6 th International Conference on Advances in periental Structural ngineering th International Workshop on Advanced Sart Materials and Sart Structures Technology August -, 05, University of Illinois, Urbana-Chapaign, United States Analysis of Tie-Frequency nergy for nvironental Vibration Induced by Metro Guangzhen Li, Xiaosong Ren, Bin Zhang 3,Gang Zong 4 Graduate student, Research Institute of Structural ngineering and Disaster Reduction, Tongji University, Shanghai 0009, China. -ail: 33857@tongji.edu.cn Professor, Research Institute of Structural ngineering and Disaster Reduction, Tongji University, Shanghai 0009, China. -ail: rs@tongji.edu.cn 3 Graduate student, Research Institute of Structural ngineering and Disaster Reduction, Tongji University, Shanghai 0009, China. -ail: 97373707@qq.co 4 Lecturer, Research Institute of Structural ngineering and Disaster Reduction, State Key Laboratory of Disaster Reduction in Civil ngineering, Tongji University, Shanghai 0009, China. -ail: zong@tongji.edu.cn ABSTRACT Tie and frequency doain analysis processing are two conventional ethods for vibration analysis. Despite the advantages in the tie or frequency doain, the cluttering of non-stationery signal cannot be effectively distinguished by tie doain analysis, and the characteristics in certain frequency segents ay be neglected in the frequency doain analysis. By the adjustent of the length of tie window, tie-frequency energy ethod can effectively show the characteristics of energy in tie doain and do the coponent analysis in the specific frequency range. Hence the local characteristics of signals in the useful range can be got. Although this ethod was put forward early, it is not widely used for the restriction of the hardware and software for nuerical analysis. The ground vibration induced by etro is neither a kind of stationary vibration nor a transient ipact vibration. The energy is concentrated in the frequency range above 50Hz. This paper presents the signal processing using tie-frequency energy ethod. The signals collected in 8 points along the direction vertical to the etro line are chosen as eaples. By coparative study of signal duration, frequency band recognition and energy characterization, the vibration signal induced by non-etro aspects can effectively be separated and the propagation law for etro-induced vibration can be found. KYWORDS: Tie-frequency energy; nvironental vibration induced by etro, Signal analysis processing, Local characteristic Signal analysis processing is usually done in tie doain and frequency doain. Fro the analysis in tie doain, the characteristics, such as tie duration, the peak value, the RMS (root-ean-square) and the attenuation curve can be easily got, and the cross-correlation function can reveal the linear dependence between different signals. On the other hand, the signal analysis in frequency doain can proceed by using the Fourier Transfor. The frequency coponents can be got and hence odal identification can be done by the cobination of auto-power spectru and cross-power spectru. For non-stationary signals, tie history analysis and frequency spectru analysis see not effective and convincible. For eaple, the RMS as an energy representation is a ean value along tie ais which ay not reflect the reality for the reason of deterination of duration for weak signal, while Fourier Transfor is also not accurate enough for analyzing local characteristic of the non-stationary signal. In the 40s of last century, the concept of tie-frequency energy was put forward for the treatent of the non-stationary signals. Copared with the traditional processing ethod, tie-frequency energy shows advantage in analyzing local characteristic of frequency doain because the length of tie-window and frequency-window can be adjusted to get a balance of accuracy between tie doain and frequency doain. The tie-frequency energy is the direct reflection of signal energy and akes the energy characterization ore close to the reality, which overcoes the scattering of signal energy by using traditional ethod. The references as [-5] show that so far the signal processing is still separated in tie doain or frequency doain. The ain reason is the restriction of the hardware and software for nuerical application of tie-frequency energy.
As a typical environental vibration signal with evident local characteristics, the etro-induced ground vibration is neither a kind of stationary vibration nor a transient ipact vibration. Due to the relative gradual variety of envelope in the whole process of signal, the tie doain analysis is still suitable for signal evaluation. While fro the view on signal detail, it is ore like a transient ipact vibration signal and the paraeters in tie doain, such as the peak value, RMS, vibration acceleration level by weighted RMS cannot accurately describe the characteristics of signal. In this paper, the ground vibration induced by etro is analyzed by the eans of tie-frequency energy and through the coparison to the traditional ethod, the advantages of tie-frequency energy is presented.. INTRODUCTION OF TIM-FRQUNCY NRGY For the signal t, the frequency representation can be obtained by the Fourier transfor, j ft X f t e dt (.) where f is the signal frequency.the energy can be calculated through the integral as follows. ( t) dt X ( f ) d f (.) where t () and X( f ) are used as the representation of energy in tie doain and frequency doain. Siilarly, quadratic tie-frequency distribution is used for the instantaneous signal energy in this paper, Aong the classic quadratic tie-frequency distribution functions, the Cohen bilinear tie-frequency distribution function and Affine bilinear tie-frequency distribution function are ost coonly used for epression of tie-frequency transfor [6]. As the real part of Rihaczek tie-frequency distribution, which is one kind of Cohen bilinear tie-frequency distribution, Margenau-Hill-Spectrogra distribution is different fro other tie-frequency distribution for dealing with signal energy. It is the direct description of the discrete signal energy and has advantages to treat the cross-ter, inus energy and edge effect optiization [7]. In this paper, the tie-frequency energy analysis is based on Margenau-Hill-Spectrogra distribution. The tie-frequency distribution is defined as follows. * MHS ( t, f ) R F ( t, f ; h) F ( t, f ; g) K gh (.3) where MHS ( t, f ) is the tie-frequency distribution coefficient; R represents the real part of an iaginary nuber; F ( t, f ; h) is Rihaczek distribution in tie doain with window g ; F ( t, f ; g) is conjugate of Rihaczek * * distribution in frequency doain with window h ; Kgh h( u) g ( u) du is the integration for tie window h and conjugate of frequency window g, which is used to adjust the additive energy caused by the window. The instantaneous frequency estiation, group delay estiation, effective duration (abbreviated as T) and effective frequency band (abbreviated as BF) and so on, are used as evaluation inde of tie-frequency characteristics, as stated in Ref. [8-0]. The instantaneous frequency estiation is to describe the instantaneous frequency feature on the local tie point, while the instantaneous frequency on the whole duration reflects the tie-dependent law of frequency. The group delay deonstrates the phase rate of changes due to the frequency change rate, which is intuitively tie delay of signal wavefor envelope. The effective duration and the effective frequency band are two paraeters to show the duration range and frequency band that contribute ost to the signal energy. The effective duration is used to calculate the concentration degree of the signal and the effective frequency band is used to calculate the bandwidth of the signal frequency [0]. T t t ( ) t dt (.4)
~550 8 ~450 BF f f ( ) X f df (.5) where t and f are the tie center and frequency center, respectively. t t t dt () (.6) f f X f dv ( ) (.7) The signal can be characterized in the tie-frequency plane by its center position ( t, f ) and a doain of tie-bandwidth product as T BF. The energy value can be calculated in any tie and frequency range for concern. Here, the energy value is calculated just in the effective duration and effective frequency band. This kind of calculation eliinates the proble of scattering of signal energy over duration and the influence of other frequency. The local energy can be calculated through the integral of tie-frequency distribution coefficient as follows. t t f f P( t, f ) dtdf (.8) tt ff where P( t, f ) is the tie-frequency distribution coefficient, the sae as q. (.3) in this paper; BF f. For the discrete signal, the forula can be epressed as follows. T t P( t, v ) (.9) ti vi i i i where Pi ( ti, v i ) is the discretized value of tie-frequency distribution P( t, f ) at position ( ti, f i). It is seen that the tie-frequency energy can be used for the analysis of energy distribution of the transient ipact signal in the concerned tie length and frequency range.. ANALYTSIS FOR MTRO INDUCD GROUND VIBRATION The site test was done at 8 points on the ground, which is along a line vertical to the etro line as shown in Figure.. # Station A Metro line Road 3 3 4 6 5 5 5 0 # # 3# 4# 5# 6# 7# 8# 8 points along the line Metro train Station B (a)plan of site (b) Section of test site and etro Figure. Location of test points
Ground Acceleration(/s ) Ground Acceleration (/s ) Ground Acceleration(/s ) Ground Acceleration(/s ) The test points are located in one side of the etro line as it is an epty construction site, and any buildings are at the opposite side of these test points. The etro line is beneath the road with depth of about 8 eters. The total length of site test is 4 fours in the orning. The distance between two etro stations is about k and the test points are nearly located in the iddle. Although the etro is oving on two ways, the test points are ainly influenced by the etro train of the near side in one way. As the etro is passing away about every inutes, the signals including the etro and non-etro aspects are easured. The tie duration for etro influence is found to be about 5 seconds. In order to get the influence of etro on the ground vibration, it is necessary to distinguish the signals induced by the etro and non-etro aspects.. Saple curve of ground acceleration As the traffic vehicles oves on the road, the signals of ground vibration caused by these aspects are got fro site test and should not be considered for analysis on the influence of etro. According to the tietable of etro, the signals are separated into any segents of about inutes and the influence of etro is found to be about 5 seconds. The typical curves in test point and 4 are given in Figure. and.3. The left figures in Figure. and.3 represent the whole signals in one interval of the etro train, and the length of tie is 0 seconds. And the right figures in Figure. and.3 are part of the whole signals, which is arked as red block in left figures and represent the signals with obvious influence of only one etro train in about 5 seconds (fro 68 second to 84 second in the whole signal). 0.0 Metro induced signal 0.0 0.0 6 0-0.0-6 -0.0-0.0-0.03 0 0 40 60 80 00 0 Tie(s) 68 7 76 80 84 Tie(s) (a)in one interval of the etro train (b)with obvious influence of only one etro train Figure. Typical tie history of vibration signal at test point Metro induced signal 0.0 0.0 0.0 0.0-0.0-0.0-0.0-0.0 0 0 40 60 80 00 0 Tie(s) 68 7 76 80 84 Tie(s) (a)in one interval of the etro train (b)with obvious influence of only one etro train Figure.3 Typical tie history of vibration signal at test point 4
Aplitude Aplitude 0 saple curves with duration of about 5 seconds are chosen for analysis in this paper. The typical curves involve a short period of the ground-borne vibration with peak value as /s at the beginning and the end of the signal, which are arked by red dashed circles in the graph. Fro the site test in other quiet periods, the typical curves of the ground borne vibration is found with peak value as /s, which is consistent with the test signals at the beginning and the end. It can be found that the peak acceleration in test point and 4 are about 0.0 /s and /s, which includes the influence of the ground borne vibration. Although the ground borne vibration cannot be eliinated directly by subtracting it fro the test signals, the peak value of ground vibration at test point 4 is uch larger than that at test point. It is an interesting phenoenon of etro-induced ground vibration and will be deonstrated by other ethods later.. Analysis by traditional ethod The average value of the ten signal duration turns to be 5.9 seconds, as shown in Table.. For the reason of not restrict way for truncating data fro the whole record, the tie duration of signal ay be shorter or longer when the ground borne vibration is less or ore considered for analysis. Table. Statistic value of etro vibration signal duration Saple 3 4 5 6 7 8 9 0 average Duration(s) 6.87 5.04 5. 5. 7.9 7.96 7.54 5. 3.85 5.00 5.9 Applying Fourier transfor, the analysis in the frequency doain can be done and the energy distribution in frequency can be found. In Figure.4, the vertical ais represents the energy by q. (.). The predoinant frequency band is found to be in the range of 50Hz-70Hz. It is consistent with the general conclusion that predoinant frequency for the etro induced vibration is usually above 50Hz. By coparing the aplitude of Fourier transfor near 60Hz, the value for test point 4 is about twice the value for test point, which eans the aplification phenoenon of the ground vibration in certain area as found in tie history. The influence of low frequency as below 50Hz in the frequency spectru of the signal can also be found in the figures, which represents the influence of non-etro aspect, such as the traffic vehicles with influence in the range of 0-0Hz. As the influence of low frequency (less than 40Hz) not easy to be eliinated, frequency doain analysis is not clear and efficient for analysis of etro-induced ground vibration. 0.4 0.4 0. 0.0 0.08 Non-etro influence 0. 0.0 0.08 Non-etro influence 0.06 0.06 0.04 0.04 0.0 0.0 0 50 00 50 Frequency(Hz) 0 50 00 50 Frequency(Hz) (a) Test point (b) Test point 4 Figure.4 Frequency spectru of typical signal at test point and test point 4.3 Analysis of tie-frequency energy Local characteristics of tie-frequency energy at test point and 4 are suarized in Table. and.3. The effective duration, the tie center and the frequency center of the vibration signal at point and 4 is nearly the sae, while the effective frequency band at point is large than that at point 4, which eans the energy is ore concentrated at the frequency center as about 6Hz. The tie center and the effective duration are nearly the half and two thirds of the value in tie doain respectively.
BF Tie-frequency nergy Frequency Table. Local characteristics of tie-frequency energy at test point Saple 3 4 5 6 7 8 9 0 average t (s) 8.55 8.0 6.87 7.8 7.66 8.59 0.06 8.04 7.34 7.04 8.00 T (s).30.55 0.34.05.06.5 4.63..84 0.33.67 f (Hz) 66.7 68.3 6.56 56. 70.0 64. 4.99 66.04 60.68 65.4 6. BF (Hz) 69.00 67.0 5.0 93.8 65.0 76.85 66.35 58.6 5.86 5.8 6.57 Table.3 Local characteristics of tie-frequency energy at test point 4 Saple 3 4 5 6 7 8 9 0 average t (s) 7.79 7.37 6.38 8.36 7.66 8.55 0.06 8.69 7.85 7.35 8.00 T (s).0 0.95 9.46.08.03.08 3.05 0.63.78 9.85.0 f (Hz) 6.6 59.4 63. 60.9 6. 59.0 58.3 6. 63.4 63. 6.4 BF (Hz) 44.0 44. 7.90 34.6 37.65 44.70 40.80 4.5 3.85 35.75 38.35 The tie-frequency energy is illustrated in Figure.5, while the left figure is the power spectral density function by Fourier transfor and the upper figure is the tie history of ground vibration. Fro the bar on the left, the red and blue colour represents relative large and low energy concentration on the tie-frequency energy distribution. The energy is ainly distributed in the doain by its center position ( t, f ) and a tie-bandwidth product as T BF.The tie history curve is got fro the site test and the frequency doain analysis curve as the aplitude represents the energy on the whole process of signal. The tie-frequency energy reflects the influence of tie ais and the frequency ais along with the quantity of energy by different colour in the figure. It eans three diensional aspects of the signal of ground vibration, as the tie duration, the frequency range and the energy distribution. Tie history Spectral density by Fourier transfor T high Tie low Figure.5 Illustrative tie-frequency energy vs. the tie history and Fourier spectru The tie-frequency energy of typical signal at point and 4 are shown in Figure.6 and.7, which is ainly in the range of 70 second to 8 second on the tie ais while the effective frequency band is different with nearly the sae frequency center at about 6Hz. It is deduced the energy distributed in the doain as T BF is ainly induced by the etro.
Frequency(Hz) BF Frequency(Hz) BF 50 00 T Tie center 3 Frequency center 50 50 0 Non-etro influence 70 7 74 76 78 80 8 84 Tie(s) Figure.6 Tie-frequency energy of test point 0 00 Frequency center 50 T Tie center 0 0 Non-etro influence 70 7 74 76 78 80 8 84 Tie(s) Figure.7 Tie-frequency energy of test point 4 By coparison of the data on the left bar, which represents the relative agnitude of tie-frequency energy, the vibration level at test point 4 is uch larger than that at test point. This phenoenon of etro-induced ground vibration is also found by the tie doain analysis and frequency doain analysis as described previously in this paper, which eans the vibration in point 4 is ore influenced by the etro. Besides the iage in the doain for research interest, soe iages are also found below the lower dashed frequency line, which are arked by red dashed circles on the tie-frequency energy. The sall aount signals are distributed in 0-0Hz frequency, which represent the influence of non-etro aspects as the traffic vehicles oving through the line section by further investigation on the road. Coparatively, by Fourier transfor the 0-0Hz frequency band can be found in the spectru graph but these signals cannot be located on the tie ais. For the non-stationary signal, the effective duration and effective frequency band are ore accurate and iportant to recognize the influence of vibration than the traditional ethod of analysis in tie and frequency doain. The energy of this kind of signal concentrates in the effective duration rather than the whole tie length, and in the specific frequency range of etro influence as the frequency band. The tie-frequency energy shows the cobination of tie doain and frequency doain analytical result in one figure. This will be iportant for signal recognition and signal separation..4 Analysis of vibration propagation For an actual vibration signal, the energy value by RMS is a ean value over the whole tie range. But for an individual or a structure, the response for the vibration often depends on the concentrated energy pack in a short duration, which is different fro the peak value of the signal. Here it can be called as local energy feature, which is defined as the tie distribution of the energy on the concern frequency range. The curve of the RMS and tie-frequency energy as MHS for 8 test points are plotted in the left and right part of Figure.8.The relative value of point is set to be. The three curves arked by different sybol color represent the condition of iniu, aiu and average for 0 saples. The general attenuation tendency atches well with difference in value of the two curves. In the attenuation curve of tie-frequency energy, the relative value for point 4 is about 6.5 and is large than.6 in the left figure. Coparing with the traditional
inde RMS, the tie-frequency energy is ore evident to show the local aplification phenoenon. Relative RMS 8 Relative MHS 4# iniu 6 aiu 4# 4 average 8# 6# # # 6# 8# 80 60 40 0 0 0 0 40 60 80 Distance() Distance() Figure.8 The relative value of RMS and MHS for vibration signals in 8 test points 3. CONCLUSION This paper presents the analysis of tie-frequency energy for etro induced environental vibration by Margenau-Hill-Spectrogra distribution. The signal can be characterized in the tie-frequency plane by its center position ( t, f ) and a doain of tie-frequency band product as T BF. The evaluation inde for tie-frequency energy is calculated and further discussed. Coparing with the traditional ethods in the tie doain and the frequency doain, the tie-frequency energy can directly show the characteristics of the signal contents with convenience and reliability. By analysis of the tie-frequency energy, the influence of the non-etro aspects can be reoved and the propagation law for environental vibration induced by etro can be got effectively. The environental vibration is usually attenuated with the distance ecept local aplification not far away. Using the tie-frequency energy as the evaluation inde rather than the RMS, the aplification phenoenon in certain area is ore evident with low variance of the data. 4. ACKNWLDGMNT Financial support fro Shanghai National Science Foundation (No. 3ZR444800) is gratefully acknowledged. RFRNCS. Mao,Y.Q. (987).Characteristics and Attenuation of Ground Vibration Caused by Traffic Vehicle. Journal of Building Structures.8:,54-60. (in Chinese). Cui,G.H., Tao,X.X. and Chen,X.M.(008).Actual Measureent and Analysis on Attenuation for nvironental Vibration Induced by Urban Rail Transit on Ground. Journal of Shenyang Jianzhu University(Natural Science).4:,39-43. (in Chinese) 3. Lak,M.A., Degrande, G. and Lobaert, G. (0).The ffect of Road Unevenness on the Dynaic Vehicle Response and Ground-borne Vibrations Due to Road Traffic. Soil Dynaic and arthquake ngineering. 3:0,357-377. 4. Mhanna,M., Sadek,M. and Sharhrour,I.(0).Nuerical Modeling of Traffic-induced Ground Vibration. Coputers and Geotechnics.39:,6-3. 5. Jia,B.Y.,Lou,M. L.,Zong,G. et al.(03). Field Measureents for Ground Vibration Induced by Vehicle. Journal of Vibration and Shock. 3:4,-4. (in Chinese) 6. Ge,Z.X. and Chen,Z. S.(006). Tie-frequency Analysis Technology and Its Application by MATLAB Software. Posts and Teleco Press. (in Chinese) 7. Zhang,B., Zong, G., Li, G.Z., et al.(04).tie-frequency nergy valuation Method of the Vibration Induced by Underground Subway. Hans Journal of Civil ngineering.3:6,76-88. 8. Bouale, B. (003).Tie Frequency Signal Analysis and Processing. A Coprehensive Reference. Prentice Hall. 9. Papandreou-Suppappola, A.(003). Applications in Tie-frequency Signal Processing. CRC Press. 0. Francois, A., Patrick, F., Paulo, G. and Olivier, L. (005).Tie-Frequency Toolbo for Use with MATLAB. Tutorial and Reference Guide. (http://tftb.nongnu.org)