Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 324

which yields, by noting also that Ω * m ≥ 0,

Ω ∗ mr = − η

2
m

+ ( 1 + η

m

)

2

− 4 ζ

m

η

m

, (6.251)

Ω mr must
be real, Xu and Wang (2002) obtain another condition for resonance in addition to
Eq. (6.251),

Ω mr stands for the external source frequency at resonance. Since where

1

m 2

m m m

+ η − 4 ζ η > η . (6.252)

Ω mr with

The variation of ζ m and η m is shown in Figs. 6.5 and 6.6. It is observed

Ω mr decreases as the damping parameter that ζ

m and the phase lagging parameter

m increase. Fig. 6.7 illustrates the variation of B Ω m with

Ω m and η ζ

m at

η = 1.

m

For ζ = 0 . 9, Eq. (6.252) cannot be satisfied. Therefore, there is no resonance when

m

ζ m = 0 . 9at η m = 1 (Fig. 6.7).

| Ω mr |

Fig. 6.5 Variation of with ζ m at η = 1 . 0 (after Xu and Wang, 2002) m