1. Introduction
The model developed previously, in (1) is developed to include a greater
element of realism by providing for a change into second gear. This makes use
of a 'Switch' block.
2. Assumptions
All assumptions are as in the previous model except that first gear is assumed
to have an overall reduction of 10:1 and second gear an overall reduction of 7:1.
3. Building the Model
To incorporate the gear change the model is build a little differently to the
previous one.
For the signal source, the step final value is set to 170 (the engine torque in Nm). For convenience the
Step time is set to zero.
The output engine torque is then directed into two paths, in the lower one the first
gain (Gain 2) is set to 10 (the overall first gear ratio) and in the upper path the first gain is set
to 7 (the overall second gear ratio).
A gain block in each of these paths (Gain 7 and 8) is set = 1 / wheel radius to give the total
forward driving force.
Each of these pathways then has a summing point to enable the negative effect of
the force of the air resistance to be fed back. Each path then goes through a gain block with a
gain equal to 1/equivalent inertia of the car at the wheels, (Gain 3 and 4) the output from these is the
the vehicle acceleration.
The equivalent inertia (mass) of the vehicle is:
= 4[(I/r2) + m] + M
where I is the moment of inertia of each wheel, m the mass of each wheel, r the wheel radius
and M the mass of the vehicle less the wheels.
= 4[(0.4/0.32) + 10] +1000 = 1057.8 kg
The paths then go into the switch. The switch has 1 output (acceleration in this case).
The middle switch input is for the threshold. In this case it is assumed that the gear will
be changed up at a velocity of 15 m/s (34 mph approx.). The operation of the switch is that the
signal into the bottom port (port 3) is passed through unless the threshold criterion is
reached, when this occurs the input from the top port (port 1) is passed through.
The output signal from the switch then goes through an integrator block (to give velocity)
A loop from the velocity line is fed back to the switch middle input (port 2) for comparison
with the threshold.
Two other loops are taken back from the velocity line and then fed into the dot
product block to give the v2 value which is fed into the gain block (Gain 1) near the
bottom of the diagram which contains the value of
Air density x vehicle frontal area x drag coefficient/2 = 1.25 x 2 x 0.35/2 = 0.4375 kg/m
The output from this block is the air resistance which is is taken away from the
driving force provided by the wheels at the summing points in the forward loops.
Hence the signal from the summing points is the net force available to accelerate the car.
The (velocity) output from the first integrator is fed through a second integrator (Integrator 1)
to give the distance travelled.
Running this simulation (using only 2 gears) gives the time to cover 1000m as 25 s and
the time to reach 60 mph as just over 6 seconds. These are reasonable considering the model limitations.
David J Grieve, 2 November 2004.