This paper is made of four sections。 Introduction is set in Section 1。 Section 2 focuses mainly on the external source of the gear set excitation。 Actually, it studies the influence of both rotational speed variation and the fluctuations of an electric motor torque on the dynamic behavior of a single stage geared system。 Section 3 uses the numerical simulation based on the Newmark integration method, to study the dynamic behavior of spur gear system powered, at a first stage, by an electric motor, then at a second stage, by four strokes four cylinders inline diesel engine。 Conclusions are given in Section 4。

2。 Model of a single stage spur gear transmission

A single stage spur gear transmission model with eight degrees of freedom is proposed [5] (Fig。 1)。 It is pided into two main blocks。 Block 1 includes the driving motor and the pinion connected by a shaft。 This block is supported by a bearing。 Block 2 includes the wheel and the load connected by a shaft; it is supported by a second bearing。 The Pinion has Z1 teeth and moment of inertia J11。 The wheel has Z2  teeth and moment of inertia J22。 A driving torque Cm  is applied on the

transmission loaded by a torque Cr。 The gearmesh stiffness ke(t) is modeled by linear spring acting on the line of action of

the meshing teeth。

The displacement a along the line of action is expressed by [15]

dðtÞ¼ ðx1—x2Þsinaþðy1—y2Þcosaþy12rb12 þy22rb21: ð1Þ

xi and yi are the translations of block i (i ¼ 1, 2)。 yij is the angular displacement of the component j in block i (i,j ¼ 1, 2)。 a is the pressure angle。 The base radius of the pinion and the wheel are, respectively, rb12  and rb21。

Let N1  the rotational speed of the pinion, the mesh period (in seconds) be defined by

Taking into account the Lagrange formalism, the differential equation of motion of the adopted system can be expressed by:

½M]fq€ gþ½C]fq_ gþ½KðtÞ]fqg¼ fFext ðtÞg, ð3Þ

Fig。 1。  Single stage spur gear system model [5]。

where q is the vector of the degrees of freedom。 It is expressed by:

q  ¼  fx1,y1,y11,y12,x2,y2,y21,y22gT : ð4Þ

M  represents  the mass matrix  given by:

where mi are the masses of block i (i ¼ 1, 2)。

K(t) includes the bearings stiffness kxi, kyi (i ¼ 1, 2), the shafts torsional stiffness kyi and the time varying gearmesh stiffness ke(t)。 It is expressed by [5]

2 S3keðtÞþ kx1 S5keðtÞ 0 S7keðtÞ —S3 keðtÞ —S5 keðtÞ 0 S9 keðtÞ

The terms Si (i ¼ 1–12) are given in Table 1。

The time varying gearmesh stiffness is modeled by a square waveform [5], for a contact ratio less than 2, the maximum values of stiffness correspond to two pairs in contact and the minimum values correspond to one pair in contact。 K(t) can be pide into a mean stiffness matrix ½K]  and a time varying matrix  [k(t)]

KðtÞ¼ ½K]þ½kðtÞ]: ð7Þ

[C] is the proportional damping matrix expressed by

½C]¼ 0:05½M]þ10—5½K]: