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Approaches to resolving errors in the time of flight method

MEASURING WOOD QUALITY USING ACOUSTICS

1.2 MEASURING WOOD STIFFNESS

1.2.5 Approaches to resolving errors in the time of flight method

The ToF method is currently the only method for rapidly obtaining a measure of a standing tree’s stiffness. This makes it a valuable tool for breeding programs. For example, early selection of the best wood based on stiffness has been suggested [Apiolaza 2009, Huang et al. 2003, Matheson et al. 2008]. However, a problem with the ToF method is that it consistently overestimates the MoE, as shown in Section 1.2.4. This overestimation is demonstrated in Figure 1.5, which shows the result of one study which compared ToF and resonance measurements [Chauhan and Walker 2006].

Some authors have modelled the discrepancy between the two methods as a ratio, known as thevelocity ratio,

kr= d

ctof d

cres

, (1.13)

wherecdtof is the mean ToF velocity for a particular experiment and cdres is the mean resonance velocity measured on the same wood samples. Wang [2013] provided a review of studies which quantified the velocity ratio for several species. The studies that were cited foundkr values ranging from 1.09 to 1.36. A number of authors have attempted to resolve the discrepancy between methods using linear regression models. Wang et al. [2007a] provided an overview of this type of model, as applied to non-destructive testing in forestry applications. The basic form of the model, with two parameters, is

1.2 MEASURING WOOD STIFFNESS 21

where A and B are the regression parameters, which are fitted to the data using a numerical method such as least-squares. Ross and Wang [2005] applied this model to a number of different species and provided values forA, B, and the resultant coefficient of determination, r2. Baar et al. [2012] applied this model to the heartwood of 5 different tropical hardwoods, and found a strong correlation (r2= 0.81) for this type of model.

Other authors have attempted to incorporate the dimensions of the tree and stem into the regression model. Wang et al. [2004a] developed an empirical model with a dependency on stem diameter. Their model is given by

E0 =Actof2ρ B

DC, (1.15)

where E0 is the adjusted MoE, for comparison to the resonance velocity; ctof is the ToF velocity; Dis the log diameter; andA, B, andC, are the fitted model parameters, which are determined numerically. They found that (1.15) produced a closer fit to the resonance method’s estimate of the MoE, when compared to the simpler model of (1.14). This model was also applied to data presented by Wang et al. [2007b].

Researchers have suggested several explanations why the ToF method overestimates the stiffness when compared to other methods. It is understood that wood stiffness increases radially outwards from pith to bark8 [Chauhan et al. 2006a]. Authors hy- pothesise that static and resonance methods provide an average measurement of the stiffness throughout the sample [Carter et al. 2007]. In contrast, the ToF method may only measure a wave in the high-stiffness outerwood. Chauhan and Walker [2006], and Lasserre et al. [2007] suggested that the discrepancy between methods is due to the ToF stress wave “flight path” transiting the outerwood.

Studies have quantified the discrepancy between corewood and outerwood stiffness by measuring both ToF and resonance on stems, and also boards cut from the stems [Baar et al. 2012, Grabianowski et al. 2006]. This type of experiment seeks to homogenise the quality of the wood on which an experiment is performed by cutting boards at fixed radial distances from the pith. Grabianowski et al. [2006] hypothesised that the over estimation of the ToF was due to the waves propagating through the stiffer outerwood, while the resonance method measures a weighted cross section throughout the log. To test this, they measured the ToF and resonance of several Pinus radiata logs of ages 8, 16, and 26, and then sawed them into boards, which were also tested for ToF and resonance. Some of the results of this study are shown in Table 1.1. The results show that the mean ToF velocity of the outerwood boards was close to the ToF of the log, while the ToF of the corewood was significantly lower. They also show that the ToF method measured a significantly greater velocity than the resonance method on the complete logs. These finding support the idea that the ToF method predominantly measures the propagation through the outerwood.

Table 1.1 Average velocity of 43 stems, using both the time-of-flight and resonance methods.

The stems were cut into outerwood and corewood boards, from which the velocity was measured [Grabianowski et al. 2006]. ToF was measured using Fakopp 2D [Fakopp Bt 2001]; resonance was measured using WoodSpec [Harris et al. 2002].

ToF (m/s) Resonance (m/s)

Mean Range Mean Range

Logs 2,466 1,739–2,929 2,202 1,655–2,603 Green outerwood boards 2,505 1,852–2,966 2,099 1,665–2,600 Green corewood boards 2,223 1,598–2,685 1,931 1,445–2,360

However, the results shown in Table 1.1 also leave some questions unanswered. The mean ToF velocity in the outerwood boards (2505 m/s) is significantly greater than the resonance velocity on the same boards (2099 m/s). A similar discrepancy was also observed in the corewood boards. It is possible that there is some additional effect in either of the methods causing this discrepancy, which appears to be present even when the wood samples have been homogenised.

Andrews [2002] suggested that the discrepancy between the methods is related to stem diameter. This type of wave phenomenon was referred to by Kolsky [1963] as bar waves. The theory predicts that a cylindrical bar, such as a tree stem, acts like a waveguide, causing the majority of energy to propagate at the speed given by (1.6), provided that the wavelength is significantly longer than the diameter of the log. Andrews [2002] supposed that measurements made using the resonance method propagate at this bar speed, provided that the wavelength is sufficiently long. Kolsky [1963] suggested that a small portion of energy propagates at the dilatational speed, given by c = p

C11(see Section 2.7.1 for the definition of the dilatational speed). This speed is always greater than the bar wave speed. Andrews [2002] suggested that the ToF method measures the dilatational speed. This would explain the consistently greater speed which is measured. An overview of this theory, as applied to wood testing is described by Wang et al. [2005] and Wang et al. [2007b] (see Appendix B for a description of the theory). Mora et al. [2009] used the bar wave theory to estimate Poisson’s ratio in Pinus radiata trees. They found that the bar wave theory produced an effective correction factor for the discrepancy between ToF and resonance methods. The correction factor reduced the mean discrepancy from 32% to 0.02%, withr2 = 0.98. Though it is unclear whether this model is generalisable to other species or environmental conditions.

Several authors have evaluated the effect of log diameter on wave speed. Wang et al. [2004a] measured the MoE both statically and using the ToF method for 201 small logs from four different species. They found that for larger diameter samples, the discrepancy between static and ToF methods tends to increase. Other authors have also evaluated the effect of diameter on resonance. Chauhan and Walker [2006] noted that

1.2 MEASURING WOOD STIFFNESS 23

the difference between ToF and resonance was greatest in older, larger diameter trees. Andrews [2002] measured the ToF and resonance on a number of logs and cylindrical metal samples. He argued that the speed discrepancy between methods was dependent on the ratio of length to diameter of the sample. However, some studies have not found any systemic difference between ToF and resonance with respect to diameter [Lasserre et al. 2007, Mora et al. 2009]. Gonçalves et al. [2011] found that the dependence of velocity on diameter was only observed in some species. Zhang [1995] observed that within a stand the bigger trees generally produce a wood of lower density and stiffness. Lasserre et al. [2005] observed the same trend by influencing the DBH by controlling planting density. This effect is likely due to a reduction in density as the DBH is increased, i.e., the tree contains the same volume of cell wall material, but cell walls are thinner and cell lumens are larger, resulting in a larger stem diameter.

Some authors have observed that the presence of bark on a tree can cause the resonance velocity to be significantly lower than the same tree when de-barked, [Chauhan and Walker 2006, Grabianowski et al. 2006, Lasserre et al. 2007]. Lasserre et al. [2007] found that removing bark from Pinus radiata caused the resonance velocity to increase by 8.3% on average. The increase was consistently observed amongst sampled stems. Hsu [2003] sampled bolts cut from different heights up a Pinus radiata tree and found that the debarked velocity on the bottom bolt was 7.2% higher. The debarked velocity at the top of the tree increased by 22.6%. These are surprising results because bark’s low density and stiffness would be expected to prevent a large amount of acoustic energy from being coupled into it. These results seem to support the idea that the resonance method provides an average of the cross-sectional stiffness throughout the stem. The effect of bark on ToF measurements has also been examined. Grabianowski et al. [2006] found that debarkingPinus radiata did not have a significant effect on the ToF measurements. They note that this finding supports the hypothesis that the ToF stress wave propagates primarily through the outerwood. Their finding suggests that the ToF velocity is not an average of the stem’s cross-sectional stiffness.

Some authors have found that the presence of both branches and knots reduces the measured velocity. Lasserre et al. [2007] found that branch removal increased resonance velocity by 5.4% on average and up to 24% in low MoE samples, but this increase did not always occur. This finding is consistent with the cross-sectional average theory of the resonance method, as the branches increase the overall cross-section, but do not contribute significantly to the stiffness. Hsu [2003] measured the ToF velocity between clearwood sections (internodes) and compared that to ToF measurements taken across an entire stem. It was found that an area containing knots decreases the acoustic velocity by approximately 5%. These findings are to be expected, as the direct path through a knot takes the stress wave across the grain.

Attempts have been made to correct for the discrepancy between ToF and resonance by placing the two probes on opposite sides of the tree, thus measuring across the

grain [Dickson et al. 2003, 2004, Mahon 2007, Mahon et al. 2009, Matheson et al. 2002]. These studies measured a slower speed for the across-grain velocity than the direct-path ToF velocity. A problem with this approach is that it underestimates the MoE. This is likely because the authors did not incorporate the lower across-grain stiffness and lower heartwood stiffness into their analyses.

Some authors have observed that the frequency of ToF stimulation has an effect on the measured velocity. Haines et al. [1996] observed that dynamic bending tests overestimate the MoE of boards compared to static bending. They hypothesised that this was because the wood was actingviscoelastically. This causes the effective stiffness to increase with the frequency of stimulation. Ouis [2002] explained that the viscoelasticity of wood leads to an acoustic wave speed which is frequency dependent. This is known as

dispersion. O’Donnell et al. [1980] explained that a Kramers-Kronig relationship may be used to show that a medium which exhibits attenuation is also dispersive. The dispersive and attenuative qualities of generalised solids are explored in more detail by Kolsky [1963]. Divos et al. [2005] analysed the speeds of resonant modes in a spruce specimen. They found that the speed of the modes tends to increase with frequency. Bucur and Feeney [1992] measured the attenuation and stress wave velocity in horse chestnut wood using a continuous transmission technique. They found that the longitudinal velocity increased significantly with frequency. Salmi et al. [2013] measured the dynamic MoE in Norway spruce samples at several ultrasonic frequencies. They found a general trend of increasing MoE as frequency increases. Chiu et al. [2013] measured acousic velocity in Taiwan incense cedar using both ultrasonic ToF and hammer-excited ToF methods. They found that in general the ultrasonic velocity was greater than the lower-frequency hammer-hit excited velocity. These results are consistent with the behaviour of viscoelastic models of solids, which predict that MoE and acoustic velocity increase as frequency increases.

1.3 FACTORS WHICH AFFECT STIFFNESS

Gibson [2005] explains that the structure of wood may be modelled, to a first approx- imation, as a honeycomb. Wood, like a honeycomb, is stiffest in the direction of the alignment of the cells. The honeycomb model suggests that the longitudinal stiffness varies linearly with density. While the stiffness across the grain varies with the cube of the density. It should be noted that the density of dry cell-wall material is approximately 1500 kg/m3 across all species [Gibson 2005, Walker 2006c]. The MoE and MoR of cell-wall material in the direction of the microfibrils is approximately 35 GPa and 350 MPa, respectively [Cave 1968, 1969]. The density and stiffness of the material which constitutes the cell wall is relatively uniform. The density and stiffness of the resultant honeycomb structure, however, is largely determined by the cell wall thickness and the alignment of the cell wall fibres with the cell’s axis (MFA).