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Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions
Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
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Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions Wout Weijtjens a, John Lataire b, Christof Devriendt a, Patrick Guillaume a a b
Vrije Universiteit Brussel, Department of Mechanical Engineering, Acoustics and Vibration Research Group (AVRG), Belgium Vrije Universiteit Brussel, Department of Fundamental Electricity and Instrumentation (ELEC), Belgium
a r t i c l e i n f o
abstract
Article history: Received 17 September 2013 Accepted 15 April 2014
Periodical loads, such as waves and rotating machinery, form a problem for operational modal analysis (OMA). In OMA only the vibrations of a structure of interest are measured and little to nothing is known about the loads causing these vibrations. Therefore, it is often assumed that all dynamics in the measured data are linked to the system of interest. Periodical loads defy this assumption as their periodical behavior is often visible within the measured vibrations. As a consequence most OMA techniques falsely associate the dynamics of the periodical load with the system of interest. Without additional information about the load, one is not able to correctly differentiate between structural dynamics and the dynamics of the load. In several applications, e.g. turbines and helicopters, it was observed that because of periodical loads one was unable to correctly identify one or multiple modes. Transmissibility based OMA (TOMA) is a completely different approach to OMA. By using transmissibility functions to estimate the structural dynamics of the system of interest, all influence of the load-spectrum can be eliminated. TOMA therefore allows to identify the modal parameters without being influenced by the presence of periodical loads, such as harmonics. One of the difficulties of TOMA is that the analyst is required to find two independent datasets, each associated with a different loading condition of the system of interest. This poses a dilemma for TOMA; how can an analyst identify two different loading conditions when little is known about the loads on the system? This paper tackles that problem by assuming that the loading conditions vary continuously over time, e.g. the changing wind directions. From this assumption TOMA is developed into a time-varying framework. This development allows TOMA to not only cope with the continuously changing loading conditions. The time-varying framework also enables the identification of the modal parameters from a single dataset. Moreover, the time-varying TOMA approach can be implemented in such a way that the analyst no longer has to identify different loading conditions. For these combined reasons the time-varying TOMA is less dependent on the user and requires less testing time than the earlier TOMA-technique. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Operational modal analysis (OMA) Transmissibility functions Harmonics Periodical loads System identification Damping
1. Introduction to dealing with periodical loads In mechanical engineering, operational modal analysis (OMA) is used to retrieve the modal parameters of a structure that is subjected to loads associated with its proper use [1]. For instance a bridge excited by wind and traffic loads [1] or an
E-mail address: [email protected] (W. Weijtjens). http://dx.doi.org/10.1016/j.ymssp.2014.04.008 0888-3270/& 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
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W. Weijtjens et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
Fig. 1. (a) If the loads have a flat spectrum (i.e. white noise) all dynamics present in the measurements are caused by the system of interest. (b) For periodic loads this property no longer holds. Without any additional information about the loads, it is impossible to know whether certain dynamics are caused by the structure or by the load.
Fig. 2. Periodic loads can be modeled as the response of an input-filter to white noise. While OMA techniques will identify a combined system (inputfilterþ system of interest), TOMA-techniques are able to identify the system of interest directly no matter the input-filter.
offshore windturbine excited by wind and waves [2]. However, it is nearly impossible to correctly measure all loads that excite an operational structure. OMA therefore considers no known loads and operates purely on measured vibrations, i.e. the system response to the unknown loads. The identification of the modal parameters is made possible by assuming that the loads have a flat frequency spectrum, i.e. white noise [1]. This assumption is necessary to assure that the output spectrum resembles the transfer function of the system of interest, Fig. 1a. Therefore, all dynamics in the measurement spectrum are caused by the system of interest which then can be identified. Because wind loads approximate such a flat frequency spectrum, OMA can be easily applied to bridges and towerstructures dominantly excited by wind [1,3]. However, some loads act periodically, e.g. waves and rotating machinery, and contain more energy at given frequencies (e.g. harmonics). This poses a problem for OMA, when nothing is known about the load-spectrum one cannot guarantee that all dynamics in the measurement-spectrum are related to the system of interest itself, Fig. 1b. Several strategies are possible to still perform OMA in the presence of periodical loads. To illustrate them, one can model the non-flat spectrum of the load as the response of an unknown input filter to white noise, Fig. 2. A first strategy is to apply any of the classic OMA-algorithms [3–5] to the measurement data, without making any modifications to the algorithms. As a result the modal parameters of a system, that is the combination of the input filter and the system of interest, are identified. Afterwards the analyst can reject certain modes based on prior knowledge of the loads, e.g. rotational speeds [6], or the system itself. An alternative strategy is to pre-process the measurement data to reduce the influence of the load as much as possible, before using the data into the preferred OMA-algorithm. Examples are time-synchronous averaging [7], cepstrum analysis [8] or interpolation in the frequency domain [9]. All previously mentioned strategies require that the peaks in the load spectrum are well-distanced from the actual structural dynamics, which unfortunately is not always the case. The solution to this issue can be achieved by adapting the OMA-algorithms to incorporate an input-filter next to the system of interest. In essence such an approach still identifies the combined system (input-filterþsystem of interest). However, as these algorithms include the input filter into the model, they are able to distinguish which dynamics are linked to the input filter and which dynamics are part of the system of interest. This allows for a better identification of the system of interest, especially when the periodical forces have frequencies close to structural modes [10–12]. However, all these methods assume a certain model of the input-filter and therefore require additional information about the load spectrum. This paper will continue the work done in transmissibility based modal analysis, or TOMA [13,14]. TOMA-techniques solve the issue of periodical loads in a completely different manner. As discussed in [13] and [15] transmissibility functions can become completely independent of the input spectrum. This implies that TOMA is uninfluenced by the input filter and is able to identify the system of interest directly. Unlike all previous techniques the analyst does not require any additional information about the load spectrum to correctly identify the system of interest. The next section will recapitulate on TOMA and introduce a modification to the existing algorithms.
2. Transmissibility based OMA (TOMA) In this section we will recapitulate the state of the art in TOMA, introduce the basic concept of time-varying TOMA and discuss its advantages. Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
W. Weijtjens et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
3
Fig. 3. Illustration of TOMA on a system with two poles. (a) Two different loading conditions are simulated by changing the input location of the load. (b) Transmissibility functions are dependent on the load location (dotted vertical lines: resonance frequencies of the system). (c) Transmissibility functions intersect in the system poles ð d1 7 i2πf 1 ; d2 7 i2πf 2 Þ.
2.1. State of the art in TOMA In [13] it was shown that in the case of a single (distributed) load the transmissibility function is independent of the load spectrum.1 This independence implies that any dynamics present in the input filter will not influence the transmissibility function, nor will they influence modal parameters obtained through transmissibility functions. Unfortunately, the modal parameters cannot be derived from a single transmissibility function, Fig. 3b. The solution to find modal parameters from transmissibility functions comes from a different property of the transmissibility function. While a transmissibility function is independent of the load spectrum, it is dependent on the load location. Basically, if the load changes location (or distribution) the transmissibility function will change, Fig. 3a and b. An interesting property shows up in Fig. 3b and c, where it is shown that transmissibility functions of different loading conditions intersect in the system poles. Over recent years several strategies on how to obtain the modal parameters from transmissibility functions based on this idea have been published [13,16]. Such an approach, with different loading conditions, implies that in order for TOMA to work at least two loading conditions have to be considered. But, how does an analyst distinguish two different loading conditions if little is known about the loads?. A first solution was proposed in [17], this method based on power spectrum density transmissibilities requires only a single loading condition. However, the proposed method only works when multiple uncorrelated forces act on the structure. In particular cases such as the presence of correlated periodical loads, as caused by a single rotating machine, this can again lead to errors with misinterpreted periodical loads. The methodology of [17] was expanded in [18] into a SVD-driven algorithm in a similar fashion as the expansion proposed in [16]. While both [17,18] are very interesting approaches to TOMA, they currently lack the ability to estimate damping. In this paper we propose a different solution to this problem without losing the load spectrum independency and yielding a damping estimate. 2.2. A time-varying approach to TOMA If we look at Fig. 3 we notice how the two discrete loading conditions can help us to find the system poles. In a real world application it is rather unlikely that a system subjected to an operational load suddenly is excited in a completely different way. Even if such a sudden transition occurs it is very hard to pinpoint the exact moment of the transition without additional information on the load itself. It seems far more realistic to assume that the loads on a system vary continuously over time. For example the wind direction does not jump from e.g. East to North-East instantaneously. In fact, the wind load will vary continuously with minor changes in direction and velocity over time. Waves, traffic or the controls of a turbine may be considered in a similar way as they also vary constantly over time. As discussed earlier the transmissibility function is related to the load location/distribution. So, when the loads vary continuously over time so do the transmissibility functions. To demonstrate let us reconsider the example of Fig. 3. But this time a single load slowly moves its location, Fig. 4a. TOMA until now assumed that in a measurement record the loading condition is constant. From Fig. 4b it is evident that when that assumption is violated the transmissibility function loses its load spectrum independency and is no longer deterministic. At each instance of time there is actually an instantaneous loading condition. For instance at the first time instance, t0, the load is only applied at point 1. At the final time instance, te, a load is only applied at point 2. So at t0 and te the system is for an instant of time subjected to the same loading conditions as the system of Fig. 3. Furthermore, at each of the intermediate 1 Recently, the idea of TOMA was extended to pTOMA, [14]. In which multi reference transmissibility functions are used to retrieve the modal parameters. This extension allows to become independent of multiple input loads. In this paper we will limit us to the scenario of a single load, however it is believed that the presented time-varying approach can be extended to a multi-reference implementation.
Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
W. Weijtjens et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
4
Fig. 4. Illustration to time-varying TOMA. (a) Two different loading conditions are simulated by changing the input location of the load gradually over time. (b) Transmissibility functions calculated as in Fig. 3 are no longer behaving as expected. (c) Instantaneous transmissibility functions show the same behavior as classic transmissibilities.
instances of time, t 0 o t ot e , the system is subjected to a unique but instantaneous loading condition. For instance at the middle time instance a distributed load is applied equally to both points. Each of these instantaneous loading conditions has an associated instantaneous transmissibility function. Some of these instantaneous transmissibility functions are plotted in Fig. 4c. However, as these transmissibility functions only exist for an instant, they cannot be estimated with the approaches used before. To estimate these instantaneous functions, the timevarying behavior of the transmissibility function has to be included into the model. Once the time-varying transmissibility functions are estimated, the system poles can be derived from them in a manner similar to TOMA using two or more instantaneous transmissibility functions. Furthermore, the analyst is no longer required to distinguish two different loading conditions. Effectively, each instance of time is linked to a unique instantaneous loading condition, so the analyst only has to consider different instances of time to consider different loading conditions.
3. Using time-varying transmissibility functions for OMA In this section, we will firstly define the system of interest and the transmissibility function. Then the effects of a timevarying load will be discussed. Next, the procedure to estimate time-varying transmissibility functions will be explained. Once the time-varying transmissibilities are estimated we show how the system poles can be derived from them. Finally, it will be shown that for linear time variations the analyst does not longer have to choose different loading conditions.
3.1. Basic assumptions and definitions Assume that the system of interest is linear and time-invariant. The system transfer function, HðsÞ A CNo Ni , will relate the ! ! load, F ðsÞ A CNi , with the outputs, X ðsÞ A CNo . The transmissibility function, T(s), is defined as the function that relates the ! ! so-called reference outputs, X r ðsÞ A CNref , to the non-reference outputs, X l ðsÞ A CNo Nref , [15]: ! ! X ðsÞ ¼ HðsÞ F ðsÞ
ð1Þ
! ! X l ðsÞ ¼ TðsÞ X r ðsÞ:
ð2Þ
In this paper, we will use only a single reference and non-reference output. The single-reference transmissibility function is also defined as T lr ðsÞ ¼ X l ðsÞ=X r ðsÞ. When only a single (distributed) load acts on the system, the single reference transmissibility, Tlr(s), is load-spectrum independent. However, keep in mind that when multiple uncorrelated loads act on the system a single reference is no longer sufficient to achieve load spectrum independency [14].
3.2. Time varying loads Previous research on TOMA assumed a loading condition to be constant within a single measurement record. In practical applications it is far more likely that a load will constantly vary its distribution, e.g. the shifting wind, within a single measurement record. ! Let us assume a single distributed force f ðtÞ A RNi that is equal to the product of a time-varying distribution vector, ! N f d ðtÞ A R i , and a single source of the force μðtÞ A R: ! ! f ðtÞ ¼ f d ðtÞμðtÞ:
ð3Þ ! f d ðtÞ,
If the distribution vector, is time-invariant a single reference transmissibility function will become fully load-spectrum ! independent, as there is only a single distributed force. But what happens when f d ðtÞ is a function of time? Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
W. Weijtjens et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]] !
5 ! i d;i t ,
t Let us assume that the distribution vector can be modeled as a polynomial in t, f d ðtÞ ¼ ∑ni ¼ 0f ! load variation in time. The system outputs in the time domain, x ðtÞ A RNo , are provided by ! nt ! ! 1 i x ðtÞ ¼ L fHðsÞgn ∑ f d;i t μðtÞ
with nt the order of the
ð4Þ
i¼0
with Lfg the Laplace-transform of and n is the convolution operator. The Laplace transform of Eq. (4) is ( ) nt ! ! X ðsÞ ¼ HðsÞL ∑ f d;i t i μðtÞ
ð5Þ
i¼0
nt
¼ HðsÞ ∑
i¼0 nt
¼ HðsÞ ∑
i¼0
! i f d;i Lft μðtÞg
ð6Þ
! i ðiÞ f d;i ð 1Þ L fμðtÞg
ð7Þ
in which LðiÞ fg is the i-th derivative of Lfg with respect to s. In general Lfg will not be fully correlated with any of its derivatives LðiÞ fg. Therefore, all ðnt þ 1Þ derivatives of LfμðtÞg are to be considered as uncorrelated sources and at least ðnt þ 1Þ references are required to get a load-spectrum independent transmissibility function [15]. Furthermore, this will increase the number of required sensors, at least ðnt þ2Þ, and potentially the number of required loading conditions [14]. 3.3. Estimation of time-varying transmissibility functions The goal of the paper is to expand the transmissibility functions to a time-varying form. Such a time-varying transmissibility function can then be evaluated at a time instance, tn, to retrieve the instantaneous transmissibility functions ! associated with an instantaneous load distribution f d ðt n Þ. It is assumed that a time-varying single reference transmissibility function2 can be modeled in the time interval ½t 0 ; t e as a linear ordinary differential equation with time-varying coefficients i
na
∑ αi ðt Þ
i¼0
d xl ðtÞ dt
i
i
nb
d xr ðtÞ
i¼0
dt
¼ ∑ β i ðt Þ
ð8Þ
i
with na ; nb the model orders of the structural dynamics. The time-varying coefficients are modeled as polynomials in t; nt j j t αi ðtÞ ¼ ∑nj ¼ 0 αi;j t , βi ðtÞ ¼ ∑j ¼ 0 βi;j t . The time-varying transmissibility functions are found by estimating the unknown coefficients αi;j A R and βi;j A R from Eq. (8). In [19] a similar identification problem, for time-varying transfer functions, is discussed. This paper will translate that approach to estimate time-varying transmissibility functions and summarize all results of [19] relevant to this paper. For a more in-depth discussion on the estimation itself we refer back to [19]. One of the main advantages of an estimation algorithm in the frequency domain is the ease at which a frequency-band of interest can be defined. Eq. (8) is therefore transformed into the frequency domain: ( ) i i nb na nt nt d x ðtÞ d xr ðtÞ L ∑ ∑ αi;j t j l i ∑ ∑ βi;j t j ¼0 ð9Þ i dt dt i¼0j¼0 i¼0j¼0 na
nt
(
∑ ∑ αi;j L t
i¼0j¼0
jd
i
xl ðtÞ
dt
i
)
nb
(
nt
∑ ∑ βi;j L t i¼0j¼0
jd
i
xr ðtÞ dt
i
) ¼0
ð10Þ
As discussed in [19] a Laplace transform is preferred over a differentiation and a multiplication is preferred over a convolution. For this reason the following expression, equivalent to Eq. (10), is preferred over Eq. (10): na
nt
nb
nt
∑ ∑ αi;j Lft j L1 fsi Lfxl ðtÞggg ∑ ∑ βi;j Lft j L1 fsi Lfxr ðtÞggg ¼ 0:
i¼0j¼0
ð11Þ
i¼0j¼0
It is shown in [19] that for finite-length and sampled data the model must be extended with an additional transient a ;nb Þ 1 polynomial τðsÞ ¼ ∑imaxðn τi si . This additional term allows to capture errors caused by the difference in initial and final ¼0 conditions, and errors caused by aliasing. Because of τðsÞ, the actual system dynamics are well separated from these signalprocessing effects. Eq. (11) then becomes 2 For ease of notation all transmissibility functions discussed from this point are single reference transmissibility functions unless mentioned otherwise.
Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
W. Weijtjens et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
6
na
nb
nt
nt
∑ ∑ αi;j Lft j L1 fsi Lfxl ðtÞggg ∑ ∑ βi;j Lft j L1 fsi Lfxr ðtÞggg þ
i¼0j¼0
i¼0j¼0
maxðna ;nb Þ 1
∑
i¼0
τi si ¼ 0
ð12Þ
the unknown parameters τi A R are part of the set of parameters to be estimated. Eq. (12) is linear in all unknown parameters and can be rewritten as ! K Θ Θ ¼ 0; with ð13Þ K Θ ¼ fK l K r K τ g
ð14Þ
and
! Θ ¼ fα0;0 ; α0;1 ; …; αn;p ; …; βn;p ; …; τ0 ; τ1 ; …; τmaxðna ;nb Þ 1 gT
ð15Þ
for which the k-th element in the column of Kl associated with αn;p is given as K l;n;p ¼ ðLft p L1 fðjωk Þn Lfxl ðtÞgggÞ
ð16Þ
similarly for the column of Kr associated with βn;p : K r;n;p ¼ ðLft p L1 fðjωk Þn Lfxr ðtÞgggÞ
ð17Þ
and of which the k-th row of K τ is defined as f1 ðjωk Þ ðjωk Þ2 … ðjωk Þmaxðna ;nb Þ 1 g:
ð18Þ
Each row of K Θ corresponds to one frequency ωk =2π of the Nf frequency lines in the frequency-band of interest. For sampled signals, with sampling time Ts and sampled time ts, these equations are approximated by their discrete Fourier transform (DFT) and their inverse discrete Fourier Transform (IDFT), with equal to r or l: K ;n;p ¼ T s ðDFTfðt s Þp IDFTfjωnk DFTfx ðt s ÞgggÞ:
ð19Þ
The previous derivation is made for a continuous time model. In [20] it is illustrated how to apply this idea for discrete-time models. The current model also considers a polynomial model in both time as well as in frequency domain. Especially for high model orders, na ; nb or nt, this might result in an ill-conditioned observation matrix K Θ . The method can be fully adapted to use Legendre-polynomials in the time and/or the frequency domain, [19], which can improve the conditioning of K Θ. ! In this paper the overdetermined set of Eq. (13) is solved, and the parameters in Θ are found, by calculating the Singular Value Decomposition (SVD) of K Θ : K Θ ¼ USV H :
ð20Þ
The Total Least Squares solution of Eq. (13) is found as the singular vector associated with the smallest singular value, i.e. the last column of the matrix V. In this paper we only used the non-consistent Total Least Squares. Nonetheless, it is also possible to incorporate information about the noise into the solution of Eq. (13) in order to obtain a consistent estimator [19]. 3.4. Finding the modal parameters The final step in the time-varying TOMA approach is to retrieve the modal parameters from the time-varying transmissibility functions. The instantaneous transmissibility function at time tn, T lr ðs; t n Þ, is defined as j
na nt a ^ n i ∑i ¼ ∑ni ¼ ^ i ðt n Þsi n i 0 ∑j ¼ 0 α i;j t s n j i 0α t s ¼ ðt Þs T lr s; t n ¼ nb nb t ^ i;j ^ ∑i ¼ 0 ∑nj ¼ ∑ β 0 i ¼ 0β i
ð21Þ
in which α^ i;j and β^ i;j are the coefficients estimated in the previous section. It is assumed that for slow variations of the load (i.e. much slower than the actual structural dynamics) these instantaneous transmissibility functions are equal to the nonN time-varying transmissibilities. So, in the system pole λm A C, with mode shape ! ϕ m A C o , the following property [13] holds for each time instance in the considered measurement record: ϕ T lr λm ; t n ¼ l;m ϕr;m
8 t n A ½t 0 ; t e
ð22Þ
ϕ m , respectively, associated with the reference and nonin which ϕr;m and ϕl;m are the elements of the m-th mode shape, ! reference output. In essence Eq. (22) states that all instantaneous transmissibility functions intersect in the system poles. Eq. (22) can also be interpreted as follows: for all instances of time, T lr ðλm ; t n Þ has the same value. This is equivalent to
δT lr ðλm ; tÞ t ¼ tn ¼ 0 δt
8 t n A ½t 0 ; t e :
ð23Þ
Because of the strong similarity between these instantaneous transmissibility functions and the classic transmissibility functions we can again rely on classic TOMA techniques to find the modal parameters. For instance one possible solution, Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
W. Weijtjens et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
7
similar to the one proposed in [13], is to calculate a base function using two instantaneous transmissibility functions for arbitrary time instances t n1 and t n2 : Δ 1 T lr ðsÞ ¼
1 T lr ðs; t n2 Þ T lr ðs; t n1 Þ
ð24Þ
and to feed this base function into an existing curve-fitter for Frequency response functions, e.g. LSCF. The system poles are then found as the poles of Δ 1 T lr ðsÞ. This approach will be used in Section 4. 3.5. Discussion on the choice of the model order of the time-varying coefficients While the previous approach works correctly, there is still some discussion possible on which instances of time the instantaneous transmissibility functions should be computed. The classic task in TOMA of identifying two measurement records that both correspond to two different loading conditions has been replaced by choosing two time instances t n1 and t n2 . The analyst can be relieved from this choice if the time-varying model order nt is chosen equal to one. This can be understood by rewriting Eq. (21) as j na nt nj t ^ ^ i;j si t n ∑nj ¼ ∑i ¼ 0 ∑j ¼ 0 α 0 a j ðsÞt T lr s; t n ¼ n ¼ n j n nj b t t ^ ^ i n ∑i ¼ ∑j ¼ 0 ∑j ¼ 0 β i;j s t 0 b j ðsÞt
ð25Þ
a b ^ i ^ i;j si and b^ j ðsÞ ¼ ∑ni ¼ with a^ j ðsÞ ¼ ∑ni ¼ 0α 0 β i;j s . The partial derivative to t of Eq. (25) for nt ¼1 is
a^ 1 ðsÞðb^ 1 ðsÞt n þ b^ 0 ðsÞÞ b^ 1 ðsÞða^ 1 ðsÞt n þ a^ 0 ðsÞÞ δT lr ðs; tÞ ¼ n δt t¼t ðb^ 1 ðsÞt n þ b^ 0 ðsÞÞ2 ¼
ð26Þ
a^ 1 ðsÞb^ 0 ðsÞ b^ 1 ðsÞa^ 0 ðsÞ ðb^ ðsÞt n þ b^ ðsÞÞ2 1
ð27Þ
0
Because nt is set to one, the numerator of Eq. (27) is no longer a function of the chosen time instance tn. Moreover, when in sn the partial derivative of T lr ðsn ; tÞ to t is zero: δT lr ðsn ; tÞ ¼0 ð28Þ δt t ¼ tn the denominator, and therefore all instances of tn, can be eliminated: a^ 1 ðsn Þb^ 0 ðsn Þ b^ 1 ðsn Þa^ 0 ðsn Þ δT lr ðsn ; tÞ ¼ ¼0 δt n ðb^ 1 ðsn Þt n þ b^ 0 ðsn ÞÞ2 t¼t
ð29Þ
a^ 1 ðsn Þb^ 0 ðsn Þ b^ 1 ðsn Þa^ 0 ðsn Þ ¼ 0:
ð30Þ n
n
The equation above states that if the partial derivative of T lr ðs ; tÞ to t is equal to zero for one instance of t , then it is equal to zero for all instances of tn. It also implies that T lr ðsn ; tÞ is not a function of time. Basically, only by choosing nt ¼1 the analyst forces the model to include some values sn for which T lr ðsn ; tÞ is not a function of time. From Eq. (23) we learn that such values sn are in fact the system poles, λm, which gives a physical meaning to the choice of nt ¼1. Constraining nt to one does not only impose this physical constraint to the model, it also implies that T lr ðλm ; tÞ is not a function of time. Therefore, the choice of t n1 and t n2 in Eq. (24) does no longer have any influence on the estimation of the system poles. In essence for nt ¼ 1 the analyst is relieved from the task of identifying two different loading conditions, as any arbitrary choice will yield the same results. Estimated poles that still vary with the choice of t n1 and t n2 are most likely intersections of two transmissibility functions that are not linked with a system pole. This phenomena was discussed in [21]. Imposing nt ¼1 restricts the model to linearly time-varying transmissibilities. In some scenarios this model order might not be sufficient to cover all the variations present in the load. The analyst can detect this under-modelling by checking the ! quality of fit of the estimated model, e.g. by evaluating the remaining error, e A CNf : 9 9 9 8 8 8 β0;0 > τ0 α0;0 > > > > > > > > > > > > > > > > > > > > > > > = < α0;1 = < β0;1 = < τ1 ! ! Kr þ Kτ ¼ K l! α K r β þK τ ! τ: ð31Þ e ¼ Kl ⋮ ⋮ ⋮ > > > > > > > > > > > > > > > > > > > > > > > > : αn ;n ; : βn ;n ; : τmaxðn ;n Þ 1 ; a
t
b
t
a
b
However, because of the advantages of restricting nt to one it is advised to not increase nt, but rather to reduce the length of the measurement record. Within the shortened measurement record the varying load might be better approximated, indicated by an improved quality of fit, by the linear time-varying model. Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
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Fig. 5. The system of interest is an aluminum beam, excited by two shakers (S1 and S2). Vibrations are measured at two locations using one Laser Doppler Vibrometer (LDV) and a piezo-electric accelerometer (ICP). The input signal was generated as a response of a SDOF system to white noise, this signal was split and crossfaded from one shaker to the other. The forces applied to the system were measured but not used in the estimate.
F 1
|F|
F 2
B1 IF
B2
T1
B3
Fig. 6. (a) Output spectra obtained with the two sensors. The first three bending modes (B1–B2–B3) are indicated along with the first torsional mode (T1). The resonance frequency of the input filter is highlighted with a red dashed line (IF). (b) Input forces measured. The crossfade from one shaker to the other was initiated after several seconds to avoid startup effects. The highlighted area indicates the crossfading. (c) Force spectra during the crossfading show the presence of the input filter. Additional spikes e.g. around B3 are due to inertial forces exerted by the beam. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
4. Laboratory experiment 4.1. Experimental setup The system of interest in this experiment was a suspended aluminum beam. The beam is excited by two shakers placed at the outer limits of the beam. We measured the vibrations at two locations using a piezo-electric accelerometer (ICP) and a Laser Doppler Vibrometer (LDV). As a reference the applied forces were also measured but are not used in the estimation itself. The load was generated as the response of a single degree of freedom model to white noise. This signal was crossfaded from one shaker to another to obtain a time-varying load, Fig. 5. The laser-dot and accelerometer were deliberately put at random locations to illustrate that even an unfortunate choice of one of both sensors does not result in a complete failure of the method. For instance the laser dot is aimed at a nodal point of the first torsional mode (T1). The ICP was placed close to a nodal point of the third bending mode (B3). In Fig. 6a the output spectra are shown for both sensors. To avoid startup effects one of both shakers was already exciting the structure for several seconds before the crossfade was initiated. In Fig. 6b the forces measured at the inputs are shown and the crossfade is highlighted. The forces measured at shaker 2 before the crossfade is initiated are the result of the vibrations in the beam being transmitted back into the shaker. The crossfade itself only takes 0.8 s to complete, with the measurements sampled at 10.2 kHz. It is important that the estimation does not include the actual start and ending of the crossfade. Inclusion of these points would lead to a poor
Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
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Fig. 7. The time-varying model is able to describe the dynamics present within the data. (a) The error as defined in Eq. (31) is significantly smaller than the ! composing components. (b) Zoomed view at the overlay of jK r β j and jK l ! α þ Kτ ! τ j.
Fig. 8. (a) Instantaneous transmissibility functions for the considered dataset. (b) Using a standard base function, transmissibility functions are molded into something that resembles an FRF. The base function does not have a peak at the frequency of the input filter (IF), indicating that the estimate was not influenced by the load spectrum. (a) Instantaneous transmissibility functions and (b) 1/|T12(s,t0)-T12(s,te)|. Table 1 Results of both TOMA and classic OMA, the input filter (IF) is not part of the structure but was nonetheless estimated by OMA. Mode
B1 IF B2 T1 B3
TOMA
OMA (same data)
OMA (more data)
fres (Hz)
ξ ð%Þ
fres (Hz)
ξ ð%Þ
fres (Hz)
ξ ð%Þ
260.1 – 705.2 1258.6 1371.6
0.80 – 0.10 0.11 0.10
260.2 301.6 711.7 1254.1 1374.3
0.35 1.12 0.50 0.17 0.12
259.2 300 709.1 1257.0 1375.0
0.53 1.19 0.22 0.21 0.17
estimate as at these points the transmissibilities rapidly change from being time-invariant to time varying and the model is only able to capture this rapid transition for large values of nt. In the estimation 6144 data points were used, that are located in the middle of the crossfade. The two spectra of these data points are shown in Fig. 6a. 4.2. Results In this example the first and final instantaneous transmissibility functions are used in the base function (Eq. (24)). This base function, that resembles an FRF, is then fed into a frequency domain curve-fitter with a numerator–denominator model (LSCF) [13]. The instantaneous transmissibility functions are estimated for a structural dynamics model order of 10 ðna ¼ nb ¼ 10Þ and a time-varying model order equal to one (nt ¼1). The frequency band of interest was set from 0 Hz up to 1500 Hz, this frequency band contains the three first bending modes (B1–B3) and the first torsional mode (T1). Legendre polynomials were used for the frequency domain model to improve numerical conditioning. In Fig. 7 the quality of fit is checked using the error defined in Eq. (31). The structure is lightly damped and it was observed that for higher structural model orders (na, nb) the damping was poorly estimated, sometimes even estimating the poles as unstable. Best practice remains to start at small structural model orders (na,nb), identifying the structural dynamics and finally increase na and nb to improve the preliminary results. Over-modeling will result in very complex instantaneous transmissibility functions, that show much more dynamics than the number of poles (Fig. 8). The estimated resonance frequencies and damping ratios are given in Table 1. As comparison the same data is also used for a classic OMA. In this example we used a Correlogram approach with Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
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model order
10
B1 IF
B2
T1 B3
frequency (Hz) Fig. 9. The stabilization diagram of the correlogram-LSCF (OMA) approach shows that the input filter is misinterpreted as a structural pole.
the LSCF estimator. In Fig. 9 the stabilization diagram that is the result of this analysis is provided. The large relative variation within the damping estimates between the different approaches can be attributed to the low damping inherent to the aluminum beam. The absolute differences between damping estimates are acceptable given the data and the LSCF estimator. 5. Conclusions and future work This paper presented a novel approach to deal with periodical loads in operational modal analysis using time-varying transmissibilities. The proposed method starts from estimating single reference time-varying transmissibility functions. Like regular transmissibility functions, time-varying transmissibility functions become independent of the load spectrum in the case of a single (distributed) load. Modal parameters found through the transmissibility functions, with a TOMA, are therefore solely related to the system of interest and not influenced by the load spectrum. This is especially interesting in the presence of periodical or harmonic loads as these are often misinterpreted by OMA-techniques as structural modes. Moreover, the time-varying TOMA has several advantages over classic time-invariant TOMA. In time-invariant TOMA the analyst is required to measure the system responses to at least two different, and stationary, loading conditions. This poses two problems to the analyst: how can he/she identify two different loading conditions if little is known about the loads on the system? Furthermore, how can the analyst be certain that the loading condition was effectively stationary? The presented time-varying TOMA solves these issues by relaxing the assumption of stationary loading conditions. The assumption for continuously varying loading conditions is made. It is shown that as a consequence of this assumption multiple loading conditions are present in a single dataset. In this paper we showed that it is possible to obtain the modal parameters from such a single dataset by expanding the transmissibility function to a time-varying framework. Because only a single dataset is required, the testing time can be greatly reduced in comparison to time-invariant TOMA. In a particular implementation of the algorithm it is also possible to relieve the analyst from his task to choose two different loading conditions within that single dataset, resulting in a more user-friendly and reliable approach to TOMA. The time-varying TOMA approach was demonstrated using a lab experiment on an aluminum beam. The current time-varying model is restricted to a single reference, future research will try to extend this to a multiple reference form as presented in [14].
Acknowledgments The financial support by the Agency for Innovation by Science and Technology in Flanders (IWT) is greatly acknowledged. References [1] F. Magalhães, Á. Cunha, Explaining operational modal analysis with data from an arch bridge, Mech. Syst. Signal Process. 25 (5) (2011) 1431–1450. [2] C. Devriendt, P.J. Jordaens, G.D. Sitter, P. Guillaume, Damping estimation of an offshore wind turbine on a monopile foundation, Renew. Power Gener., IET 7 (4) (2013) 401–412.
Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
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[3] B. Peeters, G. de Roeck, Reference-based stochastic subspace identification for output-only modal analysis, Mech. Syst. Signal Process. 13 (6) (1999) 855–878. [4] E. Parloo, Application of frequency-domain system identification techniques in the field of operational modal analysis (Ph.D. thesis), Vrije Universiteit Brussel, 2003. [5] R. Brincker, L. Zhang, P. Andersen, Modal identification of output-only systems using frequency domain decomposition, Smart Mater. Struct. 10 (June (3)) (2001) 441–445. [6] M. Gagnon, S.-A. Tahan, A. Coutu, M. Thomas, Analyse modale opérationnelle en présence d'excitations harmoniques: étude de cas sur des composantes de turbine hydroélectrique, in: Proceedings of the 24th Seminar on Machinery Vibration, 2006, Montréal, pp. 320–329. [7] B. Peeters, B. Cornelis, K. Janssens, and H. Van der Auweraer, Removing disturbing harmonics in operational modal analysis, in: Proceedings of the International Operational Modal Analysis Conference, Copenhagen, Denmark, 2007. [8] R.B. Randall, Cepstral methods of operational modal analysis, Encyclopedia of Structural Health Monitoring, 2009. [9] N.J. Jacobsen, P. Andersen and R. Brincker, Eliminating the influence of harmonic components in operational modal analysis, in: Proceedings of the IMAC XXV Conference, 2007. [10] P. Mohanty, D.J. Rixen, Operational modal analysis in the presence of harmonic excitation, J. Sound Vib. 270 (1–2) (2004) 93–109. [11] R. Pintelon, B. Peeters, P. Guillaume, Continuous-time operational modal analysis in the presence of harmonic disturbances, Mech. Syst. Signal Process. 22 (5) (2008) 1017–1035. [12] R. Pintelon, B. Peeters, P. Guillaume, Continuous-time operational modal analysis in the presence of harmonic disturbances-the multivariate case, Mech. Syst. Signal Process. 24 (January (1)) (2010) 90–105. [13] C. Devriendt, G. De Sitter, S. Vanlanduit, P. Guillaume, Operational modal analysis in the presence of harmonic excitations by the use of transmissibility measurements, Mech. Syst. Signal Process. 23 (April (3)) (2009) 621–635. [14] W. Weijtjens, G. De Sitter, C. Devriendt, P. Guillaume, Operational modal parameter estimation of mimo systems using transmissibility functions, Automatica 50 (February (2)) (2014) 559–564. [15] N.M.M. Maia, J.M.M. Silva, A.M.R. Ribeiro, The transmissibility concept in multi-degree-of-freedom systems, Mech. Syst. Signal Process. 15 (January (1)) (2001) 129–137. [16] C. Devriendt, W. Weijtjens, G. De Sitter, P. Guillaume, Combining multiple single-reference transmissibility functions in a unique matrix formulation for operational modal analysis, Mech. Syst. Signal Process. 40 (1) (2013) 278–287. [17] W.J. Yan, W.X. Ren, Operational modal parameter identification from power spectrum density transmissibility, Comput.-aided Civil Infrastruct. Eng. 27 (March (3)) (2012) 202–217. [18] I.G. Araújo, J.E. Laier, Operational modal analysis using svd of power spectral density transmissibility matrices, Mech. Syst. Signal Process. 46 (1) (2014) 129–145. [19] J. Lataire, R. Pintelon, Frequency-domain weighted non-linear least-squares estimation of continuous-time, time-varying systems, Control Theory Appl., IET 5 (7) (2011) 923–933. [20] J. Lataire, R. Pintelon, Frequency domain errors-in-variables identification of a time-varying, discrete time system?, in: Proceedings of the 18th IFAC World Congress, Milan, Italy, 2011, vol. 18, pp. 13110–13114. [21] G. De Sitter, C. Devriendt, P. Guillaume, Transmissibility-based operational modal analysis: enhanced stabilisation diagrams, Shock Vib. 19 (5) (2012) 1085–1097.
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions Wout Weijtjens a, John Lataire b, Christof Devriendt a, Patrick Guillaume a a b
Vrije Universiteit Brussel, Department of Mechanical Engineering, Acoustics and Vibration Research Group (AVRG), Belgium Vrije Universiteit Brussel, Department of Fundamental Electricity and Instrumentation (ELEC), Belgium
a r t i c l e i n f o
abstract
Article history: Received 17 September 2013 Accepted 15 April 2014
Periodical loads, such as waves and rotating machinery, form a problem for operational modal analysis (OMA). In OMA only the vibrations of a structure of interest are measured and little to nothing is known about the loads causing these vibrations. Therefore, it is often assumed that all dynamics in the measured data are linked to the system of interest. Periodical loads defy this assumption as their periodical behavior is often visible within the measured vibrations. As a consequence most OMA techniques falsely associate the dynamics of the periodical load with the system of interest. Without additional information about the load, one is not able to correctly differentiate between structural dynamics and the dynamics of the load. In several applications, e.g. turbines and helicopters, it was observed that because of periodical loads one was unable to correctly identify one or multiple modes. Transmissibility based OMA (TOMA) is a completely different approach to OMA. By using transmissibility functions to estimate the structural dynamics of the system of interest, all influence of the load-spectrum can be eliminated. TOMA therefore allows to identify the modal parameters without being influenced by the presence of periodical loads, such as harmonics. One of the difficulties of TOMA is that the analyst is required to find two independent datasets, each associated with a different loading condition of the system of interest. This poses a dilemma for TOMA; how can an analyst identify two different loading conditions when little is known about the loads on the system? This paper tackles that problem by assuming that the loading conditions vary continuously over time, e.g. the changing wind directions. From this assumption TOMA is developed into a time-varying framework. This development allows TOMA to not only cope with the continuously changing loading conditions. The time-varying framework also enables the identification of the modal parameters from a single dataset. Moreover, the time-varying TOMA approach can be implemented in such a way that the analyst no longer has to identify different loading conditions. For these combined reasons the time-varying TOMA is less dependent on the user and requires less testing time than the earlier TOMA-technique. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Operational modal analysis (OMA) Transmissibility functions Harmonics Periodical loads System identification Damping
1. Introduction to dealing with periodical loads In mechanical engineering, operational modal analysis (OMA) is used to retrieve the modal parameters of a structure that is subjected to loads associated with its proper use [1]. For instance a bridge excited by wind and traffic loads [1] or an
E-mail address: [email protected] (W. Weijtjens). http://dx.doi.org/10.1016/j.ymssp.2014.04.008 0888-3270/& 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
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Fig. 1. (a) If the loads have a flat spectrum (i.e. white noise) all dynamics present in the measurements are caused by the system of interest. (b) For periodic loads this property no longer holds. Without any additional information about the loads, it is impossible to know whether certain dynamics are caused by the structure or by the load.
Fig. 2. Periodic loads can be modeled as the response of an input-filter to white noise. While OMA techniques will identify a combined system (inputfilterþ system of interest), TOMA-techniques are able to identify the system of interest directly no matter the input-filter.
offshore windturbine excited by wind and waves [2]. However, it is nearly impossible to correctly measure all loads that excite an operational structure. OMA therefore considers no known loads and operates purely on measured vibrations, i.e. the system response to the unknown loads. The identification of the modal parameters is made possible by assuming that the loads have a flat frequency spectrum, i.e. white noise [1]. This assumption is necessary to assure that the output spectrum resembles the transfer function of the system of interest, Fig. 1a. Therefore, all dynamics in the measurement spectrum are caused by the system of interest which then can be identified. Because wind loads approximate such a flat frequency spectrum, OMA can be easily applied to bridges and towerstructures dominantly excited by wind [1,3]. However, some loads act periodically, e.g. waves and rotating machinery, and contain more energy at given frequencies (e.g. harmonics). This poses a problem for OMA, when nothing is known about the load-spectrum one cannot guarantee that all dynamics in the measurement-spectrum are related to the system of interest itself, Fig. 1b. Several strategies are possible to still perform OMA in the presence of periodical loads. To illustrate them, one can model the non-flat spectrum of the load as the response of an unknown input filter to white noise, Fig. 2. A first strategy is to apply any of the classic OMA-algorithms [3–5] to the measurement data, without making any modifications to the algorithms. As a result the modal parameters of a system, that is the combination of the input filter and the system of interest, are identified. Afterwards the analyst can reject certain modes based on prior knowledge of the loads, e.g. rotational speeds [6], or the system itself. An alternative strategy is to pre-process the measurement data to reduce the influence of the load as much as possible, before using the data into the preferred OMA-algorithm. Examples are time-synchronous averaging [7], cepstrum analysis [8] or interpolation in the frequency domain [9]. All previously mentioned strategies require that the peaks in the load spectrum are well-distanced from the actual structural dynamics, which unfortunately is not always the case. The solution to this issue can be achieved by adapting the OMA-algorithms to incorporate an input-filter next to the system of interest. In essence such an approach still identifies the combined system (input-filterþsystem of interest). However, as these algorithms include the input filter into the model, they are able to distinguish which dynamics are linked to the input filter and which dynamics are part of the system of interest. This allows for a better identification of the system of interest, especially when the periodical forces have frequencies close to structural modes [10–12]. However, all these methods assume a certain model of the input-filter and therefore require additional information about the load spectrum. This paper will continue the work done in transmissibility based modal analysis, or TOMA [13,14]. TOMA-techniques solve the issue of periodical loads in a completely different manner. As discussed in [13] and [15] transmissibility functions can become completely independent of the input spectrum. This implies that TOMA is uninfluenced by the input filter and is able to identify the system of interest directly. Unlike all previous techniques the analyst does not require any additional information about the load spectrum to correctly identify the system of interest. The next section will recapitulate on TOMA and introduce a modification to the existing algorithms.
2. Transmissibility based OMA (TOMA) In this section we will recapitulate the state of the art in TOMA, introduce the basic concept of time-varying TOMA and discuss its advantages. Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
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Fig. 3. Illustration of TOMA on a system with two poles. (a) Two different loading conditions are simulated by changing the input location of the load. (b) Transmissibility functions are dependent on the load location (dotted vertical lines: resonance frequencies of the system). (c) Transmissibility functions intersect in the system poles ð d1 7 i2πf 1 ; d2 7 i2πf 2 Þ.
2.1. State of the art in TOMA In [13] it was shown that in the case of a single (distributed) load the transmissibility function is independent of the load spectrum.1 This independence implies that any dynamics present in the input filter will not influence the transmissibility function, nor will they influence modal parameters obtained through transmissibility functions. Unfortunately, the modal parameters cannot be derived from a single transmissibility function, Fig. 3b. The solution to find modal parameters from transmissibility functions comes from a different property of the transmissibility function. While a transmissibility function is independent of the load spectrum, it is dependent on the load location. Basically, if the load changes location (or distribution) the transmissibility function will change, Fig. 3a and b. An interesting property shows up in Fig. 3b and c, where it is shown that transmissibility functions of different loading conditions intersect in the system poles. Over recent years several strategies on how to obtain the modal parameters from transmissibility functions based on this idea have been published [13,16]. Such an approach, with different loading conditions, implies that in order for TOMA to work at least two loading conditions have to be considered. But, how does an analyst distinguish two different loading conditions if little is known about the loads?. A first solution was proposed in [17], this method based on power spectrum density transmissibilities requires only a single loading condition. However, the proposed method only works when multiple uncorrelated forces act on the structure. In particular cases such as the presence of correlated periodical loads, as caused by a single rotating machine, this can again lead to errors with misinterpreted periodical loads. The methodology of [17] was expanded in [18] into a SVD-driven algorithm in a similar fashion as the expansion proposed in [16]. While both [17,18] are very interesting approaches to TOMA, they currently lack the ability to estimate damping. In this paper we propose a different solution to this problem without losing the load spectrum independency and yielding a damping estimate. 2.2. A time-varying approach to TOMA If we look at Fig. 3 we notice how the two discrete loading conditions can help us to find the system poles. In a real world application it is rather unlikely that a system subjected to an operational load suddenly is excited in a completely different way. Even if such a sudden transition occurs it is very hard to pinpoint the exact moment of the transition without additional information on the load itself. It seems far more realistic to assume that the loads on a system vary continuously over time. For example the wind direction does not jump from e.g. East to North-East instantaneously. In fact, the wind load will vary continuously with minor changes in direction and velocity over time. Waves, traffic or the controls of a turbine may be considered in a similar way as they also vary constantly over time. As discussed earlier the transmissibility function is related to the load location/distribution. So, when the loads vary continuously over time so do the transmissibility functions. To demonstrate let us reconsider the example of Fig. 3. But this time a single load slowly moves its location, Fig. 4a. TOMA until now assumed that in a measurement record the loading condition is constant. From Fig. 4b it is evident that when that assumption is violated the transmissibility function loses its load spectrum independency and is no longer deterministic. At each instance of time there is actually an instantaneous loading condition. For instance at the first time instance, t0, the load is only applied at point 1. At the final time instance, te, a load is only applied at point 2. So at t0 and te the system is for an instant of time subjected to the same loading conditions as the system of Fig. 3. Furthermore, at each of the intermediate 1 Recently, the idea of TOMA was extended to pTOMA, [14]. In which multi reference transmissibility functions are used to retrieve the modal parameters. This extension allows to become independent of multiple input loads. In this paper we will limit us to the scenario of a single load, however it is believed that the presented time-varying approach can be extended to a multi-reference implementation.
Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
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Fig. 4. Illustration to time-varying TOMA. (a) Two different loading conditions are simulated by changing the input location of the load gradually over time. (b) Transmissibility functions calculated as in Fig. 3 are no longer behaving as expected. (c) Instantaneous transmissibility functions show the same behavior as classic transmissibilities.
instances of time, t 0 o t ot e , the system is subjected to a unique but instantaneous loading condition. For instance at the middle time instance a distributed load is applied equally to both points. Each of these instantaneous loading conditions has an associated instantaneous transmissibility function. Some of these instantaneous transmissibility functions are plotted in Fig. 4c. However, as these transmissibility functions only exist for an instant, they cannot be estimated with the approaches used before. To estimate these instantaneous functions, the timevarying behavior of the transmissibility function has to be included into the model. Once the time-varying transmissibility functions are estimated, the system poles can be derived from them in a manner similar to TOMA using two or more instantaneous transmissibility functions. Furthermore, the analyst is no longer required to distinguish two different loading conditions. Effectively, each instance of time is linked to a unique instantaneous loading condition, so the analyst only has to consider different instances of time to consider different loading conditions.
3. Using time-varying transmissibility functions for OMA In this section, we will firstly define the system of interest and the transmissibility function. Then the effects of a timevarying load will be discussed. Next, the procedure to estimate time-varying transmissibility functions will be explained. Once the time-varying transmissibilities are estimated we show how the system poles can be derived from them. Finally, it will be shown that for linear time variations the analyst does not longer have to choose different loading conditions.
3.1. Basic assumptions and definitions Assume that the system of interest is linear and time-invariant. The system transfer function, HðsÞ A CNo Ni , will relate the ! ! load, F ðsÞ A CNi , with the outputs, X ðsÞ A CNo . The transmissibility function, T(s), is defined as the function that relates the ! ! so-called reference outputs, X r ðsÞ A CNref , to the non-reference outputs, X l ðsÞ A CNo Nref , [15]: ! ! X ðsÞ ¼ HðsÞ F ðsÞ
ð1Þ
! ! X l ðsÞ ¼ TðsÞ X r ðsÞ:
ð2Þ
In this paper, we will use only a single reference and non-reference output. The single-reference transmissibility function is also defined as T lr ðsÞ ¼ X l ðsÞ=X r ðsÞ. When only a single (distributed) load acts on the system, the single reference transmissibility, Tlr(s), is load-spectrum independent. However, keep in mind that when multiple uncorrelated loads act on the system a single reference is no longer sufficient to achieve load spectrum independency [14].
3.2. Time varying loads Previous research on TOMA assumed a loading condition to be constant within a single measurement record. In practical applications it is far more likely that a load will constantly vary its distribution, e.g. the shifting wind, within a single measurement record. ! Let us assume a single distributed force f ðtÞ A RNi that is equal to the product of a time-varying distribution vector, ! N f d ðtÞ A R i , and a single source of the force μðtÞ A R: ! ! f ðtÞ ¼ f d ðtÞμðtÞ:
ð3Þ ! f d ðtÞ,
If the distribution vector, is time-invariant a single reference transmissibility function will become fully load-spectrum ! independent, as there is only a single distributed force. But what happens when f d ðtÞ is a function of time? Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
W. Weijtjens et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]] !
5 ! i d;i t ,
t Let us assume that the distribution vector can be modeled as a polynomial in t, f d ðtÞ ¼ ∑ni ¼ 0f ! load variation in time. The system outputs in the time domain, x ðtÞ A RNo , are provided by ! nt ! ! 1 i x ðtÞ ¼ L fHðsÞgn ∑ f d;i t μðtÞ
with nt the order of the
ð4Þ
i¼0
with Lfg the Laplace-transform of and n is the convolution operator. The Laplace transform of Eq. (4) is ( ) nt ! ! X ðsÞ ¼ HðsÞL ∑ f d;i t i μðtÞ
ð5Þ
i¼0
nt
¼ HðsÞ ∑
i¼0 nt
¼ HðsÞ ∑
i¼0
! i f d;i Lft μðtÞg
ð6Þ
! i ðiÞ f d;i ð 1Þ L fμðtÞg
ð7Þ
in which LðiÞ fg is the i-th derivative of Lfg with respect to s. In general Lfg will not be fully correlated with any of its derivatives LðiÞ fg. Therefore, all ðnt þ 1Þ derivatives of LfμðtÞg are to be considered as uncorrelated sources and at least ðnt þ 1Þ references are required to get a load-spectrum independent transmissibility function [15]. Furthermore, this will increase the number of required sensors, at least ðnt þ2Þ, and potentially the number of required loading conditions [14]. 3.3. Estimation of time-varying transmissibility functions The goal of the paper is to expand the transmissibility functions to a time-varying form. Such a time-varying transmissibility function can then be evaluated at a time instance, tn, to retrieve the instantaneous transmissibility functions ! associated with an instantaneous load distribution f d ðt n Þ. It is assumed that a time-varying single reference transmissibility function2 can be modeled in the time interval ½t 0 ; t e as a linear ordinary differential equation with time-varying coefficients i
na
∑ αi ðt Þ
i¼0
d xl ðtÞ dt
i
i
nb
d xr ðtÞ
i¼0
dt
¼ ∑ β i ðt Þ
ð8Þ
i
with na ; nb the model orders of the structural dynamics. The time-varying coefficients are modeled as polynomials in t; nt j j t αi ðtÞ ¼ ∑nj ¼ 0 αi;j t , βi ðtÞ ¼ ∑j ¼ 0 βi;j t . The time-varying transmissibility functions are found by estimating the unknown coefficients αi;j A R and βi;j A R from Eq. (8). In [19] a similar identification problem, for time-varying transfer functions, is discussed. This paper will translate that approach to estimate time-varying transmissibility functions and summarize all results of [19] relevant to this paper. For a more in-depth discussion on the estimation itself we refer back to [19]. One of the main advantages of an estimation algorithm in the frequency domain is the ease at which a frequency-band of interest can be defined. Eq. (8) is therefore transformed into the frequency domain: ( ) i i nb na nt nt d x ðtÞ d xr ðtÞ L ∑ ∑ αi;j t j l i ∑ ∑ βi;j t j ¼0 ð9Þ i dt dt i¼0j¼0 i¼0j¼0 na
nt
(
∑ ∑ αi;j L t
i¼0j¼0
jd
i
xl ðtÞ
dt
i
)
nb
(
nt
∑ ∑ βi;j L t i¼0j¼0
jd
i
xr ðtÞ dt
i
) ¼0
ð10Þ
As discussed in [19] a Laplace transform is preferred over a differentiation and a multiplication is preferred over a convolution. For this reason the following expression, equivalent to Eq. (10), is preferred over Eq. (10): na
nt
nb
nt
∑ ∑ αi;j Lft j L1 fsi Lfxl ðtÞggg ∑ ∑ βi;j Lft j L1 fsi Lfxr ðtÞggg ¼ 0:
i¼0j¼0
ð11Þ
i¼0j¼0
It is shown in [19] that for finite-length and sampled data the model must be extended with an additional transient a ;nb Þ 1 polynomial τðsÞ ¼ ∑imaxðn τi si . This additional term allows to capture errors caused by the difference in initial and final ¼0 conditions, and errors caused by aliasing. Because of τðsÞ, the actual system dynamics are well separated from these signalprocessing effects. Eq. (11) then becomes 2 For ease of notation all transmissibility functions discussed from this point are single reference transmissibility functions unless mentioned otherwise.
Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
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6
na
nb
nt
nt
∑ ∑ αi;j Lft j L1 fsi Lfxl ðtÞggg ∑ ∑ βi;j Lft j L1 fsi Lfxr ðtÞggg þ
i¼0j¼0
i¼0j¼0
maxðna ;nb Þ 1
∑
i¼0
τi si ¼ 0
ð12Þ
the unknown parameters τi A R are part of the set of parameters to be estimated. Eq. (12) is linear in all unknown parameters and can be rewritten as ! K Θ Θ ¼ 0; with ð13Þ K Θ ¼ fK l K r K τ g
ð14Þ
and
! Θ ¼ fα0;0 ; α0;1 ; …; αn;p ; …; βn;p ; …; τ0 ; τ1 ; …; τmaxðna ;nb Þ 1 gT
ð15Þ
for which the k-th element in the column of Kl associated with αn;p is given as K l;n;p ¼ ðLft p L1 fðjωk Þn Lfxl ðtÞgggÞ
ð16Þ
similarly for the column of Kr associated with βn;p : K r;n;p ¼ ðLft p L1 fðjωk Þn Lfxr ðtÞgggÞ
ð17Þ
and of which the k-th row of K τ is defined as f1 ðjωk Þ ðjωk Þ2 … ðjωk Þmaxðna ;nb Þ 1 g:
ð18Þ
Each row of K Θ corresponds to one frequency ωk =2π of the Nf frequency lines in the frequency-band of interest. For sampled signals, with sampling time Ts and sampled time ts, these equations are approximated by their discrete Fourier transform (DFT) and their inverse discrete Fourier Transform (IDFT), with equal to r or l: K ;n;p ¼ T s ðDFTfðt s Þp IDFTfjωnk DFTfx ðt s ÞgggÞ:
ð19Þ
The previous derivation is made for a continuous time model. In [20] it is illustrated how to apply this idea for discrete-time models. The current model also considers a polynomial model in both time as well as in frequency domain. Especially for high model orders, na ; nb or nt, this might result in an ill-conditioned observation matrix K Θ . The method can be fully adapted to use Legendre-polynomials in the time and/or the frequency domain, [19], which can improve the conditioning of K Θ. ! In this paper the overdetermined set of Eq. (13) is solved, and the parameters in Θ are found, by calculating the Singular Value Decomposition (SVD) of K Θ : K Θ ¼ USV H :
ð20Þ
The Total Least Squares solution of Eq. (13) is found as the singular vector associated with the smallest singular value, i.e. the last column of the matrix V. In this paper we only used the non-consistent Total Least Squares. Nonetheless, it is also possible to incorporate information about the noise into the solution of Eq. (13) in order to obtain a consistent estimator [19]. 3.4. Finding the modal parameters The final step in the time-varying TOMA approach is to retrieve the modal parameters from the time-varying transmissibility functions. The instantaneous transmissibility function at time tn, T lr ðs; t n Þ, is defined as j
na nt a ^ n i ∑i ¼ ∑ni ¼ ^ i ðt n Þsi n i 0 ∑j ¼ 0 α i;j t s n j i 0α t s ¼ ðt Þs T lr s; t n ¼ nb nb t ^ i;j ^ ∑i ¼ 0 ∑nj ¼ ∑ β 0 i ¼ 0β i
ð21Þ
in which α^ i;j and β^ i;j are the coefficients estimated in the previous section. It is assumed that for slow variations of the load (i.e. much slower than the actual structural dynamics) these instantaneous transmissibility functions are equal to the nonN time-varying transmissibilities. So, in the system pole λm A C, with mode shape ! ϕ m A C o , the following property [13] holds for each time instance in the considered measurement record: ϕ T lr λm ; t n ¼ l;m ϕr;m
8 t n A ½t 0 ; t e
ð22Þ
ϕ m , respectively, associated with the reference and nonin which ϕr;m and ϕl;m are the elements of the m-th mode shape, ! reference output. In essence Eq. (22) states that all instantaneous transmissibility functions intersect in the system poles. Eq. (22) can also be interpreted as follows: for all instances of time, T lr ðλm ; t n Þ has the same value. This is equivalent to
δT lr ðλm ; tÞ t ¼ tn ¼ 0 δt
8 t n A ½t 0 ; t e :
ð23Þ
Because of the strong similarity between these instantaneous transmissibility functions and the classic transmissibility functions we can again rely on classic TOMA techniques to find the modal parameters. For instance one possible solution, Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
W. Weijtjens et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
7
similar to the one proposed in [13], is to calculate a base function using two instantaneous transmissibility functions for arbitrary time instances t n1 and t n2 : Δ 1 T lr ðsÞ ¼
1 T lr ðs; t n2 Þ T lr ðs; t n1 Þ
ð24Þ
and to feed this base function into an existing curve-fitter for Frequency response functions, e.g. LSCF. The system poles are then found as the poles of Δ 1 T lr ðsÞ. This approach will be used in Section 4. 3.5. Discussion on the choice of the model order of the time-varying coefficients While the previous approach works correctly, there is still some discussion possible on which instances of time the instantaneous transmissibility functions should be computed. The classic task in TOMA of identifying two measurement records that both correspond to two different loading conditions has been replaced by choosing two time instances t n1 and t n2 . The analyst can be relieved from this choice if the time-varying model order nt is chosen equal to one. This can be understood by rewriting Eq. (21) as j na nt nj t ^ ^ i;j si t n ∑nj ¼ ∑i ¼ 0 ∑j ¼ 0 α 0 a j ðsÞt T lr s; t n ¼ n ¼ n j n nj b t t ^ ^ i n ∑i ¼ ∑j ¼ 0 ∑j ¼ 0 β i;j s t 0 b j ðsÞt
ð25Þ
a b ^ i ^ i;j si and b^ j ðsÞ ¼ ∑ni ¼ with a^ j ðsÞ ¼ ∑ni ¼ 0α 0 β i;j s . The partial derivative to t of Eq. (25) for nt ¼1 is
a^ 1 ðsÞðb^ 1 ðsÞt n þ b^ 0 ðsÞÞ b^ 1 ðsÞða^ 1 ðsÞt n þ a^ 0 ðsÞÞ δT lr ðs; tÞ ¼ n δt t¼t ðb^ 1 ðsÞt n þ b^ 0 ðsÞÞ2 ¼
ð26Þ
a^ 1 ðsÞb^ 0 ðsÞ b^ 1 ðsÞa^ 0 ðsÞ ðb^ ðsÞt n þ b^ ðsÞÞ2 1
ð27Þ
0
Because nt is set to one, the numerator of Eq. (27) is no longer a function of the chosen time instance tn. Moreover, when in sn the partial derivative of T lr ðsn ; tÞ to t is zero: δT lr ðsn ; tÞ ¼0 ð28Þ δt t ¼ tn the denominator, and therefore all instances of tn, can be eliminated: a^ 1 ðsn Þb^ 0 ðsn Þ b^ 1 ðsn Þa^ 0 ðsn Þ δT lr ðsn ; tÞ ¼ ¼0 δt n ðb^ 1 ðsn Þt n þ b^ 0 ðsn ÞÞ2 t¼t
ð29Þ
a^ 1 ðsn Þb^ 0 ðsn Þ b^ 1 ðsn Þa^ 0 ðsn Þ ¼ 0:
ð30Þ n
n
The equation above states that if the partial derivative of T lr ðs ; tÞ to t is equal to zero for one instance of t , then it is equal to zero for all instances of tn. It also implies that T lr ðsn ; tÞ is not a function of time. Basically, only by choosing nt ¼1 the analyst forces the model to include some values sn for which T lr ðsn ; tÞ is not a function of time. From Eq. (23) we learn that such values sn are in fact the system poles, λm, which gives a physical meaning to the choice of nt ¼1. Constraining nt to one does not only impose this physical constraint to the model, it also implies that T lr ðλm ; tÞ is not a function of time. Therefore, the choice of t n1 and t n2 in Eq. (24) does no longer have any influence on the estimation of the system poles. In essence for nt ¼ 1 the analyst is relieved from the task of identifying two different loading conditions, as any arbitrary choice will yield the same results. Estimated poles that still vary with the choice of t n1 and t n2 are most likely intersections of two transmissibility functions that are not linked with a system pole. This phenomena was discussed in [21]. Imposing nt ¼1 restricts the model to linearly time-varying transmissibilities. In some scenarios this model order might not be sufficient to cover all the variations present in the load. The analyst can detect this under-modelling by checking the ! quality of fit of the estimated model, e.g. by evaluating the remaining error, e A CNf : 9 9 9 8 8 8 β0;0 > τ0 α0;0 > > > > > > > > > > > > > > > > > > > > > > > = < α0;1 = < β0;1 = < τ1 ! ! Kr þ Kτ ¼ K l! α K r β þK τ ! τ: ð31Þ e ¼ Kl ⋮ ⋮ ⋮ > > > > > > > > > > > > > > > > > > > > > > > > : αn ;n ; : βn ;n ; : τmaxðn ;n Þ 1 ; a
t
b
t
a
b
However, because of the advantages of restricting nt to one it is advised to not increase nt, but rather to reduce the length of the measurement record. Within the shortened measurement record the varying load might be better approximated, indicated by an improved quality of fit, by the linear time-varying model. Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
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8
Fig. 5. The system of interest is an aluminum beam, excited by two shakers (S1 and S2). Vibrations are measured at two locations using one Laser Doppler Vibrometer (LDV) and a piezo-electric accelerometer (ICP). The input signal was generated as a response of a SDOF system to white noise, this signal was split and crossfaded from one shaker to the other. The forces applied to the system were measured but not used in the estimate.
F 1
|F|
F 2
B1 IF
B2
T1
B3
Fig. 6. (a) Output spectra obtained with the two sensors. The first three bending modes (B1–B2–B3) are indicated along with the first torsional mode (T1). The resonance frequency of the input filter is highlighted with a red dashed line (IF). (b) Input forces measured. The crossfade from one shaker to the other was initiated after several seconds to avoid startup effects. The highlighted area indicates the crossfading. (c) Force spectra during the crossfading show the presence of the input filter. Additional spikes e.g. around B3 are due to inertial forces exerted by the beam. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
4. Laboratory experiment 4.1. Experimental setup The system of interest in this experiment was a suspended aluminum beam. The beam is excited by two shakers placed at the outer limits of the beam. We measured the vibrations at two locations using a piezo-electric accelerometer (ICP) and a Laser Doppler Vibrometer (LDV). As a reference the applied forces were also measured but are not used in the estimation itself. The load was generated as the response of a single degree of freedom model to white noise. This signal was crossfaded from one shaker to another to obtain a time-varying load, Fig. 5. The laser-dot and accelerometer were deliberately put at random locations to illustrate that even an unfortunate choice of one of both sensors does not result in a complete failure of the method. For instance the laser dot is aimed at a nodal point of the first torsional mode (T1). The ICP was placed close to a nodal point of the third bending mode (B3). In Fig. 6a the output spectra are shown for both sensors. To avoid startup effects one of both shakers was already exciting the structure for several seconds before the crossfade was initiated. In Fig. 6b the forces measured at the inputs are shown and the crossfade is highlighted. The forces measured at shaker 2 before the crossfade is initiated are the result of the vibrations in the beam being transmitted back into the shaker. The crossfade itself only takes 0.8 s to complete, with the measurements sampled at 10.2 kHz. It is important that the estimation does not include the actual start and ending of the crossfade. Inclusion of these points would lead to a poor
Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
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Fig. 7. The time-varying model is able to describe the dynamics present within the data. (a) The error as defined in Eq. (31) is significantly smaller than the ! composing components. (b) Zoomed view at the overlay of jK r β j and jK l ! α þ Kτ ! τ j.
Fig. 8. (a) Instantaneous transmissibility functions for the considered dataset. (b) Using a standard base function, transmissibility functions are molded into something that resembles an FRF. The base function does not have a peak at the frequency of the input filter (IF), indicating that the estimate was not influenced by the load spectrum. (a) Instantaneous transmissibility functions and (b) 1/|T12(s,t0)-T12(s,te)|. Table 1 Results of both TOMA and classic OMA, the input filter (IF) is not part of the structure but was nonetheless estimated by OMA. Mode
B1 IF B2 T1 B3
TOMA
OMA (same data)
OMA (more data)
fres (Hz)
ξ ð%Þ
fres (Hz)
ξ ð%Þ
fres (Hz)
ξ ð%Þ
260.1 – 705.2 1258.6 1371.6
0.80 – 0.10 0.11 0.10
260.2 301.6 711.7 1254.1 1374.3
0.35 1.12 0.50 0.17 0.12
259.2 300 709.1 1257.0 1375.0
0.53 1.19 0.22 0.21 0.17
estimate as at these points the transmissibilities rapidly change from being time-invariant to time varying and the model is only able to capture this rapid transition for large values of nt. In the estimation 6144 data points were used, that are located in the middle of the crossfade. The two spectra of these data points are shown in Fig. 6a. 4.2. Results In this example the first and final instantaneous transmissibility functions are used in the base function (Eq. (24)). This base function, that resembles an FRF, is then fed into a frequency domain curve-fitter with a numerator–denominator model (LSCF) [13]. The instantaneous transmissibility functions are estimated for a structural dynamics model order of 10 ðna ¼ nb ¼ 10Þ and a time-varying model order equal to one (nt ¼1). The frequency band of interest was set from 0 Hz up to 1500 Hz, this frequency band contains the three first bending modes (B1–B3) and the first torsional mode (T1). Legendre polynomials were used for the frequency domain model to improve numerical conditioning. In Fig. 7 the quality of fit is checked using the error defined in Eq. (31). The structure is lightly damped and it was observed that for higher structural model orders (na, nb) the damping was poorly estimated, sometimes even estimating the poles as unstable. Best practice remains to start at small structural model orders (na,nb), identifying the structural dynamics and finally increase na and nb to improve the preliminary results. Over-modeling will result in very complex instantaneous transmissibility functions, that show much more dynamics than the number of poles (Fig. 8). The estimated resonance frequencies and damping ratios are given in Table 1. As comparison the same data is also used for a classic OMA. In this example we used a Correlogram approach with Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
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model order
10
B1 IF
B2
T1 B3
frequency (Hz) Fig. 9. The stabilization diagram of the correlogram-LSCF (OMA) approach shows that the input filter is misinterpreted as a structural pole.
the LSCF estimator. In Fig. 9 the stabilization diagram that is the result of this analysis is provided. The large relative variation within the damping estimates between the different approaches can be attributed to the low damping inherent to the aluminum beam. The absolute differences between damping estimates are acceptable given the data and the LSCF estimator. 5. Conclusions and future work This paper presented a novel approach to deal with periodical loads in operational modal analysis using time-varying transmissibilities. The proposed method starts from estimating single reference time-varying transmissibility functions. Like regular transmissibility functions, time-varying transmissibility functions become independent of the load spectrum in the case of a single (distributed) load. Modal parameters found through the transmissibility functions, with a TOMA, are therefore solely related to the system of interest and not influenced by the load spectrum. This is especially interesting in the presence of periodical or harmonic loads as these are often misinterpreted by OMA-techniques as structural modes. Moreover, the time-varying TOMA has several advantages over classic time-invariant TOMA. In time-invariant TOMA the analyst is required to measure the system responses to at least two different, and stationary, loading conditions. This poses two problems to the analyst: how can he/she identify two different loading conditions if little is known about the loads on the system? Furthermore, how can the analyst be certain that the loading condition was effectively stationary? The presented time-varying TOMA solves these issues by relaxing the assumption of stationary loading conditions. The assumption for continuously varying loading conditions is made. It is shown that as a consequence of this assumption multiple loading conditions are present in a single dataset. In this paper we showed that it is possible to obtain the modal parameters from such a single dataset by expanding the transmissibility function to a time-varying framework. Because only a single dataset is required, the testing time can be greatly reduced in comparison to time-invariant TOMA. In a particular implementation of the algorithm it is also possible to relieve the analyst from his task to choose two different loading conditions within that single dataset, resulting in a more user-friendly and reliable approach to TOMA. The time-varying TOMA approach was demonstrated using a lab experiment on an aluminum beam. The current time-varying model is restricted to a single reference, future research will try to extend this to a multiple reference form as presented in [14].
Acknowledgments The financial support by the Agency for Innovation by Science and Technology in Flanders (IWT) is greatly acknowledged. References [1] F. Magalhães, Á. Cunha, Explaining operational modal analysis with data from an arch bridge, Mech. Syst. Signal Process. 25 (5) (2011) 1431–1450. [2] C. Devriendt, P.J. Jordaens, G.D. Sitter, P. Guillaume, Damping estimation of an offshore wind turbine on a monopile foundation, Renew. Power Gener., IET 7 (4) (2013) 401–412.
Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i
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Please cite this article as: W. Weijtjens, et al., Dealing with periodical loads and harmonics in operational modal analysis using time-varying transmissibility functions, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j. ymssp.2014.04.008i