Page 41 - Mechatronics with Experiments

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INTRODUCTION 27
rate decreases from its maximum value, the lug curve does not necessarily scale linearly
with it. At very low fuel rates, the combustion process works in such a way that the shape
of the lug curve becomes quite different (Figure 1.22). The best fuel rate is accomplished
when the engine speed is around the middle of low and high idle speeds.
The torque defined by the lug curve is the mean effective steady-state torque capacity
of the engine as a function of speed at maximum fuel rate. This curve does not consider
the cyclic oscillations of net torque as a result of the combustion process. The actual torque
output of the engine is oscillatory as a function of the crankshaft angle within each four-
stoke (intake-compression-combustion-exhaust) per two-revolution cycle. The role of the
flywheel is to reduce the oscillations of engine speed. Each cylinder has one cycle of net
torque contribution (which has both positive and negative portions) per two revolutions,
and the torque functions of each cyclinder are phase shifted from each other.
A given mechanical engine size defines the boundaries of this curve:
1. maximum engine speed (high idle speed) determined by the friction in the bearings
and combustion time needed,
2. lug portion of the curve corresponding to torque-speed relation when the maximum
fuel rate is injected to the engine. This is determined by the heat capacity of the
engine as well as the injector size.
lug lug max
T = f (w )for u = u (1.2)
eng o eng fuel fuel
Once the limits of the engine capacity are defined by mechanical design and sizing of the
components, we can operate the engine at any point under that curve with the governor –
the engine controller. Any point under the engine lug curve corresponds to a fuel rate and
engine speed. Notice that the left side of the lug curve (speeds below the peak torque) is
unstable since the engine torque capacity decreases. At speeds below this point, if the load
increases as the speed decreases, the engine will stall. The slope of the high idle speed
setting can be 3–7% of speed change in mechanically controlled engines, or it can be
made perpendicular (almost zero speed change) in electronically controlled engines. The
governor (engine controller) cuts down the fuel rate in order to limit the maximum speed
of the engine. The sharp, almost vertical, line on the lug curve (steady-state torque-speed
curve) is the result of the governor action to limit the engine speed by changing the fuel rate
from its maximum value at the rated speed down to smaller values. Hence, electronically
controlled engines can maintain the same speed as the load conditions vary from zero to
maximum torque capacity of the engine.
The parameterized version of the lug curve (torque versus engine speed), where there
is a curve defined for each value of fuel rate, is called the torque map. The engine is modeled
as a torque source (output: torque) as function of two input variables: fuel rate and engine
speed. Such a model assumes that the necessary air flow is provided in order to satisfy the
fuel rate to torque transfer function (Figures 1.23, 1.22).
T eng = f (u fuel , w eng ) (1.3)
0
Notice that, when u fuel = u max , the function f (⋅, ⋅) represents the lug curve. In order to
0
fuel
define the points below the lug curve, data points need to be specified for different values of
fuel rate (Figure 1.22). An analytical representation of such a model can be expressed as
( )
u fuel ( max )
T = ⋅ f u , w (1.4)
eng max 0 fuel eng
u
fuel
Clearly, if the engine capacity limits are to be imposed on the model, the lug curve
limits can be added as a nonlinear block in this model. The standard “lug curve” defines the