Design and Stress Analysis of Extraterrestrial ... - The Black Vault

The question of system stability can be solved using the Routh-
Hurwitz criterion. For system stability it is necessary and
sufficient that a matrix made up of the coefficients of the frequency
equation and the even-order diagonal minors of this matrix be
positive. The Routh-Hurwitz matrix, in this case, has the form
-2, w -, own2 0 0
0 (,2+W 2 0 -(02(02 0 0
1 2 - 1 2
0o OMI 021t O 1020
2 0
0 0 w+0) 2 0 -0 2 0--V
2 1'2 0
0 0 -ai -6 1 O* n (on,(d
12
For system stability the foll*owing conditions are necessary
and sufficient:
(w.24) >0; ,n(.,4,, O.,>0;
2 • "2(,- 1 +2) >O
- w) 2 h2(.)8 (02+ W;) +n 2 e4Oci)>0,~2''
then the system is
stable when
,t>O, WK n, (3.165)
For the self-excitation rate w* = (n)/(nl)wo 0 , in all roots
s of the characteristic equation n > 0,
the vibrations described
by expression (3.163) will be attenuating and only forced vibrations
will exist in the system. When w > wc in the system, along with
c
forced vibrations there will also be autovibrations described by
expression (3.163).
The quantity s on the boundary of the autovibrations can be
found from equation (3.164) with allowance for the fact that
on the boundary wn 1 = W0n:
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