Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 316

orthogonality of sin β m x ( m = 1 , 2 ,... .),

τ q ¨

2
m

˙

2
m

Γ m +( 1 + ατ T β ) Γ m + αβ Γ m = 0 , (6.213)

˙

Γ m ( 0 )= φ m , Γ m ( 0 )= ψ m . (6.214)

Introduce the damping coefficient f m by

1
τ 0 + τ

m

f m = ω

T

and the natural frequency coefficient ω m by

m
τ 0 .

= αβ

2
m

ω

Eq. (6.213) reduces to

¨

f m ˙

2
m

Γ m + Γ m + ω Γ m = 0 . (6.215)

The solution of Eq. (6.215) can be readily obtained by the method of undetermined
coefficients as

be λ t Γ m ( t )= (6.216)

with λ as a coefficient to be determined. Substituting Eq. (6.216) into Eq. (6.215)

leads to

2 2

m

λ + f m λ + ω = 0 , (6.217)

which has solution λ 1 , λ 2

√

f m
2 ±

λ 1 , 2 = − Λ . (6.218)

Here Λ is the discriminate of Eq. (6.217) and is defined by

f m
2

2

2
m

Λ = − ω .

Therefore, a positive, negative and vanished discriminate yields two distinct real
numbers λ 1 and λ 2 , two complex conjugates λ 1 and λ 2 , and two equal real num-

bers λ 2 , respectively. The critical damping coefficient f mc is the damping

coefficient at a fixed

λ 1 and

ω m and Λ = 0. Therefore,

f mc = 2 ω m . (6.219)

The nondimensional damping ratio, ζ m , is defined as the ratio of f m over f mc ,

f m
f mc =

f m

τ 0 ω m + τ T ω m

1

=

ω m =

.

ζ (6.220) m

2 2 2

The system is at underdamped oscillation, critically-damped oscillation or over-
damped oscillation, respectively, when ζ m < 1, ζ m = 1or ζ m > 1. By Eqs. (6.218)