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ª
º
2
2 · « ¨ ' » ' ¸ ¨ ¸ « » © ' ¹ ¬ ¼ 2 π π 1 π §
(3)
" E E
d
where E d is the dynamic MOE of the panel (Pa) and Δ is the logarithmic decrement of the panel. According to the transverse vibration of a rectangular orthotropic plate, neglecting shear deformation and rotary inertia, the E d of the panel can be calculated using Eq. 4 (Zhou and Chui 2015), 2 3 2 1 2 d 2 48 π (1 ) 500.6 XX ML f E bh (4) where f is the natural first-order vibration frequency in bending along the length direction of the panel (Hz) without damping for this “free - free” support state, M is the weight of the panel (kg), v 1 and v 2 are the Poisson ratios of the panel, and L , b , h are the length, width, and thickness of the panel (m), respectively. The Poisson ’s ratios of the panel have a very small influence on the first resonant frequency (Sobue and Kitazumi 1991; Schulte et al. 1996). Hence, substituting a constant value for the Poisson ratios will have a small influence on the calculated properties. Here v 1 v 2 was assumed to be 0.01 as used for most wood materials (Hearmon 1946). Free vibration of the panel in this support state appears as a damped sine wave. According to the damped sine wave vibration amplitude, the logarithmic decrement Δ is shown using Eq. 5 (Hunt et al. 2013),
1
2 π
A
A
f
]
=ln
ln
2 π
(5)
'
]
1
n
1 A n A
f
2
1
[
1
n
n
r
where A 1 is the first amplitude of the damped sine wave selected, A n is the n th amplitude of the damped sine wave selected, f is first natural frequency of the panel vibration without damping, f r is first natural frequency of the panel vibration tested, and ζ is the damping ratio. In Eq. 5, ζ can be calculated using the logarithmic decrement of vibrational decay ( Δ ) in Eq. 6: th amplitude of the damped sine wave selected, A n+ 1 is the (n+1)
'
=
]
(6)
2 2
4 π +
'
According to Eq. 5 and Eq. 6, first natural frequency of the panel vibration ( f ) can then be calculated using the measured first frequency ( f r ), as shown in Eq. 7.
r 2 1 ] f
f
(7)
When the panel geometry size ( L , b , h ) is given, the dynamic viscoelasticity values (storage modulus E ’ and loss modulus E ” ) for the full-size WCP are obtained by substituting these parameters into Eq. 2 and Eq. 3. This was the theoretical basis for determining dynamic viscoelasticity for the full-size WCP.
4595
Guan et al . (2016). “Dynamic viscoelasticity,” B io R esources 11(2), 4593-4604.
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