CONTRIBUTION TO THE IDENTIFICATION OF THE DYNAMIC BEHAVIOUR OF FLOATING HARBOUR SYSTEMS USING FREQUENCY DOMAIN DECOMPOSITION

Transcription

1 CONTRIBUTION TO THE IDENTIFICATION OF THE DYNAMIC BEHAVIOUR OF FLOATING HARBOUR SYSTEMS USING FREQUENCY DOMAIN DECOMPOSITION S. Uhlenbrock, University of Rostock, Germany G. Schlottmann, University of Rostock, Germany Abstract The construction of yacht harbour systems using floating docks is a common and low cost alternative to fixed docking stages. These harbours are often built by coupling several concrete docks to one-another and mooring them to the sea bottom. Due to the difficulties of generating accurate numerical models, especially on account of frequency- and clearance-dependent hydrodynamic masses, as well as uncertainties of the boundary conditions, verification of the dynamic characteristics of installed floating systems is of great importance. However, it is very difficult to guarantee an accurate estimate of the Frequency Response Function (FRF) of installed structures using classical modal analysis, because input loads due to wind, waves and current simultaneously act on the structure. This fact leads to the necessity of applying operational modal analysis to characterize the dynamic behaviour of such structures. In this paper the results of measurements on floating wave breakers in the yacht harbour of Warnemünde in Rostock are presented. The technique employed was based on Frequency Domain Decomposition (FDD). This case study for the use of FDD for modal analysis of floating structures presents the identified resonant frequencies, mode shapes and modal damping ratios for the first modes. Also, a comparison with Single Input / Multiple Output (SIMO) modal analysis is presented, pointing out the strengths of the FDD technique in such applications. 1 Introduction The prediction of the dynamic behaviour of floating structures is usually done numerically. Generally, the aim of numerical analysis is to calculate natural frequencies in order to verify resonant effects and to determine vibration amplitudes. However, prediction of natural frequencies of floating systems is challenging because effective hydrodynamic masses are dependent on water depth. Even more difficult is the determination of wave drift forces, that also change the stiffness of the mooring system. Another problem with Fluid-Structure Interaction (FSI) computations using numerical models is their common foundation on potential theory, which neglects viscous fluid damping. Thus, accurate prediction of resonant amplitudes is difficult because no damping parameters are known for these kinds of structures. Summarized, a comparison of numerical calculation and experimental data sets is essential. Within the scope of this project, experimental modal analysis of a real structure was applied, first to validate numerical models and second, to gain damping factors.

2 When using classical input/output based modal analysis the main problem is to achieve a high input load and to filter out the forced vibration due to wave loads. An alternative is operational modal analysis. In this study results using frequency domain decomposition are presented. The problems due to regular wave excitation are pointed out for both kinds of analysis techniques and the usability of FDD in low frequency ranges for coastal structures is investigated. 2 Analysed Structure and Data Processing Equipment The analysed structure was a floating wave breaker system built in front of the yacht harbour in Warnemünde, Rostock [1]. Generally, these kinds of structures are coupled by stainless steel wires and rubber bearings. The mooring system comprises of steel chains and an anchor. The wave breaker system in Warnemünde consists of two moored structures, each structure composed of two coupled rigid bodies. The measurements were carried out on one of these structures. The dimensions of each body are 20x3x1.8 meter with a weight of approximately 45 tons. Figure 1 Floating wave breaker system at the Yacht Harbour in Rostock Warnemünde Figure 1 shows the structure and the equipment used during one measurement. Measurements were made with sensitive accelerometers (10V/g) and a VXI measurement system from HP with the software I-DEAS. Classical input/output based modal analysis was I-DEAS and evaluated up to 14 Hz, while FDD was done using ARTeMIS Extractor and Matlab and evaluated up to 6 Hz. In parallel with the modal analysis, a spectral analysis of exciting waves was accomplished. For FDD the measurement was made on three different days. Each four hour measurement duration was divided into 20 min. blocks. When doing classical modal analysis a very high excitation force is necessary, allowing waveexcitation forces to be neglected. At the department of Technical Mechanics, University of Rostock, there is experience with large impulse hammers (up to 600 kg). However, this kind of excitation was not possible due to potential damage of the structure due to the large mass. Possibilities of harmonic excitation of the structure were also unfeasible, mainly because rotational exciters do not generate sufficient force within the relevant frequency range and also because there was no accessible fixed point for the installation of shakers available. Due to the fact, that the floating structures have to be reached by boat and there was no crane available, a portable impulse hammer was chosen. An impulse hammer of approximately 30 kg was used and twelve measurements were averaged to approximate the FRF. The disadvantage of impulse excitation with a relatively small force in a regular (harmonic) swell involves some problems that will be discussed later. 3 Modal Analysis Approach To determine the most appropriate method regarding the effort necessary to obtain the modal parameters, operational modal analysis using frequency domain decomposition (FDD) was used, as well as classical modal analysis. The FDD works in the frequency domain and was presented by

3 Brincker [2-6]. For linear multi-input / multi-output systems it is possible to represent the response of a linear mechanical system in the frequency domain (the spectral matrix of the responses S ( jω) ) by the spectral matrix of the excitation S ( jω) and the frequency response matrix H( jω), * ( jω ) = ( jω) ( jω) T ( jω) xx xx S H S H. (1) The superscript * denotes the complex conjugate and T the transpose operation. By FDD the excitation is assumed to be multi-input white noise in the frequency range analysed. Brinker shows, that this assumption leads to an expression which represents the spectral matrix of the responses by a matrix with frequency-independent constant terms. In the vicinity of a natural frequency the response of the system at one measurement point can then be written down as, The variable d m S d d ( jω ) = φφ + φ φ. T * * T* m m m m m m * m jω λ jω λ m m is a constant that summarizes the contribution of white noise and the participation vector of each mode. If the input loads are bandpass white noise, the formula is applicable within the frequency band. For the derivation of FDD the response of a generally damped linear mechanical system is transferred to the frequency domain by modal superposition. In this way an expression is obtained, that links the spectral matrix of the physical coordinates S ( j Ω) with the spectral matrix of the modal coordinates S ( jω) by means of the complex conjugate of the modal matrix Ψ and its transpose, qq * T S ( jω ) = Ψ S ( jω) Ψ. (3) qq The modal matrix contains orthogonal vectors and the spectral matrix in modal coordinates is diagonal since modal coordinates do not correlate. The decomposition of the spectral matrix in modal coordinates with orthogonal eigenvectors is shown to be identical to the singular value decomposition of the spectral matrix. Spectral matrices are generally complex and symmetric, such that the singular value decomposition simplifies to The matrix Σ S ( jω ) = U( jω) Σ( jω) U H ( jω). (4) is a diagonal matrix containing singular values. The matrix U is a unitary matrix H containing singular vectors. The matrix U is the adjoint of U. In resonance the first singular vector corresponds to the unscaled eigenvector. If the singular value decomposition of the response spectral matrix is accomplished at each measured frequency Ω, the run of the first singular value curve over the frequency corresponds to the run of the autospectral density function. For well separated modes the k-th mode is approximated by the frequency at maximum singular value and the first singular vector. Within this region the first singular vector should not change, so that all singular vectors should have high values of Modal Assurance Criteria (MAC). The natural frequencies and the damping of each mode are extracted by transforming the autospectral density function into the time domain. As the autospectral density function is only well-approximated in the vicinity of the natural frequency, only the singular values in this region (2)

4 should be selected. The limits for the investigated range can be defined by comparison of MAC values. For each selected mode only the singular vectors with high MAC values were used (> 0.8). As a result the autocorrelation function is then obtained. If the autocorrelation function corresponds to a well-approximated mode shape it represents a logarithmically decrementing curve. In this way very simple procedures can be used to identify the damping and the natural frequency. For example, it is possible to determine the logarithmic decrement and the period of duration of the oscillation. 4 Characteristics of the Exciting Waves in the Frequency Domain The evaluation of the wave spectra gives an important indication of the problems that have to be faced while doing modal analysis on this kind of floating structure. A truly multi-input excitation due to wave loads is expected for measurements on sea-technical constructions. Thus a reduction in the order of the response spectral matrix, and therefore zero singular values due to single input loads, are not expected. The main problem is, that for Onshore Structures the angle of incidence of waves varies less than that for Offshore Structures. During the measurement of structural accelerations, the wave height was continuously measured by the Institute of Hydraulic and Coastal Engineering of the University of Rostock. The spectrum that results from the wave-peak time series is of particular interest, because it correlates to the spectrum of the wave exciting forces. In Figure 2 the averaged spectrum for the whole measurement day in logarithmical representation together with the first two singular values are presented. It is observed that water waves build a narrowband spectrum with high amplitudes between 0.1 Hz and 0.9 Hz (critical frequency range). Above 0.9 Hz the waves can be approximated by white noise. Concerning the FDD the basic assumptions are sustained above 0.9 Hz. One of the resonant peaks can be recognized at 1.6 Hz. Within the critical frequency range it should be proved if excitation is random enough to apply FDD. At 0.8 Hz a resonant peak is observed. Another resonant peak could be assumed at 0.35 Hz, between the maxima of the wave spectrum. For water waves there is usually a relationship between wave length and frequency. This is problematic due to the fact that a monochromatic wave with arbitrary frequency and corresponding wave length exciting a floating structure will induce a deflection shape corresponding to the load distribution and not to the mode shape. However, real water waves have a random phasedistribution and, depending on the adjacency, a cos² directional distribution occurs for windinduced waves. Thus, for water wave excitation it is important that the direction of waves is random enough to apply FDD. By measuring the wave height at different points it is possible to measure the mean wave direction assuming a time delay between the signals. So, if the variance of the phase shift between two wave measurement points is analysed, this parameter represents the directional distribution of waves. Low variance means a quasi mono directional wave, high variance means that waves come from many directions. In Figure 3 the variance of phases for a complete measurement of approximately four hours is illustrated. This time it is possible to distinguish the waves with random directional distribution, from the waves with predictable direction. The wave spectrum consists mainly of two superposed spectra. The first with maximal amplitudes between 0.2 Hz and 0.3 Hz, the second with maximal amplitudes between 0.3 Hz and 0.6 Hz. The first one is due to swell and is therefore permanent and with low variance in direction. The second spectrum is also a superposition of local wind-wave spectra and ship-induced waves. The last spectrum is less constant in direction. The analysis of variance shows, that the limit for applying FDD will be between 0.4 Hz and 0.5 Hz. In practice this means that during measurements on a day with less wind one achieves better results than on a windy day when swell is directional. It also means, that ship traffic does not necessarily affect the applicability of FDD.

5 f= 1/ Hz Wave Spectrum [cm] f= 1/ Hz 1 st Singular Value Variance [Rad] 2 nd Singular Value Figure 2 Wave Spectrum and Singular Values Figure 3 Phase variance of Waves 5 Measurement results Using classical modal analysis, the biggest problem is to distinguish the motion due to wave loads from the free vibration response due to impact. Signal windowing can be used to achieve this goal. In this case the measured data are multiplied with an exponential window. The intention is to measure only the accelerations during the initial impact, where the impulse generates a greater response than the wave loads. Modes of free vibration can be recognised and the additional damping due to the exponential window can be subtracted. However, it should be noted that within the frequency range of the waves a rough error in damping and especially in the mode shape are to be expected. Implementation of the procedure resulted in an improvement of the FRFs. Thus it was possible to identify modes down to 1.5 Hz (mode 3) with classical modal analysis. However this method does not help in the frequency ranges were wave loads are of high influence. Table 1: Identified modal parameters Mode. Damping [%] FDD Classical Analysis FDD Classical Analysis In Figure 4 selected FRF s are represented. The resonance peaks and phase shifts for modes above 1.0 Hz can be clearly observed. Figure 5 shows the run of the first two singular value curves. The higher frequency range (above 6 Hz) was not analysed with FDD. To distinguish resonant peaks in FDD the MAC values of the obtained singular vectors were calculated. Figure 6 shows the run of MAC values over frequency for five modes and the corresponding singular values. The final result check was done by comparing the singular value course with the spectrum of the waves. A mode at 0.35 Hz could not be confirmed. While MAC-Values for the first Singular Vector are good, the corresponding autospectral density function is not logarithmically decaying. Thus, no mode could

6 be detected with confidence. In Table 1 identified modal parameters for both techniques are listed. The table shows that identified modal parameters are similar for both identification techniques within the overlapping frequency range. Mode 1 could only be identified by FDD. But it should be noted that from computations the identified mode shape is expected at 0.36 Hz and not as FDD infers at 0.77 Hz. Mode shapes are illustrated in Figure 7. For implementation into numerical models, the modal damping factors must be reduced to linear damping coefficients. This can be done because the modal damping values are strongly dependent on the mode shapes. Rigid body modes without strong deformation of the coupling elements infer low damping (1,7 % for mode 1). The main part of the damping is hydrodynamic. Rigid body modes with flexible deformation of the coupling elements (mode 2 and mode 3) are characterised by additional damping (1-2 %). Within elastic modes the material damping of the structure acts additionally. The contribution is approximately 1-2 %. For the durability of the coupling elements the deflection shapes of mode 2 and mode 3 are particularly critical. Relative rolling or yawing of the pontoons within larger plants can cause the destruction of the coupling elements. This is especially relevant if this mode shape sags to the critical swell frequency range due to larger masses, whereby local cracking of concrete structures around the coupling elements can be caused. Phase [ ] f= Hz 1 st Singular Value f= Hz Amplitude [mm s -2 /N] 2 nd Singular Value Figure 4 FRF by classical Modal Analysis Figure 5 Singular Values 6 Conclusion The use of classical modal analysis with an exponential window helps to achieve results down to 1.5 Hz. Below 1.5 Hz the exponential window cannot eliminate the effect of the disturbing loads due to waves. Here the FDD provides assistance. The study shows that FDD is a good alternative to classical modal analysis for frequencies where water waves are directionally random. Here applicability of FDD is sound, and compared to classical modal analysis much easier to accomplish. The identified modal parameters are similar using classical modal analysis and FDD within the overlapping frequency range. For low frequencies modal parameter identification is more difficult. Analysis of wave spectra shows that the application of FDD is dependent on the measurement conditions and that swell causes particular problems using this method. To minimise errors due to directional waves, measurements should be made on days with less wind so that swell is evaded. New measurements on other platforms should indicate whether longer measurement times can improve the results further.

7 7 References [1] Uhlenbrock, S., Schlottmann, G., Report Entwurfsgrundlagen für schwimmende Steg- und Auslegersysteme im Flachwasser, University of Rostock, [2] Brincker, R., Zhang, L., Anderson, P. Modal Identification from Ambient Responses using Frequency Domain Decomposition, Proc. 18 th International Modal Analysis Conference, 2000 [3] Brincker, R., Anderson, P., Moller, N. An Indicator for Separation of Structural and Harmonic Modes in Output-Only Modal Testing, Proc. 18 th International Modal Analysis Conference, 2000 [4] Brincker, R., Ventura, C.E., Anderson, P. Damping Estimation by Frequency Domain Decomposition, Proc. 19 th International Modal Analysis Conference, 2001 [5] Brincker, R., Ventura, C.E., Anderson, P. Why Output-Only Modal Testing is a Desirable Tool for a Wide Range of Practical Applications, Proc. 21 st International Modal Analysis Conference, 2003 [6] Brincker, R., Anderson, P. A Way of Getting Scaled Mode Shapes in Output Only Modal Testing, Proc. 21 st International Modal Analysis Conference, 2003

8 1 st Singular Value 2 nd Singular Value Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 MAC-Value / Figure 6 Singular Values and MAC Values for Mode 1 until Mode 5 Mode 1 (0.77 Hz) Mode 2 (1.59 Hz) Mode 3 (2.01 Hz) Mode 4 (3.62 Hz) Mode 5 (4.64 Hz) Figure 7 Modes of the coupled wave breaker system