Dynamic Model of a Diesel Engine for Diagnosis and Balancing

Dynamic Model of a Diesel Engine for Diagnosis and Balancing

PER

HILLERBORG

Master's Degree Project Stockholm, Sweden 2005

IR-RT-EX-0515

Abstract To monitor and control the combustion in a diesel engine one can study the speed signal from the flywheel. The idea is that if individual cylinders give different amount of torque this will lead to variations in the flywheel speed. A model which describes the cylinder torque based on flywheel speed can be used to estimate the torque from individual cylinders. With this new knowledge of the individual performance of each cylinder the engine can be balanced. The balancing aim at making the speed of the flywheel more even but also required a model with estimated cylinder torque as input. This model may also be used for testing new control algorithms easily and gaining understanding of the dynamics. In this thesis a time dissolved model is constructed to describe the cylinder pressure-, crankshaft-, flywheeland damper dynamics. The model is based on a physical point of view by approximating the system into nodes containing mass, stiffness and friction. The inputs into the model are injection data from the engine management system (EMS) and a torque from a drive line. Ways to reduce the complexity of the model are investigated in order to invert the model to estimate the injection data based on flywheel speed measurements. Measurements are done in a test bed to receive data required for model simulation and validation. The result is that the main behavior of the dynamics is caught. The self oscillation behaviors in some operating points are however not caught which indicates that the model can not explain all behaviors. A reduced model works almost as well but of course looses more of the non stiffness behavior. As expected, the model equations can not be solved in real time. The result of the inverted reduced model depends on the flywheel signal. When the signal contains little non stiffness behavior the result is good. An observer model based on the reduced model is suggested and tested in order to estimate the indicated torque from flywheel data. The observer manages to detect errors in the injection. Keywords: combustion supervision, cylinder balancing, physical model, engine model, cylinder pressure, flywheel speed, crankshaft, observer, cylinder injection

Preface This report is a master’s thesis at the Department of Signals, Sensors and Systems, Kungliga Tekniska H¨ogskolan. The work was carried out during February to August 2005 on Scania, S¨odert¨alje under supervision of Anna Pernest˚al and Henrik Pettersson.

Thesis outline Chapter 1 gives a background to the thesis and outlines the thesis objectives. Chapter 2 explains the model used in this thesis. Chapter 3 describes the measurements which were accomplished in a test bed at Scania, S¨odert¨alje. Chapter 4 gives details of the simulation and its result. Chapter 5 summarizes the conclusions of the previous chapters. Chapter 6 discusses future work. Appendix A demonstrates one way to derive the connecting rod kinematics. Appendix B contains more figures of the simulation result.

Acknowledgment I would like to thank my supervisors’ student Anna Pernest˚al and Henrik Pettersson at Scania. Anna and Henrik have throughout the project given me support and contributed with valuable knowledge, which has been greatly appreciated. I’m thankful for the assistance given in the test bed environment. I also would like to give my recognition to Andreas Renberg who contributed with the pressure modeling and Anders Flor´en who supported in the optimization process. I give my general appreciation to all people at Scania who assisted me at my work during my stay in S¨odert¨alje. A special thanks to my examiner professor Bo Wahlberg at KTH and people at NEE, for giving up time to answer questions related to my work. Finally I would like to thank my family for your constant love and support.

iv

Contents Abstract

ii

Preface and Acknowledgment

iii

Notation & Glossary

vii

1

Background & Objectives 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Thesis objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

The Model 2.1 Engine introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Modeling pressure in one cylinder . . . . . . . . . . . . . . . . . 2.2.1 The combustion cycle . . . . . . . . . . . . . . . . . . . 2.2.2 Derivation of the engine pressure . . . . . . . . . . . . . 2.2.3 Derivation of combustion pressure . . . . . . . . . . . . . 2.3 Modeling the engine dynamics. . . . . . . . . . . . . . . . . . . . 2.3.1 Derivation of the gas torque . . . . . . . . . . . . . . . . 2.3.2 Torque due to motion of the connecting rod and the piston 2.3.3 Modeling the crankshaft torque . . . . . . . . . . . . . . 2.3.4 Friction related to the cylinder system . . . . . . . . . . . 2.4 The complete model . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 The torque balancing equation . . . . . . . . . . . . . . . 2.4.2 Time domain state space model . . . . . . . . . . . . . . 2.4.3 Adding a drive line . . . . . . . . . . . . . . . . . . . . . 2.5 Model simplifications . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Reduce order . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Simplification in the mass torque . . . . . . . . . . . . . . 2.6 Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Pressure torque observer model . . . . . . . . . . . . . . 2.6.2 Extended observer model . . . . . . . . . . . . . . . . . . 2.7 Model error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Dynamic analysis . . . . . . . . . . . . . . . . . . . . . .

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2 2 3 3 5 5 5 5 6 6 7 8 8 9 10 11 11 11 11 11 12 13 13

Measurement 3.1 Measurement setup . . . . . . . . 3.1.1 Measuring flywheel speed 3.1.2 Measuring pressure . . . . 3.1.3 Measuring other variables

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Simulation 4.1 Matlab . . . . . . . . . . . . . . 4.2 Result of the pressure model . . 4.2.1 Sensitivity analysis . . . 4.3 Result of the mechanical model .

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1 1 1

4.4 4.5 4.6

4.3.1 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Comparison between measured and simulated flywheel speed Simulating the inverted reduced model . . . . . . . . . . . . . . . . . Simulated error in injection . . . . . . . . . . . . . . . . . . . . . . . Result of the extended observer . . . . . . . . . . . . . . . . . . . . .

5 Conclusions 5.1 Discussion . . . . . . . . . . . . . . . . . 5.1.1 Discussion of the pressure model 5.1.2 Discussion of the full model . . . 5.1.3 Discussion of inverting the model 5.1.4 Discussion of the observer . . . . 5.2 Conclusions of the objectives . . . . . . .

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6 Future Work

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References

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A Derivation of the connecting rod kinematics

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B Simulation figures

39

Notation & Glossary Symbols used in the thesis report.

Variables and parameters c j k l m N pg pengine pinl pmax r s s vf uel

absolute damping moment of inertia stiffness connecting rod length mass gas molecules cylinder pressure - atmosphere pressure pressure due to engine kinetics inlet pressure to cylinder maximum pressure inside a cylinder crankshaft radius cylinder displacement relative damping fuel flow

ms [ Nrad ] [kgm2 ] [Nm/rad] [m] [kg] [mol] [Pa] [Pa] [Pa] [Pa] [m] [m] ms [ Nrad ] mg [ CAD ]

Ap J R S T Tg Tload Tm Tf ric V

cylinder Area moment of inertia matrix universal gas constant piston node position matrix temperature gas torque brake load torque engine kinetics torque friction torque cylinder volume

[m2 ] [kgm2 ] J [ mol·K ]

δ θ θ˙ θ¨ η θinji θIV C θIV O θSOC λ

fuel/stoke crankshaft angle angular velocity angular acceleration material constant injection duration inlet valve closing inlet valve opening start of combustion r/l

mg ] [ stroke [CAD] [ CAD s ] [ CAD s2 ] [1] [CAD] [CAD] [CAD] [CAD]

[K] [Nm] [Nm] [Nm] [Nm] [m3 ]

Special functions gi Gi

geometrical function matrix geometrical function vii

Glossary CAD EMS EVO IVO OBD SOI

Crank angle degree Engine management system Exhaust valve closing Inlet valve closing On board diagnostics Start of injection

Chapter 1

Background & Objectives 1.1

Background

There are laws on emissions and how much noise a heavy duty truck may do. The drivers of heavy duty truck want an engine which offers both reduced fuel consumption and comfort. To full fill these laws, and demands a better understanding of the engines is required. In the diesel engine, fuel is electronically injected into the cylinders at a desired angle. The fuel combusts and the released energy is transformed into mechanical force which forces the crankshaft to rotate. The crankshaft has a flywheel attached on the side closest to the drive line. A sensor measures the rotating speed of the flywheel. The flywheel speed oscillates and its behavior depends on several factors where the cyclic torque from the cylinders is considerate. By changing the injection on individual cylinders, the behavior of the flywheel speed oscillation alters. This makes it possible to balance the engine by controlling the electronic injection with feedback from the speed signal. Thus small injection errors in individual cylinders can be corrected as well as unwanted orders1 of the oscillations can be removed. At present engines, the engine management system (EMS) filters out a few interesting orders of the oscillating speed signal. The unwanted orders, typically the half and the first engine order, are used as feedback to the injectors in order to balance the engine. Laws are also coming on on board diagnostics (OBD) where the EMS should be able to diagnose its condition. The cylinders diagnose aims at discovering if one cylinder is broken. This requires a model of the cylindercrankshaft dynamics which can observe the unmeasured cylinder torque.

1.2

Thesis objectives

As stressed in the background, models which describe the engine dynamics are important. With models the control algorithms can be improved, parts of the engine which are not measurable may be estimated and knowledge may be gained. Models also allow for control algorithms to be tested before running them on an engine. This thesis focuses in the dynamics from the cylinder injection to flywheel speed. The objectives are 1. Constructing a time dissolved model which can calculate the flywheel speed based on the fuel injection data. 2. Investigating the possibilities of simplifying the model to be able to invert it and thus being able to calculate the torque of the individual cylinders based on flywheel speed.

1 The natural engine order is the ignition order which is 3 on a 6 cylinder engine, since three cylinders ignites per crankshaft revolution. The other orders can arise due to for example mass torques, self oscillation and variation in individual cylinder injection. The low order orders are those which are most unwanted since they feel unpleasant.

1

Chapter 2

The Model In this chapter a model describing the cylinder pressure-, crankshaft-, flywheel- and damper dynamics with inputs from injection data and drive line torque is presented. The model is based on a physical point of view and results in ordinary differential equations. The model used in this thesis is based on the model presented by Schagerberg in [17] with an extension in the gas modeling. Model simplifications to reduce calculation time are suggested. Observers which are based on a simplified model are also suggested. In the end of the chapter a brief dynamic analysis is done.

Figure 2.1: The engine parts described in the engine introduction.

2.1 Engine introduction The main parts in an engine are cylinders, crankshaft, flywheel, damper and the connecting rods which can all be seen the figure above. The combustion takes part inside the cylinders forces the crankshaft to rotate via the connecting rods which connects them. Since the torque from the cylinders oscillates a damper wheel is connected to the crankshaft to dampen the speed fluctuations. The flywheel is connected on the other side of the crankshaft relative to the damper. It is a wheel which has a high moment of inertia in order to smooth the torque which is going to be passed on to the drive line. 2

3

2.2

2.2. Modeling pressure in one cylinder

Modeling pressure in one cylinder

The chemical reaction when transforming fuel and oxygen into carbon dioxide and water provides energy to the engine. This takes part inside the cylinders. Heavy duty trucks usually have 4, 5, 6 or 8 cylinders which delivers a steady amount of power. In this section the combustion cycle is described and equations for modeling the pressure inside the cylinder are suggested.

2.2.1

The combustion cycle

The combustion cycle takes part inside the cylinders and can in a four-stroke diesel engine be divided into four phases: intake, compression, expansion and exhaust, see figure 2.2. The phases are controlled mechanically by the camshaft which controls the inlet- and the outlet valve. The fuel injection is done electronically and can be controlled to optimize performance. To finish all phases the crankshaft requires two revolutions.

Figure 2.2: A four-stroke engine can be divided into four phases. Figure taken from [9].

Intake During the air intake phase new air flows into the cylinder through the inlet valve sucked by the movement of the piston and pushed by the turbo. The pressure in the cylinder is approximately equal to the turbo pressure. Compression During the compression both valves are closed and the piston is moving upward which compresses the air trapped inside the cylinder which increases the pressure. Expansion Approximately at the top piston position, diesel is injected and starts to combust due to the high temperature and pressure from the compression. The piston is starting to move downwards and is pushed by the extra pressure which forces the piston to accelerate further. The phase is called expansion since the cylinder volume is expanding. Exhaust At the bottom piston position the outlet valve is opened and the burned gases exit the cylinder during the exhaust phase. To start the chemical reaction between fuel and oxygen, a gasoline engine (Otto engine) uses a spark, and a diesel engine uses high temperature and pressure which result in a spontaneous ignition. A key number defining the engine is the compression ratio. The compression ratio is defined as the relation between minimum cylinder volume and the present cylinder volume. The compression ratio determines how much the air inside the cylinder can be compressed. If higher compression pressure is wanted more air has to pushed into the cylinder. This can be done with a turbo. The turbo compresses the air on its way towards the cylinder. If the turbo compresses the air to two bar compared to one bar the compression pressure in the cylinder will approximately double up which will shorten the combustion time. This is a very fuel efficient way to gain more power from the

Chapter 2. The Model

4

engine and is used in most diesel engines. The turbo is driven by the gas which exits the cylinder in the exhaust phase. More information can be found in [2]. To summarize the pressure during all phases the pressure in a cylinder can be described as (2.1) p(θ) = pengine (θ) + pcomb (θ), where θ is the crankshaft angle, pengine (θ) is the pressure due to the engine kinetics controlled by the camshaft valves and pcomb (θ) is the resulting pressure due to combustion. When valves are opened pengine (θ) is approximately equal to turbo pressure. A plot of the pressure can be seen in figure 2.3.

Figure 2.3: The total pressure in the cylinder during one combustion cycle. The lower curve shows the engine pressure pengine during the combustion. The plot is taken from [9].

Figure 2.4: A P-V diagram describing the relation between pressure and volume. The highest pressure (here 35 bar) is when the piston is at it’s top position. Then the pressure drops when the piston moves downward during the expansion (here to 4 bar). Then the gases exits the cylinder as the piston moves upward (volume decreases, pressure ≈constant. New air is sucked into the cylinder (volume increases, pressure ≈constant). The valves close and the air starts to be compressed as the volume decreases. Finally fuel is injected when the volume is at its minimum and pressure is rapidly increased. The plot is taken from [9].

5

2.3. Modeling the engine dynamics.

2.2.2

Derivation of the engine pressure

pengine (θ) is denoted by the pressure due to the engine kinetics controlled by the camshaft valves. A simple way to calculate the pengine is to use the ideal gas law equation see [9] or [2]. pengine (θ)V (θ) = N (θ)kT (θ),

(2.2)

where pengine (θ) is the pressure, V (θ) the cylinder volume, N (θ) the number of gas molecules inside the cylinder, k is the Boltzmann constant and T (θ) the temperature. Since no external heat or energy is considered to be brought to the cylinder system an adiabatic equation should be used. The cylinder volume can be described as V (θ) = Ap s(θ). (2.3) Here Ap is the combustion chamber area and s(θ) is the piston position. When the valves are open the pressure pengine (θ) is approximately equal to the inlet pressure pinl from the turbo. When both valves are closed, at inlet valve closing (IV C), the amount of particles N (θ) is constant which makes the pressure and the temperature increase as the volume is decreasing. Since the combustion chamber is not perfectly isolated one can assume that some pressure is lost because of leakage of gas molecules and of loss of heat, hloss (θ, θIV C ). It can be approximated as a linear function hloss (θ, θIV C ) = q(θ − θIV C ). The pressure due to engine kinematics can be calculated as When the valves are not closed pengine (θ) ≈ pinl . (2.4) When both valves are closed and air is compressed

2.2.3

pengine (θ)

=

N (θIV C )

=

hloss (θ, θIV C )N (θIV C )kT (θ) V (θ) pinl V (θIV C ) . RT (θIV C )

(2.5) (2.6)

Derivation of combustion pressure

The start of injection angle (θSOI ) is the crank angle degree (CAD) where the diesel starts to be injected into the combustion chamber. This occurs approximately when the piston is at its top position in the beginning of the expansion phase. There is a short delay until the diesel fully starts to combust into gas. The combustion pressure can be expressed as ˙ T, pengine (θ)), pcomb (θ)f (θinj , vf uel , θSOI , θ,

(2.7)

mg ], θ˙ is the speed of the flywheel, T is the where θinj is the injection duration [CAD], vf uel the fuel flow [ CAD temperature [K] and pengine the pressure that arise from the compression. The combustion pressure equation pcomb (θ) is solved in two steps. The first step is to transform the fuel flow into a ”heat release”. The equations involved in that part can be found in articles [5], [6] and [7]. The second step is to transform the heat release to pressure. The equations for the last step can be found in [9].

2.3

Modeling the engine dynamics.

In this section torques due to pressure, mass and friction is derived. The ”torque-balancing equation” for one cylinder is also derived and explained.

2.3.1

Derivation of the gas torque

To compute the force working along the cylinder axis the piston area is multiplied with the relative cylinder pressure, pg (θ). The relative pressure is the difference in pressure inside and outside the cylinder. The resulting gas torque is then described as ds ds = pg (θ)Ap (2.8) Tg (θ) = Fg dθ dθ

Chapter 2. The Model

where pg is

6

pg (θ) = pengine (θ) + pcomb − p0 .

(2.9)

Here p0 is the pressure outside the combustion chamber (≈1 bar). pengine and pcomb can be calculated using (2.5) and (2.7). The derivation of the equation can be found in [17].

2.3.2 Torque due to motion of the connecting rod and the piston The connecting rod connects the piston to the crankshaft and transfers torque. The piston is moving up and down along the cylinder axis while the crankshaft is rotating. This leads to that the connecting rod undergoes both a translational- (from the piston movement) and a rotational movement (from the crankshaft). This movement of mass of the connecting rod and the piston results in a torque. To simplify torque calculations the connecting rod is divided into two point masses, mA and mB . mA represent the mass of the piston plus the oscillating mass of the connecting rod. mB is the part of the connecting rod which undergoes a rotational motion. See figure 2.5. s

A

r+l

l

phi

theta

B

r

Figure 2.5: The mass of the connecting rod and the piston is divided into two point masses: one rotating (mB ) and one oscillating (mA ). They are located according to the figure. ¨ θ, ˙ θ) of the rotating and oscillating masses is the derivative of the kinetic energy Em . The mass torque, Tm (θ, ˙ The derivation of the mass torque can be done as The kinetic energy is the calculated as 12 J(θ)θ.


2π

1 J(θ)θ˙2 2 0  dEm 1 dJ(θ) ˙2  ˙ = Tm θ˙ = θ θ J(θ)θ¨ + dt 2 dθ ¨ θ, ˙ θ) = J(θ)θ¨ + 1 dJ(θ) θ˙2 ⇒ Tm (θ, 2 dθ Em



=

J(θ) J(θ) dθ

Tm dθ

=

(2.10)

ds 2 ) = mB r2 + mA ( dθ 2 d s ds = 2mA dθ2 dθ .

The mass torque usually works in opposite direction of the gas torque and thus reduces speed variations. The expressions for the piston displacement are described in appendix A.

2.3.3 Modeling the crankshaft torque The crankshaft is made out of steel and its design depends on the number of cylinders and their positions e.g. a V, in-line or a opposed (boxer) arrangement. A common way to model the crankshaft is to divide it into a few nodes placed at strategic positions. All nodes has a lumped mass which is connected to their neighboring nodes with a torsion spring. There is also a relative and a absolute friction acting on the system. The relative friction

7

2.3. Modeling the engine dynamics.

dampens the difference in speed between two masses and the absolute friction comes from friction in bolts and bearings. The model used in this thesis has one mass in the damper one in the damper ring, one at each cylinder connection and one for the flywheel. See figure 2.6. For a six cylinder engine this means 9 masses. A drive line can also be added with more masses which will be investigated in section 2.4.3. The advantage of using this model is that the model parameters are well known for each specific crankshaft and are easy to estimate. Flywheel

J_9 Damper

Cyl 1

Cyl 2

Cyl 3

Cyl 4

Cyl 5

Cyl 6

Free end J_4

J_3

J_1

J_5

J_6

J_7

J_8

J_2 k_1

k_2

k_3

k_4

k_5

k_6

c_12

c_23

c_34

c_45

c_56

c_67

c_3

11 0 01 0 01 1 01 1 0 1 0 1 0 0

c_4

c_5

11 0 01 0 01 1 01 1 0 1 0 1 0 0

11 0 01 0 01 1 01 1 0 1 0 1 0 0

k_7

c_78

c_6

11 0 01 0 01 1 01 1 0 1 0 1 0 0

k_8

c_89

c_7

11 0 01 0 01 1 01 1 0 1 0 1 0 0

c_8

11 0 01 0 01 1 01 1 0 1 0 1 0 0

Figure 2.6: Node model for a six cylinder engine. The first node is at the damper wheel, the second at the damper ring, third-eighth are at the connecting rods and the ninth is at the flywheel. The absolute friction is considered to primary come from the cylinder-crankshaft dynamics see section 2.3.4 and is set to zero for the crankshaft. The crankshaft torque, TCi , from one node of the crankshaft can thus be modeled as T Ci

= Ji θ¨ + ki−1 (θi − θi−1 ) + ci−1 (θ˙i − θ˙i−1 ) − ki (θi+1 − θi ) − ci (θ˙i+1 − θ˙i ),

(2.11)

where ki is the stiffness between node i and node i + 1, ci is the relative friction between node i and node i + 1. Note that the crankshaft moment of inertia is constant. The stiffness and inertia parameters was received from [1]. The relative damping parameters were estimated as ci = ηdamp

ki , 2πf0

(2.12)

where ηdamp is a material constant and f0 is a resonance frequency.

2.3.4

Friction related to the cylinder system

The most friction in the engine system can be found in the cylinder - connecting rod - crankshaft system. Some frictions are easy to realize as friction in the connecting rod bolts and the crankshaft bearings while others as air friction is more complex. To model each friction component by it self requires much work and component knowledge and will probably not be very accurate in the end. Instead of summing up all individual friction components the approach is to try to model a friction so it fits data. It is easy to calculate the energy loss due to friction since it is the input energy minus the output. Tf riction

1  = 4πn




θ0 +4πn

(Tg (θ) − Tload )dθ

 (2.13)

θ0

Where Tf riction is the friction torque and n is a positive integer describing the numbers of crankshaft revolutions. The next step is to try to make a friction model so that the error gets as small as possible. A model describing the friction torque on one cylinder-crankshaft connection could for example be a function with angle positions, velocities or acceleration. ¨ θ, ˙ θ) Tabsf ric ≈ f (θ,

(2.14)

Chapter 2. The Model

8

This friction is called absolute friction since it only depends on absolute1 velocity of the crankshaft and not relative velocity due to a non stiff crankshaft. More information of different friction types can be found in [16].

2.4 The complete model In this section the torque equations from the previous section are put together into the torque balancing equation.

2.4.1 The torque balancing equation Putting all torque contributions together and extending it into a multi-body system the torque equation reads ¨ θ, ˙ θ) − Tload − Tabsfric (θ). ˙ Jθ¨ + Cθ˙ + KθTg (θ) − Tm (θ,

(2.15)

Here θ is now a vector containing the angles for all nodes. J, C, K are symmetric matrices of size N × N where N is the number of lumped masses. The vector Tload is the torque from a drive line and is considered to only work on the last node. During the measurement Tload could be considered constant. Note that the left side of the equation is the crankshaft torque. The N × 1 angle vector θ= The N × N moment of inertia matric

 ⎡

⎢ ⎢ ⎢ J=⎢ ⎢ ⎣ The N × N stiffness matric

⎡

⎢ ⎢ ⎢ K=⎢ ⎢ ⎣ The N × N damping matric

θ1

θ2

θ3

...

θN

J1 0 0 .. .

0 J2 0 .. .

0 0 J3 .. .

... ... ... .. .

0

0

0

0

T

.

(2.16)

⎥ ⎥ ⎥ ⎥. ⎥ 0 ⎦ JN

(2.17)

0 0 0

k1 −k1 0 .. .

−k1 k1 + k2 −k2 .. .

0 −k2 k2 + k3 .. .

... ... ...0 .. .

0

0

0

−kN −1

⎤

0 0 −kN −1 kN −1

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦

⎤ −c1 0 ... 0 c1 ⎢ −c1 c1 + c2 −c2 ... 0 ⎥ ⎥ ⎢ ⎢ 0 c + c . . . 0 ⎥ −c 2 2 3 C=⎢ ⎥. ⎥ ⎢ .. .. .. .. ⎣ . . −cN −1 ⎦ . . 0 0 0 −cN −1 cN −1

(2.18)

⎡

(2.19)

To keep track of which nodes the cylinders are connected to a N × NC matrix S is introduced. NC is the number of cylinders. For a six cylinder engine with 9 nodes S becomes ⎤ ⎡ 0 0 0 0 0 0 ⎢ 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ 1 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ 0 1 0 0 0 0 ⎥ ⎥ ⎢ ⎥ S=⎢ (2.20) ⎢ 0 0 1 0 0 0 ⎥. ⎢ 0 0 0 1 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 0 0 1 0 ⎥ ⎢ ⎥ ⎣ 0 0 0 0 0 1 ⎦ 0 0 0 0 0 0 1 Absolute

friction can be thought of as a grounded friction which is not the case with the relative friction.

9

2.4. The complete model

The first row correspond to the damper wheel, the second to the damper ring, the third-eighth are each connected to one cylinder. The ninth correspond to the flywheel node. If the engine has four cylinders and the system was modeled with four nodes with a damper wheel in node one, cylinder one and two in node two, cylinder three and four in node three and a flywheel in node four, S would look like this ⎡ ⎤ 0 0 0 0 ⎢ 1 1 0 0 ⎥ ⎥ S=⎢ (2.21) ⎣ 0 0 1 1 ⎦. 0 0 0 0 The next step is to simplify notations by rewriting the expressions involved in the pressure torque equation (2.8) and mass torque equations (2.10). Three geometrical functions for a single cylinder are defined g1 (θ) = g2 (θ) = g3 (θ) =

ds dθ 1 J(θ) 2 dθ

Ap

J(θ)

(2.22) d2 s ds dθ2 dθ ds 2 mA ( ) . dθ

mA =

(2.23) (2.24)

In a multi cylinder engine the cylinders have a specified fire sequence. In the six cylinder engine which is investigated in this thesis, the fire sequence is 1-5-3-6-2-4 where cylinder 1 is closest to the damper wheel. That means that the first cylinder fire, then the fifth cylinder and so on. One cylinder fires every 120 CAD, meaning that it takes 720 CAD for all cylinders to fire once. The phase between the cylinders is defined as Ψ=



ψ1

ψ2

...

ψNC

T

.

(2.25)

The single cylinder geometrical functions gi (θ) can now be translated into multi cylinder geometrical functions T Gi (ST θ − Ψ)diag(gi (sT 1 θ − ψ1 ), . . . , gi (sNC θ − ψN )),

(2.26)

where si correspond to the i:th column in the S matrix. The gas torque in the multi body model may now be written as Tg (θ) = SG1 (ST θ − Ψ)pg (ST θ).

(2.27)

Here pg (·) is a NC × 1 vector of the pressure in each cylinder. The extension of the mass torque defined in equation (2.10) becomes ¨ θ, ˙ θ)(SG3 (ST θ − Ψ)ST + mB r2 SST )θ¨ + SG2 (ST θ − Ψ)ST θ˙  θ. ˙ Tm (θ,

(2.28)

The  means the elementwise multiplication.

2.4.2

Time domain state space model

The model equation (2.15) is a second order differential equation with N states. By expanding the states to 2 × N states the equation can be reformulated as a first order differential equation. The new states are the angle positions and the angle velocities T T xT (2.29) θ˙T )T . x = (xT 1 2 ) = (θ ¨ θ, ˙ θ), in equation (2.28) can be separated into two parts, one that is independent of The mass torque, Tm (θ, ¨ ˙ θ). By using the notation in the previous speed Tm,1 (θ, θ) and one that is independent of acceleration Tm,2 (θ, section the mass torque reads ¨ θ) = (SG3 (ST θ − Ψ)ST + mB r2 SST )θ¨ Tm,1 (θ, ˙ θ) = SG2 (ST θ − Ψ)ST θ˙  θ. ˙ Tm,2 (θ,

(2.30) (2.31)

All terms which are dependent of θ¨ are gathered in a equation often referred to as the ”varying inertia”. J(θ) = J + SG3 (ST θ − Ψ)ST + mB r2 SST

(2.32)

Chapter 2. The Model

10

With these rearrangements the torque-balancing equation may be reformulated as ˙ θ) + Tg (θ) − Tload − Tabsfric (θ). ˙ J(θ)θ¨ = −Kθ − Cθ˙ − Tm,2 (θ,

(2.33)

By using the new state vectors on equation (2.33) and multiplying both sides with J(θ)−1 the state space model can be written as  

x1 0 I + x˙ = x2 −(J(x1 ))−1 K −(J(x1 ))−1 C 

0 + −(J(x1 ))−1 (SG2 (S T x1 − Ψ)S T x2  x2 + S · 1N · Tabsf ric (x2 )) 

0 . (2.34) (J(x1 ))−1 (SG1 (S T x1 − Ψ)pg (S T x1 ) − Tload ) The first term in equation (2.34) describes the dynamics of the crankshaft, the second the dynamics of the piston-crank inertia and friction and the last term describes the dynamics of the input signals.

2.4.3 Adding a drive line The package behind the engine is called the drive line. In heavy duty trucks it basically consists of shafts and gearwheels and compared to the engine parameters the torsion stiffness is relatively low and the moment of inertia is very large. When engine measurements were conducted in the test bed the engine was connected to a brake. This connection is stiffer than on a truck while its moment of inertia is lower. The drive line can be modeled as the engine with nodes containing moment of inertia J, stiffness k and relative damping c. Since the brake influence the engine a model for the brake should be used when comparing simulations against measurement. A model containing 5 nodes and a reduced model containing 1 node is used.

11

2.5 2.5.1

2.5. Model simplifications

Model simplifications Reduce order

Depending on how accurate the solution requirements needs to be some simplifications may be done in the model. One example is to reduce the number of nodes by putting nodes together. One special case is to consider the crankshaft as completely stiff. This is a fairly good approximation when the crankshaft does not experience any resonance but less good when it does. By doing this approximation only one node has to be used for the crankshaft which greatly reduces the order of the model. The torques just has to be summed leading to the equation J θ¨ + C θ˙ + Kθ =

N 

¨ θ, ˙ θ) − Tabsf ric )i − Tload . (Tg (θ) − Tm (θ,

(2.35)

i=1

Here J is the total moment of inertia of the crankshaft, damper and the flywheel, C is an absolute friction and N the number of pistons. The damper and the flywheel could be considered stiff as well which makes θ in (2.35) to a scalar and C = K = 0. To calculate the gas torque Tg for each cylinder is very easy now, one just have to rewrite the equation to N 

Tg (θ)i = I θ¨ + C θ˙ + Tload +

i=1

N  ¨ θ, ˙ θ) + Tabsf ric )i . (Tm (θ,

(2.36)

i=1

Since the pressure peak from each cylinder is 120 CAD apart it is possible to estimate the torque for all individual cylinders, given data of the flywheel speed. The torque due to stiffness has been removed and may cause errors in the Tg estimation even though the mean stiffness torque is zero.

2.5.2

Simplification in the mass torque

The mass torque equation reads (see section 2.3.2 on page 6) 2 2 ¨ θ, ˙ θ) = (mA ds + mB r2 )θ¨ + mA d s ds θ˙2 . (2.37) Tm (θ, dθ dθ2 dθ According to A.S. Rangwaka in [15] and S. Schagerberg in [17] the mass torque may be approximated as

d2 s ds 2 mA 2 ¨ w, ¯ θ) = ( r + mB r2 )θ¨ + mA 2 w ¯ , T¯m (θ, 2 dθ dθ where θ˙ is estimated as a average speed w, ¯ and

ds 2 dθ

¯2≈ ds is approximated as dθ

(2.38)

r2 2 .

This approximation differs 2-3% in torque contribution from (2.37) on a six cylinder engine but differs more on a four cylinder engine. Note that the terms multiplied with θ¨ in equation (2.38) is constant. This simplified mass torque is often refereed to as the constant inertia equation.

2.6

Observers

In this section two observers are set up. The first observer model introduced is a suggestion from [3]. The other observer has one more state added and is named the extended observer.

2.6.1

Pressure torque observer model

The only measured signal in the model is the flywheel speed. The injections data signal from the engine management system (EMS) is also know. What happens between input signals and flywheel measurements is interesting to know, especially the things that occur in the cylinders. By modeling the engine and using the flywheel measurements as inputs the unmeasured dynamics may be estimated. The accuracy depend on the model, the measured signal and the observer feedback design.

Chapter 2. The Model

12

Xhat(n)

y

Xhat(n+1)=A(n)Xhat(n)-L(n)(yhat(n)-y(n)) yhat(n)=CX(n)

yhat(n)

Figure 2.7: The observer estimates the torque with the angular velocity as inputs. A time-varying linear observer for torque observation is suggested in a paper by [3]. The engine is considered to be completely stiff and the torque due to stiffness is not modeled. An approximation is made that the torque variation due to the pressure, the friction and the load during one angular step is small. This two states engine model writes  X(n + 1) = A(n)X(n) (2.39) y(n) = CX(n) with X(n) A(n) C



θ˙2 (n) Tg (n) − Tload (n) − Tf ric (n)  2∆θ 2∆θ  f (θn ) J(θ 1 − J(θ ) n n) = 0 1   1 0 , = =

T (2.40)

where ∆θ is the angle path and f (θn ) and J(θn ) are mass torque equations (2.41). The measurements conducted on the flywheel are received for every angle step ∆θ which is a fix value somewhere between 1-10 CAD depending on flywheel design. More about the measurements will be found in next chapter.  NC ds(θn )j 2 J(θ) = Jcrankshaf t + mB r2 + mA j=1 ( dθ ) (2.41) NC ds(θn )j d2 s(θn )j dJ(θn ) f (θ) = dθ = 2mA j=1 dθ dθ 2 Depending on if a drive line is used the moment of inertia, Jcrankshaf t , may have to be changed. An observer with linear feedback on the squared angular speed is suggested with pole placement.  ˆ + 1) = A(n)X(n) ˆ X(n − L(n)(ˆ y (n) − y(n)) (2.42) ˆ yˆ(n) = C X(n) Stability and observability for the system is proven in [3]. Observers with different design can also be found in [3] and [4].

2.6.2 Extended observer model The torque due to the gas pressure is cyclic. This information may be used to better describe the torque variation. The torque curve is steeper on the compression side than on the combustion side and they are close to dT linear on both side. The derivative of the gas torque curve w(θ) ≈ dθg could be approximated as a square wave which is illustrated in figure 2.8. A new state x3 which describes the size of the pressure torque amplitude is introduced. The discrete indicated torque equation can now be estimated as T (n + 1) = T (n) + w(θn )∆θx3 . The new 3 states model becomes  X(n + 1) = A(n)X(n) . (2.43) y(n) = CX(n) with

13

2.7. Model error

1 T

g

0.8

w=dT /dθ g

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8

0

100

200

300

400 CAD

500

600

700

Figure 2.8: The gas torque is shown along with its approximated derivative w(θ). The sizes of both curves are scaled.

X(n) A(n) C

=



θ˙2 (n)

Tg (n) − Tload (n) − Tf ric (n) max(Tg ) − min(Tg ) ⎤ ⎡ 2∆θ 2∆θ 0 1 − J(θn ) f (θn ) J(θ n) = ⎣ 0 1 w(θ) ⎦ 0 0 1   1 0 0 . =

T

(2.44)

Here w(θ) is the approximated gas torque derivative function seen in figure 2.8. An observer with linear feedback with the squared angular velocity can be applied.

2.7

Model error

The model is a simplification of the real system. An example is that the crankshaft is transformed into few nodes containing moment of inertia, friction, and stiffness. If more nodes would be used the solution would probably be better but the time to solve the ODE would grow and more parameters has to be estimated.

2.7.1

Dynamic analysis

The speed of the flywheel depends on the contributions from of all torque acting on the system. The main torque contributions comes from: the gas pressure torque Tg (θ), the brake load Tload and torque due to moving ˙ θ). Tload is in static cases just a constant. The gas torque is dependent of the fuel injection mass J(θ)θ¨ + Tm (θ, while the mass torque is dependent of the angular acceleration θ¨ and the angular velocity squared θ˙2 . This means that at low speeds the total torque mostly consists of the gas torque while at higher speeds the torque due to mass is considerable. These phenomena will be seen clearly in the simulation result.The size of the gas torque depends on the load that has to be pulled. Higher load requires higher pressure torque.

Chapter 2. The Model

14

fm1500varv0procref 2000

fm1500varv100procref Tg−Jprim −Jprim Tg

1500

8000 T −Jprim g

−Jprim Tg

6000

1000 4000

Nm

Nm

500 2000

0 0 −500 −2000

−1000

−1500

0

100

200 CAD

300

−4000

0

100

200 CAD

300

Figure 2.9: Jprim is the mass torque term from equation (2.10) which is dependent only the velocity and the angle. Since the speeds are 1500 rpm for both figures the Jprim torque is the same. The right figure has a higher load to pull and which requires a higher gas torque. Note that the main oscillation is 6 oscillations per revolution (150 Hz) for the left figure while the right has 3 (75 Hz).

Chapter 3

Measurement This chapter contains the measurements which were carried out during the thesis in order to verify and tune the model. The main model which describes the speed of the flywheel based on fuel injection data is two individual models put together: the pressure model and the mechanical model. Measurement data are extracted so all models could be investigated separately to reduce model errors. All measurements were conducted in a test bed at Scania, S¨odert¨alje, on a DLC6 engine.

Figure 3.1: The test bed setup. Measured signals are θ˙ (≈rpm), cylinder pressure and brake torque. Injection data are also saved.

3.1

Measurement setup

The cylinder pressure signal and the flywheel speed signal was sampled in 5Mhz for 0.5 seconds in a DL750P, which is a scope and chart recorder manufactured by Yokogawa. The cylinder pressure signal was a voltage signal and the flywheel speed signal was a square wave signal where each square indicates 1 crank angle degree (CAD). Data was also acquired by the test bed equipment and the engine management system (EMS). The thirteen operating points which were measured can be seen in the table below. In all operating points the injection was changed for one or three cylinders to represent errors in the injection. 15

Chapter 3. Measurement

16

rpm \ Brake load 500 1000 1500 1800

0% x x x

20% x x x x

50% x x x

100% x x x

3.1.1 Measuring flywheel speed The flywheel speed signal is a square wave signal where each square indicates one tooth on the flywheel. In the test bed the flywheel has one CAD between each tooth and in a standard engine the distance is six CAD. The signal is in the time domain and thus can be transformed into flywheel speed which was done directly in the DL750P scope. The scope also takes a mean over a short interval which results in that the speed signal changes value each two CAD. A CAD vector is created to keep track of the angle position. A problem when measuring the position of the flywheel in many small intervals is that imperfections in the construction of the flywheel can result in poor measurement. According to [8] a tooth can be as much as one CAD misplaced on a standard flywheel with 120 teeth. This result in a 16% peak change of speed. To avoid these errors the signal is filtered which reduces measurement error but also some of the natural speed fluctuations will vanish. In this thesis the signal was digitally filtered without phase shifts to avoid time delays. The filtering was done to the 20th order with a butter filter.

3.1.2 Measuring pressure The pressure measurement was carried out on cylinder 6 which is the cylinder closest to the flywheel. The received signal to the DL750P scope was an analogous voltage signal. Other interesting signals measured by the test bed equipment is the maximum pressure (pmax ), its CAD location (θpmax ) and the inlet pressure (pinl ). The raw pressure curve needs some adjustments. First the signal is digitally filtered with a butter filter to reduce measurement noise. The filtering was done so no phases were shifted. Then the pressure needs to be translated from the time domain into the CAD domain. This translation is done almost as with the flywheel speed. Since the pressure curve has larger transients than the speed the pressure needs to be sampled more often. Five pressure samples are taken for each speed sample which results in 2.5 pressure samples per CAD. The data is equidistantly chosen over the 2 CAD intervals, with the assumption that the speed is considered to be constant over the two CAD sample period. The next step is to adjust the pressure measurement both vertically and horizontally. The vertical adjustment is due to that the voltage signal V (·) has to be transformed into a pressure signal. To do this the inlet pressure pinl was to be considered as the minimum pressure and pmax the maximum pressure. The voltage signal was then adjusted with this knowledge.   p(·) = V (·) − VL (·)

pmax − pinl + pinl max(V (·)) − VL (·)

(3.1)

Where VL (·) is a mean of the voltage curve somewhere between the peaks, and describes the minimum value of the curve. The angle position could only be measured in a relative manor. The distance between the flywheel teeth was measured but no reference to the revolution position was measured. The model assumes that the top dead center (TDC) is at approximately 0 CAD. The measured pmax position θpmax is used as a reference to where TDC is and both position vectors are corrected.

Pressure validation The pressure is assumed to be cyclic over 720 CAD and should in an operating point deliver about the same pressure every cycle. To verify this assumption cylinder pressure in different cycles has been drawn in the same plot in figure 3.3 for 1500 rpm and a brake load of 1045 Nm. The same comparison has been done on all different measured operating points in brake load and rpm. The conclusion is that the assumption holds. The small error could be caused by measuring error and/or small variance in injection data.

17

3.1. Measurement setup

Figure 3.2: The voltage curve (red) needs to be reshaped and has its angle position to the blue curve. fm1500varv50procref

fm1500varv50procref 1

1 0.9

0.9

Pressure normalized to 1

Pressure normalized to 1

0.8 0.7 0.6 0.5 0.4 0.3 0.2

0.8

0.7

0.6

0.5

0.1 0 −90

−45

0 CAD

45

90

0.4 −5

0

5

10 CAD

15

20

25

Figure 3.3: The measured pressure does not differs much between each cycle which can be seen in the left plot. The variance in pressure due to compression is almost zero while the pressure due to combustion is a bit larger which can be seen in the plot to the right which is a zoomed version of the left figure.

3.1.3

Measuring other variables

The EMS handles the data gathered from the standard engine sensors and can be received by the software Vision. The interesting parameter data collected by Vision is listed and described in table (3.1). The test bed equipment is able to do many engine measurements which are not done on a standard truck engine. The interesting parameter data gathered from the test bed is listed and described in table (3.2).

Parameter name pinl θSOI tinj ∆tinj

explanation The inlet pressure.[bar] Start of injection.[CAD] The duration of the injection.[ms] The added duration of thr injection.[ms]

Table 3.1: Parameter data gathered by the EMS with software Vision.

The gathered data was used in the models. The mechanical model needs Tload and the combustion model needs δ, pinl , θSOI , θinj and ∆θinj . Note that data describing θSOI and θinj ∼ tinj was gathered from both the test bed equipment and Vision. The data differed somewhat between the test bed and the EMS, e.g. θSOI occur at approximately two CAD earlier according to the test bed compared to the EMS. Since the EMS controls both θSOI and θinj its data was saved and the corresponding test bed data were discarded. The rest of the

Chapter 3. Measurement

Parameter name δ pinl θSOI θinj pmax θpmax Tload

18

explanation mg Injected fuel during one stroke in one cylinder.[ stroke ] The inlet pressure.[bar] Start of injection.[CAD] The mean CAD duration of the injection.[CAD] The maximum pressure.[bar] The CAD of the maximum pressure.[CAD] The torque load of the brake.[N m]

Table 3.2: Parameter data gathered by the test bed equipment.

parameters, pmax , θpmax and pinl , was used to calibrate and set the pressure and reference angles as discussed previously in this section. The test bed engine was not fully calibrated when the measurements were conducted so some variables as δ could not be measured properly by the EMS.

Chapter 4

Simulation In this chapter short information about the simulation is presented along with the result of predicting the flywheel speed based on injection data. Since the model is built as two separate models: the mechanical model and the chemical pressure model, they are also evaluated individually.

4.1

Matlab

To solve the ordinary differential equation (2.34) describing the model, Matlab 7.0.1.24704 was used. The model program is built upon the program written in simulink/C++ by [13] which is based on the theory of [17]. A quick way to solve the nonlinear differential equation in Matlab was to use the built in solver ode45 with a maximum time step of 0.0002 sec/step. This results in that the time to calculate a solution depend on the speed of the flywheel, since the CAD steps is larger with higher speeds since the time step is fix. The solver spends about 50% of the time interpolating gas curves. Less pressure data reduces solving time. The time it took to solve the problem using a Pentium 4 CPU 1700 MHz with 256 Mb ram1 is found in table 4.1. Rpm 500 1000 1500 1800

time 7200 sample [s] 76 46 25 21

time 1800 sample [s] 47 24 15 12

Table 4.1: Mean time to calculate the model equations for 720 CAD. One model used pressure data which had 7200 samples while the others had 1800 samples. Less samples reduces solving time but also reduces pressure accuracy. It is important that the time to solve the differential equation is short in order to save time when validating and simulating. However it is not required that the solver should be able to solve in real time since it is not likely to use it in the engine control unit without implementing great simplifications in the model. The idea is to use the model for gaining understanding of the systems and having a quick and easy way to evaluate new control algorithms. To further reduce solver time, other programming languages as C++ could be used.

1 More memory is recommended to reduce simulation time since matlab and windows uses 140 % of the ram memory capacity before loading the model.

19

Chapter 4. Simulation

20

4.2 Result of the pressure model The measured data from the test bench are compared to the simulated data with inputs from the test bench. The inputs to the pressure model are start of injection θSOI , injection duration θinj , inlet pressure pinl , injection ˙ A comparison between data can be seen for different operating points in speed vf uel and the flywheel speed θ. figures 4.1 - 4.3 and in B.1. fm500varv20procref 1 0.9

Pressure normalized to 1

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −180

−90

0 CAD

90

180

Figure 4.1: Normalized pressure in cylinder 6 during one combustion cycle. The injection is centered on 0 CAD. The red curve (bold) is the measured pressure from the test bench, the blue (dotted) is the simulated pressure and the green (dashed) is the simulated pressure due to compression.

fm1000varv20procref

Pressure normalized to 1

0.8 0.6 0.4 0.2 0 −180

−90

0 90 CAD fm1000varv50procref

180

1

Pressure normalized to 1

Pressure normalized to 1

Pressure normalized to 1

fm1000varv0procref 1

0.8 0.6 0.4 0.2 0 −180

−90

0 CAD

90

180

1 0.8 0.6 0.4 0.2 0 −180

−90

0 90 CAD fm1000varv100procref

180

1 0.8 0.6 0.4 0.2 0 −180

−90

0 CAD

90

180

Figure 4.2: Normalized pressure in cylinder 6 during one combustion cycle. The injection is centered on 0 CAD. The red curve (bold) is the measured pressure from the test bench, the blue (dotted) is the simulated pressure and the green (dashed) is the simulated pressure due to compression.

21

4.2. Result of the pressure model

fm1500varv20procref

Pressure normalized to 1

0.8 0.6 0.4 0.2 0 −180

−90

0 90 CAD fm1500varv50procref

180

1

Pressure normalized to 1

Pressure normalized to 1

Pressure normalized to 1

fm1500varv0procref 1

0.8 0.6 0.4 0.2 0 −180

−90

0 CAD

90

180

1 0.8 0.6 0.4 0.2 0 −180

−90

0 90 CAD fm1500varv100procref

180

1 0.8 0.6 0.4 0.2 0 −180

−90

0 CAD

90

180

Figure 4.3: Normalized pressure in cylinder 6 during one combustion cycle. The injection is centered on 0 CAD. The red curve (bold) is the measured pressure from the test bench, the blue (dotted) is the simulated pressure and the green (dashed) is the simulated pressure due to compression.

Chapter 4. Simulation

22

The overall pressure simulations are fairly similar to the measured pressure in most operating points. The modeled pressure due to compression pmotor appear to be accurate except at one operating point at 1000rpm, 100% brake load. This error seems to be due to incorrect measuring rather than model error. Another error that turns up is that the whole pressure curve appear to be phase shifted some CAD. This error is most likely due to a miss placed measured curve, see equation (3.1), and the simulated curve should have a more correct appearance in this aspect. The simulated pressure due to compression pmotor is almost identical to measured pressure. The pressure due to combustion pcomb seems to have a small error in the initial phase of the combustion around 0 CAD. This is easiest visualized when the brake load is low. The reason seems to be incorrect modeling of the initial phase of the combustion where the pressure curve raises steeper than the simulated curve. This error could be decreased by assuming that the injected fuel speed was larger in the beginning of the injection and smaller in the end2 . Note that this error usually decreases with increasing brake load. This part of the model could be improved. The final value of the pressure usually gets correct which is far more important when calculating pressure torque. To understand how much the error in the simulated pressure matters, the pressure torque was calculated for each operating point. Since the simulated pressure had its largest error close to the top cylinder position and the pressure torque is weighted as Ap ds(θ) dθ which is zero at the top cylinder position the error in that interval will decrease, see equation (2.8) and figure A.2. The pressure in the intervals ±[15 165] CAD are weighted as more vital and errors here will be magnified. This point out that the model must be accurate in this domain in order to get good torque estimation. The operating point 500rpm, 20% brake load, has a large error around 0 CAD and was used to visualize this phenomenon. This can be viewed in figure 4.4. The wrongly estimated pressure at 0 CAD completely disappear while the small error centered at 45 CAD is magnified. fm500varv20procref 1

Pressure torque normalized to 1

0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −360

−180

0 CAD

180

360

Figure 4.4: Normalized pressure torque in cylinder 6 during one combustion cycle. The injection is centered on 0 CAD. The red curve (bold) is the measured pressure from the test bench and the blue (dotted) is the simulated pressure. In figure 4.5 the pressure torques for different brake loads are shown. All four figures appear to be similar but remind that the torque in the 100% brake load is nearly 5 times as large as the one with 0% brake load; still they have very similar appearance. The major difference between the curves is the relation between the compression torque [ -180 - 0] CAD and the compression + combustion torque [0 - 180] CAD. Torque figures for the other measured operating points can be found in appendix B and look much like the one shown at 1500 rpm.

2 In energy perspective it is good that the fuel is transformed into gas as quick as possible after that the cylinder reaches its top position. This way the energy output can be maximized. Two drawbacks are that quick rises in pressure makes unwanted noise, and also results in a high maximum pressure which puts heavy constraints on the manufacturing of the cylinders. The first drawback can be decreased by using a pre-injection which smoothes the quick pressure rise reducing noise. The second can be avoided by controlling the injection speed and the inlet pressure.

fm1500varv0procref 1 0.5 0 −0.5 −1 −360

−180

0 180 CAD fm1500varv50procref

360

1 0.5 0 −0.5 −1 −360

−180

0 CAD

180

360

Pressure torque normalized to 1 Pressure torque normalized to 1

4.2. Result of the pressure model

Pressure torque normalized to 1 Pressure torque normalized to 1

23

fm1500varv20procref 1 0.5 0 −0.5 −1 −360

−180

0 180 CAD fm1500varv100procref

360

−180

360

1 0.5 0 −0.5 −1 −360

0 CAD

180

Figure 4.5: Normalized pressure torque in cylinder 6 during one combustion cycle. The injection is centered on 0 CAD. The red curve (bold) is the measured pressure from the test bench and the blue (dotted) is the simulated pressure.

Chapter 4. Simulation

24

4.2.1 Sensitivity analysis To analyze the input variables for the pressure model a sensitivity analysis is done. One input is changed and the output is compared to the model with the reference inputs. Start of combustion, θSOI θSOI is the name of the variable when the ”start of injection” occurs. If θSOI transpire earlier/later than estimated the combustion pressure will start to increase earlier/later. If the injection is moved from -5.2 CAD to -1 CAD the total work is decreased by 3%. This example is shown in figure 4.6. fm1500varv100procref 1

0.2 0.15

0.6

Error in normalized pressure torque

Pressure torque normalized to 1

0.8

0.4 0.2 0 −0.2 −0.4 −0.6

0.05 0 −0.05 −0.1 −0.15

−0.8 −1 −360

0.1

−180

0 CAD

180

−0.2 −360

360

−180

0 CAD

180

360

Figure 4.6: L.F: The red curve (bold) is the reference simulated pressure torque and the blue (dotted) is the simulated pressure torque with θSOI changed from -5.2 CAD to -1 CAD. R.F: This result in a maximum error of 9.5% at 12 CAD which leads to that the total work done by the gas torque is decreased by 3%.

Inlet pressure, pinl The inlet pressure, pinl , denotes the variable describing the inlet pressure. This variable is considered as a constant during an engine cycle which holds if no substantial acceleration in the engine dynamics is done. This variable describes the pressure that is going to be compressed. A figure with a 5% change in inlet pressure can be seen in figure 4.7. The total work is not changed much since integrating the antisymmetric error would be zero3 . fm1500varv100procref 1

0.2 0.15

0.6

Error in normalized pressure torque

Pressure torque normalized to 1

0.8

0.4 0.2 0 −0.2 −0.4 −0.6

0.05 0 −0.05 −0.1 −0.15

−0.8 −1 −360

0.1

−180

0 CAD

180

360

−0.2 −360

−180

0 CAD

180

360

Figure 4.7: L.F: The red curve (bold) is the reference simulated pressure and the blue (dotted) is the simulated pressure with pinl increased 5%. R.F: This result in a maximum absolute error of 2.2% at ±18 CAD. The work done by the gas torque is increased by 0.1% due to more efficient combustion. 3 Very little energy is lost due to friction by raising the inlet pressure. Higher inlet pressure leads to higher compression pressure which makes the combustion more efficient. This way the effect of the engine can be increased.

25

4.2. Result of the pressure model

Injection duration, θinj The injection duration, θinj , denotes the variable describing the duration of injection. By increasing the duration of the injection more fuel is injected which should result in higher torque. The reference simulation has its injection duration increased by 5% which results in 4.5 more torque work. This is expected since more fuel means more energy. This example is shown in figure 4.8. fm1500varv100procref 1

0.2 0.15

0.6

Error in normalized pressure torque

Pressure torque normalized to 1

0.8

0.4 0.2 0 −0.2 −0.4 −0.6

0 −0.05 −0.1 −0.15

−0.8 −1 −360

0.1 0.05

−180

0 CAD

180

−0.2 −360

360

−180

0 CAD

180

360

Figure 4.8: L.F: The red curve (bold) is the reference simulated pressure and the blue (dotted) is the simulated pressure with θinj increased 5%. R.F: This result in a maximum absolute error of 2.6% at 30 CAD. The work done by the gas torque is increased by 4.5%.

fuel flow, vf uel The fuel flow, vf uel , denotes the variable describing the injection fuel flow. By increasing the flow of the injection more fuel is injected which should result in higher torque. The reference simulation has its injection flow increased by 5% which is shown in figure 4.9. That the same amount of fuel is injected as in the θinj case and the torque work is almost the same. fm1500varv100procref 1

0.2 0.15

0.6

Error in normalized pressure torque

Pressure torque normalized to 1

0.8

0.4 0.2 0 −0.2 −0.4 −0.6

0 −0.05 −0.1 −0.15

−0.8 −1 −360

0.1 0.05

−180

0 CAD

180

360

−0.2 −360

−180

0 CAD

180

360

Figure 4.9: L.F: The red curve (bold) is the reference simulated pressure and the blue (dotted) is the simulated pressure with vf uel increased 5%. R.F: This result in a maximum absolute error of 2.8% at 27 CAD. The work done by the gas torque is increased by 4.8%.

Chapter 4. Simulation

26

4.3 Result of the mechanical model The simulated pressure torque was used as inputs to the mechanical model.

4.3.1 Friction The total friction is estimated according to equation (2.13) in the model chapter and can be seen in figure 4.10 and 4.12. Both measured and simulated pressure torque is investigated. There seems to be some relation in velocity as higher velocities results in higher friction work. To further evaluate friction calculations the errors in the measurements has to be considered. The result of the pressure measurements and the pressure simulations indicated that both pressure curves had errors. This leads to that the corresponding torque could be wrongly estimated. This hint to that the friction work estimation will be incorrect. The brake load torque parameter is also an important factor. If the brake load torque parameter is estimated 5% incorrect in the operating point 1500 rpm 100% brake load, this result in that friction work is changed from 330 joule to 1700 joule. The same result is derived when studying the pressure torque. In a worst case situation both pressure and brake load sum up and results in bad friction estimation. A small measurement error results in relatively large error in the estimation of the friction. This inaccuracy in the measurement results limits the result in finding a good friction model. Estimated frictionwork in different brakeload cases (measured pressure) 4000 0% 20% 50% 100%

3500

Friction work [Joule]

3000 2500 2000 1500 1000 500 0

0

200

400

600

800

1000 rpm

1200

1400

Figure 4.10: The friction work is estimated as Wf ric

1600

1800

 θ+4π θ0

Tg dθ − 4π · Tload

Estimated frictionwork in different brakeload cases (simulated pressure) 5000 0% 20% 50% 100%

4500 4000

Friction work [Joule]

3500 3000 2500 2000 1500 1000 500 0

0

200

400

600

800

1000 rpm

1200

1400

1600

1800

Figure 4.11: Two different approaches are used to solve this dilemma. The first approach uses the measurements to estimate the friction. By using equation (2.13) with a linear absolute friction in velocity, the friction can be set so that the measured mean speed gets correct. The drawback is that the solutions only are valid near their operating points.

27

4.3. Result of the mechanical model

The second approach is to minimize the error in friction data by trying to fit a function to data (see equation (2.14)) in for example a least square manor. This results in a model which can be run with optional choices of gas pressure and brake load but will result in a slightly incorrect mean flywheel speed. Using the first approach as a reference the mean relative error over all operating points was 26% (13% wrong if removing the worst operating point at 1500rpm 100% load). The friction in the worst case has a relative error of 184% which leads to that the error in mean speed gets almost as large. The estimation of the absolute friction work with both approaches is shown in figure 4.12. If pressure data is simulated the mean error decreases somewhat which is important for non static simulations. This also indicates that the friction error might depend on errors in the pressure measurements. absolute frictionwork 30 25

absolute Friction work

20 15 10 5 0 −5 −10 −15 −20 −2500

−2000

−1500

−1000 Brake torgue

−500

0

500

Figure 4.12: The ∗ are the estimated friction work according to the first approach and the ◦ are from the second ˙ approach. The second approach uses a function of nine terms where the most important are: θ, θ˙ and θ˙2

4.3.2

Comparison between measured and simulated flywheel speed

The results of three different models are compared to the measured flywheel speed. The result of two speeds 1000 rpm and 1500 rpm are shown in figure 4.13 and 4.14 and the rest can be found in appendix B along with harmonic analysis of the speed curves. M odel M odel + brake Reduced model + brake

-

The 9 node system explained in the model chapter. The 9 node system with a 1 node brake system attached. A 3 node system (damper, crankshaft, flywheel) with a 1 node brake attached.

Chapter 4. Simulation

28

fm1000varv0procref 1010

RPM

1005 1000 995 990

0

100

200

300

400 CAD

500

600

700

800

600

700

800

600

700

800

fm1000varv20procref

RPM

1010 1000 990 980

0

100

200

300

400 CAD

500

fm1000varv50procref

RPM

1020 1000 980 960

0

100

200

300

400 CAD

500

fm1000varv100procref

RPM

1050

1000

950

0

100

200

300

400 CAD

500

600 700 Measured rpm

800

Model Model+brake Reduced model+brake

Figure 4.13: Three models are compared to measured flywheel speed. The three models gives similar results and all has problem following the measured speed at low brake loads.

fm1500varv0procref 1510

RPM

1505 1500 1495 1490

0

100

200

300

400 500 CAD fm1500varv20procref

600

700

800

0

100

200

300

600

700

800

0

100

200

300

600

700

800

1510

RPM

1505 1500 1495 1490

400 500 CAD fm1500varv50procref

1520

RPM

1510 1500 1490 1480

400 CAD

500

fm1500varv100procref

RPM

1550

1500

1450

0

100

200

300

400 CAD

500

600 700 800 Measured rpm Model Model+brake Reduced model+brake

Figure 4.14: Three models are compared to measured flywheel speed. The three models gives similar results and all has problem following the measured speed at low brake loads. 1500 is a resonance frequency from for the test bed engine.

29

4.4. Simulating the inverted reduced model

4.4

Simulating the inverted reduced model

The inverted reduced model (2.36) with only one node is used as a model to describe the pressure torque with flywheel position and speed as inputs. The result depend on how much the stiffness dynamics affects the flywheel speed. The result from two operating points can be seen in figure 4.15. The operating point at 1000 rpm has a speed signal which is not affected much by the stiffness dynamics which makes the estimation more accurate. The operating point at 1500 rpm experience much of the stiffness dynamics and the estimation gets less accurate. fm1000varv100procref

fm1500varv20procref 1

Nm (Normalized)

Nm (Normalized)

1

0.5

0

0

120 240 360 480 600 720 CAD fm1000varv100procref

1020

1515

1010

1510

1000

rpm

rpm

0

0.5

990

0

120 240 360 480 600 720 CAD fm1500varv20procref

0

120 240 360 480 600 720 CAD

1505 1500

980 1495

970 0

120 240 360 480 600 720 CAD

Figure 4.15: The inverted reduced model describes the pressure torque. The left plots are from the operating point 1000 rpm 100% load and the two to the right are from 1500 rpm 20% load. The top plots are pressure torques where the blue (bold) is simulated, the red (dashdot) is a measured reference and the black (dotted) is the compression pressure. The lower plots are the corresponding flywheel speed. The result of the estimation depends on the flywheel speed signal.

Chapter 4. Simulation

30

4.5 Simulated error in injection The model is tested with simulated error in the injection and compared to the corresponding measurement. One simulation can be seen in figure 4.16 where 6% more fuel has been injected in cylinder 6. 1030

measured simulated

1020 1010

RPM

1000 990 980 970 960 950

0

200

400

600

800 CAD

1000

1200

1400

Figure 4.16: An error in the injection duration (6% less in cylinder 6) is simulated and compared to the measured flywheel measurement.

4.6 Result of the extended observer Both observer models were tested and compared to measured data from the test bed. The extended observer seemed to estimate the gas torque better than the observer suggested in [3] since the lag effect could be reduced. The result of the extended observer can be seen in figure 4.17 for a case when 10% more fuel has been injected in cylinder 4, 5 and 6. The observer indicates that the torque gets higher from the expected cylinders. Flywheel data from an engine was also tested on flywheel speed data from a truck. No torque could be measured so data from the test cell was used as a reference. The observer result for 1500 rpm 0% load is shown in figure 4.18.

31

4.6. Result of the extended observer

observed speed measured speed

0.0765842 0.687048 0.986368 158

rad/s

157.5

157

156.5

2200

2400

2600

2800 3000 CAD

3200

3400

3600

x2 x3 measured pressure torque 3000

2000

Nm

1000

0

−1000

−2000

2200

2400

2600

2800 3000 CAD

3200

3400

3600

Figure 4.17: The extended observer with more fuel injected in cylinder 4, 5 and 6. The cylinder fire sequence is 1-5-3-6-2-4. The angle step is 6 CAD.

observed speed 0.0345649 0.689014 0.986421

measured speed

158.5

RPM

158 157.5 157 156.5 156

2200

2400

2600

2800 3000 CAD

3200

2200

2400

2600

2800 3000 CAD

3200

3400 3600 x2 x3 T2max measured pressure torque

2500 2000 1500 Nm

1000 500 0 −500 −1000 −1500 3400

3600

Figure 4.18: Flywheel speed data from a heavy duty truck. The angle step is 6 CAD.

Chapter 5

Conclusions In this chapter the results of the previous chapters are discussed and conclusions are drawn. Conclusions of the objectives of the thesis are also stated.

5.1 Discussion 5.1.1 Discussion of the pressure model The errors in the pressure model were often very small. The largest errors occurred at the beginning of the combustion. Except at the beginning of the combustion the curve was correct which is more important when calculating the pressure torque.

5.1.2 Discussion of the full model The simulated models lack accuracy in certain operating points. One important reason is probably due to the stiffness dynamics. Making small changes on the stiffness parameters may result in different solutions depending on the operating point. One example of this is the model with nine nodes plus one brake node which clearly is erroneous at 500 rpm 20% brake load (see figure B.4). At the other operating points the same model works as well as the others. Another example of the stiffness dynamics is found at 1500 rpm. This is a resonance frequency for the engine with the test bed setup. Here can not only the large contribution in the normal third engine order be found, but also a large contribution from the half and first engine order can be found, (also see [11]). The half and first engine order should not be there according to the model and might be explained as part of the stiffness dynamics in the engine with the test bed setup. All models lacks this dynamics at 1500 rpm but it will most likely be found at other velocities. Since the model was not tested on a truck no conclusions can be drawn there. Much time was spent on optimizing the friction model but the result was poor. The conclusions is that measurements have to be done more accurately and in more operating points to get a better result. The torque due to self oscillation should also be measured. The damper wheel parameters are also difficult to estimate and are frequency dependent. The model works satisfying if the objective is to test control algorithms. A drawback is that the current time to solve the model equations is rather long. A reduced model would perhaps be enough if the goal is to verify an observer or a control algorithm. The model was found to work satisfying at describing errors in the injection. The effect is that control algorithms may be tested to stabilize the model and hopefully also the real engine.

5.1.3 Discussion of inverting the model The full model (without drive line) has 18 states which only two can be measured. It is impossible to invert the full model to get cylinder pressure without simplifications. On the other hand a model with more nodes does not do much better than a reduced model which was shown in the model simulations. The non stiffness dynamic which the larger models try to include is difficult to model correctly. The simple solution is to consider the whole model as stiff which makes it easy to estimate the pressure torque. The drawback with this model is that 32

33

5.2. Conclusions of the objectives

the stiffness dynamics results in very bad estimation when the engine dynamics starts to self-oscillate at certain speeds.

5.1.4

Discussion of the observer

If an engine with five cylinders has 20% more fuel in one cylinder the total torque will increase only 20/5=4%. If an engine with eight cylinders has 20% more fuel in one cylinder the total torque will increase only 20/8=2.5%. This results in that the engine with more cylinders will have smaller variation on the total torque which makes an error harder to find. If a cylinder gets 20% more fuel than expected the emissions will change as well. If the cylinder always gets too much fuel it will also be worn out faster. This is not good and should be compensated. The extended observer seems to estimate the gas torque fairly acceptable in the measured operating points and without lag effects. Tested errors in the injection of (6% error) could be found by the observer. The stiffness dynamic is still a problem which may cause trouble. The result deteriorate when the the non stiffness torque was large. A more advanced observer strategy than pole placement will probably do better result.

5.2

Conclusions of the objectives

The objectives were 1. Constructing a time dissolved model which can calculate the flywheel speed based on the fuel injection data. 2. Investigating the possibilities of simplifying the model to be able to invert it and thus being able to calculate the torque of the individual cylinders based on flywheel speed. The first objective was completed. The options are to simulate the model with either measured cylinder pressure or fuel injection data. Drive line models can also be included. The model catches the behavior of the flywheel speed in most operating points but lacks some of the dynamic at the test bed resonances. The model should be sufficient to evaluate new control and observer algorithms. In some operating points the model lacked parts of the dynamics compared to the test bed measurements. There seems to be no point in inverting a model which is not valid in certain operating points. The full model has 18 nodes which only two are measurable. A reduced model where the 18 nodes are put together to one makes almost as good result in catching the main behavior. In this reduced model it is easy to estimate the torque from each cylinder with few calculations. Just divide the speed signal into intervals and match it with the cylinder ignition sequence. This one-node model was tested with an observer which managed to observe the indicated torque well enough to notice small injection errors.

Chapter 6

Future Work A user’s manual will be written about the simulation program in Matlab. Measurements on a heavy duty truck with the clutch down are planed to verify if the model works better there. If it works satisfying a model of the trucks drive line may be connected and simulated and compared. The observer models will be more investigated with different feedback designs to try to make them better. The hope is that the observer can estimate the pressure torque well enough to be used for on board diagnostics and for controlling the injections. Perhaps more measured signals can be used in the observer model. The cylinder inlet pressure for example has much information of the size of the compression torque which may be used. This will be more investigated. Perhaps making a model working in the frequency domain would be something to investigate. One advantage will be faster calculations.

34

References [1] Engine parameters used in the model was received from Johan Lundqvist, Scania, department NMKA [2] Automotive Handbook . Bosch, 2000.30.09, 5th edition [3] J. Chauvin, G. Corde, P Moulin, M, Castagn´e, N. Petit, P. Rouchon, Time-varying linear observer for ´ torque balancing on a DI engine. centre Automatique et Syst`emes,Ecole des Mines de Paris and Insitut Francais du P´etrole, France [4] J. Chauvin, G. Corde, P Moulin, M, Castagn´e, N. Petit, P. Rouchon, Real-time combustion torque estimation on a diesel engine test bench using time-varying kalman filtering. in Proc. of 43rd IEEE Conference on Decision and Control, 2004, Paradise Island, Bahamas. dec [5] F. Chmela, G. Orthaber, W. Schuster, Die vorausberechnung des brennverlaufs von dieselmotoren mit direkter einspritzung der basis des einsspritzverlauf. MTZ, Motortechnische Zeitschrift 59 (1998) 7/4 [6] F. Chmela, G. Orthaber, Rate of heat release predictions for direct injection diesel engines based on purely mixing controlled combustion. SAE paper 1999-01-0186. (1999) [7] F. Chmela, G. Orthaber, M. Eng´elmayer, Integrale induziertechnik am DI-dieselmotor zur vertiefen verbrennungsanalyse und als simulationsbasis. 4 internationales symposium f¨ur verbrennungsdiagnostik. (2000) [8] M. Hellstr¨om, Engine speed based estimation of the indicated engine torque. Department of Vehicular systems, Link¨opings Universitet, 2005. Master’s thesis LiTH-ISY-EX-3569-2005,Link¨oping, Sweden, January. [9] J.B. Heywood, Internal Combustion Engine Fundamentals. McGraw-Hill International Editions, 1988. ISBN 0-07-100499-8, Singapore, international edition. [10] R. Johnsson Indirect measurements for control and diagnostics of IC engines. Department of human work sciences, division of sound and vibrations, Lule˚a University of technology, 2004, doctorial thesis, Lule˚a University, Sweden, 04 [11] E Jorpes, Ber¨akning av torsionsfenomen i ny motorprovcell. NMBP,2002/1, M22/053 [12] E Jorpes, Ber¨akningsrapport torsionssimulering av DL. NMBP,2001/5, M22/049 [13] M. Nilsson, Modeling Flywheel-Speed Variations Based On Cylinder Pressure. Department of Vehicular systems, Link¨opings Universitet, 2004. Master’s thesis LiTH-ISY-EX-3584-2004,Link¨oping, Sweden, March. [14] T. Petterson, Torsionsanalys av vevaxeln, kamaxeln och motortranmissioen p˚a Scanias D12:a. Department of solid mechanics, Lule˚a university of technology, 2001. Master’s thesis LTU-EX–01/341–SE, Lule˚a, Sweden. [15] A.S. Rangwala, Reciprocating Machinery dynamics, Design and Analysis. Marcel Dekker, Inc., 2001. [16] K-E Rydberg, Matematiska verkningsgradsmodeller f¨or fasta och variabla hydralmaskiner. Department of hydraulics, Link¨opings Universitet , 1980. LiTH-IKP-S-171, Link¨oping, Sweden, august. [17] S. Schagerberg, Torque Sensors for Engine Applications. Department of signals and systems, School of Electrical Engineering, Chalmers University of technology, 2003. Licentiate thesis NO 472L,G¨oteborg, Sweden. 35

Appendix A

Derivation of the connecting rod kinematics This derivation of the connecting rod kinematics is a copy of the derivation done by Schagerberg [17] but several similar derivations can be found. s

A

r+l

l

phi

theta

B

r

Figure A.1: The connecting rod mechanism The point A in figure A.1 may be described by the complex number ζA = rejθ + le−jφ ,

(A.1)

where the minus sign of the argument of the second term comes from the definition of the angle directions. The relation between θ and φ is via the piston pin offset1 , defining the equation yof f s = Im{ζA },

(A.2)

with the solution for sinφ = rsinθ − lsinφ ⇔ rsinθ − yof f s sinφ = . (A.3) l For the derivation of indicated and mass torques the first and second derivative of the piston displacement, s = xmax − Re{ζA }, see figure A.1, w.r.t θ are needed. Start by finding the derivative yof f s

∂φ ∂ζA = jrejθ − jle−jφ , ∂θ ∂θ

(A.4)

1 The piston pin may be offset slightly to reduce piston slap, i.e. slamming the cylinder bore. Typically, the piston pin offset as defined by figure A.1 is negative. For crankshaft torsional vibrations, the piston pin offset may very well be neglected.

36

37

and the second derivative ∂ 2 ζA ∂θ2

The derivative

∂φ ∂θ

∂φ  ∂  jθ jre − jle−jφ = ∂θ ∂θ  ∂φ 2 ∂2φ = −rejθ − le−jφ − jle−jφ 2 . ∂θ ∂θ =

(A.5)

in (A.4) and (A.5) is found by taking the derivative of (A.3) w.r.t θ

cosφ

∂φ ∂θ ∂φ ∂θ

= =

r cosθ l rcosθ lcosφ

⇒ 

= l

1−



rcosθ rsinθ−yof f f s l

2 ,

(A.6)

where cosφ was expressed using the trigonometric unity  2  rsinθ − y  of f s . cosφ = 1 − (sinφ)2 1 − l

(A.7)

The second derivative of φ w.r.t θ in (A.5) is ∂ 2 φ ∂  rcosθ  rcosθsinφ ∂φ rsinθ = · − . ∂θ2 ∂θ lcosφ lcos2 φ ∂θ lcosφ

(A.8)

The crank lever, i.e. the derivative of s w.r.t θ, is found by combining the real part of (A.4), (A.3) and (A.6) ∂s ∂θ

= =

 ∂ζ  ∂ A = (xmax − Re{ζA }) = −Re ∂θ ∂θ ∂φ rcosθ(rsinθ − yof f s ) rsinθ + lsinφ · rsinθ +  2 ,  ∂θ rsinθ−yof f s l 1− l

(A.9) (A.10)

where the second equality follow since derivation is linear operation and xmax is a constant. For the second derivative of s, take the real part of (A.5) ∂2s ∂θ2

= =

 ∂2ζ  ∂2 A (xmax − Re{ζA }) = −Re 2 ∂θ ∂θ2  ∂φ 2 ∂2φ rcosθ + lcosφ · + lsinφ 2 , ∂θ ∂θ

=

where insertion of equation (A.7), (A.6), (A.3) and (A.8) will yield the full expression. ———————————————————————-

(A.11) (A.12)

Appendix A. Derivation of the connecting rod kinematics

38

0.1 0.08 0.06 0.04 0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1 −400

Figure A.2:

∂s(θ) ∂θ

−300

−200

−100

0

100

200

300

400

over a engine cycle. The combustion is assumed to take place at approximately 0 CAD.

Appendix B

Simulation figures fm1800varv20procref

Pressure normalized to 1

0.8 0.6 0.4 0.2 0

−90

0 90 CAD fm1800varv50procref

1

Pressure normalized to 1

Pressure normalized to 1

Pressure normalized to 1

fm1800varv0procref 1

0.8 0.6 0.4 0.2 0 −180

−90

0 CAD

90

180

1 0.8 0.6 0.4 0.2 0 −180

−90

0 90 CAD fm1800varv100procref

180

1 0.8 0.6 0.4 0.2 0 −180

−90

0 CAD

90

180

Figure B.1: Normalized pressure in cylinder 6 during one combustion cycle. The injection is centered on 0 CAD. The red curve (bold) is the measured pressure from the test bench, the blue (dotted) is the simulated pressure and the green (dashed) is the simulated pressure due to compression.

39

40

fm1000varv0procref 1 0.5 0 −0.5 −1 −360

−180

0 180 CAD fm1000varv50procref

360

1 0.5 0 −0.5 −1 −360

−180

0 CAD

180

360

Pressure torque normalized to 1 Pressure torque normalized to 1

Pressure torque normalized to 1 Pressure torque normalized to 1

Appendix B. Simulation figures

fm1000varv20procref 1 0.5 0 −0.5 −1 −360

−180 0 180 CAD fm1000varv100procref

360

−180

360

1 0.5 0 −0.5 −1 −360

0 CAD

180

fm1800varv0procref 1 0.5 0 −0.5 −1 −360

−180

0 180 CAD fm1800varv50procref

360

1 0.5 0 −0.5 −1 −360

−180

0 CAD

180

360

Pressure torque normalized to 1 Pressure torque normalized to 1

Pressure torque normalized to 1 Pressure torque normalized to 1

Figure B.2: Normalized pressure torque in cylinder 6 during one combustion cycle. The injection is centered on 0 CAD. The red curve (bold) is the measured pressure from the test bench and the blue (dotted) is the simulated pressure.

fm1800varv20procref 1 0.5 0 −0.5 −1 −360

−180 0 180 CAD fm1800varv100procref

360

−180

360

1 0.5 0 −0.5 −1 −360

0 CAD

180

Figure B.3: Normalized pressure torque in cylinder 6 during one combustion cycle. The injection is centered on 0 CAD. The red curve (bold) is the measured pressure from the test bench and the blue (dotted) is the simulated pressure.

41

Measured rpm Model Model+brake Reduced model+brake

fm500varv20procref 550 540 530 520 510 500 490 480 470 460 450

0

100

200

300

400

500

600

700

Figure B.4:

fm500varv20procref

Speed (rpm)

150 Measured rpm Model Model+brake Reduced model+brake

100

50

0

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5

5

5.5

phase (deg)

180 90 0 −90 −180

Figure B.5: Harmonic analysis of the simulations at 500 rpm

800

Appendix B. Simulation figures

42

phase (deg)

Speed (rpm) phase (deg)

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order Harmonic analysis

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order Harmonic analysis

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order Harmonic analysis

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5 5 5.5 Measured rpm

40 20

0 180 90 0 −90 −180

100 50

0 180 90 0 −90 −180

Speed (rpm) phase (deg)

10

0 180 90 0 −90 −180

Speed (rpm)

phase (deg)

Speed (rpm)

Harmonic analysis 20

200 100

0 180 90 0 −90 −180

Model Model+brake Reduced model+brake

Figure B.6: Harmonic analysis of the simulations at 1000 rpm

43

phase (deg)

Speed (rpm) phase (deg)

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order Harmonic analysis

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order Harmonic analysis

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order Harmonic analysis

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5 5 5.5 Measured rpm Model Model+brake Reduced model+brake

20 10

0 180 90 0 −90 −180

40 20

0 180 90 0 −90 −180

Speed (rpm) phase (deg)

10

0 180 90 0 −90 −180

Speed (rpm)

phase (deg)

Speed (rpm)

Harmonic analysis 20

100 50

0 180 90 0 −90 −180

Figure B.7: Harmonic analysis of the simulations at 1500 rpm

Appendix B. Simulation figures

44

fm1800varv0procref

RPM

1810 1800 1790 1780

0

100

200

300

400 500 CAD fm1800varv20procref

600

700

800

0

100

200

300

400 500 CAD fm1800varv50procref

600

700

800

0

100

200

300

600

700

800

RPM

1820 1810 1800 1790

1820

RPM

1810 1800 1790 1780

400 CAD

500

fm1800varv100procref

RPM

1850

1800

1750

0

100

200

300

400 CAD

500

600 700 Measured rpm

800

Model Model+brake Reduced model+brake

Figure B.8:

45

phase (deg)

Speed (rpm) phase (deg)

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order Harmonic analysis

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order Harmonic analysis

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order Harmonic analysis

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5

5

5.5

0.5

1

1.5

2

2.5 3 3.5 Engine order

4

4.5

5

5.5

20 10

0 180 90 0 −90 −180

20 10

0 180 90 0 −90 −180

Speed (rpm) phase (deg)

20

0 180 90 0 −90 −180

Speed (rpm)

phase (deg)

Speed (rpm)

Harmonic analysis 40

100 50

0 180 90 0 −90 −180

Measured rpm Model Model+brake Reduced model+brake

Figure B.9: Harmonic analysis of the simulations at 1800 rpm