Chapter 4 TORSIONAL VIBRATION ANALYSIS AND EXPERIMENTAL INVESTIGATIONS

Transcription

1 Chapter 4 TORSIONAL VIBRATION ANALYSIS AND EXPERIMENTAL INVESTIGATIONS In this chapter computerized Holzer method [Nestrides, 1958, B. Challen, 1999] is developed to determine the natural frequencies of torsional vibrations, mode shapes, critical speed and stresses of six cylinder diesel engines. In order to perform torsional vibration analysis modeling of reciprocatng deisel engine is essential. Finite element analysis is performed to study the effect of inertia forces in more than one cylinder. Experimental work is carried out on two six cylinder diesel engines to investigate the effect of misfuel manufactured by Kirloskar Oil Engines Ltd, Pune and Greaves Cotton Ltd, Pune. This chapter is grouped in the following catagories: 1. Modeling of Reciprocatng Deisel Engine 2. Overview on different methods of torsional vibration analysis 3. Torsional vibration analysis of six cylinder diesel engines 4. Harmonic frequency analysis for detection of imbalance 5. Method for detecting misfuel in six cylinder engine 4.1 Modeling of Reciprocatng Deisel Engine In order to analyze the cylinder-wise torque contribution the applied torque on the flywheel should be reconstructed using an appropriate engine model. The main applications for medium-speed reciprocating diesel engines are automotive and power generation. While power systems always include an engine, a flexible coupling and a generator typically connected to an infinite bus. The proposed Cylinder (Power) imbalance detecting and correcting method will be applicable to power generation applications.in this work the focus is on an engine-generator set and the engine model is developed for this case. Modeling of a Six-cylinder Engine: The dynamic environment in which the power plant exists is somewhat different from that of automotive applications. As this diesel engine balancing has relevance on the torsional vibration control problem due engine speed variation, hence it is investigated in more detail. An engine-generator set can be modeled in terms of a single-machine infinite bus system [Sadikovic, et al., 2003; Kundur, 1994], which includes the dynamic influence of the transformer, transmission lines and the grid [Kundur, 1994], as shown Fig According to Kundur [1994] the dynamics between the generator rotor and the grid imposed by the transformer and the transmission lines can be represented by resistance (R e ) and impedance 31

2 (X e ) as shown in Fig Since it is in practice impossible to induce other frequencies than the nominal frequency of the grid, all torsional oscillations will have a node in the grid, i.e. the grid can be represented as a wall in a lumped-mass system.the system considered here consists of a six cylinder diesel engine, a flexible coupling and a generator (G) connected to an infinite bus. It is assumed that the engine is operated at the constant speed of 1500 rpm. Six Cylinder Diesel Engine G E t R e + jx e Fig. 4.1 Schematic diagram of an engine-generator set connected to an infinite bus E B Infinite Bus Fig.4.2 illustrate an equivalent mass-elastic model of a four stroke six cylinder inline diesel engine system model in which the engine is coupled to drive a generator connected to an electric grid. There is a flywheel on the engine shaft. The main purpose of the flywheel is to even the rotational speed of the engine shaft. The engine shaft and the generator shaft are interconnected through a coupling. The connection between generator and the electric grid is illustrated as a dynamic link. The circles in Fig.4.2 represent masses rotating in relation to the shaft. For example, the mass of the piston/crank mechanism of each cylinder is illustrated by circles. The fixed parts of the model, such as the electric grid are shown by diagonal lines. Dampings are represented by the box/plate symbol. The coupling between the generator and the engine is flexible. Cyl.6 Engine Cyl.1 Flywheel Alternator Pulley K1 K2 C 1 C 2 Fig.4.2 Mass-elastic model of a 6-cylinder engine genset connected to an infinite bus As the flexible coupling is significantly softer than the other components of the crankshaft system, the first torsional node is located in the flexible coupling. This means that the engine 32

3 load model includes an additional mass which describes the influences of the load. In Fig.4.3 where φ 1 and φ 2 represent the angular deflections of the engine and generator, K 1 and C 1 are determined by the flexible coupling and K 2 and C 2 are given by the dynamics between the generator rotor and grid. φ 1 φ 2 K 1.K 2 C 1 C 2 Fig. 4.3 Equivalent system of a medium speed genset Flexible couplings have typically very nonlinear characteristics, where the stiffness and damping are strongly dependent on the vibration frequency. Since estimation of torsional vibration levels is vital in the design of power systems, manufacturers often provide the stiffness and damping values of flexible couplings The assumption that the engine is decoupled from the load is usually valid in automotive applications, since the nominal rotational speed of the engine is typically significantly higher than the first natural frequencies of the crankshaft. This is, however, usually not the case with medium-speed diesel engines for power plant applications. As the rotational speed for these engines is usually in the range rpm, the lowest orders of the tangential torque are close to the first natural frequencies of the rotating shaft system. The implication is that the dynamics of the engine are significantly affected by the rotational behaviour of the rotor shaft, which should therefore be taken into account in the engine model. It has been shown that, given a rigid crankshaft, an engine model comprising of two masses and an ideal shaft with a given stiffness and damping represents medium-speed power engines sufficiently well for the purpose of balancing the torque contributions. For engine-generator set where the generator is coupled via a transformer to an infinite bus, the engine model is supplemented with an additional ideal shaft between the generator and infinite bus representing the dynamics between the rotor and bus, Fig The engine-generator set depicted in Fig.4.3 is described by the Eq. 4.1 and Eq. 4.2 as; 1 dj ( ϕ ) J ( ϕ )&& ϕ D & ϕ + C (& ϕ & ϕ ) + K ( ϕ ϕ ) = T dϕ i [4.1].. 33

4 J && ϕ + C & ϕ + K ϕ + C ( & ϕ & ϕ ) + K ( ϕ ϕ ) = T L [4.2] Where J 1 is the sum ofmoments of inertia on engine side and J 2 is the moments of inertia of the load side; K 1 and K 2 are the stiffness between the engine and load and the load and grid, respectively; C 1 and C 2 are the viscous damping between engine and the rotor and the rotor and grid, respectively; D 1 is the absolute damping of the engine; ϕ&&, ϕ&, ϕ are the angular acceleration, speed and position of the load respectively and T L the external load torque affecting the load side of the flexible coupling. Note that the parameters K 1 and C 1 of the flexible coupling usually depend on the nominal torque level or the vibratory frequency, but this dependency has been suppressed in the equations. For the engine model in Fig. 4.2, the oscillating torque ΔT applied on the flywheel is given by Eq.4.3; J && ϕ + D & ϕ + C ( & ϕ & ϕ ) + K ( ϕ ϕ ) = ΔT [4.3]... 34

5 4.2 Overview on Different Methods of Torsional Vibration Analysis Over the last two decades, many discretization techniques have been developed with the aim of replacing the equation of motion consisting of partial derivative differential equations (with derivatives with respect to time and space coordinates) with a set of linear ordinary differential equations containing only derivatives with respect to time. The resulting set of equations, generally of the second order, is of the same type as seen for discrete systems (hence the term discretization). The discretization techniques extensively used for torsional vibration analysis are as follows; Assumed-Modes Methods, Lumped-Parameters Method Methods based on Transfer Matrices Finite Element Method Dynamic Stiffness Method Holzer s Method The first class is that of the methods in which the deformed shape of the system is assumed to be a linear combination of n known functions of the space coordinates, defined in the whole space occupied by the body. These methods could be labelled as Assumed- Modes Method, owing to the similarity of these functions, which are arbitrarily assumed, with the eigen functions (i.e., the mode shapes) of the system. The second class is that of the so called Lumped-Parameters Method. The mass of the body is lumped in a certain number of rigid bodies (sometimes simply point masses) located at given stations in the deformable body. These lumped masses are then connected by mass less fields that possess elastic, and sometimes damping, properties. Usually the properties of the fields are assumed to be uniform in space. Because the degrees of freedom of the lumped masses are used to describe the motion of the system, the model leads intuitively to a discrete system. While the mass matrix of such systems is easily obtained, it is often quite difficult to write the stiffness matrix, or alternatively, its compliance matrix. To avoid such difficulty, together with that linked to the solution of large Eigen problems, an alternative approach can be followed. Instead of dealing with the system as a whole, the study can start at a certain station and proceed station by station using the so-called transfer matrices. Methods based on Transfer Matrices were very common in the recent past, because they could be worked out with tabular manual computations or implemented on very small computers. Their limitations are now making them yield to the finite element 35

6 method. This change between the two approaches is not yet complete, and many computer codes based on transfer matrices are still in use. A separate class can be assigned to the Finite Element Method (FEM). The use of the FEM can be limited to writing the stiffness matrix to be introduced into a lumped-parameters approach; alternatively, it can be used to write the mass matrix too. In this case, the mass matrix is said to be consistent, because it is obtained in the same way as the stiffness matrix.although often considered a separate approach, the Dynamic Stiffness Method will be regarded here as a particular form of the FEM in which the shape functions are obtained from the actual deflected shape in free vibration. The facts that it can be used only for free vibration or harmonic excitation and that the dynamic stiffness matrix it yields is a function of the frequency have prevented it from becoming very popular.the FEM is gaining popularity for the unquestionable advantage of being implemented in the form of general-purpose codes, which can be used for static and dynamic analysis and interfaced with CAD and CAM codes. The Holzer s Method [details given in appendix A.1] is well known standard procedure for the calculation of natural frequencies of multi-mass systems. The analysis is generally presented and justified on a physical basis. It is useful not only for frequency determination but also for a number of other data, such as relative amplitudes, which are required in torsional vibration investigations. This method can be applied to both free and forced vibrations. Hence Holzer method is used to investgate torsional vibrations for the engines under study. 36

7 4.3 Torsional Vibration Analysis of Six Cylinder Diesel Engine Torsional vibration analysis is carried out with followingsteps to determine the natural frequencies, mode shapes, critical speed and stresses for Kirloskar and Geaves diesel engines. A. Natural frequency calculation by holzer tabulation method B. Representation of mode shapes C. Representation of phase vector diagram and calculations for critical speed D. Calculations for amplitudes E. Calculations for stresses Case Study- I. Kirloskar Six Cylinder Diesel Engine The following information pertains to the Kirloskar four strokes, six cylinder diesel engine; Engine Basic Data: a) Type: SL90 Engine-(Kirloskar Made) b) Operating speed range: 750 to 2200 RPM c) Cylinder diameter : 118mm d) Piston stroke : 135mm e) Maximum cylinder pressure (Gauge) : 165bar (at 180 from TDC) f) Cylinder air inlet pressure (Gauge) : 2.2 bar g) Compression ratio : 15.5 h) Mass of piston, piston rod & cross head : kg i) Mass of connecting rod (incl. bearing) : kg j) Length of connecting rod (between bearing) : 222.5mm k) C.G of connecting rod from large end : mm l) Compression index : 1.35 m) Expansion index : 1.35 Coupling Specifications: n) Type : Flexible coupling o) Model: VULASTIK-L-COUPLING Mass Elastic Data: Table 4.1Mass elastic system data Engine Mass-Elastic System Cylinder Mass Pulley Flywheel Inertia Kg-m Stiffness MNm/rad

8 A. Natural frequency calculation by Holzer tabulation method Table 4.2, Table 4.3, Table 4.4, Table 4.5, Table 4.6 and Table 4.7 show the calculation of natural frequency for Kirloskar engine. In Table 4.2, Table 4.3, Table 4.4, Table 4.5, Table 4.6 and Table 4.7 column 1 shows the positions of masses as indicated in Fig 4.2 and column 3 and column 8 show the values of moment of inertia and shaft stiffness taken from Table Mass Moment No. of inertia J Table.4.2 Natural frequency of whole system = Hz 4 Torque per unit deflection Jω Torque in Total Shaft plane of torque stiffness mass Jω 2 θ K Jω 2 θ Deflection in plane of mass Θ Change in deflection θ Kg m 2 MN m ± rad MN m MN m MN Rad m/rad Pulley Cyl Cyl Cyl Cyl Cyl Cyl Mass No. Moment of inertia J Table.4.3 Natural frequency of whole system = Hz Torque per unit deflection Jω 2 Deflection in plane of mass θ Torque in plane of mass Jω 2 θ Total torque Jω 2 θ Shaft stiffness K Change in deflection θ Kg m 2 M N m ± rad MN-m MN-m MN m/rad Rad Pulley Cyl Cyl Cyl Cyl Cyl Cyl Flywheel Flywheel

9 Mass No. Table.4.4 Natural frequency of whole system = Hz Moment of inertia J Torque per unit deflection Jω 2 Deflection in plane of mass θ Torque in plane of mass Jω 2 θ Total torque Jω 2 θ Shaft stiffness K Change in deflection θ Kg m 2 M N m ± rad MN-m MN-m MN m/rad Rad Pulley Cyl Cyl Cyl Cyl Cyl Cyl Mass No Moment of inertia J Table.4.5 Natural frequency of whole system = Hz Torque per unit deflection Jω 2 Deflection in plane of mass θ Torque in plane of mass Jω 2 θ Total torque Jω 2 θ Table.4.6 Natural frequency of whole system = 1148 Hz Shaft stiffness K Change in deflection θ Mass No. Moment of inertia J Torque per unit deflection Jω 2 Deflection in plane of mass θ Torque in plane of mass Jω 2 θ Total torque Jω 2 θ Shaft stiffness K Change in deflection θ Flywheel Kg m 2 M N m ± rad MN m MN m MN Rad m/rad Pulley Cyl Cyl Cyl Cyl Cyl Cyl Flywheel Kg m 2 M N m ± rad MN m MN m MN Rad m/rad Pulley Cyl Cyl Cyl Cyl Cyl Cyl Flywheel

10 Mass No. Moment of inertia J Table.4.7 Natural frequency of whole system = 2379Hz Torque per unit deflection Jω 2 Deflection in plane of mass θ Torque in plane of mass Jω 2 θ Total torque Jω 2 θ Shaft stiffness K Change in deflection θ Kg m 2 M N m ± rad MN m MN m MN Rad m/rad Pulley Cyl Cyl Cyl Cyl Cyl Cyl Flywheel Table 4.2, Table 4.3, Table 4.4, Table 4.5, Table 4.6 and Table 4.7 show the calculations of natural frequencies and are found to be Hz, Hz, Hz 662.5Hz, 1148 Hz, 2379Hz. B. Representation of mode shapes Hz Hz Hz Amplitude [rad.] Mass No Hz 1148 Hz 2379Hz -3.5 Fig.4.4 Normal mode shape curves Fig. 4.4 shows the normal mode shape of Kilroskar six cylinder diesel engine. It is seen that first mode corresponding to natural frequency 1148Hz occurs in between the masses no. 6 and 7 and second mode corresponding to natural frequency 2379Hz occurs in between mass no. 7 and 8. Knowing the natural frequencies and the mode shapes, the limitations in use of the engine operating speeds can be fixed so as to avoid occurrence of resonance and the unwanted damage of thecrankshaft. 40

11 C. Representation phase vector diagram and calculations for critical speed A phase vector diagram as shown in Fig. 4.5 is drawn for the Kirloskar engineto analyze the minor and major critical harmonic orders in torsional vibration analysis. It is seen that the third order is the major critical orders of excitation. The corresponding critical speeds of engine operation are; For sixth order excitation; = Hz (60/2π) /6 = 291rpm = Hz (60/2π) /6 = rpm = Hz (60/2π) /6 = rpm = 662.5Hz (60/2π) /6 =1055 rpm = 1148Hz (60/2π) /6 =1827 rpm = 2379Hz (60/2π) /6 = rpm For third order excitation; = Hz (60/2π) /3 = rpm = Hz (60/2π) /3 = rpm = Hz (60/2π) /3 = rpm = 662.5Hz (60/2π) /3= rpm = 1148Hz (60/2π) /3 = rpm = 2379Hz (60/2π) /3 = rpm Out of the above calculated speeds rpm is the critical speed for the third order excitation corresponding to frequecncy Hz which falls nearer to operating speed of 1500 rpm, hence the forced vibration analysis is performed for this excitation frequency only. Fig.4.5 Phase vector diagram 41

12 D. Calculations for amplitudes Table 4.8 shows the calculations of amplitudes of torsional vibrations for frequencies Hz, Hz, Hz, 662.5Hz, 1148 Hzand 2379Hz.Table 4.9 shows vector summation (for 3 rd order) for frequency Hz based on which critical speed is decided. Table.4.8 Amplitude calculation Mass No. Moment of Inertia Kg m 2 One-node Mode (Frequency=182.77Hz) Two-node Mode (Frequency =435.73Hz) Three-node Mode (Frequency=662.44Hz) J θ Jθ 2 θ Jθ 2 θ Jθ 2 1 Pulley Cyl Cyl Cyl Cyl Cyl Cyl Flywheel Σ Jθ Firing Order Table.4.9 Vector summation (for 3 rd order) Specific Amplitude (A n ) Phase Angle (for 3 rd order) [Deg.] A n sin (Phase angle) A n cos (Phase angle) Cyl Cyl Cyl Cyl Cyl Cyl ΣA n E. Stress calculations The necessary data required to do the stress calculation for 3 rd order is presented in the Table.4.8 and Table.4.9. Table.4.10 Stress calculation Order No. (k) Critical Speed, N Torque, T n = t AR n (N/mm 2 - m) ΣA n T n ΣA n Equilibrium Amplitude, T n An θ e = 2 2 ω ( Jθ ) (deg) Stress per/deg amplitude at Mass no. σ** (N/m 2 /degree) Equilibrium Stress, σ v = σ θ e (N/m 2 ) Dynamic Magnifier***, M d, 42 Vibratory Stress, σ v = M d σ v (MPa) * E6 1.87E *t n = 0.30 (N/mm 2 -m)is as per Appendix [A.5] ** Calculated from Holzer Table ***Known data

13 Table.4.10 shows stress calculation for equilibrium Stress and vibratory Stress. The maximum stress occurs in the shaft connecting 6 th cylinder and Flywheel and its value is 50.96MPa. 43

14 Effect of inertia forces in more than one cylinder of SL90 Type engine In order to study the effect of inertia forces in more than one cylinder finite element analysis [FEA] approach was used to study the dynamic behavior of the engine and to investigate the early faulty cylinder. Fig. 4.6 shows wireframe model of crankshaft of SL90 Type diesel engine. A solid model meshed with tetrahedral element as shown in Fig. 4.7 and Fig a= Ø Fig.4.6 Wireframe model of crankshaft of SL90 Typediesel engine Fig.4.7 Solid model of crankshaft of SL90 Type diesel engine 44

15 Type of Element- Tetrahedral No. of Elements Fig.4.8 Solid meshed model of crankshaft of SL90 Type diesel engine Fig. 4.9 Multi-body model of SL90 Type diesel engine Fig. 4.9 shows multi-body model of SL90 Type diesel engine developed in LAB View. Rigid crank train model is created using mass/inertia input data and P-θ curve. Then multi-body analysis is carried out for one complete cycle of to analyse the effect of misfuel in more than one cylinder. 45

16 Fig Combustion and bearing reaction forces (Firing Order ) Fig.4.11 Force data when cylinder 1 misfires 46

17 Fig Force data when cylinder 3 misfires Fig Force data when cylinder 5 misfires (Firing Order ) From Fig. 13 it is seen that, when cylinder no.5 cutoff completely reaction forces on bearing increases which has significant impact on engine performance which leads to increase in level of torsional vibrations due to inertia forces as shown in Fig Whereas when Cylinder 47

18 No.1 and Cylinder No.3cutoff completely reaction forces on bearing has very less impact on engine performance which shows very small rise in level of torsional vibrations as shown in Fig and Fig Hence in this research work only the effect of cylinder 5 of six cylinder diesel engines is investigated in details in view to bring in focus the significance of torsional vibration. 48

19 Experimental Set up and Measurements In order to investigate the misfuel condition in operating diesel engines, experiments were conducted on Six Cylinder DI Diesel engine (SL90) mounted on the test bed. Measurement of angular speed of multi-cylinder six cylinders diesel engine: There are two ways to measure angular speed; 1. In this method the flywheel of the diesel engine is marked with 36 divisions equidistantly on the periphery. A built-in hardware can be used to measure time difference between marks as they pass a sensor. An array of time difference may be extracted, where the first element corresponds to time difference between top dead centre from cylinder 1 and mark 10 thereafter. From the time stamp array of analyser, flywheel speed as a function of flywheel position may be calculated. 2. In this method agear with 36 teeth is attached next to the flywheel. The test bed is set up such that the analyser pulse- train signal can be extracted where each pulse corresponds to a passed tooth.the gearthat holds trigger teeth that (when passed) can be used as a reference to keep track of what angle we are on. We may create an array of time differences between adjacent marks and there after calculate flywheel speed as a function of flywheel position. For experimentation the second method is used. Experimental set up: Flywheel Cylinder No. Sensor FFT Spectrum Analyzer Six Cylinder Diesel Engine Alternator Gear wheel Fig.4.14 Schematic layout of test set up for measurement of flywheel speed Fig 4.14 shows schematic layout of test set up developed for the measurement of engine speed to analyse and measure the variations in engine speed and time.fig shows position of a six cylinder engine, flywheel with alternator, gear wheel, sensor and FFT spectrum analyser. Crankshaft angular speed of internal combustion engines is usually measured by means of a 49

20 gearor measurement disk and a speed-pickup. As the gear wheele rotates the tooth on gear or mark passes the sensor, a step formed voltage is generated, called pulse train, which is used for calculating the angular speed of flywheel. The method proposed is used to calculate the angular velocity at every edge of the gear wheel signal as it rotates. The crank shaft angle was measured at every ten degrees, i.e. when a new edge on the gear is sensed by sensor.the measured speed response as shown in Fig and Fig are used to calculate angular velocity to decide the position of the crank shaft. The angular velocity is calculated, when a positive edge appears, using the differential Eq. 4.4 Δϕ ω = Δt [4.4] where, 2π Δφ= Z is the known sector angle described by the set of pulses for which the engine speed is measured and t is the measured time, Z is the number of teeth on gear.the time is measured by a digital timer set in FFT Spectrum Analyzer which is controlled by the zero crossings of pulse signal. Fig shows the gear wheel mounted on crankshaft next to flywheel when engine in stationary position. Fig shows the gear wheel mounted on crankshaft next to flywheel with position of hall effect sensor when engine in rotating position with speed of 1500rpm. Here speed was measured by digital display mounted on engine housing as well digital tachometer. Fig shows the schematic layout of gear wheel with 36 teeth spaced equidistantly on the periphery. Fig shows the measurement of engine speed in term of the speed step response in time domain. Here FFT spectrum analyser is used to plot the harmonic spectrum of speed step response corresponding to engine harmonic order. Fig Stationary position 50

21 Fig Rotating position Fig Gear wheel Fig Recording result on FFT Spectrum Analyser 51

22 Experimental Test Procedure and Results Experiments were conducted on a four-stroke six cylinder direct injection diesel engine (Kirloskar SL90) in I. C. Engine laboratory of M.E.S. College of Enginering, Pune. The engine was operated at constant speed and no load condition for uniform and non-uniform operation [i.e. when cylinder 5 cutoff]. By varying the fuel supplyto the cylinder, it was possible to reduce gradually the amount of fuel injected into the cylinder from the rated value to zero, to simulate a wide range of malfunctions up to a complete misfuel. The pressures in the cylinders were measured by piezoelectric pressure transducers as shown in Fig The 9measured speed is then subjected to a Discrete Fourier Transform (DFT) to determine the amplitudes and phases of their harmonic components. The time of rotation between successive teeth was measured during uniform and non-uniform working of engine. Then measured time is converted into crank angle domain by using Eq.4.4. Pressure (p) in bar Crank Angle (θ) in deg Fig.4.19 Actual cylinder pressure curve 52

23 Speed response for six cylinder engine: Fig.4.20 shows the speed step response for normal working of six cylinder diesel engine. Fig.4.21 shows the speed step response of six cylinder diesel engine when Cylinder 5 cutoff. The measured speed responses as shown in Fig and Fig are used to determine time between two edges of the gear tooth signal as it rotates to calculate angular velocity crank shaft. Fig Measured speed step response for six cylinder engine (uniform operation) Fig.4.21 Measured speed response for six cylinder engine (cyl. 5 cut off) 53

24 Table 4.11 shows measured time for six cylinder diesel engine foruniform operation and misfuel in cylinder no.5 measured from speed response of six cylinder engine. Sr. No. Crank Angle (Deg.) Table 4.11 Measured time Measured Time per cycle Uniform Operation Non- Uniform Operation Variation in Measured Angular Engine Speed Non- Uniform Uniform Operation Operation (seconds) (seconds) (rpm) (rpm)

25 Fig.4.22 shows measured time for six cylinder diesel engine for normal working and misfuel in cylinder no.5 plotted by using data given in Table Uniform operation Non-uniform operation Measured Time (s) Crank Angle (Degree) Fig Crank angle verses measured time for normal working and misfuel in cylinder5 Fig.4.23 and Fig shows the comparison of the speed signal for the normal operation and non-uniform operation [when cylinder 5 is cut off] in Cartesian and polar coordinates, respectively. This shows that average engine speed is rpm for normal working and rpm for misfuel in cylinder no.5. 55

26 Normal working 1530 Imbalance in Cyl.5 (Cyl.5 Cut off) 1520 Speed (rpm) cyl.1 cyl. 5 cyl.3 cyl.6 cyl.2 cyl.4 cyl.1 Crank Angle (Degree) Fig Crank angle verses speed for for normal working and misfuel in cylinder no Normal Working Imbalance in Cyl.5 ( Cyl.5 Cut off) Crank Angle (Degree) (0-720 RPM) -- Speed Variation (RPM) Firing order Cyl.1= 0 0 Cyl.5=120 0 Cyl.3=240 0 Cyl.6=360 0 Cyl.2=480 0 Cyl.4= Fig Polar coordinate comparison of engine speed for a six-cylinder diesel engine with uniform and non-uniform operation atmean engne speed 1500 rpm 56

27 Harmonic Analysis of Six Cylinder Engine Now using FFT spectrum analyserharmonic analysis of engine speed step response [as shown in Fig and Fig. 4.21] is carried out to detect misfuel in six cylinder engine. Fig.4.25 [A] shows the spectrum of torque verses engine order with uniform working and non-uniform working [whencylinder 5 cut off]. Fig.4.25 [B] shows spectrum of engine speed verses engine order with uniform working and non-uniform working Uniform Operation Non-uniform Operation Torque (Nm) Harmonic Order (A) Fig.4.25 [A] Spectrum of gas torque verses engine order Uniform Operation Non-uniform Operation Speed (1/s) Harmonic order (B) Fig.4.25 [B] Spectrum of engine speed verses engine order It is observed that when the cylinders are uniformly contributing to the total engine torque, the first three harmonic orders (k=½, 1, 1½) play an insignificant role in the frequency spectrum of the total gas-pressure torque and, consequently, appear with a very low contribution in the frequency spectrum of the crankshaft speed as shown in Fig [A] and 57

28 Fig. 4.25[B]. If the frequency spectrum of the crankshaft speed corresponding to uniform cylinders operation is compared to the spectrum corresponding to a misfueled cylinder, one may see that the major difference is produced by the amplitudes of the first three harmonic orders [Fig. 4.25[B]]. As far as the cylinders operate uniformly, these amplitudes are maintained under a certain limit. Once a cylinder starts to reduce its contribution, the amplitudes of the first three harmonic orders start increasing. These amplitudes may be used to determine the degree by which a cylinder reduces its contribution to the total gas pressure torque. The identification of the misfueled cylinder is achieved by analyzing the phases of the lowest three harmonic orders. Fig.4.26 shows the phase angle diagram for six cylinder engine with non-uniform engine operation. Phase (Deg.) Engine order Fig.4.26 Phase verses engine order measured by FFT spectrum analyser 58

29 Method for Detecting Misfuel in Six Cylinder Engine It is observed that when the cylinders are uniformly contributing to the total engine torque, the first three harmonic orders (k=½, 1, 1½) play an insignificant role in the frequency spectrum of the total gas-pressure torque and, consequently, appear with a very low contribution in the frequency spectrum of the crankshafts speed. If the frequency spectrum of the crankshafts speed corresponding to uniform cylinders operation is compared to the spectrum corresponding to a misfueled cylinder, one may see that the major difference is produced by the amplitudes of the first three harmonic orders. As far as the cylinders operate uniformly, these amplitudes are maintained under a certain limit. Once a cylinder starts to reduce its contribution, the amplitudes of the first three harmonic orders start increasing. These amplitudes may be used to determine the degree by which a cylinder reduces its contribution to the total gas pressure torque. Here the identification of the misfueled cylinderis achieved by analyzing the phases of the lowest three harmonic orders Fig [A] is plotted by reconstructing the pressure traces of the six cylinders in a sequence corresponding to the firing order ( ) with cylinder 5 is cutoff. Fig [B, C, D] shows the lowest three harmonic orders of the measured speed, respecting the measured amplitudes and phases. It is seen that, only for the expansion stroke of cylinder 5, all three harmonic curves have, simultaneously, a negative slope. In the phase angle diagrams of these orders [Fig.4.26], the vector corresponding to the harmonic component of the measured speed is also represented. One may see that, for each of the three considered orders, the vectors are pointing toward the group of cylinders that produces less work. The cylinder that is identified three times among the less productive cylinders is the misfueled, as shown in Fig

30 Pressure(p) in bar A Cyl6 Cyl2 Cyl4 Cyl1 Cyl Crank Angle (θ) in Degree 1 B Amplitude (1/s) Cy. 6 Cy.2 Cy 4 Cy.1 Cy.5 Crank Angle (Degree) Cy.3 Cy C Amplitude (1/s) Cy.6 Cy.2 Cy. 4 Cy.1 Cy.5 Crank Angle (Degree) Cy.3 Cy D 0.2 Amplitude (1/s) Cy.6 Cy.2 Cy. 4 Cy.1 Cy.5 Crank Angle (Degree) Cy.3 Cy.6 Fig Detection of a misfuel in cylinder from the phases of the lowest three harmonic orders B) ½ order C) 1 order D) 1½ order 60

31 ½ 1 1½ [E] [F] [G] Fig.4.28 Phase angle diagram Based on the above observations, the following method is developed to detect misfuel; The phase-angle diagramsare drawn for the lowest three harmonic orders placing in the cylinder that fire at 0 0 inthe top dead center (TDC) position in the cycleas per the firing order of the engine. On these phase angle diagrams, the corresponding vectors of the measured speed are represented in a system of coordinate axes having on the vertical the cosine term and on the horizontal the value of the sine term as shown in Fig [E, F, G]. The cylindes nearer to the vector indicate that, the cylinders are the less contributors. The less contributing cylinders for all three harmonic orders are marked with a - sign. The cylinders that receive a - sign for all three harmonic orders are clearly identified as less contributors to the engine total output. This procedure is presented in Table 4.12 for the case shown in Fig This method is able to identify a misfueled cylinder as soon as its contribution dropped below 20-25% with respect to the contribution of the other cylinders. Table 4.12 Identification of the misfueled cylinder Cylinders K ½ ½ I From Table 4.12 it is concluded that the here cylinder 5 is identified as a low torque contributor, i.e. generating less power. 61

32 4.3.2 Case Study-II. Greaves Six Cylinder Diesel Engine The following information pertains to the Greaves four strokes, six cylinder diesel engine; Engine Basic Data: a. Power Speed: 1500 RPM b. Operating speed range: 750 to1500 RPM c. Cylinder Diameter : 120 mm d. Piston Stroke : 130 mm e. Maximum Cylinder Pressure (Gauge) : 110 bar (at 180 from TDC) f. Mean Indicated Pressure at 1500 rpm : bar g. Cylinder Air Inlet Pressure (Guage) : 2.2 bar h. Compression Ratio : 14.6 i. Mass of Piston, Piston Rod & Cross Head : kg j. Mass of Connecting Rod (inclusive bearing) : kg k. Length of Con. Rod (between bearing) : 225 mm l. Centre of Gravity of Connecting Rod from Large end : 85 mm m. Radius of Gyration of Connecting Rod : 84 mm CouplingSpecifications: n. Type : Flexible Coupling o. Model: VULASTIK-L-COUPLING Mass Elastic Data: Table 4.13 Engine mass elastic system data Moment of Inertia Stiffness Mass Pulley Engine Mass-Elastic System Data Cylinder Flywheel Kg-m MNm/ Rad Alternator 62

33 A. Natural frequency calculation by Holzer Tabulation Method Table 4.14, Table 4.15 and Table 4.16 shows the calculations of natural frequency for Greaves engine. Engine model shown in Fig 4.2 is used to show the position of masses shown in Column 1 of Table 4.14, Table 4.15 and Table Colunm 3 and column 8 shows the values of moment of inertia and shaft stiffness taken from Table Table 4.14 Natural frequency of whole system = Hz Mass No. Moment of inertia J Torque per unit deflection Jω 2 Deflection in plane of mass θ Torque in plane of mass Jω 2 θ Total torque ΣJω 2 θ Table 4.15 Natural frequency of whole system = Hz Shaft stiffness K Change in deflection θ Kg m 2 N m /rad ± rad N m N m N m/rad rad Pulley Cyl Cyl Cyl Cyl Cyl Cyl Flywheel Alternator Mass No. Moment of inertia J Torque per unit deflection Jω 2 Deflectio n in plane of mass θ Torque in plane of mass Jω 2 θ Total torque ΣJω 2 θ Shaft stiffness K Change in deflection θ Kg m 2 N m /rad ± rad N m N m N m/rad rad Pulley Cyl Cyl Cyl Cyl Cyl Cyl Flywheel Alternator

34 Table 4.16 Natural frequency of whole system = Hz Mass No. Moment of inertia J Torque per unit deflection Jω 2 Deflectio n in plane of mass θ Torque in plane of mass Jω 2 θ Total torque ΣJω 2 θ Shaft stiffness K Change in deflection θ Kg m 2 N m /rad ± rad N m N m N m/rad rad Pulley Cyl Cyl Cyl Cyl Cyl Cyl Flywheel Alternator Table 4.14, Table 4.15 and Table 4.16 shows the calculated natural frequencies and are found to be Hz, 827.5Hz, and Hz. B. Representation of mode shapes Amplitude Mass No Hz Hz Hz Fig Normal mode shape curves Fig shows the normal mode shape of Greaves six cylinder diesel engine. It is seen that first mode corresponding to natural frequency 186.9Hz occurs in between the mass no. 8 and 9; second mode corresponding to natural frequency Hz occurs in between mass no. 6 and 7 and third mode corresponding to natural frequency Hz occurs in between mass no. 7 and 8. 64

35 Knowing the natural frequencies and the mode shapes, the limitations in use of the engine operating speeds can be fixed so as to avoid occurrence of resonance and the unwanted damage of the crankshaft. C. Representation phase vector diagram and calculations for critical speed A similar phase vector diagram as shown in Fig.4.5 can be drawn for the Greaves engine to analyze the minor and major critical harmonic orders It is seen that third and sixth orders are the major critical orders of excitation.the corresponding critical speeds of engine operation are; For sixth order excitation: = (60/2π) /6 = rpm = (60/2π) /6 = rpm = (60/2π) /6 = rpm For third order excitation: = (60/2π)/3 = 595 rpm = (60/2π)/3 = rpm = (60/2π)/3 = rpm Out of the above calculated speeds 1317 rpm is the critical speed for the sixth order excitation of second mode of vibration which falls nearer to operating speed of 1500 rpm, hence the forced vibration analysis is performed for this excitation frequency only. D. Calculations for amplitudes Mass No. Table 4.17 shows the calculations of amplitudes of torsional vibrations for frequencies Hz, Hz, Hz. Table 4.18 shows vector summation (for 6 th order) for frequency Hz based on which critical speed is decided. Table 4.17 Calculations of amplitudes Moment One-node Mode Two-node Mode Three-node Mode of (Frequency = Hz (Frequency = Hz (Frequency = Inertia Hz Kg m 2 J θ Jθ 2 θ Jθ 2 θ Jθ 2 1 Pulley Cyl E-05 3 Cyl Cyl Cyl Cyl Cyl Flywheel Alternator E-06 Σ Jθ

36 Table 4.18 Vector summation (for 6 th order for Freuency Hz) Firing Order Specific Amplitude (A n ) Phase Angle (for 6 th order) [Deg.] A n sin(phase angle) A n cos(phase angle) Cyl Cyl Cyl Cyl Cyl Cyl ΣA n E. Stress calculations For 6 th order, the torque value obtained is * Nm. The necessary data required to do stress calculation is presented in the Table 4.17 and Table Table 4.19 Stress calculation Order No. (k) Critical Speed, N (RPM) Torque, T n=t nar Nm ΣA T n A n Equilibrium Amplitude, T n An θ e = 2 2 ω ( Jθ ) Stress per/deg amplitude at Mass no. σ** (N/m 2 /degree) Equilibrium Stress, σ v = σ * θ e (N/m 2 ) Dynamic Magnifier***, M d, Vibratory Stress, σ v= M d*σ v (MPa) (deg.) Engine crankshaft * E E Alternator shaft * E E *t n = N/mm 2 -m, As per Appendix A.6 ** Calculated from Holzer Table ***Known data Table 4.19 shows stress calculation for equilibrium stress and vibratory stress. The maximum stress occurs in the shaft connecting 6 th cylinder and flywheel and its value is MPa. 66

37 Harmonic Frequency Analysis for Detection of Imbalance In this section measurement of cylinder pressure, gas torque and imbalance detection for operating Greaves six cylinder diesel engine is explained in details. Data required for analysis is obtained by performing trial on six cylinder diesel engine at Greaves Cotton Ltd, Pune. The engine coupled to a dynamometer as shown in Photo.2 was run at constant speed and load. To simulate a faulty cylinder, the nut connecting the high-pressure fuel line to the corresponding element of the injection pump was slightly unscrewed and a leakage was introduced in the fuel supply of the cylinder. Controlling the tightness of the high-pressure fuel line, the amount of fuel injected into the cylinder 5 was reduced gradually from the rated value to zero to simulate a wide range of non-uniformities up to a complete misfuel. The pressure was measured in cylinders by piezoelectric pressure transducers via charge amplifiers. The crankshaft speed variation was measured by using a high precision hollow shaft encoder. An experimental data was sampled and recorded by 8-channel data acquisition system (AVL INDI-MODULE 621) having an internal clock with a 10MHz frequency. The maximum pressures of each cylinder were calculated from the pressure traces and the measured speed was subjected to a Discrete Fourier Transform (DFT) to determine the amplitudes and phases of its harmonic components. 8-Channels Photo.1 AVL Indi-modul 621 System (Courtesy Greaves Cotton Ltd,Pune-19) 67

38 6 Cylinder Diesel Engine Coupling Dynamometer Photo. 2 Engine set up (Courtesy Greaves Cotton Ltd, Pune-19) Measured cylinder pressure and gas torque: The measured cylinder pressure verses crank angle diagram is obtained for Greaves six cylinder diesel engine from test bed and is shown in Fig Similarly, torque verses crank angle diagram is obtained as shown in Fig From Fig.4.30, it is observed that the highest peak value of cylinder pressure is 110 bars occur at 40 0 crank angles. From Fig.4.31, it is noticed that the highest peak value of torque output is approximately 38 times greater than the mean torque output of the engine (the torque which the dynamometer measures) corresponding to peak value of pressure as 110 bars at 40 0 crank angles. Also it is observed that the torque curve contains a negative peak (valley) which is nearly 9.5 times the mean engine torque. From the resulting torque curve, the torque values for different orders are obtained by harmonic analyser integrated in AVL Indi-modul 621 system. 68

39 Fig Measured cylinder pressure verses crank angle diagram Fig. 4.31Torque verses crank angle diagram 69

40 Condition-1. When Cylinder No. 5 cut off A. HarmonicStructure of the Resultant Gas Pressure Torque with Uniform and Non-Uiniform Engine Operation The resultant gas pressure torque was calculated from the pressure variation measured in each cylinder and then subjected to a DFT to obtain its spectrum for two operating conditions as uniform and non-uiniform engine operation when cylinder No. 5 cut off. Fig shows the spectrum of the resultant gas pressure torque.when all the cylinders operating fairly uniform the harmonic components of the resultant gas pressure torque are basically the major harmonic orders of the six cylinder engine (k=3, 6, 9). Small differences in the cylinders operation result in negligible contributions of the non-major harmonic orders. Compared to the amplitude of the third harmonic order (k=3) the maximum amplitude of a non-major order (k=1) is only about below 2%. Torque (Nm) Uniform Operation Non-uniform operation Harmonic Order Fig Frequency spectrum of the resultant gas-pressure torque for six-cylinder engine A large non-uniformity is introduced when cylinder no.5 is cut off, the lower nonmajor orders play a significant role in the frequency spectrum of the resultant gas-pressure torque. The analysis of Fig shows that, both for uniform and non-uniform contribution of the cylinders, the dominant component of the spectrum is the third harmonic order, which is the first major order of this engine. At the same time, the values of the amplitudes of the third harmonic order reflect the differences in the average engine torque for the two operating conditions. 70

41 B. Harmonic Structure of the Measured Crankshafts Speed Uniform operation Non-uniform operation Speed (1/s) Harmonic order Fig.4.33 Frequency spectrum of the crankshaft speed for six cylinder engine If the lowest major order has a frequency that is far from the first natural frequency of the shafting, the amplitude of this order, in the crankshafts speed spectrum, will be proportional to the amplitude of the corresponding order in the resultant gas-pressure torque spectrum. This dependency could be used to determine a quantitative relationship between the amplitude of the first major order of the engine, calculated by a DFT of the measured speed and the average IMEP or gas-pressure torque. In Fig. 4.33, the frequency spectrum of the measured speed is presented for the two cases considered in Fig There is a very close relationship between the crankshafts speed and gas pressure torque spectrum in the both cases. However the major difference occurs for the orders k= 7.5 and k=8in the speed spectrum. This is due to the fact that the shafting is close to resonance with the eighth harmonic order when the engine is running at 1500 rpm. This situation points out the necessity to choose the lowest major order for the correlation to the average IMEP of the engine [as explained in chapter 6]. An interesting fact is that, even if the power contribution of each cylinder is different, the amplitude of the lowest major harmonic order in the crankshafts speed spectrum always reflects the average value of the IMEP. 71

42 C. Method for Detecting Imbalance in Engine Operation for Cylinder 5 Cutoff When the cylinders are uniformly contributing to the total engine torque, the first three harmonic orders (k=½, 1, 1½) play an insignificant role in the frequency spectrum of the total gas-pressure torqueas shown infig and consequently, appear with a very low contribution in the frequency spectrum of the crankshaft speed as shown in Fig On the contrary, if a cylinder generates less work with respect to the others, the first three harmonic orders have a larger contribution in the frequency spectrum of the gas-pressure torque [Fig. 4.32] and determine larger amplitudes of the corresponding components in the frequency spectrum of the crankshaft s speed as whown in Fig If the frequency spectrum of the crankshaft speed corresponding to uniform cylinders operation is compared to the spectrum corresponding to a misfuel cylinder as shown in Fig. 4.33, one may see that the major difference is produced by the amplitudes of the first three harmonic orders. As far as the cylinders operate uniformly, these amplitudes are maintained under a certain limit. Once a cylinder starts to reduce its contribution, the amplitudes of the first three harmonic orders start increasing. These amplitudes may be used to determine the degree by which a cylinder reduces its contribution to the total gas-pressure torque. The identification of the faulty cylinder may be achieved by analyzing the phases of the lowest three harmonic orders. Fig indicates this situation in detail. Fig [a] shows the pressure traces of the six cylinders in a sequence corresponding to the firing order ( ) with cylinder 5 cutoff.fig [b, c, d], represent the lowest three orders of the measured speed, representing the measured amplitudes and phases. It is seen that, only for the expansion stroke of cylinder 5 all three curves have, simultaneously, a negative slope. The phase angle diagrams of these orders are shown in the Fig. 4.34[e, f, g] andthe vector corresponding to the harmonic component of the measured speed are represented by arrows. For each of the three considered orders, the vectors are pointing towards the group of cylinders that produce less work. The cylinder that is identified three times among the less productive cylinders is the misfueled one. 72

43 (e) (f) (g) Fig Detection of imbalance in cylinder from the phases a) Actual pressure traces of the cylindersb) ½ order c) 1 order d) 1½ order of the crankshaft speed Simillarly misfuel in cylinder is detected as follow; 73

44 The phase-angle diagrams are drawn for the lowest three harmonic orders placing in the cylinder that fires at 0 0 in the top dead center (TDC) positionin the cycleas per the firing order of the engine. On these phase angle diagrams, the corresponding vectors of the measured speed are represented in a system of coordinate axes having on the vertical the cosine term and on the horizontal the value of the sine term as shown in Fig [e, f, g]. The cylindes nearer to the vector indicate that, the cylinders are the less contributors. The less contributing cylinders for all three harmonic orders are marked with a - sign. The cylinders that receive a - sign for all three harmonic orders are clearly identified as less contributors to the engine total output. This procedure is presented in Table 4.20 for the case shown in Fig This method is able to identify a misueled cylinder as soon as its contribution dropped below 20-25% with respect to the contribution of the other cylinders. Table 4.20 Identification of the misfuled cylinder K Cylinders ½ ½ - - I i From Table 4.20 it is concluded that the cylinder 5 is identified as a low torque contributor, i.e. generating less power. 74

45 Condition-II. When 40% more Amount of Fuel is supplied to Cylinder No. 5 A. Harmonic Structure of the Resultant Gas Pressure Torque and Measured Crankshaft Speed Initially the engine coupled to a genset as shown in Photo.2 was run at constant speed and load. To simulate a mifueled cylinder, the nut connecting the high-pressure fuel line to the corresponding element of the injection pump was slightly unscrewed and a 40% more quantity of fuel was introduced in the fuel supply of the cylinder. Controlling the tightness of the highpressure fuel line, the amount of fuel injected into the cylinder was increases gradually from the rated value to required value, to simulate a wide range of non-uniformities up to a required misfuel. The resultant gas pressure torque was calculated from the pressure variation measured in each cylinder and then subjected to a DFT to obtain its spectrum. When all cylinders operate fairly uniform, the spectrum of the total gas-pressure torque contains mainly the major harmonic orders of the engine with insignificant contribution of other orders as shown in Fig The harmonic components of the resultant gas pressure torque are basically the major harmonic orders of the six cylinder engine (k=3, 6, 9). Small differences in the cylinders operation result in negligible contributions of the non-major harmonic orders. Compared to the amplitude of the third harmonic order (k=3) the maximum amplitude of a non-major order (k=1) is only about below 2%. Uniform Operation Non-uniform Operation 950 Torque (Nm) Harmonic Order Fig Frequency spectrum of the resultant gas-pressure torque When a large non-uniformity is introduced by increasing the fuel supply into a cylinder, the lower non-major orders play a significant role in the frequency spectrum of the 75

46 resultant gas pressure torque. Fig.4.35 shows that, both for uniform and non-uniform contribution of the cylinders, the dominant component of the spectrum is the third harmonic order, which is the first major order of this engine. At the same time, the values of the amplitudes of the third harmonic order reflect the differences in the average engine torque for the two operating conditions. If the lowest major order has a frequency that is far from the first natural frequency of the shafting, the amplitude of this order, in the crankshaft speed spectrum, will be proportional to the amplitude of the corresponding order in the resultant gaspressure torque spectrum. This dependency could be used to determine a quantitative relationship between the amplitude of the first major order of the engine, calculated by a DFT of the measured speed, and the average IMEP, or gas-pressure torque. In Fig.4.36 the frequency spectrum of the measured speed is presented for the two cases considered in Fig There is a very close relationship between the crankshafts speed and gas pressure spectrumin both cases. The major difference occurs for the orders k= 7.5 and k= 8 that have a significant contribution in the speed spectrum, and are completely insignificant in the gas pressure torque spectrum. This discrepancy is caused by the fact that the shafting is close to resonance with the eighth harmonic order when the engine is running at 1500 rpm. This situation points out the necessity to choose the lowest major order for the correlation to the average IMEP of the engine. 1.4 Uniform Operation Non-uniform Operation Speed (1/s) Harmonic order Fig.4.36 Frequency spectrum of the crankshaft speed An interesting fact is that, even if the contribution of the cylinders is very different, the amplitude of the lowest major harmonic order in the crankshafts speed spectrum always 76

47 reflects the average value of the IMEP. Based on this property the experimental six-cylinder engine was operated at 1500 rpm and both with uniform and non-uniform contribution of the cylinders. B. Method for Detecting Imbalance in Working Engine For 40% More Fuel When the cylinders are uniformly contributing to the total engine torque, the first three harmonic orders (k =½, 1, 1½) play an insignificant role in the frequency spectrum of the total gas-pressure torque [Fig.4.35] and, consequently, appear with a very low contribution in the frequency spectrum of the crankshafts speed as shown in Fig On the contrary, if a cylinder generates less/more work with respect to the others, the first three harmonic orders have a larger contribution in the frequency spectrum of the gaspressure torque [Fig. 4.35] and determine larger amplitudes of the corresponding components in the frequency spectrum of the crankshaft speed [Fig. 4.36]. If the frequency spectrum of the crankshafts speed corresponding to uniform cylinders operation is compared to the spectrum corresponding to a non-uniform cylinder as shown in Fig. 4.36, one may see that the major difference is produced by the amplitudes of the first three harmonic orders. As far as the cylinders operate uniformly, these amplitudes are maintained under a certain limit. Once a cylinder starts to increase its contribution, the amplitudes of the first three harmonic orders start changing. These amplitudes are used to determine the degree by which a cylinder increases its contribution to the total gas-pressure torque. The identification of the misfueled cylinder was achieved by analyzing the phases of the lowest three harmonic orders. 77

48 120 Pressure P (bar) Crank angle θ (Deg.) (a) Speed N (1/s) Speed N (1/s) Speed N (1/s) Crank angle θ(deg.) (b) Crank angle θ(deg.) (c) Crank angle θ (Deg.) (d) (e) (f) (g) Fig Detection of imbalance in cylinder from the phases a) Actual pressure traces of the six cylinders b) ½ order c) 1 order d) 1½ order of the crankshafts speed 78