1. Introduction

Other than hull design, propulsion plant design is demanded to ensure reliable and efficient transport solutions for growing trades in ice-bound waters(1). Propellers, shafts, transmission systems and prime movers are the main components of ice-classed vessels. For the directly coupled propulsion system, its main advantage over other configuration is the reduction in the power transmission loss(2). Likewise, numerous researches have been focused on propeller-ice interaction. Garma, in his paper, concluded the importance of propeller geometry design in considering ice loads under extreme and transient operating conditions(3). Baik, on the other hand, presented a propeller boss slippage accident brought by impact load on propeller at sub-zero sea water condition(4). Yet, ice class challenges are still present and it is stated that not all aspects of design for cold climates are accounted for by Ice Rules(5,6).

This paper obtained theoretically the transient torsional vibration response of a low-speed two-stroke diesel engine due to the irregular ice impact torque in order to evaluate the design and reliability of a directly-coupled propulsion system for an ice-class vessel. Classification regulations on ice-propeller interaction impact torque are applied(7) whereas the Newmark numerical method was used to analyze the transient torsional vibration overall response of the propulsion system(8).

2. Ice and Propeller Impact Torque Calculation

Additional provisions on regulations for Polar vessels sailing the Baltic Sea includes the Finnish - Swedish Ice Class Rules and the Canadian Arctic Shipping Pollution Prevention Regulations which must be applied. Polar classed vessels are divided into seven (7) classifications from PC1 ~ PC7. This Polar ship classification is based on different estimation factors for the calculation of design ice loads of the propeller, i.e., the ice thickness (Hice) and the ice strength indexes for the blade ice force (Sice) and the propeller ice torque applied (Sqice). For the calculation of the maximum backward blade force (Fb) and the maximum forward blade force (Ff), this paper applied the Eqs. (1) to (4) respectively in reference of propeller diameter (D).

The backward blade force formula when D

whereas when D≥Dlim,

The diameter limit is given as Dlim=0.85·[Hice]1.4. Accordingly the forward blade force formula when D

and when D≥Dlim,

The Dlim is given as The calculation for blade spindle torque is given in Eq. (5).

The calculations for the maximum propeller ice torque applied on open propeller are shown in Eqs. (6) and (7). If D

and if D≥Dlim,

The diameter limit for the calculation of the maximum propeller ice torque is defined as Dlim = 1.81 · Hice.

Shaft line dynamic analysis of the propeller torque excitation is described by the sequence of blade impacts being half sine shape and occurring at the blade. The torque due to a single blade impact as a function of the propeller rotation angle is represented in Eqs. (8) and (9).

In Table 1, the parameters Cq and αi are shown. By taking into account the phase shift 360˚/Z, the total ice torque is attained by summing the torque of single blades. Equation (10) is the formula for obtaining the number of propeller revolutions during a milling sequence.

Table 1Cq and αi parameters

The number of impact is implied as Z·NQ.

The calculated torque variation in relation to the 6th order torsional vibration resonance of PL1 ice class subject vessel propulsion shafting at 52 RPM. is shown in Fig. 1.

Fig. 1Three cases of total ice impact torque for 4 blade propeller and 52 RPM of PC1 ice class

Propeller assembly by shrink fitting using hydraulic equipment (fixed-pitch propeller blade) or bolts (variable pitch propeller blade) multiplied by safety factor should be greater than any torque fluctuations.

Therefore, the torque required for assembly in the full speed range (Tc1) must satisfy Eq. (11) in the friction force 35 ℃. The minimum value for Tc1 is 2.8·T0.

Below is the application of ice factor due to impact load, as a minimum requirement of IACS (International Association of Classification Society) UR (Unified Requirements)_K, on shaft transmitted torque at maximum continuous rating.

3. Theoretical Analysis and Result

The diesel engine and propulsion system particulars of a 154 k tanker as the subject vessel of this paper are given in Table 2. The torsional vibration measurement was carried out and the transient torsional vibration was analyzed in time domain as represented by the mass-spring system consisting of twelve (12) masses in Fig. 2. The propulsion shafting system excitation analysis was divided in two parts: the ice-propeller impact torque and the diesel engine excitation torque attributable to the cylinder gas pressure and reciprocating mass force of piston. Likewise, the analysis method was done utilizing a specialized software developed by the authors(9). The excitation torque curve at critical speed 52 RPM of 6th order torsional vibration resonance, as shown in Fig. 3, of a six-cylinder engine is relatively higher when compared to Fig. 1 case 2 ice impact torque. Figure 4 illustrates the vibratory torque due to engine excitation at 52 RPM whereas Fig. 5 is the vibratory torque due ice-propeller impact torque. The total vibratory torque due to ice-propeller impact torque and engine excitation of PC1 ice class vessel at 52 RPM critical is shown in Fig. 6. In this case, the engine total vibration torque was analyzed in the time domain. However, the results have shown no significant differences with the results using the frequency domain order analysis. As such, actual vibration measurement was done during sea trial and the result confirms the vibratory resonance to be slightly higher than the theoretical analysis (Fig. 7)(10). The torsional vibration result confirms the full compliance of the design and assembly in accordance with the classification rule. Engine and propeller blade excitation torques are relatively low and higher resonance does not occur. However, the maximum and minimum excitation torque acting simultaneously in Fig. 6 is significantly higher compared to Fig. 4. This analysis applied the Classification Society rules on vibration response of the system during the initial phase but it should be considered in the subsequent phase that the occurrence of the ice impact torque will likely increase the actual vibration value. Hence, the result of this theoretical analysis considers simultaneously the torsional vibratory torque of the engine and the ice impact torque.

Table 2Specification for ship and propulsion shafting

Fig. 2Mass-spring system for torsional vibration analysis

Fig. 3Excitation torque due to gas pressure of cylinder and reciprocation mass of piston at the critical speed of engine cylinder number (1 node 6th order and 52 RPM)

Fig. 4Vibratory torque of intermediate shaft due to engine excitation at the critical speed of engine cylinder number (1 node 6th order and 52 RPM)

Fig. 5Vibratory torque of intermediate shaft due to ice impact torque of PC1 class at the critical speed of engine cylinder number (1 node 6th order and 52 RPM)

Fig. 6Vibratory torque of intermediate shaft due to engine excitation and ice impact torque of PC1 ice class at the critical speed of engine cylinder number (1 node 6th order and 52 RPM)

Fig. 7Measured torsional vibration stress at intermediate shaft (scale : N/mm2 = 44.57 kN-m)

In Fig. 8 above, the excitation torque curve of the propeller blades at critical speed 78.4 RPM is shown and the value was twice as that in comparison with Fig. 3. Figure 9 shows the vibratory torque mainly due to the engine excitation to be lower than the resonance limit. The lowest e PC7 ice class grade was applied on the calculation of the ice impact vibratory torque of the propeller as seen in Fig. 10. The vibratory torque was seen to significantly increase compared to Fig. 5. Thus, the collected ice impact torque data information operating at prohibited range should be set, with the scope on the safety of the whole system and not only of the propulsion shaft. Figure 11 illustrates the increased total vibratory torque reflecting both the engine torque and the ice impact torque in comparison with Fig. 11. Hence, if the results of these ice vessels ice impact torque will affect the strength of the propeller, the propulsion and ice impact torque being far from the prohibited range should be taken into account for propeller shrink fitting assembly and should be considered both in the design and assembly stage.

Fig. 8Excitation torque due to gas pressure of cylinder and reciprocation mass of piston at the critical speed of propeller blade (1 node 4th order and 78.4 RPM)

Fig. 9Vibratory torque due to engine excitation at the critical speed of propeller blade number (1 node 4th order and 78.4 RPM)

Fig. 10Vibratory torque due to ice impact torque of polar class PC7 at the critical speed of propeller blade number (1 node 4th order and 78.4 RPM)

Fig. 11Vibratory torque due to engine excitation and ice impact torque of PC7 ice class at the critical speed of propeller blade number (1 node 4th order and 78.4 RPM)

3. Conclusion

In this paper, a theoretical analysis was performed to obtain the transient torsional vibration response due to ice impact torque in accordance with the Classification Rules. This was carried out in order to review the safety of propulsion systems directly coupled to main engine. The results are as follows:

(1) For low-speed two-stroke diesel engine having seven (7) cylinders or less, engine operation at barred speed range (critical speed) is prohibited due to torsional vibration limits. This vibratory torque increases further if the ice impact torque will act simultaneously on the propulsion shafting. The engine and propulsion excitation torque characteristics should be considered at the same time. In particular, the vibratory torque increased significantly when the engine is passing in the reverse rotation owing to the backward force on the propeller compared to the forward force. This force is the opposite action

(2) The new operation guide on ice impact torque is needed to protect ice-classed vessels. Even applying the lowest ice class grade factor, increased torsional vibratory torque on the resonance range acting on the propeller shaft and engine was confirmed.

Considering the IACS regulation and the Korean Register, it has been recommended to avoid resonance and propeller blades resonance should be within ±20 % range of the maximum continuous operating speed.

(3) For ice class propulsion systems directly coupled to a two-stroke low-speed marine engine, the resonance due to engine's number of cylinders should be considered alongside with the number of propeller blades. A smaller 8-cylinder engine can be employed in view of lower natural frequency of torsional vibration and considered advantageous to the main propulsion design and safety.

Nomenclature

c0.7 : Propeller blade chord length(m) at 0.7R D : Propeller diameter(m) d : Propeller hub diameter(m) EAR : Expanded propeller blade area ratio Fb : Maximum backward blade force(kN) Ff : Maximum forward blade force(kN) Hice : Thickness for machinery strength design(m) KAice : Application factor due to ice shock lim : Limit n : Rotational speed at bollard condition(r/s) P0.7 : Propeller pitch at 0.7R(m) Q : Torque(kNm) R : Propeller radius S : Strength index t0.7 : Maximum thickness at 0.7R T : Torque T0 : Maximum continuous torque(kN·m) Z : Number of propeller blades