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Application of a bending vibration method without weighing specimens to the practical wooden members conditions
Journal of Wood Science volume 68, Article number: 39 (2022)
Abstract
A vibration test to measure the mass of a specimen without weighing it using the difference between the resonance frequency with an additional mass and that without it (vibration method with additional mass, VAM) was applied to small, clear, and wooden specimens assuming practical situations. An apparatus that provided various end conditions by compressing the ends of the specimens was used in the tests. Bending vibration tests were performed on the specimens installed in the apparatus with/without an additional mass. Because the mass ratio (estimated mass by VAM/measured mass), namely, M_{VAM}/M_{0}, which represents the estimation accuracy of VAM, was stable when the compressive strength was sufficiently large, VAM could be adequately used when the specimen was properly installed in the apparatus. The end condition at maximum compressive stress by the apparatus was an imperfect fixed condition. The M_{VAM}/M_{0} value greatly deviated from 1 in the specimens with a large height. The deviation in M_{VAM}/M_{0} from 1 was due to the introduction of the measured resonance frequencies with/without a concentrated mass into the frequency equation that required resonance frequencies under an ideal fixed–fixed condition. A correction method for M_{VAM}/M_{0} was then proposed.
Introduction
Vibration test is a simple and nondestructive testing method for measuring the Young’s modulus. Density is needed to calculate the Young’s modulus. Weighing a specimen requires much time in some cases, such as weighing of each piled lumber and each round bar that is fixed to a post of a timber guardrail.
The mass of a specimen can be measured without weighing the specimen using the difference between the resonance frequency with an additional mass and that without it [1,2,3,4,5]. In the present study, this method is referred to as the “vibration method with additional mass (VAM)”.
To analyze the potential practical application of VAM, a series of parameters was investigated, including the suitable mass ratio (additional mass/specimen) [6], connection between the additional mass and specimen [6], crossers’ position of piled lumber [7], moisture content of specimen [8], bending vibration generation method [9], and effects of shear and rotatory inertia in bending [10]. VAM can be applied to a timber guardrail crossbeam [11], piled lumber [12], and piled round bars [13].
The end conditions of the specimens in the abovementioned studies were simply supported, free, and fixed. However, the end conditions of actual wooden members do not represent these “ideal” conditions. For example, the end condition of a timber guardrail crossbeam is believed to be between the simply supported and fixed conditions, i.e., semirigid [14, 15]. Although the accurate mass of a timber guardrail crossbeam, which can reflect its deterioration, can be estimated using VAM when its attachment to the post is loose [11], the deterioration of the timber guardrail crossbeam can be more efficiently assessed without loosening the attachment. In addition, the underground end condition of wooden piles is semirigid [16]. The simply supported–simply supported condition has been used to estimate the density in traditional timber buildings using nondestructive tests [17].
The effect of the nonideal conditions of specimens on VAM should be clarified for applying VAM to the wooden members that are practically used. This matter was examined by changing end conditions of specimens during vibration tests.
Theory
The bending vibrations under the simply supported–simply supported and fixed–fixed conditions are considered. In the case of a thin beam with a constant cross section, the effect of deflections due to shear and rotatory inertia involved in the bending vibrational deflection is negligible, and the Euler–Bernoulli elementary theory of bending can be applied to the bending vibration.
The resonance frequency, which is denoted by f_{n0} (n refers to the resonance mode number and “0” denotes the value without an additional mass), can be expressed as follows:
where l, E, ρ, I, and A are the span, Young’s modulus, density, second moment of area, and crosssectional area, respectively. m_{n0} is a constant, which is expressed as follows:
The resonance frequency is experimentally reduced by attaching an additional mass while the dimensions, density, and Young’s modulus are not altered. Hence, we can state that m_{n0} changes to m_{n}. Thus, the resonance frequency after attachment of the additional mass is expressed as follows:
The frequency equation for the bending vibration with concentrated mass M placed at position x = al (x: distance along the bar, 0 ≤ a ≤ 1, a + b = 1) on a bar (Fig. 1) can be expressed as follows:
[18].
where μ is the ratio of the concentrated mass to the mass of the bar and can be defined as follows:
To calculate m_{n}, the measured resonance frequencies (f_{n0} and f_{n}) are substituted into Eq. (4). Calculated m_{n} is substituted into Eqs. (5a) and (5b) to calculate μ. The specimen mass and density can be obtained by substituting calculated μ, the concentrated mass, and dimensions of the bar into Eq. (6). The Young’s modulus can be calculated by substituting the estimated density, resonance frequency without a concentrated mass, and dimensions of the bar into Eq. (1) [1,2,3,4,5]. The specimen mass is not required in these calculations.
These calculations illustrate the VAM procedure. In the present study, the estimation accuracy of VAM is expressed by the ratio of the specimen mass estimated using VAM (M_{VAM}) to the measured specimen mass (M_{0}). Here, M_{VAM} represents the product of the estimated mass using the VAM process and lengthtospan ratio. The estimation accuracy of VAM in this study is considered to be sufficiently high at 0.9 ≤ M_{VAM}/M_{0} ≤ 1.1.
Materials and methods
Specimens
Sitka spruce (Picea sitchensis Carr.) rectangular bars with a width of 25 mm (radial direction, R), heights of 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, and 25 mm (tangential direction, T), and a length of 300 mm (longitudinal direction, L) were used as specimens. The surface of each specimen was finished using a tip saw. Three specimens at each dimension were used. The specimens were conditioned at 20 °C and 65% relative humidity until weight became constant. All tests were conducted under the same conditions.
Free–free bending vibration test
To obtain the Young’s modulus by bending, free–free flexural vibration tests were conducted according to the following procedure [19]. The test bar was suspended using two threads at the nodal positions of the free–free vibration corresponding to its resonance mode. The bending vibration was initiated by hitting the LRplane of the specimen at one end using a wooden hammer (hammer head size: 5 mm × 6 mm × 10 mm, 0.85 g), whereas the bar motion was monitored using a microphone (PRECISION SOUND LEVEL METER 2003, NODE Co., Ltd., Tokyo, Japan) at the other end. The signal was processed using a fast Fourier transform (FFT) digital signal analyzer (MultiPurpose FFT Analyzer CF5220, OnoSokki, Co., Ltd., Yokohama, Japan) to obtain highresolution resonance frequencies (Fig. 2).
A vibration test was conducted on a specimen without a concentrated mass, and the resonance frequencies in the 1st to 6th modes were measured. Young’s modulus (E_{TGH}) and shear modulus of the LTplane (G_{LT}) were calculated using the Goens–Hearmon regression method based on the Timoshenko theory of bending (TGH method) [20,21,22]. The Young’s modulus in the absence of the effects of shear and rotatory inertia and shear modulus could be obtained using the TGH method for bending vibration.
Bending vibration test under a semirigid condition
The bending vibration tests under the semirigid condition were conducted according to the following procedure [14]. The apparatus (Takachiho Seiki Co., Ltd., End condition controller KS200) shown in Fig. 3 was used to provide different end conditions. The 25 mm (L) × 25 mm (R) regions from both ends were supported by the posts of the apparatus whose cross section was 25 mm × 25 mm. Consequently, the span was 250 mm. The test bar was compressed by screwing a bolt attached to a load cell. It is thought that the vertical displacement was almost 0 and the rotation was partially restrained at both ends. The compression load was measured by the load cell and recorded using a data logger. Bending vibration was initiated by hitting the LRplane of the specimen in the vertical direction at the center part using the aforementioned wooden hammer. The bar motion was detected by the aforementioned microphone installed at the center part. The signal was processed using the aforementioned FFT digital signal analyzer to generate highresolution resonance frequencies. Because the stress relaxation was observed for each set compressive stress, the vibration test was conducted when the load change became sufficiently small.
Vibration tests were conducted on the specimen with and without a steel plate, which was used as the concentrated mass (1.29–7.55 g, as listed in Table 1) and the resonance frequency of the 1st mode was measured. The steel plate was bonded at x = 0.5 l on the LRplane using a doublesided tape. The bending vibration test was performed using the same specimen without the concentrated mass during the increase in the compressive stress. After the bending vibration test at maximum compressive stress, which means the stress when loading was stopped in this study, the stress was reduced to 0. Subsequently, the concentrated mass was attached to the same specimen, and the bending vibration test was performed using the same specimen with the concentrated mass during the increase in the compressive stress. The compressive stress during the resonance frequency measurement with the concentrated mass and that without the concentrated mass were made to have the same value as much as possible.
Results and discussion
The mean and standard deviation of the density were 411 and 26 kg/m^{3}, respectively, whereas those of the Young’s modulus and shear modulus of the LTplane according to the TGH method were 11.52 and 0.97 GPa, 0.78 and 0.15 GPa, respectively.
Figure 4 shows an example of the changes in the estimation accuracy of VAM during the increase in compressive stress. The changes in M_{VAM}/M_{0} were large at the early stage of loading specimens and M_{VAM}/M_{0} became stable under the larger compressive stress. Because the compression set caused by the partial compression around the ends of the specimens was observed in the case of specimens with a larger height during the preliminary experiments, the maximum compressive stress for the specimens with a larger height was smaller than that for the specimens with a smaller height.
Figure 5 shows an example of the changes in the measured resonance frequency without the concentrated mass (f_{10M}) during the increase in the compressive stress. The f_{10M} value increased with the increase in compressive stress and approached the value of ideal fixed ends for the 1st resonance mode (f_{FF}) (subscript FF: fixed–fixed). Here, f_{FF} was calculated by introducing E_{TGH} into Eq. (1).
Comparing Fig. 4 with Fig. 5, when M_{VAM}/M_{0} was stable (larger than 0.5 MPa for height = 5 mm, larger than 1.7 MPa for height = 25 mm), f_{10M} was stable. In other words, VAM could not be adequately used when the specimen was not properly installed in the apparatus. Hence, M_{VAM}/M_{0} at the maximum compressive strength was discussed hereafter.
Because f_{10M} was close to f_{FF} and was less than f_{FF} in the case of the specimen with a smaller height as shown in Fig. 5, the end condition was considered as an imperfect fixed condition in this study. Hence, the frequency equation under the fixed–fixed end condition, i.e., Eq. (5b) was used to estimate the specimen mass by VAM hereafter.
According to Fig. 6, M_{VAM}/M_{0} increased with the increase in the specimen height (h), i.e., greatly deviated from 1 in the specimens with a larger height. The M_{VAM}/M_{0} value was experimentally approximated as follows:
We assume that the increase in M_{VAM}/M_{0} shown in Fig. 6 is related to the result of f_{10M} < f_{FF} shown in Fig. 5. It is thought that f_{10M} < f_{FF} was caused by the deflections due to shear and rotatory inertia involved in the apparent deflection in the bending vibration [20] and semirigid condition.
The frequency equation taking into account the effects of shear and rotatory inertia under the fixed–fixed condition for a rectangular bar without the concentrated mass is expressed by Eq. (8) based on the boundary condition [23]:
k_{n} and s are a constant and shear factor (= 1.2 for rectangular cross section). Substituting the measured E_{TGH}, G_{LT} and l/h into Eq. (8), k_{n} can be obtained. Mathematica 10.4J software (Wolfram Research Co., Ltd.) was used in the calculation. The relationship of Eq. (15) exists.
The frequency ratios of f_{10SR}/f_{FF} and f_{10M}/f_{FF} were plotted against l/h in Fig. 7. The f_{10M}/f_{FF} value was smaller than f_{10SR}/f_{FF}. It is thought that the difference between f_{10SR}/f_{FF} and f_{10M}/f_{FF} were caused by the semirigid condition at the end. The difference increased with the decrease in l/h.
In the previous study, M_{VAM}/M_{0} decreased with the decrease in l/h under the free–free condition [10] while the opposite tendency is observed in Fig. 6. We think that semirigid condition more strongly affected M_{VAM}/M_{0} rather than the effects due to shear and rotatory inertia.
The M_{VAM}/M_{0} value that was larger than 1 was discussed. The VAM procedure expressed by Eqs. (1)–(5) are based on the assumption that a specimen is under the ideal conditions, however, the measured resonance frequencies with and without the concentrated mass subjected to the effect of the nonideal condition (f_{1M}, f_{10M}) were used as f_{1} and f_{10} for VAM (Eq. (4)), respectively, in the “Results and discussion” Section. Because m_{10} = 4.73 for the ideal condition was substituted into Eq. (4), the resonance frequency ratio f_{1M}/f_{10M} was discussed.
The relationship between f_{n}/f_{n0} and f_{nM}/f_{n0M} was experimentally investigated. Ideal m_{n}/m_{n0} could be calculated by substituting the measured mass ratio = (measured concentrated mass)/(measured specimen mass) into Eq. (5b). Mathematica 10.4J software was used in the calculation. By substituting the obtained m_{n}/m_{n0} into Eq. (4), ideal f_{n}/f_{n0} can be calculated.
Here, degree of ideality (DI) was defined as:
where DI = 1 indicates the ideal condition of specimens and DI < 1 indicates that a specimen deviates from the ideal condition. According to Fig. 7, DI was experimentally approximated as follows:
The ratio (f_{1M}/f_{10M})/(f_{1}/f_{10}) was plotted against DI (Fig. 8), and a liner relationship existed at 1% significance level as follows:
Since f_{1M}/f_{10M} > f_{1}/f_{10} as shown in Fig. 8, m_{1} obtained by substituting f_{1M}/f_{10M} into Eq. (4) was larger than m_{1} under the ideal condition. According to the previous study [4], m_{n} theoretically decreases with the increase in μ under the fixed–fixed condition. The M_{VAM} value increases with the decrease in μ from Eq. (6). Therefore, M_{VAM}/M_{0} under the nonideal condition was larger than that under the ideal condition.
The M_{VAM} value could be corrected by according to the following procedure. Measured l/h was substituted into Eq. (18) to estimate DI. Substitution of the estimated DI and the measured f_{1M} and f_{10M} into Eq. (19) yielded the estimation of f_{1}/f_{10}. The m_{1} value was estimated by substituting the estimated f_{1}/f_{10} and m_{10} = 4.73 into Eq. (4). The μ value was estimated by substituting the estimated m_{1} and the position of the concentrated mass into Eq. (5b). The M_{VAM} value was corrected with the substitution of the estimated μ into Eq. (6). The corrected M_{VAM}/M_{0} is shown in Fig. 9. Many results could be improved using this correction method.
Conclusions
VAM was applied to small clear specimens under conditions where wooden members are expected to be practically used, and the following results were obtained:

1.
VAM could not be adequately used when the specimen was not properly installed in the apparatus.

2.
Because f_{10M} was close to f_{FF} and was less than f_{FF} in the case of the specimen with a smaller height, the end condition at the maximum stress was considered as an imperfect fixed condition in this study.

3.
M_{VAM}/M_{0} increased with the increase in the specimen height and greatly deviated from 1 in specimen with a larger height.

4.
The deviation in M_{VAM}/M_{0} from 1 was due to the substitution of the measured resonance frequencies into the frequency equation that required the resonance frequencies under the ideal fixed–fixed condition.

5.
A correction method for M_{VAM}/M_{0} was subsequently proposed.
Availability of data and materials
All data generated or analyzed during this study are included in this published article.
Abbreviations
 VAM:

Vibration method with additional mass
 L:

Longitudinal direction
 R:

Radial direction
 T:

Tangential direction
 FFT:

Fast Fourier transform
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Acknowledgements
This study was supported by Research grant #201805 of the Forestry and Forest Products Research Institute and JSPS KAKENHI Grant Number JP20H03052.
Funding
This study was supported by Research grant #201805 of the Forestry and Forest Products Research Institute and JSPS KAKENHI Grant Number JP20H03052.
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All authors designed the experiments. YK performed the experiments, analyzed the data, and was a major contributor in writing the manuscript. All authors contributed to interpretation and discussed results. All the authors read and approved the final manuscript.
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Kubojima, Y., Sonoda, S. & Kato, H. Application of a bending vibration method without weighing specimens to the practical wooden members conditions. J Wood Sci 68, 39 (2022). https://doi.org/10.1186/s10086022020461
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DOI: https://doi.org/10.1186/s10086022020461
Keywords
 Additional mass
 Fixed–fixed condition
 Mass
 Rotatory inertia
 Semirigid
 Shear
 Vibration test