
k 1t k=2
exp
1 t 2
tk 1 k 1
t0k
2
(12)
1
(13)
Fig. 3. Illustration of accumulative energy-based sampling algorithm.
It should be noted that w shown in Eq. (12) is Gaussian function, which is frequently used in time-frequency analysis as a window function. When the IF of the de-chirplet operator is equal to the real IF of the analytical signal z (t ) , that is, d (t )/dt = f (t ) , the PZT could produce a high-quality time-frequency distribution for z (t ) and the result can have good concentration. Fig. 2 shows the algorithm to evaluate the IF trajectory: in the initial step, a predefined threshold, , and the polynomial order, n should be set and the polynomial coefficients are assumed as zero; for the ith step, the ith PCT is calculated and the ith IF can be obtained by picking the peaks in PCT, and then the polynomial coefficients can be calculated by weighted least square method; once the difference between the evaluated IFs in two constitutive steps is less than the predetermined threshold, the final estimation of IF trajectory can be determined. It should be noted that when calculating the coefficients in the ith step, t 0 and te indicate the start time and end time of the travelling, respectively, while f0 and fe are the natural frequencies of the interaction system for the lumped mass standing at the start-point and the endpoint, respectively. This is because when the lumped mass is just standing on the beam, the frequency of the interaction system can be identified accurately.
2.3. Accumulative energy-based sampling algorithm Usually, the measurement points are sampled uniformly; however, some points may have more significant measurement noise, especially for those near the nodal points. Therefore, an accumulative energybased sampling algorithm is proposed. Considering that the amplitude in PCT at the resonance frequency usually represents the vibration energy, the accumulative energy at a certain time, tc, is defined as tc
AE (tc ) =
0
Amp (t )dt
(14)
The basic idea of this sampling algorithm is that more data can be sampled at the time interval with higher accumulative measured energy and Signal-to-Noise Ratio (SNR) and vice vasa, which can be seen in Fig. 3; therefore, the extracted mode shapes are more robust to noise. If the total accumulative energy is AE and N points are required, the data will be sampled at time t1, t2, …, tN, which can be determined by:
AE (ti ) =
(i N
1) AE , i = 1, 1
,N
Fig. 2. Flowchart of weighted polynomial chirplet transform based IF estimation algorithm. 656
(15)
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
3. Numerical simulations
Before extracting the mode shapes by the proposed method, the signal is filtered by a band-pass filter so that only one single frequency component is contained. In numerical simulations, three bands, 1–10 Hz, 20–30 Hz, 40–60 Hz are used for the first three modes. In this study, the value of σ is set as 0.3, which is identical to that in Ref. [26]. It should be noted that when σ is in the range of 0.2–0.5, it shows little influence on the identified IFs and re-constructed mode shapes. The polynomial order, n, is also important: it should be neither too small nor too large, otherwise, underfitting or overfitting can be observed. It can be determined as follows: the number of half-sine pulses that are contained in one specific mode shape should be evaluated by prior knowledge, and n can then be calculated by multiplying this number by 4 because quadric function can usually fit a half-sine pulse well. Therefore, n can be obtained as 4, 8,12 for the first three mode shapes. Moreover, is set as 0.001 herein. Fig. 5 shows the initial PCTi=1 of the acceleration at t0 = 1 s and t0 = 5 s where polynomial coefficients are zero, and the locations of peaks (IFi=1) can be obtained by searching the maximum values which are also defined as Ampi=1(t0). For example, for the first mode, IFi=1(t0 = 1) = 5.86 Hz, Ampi=1(t0 = 1) = PCTi=1(t0 = 1, ω = 5.86 × 2π, α = 0) = 0.66, IFi=1(t0 = 5) = 5.37 Hz, Ampi=1 (t0 = 5) = PCTi=1(t0 = 5, ω = 5.37 × 2π, α = 0) = 0.21. Fig. 6 shows the extracted mode shapes by the proposed method, in which the solid line represents the analytical solution and the circle indicates the extracted mode shapes sampled by using the sampling algorithm based on accumulative energy. For each vibrational mode, 200 points are sampled. It is observed that the first three mode shapes are identified with good accuracy where fewer points are sampled near the nodal points.
3.1. FE model FE model is constructed by using general purpose commercial software ABAQUS in this study. The simply supported Euler beam has length of 20 m, cross-sectional area of 1.8 m2, moment of inertia of 0.3375 m4, density of 2400 kg/m3, and elastic modulus of 30 GPa. Therefore, the first three natural frequencies of the beam are 6.012, 24.048 and 54.108 Hz. The beam is modeled by 2-node plane beam elements in length of 0.5 m. The lumped mass is 10,000 kg and it passes over the beam at 2 m/s. The time step of the analysis is 0.001 s. It is noteworthy that the parameters of the FE model are selected to simulate a heavy road roller with high stiffness travelling over a short-span bridge, so that the proposed method is possible to be used in full scale structures. 3.2. Extraction of mode shapes Fig. 4 shows the time history of vertical acceleration of the lumped mass and its corresponding time frequency representation by using STFT in which “Hanning” window with length of 2048 is used. It can be seen clearly that the frequency resolution is quite low by using STFT due to limited window length and therefore it is almost impossible to extract accurate IF with high resolution.
3.3. Effect of measurement noise Because the measurement noise due to either surface roughness or equipment makes the identification more difficult, it is necessary to examine the resistance of the proposed method to noise. Three levels of Gaussian noise, 5%, 10%, and 20% are considered herein which are added directly to the acceleration response as follows:
Sn (t ) = S (t ) + nl random(t )max[S (t )]
(16)
where S (t ) and Sn (t ) are original and polluted signals, respectively, nl is the noise level and random(t ) is a function to generate random number between −1 and 1. Fig. 7 shows the original acceleration signal and polluted signals with different noise levels. Then the extracted mode shapes by applying the proposed method to the contaminated signals are shown in Figs. 8–10. The proposed method performs quite well when the noise level is up to 10% and almost no deviation can be found between the extracted mode shapes based on accelerations with and without noise. When the noise increases to 20%, the first two mode shapes can be identified well; however, the extracted third mode shape is less accurate near the nodal points. It is noteworthy that the number of points with less accuracy are comparatively fewer than others with higher accuracy due to the proposed accumulative energy-based sampling algorithm. 3.4. Effect of weight of lumped mass It is found in Eq. (7) that the IFs change more significantly if the lumped mass is heavier and they may be time independent constants if the weight of lumped mass can be neglectable; therefore, it is necessary to investigate the effect of weight of lumped mass on the proposed method. Six more lumped masses are used herein for parametric study: 200 kg, 1000 kg, 2000 kg, 5000 kg, 15,000 kg and 20,000 kg. By following the same procedure, the first three mode shapes can be extracted for each lumped mass. Table 1 lists the MAC values between the extracted mode shape and theoretical ones. Generally, the proposed method performs best when the lumped mass of 10,000 kg (mv/ ml = 11.6%) is used and it performs quite well if the mass is in the range of 5000 kg (mv/ml = 5.8%) to 15,000 kg (mv/ml = 17.4%); however, it fails if the mass is less than 1000 kg (mv/ml = 1.16%).
Fig. 4. (a) Vertical acceleration response of the traveling lumped mass and (b) its corresponding time frequency representation (STFT). 657
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 5. PCT at t0 = 1 s and t0 = 5 s (a) the first mode, (b) the second mode, (c) the third mode.
Fig. 6. Normalized mode shapes, (a) the first mode, (b) the second mode, (c) the third mode.
When the mass is only 200 kg, the IFs are almost constant; therefore, the mode shapes cannot be identified, and the MAC values are not available. When mv increases to 1000 kg, the first two mode shapes can be extracted but the MAC values are relatively low, and the third mode shape cannot be identified because higher vibrational modes cannot be excited. All the first three mode shapes can only be identified when mv increases to 2000 kg; however, the MAC values are still not high. When the lumped mass of 5000 kg is used, the identified first three mode shapes shown in Fig. 11 are quite accurate and the MAC values are all over 0.997. The first three mode shapes match best with theoretical ones when mv is 10,000 kg and the MAC values are over 0.998. The proposed method performs quite well in extracting the first mode shape when mv increases to 15,000 kg and 20,000 kg (Fig. 12), while the reconstructed higher mode shapes show slight deviations concentrated at the nodal points. This may be because the proposed method assumes linear vibration and the mode shapes of the beam do not change due to the travelling lumped mass, which may not be true when the lumped mass is too heavy.
Fig. 7. Original acceleration signal and polluted signals with different noise levels (5–5.5 s). 658
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 8. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode by using the acceleration with 5% noise.
3.5. Effect of travelling speed
frequencies are 3.87, 15.27 and 34.40 Hz. First, a lumped mass of 5000 kg is used to extract the first three mode shapes, which are shown in Fig. 15. The re-constructed mode shapes match quite well with the theoretical ones. Then a test vehicle which is modeled as a single degree of freedom mass-spring system is adopted to estimate the first three mode shapes. It has mass of 5000 kg and stiffness of 5 × 106 N/m. Fig. 16 shows the estimated first two mode shapes: the first one can be re-constructed quite well but the second one show slight deviations near the end nodes. However, it should be admitted that the third mode shape cannot be identified by applying the proposed method to the acceleration of the test vehicle, because the vehicle’s frequency is much lower than the bridge’s third frequency and it performs like a low-pass filter so that the higher frequency component is filtered. This is also the reason that a lumped mass which has very high stiffness is recommended for the proposed method, for example, a heavy road roller with high stiffness. However, considering that a general test vehicle is more valuable in practice, how to extract higher order mode shapes by using the dynamic responses of the test vehicle should be further investigated in the future.
The travelling speed is also quite important for the proposed method because it saves more time if the travelling speed is higher. In the case of baseline, the velocity of 2 m/s is considered, while velocities of 1 m/s and 4 m/s are investigated herein. The estimated first three mode shapes for these two cases are shown in Figs. 13 and 14, respectively. As expected, when the travelling speed is lower, the proposed method performs better; while it is also observed that when the lumped mass passes over the beam at 4 m/s, the identified mode shapes are almost as accurate as those extracted by the acceleration of lumped mass with velocity of 2 m/s. This is an advantage of the proposed method that it still performs well when the lumped mass travels over the whole beam by using 5 s. However, it is admitted that when the travelling speed increases to 10 m/s, the proposed method can hardly identify the higher mode shapes due to very limited data. 3.6. Example of a bridge inspected by a lumped mass and a test vehicle A bridge with lower frequencies is adopted for further demonstration. The parameters used are identical to those used in Ref. [20]: the bridge has length of 30 m, moment of inertia of 0.175 m4, mass density of 1000 kg/m, and elastic modulus of 27.5 GPa; the first three natural
3.7. Effect of accuracy of f0 and fe Two frequencies, f0 and fe, are quite important in the proposed
Fig. 9. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode by using the acceleration with 10% noise. 659
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 10. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode by using the acceleration with 20% noise.
acceleration of the lumped mass to estimate the IFs, the re-constructed first three mode shapes shown in Fig. 18 match the theoretical ones well. However, when it is applied on the acceleration with 10% noise, only the extracted first mode shape matches the theoretical one well and large deviations concentrated at the nodal points can be found in the second and third mode shapes. Similar conclusion can be obtained when HHT is used (Fig. 19). In fact, HHT performs quite well if the signal is noise-free and the estimated IFs and the re-constructed mode shapes are accurate; however, it cannot estimate the IFs accurately if the signal is subject to 10% noise. Therefore, specific filtering technique is required before PCT or HHT can be used to extract the IFs and reconstruct mode shapes.
Table 1 MAC of the first three modes for different lumped masses. mv (kg)
200
1000
2000
5000
10,000
15,000
20,000
Mode 1 Mode 2 Mode 3
– – –
0.9914 0.9963 –
0.9916 0.9980 0.9928
0.9998 0.9991 0.9972
0.9999 0.9993 0.9981
0.9998 0.9972 0.9943
0.9998 0.9968 0.9934
method, and therefore, the effect of accuracy of f0 and fe is investigated herein. For each mode, the two frequencies are assumed to have 5% errors, that is, f0 = fe = 5.71 Hz for the first mode, f0 = fe = 22.85 Hz for the second mode and f0 = fe = 51.40 Hz for the third mode. Fig. 17 shows the extracted first three mode shapes when f0 and fe have 5% errors. The proposed method still performs well but large deviations can be observed near the end nodes, because the frequencies, f0 and fe have quite large errors. Fortunately, f0 and fe can usually be measured accurately since they are measured when the lumped mass stops on the beam.
3.9. Effect of boundary condition A fixed-fixed beam is investigated herein to verify the availability of the proposed method for different boundary conditions. The beam is identical to that used in Section 3.2 and the first three frequencies increase to 13.63, 37.57, and 73.65 Hz due to the fixed-fixed boundary condition. Fig. 20 shows the extracted first three mode shapes and they match very well with those obtained from numerical simulations.
3.8. Comparison of WPCT, PCT and HHT Two more IF estimation algorithms, PCT and HHT are used herein for better demonstration. When PCT is applied on the noise-free
Fig. 11. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode by using the 5000 kg lumped mass. 660
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 12. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode by using the 20,000 kg lumped mass.
Fig. 13. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode when the lumped mass passes through the beam by 1 m/s.
Fig. 14. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode when the lumped mass passes through the beam by 4 m/s.
661
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 15. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode when the lumped mass passes through the bridge.
Fig. 16. Estimated mode shapes (a) the first mode, (b) the second mode when the test vehicle passes through the bridge.
3.10. Example of two-span continuous beam
differentiate these two modes, indicating the limitation of the method in distinguishing closed modes. Therefore, how to improve the performance of the proposed method on multi-span beams and closed modes should be further investigate in the future.
Since the multi-span continuous bridge is widely used in practice, it is necessary to investigate whether the proposed method is still applicable for this case. A simply supported two-span continuous beam having length of 40 m is adopted herein, and other properties are identical to those in Section 3.2. The first four frequencies are 6.01, 9.39, 24.05 and 30.43 Hz, respectively. It is noteworthy that the third frequency (24.05 Hz) is nearly 80% of the fourth frequency (30.43 Hz), which can be approximated as closed modes. Fig. 21 shows the identified first four mode shapes, coinciding with the FEM simulations well; nevertheless, it is admitted that slight deviations can be observed near the nodal points. Moreover, the proposed method has also been applied to three-span continuous beam, and the corresponding first five frequencies are 6.01, 7.67, 11.28, 24.05, and 27.27 Hz, respectively. The fourth frequency is 88% of the fifth frequency, which are two closer modes than the previous case. However, the proposed method fails to reconstruct the fourth and fifth mode shapes since it is difficult to
4. Experimental study Fig. 22 shows the experimental setup: a thin steel beam (2.4 kg) with dimension of 120 cm × 4.4 cm × 0.6 cm clamped at two ends, a linear motion unit pulling a lumped mass to travel at the top surface with a constant speed, and an accelerometer attached on the lumped mass. The vertical acceleration was recorded at sampling frequency of 1000 Hz. Four cases including two travelling speeds and two lumped masses were investigated in this study. For Case 1 and Case 2, the lumped mass is 0.5 kg and the travelling speeds are 5 cm/s and 10 cm/s, respectively; while for Case 3 and Case 4, the lumped mass is 1 kg and the travelling speeds are 5 cm/s and 10 cm/s. The first three frequencies of the beam are 24.5, 59.6 and 116.1 Hz, respectively. The first three 662
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 17. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode when f0 and fe have 5% errors.
Fig. 18. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode when PCT is applied.
Fig. 19. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode when HHT is applied.
663
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 20. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode when the beam is fixed at both ends.
Fig. 21. Estimated mode shapes (a) the first mode, (b) the second mode, (c) the third mode, (d) the fourth mode of a two-span continuous beam.
Fig. 22. Experimental setup.
664
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 23. Time history of vertical accelerations (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4.
mode shapes were reconstructed by the proposed method and 200 points were sampled for each mode. For better comparison, the mode shapes were also identified by the conventional method based on Frequency Response Functions to perform as baseline (25 points per mode). Fig. 23 shows the measured vertical accelerations for these four cases with different mass weights and travelling speeds. Generally, the higher the speed, the larger the amplitude of acceleration. The vibration is mainly caused by the sliding friction due to uneven surface roughness; and a few sudden peaks can be found because of the impacts between the lumped mass and the beam. Fig. 24 shows the time-frequency analysis of the acceleration signals in which the “Hanning” window with length of 2048 is used. The first three time-varying IFs of the interaction system can be observed in all four cases. Each impact between the lumped mass and the beam shown in the time history introduces an obvious horizontal “line” in the timefrequency representation, because the delta function in time domain is uniform in frequency domain. The IF trajectories are wider if the travelling speed is higher, making the IF extraction more difficult, although higher traveling speed is more expected in practice. Moreover, it is observed that for each mode, the corresponding IF trajectory becomes blur and undetectable near the nodal points. Three band-pass filters with bands of 10–30 Hz, 30–70 Hz, 70–120 Hz are used for the first three modes. Figs. 25–28 show the extracted first three mode shapes for the four cases. In comparison with the mode shapes constructed by the conventional method, those extracted by the proposed method show a high level of accuracy for all four cases and more importantly, they consist of more sampled points. In particular, it is observed that the heavier lumped mass is preferable because the IFs may change more significantly which benefits the proposed method. However, it should be noted that the lumped mass
cannot be too heavy because the assumption of the proposed method may be invalid. It is also found that even the travelling speed increased to 10 cm/s and the lumped mass passed through the beam by only 10 s (Case 2 and Case 3), the accuracy of identified mode shapes is still high. 5. Conclusion A novel method to extract the mode shapes of beam like structure by using a moving lumped mass is proposed in this study. Through the analytical study, extracting mode shapes becomes a typical problem of time varying IF estimation of the dynamic response of the travelling lumped mass. A modified time-frequency analysis method which is more accurate than conventional time-frequency analysis method like STFT and CWT is proposed to evaluate the time varying Ifs, and a measured energy-based sampling algorithm is proposed to reconstruct the mode shapes, which samples points with higher SNR; therefore, it is more robust to noise. Numerical simulation has been conducted, and it is observed from the numerical results that the proposed method performs quite well in reconstructing the mode shapes. Parametric study has also been carried out: even if 20% noise is added to the original signal, the proposed method still performs well especially for low order mode shapes; the lumped mass should be selected carefully because the variation of IFs cannot be detected if the lumped mass is too light and the proposed method may fail if the lumped mass is too heavy; although lower travelling speed is preferable, higher travelling speed which is more practical for this method can also give satisfactory results. Laboratory experimental study is carried out to validate the proposed method, even if the lumped mass is as light as 20% of the beam mass and the total travel time is as short as 10 s. The proposed method is more convenient because only a lumped 665
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 24. Time frequency analysis (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4.
Fig. 25. Estimated mode shapes for Case 1 (a) the first mode, (b) the second mode, (c) the third mode.
666
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
Fig. 26. Estimated mode shapes for Case 2 (a) the first mode, (b) the second mode, (c) the third mode.
Fig. 27. Estimated mode shapes for Case 3 (a) the first mode, (b) the second mode, (c) the third mode.
Fig. 28. Estimated mode shapes for Case 4 (a) the first mode, (b) the second mode, (c) the third mode.
667
Engineering Structures 180 (2019) 654–668
Y. Zhang et al.
mass with a single accelerometer is required, and it is more practical because external exciter is not required with the roughness being the source of excitation. Moreover, it is more robust to noise because only points with higher SNR are sampled.
[11]
Acknowledgement
[12]
The authors would like to thank the School of Civil and Environmental Engineering at Nanyang Technological University, Singapore for kindly supporting this research topic.
[13]
[14]
Appendix A. Supplementary material
[15]
Supplementary data to this article can be found online at https:// doi.org/10.1016/j.engstruct.2018.11.074.
[16]
References
[17] [19]
[1] Altunisik AC, Bayraktar A, Ozdemir H. Seismic safety assessment of eynel highway steel bridge using ambient vibration measurement. Smart Struct Syst 2012;10(2):131–54. [2] Fan W, Qiao PZ. Vibration based damage identification methods: a review and comparative study. Struct Health Monit 2011;10:83–111. [3] Khiem N, Toan L. A novel method for crack detection in beam like structures by measurements of natural frequencies. J Sound Vib 2014;333:4084–103. [4] Dilena M, Limongelli M, Morassi A. Damage localization in bridges via the FRF interpolation method. Mech Syst Sig Process 2015;52:162–80. [5] Caldwell RA, Feeny BF. Output-only modal identification of a nonuniform beam by using decomposition methods. J Vib Acoust 2014;136(4):041010. [6] Yang YB, Yang JP. State-of-the-art review on modal identification and damage detection of bridges by moving test vehicles. Int J Struct Stab Dyn 2018;18(2):1850025. [7] Yang YB, Lin CW, Yau JD. Extracting bridge frequencies from the dynamic response of a passing vehicle. J Sound Vib 2004;272:471–93. [8] Lin CW, Yang YB. Use of a passing vehicle to scan the fundamental bridge frequencies: an experimental verification. Eng Struct 2005;27:1865–78. [9] Yang YB, Chen WF, Yu HW, Chan CS. Experimental study of a hand-drawn cart for measuring the bridge frequencies. Eng Struct 2013;57:222–31. [10] Oshima Y, Kobayashi Y, Yamaguchi T, Sugiura K. Eigenfrequency estimation for
[20] [21] [22] [23] [24] [25] [26]
668
bridges using the response of a passing vehicle with excitation system. Proceedings of the 4th International Conference on Bridge Maintenance, Seoul, Republic of Korea. 2008. Kim CW, Isemoto R, Toshinami T, Kawatani M, McGetrick PJ, O’Brien EJ. Experimental investigation of drive-by bridge inspection. Proceedings of the 5th International Conference on Structural Health Monitoring of Intelligent Infrastructure, Cancun, Mexico. 2011. Siringoringo DM, Fujino Y. Estimating bridge fundamental frequency from vibration response of instrumented passing vehicle: analytical and experimental study. Adv Struct Eng 2012;15(3):417–33. Nagayama T, Reksowardojo A, Su D, Mizutani T, Zhang C. Bridge natural frequency estimation by extracting the common vibration component from the responses of two vehicles. 6th international conference on advances in experimental structural engineering, Urbana-Champaign, USA. 2015. Gonzalez A, Obrien EJ, McGetrick PJ. Identification of damping in a bridge using a moving instrumented vehicle. J Sound Vib 2012;331:4115–31. Keenahan J, OBrien EJ, McGetrick PJ, González A. The use of a dynamic trucktrailer drive-by system to monitor bridge damping. Struct Health Monit 2014;13:143–57. Ewins DJ. Modal testing: theory, practice and application. 2nd ed England: Research Studies Press Ltd; 2000. Wenzel H, Pichler P. Ambient vibration monitoring. England: John Wiley & Sons Ltd.; 2005. Zhang Y, Wang LQ, Xiang ZH. Damage detection by mode shape squares extracted from a passing vehicle. J Sound Vib 2012;331:291–307. Yang YB, Li YC, Chang KC. Constructing the mode shapes of a bridge from a passing vehicle: a theoretical study. Smart Struct Syst 2014;13:797–819. Malekjafarian A, Obrien EJ. Identification of bridge mode shapes using short time frequency domain decomposition of the responses measured in a passing vehicle. Eng Struct 2014;81:386–97. Obrien EJ, Malekjafarian A. A mode shape-based damage detection approach using laser measurement from a vehicle crossing a simply supported bridge. Struct Control Health Monit 2016;23:1273–86. Malekjafarian A, Obrien EJ. On the use if a passing vehicle for the estimation of bridge mode shapes. J Sound Vib 2017;397:77–91. Kong X, Cai CS, Kong B. Numerically extracting bridge modal properties from dynamics responses of moving vehicles. ASCE J Eng Mech 2016;142(6):04016025. Qi ZQ, Au FTK. Identifying mode shapes of girder bridges using dynamic responses extracted from a moving vehicle under impact excitation. Int J Struct Stab Dyn 2017;17(8):1750081. Peng ZK, Meng G, Chu FL, Lang ZQ, Zhang WM, Yang Y. Polynomial chirplet transform with application to instantaneous frequency estimation. IEEE Trans Instrum Meas 2011;60(9):3222–9.