BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
buletinul institutului politehnic din iaşi - Universitatea Tehnică ...
buletinul institutului politehnic din iaşi - Universitatea Tehnică ...
- No tags were found...
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
90 Florin Popa et al.<br />
the notations have the significance of Fig. 3.<br />
Accepting a linear variation of characteristics M m and M r in the vicinity<br />
of the operation a , which involves taking only the terms containing derivatives<br />
of order zero and first order of Taylor development and, second, entering the<br />
following variable transformations<br />
ω= ωa<br />
+ Ω,<br />
χi<br />
= χχia<br />
+ Χ ,<br />
(8)<br />
and notation<br />
ψ = ψ + Ψ ,<br />
i<br />
ia<br />
Δ = tan δ . (9)<br />
kl<br />
Narrowing the difference Δrω −Δ<br />
mω<br />
=Δ, Eq. (5) becomes successively<br />
simplified forms<br />
dΩ I +ΔΩ= ... +Δ<br />
mχ<br />
Χ ...,<br />
i i −Δ<br />
rψΨ<br />
i j + (10)<br />
dt<br />
or still<br />
d<br />
I<br />
Ω + ΔΩ = C , dt<br />
(11)<br />
where C is denoted by the algebraic sum of terms on the right, that is<br />
kl<br />
= ... +Δ Χ −Δ Ψ + ..., (12)<br />
C<br />
χ ψ<br />
m i i r i i<br />
which means, in fact, the system disturbance occurs.<br />
3. System Response Shown by the Mathematical Model<br />
Solving differential Eq. (11) is based on the constant values I, Δ , C. But<br />
since, obviously, equivalent mechanical model is non-zero mass, we have<br />
I ≠ 0 . In this situation, to solve the equation will consider the following cases<br />
further developed.<br />
1. If Δ ≠ 0 and C ≠ 0 , above equation becomes<br />
dΩ dt = , (13)<br />
C−ΔΩ<br />
I<br />
with solution<br />
Δ<br />
C ⎛ −<br />
I<br />
1 e t ⎞<br />
Ω= ⎜ − ⎟. (14)<br />
Δ ⎝ ⎠<br />
2. When Δ = 0 and C ≠ 0 equation takes the form<br />
dΩ<br />
I = C , (15)<br />
dt<br />
with solution