ISSN 1068798X, Russian Engineering Research, 2012, Vol. 32, No. 7–8, pp. 523–525. © Allerton Press, Inc., 2012. Original Russian Text © E.V. Zubkov, A.A. Novikov, 2012, published in Vestnik Mashinostroeniya, 2012, No. 6, pp. 7–9.
Regulation of the Crankshaft Speed of a Diesel Engine with a Common Rail Fuel System E. V. Zubkov and A. A. Novikov Kamsk State Engineering and Economics Academy, Naberezhnye Chelny email:
[email protected] Abstract—A mathematical model is proposed for the regulation of the crankshaft speed of a diesel engine with a Common Rail fuel system by control of the injection time and the load. DOI: 10.3103/S1068798X12060317
In developing an automated test system for diesel engines, a mathematical model of the engine is required for adjustment of the test system’s parame ters. The dynamic properties of the diesel engine must always be taken into account in developing automatic control systems. In addition, the mathematical model may be used for computer control of the diesel engine’s operation in bench tests [1]. For a turbocharged diesel engine, equipped with the Common Rail fuel system, we may propose the following mathematical model on the basis of the model in [2–4] 2
2 n ( t) + T dn (t) dθ ( t) + K θ ( t ) T2 d 1 + T 0 n ( t ) = T θ θ 2 dt dt dt (1) dM C ( t ) – T M – K M M C ( t ), dt
one of the input parameters is held constant in tests of diesel engines [5]. In Fig. 1, we show the time depen dence of the crankshaft speed. Points 1–7 correspond to transitions from one state to another [6]. MATHEMATICAL MODEL FOR CONTROL OF THE CRANKSHAFT SPEED BY ADJUSTING THE INJECTION TIME AT CONSTANT LOAD When regulating the crankshaft speed n(t), the die sel engine may be described by Eq. (1). We now solve Eq. (1) for θ at constant load. Since Mc = const, we know that dMc(t)/dt = 0. Hence, Eq. (1) takes the form
where n(t) is the crankshaft speed; θ(t) is the injection time; Mc(t) is the load; t is the time; T0, T1, T2, Tθ, Kθ, TM, and KM are design constants of the motor. Bench tests of the KAMaz 760.64420 diesel engine confirm the model and permit the determina tion of the coefficients in Eq. (1). Hence, when Mc = const, the model takes the form
2
2 d n(t) dn ( t ) dθ ( t ) T 2 + T 1 + T 0 n ( t ) = T θ + K θ θ ( t ) 2 dt dt (3) dt – K M M c ( t ).
n, min 5
6
2
d n(t) dn ( t ) 0.0024 + 0.04 + n ( t ) 2 dt dt dθ ( t ) = 277.3 + 5555.6θ ( t ) – 10.2M c ( t ). dt
(2)
7 3 4
According to Eq. (2), the crankshaft speed may be varied by means of two input parameters: the injection time θ(t) and the load Mc(t) on the shaft. Hence, we may calculate the values of θ(t) and Mc(t) required for specified crankshaft speed. Mathematical models for the control of θ(t) and Mc(t) may be based on the calculations for motors with highpressure fuel pumps of rack type in [4]. As a rule, 523
2 1
t1
t2 t3
t4
t5 t6
t, s
Fig. 1. Dependence of the crankshaft speed n on the time.
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ZUBKOV, NOVIKOV
For the linear sections of the n(t) curve (Fig. 1), we n – ni t – ti may write = ; where i = 1, m , m is the ni + 1 – ni ti + 1 – ti number of points. On that basis ni + 1 – ni ni ti + 1 – ni + 1 ti n ( t ) = t + . ti + 1 – ti ti + 1 – ti
By inverse Laplace transformation, we obtain Kθ
– t A t + ⎛ B – T B⎞ e Tθ . θ A⎞ θA ⎛T θ ( t ) = + – K θ ⎝ K θ K 2θ ⎠ ⎝ K 2θ K θ⎠
Finally, we may write T ni + 1 – ni T0 ni ti + 1 – ni + 1 ti KM θ ( t ) = ⎛ 1 + + M c ⎝ Kθ ti + 1 – ti Kθ ti + 1 – ti Kθ
ni + 1 – ni ni ti + 1 – ni + 1 ti Let = k; = b. ti + 1 – ti ti + 1 – ti
(6)
Kθ
Then n ( t ) = kt + b.
(4)
ni + 1 – ni dn ( t ) . Note also that = k = ti + 1 – ti dt For the sake of simplicity, we may omit the small 2 2 d n(t) in Eq. (3), to obtain term T 2 2 dt dθ ( t ) T 1 k + T 0 ( kt + b ) = T θ + K θ θ ( t ) – K M M c ( t ). dt If we group the terms that contain θ on the left side, we obtain dθ ( t) + K θ(t) = T k + T ( kt + b ) + K M ( t ). Tθ θ 1 0 M c dt Laplace transformation yields 1 1 ( T θ p + K θ )θ ( p ) = T 0 k 2 + [ T 1 k + T 0 b + K M M c ( p ) ] . p p Let T0k = A, T1k + T0b + KMMc(p) = B. Then
– t T θ⎞ T ni + 1 – ni T0 Tθ ni + 1 – ni ⎞ ⎛ – – 1 e t. ⎜ ⎟ + 0 2 t ⎠ ⎠ Kθ ti + 1 – ti Kθ i + 1 – ti ⎝
From Eq. (6), which expresses the dependence of the injection time on the crankshaft speed and the load, we may determine the injection time corre sponding to the required crankshaft speed at specified load. MATHEMATICAL MODEL FOR CONTROL OF THE CRANKSHAFT SPEED BY ADJUSTING THE LOAD AT CONSTANT INJECTION TIME Analogously, we may obtain a model corresponding to regulation of the crankshaft speed by adjusting the load at constant injection time. First, we solve Eq. (1) for Mc. Since θ(t) = const, we know that dθ(t)/dt = 0. Hence, Eq. (1) takes the form 2
2 d n(t) dn ( t ) T 2 + T 1 + T 0 n ( t ) = K θ θ ( t ) 2 dt dt
(7)
dM C ( t ) – K M M c ( t ). – T M dt
A B θ ( p ) = 2 + . T ( p θ + K θ )p ( T θ p + K θ )p
(5) Substituting Eq. (4) into Eq. (7), we obtain
By means of undetermined coefficients, we may expand the fractions on the right side of Eq. (5) in terms of simple fractions 2
Tθ A Tθ A 1 A 1 A 1 2 = – + ; 2 T p+K 2 2 θ ( T θ p + K θ )p Kθ θ Kθ p Kθ p
dM c ( t ) + K M M c ( t ) = K θ θ ( t ) – T 1 k – T 0 ( kt + b ). T M dt By Laplace transformation 1 ( T M p + K M )M c ( p ) = – T 0 k 2 p + ( K θ θ ( p ) – T 1 k – T 0 b ) 1 . p
BT B 1 B1 = – θ + . ( T θ p + K θ )p Kθ Tθ p + Kθ Kθ p Grouping similar terms, we may write Eq. (4) in the form A 1 B T θ A⎞ 1 ⎛ T θ A B ⎞ 1 θ ( p ) = 2 + ⎛ – + – . ⎝ Kθ p K θ K 2θ ⎠ p ⎝ K 2θ K θ⎠ K p + θ Tθ
Let T0k = A and Kθθ(p) – T1k – T0b = B. Then B A M c = 2 + . T ( p M + K M )p ( T M p + K M )p
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REGULATION OF THE CRANKSHAFT SPEED OF A DIESEL ENGINE
We may expand the fractions on the right side of Eq. (8) in terms of simple fractions
n, min–1
525
(a)
1600
A 1 B T M A⎞ 1 M c ( p ) = – 2 + ⎛ + K M p ⎝ K M K 2M ⎠ p
1200
B + T 1 . M A⎞ – ⎛ ⎝ KM K2 ⎠ KM M p + TM
800 400
By inverse Laplace transformation, we obtain 0 2 n, min–1 1800
KM
– t ⎛ TM ⎞ A t + ⎛ B + T M A⎞ M c ( t ) = – 1 – e ⎜ ⎟. 2 ⎝ ⎠ KM KM KM ⎝ ⎠
K T ni + 1 – ni T0 ni ti + 1 – ni + 1 ti M c ( t ) = ⎛ θ θ – 1 – ⎝ KM KM ti + 1 – ti KM ti + 1 – ti T ni + 1 – ni t. ⎟ – 0 ⎠ Kθ ti + 1 – ti
1000 368.2
12
14 t, s
(b)
1000 800
1400 1.17
1400 407.4
1800 446.6
In Fig. 2b, we plot the specified n(t) dependence and the simulation. The relative error in simulation, calculated as the mean square deviation, is 0.53%. This confirms the precision of the model. Thus, in regulating the crankshaft speed of a turbo charged diesel engine with a Common Rail fuel system by control of the input parameters, the proposed
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2
4
6
8
10
12
14 t, s
Fig. 2. Simulated (continuous curves) and specified (dashed curves) time dependences of n at constant load Mc = 100 N m (a) and constant injection time θ = 1 ms (b).
model permits calculation of the injection time and the load corresponding to the specified crankshaft speed. Simulation confirms the validity of the model. REFERENCES
1800 1.242
In Fig. 2a, we plot the specified n(t) dependence and the simulation. The relative error in simulation, calculated as the mean square deviation, is 0.98%. Verification of Eq. (9) is based on simulation at constant injection time θ = 1 ms. We obtain the fol lowing results n, rpm Mc, N m
10
1200
(9)
KM – t TM ⎞
VERIFICATION OF THE MODEL By simulation of diesel operation on the basis of SciLab software according to Eq. (2), we may verify Eq. (6) at constant load Mc = 100 N m. We obtain the following results 1200 1.134
8
1400
From Eq. (9), which expresses the dependence of the load on the crankshaft speed and the injection time, we may determine the load corresponding to the required crankshaft speed at specified injection time.
n, rpm θ, ms
6
1600
Finally, we may write
T0 TM ni + 1 – ni ⎞ ⎛ + ⎜ 1 – e 2 KM ti + 1 – ti ⎠ ⎝
4
1. Dvigateli vnutrennego sgoraniya. Kn. 3. Komp’yuternyi praktikum. Modelirovanie protsessov v DVS (Internal Combustion Engines, Vol. 3: Computer Practicum: Simulation of Processes in Internal Combustion Engines), Lukanin, I.N. and Shatrov, M.G., Eds., Moscow: Vysshaya Shkola, 2005. 2. Lawson, C.L. and Hanson, R.J., Solving Least Squares Problems, Englewood Cliffs, New Jersey: Prentice Hall, 1974. 3. Krutov, V.I., Avtomaticheskoe regulirovanie i upravlenie dvigatelei vnutrennego sgoraniya (Automatic Control of Internal Combustion Engines), Moscow, 1989. 4. Zubkov, E.V., Makushin, A.A., and Bakhvalova, V.S., Simulation of the Operation of Internal Combustion Engines to Determine Their Transient Characteristics, Avto. Prom., 2009, no. 5, pp. 37–39. 5. GOST (State Standard) 14846–81: Auto Engines: Bench Testing Methods, 2003. 6. Makushin, A.A., Zubkov, E.V., and Novikov, A.A., Optimizing the Tests of Internal Combustion Engines, Sborka Mashinostr., Priborostr., 2010, no. 4, pp. 38–42.
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