BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
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Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 89<br />
Since the mass moment of inertia I variation depending on the angle of<br />
rotation ϕ is small, can introduce approximation,<br />
dI<br />
0<br />
dφ = . (4)<br />
Eq. (3) becomes<br />
dω<br />
I = Mm<br />
−M<br />
r. (5)<br />
dt Analyzing this equation shows that the propulsion system operation may<br />
occur several situations (Heisler, 2002 & Heywood, 1988), depending on the<br />
relationship in which the two moments, M m and M r . Thus, if the M m = M r<br />
resulting from Eq. (5) that the angular acceleration dω/dt = 0, so the motor shaft<br />
angular velocity is constant. In this situation, powertrain operation is performed<br />
in a steady or stable. Equality of torque and moment resistant is done only at<br />
the intersection point a, between the output characteristics of the engine and<br />
transmission input characteristics, as shown in Fig. 3.<br />
Fig. 3 – Coordinates defining the operating point of the propulsion system.<br />
Coordinates of this point, called point of operation (Rakoşi et al., 2006),<br />
is the parameters of the engine required power, represented by the torque M a<br />
and the angular velocity ω a during the stationary regime<br />
Pa<br />
= Maω a<br />
(6)<br />
Operating point stability analysis by analytical mechanical model<br />
involves studying evolution during transient processes. This development is<br />
defined by the solution of differential Eq. (5). Normally, to solve this equation<br />
requires knowledge of the functions that describe the output characteristics of<br />
the engine and the transmission input characteristics. But using the Taylor series<br />
development needs can be assessed in the general case, the performance of<br />
these functions in the vicinity of the operation, obtaining<br />
∂M m( ωa,... χia,... ) ω−ω ∂M m( ωa,... χia,...<br />
) χ<br />
( ,..., ,...) ( ,..., ,...)<br />
a<br />
i−χia<br />
Mm ω χi = Mm ωa χia<br />
+ +<br />
+ ... ,<br />
∂ω<br />
1! ∂χi<br />
1!<br />
respectively, (7)<br />
∂M r( ωa,... Ψia,... ) ωω − ∂M r( ωa,... Ψia,...<br />
) Ψ−Ψ<br />
M ( ,..., ,...) ( ,..., ,...)<br />
a i ia<br />
r<br />
ω Ψi = Mr ωa Ψia<br />
+ + + ...,<br />
∂ω<br />
1! ∂Ψ<br />
1!<br />
i