Combustion engine creates torque to spin the crankshaft。 This engine torque Cm is the sum of combustion moment Cvg4

and engine moment of inertia Cvi4 [19]。

Cm4  ¼ Cvg4 þCvi4: ð17Þ

Index 4 denotes a four cylinder engine。 The combustion moment expression is given by the next  formula:

C C Pmax V 0:46sin 2a 0:24 sin 4a 0:03 sin 6a , 18

192

where ac is the crankshaft angular position, Pmax is the maximum gas pressure inside cylinders, Cm4 is the average friction torque of the engine and Vcyl  is the engine cubic capacity。

3。 Numerical simulations

Numerical simulation based on Newmark integration method is used to study the dynamic behavior of spur gear system powered in the first place by an electric motor, then in the second place by four strokes four cylinders inline diesel engine。

The parameters of the spur gear transmission are given in Table 2。

Using the mean values of stiffness presented in Table 2, the eigenfrequencies of the system are computed and presented in Table 3。

Table 2

Parameters of the spur gear [5]。

Pinion Wheel

Teeth numbers 20 40

Mass (Kg) 0。6 2。5

Inertia moment (Kg m2) 2。6 ~ 10— 4 0。0045

Base circle (m) 0。05 0。11

module (m) 0。003

Bearing stiffness (N/m) kx1 ¼ ky1 ¼ kx2 ¼ ky2 ¼ 108

Torsional stiffness (N rd/m) ky1 ¼ ky2 ¼ 105

Pressure angle a¼ 201

Teeth width (m) 0。023

Contact ratio c12 ¼ 1。6

Table 3Eigenfrequencies of the system。

Eigenfrequencies (Hz) f1 f2 f3 f4 f5 f6 f7 f8

0 820 1007 1038 1577 2068 3441 5771

Table 4Motor specifications [16]。

Motor type ABB-MT 90L

Electrical characteristics 4poles, 50 Hz 3 phase, 415  V

Output power (KW) 1。5

Full load speed (rpm) 1420

Full load torque Tf (N m) 10

Torque starting Ts/Tf