Journal of Mechanical Science and Technology 29 (4) (2015) 1493~1500 www.springerlink.com/content/1738-494x
DOI 10.1007/s12206-015-0322-8
A mathematical model to investigate the effects of misfire and cyclic variations on crankshaft speed fluctuations in internal combustion engines† Hamit Solmaz* and Halit Karabulut Automotive Engineering Department, Faculty of Technology, Gazi University, Ankara, 06500, Türkiye (Manuscript Received October 16, 2014; Revised December 18, 2014; Accepted December 22, 2014) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Abstract In internal combustion engines the engine vibrations come up as block vibrations and crankshaft speed fluctuations. Crankshaft speed fluctuations indicate the unstable running of the engine. Crankshaft speed fluctuations are caused by several sources. Main factor affecting the crankshaft speed fluctuations is in-cylinder pressure. Changes in-cylinder pressure caused by cyclic variations and misfire result as speed fluctuations on a crankshaft. In this study the effects of the cyclic variations and misfire problem on the engine crankshaft speed fluctuations have been investigated. A mathematical model including engine kinematics has been developed for a four cylinder diesel engine. In-cylinder pressure profiles used in the mathematic model were obtained experimentally. Two pressure profiles including 11 cycles and averages of these cycles were used in analysis. Pressure profiles were expressed mathematically by Fourier series having 1001 term. Although the indicated mean effective pressure values of pressure profiles were stable, the crankshaft speed fluctuations were determined as 5.5% and 11.1% at 230 rad/s for 5.15% and 12.92% COVimep values. When single misfire take place in the third cylinder, average crankshaft speed decreased 6.6 rad/s. Also, in case of continuous misfire the crankshaft speed fluctuations increased from 4.3% to 8%. Keywords: Crankshaft dynamic; Crankshaft speed fluctuations; Cyclic variations; Engine dynamic; Engine vibration; Internal combustion engine; Misfire ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1. Introduction One of the major problems of vehicle technology is minimization of vibration and noise. Vibration and noise influence both driving safety and journey comfort. In vehicles one of most dominant vibration sources is the engine. Engine vibration can never be completely eliminated; however, it can be isolated or minimized to a certain degree. Isolation of vehicles from engine vibration dates to back 1950s [1]. In those years, investigations were concentrated especially on the flexible engine mountings [1-5]. In subsequent years, simulation and optimization of engine block vibrations became fashionable [4-7, 8]. In these studies basic approaches, disregarding the crankshaft vibrations, were used for modeling the engine block vibrations [8, 9]. In internal combustion engines, vibration is caused mainly by the in-cylinder gas pressure force, inertial forces and moments of reciprocating and rotating components, unbalances of rotating components and frictions. In internal combustion engines due to working principles of the engine, inherently, there is a fluctuation in crankshaft speed. The crankshaft speed *
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[email protected] † Recommended by Editor Yeon June Kang © KSME & Springer 2015
fluctuations caused by several faults should not be confused with the natural speed fluctuations of the crankshaft. It is accepted that the crankshaft speed fluctuations are important parameters, indicating whether the operation of the engine is stable or not. In an engine having a relatively higher crankshaft speed fluctuation, a decrease in torque and power may be observed. It is also seen that fuel consumption and exhaust emissions increase due to unstable operation [10, 11]. Variations of cyclic mean pressure, knocking and misfiring have significant effects on crankshaft speed fluctuations. Charge mixture, air/fuel ratio, heat transfer from the cylinder wall, gas leakage from the piston rings and valves vary cycle to cycle in an internal combustion engine [12]. So that in-cylinder pressure, temperature, mixture composition vary during the combustion in each cycle. Misfires are other sources for cyclic variations. Misfiring causes breakdown of the engine and increases the exhaust emissions because of instant speed and load variations. Catalytic converter can also be damaged during the long misfiring [13]. All of these parameters result in cyclic variations in the engine. The cyclic variations generally indicate with mean effective pressure. Cyclic variations in the indicated mean effective pressure show the stability of the engine. Furthermore, cyclic variations restrict the operating range, efficiency and emissions of the
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Fig. 1. Forces and moments acting on the crankshaft. Fig. 2. Displacement of crank mechanism.
engine. It is suggested that the cyclic variations in the internal combustion engine should not exceed 10% for stable running [12, 14]. In this study, a mathematical model has been developed and FORTRAN code has been written to analyze the crankshaft speed fluctuations in a four cylinder diesel engine. The developed model includes the in-cylinder gas pressure force, engine frictions and inertial effects of the parts which make linear and circular motion. Within the scope of this study, the effects of misfiring problem and cyclic variations on the crankshaft speed fluctuations have been investigated.
2. Mathematical model It was assumed that injection order of the modeled diesel engine is 1-3-4-2. Newton’s law was used to derive the motion equation of the crankshaft. The engine block is assumed to be stable. In previous studies, it was shown that gravity has no significant effect on the engine vibrations. Thus, gravity was neglected in this study. Also, cyclic variations of the engine were taken into consideration. 2.1 The motion equation of the crankshaft It is required to determine the excitation forces affecting the crank mechanism and caused by motion of the crank mechanism to derive the motion equation of the crankshaft. Fig. 1 shows that excitation forces and torques affect the crankshaft. In Fig. 1 q , b and l define the crankshaft angle, the angle between connecting rod and cylinder axis and connecting rod length, respectively. Crankshaft is accelerated at the positive direction by Fb force formed by the effect of in-cylinder gas pressure and the starter external moment M s acting only during start. As shown in Fig. 1 M q , M m and M km define the engine load, the torque caused by the friction in the crankshaft main journals and the excitation torque caused by the friction in the crankshaft rod journals, respectively. M q , M m and M km are the torques which try to decelerate the crankshaft.
The motion equation of the crankshaft affected by these forces and torques is as follows: éæ 4 ép ùö ù êç å Fbi cos ê - ( bi + qi ) ú ÷ ú ë2 ûø ú êè i =1 ê ú 4 ö r ê æ ú. q&& = - ç å Fbi cos( bi + qi ) ÷ I cr ê è i =1 ú ø ê ú 4 ê+ æ M - M - M - M öú q m ÷ú ê ç S å kmi i =1 øû ë è
(1)
There is an effect of each piston-connecting rod-crankshaft mechanism on the crankshaft motion in a four cylinder engine. For this reason, each piston-connecting rod-crankshaft mechanism must be individually defined mathematically in all cylinders. The i indicates the cylinder number in Eq. (1). Crank angles are q1 = q , q 2 = q - 3p , q3 = q - p and q 4 = q - 2p for each cylinder according to the cylinder order in the engine which ignition order is 1-3-4-2. Fb and Fb excitation forces and M m , M km excitation torques in Eq. (1) must be determined. Therefore, it is needed to obtain the kinematical relations and the motion equations of pistons. 2.2 Kinematic relations and motion equations of pistons Kinematical and kinetic relations of crank mechanism must be determined to solve the motion equation of crankshaft. Two instantaneous positions of the crank mechanism are shown in Fig. 2. While the crankshaft rotates as q rad, piston moves as y p from TDC to BDC. l and b define the connecting rod length and the angle between the connecting rod and the cylinder axis. Also r defines the crank rotation radius. The distances moved by the pistons in the cylinder axis are
y pi = r + l - ( r cos qi + l cos bi )
(2)
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2.3 Friction forces and torques In the internal combustion engines the major frictions occur in the crankshaft main bearing, crankpin bearing and between the piston and cylinder wall. There are two different friction forces on the piston which occur by dry and hydrodynamic friction. Hydrodynamic friction occurs in the area of the oil film while dry friction occurs where the oil film is weak. The dry and hydrodynamic friction forces between the piston and cylinder are,
( )
Fui = éë F¥ + Cu Fsi ùû sgn y& pi Fmi = C p y& pi Fig. 3. The forces acting on the piston and free-body diagram of the piston.
From the kinematical relations, the angles between the connecting rods and cylinder axis can be written as
ér
ù
bi = arcsin ê sin qi ú ël û.
(3)
The forces acting on the pistons should be determined in order to obtain the motion equations of pistons. Fig. 3 shows the forces acting on the piston and the piston free-body diagram. Gas force ( Fgi ), side surface force ( Fsi ), crankcase pressure force ( Fc ), connecting rod force ( Fbi ), dry friction force on piston surface ( Fu ), hydrodynamic friction force ( Fmi ) and the forces resulting from connecting rod journal frictions ( Fkm ) act on the piston in internal combustion engines. From Fig. 3, the motion equations of pistons in the X p and Yp directions are as follows, respectively.
(9)
respectively [15, 16]. The piston friction force occurs in the opposite direction to the piston motion. As used herein, signum function provides to change the direction of dry friction force according to the piston speed continuously. Cu , C p define the dry friction coefficient between the piston and cylinder and the coefficient of hydrodynamic friction respectively. F¥ is a constant friction force created by piston rings between the piston and cylinder [15, 17]. Fkm is a reaction force on the piston pin created by the torques of hydrodynamic friction in the crankshaft journal. Fkm forces can be determined with Eq. (10). i
Fkmi =
i
i
(8)
Ckm æ & d b i & ö qi ÷ ç qi + l è dq ø .
(10)
Here, Ckm is a coefficient of hydrodynamic friction. Fkmi forces cause excitation torques ( M kmi ) in the opposite direction of crankshaft motion on the crankshaft with the effects of hydrodynamic frictions. The torques caused by Fkm according to the crankshaft center can be written as i
m p && x pi = Fsi - Fbi sin b i + Fkmi cos b i
(4)
m p && y pi = Fgi - Fc - Fui - Fmi
(5)
- Fbi cos b i - Fkmi sin b i .
As the piston does not move in the X p directions, the left side of the Eq. (5) becomes zero. From Eq. (5), piston side surface force Fs can be determined as i
Fsi = Fbi sin bi - Fkmi cos bi .
(6)
The piston acceleration y pi can be determined via the second order derivatives of Eq. (2). Thus, the left side of the Eq. (5) is determined. Hence, connecting rod forces Fbi can be determined from the Eq. (5). Connecting rod forces can be expressed as follow,
Fbi =
Fgi - m p &&y pi - Fc - Fui - Fmi - Fkmi sin bi cos bi
.
M kmi = Fkmi cos b i [ r cos qi + l cos b i ] .
(11)
Connecting rod inertia force, which is in the motion equations of the crankshaft in (Eq. (1)), must be determined. The small end of the connecting rod performs linear motion with the piston in the cylinder. But the big end of the connecting rod performs the rotational motion owing to connecting to the crankshaft. It is seen that connecting rod performs the oscillating movement around the piston pin center when viewed from the piston head to the connecting rod. This oscillating movement occurs with the result of connection of crankshaft journal and the big end of the connecting rod. Because of this oscillating movement, mass moment of inertia of connecting rod formed by around of piston pin axis causes implementation of Fb i force by the connecting rod to the crankshaft journal. Fb i force can be defined with Eq. (12).
(7) Fbi =
I B && bi l
(12)
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Fig. 4. The schematic view of the experimental set-up.
Fig. 5. Comparison of pressure data and Fourier fitting.
In this equation I B defines the mass moment of inertia according to the pin center of connecting rod. 2.4 In-cylinder gas pressure force The first movement of the engine was provided by the starter torque in the dynamic model. On the other hand, incylinder gas pressure forces are needed to continue the operation of the engine and support the external loads acting on the crankshaft. For this reason, in-cylinder gas pressure should be determined in the model. In-cylinder pressures can be defined with different methods. Pressure profiles can be obtained by detailed cycle analysis of internal combustion engine by using thermodynamic expressions. Also, experimental measurement of in-cylinder pressure can be used directly. Moreover, incylinder combustion can be modeled and pressure profiles can be obtained numerically with different combustion analysis and modeling programs. But it was stated that the experimental data provide more realistic results [15, 16, 18]. In this study, pressure data were measured from Antor 6LD400, a single cylinder direct injection diesel engine at maximum brake torque speed of 2200 rpm. Experimental setup of in-cylinder pressure measurement is shown in Fig. 4. Incylinder pressures were measured using water cooled AVL 8QP500c quartz in-cylinder pressure sensor, which is coupled to the charge amplifier. Analog cylinder pressure signals were amplified using Cussons P4110 combustion analysis device. The National Instruments USB 6259 data acquisition card was used to convert analog cylinder pressure signals to digital signals. A shaft encoder mounted on the crankshaft was used as the data clocking pulses to acquire the cylinder pressure data. The shaft encoder has a timing resolution of 0.36° crank angle. 11 cycles were used to analyze the cyclic variations in the engine. Pressure profiles were transformed Fourier series in the model as follows: P (qi ) =
n A0 + å ( Ak cos kqi + Bk sin kqi ) . 2 k =1
(13)
Gas pressure forces acting on the piston were calculated using Eq. (14).
Fig. 6. Comparison of average pressure data and Fourier fitting.
Fgi (qi ) = Ap P (qi ) .
(14)
Cyclic variations are defined by the variations of the indicated mean effective pressures of cycles. In this study the coefficient of variations of indicated mean effective pressure was calculated. Cyclic variations of imep should not exceed 10% in a stable running engine [12]. Cyclic variations of imep were calculated using Eq. (15). COVimep =
s imep P
100 .
(15)
Here COVimep , s imep and P define the cyclic variations of indicated mean effective pressure, standard deviation of indicated mean effective pressures and indicated mean effective pressure, respectively. Two different pressure profiles and an accordance of Fourier series fitting are shown in Fig. 5. A Fourier series having 1001 terms was used to define in-cylinder pressure occurred from 11 cycles. It was shown that cyclic variation is 5.15% in the pressure profile A during the stable operation and the coefficient of correlation of fitting is 0.9996. During the unstable operation B, cyclic variation is 12.92%. The coefficient of
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Table 1. Coefficients for Fourier series. 6.27 bar
Number of cylinder
Ak
Bk
Ak
Bk
0
1244.8
-
1050.3
-
1
980.6
167
749.2
164.3
2
789.6
183
631
218.1
3
657.2
168.9
513.3
255.7
4
556.3
174.3
415.5
266.6
5
436.2
181
316.5
273
6
348
161.6
239.4
258.6
7
276
148
172.7
239.1
8
227.4
137.7
126.7
219.2
9
170.1
131.1
77.9
195.4
10
137.3
113.5
49.8
170.9
11
105.5
106.5
22
148
12
83.4
96
5
128.3
13
60.5
86.4
-11
106.7
14
47.9
77.3
-19.4
91.9
15
33
68.7
-27.6
75
16
26.1
60.6
-31.8
64.2
17
17
52.4
-35.3
50.7
18
13.7
45.8
-36.6
43
19
8.9
39.7
-37.2
33
20
7.7
35.6
-37.4
27
21 22
5.4
31
-36.6
19.4
5
28.1
-35.7
14.4
23
3.1
24.4
-33.6
8.8
24
2.9
21.9
-32.2
4.9
25
2.1
19.5
-29.6
0.7
correlation is 0.998 in the fitting process of the pressure profiles. The mean pressure profiles of two different 11 cycles and fitting with Fourier series from stable and unstable operation were seen in Fig. 6. A A and B defined the mean values of A and B pressure profiles. It was shown that an indicated mean effective pressure of A pressure profile is 6.43 and the coefficient of correlation is 0.999. Indicated mean effective pressure of B pressure profile is 6.27 and the coefficient of correlation is 0.997. 51 terms were used to fit the pressure profiles with Fourier series. Determined coefficients for using in this series are shown in Table 1. 2.5 Friction forces and torques Analytical solution is not possible as the equation of crankshaft motion is non-linear equation. For this reason, Taylor series method was used to solve the motion equations. It is known that Taylor series method gives good results when small step intervals and sufficient number of terms are chosen. Crank angle and crankshaft angular speed were expanded into the Taylor series as follows:
4
Crank journal (m)
Engine parameters
6.43 bar
0.0375
Piston diameter (m)
0.09
Connecting rod length (m)
0.15
Flywheel and crankshaft total mass inertia torque (m2kg)
0.08
Mass inertia torque of connecting rod according to pin axis (m2kg)
0.003
Total piston and piston pin mass (kg)
0.85
Starting torque (Nm)
50
Piston ring dry friction force (N)
Friction coefficients
Imep
Table 2. Engine parameters and friction coefficients.
55
Piston dry friction coefficient
0.05
Hydrodynamic friction coefficient between piston and cylinder (Ns/m)
2.5
Hydrodynamic friction coefficient in the crankshaft main bearing(Nms/rad)
0.01
Hydrodynamic friction coefficient in the connecting rod bearing (Nms/rad)
0.0025
q j = q j -1 + q&j = q&j -1 +
q&j -1 1! q&&
j -1
1!
Dt + Dt +
q&&j -1 2! q&&&
j -1
2!
Dt 2 + Dt 2 +
q&&&j -1 3!
q ıvj -1 3!
Dt 3 +
q ıvj -1 4!
Dt 3 + ... .
Dt 4 + ...
(16) (17)
Second and third derivatives were differentiated from Eq. (1) and derivative of the Eq. (1). Time interval was chosen as Dt = 0.00001 s in the solution. Initial conditions were defined as follows. t =0 ®q =0 t = 0 ® q& = 0 .
(18) (19)
Prediction-correction algorithm was used to determine the enforcement forces and torques of kinematical relations [15, 18-21]. Initial values of the kinematic relations, excitation forces and torques and motion equation of the crankshaft were calculated using initial conditions. In the next time interval, the crank angle and the crankshaft angular speed were calculated with Eqs. (16) and (17) after the all starting values were determined. Other required excitation forces and torques can be determined via utilizing from these values. Thus, all values were calculated for second time step interval. For all time domains this process continues and a solution is obtained. Engine parameters and friction coefficients must be inserted into the FORTRAN code. The friction coefficients used in the study were determined by taking consideration of previous works. The friction coefficients used in the study and engine parameters are given in Table 2. The friction coefficients in Table 2 were determined in the experimental works [15-17, 20, 22, 23].
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3. Results and discussions Engine does not run at instant t = 0 in the analyses performed according to time. Engine was started with a 50 Nm starter moment. Starting torque does not apply after the crank angle of 1 rad ( q > 1 ). Engine was loaded at different engine speeds with different pressure profiles and crankshaft speed fluctuations were investigated. The effects of different pressure profiles on crankshaft angular speed are shown in Fig. 7. Used pressure profiles here were determined by averaging of two different pressures for 11 cycles. A and B are the pressure profiles for one cycle. The Imep value of the A cycle is higher 0.16 bar than B cycle. When angular speed of crankshaft reached at 146.6, 230 and 314.15 rad/s, a proper external load was applied to the crankshaft to keep the mean angular speed constant. It was determined that required engine load is 86.75, 84 and 81.25 Nm when A cycle is used for 146.6, 230 and 314.15 rad/s angular speeds, respectively. Engine load is 86.5, 83.25 and 80.85 Nm, respectively, when B cycle is used. The reason for the decrease in the external loads is lower imep of the B cycle. It was also determined that crankshaft angular speed fluctuations occurred by about 14.7%, 4.3% and 1.4%, respectively, when A cycle is used. Crankshaft speed fluctuations decreased by about 12.3%, 3.4% and 1.1%, when B cycle used. The reason for the decrease of the crankshaft speed fluctuations is lower imep value when B cycle is used. Crankshaft speed fluctuations decreased as mean crankshaft speed increased in accordance with practice. But determined speed fluctuations are for one cycle because of using pressure profiles for one cycle. However, in-cylinder pressures vary cycleto cycle. Therefore, crankshaft speed fluctuates more. The effects of A pressure profiles for 11 cycles which COVimep is 5.15% are shown in Fig. 8. When the crankshaft angular speed reached to 146.6, 230 and 314.15 rad/s, 86.75, 84 and 81.25 Nm loads were applied to the engine, respectively. It was seen that load carrying capacity of the engine does not change compared to A cycle. Imep of the A cycle was calculated from A pressure profile, which consists 11 cycles. Therefore, the imep values of the A and A pressure profiles are equal. So the external moment of the engine remains stable. On the other hand the crankshaft speed fluctuations are increased according to A cycle. It was determined that crankshaft speed fluctuations are 17.9%, 5.5% and 2.4% for 146.6, 230 and 314.15 rad/s engine speeds, respectively. In addition, speed fluctuations occurred depending on the time not for cyclic because of cyclic variations. The effect of B pressure profile, which COVimep is 12.92%, on the crankshaft angular speed at different engine speeds is shown in is an indication of knocking in-cylinder pressure profiles. There is an increase in engine moment, as increasing knocking promotes the Imep. However, engine external moment does not change when B pressure profile is used because B and B pressure profiles’ Imeps are equal. On the other hand, the crankshaft speed fluctuations increased significantly.
Fig. 7. The variation of crankshaft angular speed dependent on the time for two different mean pressure profiles.
Fig. 8. The effect of A pressure profiles for 11 cycles on the crankshaft angular speed.
Fig. 9. The effect of B pressure profiles on the crankshaft angular speed.
Speed fluctuation rates of crankshaft were determined as 31.8%, 11.1% and 5% for 146.6, 230 and 314.15 rad/s engine speeds, respectively. When A and B pressure profiles are compared their imep values are too close to each other. The
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Fig. 10. The variation of gas pressure force in case of misfiring. Fig. 12. The effect of continuous misfire on the crankshaft angular speed.
Fig. 11. The effect of single misfire on the crankshaft angular speed.
reason for the higher speed fluctuations with B pressure profile is the higher COVimep of the B pressure profile. The effects of misfiring on cylinder gas pressure force acting piston are shown in Fig. 10. Crankshaft speed variations were investigated in the case of single and continuous misfire in the third cylinder. The gas pressure force is approximately halved in the case of misfire. In the case of continuous misfire, gas pressure force decreased in all cycles after the starting of the misfire cycle. The effect of single misfire on crankshaft angular speed is shown in Fig. 11. Engine speed was kept constant via loading the engine when the crankshaft angular speed reached at 230 rad/s. Mean crankshaft angular speed decreased 6.6 rad/s when single misfire occurred in the third cylinder. It is seen that 2.5 s is required to reach the crankshaft speed to average. Because, the load applied on the crankshaft does not change. In practice, vehicle speed decreases as the engine speed decreases. Resistance forces acting on the vehicle such as aerodynamic forces, rolling friction forces decrease with the decrease of vehicle speed. As a result, engine load will decrease a little. This situation can shorten the time to reach operating speed of the engine. The variations of the crankshaft angular speed are shown in the third cylinder during the continuous misfire in Fig. 12. 84 Nm external load was applied to the crankshaft when the
crankshaft angular speed reached 230 rad/s. Crankshaft angular speed decreased rapidly with the start of misfire in the third cylinder. Crankshaft angular speed became zero after about 2 s and exactly stopped after 3 s when the engine load was not decreased. However, in practice, external load on the engine was decreased by vehicle with shifting by the driver for not deceleration of the engine speed and prevented the engine from stopping. As shown in Fig. 12, external load is 72.5 Nm during the rapid decrease of the crankshaft angular speed. This provided to reach again the previous value of the engine speed at about 3.5 s. But continuous misfire caused to increase the crankshaft speed fluctuations from 4.3% to 8% in the third cylinder.
4. Conclusion A mathematical model has been developed to analyze the crankshaft speed fluctuations in a diesel engine. In the model, the in-cylinder gas pressure force, engine frictions and inertial effects of the parts which make linear and circular motion have been taken into consideration. The effects of misfiring problem and cyclic variations on the crankshaft speed fluctuations have been investigated. In the analysis two pressure profiles including 11 cycle (A and B) and average of these cycles (A and B) were used. The crankshaft speed fluctuations were determined as 4.3% and 3.4% at 230 rad/s average speed when A and B pressure profiles were used. The crankshaft speed fluctuations were determined as 5.5% and 11.1% at 230 rad/s for 5.15% and 12.92% COVimep values. The average crankshaft speed decreased by 6.6 rad/s when single misfire occurred in the third cylinder. The crankshaft speed fluctuations increased from 4.3% to 8% in case of continuous misfire.
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Hamit Solmaz received his Ph.D. at Gazi University, Turkey, in 2014. His research interests include engine vibration, Stirling engines, alternative fuels and combustion.
Halit Karabulut received his Ph.D. at Heriot-Watt University. He is currently a Professor at Gazi University. His research interests include engine vibrations, cam design and Stirling engines.
