Simulation of Dynamic Systems - MECH 337

1. Learning outcomes for this module
i. You will gain an understanding of engine management systems for spark ignition engines.
ii. You will gain an understanding of using a sophisticated simulation package - Matlab Simulink.
iii. You will gain experience of using a simulation package to model a complex dynamic system.

2 The Assignment
There is one assignment for this 10 credit module. Consequently it is expected that each student will spend about 100 hours on attending classes, background research and appropriate reading and developing the model for this assignment.
This assignment is due in at 11.00 am on Thursday 13th March 2008 Confirmed..
You should work in pairs on this assignment - both students will normally receive equal marks.

i) Each group will either be given information about a vehicle or can choose a vehicle that they wish to simulate. You will need to know / be able to find the following information about the chosen vehicle: (data in brackets is for a Porsche 944).

ii) What you have to do
For your chosen / assigned vehicle you should develop a simulation model to simulate it's velocity (for the first 10 seconds of its motion when the throttle is actuated as shown in the diagram below:

This is provided in Matlab Simulink by a 'Signal Builder' 'Source' block.

iii) The Report Must Include:

3. Matlab - Simulink
No prior knowledge of Matlab - Simulink is assumed, but you must read the Matlab - Simulink notes before attending the sessions in the computer lab (Smeaton 109).
It is very important that you attend the specified slots in the computer lab for instruction about using Simulink.

Below are some examples of simple approaches using Matlab Simulink to model car performance, they should help you develop your model:
a) The note here describes in general terms how to set about developing a model for a complex system
b) The note here describes how to set up a simple model in Matlab Simulink to simulate single degree of freedom spring - mass - damper system.
c) The note here describes how to set up a simple model to simulate the acceleration of a car driven by constant torque with air resistance proportional to velocity²
d) The note here develops the model in (c) to include simulating a gear change by using a switch element.
e) The note here develops the model in (c) to include simulating the variation in engine torque with engine speed by using a linear interpolation look up table.
e2) The note here combines the two previous models by incorporating a one dimensional look up table with gear changes.
e3) This model here introduces greater realism in the gear changing process by putting in a time delay which simulates zero torque transmission during the gear change.
f) The note here develops the model in (e) to include simulating the variation in engine torque with engine speed and throttle opening by using a linear interpolation two dimensional look up table.
For information about setting up a 2D look up table in Simulink, click here
g) In this example (fuel1) a somewhat different approach is taken using throttle position and engine rpm as the inputs to a 2d look up table whose output is assumed to be air mass flow rate.
h) In this model (fuel4) the pressure drop across the throttle, the effect this has on the air mass flow rate and the cross section area opened by the throttle for the air to flow through are all considered. Again a single gear is assumed.

i) Two different Matlab models are shown here which have different, simplistic approaches, to determining the pressure drops across the throttle plate and across the inlet valve. Using both these programmes involves a trial and error approach.

In this section (isentropic1.htm) a Matlab model is shown which takes a simple approach assuming that the air flow in the inlet tract remains at a constant temperature.

In this section (compflow3.htm) a Matlab model is shown which takes a simple approach assuming that the air flow in the inlet tract is compressible.

For the same inputs, these two approaches give different results.

j) In this Simulink model model (fuel5) the pressure drop across the throttle plate and across the inlet valve are both considered.

4. Background to Automotive Fuel Systems
For the first 80 years of the automobile, the petrol / air mixture was formed by a carburettor. Despite considerable improvements over the years it was becoming clear by the late 1970s that the carburettor was nearing the limits of its capabilities and the increasing requirements for improved economy and reduced exhaust emissions indicated that a better way of providing the mixture was needed.
Some systems exercising control by a combination of mechanical and electronic devices had been developed by the late 1970s, but it was the developments in electronics in the 1980s and 1990s which lowered the costs of these systems and led to their widespread introduction, so that now there are virtually no new petrol powered cars sold that do not have a fuel injection / engine management system.

For introductory information about fuel injection systems, have a look at the first web site in section 10, below.

5. System Basics
The primary control on the mass of fuel to be injected is the mass flow rate of air into the engine. Although the stoichiometric ratio for air to petrol is about 14.7:1 this in practice needs to be changed significantly for starting, running while the engine is warming up and accelerating (made richer) and economical cruising (made leaner) and for coasting with the throttle closed no fuel is injected until the engine revs. drop below about 1600 rpm. To achieve this a number of sensors provide data for a microprocessor and this is then compared with an operating 'map' and the signal to open the injector for the appropriate length of time is produced and sent to the injectors.
Fuel is supplied at constant pressure to the injectors, typically 266 kPa, so (ignoring very slight possible density changes) the mass of fuel injected is proportional to the time that the injector is open.

Early systems - e.g. Bosch L Jetronic - controlled by a bulky box of electronics:

used a hinged flap, lightly spring loaded, in the inlet tract to measure the air mass flow, however this in itself introduced a pressure drop which slightly lowered the engine efficency and more modern systems (e.g. Bosch LE Jetronic) use hot wire anemometry to compute the air mass flow as this does not cause a significant pressure drop. The electronics were also much more compact.

The key components of the inlet tract are shown above, figure 2.

6. Suggested Initial Assumptions - some of these figures are rounded for convenience.
i) Ambient air pressure is 100 000 N/m²
ii) The density of air is 1.25 kg/m³
iii) Energy released when petrol is burnt: 46 000 000 J/kg
iv) For 'ideal' combustion the stoichiometric ratio of air to petrol is 14.7:1
v) There is no pressure drop across the hot wire device.

7. Possible Approaches to the Modeling
i) The most basic approach is to assume that the engine torque is proportional to the throttle opening and only one gear ratio is used. This could be modeled with Simulink as shown in the diagram below:

A major problem with the above model is that the engine torque is not just proportional to throttle opening in the transient phase and for the model to be realistic this needs to be considered.

ii) To determine the build up of engine torque as the throttle is opened, it is necessary to consider the air mass flow, which will be governed by the pressure values, P1, P2 and P3 (see figure 2, above). For P3 it is assumed that the inlet valve is open and the piston is descending.
At low engine speeds the drop in pressure P2 - P3 will be small, this will increase as the engine speeds up and as a first approximation it can be assumed that the pressure drop P2 - P3 is proportional to the engine speed. From typical engine data it can be assumed that at maximum engine revs this pressure drop is about 0.2 P2, ie P3 = 0.8 P2
If it is assumed that the max. engine rpm are 5000, then for any engine speed 'N' between 0 and 5000 the relationship between P2 and P3 is given by:

(P_upstream/P_downstream)_crit = ((gamma + 1)/2)^{(gamma/(gamma - 1))}

For gamma = 1.35, the critical pressure ratio is 1.86
If it is assumed that the P1 = 100 000 N/m², then even if the pressure P2 drops below 53600 N/m², there will be no further increase in mass flow rate.

Keeping this limitation in mind, it is therefore possible to write an equation for the air mass flow in the form:

8. A suggested methodology for setting up the model
a) Look up the specified maximum engine power output and the engine rpm at which it is developed.
b) Work out how much fuel has to be burnt per second to provide this power. To do this you will have to estimate what the efficiency of the engine is at maximum power output, probably between 30 and 35 % for a petrol engine.
c) Work out the air mass flow rate to burn this fuel under perfect combustion conditions.
d) For tick over (assume 800 - 1000 rpm), when the engine is working as an air pump drawing in air, use vdp (v = the volume of air drawn in per second and dp = the pressure drop from atmospheric to the pressure of the air in the cylinders during the induction stroke) to determine the power absorbed by the pumping in of the air. Again allow for an efficiency of about 30% and calculate the required fuel and then air mass flow rates.
e) The information from (c) and (d) can be used to determine a value of Kt.

With the above information it should be possible to build a more complex and accurate model of the engine under transient conditions.
f) The steps described so far give the net output power. This can be divided by the engine angular velocity to give the torque. However this needs slight modification, a small number must be added, eg 0.1, as otherwise the calculation will fail immediately due to a 'divide by zero' error.
g) With the net torque available, this can be divided by the total equivalent vehicle inertia at the engine (using a 'Gain' block), to give the angular acceleration of the engine. Examples of how to determine the equivalent inertia of the vehicle are given in some of the module notes.
h) The angular acceleration of the engine can be passed through an 'Integration' block (from the 'Continuous' group) to give the engine angular velocity.
i) The appropriate gain factor can be applied to the engine angular velocity to give the vehicle velocity.

Further elaboration is possible by:
Using 'Switch' elements to change gear based on the vehicle speed.
Introducing a wind resistance component using the vehicle frontal cross section area, drag coefficient and speed.
Once the model starts to get at all complicated, changes in the model often have effects that are very difficult to predict. It is important to make use of lots of 'scopes to see what is happening in different parts of the model when trying to assess how realistic it is.

9. References

'Mechanical Engineering Design Notes - Theme: Automotive Engines', by D J Grieve, Published by D J Grieve, ISBN: 978-0-9560037-0-6.

'Bosch Automotive Handbook', 5th Ed., 2000, ISBN: 0-7680-0669-4

'The Motor Vehicle', by K Newton, W Steeds, T K Garrett, 9th Ed., Iliffe, 1972, ISBN: 0 592 00070 2

'Internal Combustion Engines - Applied Thermosciences', by C R Ferguson and A T Kirkpatrick, John Wiley and Sons, 2001, 2nd Ed., ISBN: 0-471-35617-4.

'Introduction to Modeling and Control of Internal Combustion Engine Systems', by L Guzzella and C H Onder, Springer, 2004, ISBN: 3-540-22274-x.

'Design and Simulation of Four Stroke Engines', by G P Blair, SAE, 1999, ISBN: 0-7680-0440-3.

'Introduction to Matlab 6 for Engineers', by W J Palm III, McGraw-Hill, 2001, ISBN: 0-07-234983-2.

Web Site How Stuff Works - Auto

10. Websites about Bosch fuel injection systems, etc.:
http://www.roversd1.nl/sd1web/injection.html
http://users.exis.net/~audserv/L.html
http://www.type2.com/library/fuel/Bosch_L-Jetronic_FI_Manual/
http://demo.cs.brandeis.edu/postscript.dump/Ltronic/Injection.html
http://homepage.mac.com/ratwell/L-JetronicTester.html

Return to module introduction

David J Grieve, 11 th September 2008.