A New Approach for Marine Propulsion Shafting Design

A New Approach for Marine Propulsion Shafting Design M. A. Mosaad , M. Mosleh, H. El-Kilani , W. Yehia Department of Naval Architecture and Marine Engineering, Suez Canal University Abstract In this paper, a new design approach for propulsion shafting system is presented. The aim is to improve the dynamic response of the shafting system concerning torsional vibrations. The approach result in raising the permissible stress limits set by the Rules of the Classification Societies, and reduces the shafting response due to engine excitation without any barred speed range. A computer program to calculate the vibration response of torsional stresses (VIBRTS) using the proposed method has been developed. A numerical example of a 2-stroke, 6 cylinder marine diesel engine is investigated and the results are compared with those obtained from the basic design approaches based on the flexible shafting system, and the rigid shafting system. Keywords: torsional vibration - propulsion shafting - flexible shafting system 1. Introduction Propulsion shafting deserves special consideration during design, manufacturing and later, operation. The design of a shafting system is, by necessity, an iterative process because the various system design parameters are, to some extent, mutually dependent [1]. The dynamic behavior of the propulsion shafting is the major factor influencing its overall reliability and efficiency. Three kinds of shafting vibrations are identified; namely, torsional, axial and lateral vibrations, each with specific sources, characteristics and consequences. In general, axial and/or lateral vibrations are less severe than torsional vibrations that are the most important factor influencing shafting design [2]. Torsional vibrations may lead to fatigue failure in the shafting system and can adversely affect the efficiency of operating machinery and auxiliaries. They are due to the periodical vibrations of the masses on the shaft in the plane of rotation, i.e., the periodical changes of the tangential forces acting on the crankshaft which result from torsional resistance of the shaft. Vibration characteristics have to be satisfactory with respect to vibration limits set by the IACS requirements [3]; it is imperative to check the propulsion shafting of a main engine for torsional vibration. However the given formulae take into account static loads only and, thus, may not be sufficient for proper shafting design. Many studies had been carried out to improve the situation in order to obtain acceptable torsional vibration stresses of the system [4-7]. The most common possible approaches are the flexible shafting system [6], and the rigid shafting system [5]. These approaches are reviewed in this paper and a new approach which combines the benefits of the two preceding methods is then presented. The basic core of this approach is increasing the shafting diameter designed by the classification societies recommended equation by a proposed correction factor. This factor depends upon the main engine operating cycles, mean effective pressure, and the number of cylinders. 1 2. Analysis of Torsional Response The equations of motions of torsional vibration systems may be written as %% - Cs% - K s ? T (t) Js .........................(1) K is the stiffness matrix, J the inertia matrix, C the matrix of external and internal damping, s the angular displacement vector and T(t) the external excitation. The first step in determining the torsional response is to calculate the torsional natural frequency of the system conforming to the homogeneous form of Equation (1). This may be carried out by means of the Transfer Matrix method (Holzer) [1], that requires the stiffness and mass inertia of the shaft components being analyzed (referred to as masselastic data)[10]. The system is thus modeled as a multi-rotor system as, for instance, that shown in Fig.(1)[2]. The model is also essential to define the dynamic magnifier of the engine and propeller required to determine the torsional vibratory stresses within the system. components resulting from the effect of inertia of the moving parts. A tangential effort caused by the cylinder gas pressure, TEg can be represented by a Fourier series consisting of a constant term and a series of harmonically varying terms having orders of 1, 2, 3, etc. The constant term is the mean tangential effort, Tm and does not excite torsional vibration. The harmonically varying terms, however, are the principal sources torsional vibration and do not contribute to the useful work output of the engine. A Fourier series representing the gas–pressure tangential efforts diagram illustrated by Fig.(2) can be written as [1, 2]: TEg ? Tm - Â (An sin nl - B n cos nl ) ? Tm - Â Tn sin(nl - d n ) … where T m? tan d n ? pm ri Tn ? An2 - B n2 . ….(2) Bn An 70 Tangential Effort, kg/cm 2 60 50 40 30 20 Mean Tangential effort Tm 10 0 -10 -20 -30 -40 Fig.1. Torsional model of a propulsion shafting system 2.1 Engine torsional excitation The torque excitation in marine diesel engines is due to two sources of variation, namely, the varying piston gas pressure and the varying inertia loads due to the cylinder reciprocating masses. A tangential–effort diagram can be developed from a cylinder gas pressure indicator card where the cylinder pressure is related to the piston crankpin angle [1]. This curve is then modified by harmonic 0 60 120 180 240 Crank Angle, deg. 300 360 Fig.1. Tangential-effort diagram with harmonic components [1] In general, in elementary calculations of shaft torsional vibrations, the inertia force harmonics are completely ignored [8]. 2.2 Propeller torsional excitation The pattern of fluid flow through the propeller can be reduced to a series of harmonic components. The frequency 2 of excitation applied to the propeller and the shaftline can be calculated as follows [9]: f ? nP . NP ……………………….(3) In general the propeller order of excitation in single-screw installation is not the same as in twin-screw installation. It also depends on the existence of odd or even number of propeller blades ZP: nP = ZP For twin-screw installation with odd or even number of blades nP = ZP For single-screw installation with an even number of blades nP =2 ZP For single-screw installation with an odd number of blades As a broad guideline, the number of propeller blades should be selected to keep the propeller excitation frequency and excitation order away from combination with engine excitation frequency and order [10]. 3. Shaft Design According to the Classification Societies [3], the minimum shaft diameters are determined according to the shaft's intended service, transmitted power, shaft speed and applied material. The diameter of the intermediate or propeller shaft is not to be less than determined by the following formula: d ? FK 3 P Ã 560 Ô Õ ……………(4) Ä MCR ÄÅ u u - 160 ÕÖ Without exception, all Classification Societies Rules design the shaftline diameters based on the transmitted power originated from mean torque, i.e. mean indicated effective pressure. The Rules diameter neglect alternating loads and vibration behavior of the shafting system. The series of harmonically varying orders of exciting torques that are superimposed with the mean torque are neglected by the Rules formula given by (3). (2) This situation may be overcome by adding a proposed correction factor (g) for the effect of the altering loads and exciting torque according to each propulsion engine characteristics, namely, the mean indicated pressure, operating cycles, and the number of engine cylinders. This factor can be estimated as c ?3 1- Tng Tm ………………… ………(5) where Tng is the resultant value of the harmonic component of gas pressure tangential effort curve for nth order of excitation for the marine internal combustion engine and can be calculated from ÊÇ 140.62 Ç10.194 pm for 2 - stroke engines -È ÍÈ 3 Ù 3Ù ÍÍÉ (50 / n ) - n Ú É 20 - n Ú Tng ? Ë ÍÇ 70.31 Ç 5.097 pm for 4 - stroke engines ÍÈ Ù3 ÍÌÉ (50 / n ) - n3 Ú ÈÉ 18 - n ÙÚ ………….(6)[9] It is considered that, the major order of excitations are those which are integer multipliers of the number of cylinders; in the case of two-stroke cycle engines these are Z, 2Z, 3Z, etc. and in the case of four-stroke cycle engines they are ½Z, Z, 1½Z, etc.. Figs.3 and 4 introduce the value of the correction factor *g) based on the mean indicated pressure pm and number of engine cylinders Z for two and four-stroke diesel engines. 3 Fig.3. Correction factor (g) of 2-stroke engines vibration stress limits are defined as follows: “In no part of the propulsion system may the altering torsional vibration stresses exceed the value of v c for continuous operation and v t for transient running”. For continuous operation the permissible stresses due to altering torsional vibration are not to exceed the values given by the following formula * Ê u u - 160 2 Í‒ 18 ck c D 3 / 2n vc ? Ë u - 160 Í ‒ u ck c D 1.38 18 Ì (7) c D ? 0.35 - 0.93d /0.2 + for for n > 0.9 0.9 ~ n ~ 1.05 …………….(8) For transient running, the permissible stresses due to altering torsional vibration are not in any case, to exceed the value given by Fig.4. Correction factor (g) of 4-stroke engines Thus, the minimum shaft diameter according to the Rules can be modified considering the effect of the exciting torque as: d ? c FK 3 P Ã 560 Ô Ä Õ …………..(7) MCR ÄÅ u u - 160 ÕÖ where g is the proposed correction factor. 4. Stress consideration Classification societies prescribe the amount of allowable torsional vibration stresses for engine crankshafts, intermediate shafts and propeller shafts. These stress limits are determined by the purpose, shape, material selected, dimensions and intended operation of shafting. Moreover, the stress limits are not constant; instead they are a function of engine speed. According to the worldwide accepted requirements [3, 6] the torsional v t ? ‒1.7v c / c k , for n > 0.8 ……..(9) Fig.5 shows the influence of the selected shaft diameters and UTS on the permissible stress due to torsional vibrations of the shaft. It shows that at certain speed ratios, an increase in the ultimate tensile strength of the selected shafting material leads to an increase in the permissible stress for continuous operation at constant diameter. This design approach is called the flexible shafting system [6]. However, it introduces some unfavorable side effects if, for instance, the shafting diameters may be reduced below the Rules diameters resulting in high torsional stresses approaching the permissible limit[5]; moreover, the need for high quality manufacturing is essential. It is noticed that the increase in shafting diameter allows higher permissible stress for the same UTS of the shafting material. This is the rigid shafting design approach. It depends 4 on increasing rationally the shaft diameter. This will change the dynamic behavior of the whole system which results in decreasing the torsional stresses; these will be less than the permissible stress limits for continuous running in the whole engine speed range [5]. The difficulty of this approach is the time consumed in the design iterations to obtain the suitable shaft diameter of the system. Max. Continuous speed 105 rpm Mean indicated effective pressure/MCR 20 bars Firing order 1-5-3-4-2-6 Oscillating mass/cylinder 5003 kg The mass-elastic data required for the torsional model adopted (Fig.1) has been computed and is given in Table1 [10]. The original shafting design using the Rules as a guide is summarized in Table 2. Table 1: Mass-elastic data of the model Fig.5. Influence of shafting material on permissible stress limit Mass Stiffness Diameter No E l e m e n t inertia 106 cm N.m.s2 N.m/rad 1720 72 1 f l a n g e 212 1370 72 2 cylinder 1 11160 1390 3 cylinder 2 11160 72 1350 4 cylinder 3 11160 72 1380 5 cylinder 4 11160 72 11160 1440 6 cylinder 5 72 1880 7 cylinder 6 11160 72 4802 2740 8 Camshaft drive 72 80 51 9 Turning wheel 4982 162 61.6 10 f l a n g e 612 11 propeller 64800 5. The Proposed Design Approach Starting from the aforementioned approaches, good benefits can be obtained by combining the advantages of the flexible and rigid design approach. The shafting can be designed from high quality steel with a large ultimate tensile strength, considering the proposed correction for the altering loads in the Rules diameter equation. The proposed approach design is illustrated by a shaft line design of a slow speed propulsion plant with the following particulars: 2-stroke, 6-cylinders marine diesel engine Cylinder bore 600 mm Length of stroke 2400 mm Connecting rod length 2460 mm Max. continuous power output 13530 kW Table 2. The Rules shaft line design Intermediate shaft material Intermediate shaft diameter Propeller shaft material Propeller shaft diameter UTS = 400 MPa 510 mm UTS = 400 MPa 616 mm Fig.6 shows the torsional vibration analysis of the original design. The resonance vibratory stress occurs at the engine speed of 60 rpm. It is excessively high and beyond all permissible limits. The flexible shafting approach using high tensile steel with ultimate tensile strength of 600 N/mm2 is illustrated in Table 3. The corresponding torsional vibration analysis of this design is shown in Fig. 7. It shows that the excessive torsional vibration stress has been decreased with 5 decreasing of resonance speed to 50 rpm. output (Fig.8) shows that thanks to the proposed approach, the final shafting design conforms to the permissible stress level set by the IACS requirement [3] without any barred speed range. The permissible stress limits raise, the resonance speed moves far from the nominal engine speed consequently, the shafting response due to engine excitation falls, and the torsional vibration stresses reduce to lower level due to better dynamic behavior. Table 4. Proposed design Fig. 6. Torsional Stress in intermediate shaft for initial design(Table 2) Table 3 : Design by flexible sytem approach Intermediate shaft material Intermediate shaft diameter Propeller shaft material Propeller shaft diameter Intermediate shaft material Intermediate shaft diameter Propeller shaft material Propeller shaft diameter Correction factor “ ” UTS = 600 MPa 500 mm UTS = 600 MPa 560 mm 1.07 UTS = 600 MPa 460 mm UTS = 600 MPa 560 mm To Fig. apply the proposed design 7. Torsional Stress in intermediate approach, the shafting factor gapproach is simply taken shaft flexible (Table 3) from Fig.2. The shafting design is summarized in Table 4. The result of the torsional vibration analysis of this design is illustrated in Fig. 8. This analysis has been carried out by the computer program VIBRTS “Vibration Response of Torsional Stresses”[10]. The main steps of this program are given in the flow chart in Fig. 9. The Fig. 8 Torsional Stress in intermediate shaft for proposed approach (Table 4) Identification of engine data [bb, pm, MCR, L, D, R, r, wr, i, Z, firing order] Identification of torsional model data [Diameters, torsional moments of inertia, torsional stiffness, number of rotors of the shafting system] Estimation of torsional natural frequencies and mode shapes based on Holzer’s Method Calculation of specific stresses per unit amplitude for each element of the 6 flexible and rigid design approaches and overcomes their defects. This has been achieved by the use of high tensile steel and a proposed correction factor to increase the Rules shafting diameter. This factor depends directly on the characteristics of the main engine, such as the mean indicated effective pressure, the number of engine cylinders, and operating cycle. The proposed approach raises the permissible stress limits and reduces the shafting response due to engine excitation without any barred speed range. The application of this approach to a propulsion shafting of a 2-stroke, 6-cylinder marine diesel engine has resulted in a reduction of about 8% in the torsional stresses compared with the flexible approach. Nomenclature D F K L MCR cylinder bore, mm factor describing shaft service factor describing shaft design stroke length, mm Maximum continuous rating, rpm Np R P TEg Tm Fig. 9. Flow chart of VIBRTS (Vibration Response of Torsional Vibration) 6. Conclusion The problem of excessive torsional vibration stresses had not been completely solved with the flexible shafting approach that results in a barred speed range for engine operation, and the rigid approach that does not specify an upper limit to the shafting diameter. The proposed shafting design approach combines the benefits of the propeller rpm crank radius, mm transmitted power at MCR, kW tangential effort, bar constant mean tangential effort, bar Tng the nth order resultant harmonic component due to gas pressure, bar Z number of engine cylinders bb brake power , kw ck factor for different shaft design cD size factor d shaft diameter, mm i 1 for 2-stroke, 2 for 4-stroke engines r connecting rod/crank radius ratio w r reciprocating mass per cylinder, kg n Order no of engine excitation np Order no of propeller excitation pm indicated mean effective 7 g n l u vc vt pressure, bar the proposed correction factor phase angle crank angle speed ratio ultimate tensile strength of shaft material, MPa permissible stress due to torsional vibration for continuous operation, MPa permissible stress due to torsional vibration for transient running, MPa References [1] Long C.L., "Propellers, Shafting and Shafting System Vibration Analysis", in: R.L. Harrington (Ed.), "Marine Engineering" SNAME, New York [1990] [2] Magazinvoi7 G., “Shafting Vibration Primer” Technical Report, CADEA, Split, Croatia [ 2002] [3] International Association of Classification Societies, ”Requirements concerning Machinery Installations” IACS Req.2005, www.iacs.org.uk. [4] “Vibration Characteristics of Two-Stroke Low Speed Diesel Engines”, 2nd Edition, MAN B&W Diesel A/S Copenhagen [1993] [5] Magazinvoi7 G., “Shafting Design Considerations of FiveCylinder Low-Speed Propulsion Plants”, Proceeding of the 13th Symposium on Theory and Practice of Shipbuilding, Croatia, September 1-3[1998] [6] Magazinvoi7 G., “Utility of HighStrength Steel for Main Propulsion Shafting Design” Proceeding of the 9th Congress IMAM 2000, Ischia, April 2-6 [2000] [7] “An Introduction to Vibration Aspects of Two-Stroke Low Speed Diesel Engines in Ships”, 2nd Edition, MAN B&W Diesel A/S Copenhagen, [2002] [8] Jenzer J. “Some Vibration Aspects of Modern Ship Installations”, Värtsilä NSD Switzerland Ltd. [1997] [9] Wilson W. KER, "Practical Solution of Torsional Vibration Problems", Volume II, London, Champman & Hall, [1963] [10] Yehia W." Vibration Aspects of marine propulsion system ", M.Sc. , pending for discussion, Suez Canal University [2006]. 8