Bridge natural frequency estimation by extracting the common vibration component from the responses of two vehicles

Engineering Structures 150 (2017) 821–829

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Bridge natural frequency estimation by extracting the common vibration component from the responses of two vehicles T. Nagayama b, A.P. Reksowardojo a,⇑, D. Su b, T. Mizutani c a

School of Architecture, Civil and Environmental Engineering, Swiss Federal Institute of Technology (EPFL), CH-1015 Lausanne, Switzerland Department of Civil Engineering, The University of Tokyo, Tokyo 113-8656, Japan c Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan b

a r t i c l e

i n f o

Article history: Received 16 September 2016 Revised 12 July 2017 Accepted 13 July 2017

Keywords: Vehicle-bridge interaction system Indirect frequency estimation Bridge assessment

a b s t r a c t Bridge natural frequencies are among fundamental properties of bridges. However, natural frequencies of most bridges remain unknown. Natural frequency identification through acceleration measurement with sensors installed on bridges is not practical if a large number of bridges are investigated. Indirect methods to detect the frequencies from the acceleration responses of a vehicle driving over bridges, on the other hand, still have difficulties. Vehicle responses contains various components, making it difficult to distinguish the bridge frequency from other frequency components. This study proposes new frequency estimation strategy utilizing two vehicles. The key idea is that bridge vibration, a common vibration component among responses of multiple vehicles, is extracted through signal processing involving cross-spectral density function (CPSD) estimation. Numerical analyses employing a vehicle-bridge interaction (VBI) model are conducted to examine the algorithm performance under various conditions. The numerical simulations have confirmed the feasibility of such indirect frequency detection. An experimental study featuring synchronized-sensing of two vehicles is then performed. The first natural frequency of the bridge has been identified under various driving speed combinations, demonstrating the performance of the proposed approach. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction While bridge natural frequencies are among fundamental properties of bridges, natural frequencies of most bridges remain unknown. Natural frequencies, considered to reflect bearing conditions and characterize resonance phenomena under periodic loading, are typically identified through acceleration measurement with sensors installed on bridges [1]. Nonetheless, sensor installation on a large number of bridges is not practical. Installation of instruments to the structure involves a costly process and often encounters site-specific technical difficulties. Studies have investigated the possibility of using a vehicle to obtain the natural frequency, such as those conducted by Yang et al. [2], McGetrick et al. [3], and Siringoringo et al. [4]. Vehiclebased measurement centers on the concept that bridge vibration influences the vehicle response as a result of vehicle-bridge interaction (VBI) process. Therefore, it is theoretically feasible to obtain bridge natural frequency from the vehicle response. The feasibility has been verified in several studies by means of closed-form solu⇑ Corresponding author. E-mail address: [email protected] (A.P. Reksowardojo). http://dx.doi.org/10.1016/j.engstruct.2017.07.040 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.

tion [4,2], finite element simulations [3–5,2] and experiments [6,4,7]. In practice, however, the application of vehicle-based measurement is difficult. Typically, the spectral peak associated with bridge frequency is low in magnitude. In some cases, peaks associated with road roughness components are prevalent [3,5], to the extent that these peaks nullify the bridge related frequency peak. Engineers are oftentimes expected to assess the natural frequency without a priori knowledge of the frequency range. The presence of non-bridge frequency peaks combined with a weak bridge frequency peak (or a complete lack thereof) potentially leads to a misleading frequency identification. Thus, there is a need for a technique to extract the bridge frequency among others. This study proposes new frequency estimation strategy utilizing two vehicles. While there are past studies utilizing a tractortrailer system and addition heavy vehicles as excitation sources [6,7], this proposed strategy utilizes two ordinary vehicles without trailers as measurement vehicles to emphasize a bridge natural frequency component among others; there is no need to prepare special vehicles. Bridge vibration, a common vibration component among responses of multiple vehicles, is extracted through signal processing. Signal processing techniques investigated include cross-spectral density function estimation, spectrum averaging

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and envelop technique, and frequency domain decomposition. In this paper, bridge natural frequency identification using the cross-spectral density function estimation, which is assessed to be most effective among the three in extracting the bridge vibration components, is proposed. Numerical analyses employing a vehicle-bridge interaction (VBI) model are conducted to examine the algorithm performance under various conditions. The numerical simulations have confirmed the feasibility of such indirect frequency detection. An experimental study featuring synchronizedsensing of two vehicles is then performed. The vehicles are of different types, leaving the bridge frequency as the remaining signal commonality. The first natural frequency of the bridge has been identified under various driving speed combinations, demonstrating the natural frequency identification performance of the proposed approach. 2. Bridge natural frequency extraction strategy Vibrations of a vehicle on a bridge has two main excitation sources (i.e., bridge vibration and road roughness) [8]. While vehicle responses associated with the road roughness can be large depending on the road profile, their frequency peaks of the spectrum are not necessarily sharp due to the random nature of road roughness as well as large damping of vehicles. On the other hand, bridge vibration components are typically not as large as the components associated with road roughness. Small vibration of bridge surface is further damped before the vibration reaches the vehicle body. Separation of bridge vibration components from the others and its extraction is usually difficult. The bridge vibration component is, however, considered to appear commonly in responses of vehicles on the bridge. The extraction of bridge natural frequency as the common vibration component is proposed herein. In terms of commonality of two signals, the vehicle response components due to road roughness are not perfectly common because of differences in drive paths and speeds. On the other hand, the bridge vibration components are approximately common to multiple vehicles on the bridge; in particular, first vertical mode vibration, where all locations of the bridge vibrate in phase, is considered to commonly appear on all vehicles on the bridge though local difference in mode shape amplitude affects signals. The measurement strategy that employs a cross-spectral density function estimation process is described in Fig. 1. Under this strategy, two vehicles are employed to simultaneously capture €A and y €B of vehicle A and the vertical acceleration responses, i.e., y B, respectively. Because vehicles are running in a consecutive

B

manner, both vehicles are excited by the bridge responses at the same moment. On the other hand, the vehicle excitations due to road roughness are expected to be less identical for reasons explained above. The bridge vibration components are extracted through cross-spectrum estimation [10]:

2 Gxy ðf Þ ¼ lim E½X  ðf ; T ÞY ðf ; T Þ T!1 T

ð1Þ

where Gxy ðf Þ is the cross-spectral density function of two random processes xðt Þ and yðtÞ as a function of frequency, f. X  ðf ; T Þ and Y ðf ; T Þ are finite Fourier Transform of xðtÞ and yðt Þ. E [] denotes the expectation operator and the asterisk ⁄ denotes complex conjugate. T is the time length of the random process to be evaluated. In practice, the spectral density function is estimated as temporal average instead of ensemble average using the ergodicity assumption. Cross-spectral density function (CPSD) and power-spectral density function (PSD) will simply be referred to as cross-spectrum and power-spectrum from hereafter. The strategy of bridge frequency estimation using crossspectrum estimation is summarized as follows: €A and y €B are obtained. 1. Two vehicles’ acceleration responses, i.e., y Only data corresponding to the time during which both vehicles are on the bridge shall be taken. In addition, both signals are time-synchronized. €A and y €B as the ran2. Cross-spectrum is computed by employing y dom processes xðt Þ and yðt Þ. 3. Prominent peaks that have appeared on the cross-spectrum are identified. The prominence of a peak is quantified by the clarity index defined herein as

I ðf i Þ ¼

1 ðf u f l Þ

Gxy ðf i Þ R fu Gxy ðf Þdf f

ð2Þ

l

where f i is the frequency value of interest. f l and f u are the lower- and upper- limits of the frequency range of interest, respectively. Peak i is considered as prominent when its associated clarity index Ii is greater than a threshold. In this paper, the threshold is empirically set as 3 through numerical simulations. 4. At these prominent peak frequencies, clarity indices of each vehicle’s power spectrum are calculated. The clarity index for vehicle A, IA ðf i Þ, is calculated by employing vehicle A’s acceleration response as the random processes xðtÞ and yðtÞ. In the same manner, IB ðf i Þ is calculated using vehicle B’s response.

A

vB yA

yB yb(x) w(x) r(x)

Fig. 1. Mechanical model of the measurement strategy.

vA

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5. The clarity index differences of vehicle A, defined as DIA ðf i Þ ¼ Iðf i Þ  IA ðf i Þ, is computed. The clarity index difference DIB ðf i Þ for vehicle B is calculated in the same manner. 6. A prominent peak with the largest value of DIðf i Þ ¼ DIA ðf i Þþ DIB ðf i Þ is treated as a frequency component with substantial commonality. This frequency is identified as the bridge natural frequency estimate.

mv b1 b2 yv

Several Vehicle-bridge interaction (VBI) analysis methods have been developed. Efficiency in terms of computational resource usage, nonetheless, varies. Element-level coupling method, as pointed out by Yang et al. [9], has the best performance in terms of computational resource. The method employed in this study consists of multiple equations of motion:

€ v g þ ½C v fu_ v g þ ½K v fuv g ¼ fF cv g ½M v f u and

Mb;i







€ b;i þ C b;i u



u_ b;i



m2

m1 y2 ct2

ct1 kt2

kt1 y1

ð3Þ

w2



    þ K b;i ub;i ¼ F cb;i

ð4Þ

Eqs. (3) and (4) represent the equations of motion associated with vehicle and bridge, respectively. ½M v , ½C v  and ½K v  are vehi      cle’s mass, damping, and stiffness matrices. Mb;i , C b;i , and K b;i are bridge’s mass, damping, and stiffness matrices. fuv g and   ub;i are vehicle’s and bridge’s displacement vectors. Note that the index i denotes the bridge element upon which vehicle is acting. Simulation of n vehicles on a bridge can be achieved simply by employing n vehicle equations, i.e., Eq. (3), and by modifying Eq. (4) to incorporate multiple vehicles. Encapsulated in the interaction load term of vehicle fF cv g is the road roughness profile, which is simulated by computing the inverse Fourier transform of a predescribed PSD curve with trigonometric sum technique [11]:

r ðxÞ ¼

k1

k1

3.1. Vehicle-bridge interaction (VBI) formulation



c1

c1

3. Numerical feasibility study

θv

N X Ai ðsin Xi x  /i Þ

w1

Fig. 2. Mechanical model of the vehicle.

The bridge is modeled as a simple beam with a length of L, a  uniform flexural rigidity of EI, and a mass per unit length of m. Damping of the bridge is approximated by mass-stiffness proportional Rayleigh damping. While bridge elements upon which vehicles directly act are modeled by so-called interaction element [8], other bridge elements are modeled as 6-DOF beam element. By employing the Newmark-b method, one can solve the assembled system of vehicle and bridge in tandem to obtain the dynamic responses of vehicles and the bridge. For unconditional stability, parameters b and c of the Newmark- b method are selected as 0.25 and 0.5, respectively.

ð5Þ 3.2. Numerical results

i¼1

where /i is uniformly distributed random phase angle, Ai is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi UðXi ÞDXp1 . UðXi Þ represents the pre-described PSD of the profile. Xi denotes angular spatial frequency and DX is the frequency resolution of the roughness power-spectrum. As shown in Fig. 2, a 4-DOF vehicle model is employed. Thus the vehicle model can represent vertical and rotational movement of the body, as well as vertical movement of the axles. The subsystem matrices of a 4-DOF model and the response vector are as follows:

2

mv 6 0 6 ½M v  ¼ 6 4 0

0

0

jv

0

0

m1

0

0

0

2

c1 þ c2 6c b  c b 1 1 6 2 2 ½C v  ¼ 6 4 c1

0 7 7 7 0 5

c 2 b2  c 1 b1 2

k1 þ k2 6k b  k b 1 1 6 2 2 ½K v  ¼ 6 4 k1

2

c1

c 1 b1  c 2 b2

c 1 b1

c 1 b1

c1 þ ct1

c2 b2

0

k2 b2  k1 b1 2 2 k1 b1  k2 b2

k1 k1 b1

k1 b1

k1 þ kt1

k2 b2

0

k2 fuv g ¼ ½ y h y1

ð6Þ

m2

c2 2

3

0

y2 

mv ¼ 16600 kg, jv ¼ 64598 kg m2 , mi ¼ 700 kg, ci ¼ 10  103 Ns=m,

ki ¼ 4  105 N=m and kti ¼ 3:5  106 N=m. While the mass is as large as truck or bus mass, this parameter set is adopted to better understand the phenomena with respect to resonance, speed, and vehicle spacing through numerical simulation with large bridge vibration. Parameters for vehicle B is same as those for vehicle A except that the body mass is reduced to 8800 kg. The two vehicles thus have different vibration characteristics. As for the bridge, its physical parameters are: EI ¼ 1:06  105 MN m2,  ¼ 4406:78 kg=m, and L = 59 m. Road profile is generated accordm ing to the power-spectrum curve of Class A surface provided in

3

c2 c2 b2 7 7 7 5 0

Numerical simulations are performed assuming a box-girder bridge used for experimental validation described in Section 4.1. The vehicle parameters are adopted from McGetrick et al. [3] with some modifications. Those parameters for vehicle A are:

ð7Þ

c2 þ ct2 3 k2 k2 b2 7 7 7 5 0

ð8Þ

k2 þ kt2 ð9Þ

Fig. 3. Simulated vertical acceleration of vehicles and bridge midspan.

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ISO8608 [12]. The time step in the simulation and the sampling time interval are both 0.005 s. Fig. 3 presents vehicle body acceleration responses of the vehi€A and y €B , and bridge acceleration response y €b . Two vehicles cles, y successively pass a bridge. The drive speed of vehicle A is 30 km/ h while that of vehicle B is 5% smaller, i.e. 28.5 km/h. The dashed lines in the figure indicate the start and end point of data included in spectral analysis, that is, the data portion where both vehicles are over the bridge. Fig. 4a shows the individual power-spectra computed from the responses of vehicle A and vehicle B. The dashed vertical line indicates fb,l = 2.21 Hz, the bridge natural frequency calculated from FEM analysis, which is consistent with the analytical solution of the beam. This frequency value is supposedly unknown in the real measurement. The bouncing and pitching frequencies of the vehicle are 1.05 and 1.00 Hz, respectively. Two axle modes are at 11.9 Hz. At this stage, it is not possible to identify a peak which is a bridge natural frequency without a priori knowledge of the value. A dominant peak is noticeable in the PSD A within the proximity of the vehicle frequency. Other peaks with possible influences from roughness also exist. In the PSD B, several dominant peaks are visible not only in the region of vehicle frequency, but also in the bridge natural frequency range. Roughness influences appear on the spectrum, too. Cross-spectrum presented in Fig. 4b has four prominent peaks whose clarity indices are greater than 3. By following the aforementioned procedure, the relative change of clarity indices Ii for all prominent peaks between those of cross-spectrum and power-spectra are computed. Table 1 presents DIðf i Þ for i ¼ f1; 2; 3; 4g. As indicated by the symbol ‘‘” in the table, peak 4 at 2.18 Hz exhibits the largest increase in clarity index; this is a strong indication of frequency commonality. In addition, peak 4 is close to the bridge natural frequency obtained from the eigenvalue analysis. Note that the cross-spectrum does not eliminate the other peaks completely; peaks due to vehicle natural frequencies and roughness effects, which are considered less common in the two signals, become less clear. In this simulation, the difference of 0.03 Hz, i.e. 1.4%, is smaller than 10%; the identification is successful. As for the higher modal frequencies (i.e., fb,2, fb,3, . . .), these peaks are unidentifiable in the individual power-spectra as well as in the cross-spectrum. 3.3. Drive speed effect In order to examine the effect of different driving speed between vehicle A and vehicle B, a series of simulations is performed. For each simulation, a scaling factor rv is introduced to differentiate driving speeds of vehicle A and vehicle B.

Table 1 Clarity index changes DIi obtained from simulation. i

1

2

3

4

f i (Hz) DIA ðf i Þ DIB ðf i Þ DIðf i Þ

0.44 3.93 4.01 7.94

0.96 5.70 12.30 18.00

1.74 0.11 0.76 0.65

2.18 17.94 16.60 34.54 d

v B ¼ rv v A

ð10Þ

where v A and v B are driving speeds of vehicles A and B, respectively. Because vehicle B follows vehicle A, v B is smaller than v A and the value of r v is always smaller than unity. Other parameters such as vehicle mass and axle stiffness remain identical between the two vehicles. A measurement is considered successful when it satisfies the following condition:

  ^  f b;1  f b;1   0:1f b;1 ð11Þ

ð11Þ

where ^f b;1 is the estimation of the first natural frequency of the bridge, f b;1 . Due to the coupling between the bridge and vehicle, the dominant frequency of the system is known to be off the eigenvalue of the bridge [14]. This frequency shift is smaller than this criterion of 10%. The same criteria is also used for experimental validation later. Table 2 presents the summary. The value of ^f represents the b;1

bridge natural frequency estimated by the algorithm described in  Section 2. DI ^f b;1 presents the clarity index increment that corre  sponds to the DI ^f b;1 . In this case, DI ^f b;1 is the largest increment among all. The symbol ‘‘” indicates a successful identification, while ‘‘s” indicates otherwise (see Eq. (11)). For comparison, another value is introduced as ~f which represents the prominent b;1

frequency peak that is the nearest to the bridge eigenvalue. ~f b;1 is  shown along with DI ~f b;1 for the sole purpose of examining the effect of speed difference to the ‘‘actual” bridge natural frequency peak. The result reveals that the effect of speed difference is apparent, but exhibiting no specific pattern. Nevertheless, in most cases simulations with slower driving speed for vehicle B yield greater visibility of bridge natural frequency. 3.4. Vehicle parameter effect An additional series of simulation is performed to examine the effect of parameter difference between vehicle A and B. Two

Fig. 4. Spectra from simulation.

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T. Nagayama et al. / Engineering Structures 150 (2017) 821–829 Table 2 Effect of driving speed difference. rv  1.0 0.9 0.8 0.7 0.6 0.5 rv  1.0 0.9 0.8 0.7 0.6 0.5

v A ¼ 10 km=h

v A ¼ 15 km=h

^f b;1 (Hz)

 DI ^f b;1

~f b;1 (Hz)

 DI ~f b;1

^f b;1 (Hz)

 DI ^f b;1

~f b;1 (Hz)

 DI ~f b;1

2.01 1.67 1.79 2.13 2.27 1.86

45.87 33.21 32.25 86.68 44.33 42.26

2.12 2.09 2.25 2.13 2.27 2.24

21.45 31.50 25.37 86.68 44.33 8.03

1.48 1.43 1.44 1.45 1.45 1.48

60.77 55.93 51.28 28.09 53.93 59.28

2.22 2.28 2.22 2.23 2.27 nil.

1.99 0.68 1.16 19.85 0.93 nil.

^f b;1 (Hz)

 DI ^f b;1

~f b;1 (Hz)

 DI ~f b;1

^f b;1 (Hz)

 DI ^f b;1

~f b;1 (Hz)

 DI ~f b;1

1.96 1.94 1.93 2.10 2.07 2.09

28.15 38.98 47.49 28.85 31.88 41.60

2.18 2.31 2.16 2.10 2.07 2.09

8.57 3.33 1.20 28.85 31.88 41.60

2.18 2.15 2.22 2.13 1.86 2.28

20.22 38.11 35.60 38.19 22.94 30.45

2.18 2.15 2.22 2.13 2.27 2.28

20.22 38.11 35.60 38.19 10.10 30.45

d s s d d s

v A ¼ 25 km=h

s s s d d d

Table 3 Effect of parameter difference (with

s s s s s s

v A ¼ 30 km=h

d d d d s d

vA = vB = 30 km/h).

(a) Mass difference

(b) Stiffness difference

~ ^f b;1 ¼ f b;1 (Hz)

  DI ^f b;1 ¼ DI ~f b;1

rk

~ ^f b;1 ¼ f b;1 (Hz)

  DI ^f b;1 ¼ DI ~f b;1

1.5 1.4 1.3 1.2 1.1 1.0

2.27 2.27 2.27 2.27 2.18 2.18

d d d d d d

28.89 30.12 30.84 31.06 31.88 32.68

1.5 1.4 1.3 1.2 1.1 1.0

2.18 2.18 2.18 2.18 2.18 2.18

d d d d d d

29.31 29.57 29.73 29.94 30.03 29.78

0.9 0.8 0.7 0.6 0.5

2.18 2.18 2.18 2.18 2.18

d d d d d

33.40 34.18 35.01 35.57 35.57

0.9 0.8 0.7 0.6 0.5

2.18 2.18 2.18 2.18 2.18

d d d d d

29.05 27.85 25.80 21.96 22.19

rm

scaling factors are introduced, rm and rk, that correspond to vehicle mass and axle stiffness, respectively.

½M v B  ¼ r m ½M v A 

ð12Þ

½K v B  ¼ r k ½K v A 

ð13Þ

where ½M v A  and ½M v B  are mass matrices of vehicles A and B, respectively. ½K v A  and ½K v B  are stiffness matrices of vehicles A and B, respectively. Driving speeds of the two vehicles are identical. Table 3 presents the summary. The result reveals that successful measurements are obtained in   all cases, hence ^f ¼ ~f ¼ DI ~f . It is clearly and DI ^f b;1

b;1

b;1

b;1

shown that decreases in vehicle B mass result in increases in bridge visibility. Decreases in the stiffness matrix of vehicle B result in decreases in bridge peak visibility. However, this is true only when rk < 1.1. 3.5. Driving speed and spacing effect Finally, the numerical simulation is repeated under different conditions with respect to the driving speeds and spacing. The influence of driving speed v within the range of 5–80 km/h, in conjunction with the influence of inter-vehicle spacing b within the range of 1–10 m are investigated. Fig. 5 presents what the authors refers to as feasible region of measurement (FRM), that is the area (i.e., the combinations of driving speed and spacing) where identification of bridge natural frequency peak is possible. The FRM of identification from power-spectrum A and power-spectrum B indi-

Fig. 5. Feasible region of measurement: ( spectrum B and ( ) Cross-spectrum.

) Power-spectrum A, (

) Power-

cates areas where bridge natural frequency peak happens to have the greatest amplitude in the spectrum. The FRM of identification from cross-spectrum indicates the area where bridge natural frequency happens to have the largest value of DI. It is demonstrated that under cross-spectrum strategy, the FRM is substantially larger than those of power-spectrum A and power-spectrum B. Such expansion indicates that under cross-spectrum strategy, not only the rate of success of bridge natural frequency identification is higher, but also the measurement is less sensitive to variation of speed v and inter-vehicle spacing b. It is also clear that the identification of the bridge natural frequency is not possible when the drive speed is too high. A possible explanation is that shorter acceleration data associated with high drive speed result in spectra,

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T. Nagayama et al. / Engineering Structures 150 (2017) 821–829

which are poor in frequency resolution. Note that these results depend on the bridge and vehicle characteristics; when vehicle and bridge parameters are different, similar analysis needs to be performed. FRMs appear to fit a certain order. Formulas derived by Yang et al. [9] defined vehicle-driven resonance as a phenomenon that takes place whenever certain conditions of driving speed and inter-vehicles spacing are met. Resonance conditions are briefly explained here. The period during which a vehicle is travelling on the bridge, assuming a constant velocity of v and bridge span length of L, will give rise to an excitation frequency corresponding to the vehicle driving frequency, which is defined as:

v

a¼

2L

ð14Þ

The existence of this frequency implies the occurrence of dynamic resonance in cases where driving frequency or its multiple coincides with the bridge natural frequencies. To demonstrate this, a non-dimensional parameter Sn is introduced:

Sn ¼

na nv ¼ f b;n f b;n 2L

ð15Þ

However, usually the velocity corresponding to integer values of Sn is impractically high. The condition is not satisfied under the driving speed considered here. A sequence that consists of several vehicles travelling in identical constant velocity of v and with a spacing between successive moving loads b0 will generate spacing-related frequency as follows:

k¼

v b

0

ð16Þ

Analogous to driving frequency, the spacing frequency will also give rise to dynamic resonance in cases where spacing frequency or its multiple coincides with the bridge natural frequencies. A nondimensional parameter of Rn is introduced.

Rn ¼

k v ¼ f b;n f b;n b0

ð17Þ

For the sake of simplicity, only resonance in the first bridge frequency is discussed. Resonance occurs whenever the first bridge frequency f b;1 coincides with the spacing frequency k. Hence R1 is equal to unity. As the spacing-related frequency is not pure sinusoidal wave, excitation component exist at integer multiples of the frequency. The equation is solved for the spacing b0 as 0

b ¼

kv f b;1

4. Experimental study 4.1. Overview An experimental study has been conducted in 2014 with a 59 m simply supported steel box-girder bridge as the test bed. The target bridge, Tsukige Bridge, is situated in Kimitsu City, Chiba Prefecture, Japan. With a width of 4 m, one can expect the bridge to behave akin to a two-dimensional simply supported beam. Fig. 6 illustrate the measurement process in the field. Two different commercial vehicles are employed, namely, Toyota Land Cruiser and Toyota HiAce for Vehicle A and Vehicle B, respectively. Though the weight of vehicles are not measured, these vehicles are considered to weight about 2 ton based on the specifications. High accuracy MEMS-type tri-axial accelerometers, JA-70SA [13], are installed on the two vehicles. The installation location near the center of gravity of the vehicles are considered advantageous in reducing the effect of pitching and rolling motion based on preliminary numerical analysis. However, sensors are installed on flat surfaces of the vehicles near their rear axles due to the practicability in sensor installation. An additional accelerometer is attached directly on the road surface to obtain the reference values of bridge natural frequencies. Using peak-picking method bridge natural frequencies are obtained as 2.17 Hz and 5.43 Hz for the 1th and 2nd mode, respectively. Sampling rates of the accelerometers are all set at 1 kHz. Data acquisition systems are PC-driven and are connected with Global Positioning System (GPS) receivers that feature 30 ns time accuracy. Utilizing these hardware devices, a LabVIEW [14] program is designed to achieve high accuracy synchronized sensing as follows. The primary role of GPS receiver within the system is to provide 1 Hz pulse triggers for data recording so that the vehicle response measurements at 1 kHz for 1000 samples are triggered synchronously every one second. In addition, GPS receiver acquires precise wall clock time information, which is later converted to JST (Japan Standard Time) and appended into the acceleration record as timestamp. In conjunction with timestamping, GPS receiver also captures velocity. Both timestamping and velocity acquisition are performed at 1 Hz. The resulting time synchronization is estimated to be able to deliver pairwise synchronization with an accuracy well below 1 ms. Free vibration tests of the test vehicles are conducted. The vehicles are manually excited by 4 persons. Free vibrations after the release are measured and analyzed. Using peak-picking method, natural frequencies of Vehicle A and Vehicle B are estimated as 1.46 Hz and 1.62 Hz, respectively. However, as most commercial vehicles typically have high damping, a full cycle free vibration response is difficult to be produced; consequently the identification of vehicle frequencies presented herein is considered to have unignorable estimation errors.

ð18Þ 4.2. Experimental results

where k denotes integer number. Superimposed on Fig. 5 are the vehicle-driven resonance criteria of k ¼ f1; 2; 3; 4g. FRMs appear to cluster around resonance criterion of k ¼ f1; 2; 3; 4g. Note that the inter-vehicle spacing b in Fig. 5 is defined as the distance between the rear axle of vehicle A and the front axle of vehicle B. The spacing between loads b0 considered above is obtained by addition the wheelbase to the intervehicle spacing b. There are large FRM near 30 km/h driving speed. This driving speed corresponds to the resonant condition involving the loading from the front and rear axles. When the spacing between load b0 is taken as the wheelbase distance, i.e. 3.75 m, the velocity satisfying Eq. (18) with k = 1 is 30 km/h.

The test was repeated multiple times with different driving speed combinations shown in Table 4. For each case, the vehicles ran over the bridge five times from both ends, i.e., 10 total observations for every case. Thus, there are 40 observations of various driving speeds in total. Note that relatively constant driving speeds were achieved in most observations according to the GPS records. Fig. 7 presents power-spectra and cross-spectrum of one observation from Case 2. Table 5 presents the corresponding DIðf i Þ for i ¼ f1; 2; . . . ; 7; 8g. Peak #4 at 2.17 Hz exhibits the largest increase in clarity index. Furthermore, Fig. 8 shows cross-spectra from four observations, each from different measurement cases.

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T. Nagayama et al. / Engineering Structures 150 (2017) 821–829

Fig. 6. Measurement and instrumentation of vehicles.

One should direct their attention, however, to the result of Case 4 in Fig. 8d; which is an example of occurrences where the strategy failed to correctly estimate the bridge frequency. Summary of clarity index changes shown in Table 6 reveals that although clarity index change of the peak #3 (the closest candidate of bridge frequency peak) is significant in value, it is undermined with that of peak #1; thus raising a misleading conclusion of 0.89 Hz as the bridge frequency. There are several possible reasons that may contribute to such identification failure,

Table 4 Measurement cases. Case # 1 2 3 4

Driving speed

v (km/h)

Vehicle A

Vehicle B

10 15 30 30

10 10 30 25

Fig. 7. Spectra from Case 2 of experiment.

Table 5 Clarity index changes DIi obtained from Case 2 of experiment. i

1

2

3

4

5

6

f i (Hz) DIA ðf i Þ DIB ðf i Þ DIðf i Þ

0.05 11.57 13.24 24.81

1.58 18.81 18.13 36.94

1.96 5.95 1.03 6.98

2.17 56.60 49.87 106.48 d

2.34 28.47 18.24 46.71

2.55 12.84 9.40 22.24

Fig. 8. Sample spectra from various measurement cases.

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T. Nagayama et al. / Engineering Structures 150 (2017) 821–829

Table 6 Clarity index changes Dli obtained from Case 4 of experiment. i

1

2

3

4

f i (Hz) DIA ðf i Þ DIB ðf i Þ DIðf i Þ

0.89 51.12 44.07 95.19 s

1.88 2.86 1.75 4.61

2.17 32.60 34.94 67.53

2.47 0.65 8.01 8.65

namely, driving speed fluctuations, excessive passenger movements, road irregularities, and synchronization error. Nonetheless, the resulting cross spectrum as shown in the figure is relatively clear, showing only two dominant peaks. It is still possible to subjectively select peak #3 as the bridge frequency, given a priori knowledge on possible range of bridge natural frequency. Additionally, if one can determine what frequency peak #1 is attributed to (e.g., vehicle’s natural frequency), peak #3 can be selected as the bridge frequency. Due to some technical difficulties, four observations from Case 3 and three observations from Case 4 were lost and hence omitted from the sample population. The success rate (see Eq. (11)) of the experiment is measured as 48.48%, i.e., 16 successes out of 33 observations. Table 7 presents the breakdown of the measurement results. It is immediately apparent that measurements conducted in lower

speed (Case 1 and 2) yield more favorable results compared to those conducted in higher speed (Case 3 and 4). Two separate summaries are presented, namely estimated bridge frequency averaged from all data, and estimated bridge frequency averaged from successful realizations only. The former better reflect an actual measurement where the actual bridge frequency is not known a priori. It is implied that measurement in slower driving speed may be beneficial should the knowledge on the bridge frequency is not available beforehand. As pointed out by several studies [6,3,4,2], frequency peak that presents on vehicle response spectra typically shifts by the vehicle’s driving speed frequency as described by Eq. (14). Thus the ‘‘apparent” bridge frequency is ^f ¼ f a. b;1

b;1

4.3. Effect of synchronization error As previously discussed, the measurement clings on the ability to time-synchronize multiple signals. Therefore, cases where GPS receivers failed to capture wall clock time with good accuracy, are examined. Synchronization errors in the form of incremental time difference e of 500 ms are introduced to the experimental results. Fig. 9 presents the frequency detectability after incremental introductions of synchronization error to samples previously presented in Fig. 8.

Table 7 Summary of measurement. Case #

Success rate (%)

1

2

3

4

60.00

60.00

16.67

42.86

All data Estimated frequency ^f b;1 (Hz) Discrepancy Df b;1 a (%)

2.32

2.02

0.99

1.59

6.92

7.06

54.23

26.65

Successful realizations Estimated frequency ^f b;1 (Hz) Discrepancy Df b;1 a (%)

2.19

2.17

2.11

2.23

1.14

0.12

2.98

2.89

0.02 0.02    ^ Df b;1 ¼ f b;1  f b;1   100%; f b;1 ¼ 2:17 Hz. Estimated by constant driving speeds of 10 km/h for Case 1 and 2; 30 km/h for Case 3 and 4.

0.07

0.07

ab (Hz) a b

Fig. 9. Effect of synchronization error to cross-spectrum.

T. Nagayama et al. / Engineering Structures 150 (2017) 821–829

Dots in Fig. 9a indicate successful measurements. Fig. 9b presents cross-spectra (Case 1) corresponding to various synchronization errors, namely 0, 1, 3, and 4 s, labeled as (1), (2), (3), and (4) respectively. With incremental introductions of synchronization error the bridge peak appears to gradually lose its clarity. The introduction of 4 s time difference is enough to render the bridge peak undetectable. With large synchronization error, an increased presence of spurious peaks is to be expected. It is also implied that with higher driving speed, the effect of synchronization error is more severe (Case 3).

829

Toward practical implementation of this technique, the applicability to other bridges and other types of vehicles need further investigation. Acknowledgement The research reported herein is possible with the assistance from Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. Also, this research was supported by JSPS KAKENHI Grant Number 25630193. References

5. Concluding remarks This paper proposes the strategy to indirectly estimate the bridge natural frequency from vibration responses of two vehicles. The strategy involves two vehicles moving across the bridge. The acceleration responses recorded for the period when each vehicle is moving on the bridge is processed to compute their individual power-spectra and cross-spectrum. From the numerical study on a 59 m box-girder bridge, the strategy is demonstrated to be feasible under a relatively low driving speed. Bridge natural frequency can be identified as clear peak in the cross spectrum even under the influences of road roughness. Similar result is obtained from the field measurement, confirming the practicality of the strategy. The average success rate of the measurement is 44.88%. The estimated bridge natural frequency is found to be in fair agreement to the values obtained from direct measurement, with the largest discrepancy measured at 2.98%. Also investigated in the study, is the effect of synchronization error. The bridge natural frequency peak becomes less clear as the synchronization error increases. However, the bridge frequencies remains detectable even with the synchronization error of 3 s.

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