Engine Modelling

ENGINE MODELLING

CHAPTER 1: INTRODUCTION

It is defined as the mathematical model of an engine, including engine inertia, friction, applied torque, fuel inputs and other variables like that which define the system as a whole. This method of modelling is one of many approaches to define the various parameters of a system model. This is just an example of modelling an engine; we can model other systems as well with the help of this system. The main reason for such a wide ranging use of mathematical models is that they provide a well-defined picture of the system.

Continuing with the modeling of an engine, there are various variables and parameters that we need to represent the system as a whole. So the first thing that needs to be done is to simplify the system and take it step by step. This is described below:

STEP 1: DRAW A SIMPLE DIAGRAM

The above diagram shows a simple relationship. It shows how the engine speed changes with changes in the fuel input.

STEP 2: IDENTIFYING VARIOUS PARAMETERS

Now that we have identified the relationship, we will now define the parameters that will have an effect on this relation. It is important to define all the parameters that will be involved. If we miss out, then obviously we won't get the correct model.

To help identify all the parameters, it is always beneficial to draw a mechanical model of the system. This is shown in the figure below:

So the various factors, in order from left to right, are

* Fuel Rate, F in %

* Applied Torque, T in Nm

* Friction Torque, TF in Nm

* Moment of Inertia, J in kgm2

* Rotational Displacement, in rad

* Rotational Speed, in rads-1

* Rotational Acceleration, in radsˉІ

STEP 3: GET THE RELATIONS/EQUATIONS

The relationships will be common physics, mathematical or engineering equations.

* : Comparable to .

* : Simple mathematical relation.

* : Simple mathematical relation.

* : Where k1 is the coefficient of friction.

* : Where k2 is a constant.

STEP 4: DRAW THE SYSTEM DIAGRAM, SHOWING INPUTS ON THE LEFT AND OUTPUTS ON THE RIGHT:

To get the complete system model, we have to take the above relations step by step, combine them and then get the final model. This will become clear as we move along. So considering the equation, the model for this particular relation will be:

Similarly, defining the model for the next equation, we get the next part of the system diagram. The next equation is . To represent this equation in the diagram, we show it as (where Tn= net torque= T-Tf)

Moving on to the next equation, we have the relation . Showing this relation in the diagram, we have

In the above, the 1/s stands for integration. Thus, the integral of angular acceleration gives us angular velocity.

The last equation is the relation . This is shown below in the figure below. It should be noted here that this is not a control loop, it is simple feedback.

Combining the above individual equation representations in diagrams, we can the system diagram as a whole. This is shown below

Thus, we see how the individual equations that we obtained helped us to get the whole system diagram or the system model.

The next step here is to put this in a spread sheet so that we can use the above to get some useful data. For this, we shall consider the example of the time taken for the engine to get to 3000 rpm and then shall carry on till 10 sec to obtain a sensible plot. For this exercise, we shall set the constants as K1= 0.0001, K2= 0.2 and J= 0.25. Also, the fuel will be set at 100%. The columns in the spread sheet are in the order as shown in the diagram from left to right.

A B C D E F G H

Time (s) Fuel (%) T Tf Tn Acc. Vel. W(rpm)

0.1 100 20 0 20 80 0 0

Hence, we see that the order of the contents of the table is from left to right, as in the diagram. The value for torque T is set at 20 Nm and the initial value for Tf = 0. Now we know that Tn= T-Tf; using this formula we can get the values for column D. To use this formula in the spread sheet, we write it as C2-D2.

Also, from the system diagram, we know acceleration (column F) = Tn/J. Thus, using this formula we can get the values for column F. This formula is written in the spread sheet as E2/0.25.

Again, we know that velocity is an integral of acceleration. To use this formula in the spread sheet, we use it as =G2+(A3-A2)*F2.

And finally, we get the engine speed by the formula

The spread sheet is shown on the next page.

(The complete spread sheet with all values is shown in Appendix 1)

We get a plot for the above values between the