202 E. Orlowitz and A. Brandt Fig. 20.3 Picture of mounting cubes used for measuring in multiple directions with single accelerometers Table 20.1 Operation conditions for the three measurements Speed (knots) Water depth (m) Shaft speed (rpm) Sea state 10 knots 10˙2 20 65 Calm 18 knots 18˙2 20 98 Calm Anchor 0 100 98 Calm The measurement time was limited to periods where nearly stationary conditions were established. In the present paper all three data sets consist of time histories of 30 min. For hardware reasons the actual sampling frequency was 1 kHz, but for conveniences the time histories were down sampled to a sampling frequency of 8 Hz before further processing. 20.2.3 Measurement Conditions As stated above three operating conditions are considered in this paper. Two conditions are identical except cruising speed at a water depth of approximating 30 m. The two cruising speeds were approximately 10 and 18 knots respectively, constant during the measurement (with variations less than 2 knots). These cruising conditions are from now on referred to as ‘ 10 knots’ and ‘ 18 knots’ respectively. The third condition is with the vessel anchored on deep water ( 100 m). For all three conditions the sea was very calm and the wind speed below 5 m/s. The rotation speed of the propeller was 98 rpm for the 18 knots and anchor condition and 65 rpm for the 10 knots condition, giving blade pass excitation of 6.5 and 4.3 Hz, respectively, which is far from the modes presented in the present paper (<2.5 Hz). The measurement conditions are summarized in Table 20.1. 20.3 Results/Experimental Modal Analysis (Output-Only) The variety of methods for OMA modal parameter estimation is large, many of them being modifications of some basic methods. In the present paper the Multiple-reference Ibrahim Time Domain (MITD) Method is used [5]. The original basis function for this method are impulse responses, which in the OMA case are replaced by correlation functions (CF’s), which are estimated by the (indirect) Welch approach, see e.g. [6] or [7]. The MITD method constructs a block Hankel matrix. For determination of system order and to suppress noise the block Hankel matrix is decomposed by singular value decomposition is computed (SVD). From the SVD compressed block Hankel matrix an eigenvalue problem can be established for different system order (number of poles) the solutions of which are plotted in a stabilization diagram, from which the poles and mode shapes can be manually selected. An example of a stabilization diagram using the data of the present work is presented in Fig. 20.4. In Fig. 20.4, around 1.6 Hz, the second mode is seen to be very close to a pole not obtaining the stability criteria. In this case this unstable pole is the propeller shaft fundamental rotation speed. In addition the damping is very low (0.04 %) which also indicates a harmonic rather than a mode.
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