where the amplitudes and phases are determinate by analytical calculations.

　

2. NUMERICAL COMPUTATION

　

According to the theoretical model proposed in the previous paragraph, an original computer code have been developed, in order to calculate the harmonic structure of the marine propeller excitations induced in the shafting system. The program was applied to the above-mentioned ship.

As a function of the instantaneous advance coefficient J, the torque and thrust coefficients (K_M, K_T) and the open water propeller efficiency η_o have been read from the propeller series diagrams.

Based on empirical formula for the wake [1], [11] and suction coefficient ( η_H ), the pitch angles ( β_k ) and the induced velocity angles (β_indk) are computed according to (23). Six values of the radius r_k (k=1,2,...6) and 12 of the radius s_i (i=1,2,..6) have been considered, the corresponding data being presented in Table 1 and Table 2.

In this situation, relation (18) represents. for the 2N=12 values of σ_i resulting from the change of variables (14) and each value of k, a set of two systems of linear equations with 24 unknowns, 12 for the tangential components (indices a), and 12 for axial components (indices a) yielding a system of 144 linear equations having as unknowns the factors I_{a, ti}.

Introducing the solutions of this system in (21), the coefficients h_a,t, are calculated and, taking as a reference θ₀ = 15^oCA and solving the linear system (22), the coefficients of the non-dimensional circulation gk, k=1,2,..6 are determined. Thus, the variations of the induced velocities, with radius and angle of rotation may be computed.

The general flow-chart of this computer code is shown in Figure 6. Table 3 presents the computed torque amplitudes and phases.

In computing these magnitudes, the harmonic orders k have been related to the period of the blade motion:

　

Table 1. Geometric Input Data

Table 2. Proeller Efficiency Input Data

　

resulting the following correlation between the propeller shaft frequency and torque/thrust frequency expressed as a correspondence between their harmonic orders:

　

k_shaft = kZ (27)

　

Thus, first harmonic order for the propeller blade (k=1) becomes the Z-th harmonic order for the propeller shaft (in our case the 4-th order), and so on.

　

3. EXPERIMENTAL INVESTIGATIONS

　

The prediction capacity of the theoretical models was check on physical model. Experiments have been carried-out at the Romanian Ship Research and Design Institute ICEPRONAV Galatz-Romania on dummy model (scale 1:30) in order to determine in the towing tank the nominal wake, the loading coefficients and the efficiency (Fig. 7 and 8).

　

Fig. 6 Flow-chart for the Calculation of the Propeller Harmonic Excitations.

BACK　　　CONTENTS　　　NEXT