Вестник МГТУ. 2018, №4.

Вестник МГТУ. 2018. Т. 21, № 4. С. 566-576. DOI: 10.21443/1560-9278-2018-21-4-566-576 During the calculation the resistance augment fraction t is assumed to be equal to zero. This is done to simplify the formulae recording and does not change the fundamental conclusions. The second equation describes the rotation of the propeller on a moving ship. The rotation of the screw is provided by the engine torque Mdv(t, n), and the torque of the screw represents the antitorque moment. For this the torque coefficient CQ is taken from the curve. Then, the torque is found by the formula: QQ = CQ ■ (v 2 + (0.7 n nD) 2 ) PP D -. (3b) 8 In order to obtain two differential equations for the linear acceleration of the vessel dv 1 and the angular acceleration of the screw dn 1 we are to: 1) divide the sum of forces by the mass of the vessel taking into account the virtual mass of entrained water MM, and 2) divide the sum of torques by the moment of the rotating parts inertia with the account of the virtual mass moment of water inertia JJ: dv1(v,n) - kV (v) • v • |v| + k T • TT (v, n ) MM (4) dn1(v, n ,t) : = k M ■ M d v ( t ' n ) - k Q ' g g ( v ' n ) In addition to specifying the differential equations themselves, it is necessary to determine the law of change in the screw rotation speed by means of the control system. For this purpose, the time-dependent function of statutory revolutions nUst(t) has been introduced. In our research, the periodic time function with the period T 0 = 1200 s (20 min) has been chosen. nUst ( t ) nU n0 • cos 2 n t T 0 (5) return nU The function is recorded in the MathCad programme function. The statutory function (5) is included in the law of changing the torque of the ship's engine Mdv(t, n) and allows to maintain the screw speed at the given set level. For the regulation law, the parameters aa = 2 100, bb = 100 are chosen. This provides the regulatory characteristic close to the vertical curve in the m - n (torque - rpm) axes. Mdv ( t , n ) : = aa ■ sign ( nUst ( t ) ) ■ nUst ( t ) + bb ■ ( nUst ( t ) - n ). (6) The first equation (5) takes into consideration the effect of diffferent hydraulic resistance while moving ahead or astern. It is done in the form of MathCad function for resistance coefficient: Kv KV 0 if v > 0 kV(v) : = Kv KV0 - 1.2 otherwise . return Kv (7) All the constants are to be established at the beginning of the integration process. Their values are given in the groups of equations (8) where the values of all the constants correlate with SI system, except for MM mass which is traditionally presented in tonnes. T 0 = 1200 n 0 = 120/60 v 0 = 4 D = 6.1 p = 1.025 a a = 2 100 bb = 100 MM = 24 000 J J = 10 000 (8) kT = 1 kQ = 1 0 . = kT • TT ( v 0, n 0) kM . = kQ ■QQ ( v 0, n 0) v 0 • Iv0| Mdv (0, n 0) Integration of the equations' system The solution of the system (4) is performed using a built-in function rkfixed() of MathCad and is shown in Fig. 4. In two initial lines, the right-hand members of the differential equations of v and n system are repeated. In MathCad syntax, these variables become components of a column vector that is shown in a transposed form. The number of integration steps is chosen to be 2 048 = 2 11 so that the fast Fourier transform (FFT) can be used later. The vector of derivatives is given by the expression P (t, y), where the components are represented by the 569