Thermal Systems and Models 427
rmCR
/
, we obtain the non-dimensional eigenvalue equation (characteristic
equation):
>@
0844
422
20
20
2
///
/
/
/
aaaa
a
HH
H
[8.25]
The roots of equation [8.25] are:
3,2,
2
48164
;
22
1
r
/ / i
aaa
a
i
H
HHH
[8.26]
As the sum of the roots /
2
and /
3
of the equation are negative and their product
is positive, the two roots are negative (a diffusive system is aperiodic).
The symmetry of the system allows us to immediately find the modes. A
symmetric mode is characterized by
21
ii
4 4 . Substituting this relation into the
homogenous system derived from [8.24], we have:
1331
44;2
iiii
a 4 4/4 4/
Eliminating 4
i3
between these relations, we find that the eigenvalues /
i
corresponding to these modes satisfy the second order trinomial of characteristic
equation [8.25]. The values of 4
i1
and 4
i3
are of the same sign or of opposite sign
depending on whether
a/ is positive or negative. Substituting –a as the value of
/ in the preceding trinomial, we find that this takes on a negative value, which
shows that the value – a is situated between the roots of the trinomial. We can easily
derive from this that the quantity
a/ is positive for the largest root /
3
, whereas
it is negative for the other root /
2
. We can verify immediately that the root
a /
1
corresponds to the anti-symmetric mode
1211
4 4 and 0
13
4 .
+ + +
/
2
(
1
(
3
(
2
+ – 0
/
1
(
1
(
3
(
2
+ + –
/
3
(
1
(
3
(
2
Figure 8.9.
Structure of normal modes classed with increasing modulus of the eigenvalues
312
rrr
(decreasing time constants)