{Fext(t)} is the external applied torques vector。 It can be written  as

fFext ðtÞg ¼ f0,0,Cm,0,0,0,—Cr ,0g  : ð9Þ

where

Z2

r  ¼ Cm: ð10Þ

1

In the two next sections the torques for an electric motor and a combustion engine will be defined and   modeled。

2。1。 Electric  motor modeling

Using the kinetic energy theorem and neglecting the power losses, the rotational velocity of an electric motor can be linked to its torque and to the receiver torque  by

Table 1Coefficients Si  of K(t)。

S1 ¼ sin a S7 ¼ rb12 sin a

S2 ¼ cos a S8 ¼ rb21 cos a

S3 ¼ sin a S9 ¼ rb21 sin a

S4 ¼ cos a S10 ¼ r2

S5 ¼ sin a cos a S11 ¼ r2

S6 ¼ rb12 cos a S12 ¼ rb12 rb21

where J is the equivalent moment of inertia of the rotating   parts。

The second term of Eq。 (11) is frequently called acceleration torque Cacc。 However, two rotating phases of the motor can be distinguished

– The starting phase during which Cacc is positive。 The motor torque must be higher than the receiver torque on the gears

i。e。 Cm 4Cr  in order to accelerate the system (dO=dt 40)。

– The steady state phase during which the motor reaches its nominal operating mode and the acceleration is nil (dO/dt  40)。

The mechanical characteristic of the motor is the relation between driving torque and speed of the motor。 It is expressed by [16]

ð1 þðsb—sÞ2ðða=sÞ—bs2ÞÞ

where sb and Tb are the slip and torque at breakdown, respectively, a and b are all constant properties of the motor and s is the proportional drop in speed given by

Ns  is the synchronous speed and N is the actual rotational speed of the motor。

Load torque characteristic Cr is chosen to be proportional to the square of the rotational speed。 This is the case for several receivers such as pumps [17]

Cr  ¼ hO2, ð14Þ

where h is a coefficient depending on the characteristics of the driven system。

Eq。 (11) is transformed into the following differential   equation:

2。2。 Combustion  engine modeling

During the power stroke, many forces arise inside combustion engine, due to gas pressure and inertia of moving parts。

The engine does not work regularly and generates velocity and torque fluctuations。

Combustion engine non-cyclic condition is the consequence of its unsteady rotations or irregular parts deflections。

Acyclism may be related to engine torque and during this loading state the rotational speed of engine is not constant but it has an average value。 For the studied gear set, a general expression of the pinion speed for rigid body conditions can be expressed as Fourier by [18]

!

OðtÞ¼ O10     1 þ Xrn ðO10ÞsinðnO10t þjn Þ   ,

n

ð16Þ

O10 is the nominal average angular velocity, n is the harmonic of the rotational speed, rn is the corresponding dimensionless amplitude and jn is the initial phase。