The resonance method

Stiffness can be determined acoustically by measuring the resonant frequency of an oscillating wood member. There are two types of resonance tests: transverse resonance, and longitudinal resonance. Transverse resonant vibrations have long been used for gauging wood stiffness of logs and timber [Jayne 1959, Marra et al. 1966, Pellerin 1965]. In this method, a wooden member is suspended at specific locations, known as nodes. A transverse vibration is excited in the member by striking it at anantinode, usually the middle of the member. A measurement device is used to record the frequency

1.2 MEASURING WOOD STIFFNESS 13

at which an antinodal location oscillates vertically. Haines et al. [1996] explains that the frequency of oscillation can be modelled using the Timoshenko beam theory. The resulting MoE is given by

E= 0.946ρfres 2_L4

D2 , (1.8)

where ρ is the wood’s density, fres is the induced resonant frequency,Lis the length of the specimen, andD is the diameter of the specimen. A derivation of (1.8) is found in Timoshenko [1955]. The transverse resonance method is not as widely used as the longitudinal method, this is possibly due to the difficulty of suspending the sample when conducting a measurement.

It has been known since the 1950s that longitudinal resonance vibrations could be used to measure the MoE of wood [Bell et al. 1954, Bodig and Jayne 1982, Galligan and Courteau 1965, Pellerin and Galligan 1973, Porter et al. 1972]. The longitudinal resonance method is usually referred to simply as the resonance method. Though it was used in laboratories for some time, the method was first used for log grading applications by Japanese researchers [Aratake et al. 1992, Arima et al. 1990, Fujisawa et al. 1994]. The resonance method is a technique used for measuring the stiffness of harvested logs (and sometimes cut timber). The method uses a specialised device called a resonance tester. This device contains a single transducer, which is placed against one end of a log. The operator strikes the end of the log with a hammer, causing a stress wave to propagate longitudinally. The wave is reflected by the end faces of the log, causing a

standing wave oscillation. The stress wave predominates at frequencies given by

fres=

ncres

2L , (1.9)

where n is a positive integer, called theharmonic number;cresis the resonant wave’s velocity in the log; andLis the length of the log. Each harmonic number corresponds to a

resonant frequencyof the system. Each harmonic is separated in frequency by an integer multiple of the fundamental. The device records the signal at the transducer and then performs a fast Fourier Transform (FFT) analysis to find the resonant harmonics. (1.9) can then be used to determine the longitudinal acoustic velocity of thenth harmonic. The velocity can then be used to estimate the MoE using (1.6), the result of combining these equations is

E = 4ρ fresL n

!2

. (1.10)

An illustration of how the measurement is conducted, and a plot of the variation in magnitude along the length of the log for the first three harmonics, is shown in Figure 1.3. Note that the amplitude is zero at the nodes, such as in the middle of the member; and maximal at the antinodes, such as at the member’s ends.

0 L 6 L 4 L 3 L 2 2L 3 3L 4 5L 6 L

Distance along log 0 1 α Amplitude n= 1 n= 2 n= 3

Figure 1.3 In the resonance method, striking the end of the log induces a standing wave. A receiving

transducer is held in contact with one end of the log. The magnitude of the stress wave varies along the length of the log.

frequencies in the low kHz range. They claimed that this is only suitable for samples longer than about 200 mm. For this reason, to test small samples they employed a

swept resonance excitation technique. This technique uses an amplified loudspeaker on one end of the sample, and a receiving transducer on the other end. The frequency of the loudspeaker is swept over a range of frequencies (typically 10 harmonics), and the spectrum of the received signal is analysed to find the resonant harmonics. Harris et al. [2002] described the WoodSpec system, which uses the swept resonance technique for measuring the velocity of small samples. They explain that the sample is stimulated using “a predefined band of frequencies in a continuous or semi-continuous fashion over an excitation period.”

Few papers have focused specifically on the implementation of the resonance method, though details may be gleaned from papers which performed experiments using the method. Harris and Andrews [1999] noted that implementations of the method generally use the second harmonic to determine the MoE, though the reason for this choice is unclear. Carter et al. [2007] stated that the Director HM200 also uses the second harmonic. Several authors have noted that the resonance peaks may not be integer- multiple harmonics of each other [Andrews 2000]. Chauhan and Walker [2006] observed that the first and second harmonics can differ in speed by as much as 11%. It is unclear which harmonic gives the closest estimate of the static stiffness. Harris et al.

1.2 MEASURING WOOD STIFFNESS 15

[2002] showed using a time-frequency analysis that the initial portion of the recorded waveform contains comparatively more high-frequency content. This was explained as a result of the “hit function” of the hammer excitation, which decays away after some time. This effect tends to bias the lower harmonics to slightly higher frequencies. They showed that by dividing the waveform into time windows, a more consistent frequency reading is obtained by discarding the first window and keeping subsequent windows. The WoodSpec system, described by Harris et al. [2002], circumvents this problem by continuously or semi-continuously exciting the sample. Ross et al. [1997] used a variant of the resonance method. Instead of performing an FFT analysis, they measured the time between peaks in the time-domain. This method was first described by Ross et al. [1994], and was also used by Wang et al. [2004b]. This method is likely less exact, as higher frequency harmonics can overlay the fundamental, which makes it more difficult to identify the exact location of the fundamental.

Several studies have compared the MoE, as calculated using (1.10), to that measured by static methods. Wang et al. [2001a] conducted resonance and static bending tests on a total of 159 Pinus resinosa (red pine) and Pinus banksiana (jack pine) logs. They found that both methods determined similar values of MoE for both species, with

r2 values 0.95 and 0.85 for red pine and jack pine, respectively. Harris et al. [2002] measured resonance velocity on 44 small internodal bolts cut from clonal stems aged 2 years. They found a close match between the static MoE of the samples and the resonance-estimated MoE (Estatic = 1.02Eres), with r2 = 0.93. Lindström et al. [2002] conducted resonance and static bending tests on sevenPinus radiata clones of age four years. Regressions calculated between the methods showed a close correspondence using both a microphone based resonance (Estatic= 1.04Eres), and a swept-resonance based approach (Estatic= 1.07Eres). Evans and Kibblewhite [2002] measured the resonance velocity on 50 small Pinus radiata samples of age 25 years. These were compared against an X-ray based estimate of the static stiffness (for details on this method, see Evans et al. [2000a] and Evans and Ilic [2001]). They found that the two methods reported similar values for the MoE of the samples, with r2 = 0.88. Lindström et al. [2004] conducted resonance and static compression tests on 22 Pinus radiataclones of age three years. Measuring at 12% MC, they found a close relationship between the two methods (Estatic = 1.02Eres), with r2 = 0.93. Ilic [2001] measured the resonance velocity of 104Eucalyptus delegatensis samples, and then compared the resonance MoE to that determined using a static technique. A strong correlation between the two methods was found (r2 = 0.95). Hodousek et al. [2017] conducted resonance and static tests on a total of 110 samples of Cupressus lusitanica andPopulus canadensis. They found strong correlations, with values of r2= 0.87, andr2 = 0.81, respectively.

Several authors have suggested that the resonance method’s close match with static tests is due to the method providing a weighted average of the log’s cross sectional stiffness [Carter et al. 2007, Chauhan et al. 2005, Grabianowski et al. 2006, Harris

et al. 2002]. Chauhan et al. [2005] measured the MoE of layered plywood which was constructed using several different layering configurations. They predicted the overall MoE of each type of plywood using an area-weighted sum of each layer’s effective MoE. They found that for most plywood configurations the difference between the resonance-predicted MoE and the weighted-average predicted MoE was significantly less than 5%. Carter et al. [2007] measured the resonance velocity in six Pinus radiata

logs aged 24 years, followed by the resonance velocity in boards sawn from the logs. They found that while there was significant variation amongst the board velocity, the average of all boards in a log was similar to the log’s overall velocity. These studies appear to confirm the hypothesis that the resonance method provides an area-weighted average measurement of a log’s MoE.