In wood drying, the moisture content of wood lumber at the end of the drying process is very important. The moisture content of the wood lumber outside of the pile can be easily determined using a commercial moisture meter. However, it is difficult to measure the moisture content of the wood lumber inside the pile; moisture meters cannot be generally attached to the cross section of the wood lumber because they are not designed to measure the cross section, and the moisture content around the end of wood lumber is too low and not representative of the whole wood lumber. Hence, it is necessary to dismantle the pile and take the wood lumber from inside the pile out for weighing. However, this process is labor-intensive and space is required to place the wood lumber when the pile is dismantled.

The mass of water that has been removed from the wood lumber due to drying can be obtained by weighing the wood lumber before and after drying. Hence, a weighing method for the wood lumber inside the pile is desired.

A vibration method for obtaining the mass, density, and Young’s modulus without weighing based on the difference of the resonance frequency when a concentrated mass is added to and subtracted from wood has been proposed [1,2,3,4,5,6,7,8,9,10,11,12]. This method is referred to as the “Vibration method with additional mass (VAM)” in the present study.

VAM has been studied from various aspects for practical use. Since the weight of the upper lumber is applied to the lower lumber through crossers, the effect of the crossers’ positions on VAM was investigated [8]. As a result, accurate density and Young’s modulus values could be determined using VAM, without the influence of the weight of the upper lumber, by placing crossers at the nodal positions of the longitudinal vibration. The estimation accuracy of VAM using bending vibration was higher than that using longitudinal vibration [9]. The accurate resonance frequency for bending vibration could be generated by tapping the cross section of the wood lumber and the bending vibration with such a method was effective for VAM [10].

In this study, we created a model of wood lumber inside the pile. The purpose of this study is to determine whether bending vibration can be generated by tapping the cross section of the wood lumber inside the pile, and to determine whether this method for generating the bending vibration is effective to VAM.

### Theory

Bending vibration under free–free condition is considered. In the case of a thin bar with a constant cross section, the effects of shear deflection and rotary inertia involved in the bending vibrational deflection are negligible, and the Euler–Bernoulli elementary theory of bending can be applied to the bending vibration.

The resonance frequency, represented by *f*_{n0} (n: resonance mode number, 0: value without the additional mass), is expressed as follows:

$$ f_{\rm {n0}} = \frac{1}{2 \uppi }\left( {\frac{{m_{\rm {n0}} }}{l}} \right)^{2} \sqrt {\frac{EI}{\rho A}} , $$

(1)

where *l*, *E*, *ρ*, *I*, and *A* are the specimen length, Young’s modulus, density, the second moment of area, and the cross-sectional area. *m*_{n0} is a constant that is expressed as follows:

$$ m_{10} = 4.730, \quad m_{20} = 7.853, \quad m_{30} = 10.996, \quad m_{\rm {n0}} = \frac{1}{2}\left( {2n + 1} \right)\uppi \quad (n > 3). $$

(2)

The resonance frequency is decreased experimentally by attaching an additional mass while the dimensions, density, and Young’s modulus are not altered. Hence, it can be said that *m*_{n0} changes to *m*_{n}. As a result, the resonance frequency after attaching the additional mass is expressed as follows:

$$ f_{\rm n} = \frac{1}{2\uppi }\left( {\frac{{m_{\rm n} }}{l}} \right)^{2} \sqrt {\frac{EI}{\rho A}} . $$

(3)

From Eqs. (1) and (3),

$$ m_{\rm n} = \sqrt {\frac{{f_{\rm n} }}{{f_{\rm n0} }}} m_{\rm n0} . $$

(4)

The frequency equation for the free–free vibration with concentrated mass *M* placed at position *x* = *al* (*x*: distance along the bar, 0 ≤ *a* ≤ 1, *a* + *b* = 1) on the bar (Fig. 1) is expressed as follows:

$$ \begin{aligned} \left( {\cos m_{\rm n} \cosh m_{\rm n} - 1} \right) - \frac{1}{2}\mu {m}_{\rm n} & \left\{ {\left( {\cos am_{\rm n} \cosh am_{\rm n} + 1} \right)\left( {\sin bm_{\rm n} \cosh bm_{\rm n} - \cos bm_{\rm n} \sinh bm_{\rm n} } \right)} \right. \\ & \left. {\; + \left( {\cos bm_{\rm n} \cosh bm_{\rm n} + 1} \right)\left( {\sin am_{\rm n} \cosh am_{\rm n} - \cos am_{\rm n} \sinh am_{\rm n} } \right)} \right\} = 0, \\ \end{aligned} $$

(5)

where *μ* is the ratio of the concentrated mass to the mass of the bar, and it is defined as

$$ \mu = \frac{M}{\rho Al}. $$

(6)

The measured resonance frequencies *f*_{n0} and *f*_{n} are substituted into Eq. (4) to calculate *m*_{n}. The calculated *m*_{n} is substituted into Eq. (5) to calculate *μ*. The specimen mass and density can be obtained by substituting the calculated *μ*, the concentrated mass, and the dimensions of the bar into Eq. (6). Young’s modulus can be calculated by substituting the estimated density, the resonance frequency without the concentrated mass and the dimensions of the bar into Eq. (1) [1,2,3,4,5,6].

A measurement of the specimen mass is not required for these calculations. The above is the calculation procedure of VAM.