Page 525 - Dynamics of Mechanical Systems

0593_C14_fm Page 506 Tuesday, May 7, 2002 6:56 AM

506 Dynamics of Mechanical Systems

Then, we obtain:

a b λ +( ab + a b − ab λ ) 2 + ab =
4
11 31 1 3 2 2 33 0 (14.7.8)
Equation (14.7.8) is identical to Eq. (14.6.52), which was obtained by the simultaneous
solution of Eqs. (14.6.40) and (14.6.41).
It happens, in infinitesimal stability procedures in the analysis of small disturbances of
a system from equilibrium, that we can generally convert the governing simultaneous
linear differential equations into a single differential equation of higher order, as Eq.
(14.7.6). Then, in the solution of the higher order equation, we generally obtain a polyno-
mial equation of the form:

−
a λ + a λ n 1 +…+ a λ + a = 0 (14.7.9)
n
−
0 1 n 1 n
From our previous analyses we see that a solution to the governing equations in the
*
form of θ in Eq. (14.7.7) is stable if (and only if) λ does not have a positive real part.
Necessary and sufficient conditions such that Eq. (14.7.9) will produce no λ (i = 1,…, n)
i
with positive real parts may be determined using the Routh–Hurwitz criteria (see Refer-
ence 14.1). Specifically, let determinants ∆ (i = 1,…, n) be defined as:
i
∆ = a
1 1
a a
∆ = 1 0
2
a a
3 2
a a 0
1 0
∆ = a a a
3 3 2 1
a a a
5 4 3
a a 0 0
1 0
a a a a
∆ = 3 2 1 0 (14.7.10)
4 a a a a
5 4 3 2
a a a a
7 6 5 4
-------------
a a 0 0 … 0
1 0
a a a a … 0
3 2 1 0
∆ = a a a a … 0
n 5 4 3 2
--- - --- --- - -- --- ---
a a --- - -- --- a
−
−
2 m 1 2 m 2 m
Then, the criteria, such that the solutions λ (i = 1,…, n) have no positive real part, are that
i
each of the ∆ (i = 1,…, n) must be positive and that each of the a (k = 0,…, n) must have
i
k
the same sign.
The above statements asserting that the a and the ∆ need to be positive for the λ to
i
i
i
have no real parts are commonly called the Routh–Hurwitz stability criteria. It is seen that