BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

90 Florin Popa et al.
the notations have the significance of Fig. 3.
Accepting a linear variation of characteristics M m and M r in the vicinity
of the operation a , which involves taking only the terms containing derivatives
of order zero and first order of Taylor development and, second, entering the
following variable transformations
ω= ωa
+ Ω,
χi
= χχia
+ Χ ,
(8)
and notation
ψ = ψ + Ψ ,
i
ia
Δ = tan δ . (9)
kl
Narrowing the difference Δrω −Δ
mω
=Δ, Eq. (5) becomes successively
simplified forms
dΩ I +ΔΩ= ... +Δ
mχ
Χ ...,
i i −Δ
rψΨ
i j + (10)
dt
or still
d
I
Ω + ΔΩ = C , dt
(11)
where C is denoted by the algebraic sum of terms on the right, that is
kl
= ... +Δ Χ −Δ Ψ + ..., (12)
C
χ ψ
m i i r i i
which means, in fact, the system disturbance occurs.
3. System Response Shown by the Mathematical Model
Solving differential Eq. (11) is based on the constant values I, Δ , C. But
since, obviously, equivalent mechanical model is non-zero mass, we have
I ≠ 0 . In this situation, to solve the equation will consider the following cases
further developed.
1. If Δ ≠ 0 and C ≠ 0 , above equation becomes
dΩ dt = , (13)
C−ΔΩ
I
with solution
Δ
C ⎛ −
I
1 e t ⎞
Ω= ⎜ − ⎟. (14)
Δ ⎝ ⎠
2. When Δ = 0 and C ≠ 0 equation takes the form
dΩ
I = C , (15)
dt
with solution