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VISCOELASTIC TRANSMISSION LINE MODELS

DISPERSION MODELS AND THEIR APPLICATION TO WOOD ACOUSTICS

4.2 VISCOELASTIC TRANSMISSION LINE MODELS

E/dz

ρdz ρdz

Figure 4.1 The one-dimensional lossless transmission-line model. The top elements are springs with

stiffness per unit lengthE/dz. The bottom elements are masses with values ofρdz.

4.2 VISCOELASTIC TRANSMISSION LINE MODELS

The wave equation for a stress-wave in a one-dimensional lossless solid is given by

2σ ∂t2 =c

22σ

∂z2, (4.5)

where σ is the stress,z is the distance along the medium, andc is the wave’s velocity [Feynman et al. 1963]. (4.5) can be derived from a one-dimensional lossless transmission- line model of the medium, such as shown in Figure 4.1. The general solution to (4.5) is given by

σ(t, z) =f(ctz) +g(ct+z), (4.6)

where f is the forward-travelling solution, and g is the backward-travelling solution. This analysis is a simplification, as real bulk-waves propagating in a material such as wood are three-dimensional, anisotropic, and spread spherically inside a volumetric space, or spread in plane when on the surface.

The one-dimensional solution, (4.6), can also be expressed as a Fourier-domain transfer function. It can be shown that a forcing function applied at the origin will result in a reproduction of that signal at distance zalong the line, delayed in time by

τ =z/c. Thus, the transfer function in the Fourier domain is given by

H(jω, z) = exp jzω c , (4.7) = exp (jzk), (4.8)

where kis the wavenumber, or angular spatial frequency, in radians per metre. In the lossless model, the wavenumber is real valued.

(4.7) is a pure phase delay with zero attenuation. To create a lossy model, the wavenumber is allowed to become complex,

H(jω, z) = exp −zIm{k}

exp jzRe{k}

. (4.9)

Im{k} is another way of expressing the attenuation coefficient4, α. Alternatively, if the stiffness, E, is specified as the complex frequency-dependent quantity, (4.7) may be expressed as H(jω, z) = exp jωz r ρ E ! , (4.10)

where (1.6) was substituted into (4.7). In sections 4.2.2 and 4.2.3, potential models for describing the complex stiffness, E, are proposed.

4.2.1 Transfer function for the time of flight method

Several variants of the ToF method were described in Section 1.2.3. In the present section, a transfer function is developed for the variant of the ToF method where a metal spike is used to excite the stress wave. This variant is depicted in Figure 1.4. Figure 4.2 shows a block diagram of a linear, time-invariant (LTI) system designed to model the ToF system. In the modelled ToF system, x(t) represents the (unknown) input signal generated by the hammer striking the spike. The output signals y1(t) and

y2(t) are received by the transducers, which are separated by a distance ∆z. The spike and the lower transducer are separated by a distancez0. The transfer function for wood, given by (4.9), is used to model the delay between the output signals. In the Fourier domain, the output signals are described by

Y1() =X()H(jω, z0)P(), (4.11)

Y2() =X()H(jω, z0+ ∆z)P(), (4.12)

whereX() is the input signal, and P() is the response of each transducer. This assumes that the two transducers are matched. These equations use convention of representing time-domain quantities with lower-case letters and frequency-domain quantities with the corresponding upper-case letters. By dividing (4.12) by (4.11),

H(jω,z) is obtained, Y2() Y1() = H(jω, z0+ ∆z) H(jω, z0) = exp (jkz) =H(jω,z). (4.13)

(4.13) shows that if H(jω,z) is known, Y2() may be obtained directly from

4More specifically,αshould be referred to as theabsorption coefficient, as it expresses the amount

of energy lost in the medium due to frictional effects. In this chapter, the term absorption is preferred, but attenuation is used in some instances, as it is used more commonly in the literature.

4.2 VISCOELASTIC TRANSMISSION LINE MODELS 73

h(t, z0) h(t,z)

x(t) y1(t) y2(t)

Figure 4.2 System diagram for the one-dimensional transmission-line model of the ToF method. In

this diagram the input and output signals are described in the time domain. The wood is modelled as an impulse response, which is a function of time and distance.

Y1(), without knowledge of the input signal. In practice,H(jω,z) is not known, and must be inferred from the measured signalsY1() and Y2(). Rather than obtaining the wood’s transfer function directly from (4.13), it is useful to first parametrise the wood’s transfer function in terms of a lumped-element viscoelastic model. This approach allows various unknown quantities such as the static stiffness and the dispersion relation to be estimated.

4.2.2 The Voigt model

The Voigt model (also known as the Kelvin-Voigt model) is a lumped-parameter mechanical model of a viscoelastic material [Kolsky 1963]. The Voigt model describes a frequency dependent relationship for the material’s effective stiffness. The model’s stiffness can be substituted for the stiffness element of the lossless transmission line (Figure 4.1), allowing a frequency-dependent transfer function to be derived. The Voigt

model is shown in Figure 4.3(a).

The Voigt model consists of two parameters, the spring element,E0, representing the material’s elasticity, and the damping element,η, representing the material’s viscosity. Each of these parameters can be quantified in terms of its effective stiffness. The spring element’s stiffness is a constant value,E =σ/. The damper element exerts a stress proportional to the rate of change of its strain, σ =ηddt. In the Fourier domain, the damper’s effective stiffness is jωη. The total effective stiffness of the Voigt model, in the Fourier domain, is

E() =E0+jωη. (4.14)

Kolsky [1963] suggested an alternative form of the Voigt model stiffness,

E() =E0(1 +jωτ), (4.15)

whereτ is therelaxation time, in seconds, given byτ =η/E0. (4.15) shows that when the frequency,f =ω/2π, is much smaller than 1, the effective stiffness is approximately

E0 η (a) E0 E1 η (b)

Figure 4.3 Viscoelastic solid models. (a) the Voigt model, (b) the Zener model.

greater than 1, the total stiffness is dominated by the damping element, and thus increases linearly with frequency.

4.2.3 The Zener model

TheZener model (sometimes referred to as the standard linear solid (SLS) model) is another lumped-parameter viscoelastic model [Zener 1948]. The model is shown in Figure 4.3(b).

For the Zener model, the effective stiffness is given by

E() =E0+

E1jωη

E1+jωη

. (4.16)

(4.16) can be rewritten in the same form as (4.15),

E() = E0(1 +jωτ0) 1 +jωτ1

4.2 VISCOELASTIC TRANSMISSION LINE MODELS 75