[雙語翻譯]--外文翻譯--連續(xù)小波變換在鋼-混凝土組合鐵路橋上的（原文）

1、Engineering Structures 33 (2011) 911–919Contents lists available at ScienceDirectEngineering Structuresjournal homepage: www.elsevier.com/locate/engstructApplication of the continuous wavelet transform on the free vibrat

2、ions of a steel–concrete composite railway bridgeMahir Ülker-Kaustell ?, Raid KaroumiThe Royal Institute of Technology, KTH, Brinellvägen 34, SE-100 44 Stockholm, Swedena r t i c l e i n f oArticle history: Rec

3、eived 10 July 2010 Received in revised form 12 November 2010 Accepted 1 December 2010 Available online 15 January 2011Keywords: Railway bridges Steel–concrete composite bridge Train induced vibrations Continuous wavelet

4、transform System identificationa b s t r a c tIn this article, the Continuous Wavelet Transform (CWT) is used to study the amplitude dependency of the natural frequency and the equivalent viscous modal damping ratio of t

5、he first vertical bending mode of a ballasted, single span, concrete–steel composite railway bridge. It is shown that for the observed range of acceleration amplitudes, a linear relation exists between both the natural f

6、requency and the equivalent viscous modal damping ratio and the amplitude of vibration. This result was obtained by an analysis based on the CWT of the free vibrations after the passage of a number of freight trains. The

7、 natural frequency was found to decrease with increasing amplitude of vibration and the corresponding damping ratio increased with increasing amplitude of vibration. This may, given that further research efforts have bee

8、n made, have implications on the choice of damping ratios for theoretical studies aiming at upgrading existing bridges and in the design of new bridges for high speed trains. The analysis procedure is validated by means

9、of an alternative analysis technique using the least squares method to fit a linear oscillator to consecutive, windowed parts of the studied signals. In this particular case, the two analysis procedures produce essential

10、ly the same result. © 2010 Elsevier Ltd. All rights reserved.1. BackgroundThe dynamic properties of railway bridges are known todepend on a rather large number of phenomena. These consist of soil–structure interacti

11、on, train–bridge interaction, interaction between the track and the bridge superstructure and the material properties of the structure. For certain bridge types, some of these phenomena give rise to more or less pronounc

12、ed non-linearities, which may have noticeable effects on the dynamic properties of the structure [1].Today, many railway owners wish to upgrade existing bridgesto meet the increasing demand on train speed and axle loads.

13、 In this context, the damping ratio is highly important and can have a large influence on theoretical estimates of the dynamic response of the structure. Also, in the design of new railway bridges for high-speed railway

14、lines according to the Eurocode [2], the vertical bridge deck acceleration is often decisive for the dynamic analysis. The vertical bridge deck acceleration must be limited in order to ensure that the wheel-rail contact

15、is maintained and to eliminate the risk for ballast instability in the case of ballasted railway bridges. For these reasons, it would be desirable to learn more about the phenomena governing the dissipation of energy in

16、railway bridges.? Corresponding author. Tel.: +46 8 7907949.E-mail address: mahir.ulker@byv.kth.se (M. Ülker-Kaustell).One approach to increasing our knowledge within this fieldwould be to establish a reliable exper

17、imental methodology to determine how the damping ratio varies with the amplitude of vibration and then use that knowledge as a basis for theoretical studies of the phenomena which are believed to govern this behavior. Fo

18、r this purpose, alternative methods should be used to verify the outcome of the experimental procedures. This paper aims at describing the application of such an alternative, namely the Continuous Wavelet Transform (CWT)

19、. This mathematical tool has traditionally been applied in quantum mechanics and signal analysis [3,4], but during later years, several authors have presented applications in system identification and to some extent also

20、 damage detection (see [5] and the references therein), though most publications describe theoretical and/or laboratory studies. Staszewski [6] used the CWT to estimate the damping of simulated linear and non-linear mult

21、i degree of freedom systems with additive noise, based on the assumption that the system is viscously damped. Slavi? et al. [7] succeeded in applying the CWT to experimental data produced in a laboratory, for a linearly

22、elastic, viscously damped beam. Le and Argoul [8] described procedures to identify the eigenfrequencies, damping ratios and mode shapes of linear structural systems from free vibration data by means of the CWT. An extens

23、ion towards applications of the CWT to identify non-linear systems was suggested by Staszewski [9] where the CWT was used to estimate the skeleton (the variation of the amplitude with time) of different signals. These co

24、ncepts were further elaborated by Ta and Lardies [10], who applied their methodology to simulated numerical data and0141-0296/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.20

25、10.12.012M. Ülker-Kaustell, R. Karoumi / Engineering Structures 33 (2011) 911–919 913time (s)frequency (Hz)0 5 10 15 20 25 30 35 –0.500.52.533.544.550 5 10 15 20 25 30 35Fig. 1. The CWT of the first bending mode of

26、the bridge at Skidträsk (see Section 3) together with its ridge (black solid line) and the boundaries (grey, dashed lines) within which the edge effect is negligible.first factor of Eq. (14), whereas in the Matlab W

27、avelet Toolbox, the (complex) Morlet wavelet is directly defined with this parameter. This factor may be varied so that the variation of the amplitude of the Morlet wavelet is stretched or contracted. The center frequenc

28、y ω0 of the Morlet wavelet is approximately bounded from below by ω0 ≥ 5 in order to fulfill the condition (3).2.3. The edge effectDue to the finite duration of the analyzed signal, there is amismatch between the wavelet

29、 function and the signal at the beginning and end of the signal. This is referred to as the edge effect and there is no known procedure by which it can be removed. However, one can determine a domain D for a and b on whi

30、ch the edge effect is negligible [7,8]. In [8], the following bounds on the circular frequency were determined2ctQµψωj ≤ bj ≤ L ? 2ctQµψωj (15)0 < ωj ≤ 2πfNyquist1 + cf 2Q(16)where ct ≥ 1 and cf ≥ 1 are para

31、meters chosen so that when t and ω are outside the intervalsIct = [tψ ? ct?tψ, tψ + ct?tψ] (17)andIcf = [ωψ ? cf ?ωψ, ωψ + cf ?ωψ] (18)respectively, the wavelet and its Fourier transform have very small values. In [8] a

32、good compromise was found in ct = cf = 5, which have also been used here. These bounds are shown in Fig. 1 using red dashed lines. Several methods to reduce the edge effect in short signals are described in [15], in the

33、present context however, the above described bounds were found to be sufficient.2.4. Asymptotic analysisFor a certain group of wavelets, referred to as analytic (orprogressive) wavelets, the analysis can be much simplifi

34、ed if the signal is asymptotic. An analytic function fa is characterized by having a Fourier transform which is zero for all negative frequencies? fa(ω) = 0, ?ω < 0. (19)A general monochromatic signal can be described

35、 in terms of an instantaneous amplitude A(t) and phase φ(t) by functions of the form [16]u(t) = A(t) cos(φ(t)). (20)Then, the instantaneous circular frequency can be defined as the time derivative of the phaseω(t) = ˙ φ(

36、t). (21)If the amplitude A(t) varies slowly compared to the phase φ(t), i.e. if the following conditions are met? ?˙ φ(t) ? ? ?? ? ? ? ˙ A(t)A(t)? ? ? ? (22)the signal is asymptotic. If the signal is asymptotic and the w

37、avelet is analytic, the CWT can be approximated by [16]? Tψ[u](a, b) ≈√a2 A(b)eiφ(b) ? ψ?(a˙ φ(b)). (23)2.5. The ridge and skeleton of the CWTAssuming that the signal consists of only one component, themaximum modulus of

38、 its CWT will be restricted to a curve in thea bFig. 2. The estimated natural frequency and the corresponding equivalent viscous damping ratio of the first bending mode of the bridge (see Section 3). (a) Without smoothin

39、g the amplitude and phase from the skeleton (grey), with smoothing (black). (b) The dashed parts of the lines illustrate the regions of the CWT-estimates which are affected by the edge effects and the part during which t