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Experimental and dynamic system simulation and optimization of a centrifugal pump-coupling-engine system. Part 2: System design F. Jiménez Espadafor *, J. Becerra Villanueva, M. Torres García, E. Carvajal Trujillo Departamento Ingeniería Energética, Universidad de Sevilla, Camino de los Descubrimientos S/N, 41092 Sevilla, Spain

a r t i c l e

i n f o

Article history: Received 22 June 2009 Received in revised form 23 June 2010 Accepted 25 June 2010 Available online 30 June 2010 Keywords: Pump failure Centrifugal pumps Torsional dynamics

a b s t r a c t A comprehensive theoretical non-linear torsional dynamic model of a pump-couplingengine assembly has been constructed, with some of the system characteristics evaluated through the ﬁnite element method. Coupling elements have non-constant stiffness, which makes the dynamic system non-linear. The resulting model provided information on the torsional vibration response of the system and therefore allowed analysis of the effect of different designs on the torsional system dynamics. Model outputs have been compared to experimental test rig results where system rotational inertia and four different couplings have been studied. The main aim of the work was to improve system design in order to guarantee reliable operation. The model facilitated a dynamic characterization of the system and has been used as predictive tool for subsequent design. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Torsional dynamic models have a large number of applications related to design, optimization and diagnosis [1–5] of rotating systems. The main parts of the irrigation system modeled in this paper are engine, coupling, shaft and pump impellers. The important frequency range for system design or diagnosis is the regime that includes the lowest resonance modes. Vibrations and noise contribute to a higher frequency range, which is not considered here [6]. The assumptions that must be made when developing a model are often limited by the dependability of the type of elements, linearity and the operating range. Thus, as the level of intricacy increases, the complexity of the model also increases along with oscillations (shaft, couplings, impellers and the engine [7]). More speciﬁcally, the coupling between the engine and the pump shaft can be a major source of problems during system operation, particularly in an excited system with a four stroke four cylinder internal combustion engine. As far as elastomeric couplings are concerned, these have both energy storage and energy dissipation capability, thus minimizing the problems associated with torsional vibrations and critical speeds. However, torsional load may affect the coupling performance and therefore their torsional characteristics may differ from those inferred from a steel shaft in torsion [4,7]. This paper concerns the development of a non-linear dynamic model and its validation for the prediction of system performance, which allows the model to be considered as a tool for improving design. 2. Torsional dynamic model: non-excited system Previous results suggest that the system operates close to the ﬁrst system resonance and, in addition, coupling does not provide sufﬁcient damping to the system. Therefore, most engine torque oscillations are transmitted to the pump shaft. In * Corresponding author. Address: Departamento Ingeniería Energética, Escuela Superior de Ingenieros, Camino de los Descubrimientos S/N, 41092 Sevilla, Spain. Tel.: +34 95 448 72 45; fax: +34 95 448 72 43. E-mail address: [email protected] (F.J. Espadafor). 1350-6307/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2010.06.006

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order to obtain a reliable system design at minimum cost, two different alternatives were addressed: (a) reduction of the coupling stiffness and (b) changes in engine inertia. The linear model developed in Part 1 was used to estimate the effect that changing the coupling stiffness had on the system resonance. Data for the original PV60 coupling are shown in Table 1 along with those for three new couplings (denoted PV50, EST3 and M4), with the design data being congruent with engine mean torque when the system is under nominal operational conditions. Stiffness is considered constant and was taken as the value provided by the manufacturer; the resulting ﬁrst three system resonances are given in Table 1. It can be observed that the effect of diminishing the coupling stiffness was to reduce system resonances, mainly the ﬁrst resonance, with the ﬁrst natural frequency moving away from the main engine torque frequency (2 2800 = 5600 rpm). The mode shapes of the ﬁrst and second natural system frequencies are shown in Fig. 1. Modal amplitude differences between degrees of freedom 1 and 2 represent coupling torsion and it can be seen that, for the ﬁrst natural frequency, the torsion increases as the coupling stiffness is reduced. In addition, torsion between degree of freedom 2 and the rest of the DOF is reduced. These observations suggest that by reducing coupling torsional stiffness, torque engine oscillations are ﬁltered out and are transmitted to the pump shaft with lower amplitude. The increase in torsion between DOF 1 and 2 suggests a reduction in the expected coupling life, although this effect can be controlled by improving coupling design.

Table 1 Torsional stiffness of the dynamic model and the ﬁrst three natural frequencies. Coupling

PV60 PV50 EST3 M4

Torsional stiffness (Nm/rad)

10,000 4000 3000 1250

Natural frequencies (rpm) Ist

IInd

IIIrd

5220 3780 3060 2220

21,120 19,500 19,020 18,600

31,560 31,140 31,020 30,960

Fig. 1. Modal amplitude of ﬁrst and second system natural frequencies.

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For the second natural frequency, reducing coupling stiffness had only a minor effect on the modal shape. Increases in engine inertia led to reductions in the lowest system resonance frequencies. This effect is discussed in the next section. 3. Torsional dynamic model of the pump – engine assembly: excited system 3.1. System model The system analysis described in Section 2 was unable to predict the system response for the excited system or for the non-linear system. In order to quantify system output from changes in system design, an excited non-linear torsional lumped dynamic model was developed with eight degrees of freedom (DOF) corresponding to the six impellers, the coupling and the engine. A schematic representation of the system considered is shown in Fig. 2. For DOF 3–8, rotational inertia is that of the impeller (which is full of water) plus that corresponding to the shaft section with the keyway; the impeller inertia was evaluated experimentally. The torsional stiffness of these DOF is that of each shaft section coupled through the keyway to every impeller. This stiffness was evaluated through a ﬁnite element code; the ﬁnite element grid of the shaft used for torsional stiffness evaluation is presented in Part 1 of this paper. DOF 2, see Fig. 2, represents the coupling. The stiffness of this element, which is made of synthetic rubber, does not usually have linear behavior [8]. Therefore, the data supplied by the manufacturer were not considered reliable and an experimental test rig was prepared. The measured stiffness of the original coupling, PV60, and the other three couplings tested as a function of applied torque are represented in Fig. 3. Raw data and polynomial ﬁtting for each coupling are shown. It can be appreciated that the stiffness changes with the torque applied in a non-linear manner. The stiffness of couplings PV60, PV50 and EST3 decreased with applied torque (elastomeric material work shear) and that of M4 increased (elastomeric material work compressed). DOF 1, see Fig. 2, represents the engine. The crankshaft is assumed to have much higher stiffness than the rest of the system and was therefore modeled as a rigid body. Engine inertia has two different components: one is constant (crankshaft,

Fig. 2. Schematic representation of the eight-DOF torsional dynamic system.

Fig. 3. Torsional stiffness of several couplings: PV60 (original), PV50, EST3 and M4.

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ﬂywheel and half coupling inertia) and the other changes with crankshaft angle and is due to the alternative movement of pistons and connecting rods. The seal elements (see Fig. 2) dissipate energy and therefore develop some damping on system oscillations. The main contribution to the torque on each crank is due to the pressure developed on the top of the piston of every cylinder in the engine, see characteristics in Table 2. This process was simulated through the development of a combustion model [9] compatible with engine power at nominal rating, 25 kW at 3000 rpm (see engine characteristics in Table 2). The indicated engine torque estimated with the procedure described in [10] for continuous and intermittent power at 2800 rpm is represented in Fig. 4; harmonic decomposition reveals the fourth engine harmonic as the most powerful. Pump

Table 2 Main engine characteristics. No. of Cylinders

4

Engine cycle Bore stroke Intake system Fuel

Four stroke compression ignition 78.0 78.4 Natural aspirated Diesel fuel

Output Net intermittent Net continuous

mm

kW/rpm kW/rpm

25.0/3000 21.7/3000

Fig. 4. Indicated engine torque on the basis of crankshaft angle and frequency.

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torque was estimated from characteristic pump curves. In addition, mechanical losses produced by friction [11] were considered. The torsional lumped system model was formulated as follows:

_ xÞÞ€h þ C h_ þ KðhÞh ¼ M fric ðh; h; _ l; xÞ þ M indicated þ M pump ðJ þ J reciproc: ðh; h;

ð1Þ

where J (kg m2): inertia matrix associated with elements that rotate in the system. All degrees of freedom have a term in this matrix. Jreciproc. (kg m2): reciprocating engine inertia changes with shaft angle and speed. x: engine dimensions. € h (rad/s2): acceleration vector for each degree of freedom. K (Nm/rad): stiffness matrix. DOF 3–8 have constant stiffness. Stiffness of DOF 2 changes with applied torque. h (rad): angular displacement vector for each degree of freedom. C (Nm s/rad): damping matrix. h_ (rad/s): velocity vector for each DOF. l (N s/m2): lubricant oil engine viscosity. Mfric (Nm): engine mechanical losses. Mpump (Nm): pump torque. Mindicated (Nm): indicated engine torque. Solving model Eq. (1) gives the instantaneous angular oscillation of each degree of freedom. 3.2. Model outputs The effect that changing engine inertia has on system dynamics was evaluated by solving the differential equation system deﬁned by Eq. (1). The instantaneous speed differences between adjacent DOF when the system is excited with the indicated engine torque (mean torque 68 Nm, see Fig. 4) and engine inertia is doubled are shown in Fig. 5. This change can be easily achieved by adding a special steel disc to the engine ﬂywheel. As can be observed, increasing engine inertia leads to decreases in the speed oscillations for all DOF. Due to the relationship between torque and speed oscillations demonstrated in Part 1, this represents a method for diminishing failure potential. The combined effect of increasing engine inertia and changing coupling stiffness in the system was analyzed at different engine speeds. The relative maximum speed oscillations between degrees of freedom 1 and 2 are shown in Fig. 6. With reference to doubling engine inertia, it can be observed that any coupling reduces maximum amplitude oscillations and the resonance also moves away from the nominal engine speed, 2800 rpm. The effect of changing the coupling design also reduces both amplitude oscillation and resonance frequency, although the system is more sensitive to changes in coupling

Fig. 5. Instantaneous DOF differences between adjacent DOF for 1 engine inertia and 2 engine inertia. Mean indicated torque 68 Nm.

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Fig. 6. Maximum speed oscillations between DOF 1 and 2 for different engine inertia and coupling designs. Engine speed range 700–3000 rpm.

stiffness. This result is also relevant because during pump start up the operation system always evolves from idle speed, 700 rpm, to nominal engine speed, 2800 rpm. A reduction in the coupling stiffness and increase in the engine inertia diminishes both speed oscillation amplitude and the number of system resonance frequencies and therefore contributes to extending the pump life. 4. Experimental study of system design modiﬁcations on pump pressure and speed The model output predictions were analyzed through experiments with the test rig presented in Part 1. The instantaneous pump outlet pressures on the basis of shaft angle, with the engine running at 2800 rpm, during more than ﬁve engine cycles (one engine cycle = 720 shaft angle) for different system conﬁgurations are represented in Fig. 7 (discharge valve closed) and

Fig. 7. Instantaneous discharge pump pressure. Valve closed.

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Fig. 8 (discharge valve open). In these ﬁgures, curves denoted M4+V signify coupling of M4 with an additional inertia disc bolted to the engine in DOF 1. It can clearly be observed in Fig. 7 that pressure oscillations diminish with coupling stiffness, although for the PV50 coupling the main engine torque frequency (2) can still be observed. In the case where the outlet valve is open, Fig. 8, this effect is not clearly observed. The spectral densities of the pump discharge pressure corresponding to Figs. 7 and 8 are shown in Figs. 9 and 10, respectively. In Fig. 9 there is a huge reduction in the energy of the pressure signal in the frequency band of the main engine torque, 2 between PV60 and the other three couplings tested. The amplitude differences between the other couplings are not so high; this ﬁnding is compatible with model results shown in Fig. 6. When the outlet valve is open, Fig. 10, the outlet pressure energy amplitude is reduced signiﬁcantly because part of the

Fig. 8. Instantaneous discharge pump pressure. Valve open.

Fig. 9. Discharge pressure spectrum for closed valve.

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Fig. 10. Discharge pressure spectrum for open valve.

Fig. 11. Speed oscillations at DOF 2 when discharge valve pressure is closed for couplings PV60 and PV50.

Fig. 12. Speed oscillation spectrum at DOF 2 when discharge valve pressure is open for all couplings tested.

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energy related to shaft oscillations is also transmitted to mass ﬂow. However, for the PV60 coupling, the main torque engine frequency and also the amplitude reduction for the rest of the couplings can be clearly identiﬁed. The system speed oscillations at DOF 2 for couplings PV60 and PV50 when the discharge valve is closed are represented in Fig. 11; the diminution in maximum amplitude as the coupling stiffness is reduced can be clearly observed. A spectrum of speed at DOF 2 for all the couplings tested is shown in Fig. 12. The huge reduction in speed oscillations at the main engine torque frequency and twice this frequency can be clearly identiﬁed; therefore, reduction of the coupling stiffness reduced both discharge pressure and shaft speed oscillations. 5. Conclusions According to modeling and experimental measurements, the system torsional dynamic for the ﬁrst resonance frequency depends very much on coupling stiffness. A reduction of the coupling stiffness leads to a decrease in the ﬁrst torsional natural frequency and therefore PV50, EST3, M4 and M4+V move the ﬁrst resonance away from the nominal engine speed, 2800 rpm. Coupling M4 reduces velocity and pressure oscillations more than the others. The conﬁguration ﬁnally adopted involved the use of coupling M4 with an inertia wheel (double the original engine inertia), which reduces amplitude oscillations under operational conditions and during the system start up. This new conﬁguration was installed in the irrigation ﬂeet and further damage was not observed. References [1] Newland DE. Mechanical vibration analysis and computation. New York: Wiley; 1989. [2] Kiencke U, Nielse L. Automotive control systems. Berlin: Springer; 2000. [3] Connolly FT, Yagle AE. Modeling and identiﬁcation of the combustion pressure in internal combustion engines: II – experimental results. Mech Sys Signal Process 1994;8:1–19. [4] Zweiri YH, Whidborne JF, Seneviratne LD. Detailed analytical model of a single-cylinder diesel engine in the crank angle domain. Proc Inst Mech Eng 2001;215:1197–216. [5] Schwibinger P, Nordman R. Torsional systems: vibration response by means of modal analysis. Rotordynamics. 2: problems in turbomachinery. New York: Springer Verlag; 1988. [6] Cempel C. Vibroacoustic condition monitoring. Chichester: Ellis Horwood Limited; 1991. [7] Martyr AJ, Plint MA. Engine testing. third ed. London: Butterworth; 2007. [8] Tadeo AT, Cavalca KL. A comparison of ﬂexible coupling models for updating rotating machinery response. J Braz Soc Mech Sci Eng 2003;25:235–46. [9] Rahnejat H. Multi-body dynamics. UK: Professional Engineering Publishing; 1998. [10] Genta G. Vibration of structures and machines. New York: Springer Verlag; 1995. [11] Rezeka SF, Henein NH. A new approach to evaluate instantaneous friction and its components in internal combustion engines. SAE paper 840179; 1985.