Modeling and simulation of a vehicle with automatic transmission

3. Plant model: Engine

The engine used for the simulation example is a Mercedes 5.0 litre V8, with the following performance parameters:

  • maximum engine power @ engine speed [kW@rpm]: 225 @ 5600
  • maximum engine torque @ engine speed Nm@rpm]: 460 @ 3000-4250

The engine torque is mapped, function of the accelerator (throttle) position and engine speed. Only the full load data points are matched against the real engine torque output. The part load torque values are extrapolated from the full load points.

Engine torque [Nm]Engine speed [rpm]
10001500200025003000350040004500500055006000
Accelerator pedal
(throttle)
position [%]
00-25-50-71-89-105-116-128-139-150-161
202411559357306-16-36-53-68-81
302832361951631371169985756761
40314326334328308275238199168150140
50329346361371373365346316276230187
60349365378387396398384357322284246
70359382400411417417409396370334299
80367399424436439440438426400368332
90373408435448451451449436414385349
100377414443456460460460450427402366

In graphical form, the engine torque output at full load and part loads looks like:

Engine torque map - 3D

Image: Engine torque map – 3D

Engine torque map - 2D

Image: Engine torque map – 2D

The engine provides positive torque during acceleration phases and negative (braking) torque in the overrun phases. The engine is modelled as a single inertia lumped parameters and a torque map.

Engine free body diagram

Image: Engine free body diagram

The governing differential equation of the engine model is:

\[J_{e} \frac{d \omega_{e} }{dt} = T_{e}-T_{i} \tag{1}\]
where:
Te [Nm] – engine torque
Ti [Nm] – impeller torque
ωe [rad/s] – engine speed
Je [kg·m2] – engine and impeller inertia

From equation (1) we can calculate the engine speed as:

\[\omega_{e} = \frac{1}{J_{e}} \int (T_{e}-T_{i}) dt \tag{2}\]

To convert the engine speed from [rad/s] into [rpm], we use the following expression:

\[N_{e} = \frac{30 \cdot \omega_{e}}{\pi} \tag{3}\]
where:
Ne [rpm] – engine speed

The engine power can be calculated as:

\[P_{e} = \frac{T_{e} \cdot \omega_{e}}{1000} \tag{4}\]
where:
Pe [kW] – engine power

Equations (2), (3) and (4) are used to define the Xcos block diagram for engine simulation.

Engine - Xcos block diagram

Image: Engine – Xcos block diagram

The calculated engine speed is limited to a maximum and a minimum value. The minimum value is considered the idle speed of the engine.

Inputs

NameValueDescription
ImplTq_NmImpeller torque [Nm]
AccrPedlPosn_prcAccelerator pedal position [%]

Parameters

NameValueDescription
EngImplJ_kgm2_C0.08Engine and impeller inertia [kg·m2]
EngMaxN_rpm_C7000Maximum engine speed [rpm]
EngMinN_rpm_C1000Minimum engine speed [rpm]
EngN_rpm_X(table)Engine speed axis [rpm]
AccrPedlPosnEng_prc_Y(table)Accelerator pedal position axis [%]
EngTq_Nm_Z(table)Engine torque map [Nm]

Outputs

NameValueDescription
EngN_rpmEngine speed [rpm]

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