# Determination of the Dynamic Modulus of Elasticity of Pine Based on the PZT Transducer

^{1}

^{2}

^{*}

## Abstract

**:**

_{d}) of pine wood, based on the transverse vibration excitation and electromechanical impedance (EMI) response of the lead zirconate titanate (PZT) transducer is proposed. The influence of the length to thickness ratio of the pine specimen on the measurement accuracy was studied through modal simulation analysis. Based on the results of the modal simulation, the size of the pine specimen was optimized, and the scanning frequency range of the EMI response was determined. On this basis, the EMI simulation and test of the pine specimen coupled with a PZT patch were carried out to verify the effectiveness of the novel method. The impedance simulation results of three kinds of pine specimens show that a unique and significant formant appears in the real part of each EMI response curve, and the maximum relative errors of the rectangular PZT patch and circular PZT patch are 1.34% and 1.81%, respectively. The impedance test results of three kinds of pine specimens indicate that the maximum relative errors of the rectangular PZT patch and circular PZT patch are 1.41% and 1.68%, respectively, compared with the corresponding results obtained by the traditional transverse vibration method. Simulation and experimental results verify the validity of the proposed method for the elastic modulus determination of pine wood.

## 1. Introduction

_{d}) of wood can be traced back to the 1950s, and it mainly includes two methods of longitudinal and transverse vibration excitation. Hearmon et al. [2] determined the elastic modulus of pine through transverse and longitudinal vibration methods, and their research showed that the result of the transverse test was 5%–8% smaller than that of the longitudinal test. Brenndorfer et al. [3] also described the difference between the modulus of elasticity obtained from transverse and longitudinal vibration methods. Haines’ research indicated that the transverse vibration method was more accurate than other dynamic methods for the determination of the elastic modulus of wood [4]. Yin et al. [5] determined the modulus of elasticity of Canadian conifers with the transverse vibration method, and the results showed that the elastic modulus based on the transverse vibration had a strong linear correlation with the static modulus of elasticity. Wang et al. [6] demonstrated that the transverse vibration method was the best dynamic technology to assess sawn timber. The transverse vibration method has been standardized under ASTM D6874-12 for solid wood [7], and many researchers have tested the modulus of elasticity of wood and wood-based materials using this method. The transverse vibration method often adopts only the first-order bending mode frequency based on the Euler–Bernoulli theory. However, this method requires transverse vibration excitation and signal reception to acquire the fundamental mode frequency. The bending modes of the beam specimen are generally excited by the external load, and its vibration response is received by the sensor. The frequency response function (FRF) can be deduced through the signals of vibration excitation and response, with which the first-order bending mode frequency can be obtained. Therefore, a two-channel signal acquisition system is usually needed. That is, the system is relatively complex, occupies a large area, and has a high cost.

## 2. Materials and Methods

#### 2.1. Materials

^{3}, 483 kg/m

^{3}, and 432 kg/m

^{3}, respectively. The size of the specimens used for impedance measurement is 250 mm × 40 mm × 5 mm (length × width × thickness), and the dimension of the large specimens for verification by the traditional transverse vibration method is 1000 mm × 40 mm × 20 mm. In addition, three kinds of pine, namely Korean pine, Scots pine (Pinus sylvestris L.), and eastern white pine (Pinus strobus), were selected for simulation analysis.

#### 2.2. E_{d} Calculation of the Transverse Vibration Method

_{d}is the dynamic modulus of elasticity along the x direction.

_{1}can be derived as follows.

_{d}of the specimen can be calculated as Equation (3) for a rectangular beam.

#### 2.3. Theoretical Basis of the EMI Method

_{a}locates between x

_{a}and x

_{a}+ l

_{a}. The PZT patch expands and contracts with the excitation of a harmonic voltage signal, which creates a longitudinal force on the surface of the beam specimen. Its effect at the neutral axis of the beam specimen is equivalent to a longitudinal force and a bending moment according to the theorem of force translation, as shown in Figure 1b.

_{31}is the electromechanical coupling factor, $\phi $ is the phase angle, and r(ω) is the stiffness ratio defined as follows:

_{p}(ω) is the quasi-static stiffness of the PZT patch and K

_{s}(ω) is the dynamic stiffness of the beam specimen, which can be calculated as follows with no consideration of the modal damping:

_{n}(x), ω

_{n}, W

_{m}(x), and ω

_{m}are the longitudinal mode shape of the beam specimen, longitudinal mode frequency, bending mode shape of the specimen, and bending mode frequency, respectively.

#### 2.4. E_{d} Detection Principle Based on the EMI Method

## 3. Results and Discussion

#### 3.1. Size Optimization of the Specimen

_{L}, E

_{R}, and E

_{T}, shear modulus G

_{LR}, G

_{RT}, and G

_{LT}, Poisson’s ratios ν

_{LR}, ν

_{RT}, and ν

_{LT}, and the density ρ. The longitudinal modulus of elasticity E

_{L}reflects the main mechanical property of pine and is much larger than the modulus of two other directions. Therefore, in this study, the length direction of the beam specimen is consistent with the longitudinal direction of the wood, with which the dynamic modulus of E

_{L}(E

_{Ld}) can be further obtained based on Equation (3). However, Equation (3) only considers the bending moment of the specimen. In an actual transverse vibration test, the flexural specimen is also affected by the rotatory inertia and shearing force, and these effects are improved with the decrease in the length to thickness ratio (LTR) of the specimen [28]. That is, the length to thickness ratio of the specimen will affect the measurement accuracy. The appropriate length to thickness ratio must be determined first prior to optimizing the size of the specimen. Therefore, bending modal analyses of the pine specimens were conducted by finite element simulation. The finite element models of free beam specimens were constructed by COMSOL Multiphysics. The parameters of three kinds of pine are from the literature [29,30,31] and are shown in Table 1. Two kinds of pine were selected for modal analysis and the parameters of the specimens are shown in Table 2 and Table 3. The physical field of solid mechanics and eigenfrequency analysis in the modal simulation were applied for the mode frequency extraction. The model was meshed at a size of 2 mm to ensure the accuracy of the modal analysis. Modal analysis results can provide modal frequencies for each vibration mode [32]. Here, only the frequency corresponding to the first-order bending mode is needed, as shown in Figure 3. The first-order bending mode frequency of each specimen was extracted, with which the E

_{Ld}was calculated based on Equation (3). Then, the relative error can be further obtained using Equation (7). All the analysis results are shown in Table 2 and Table 3.

_{Ld}is less than 1.5% when the length to thickness ratio reaches 50 for Korean pine or Scots pine. Therefore, the size of the pine specimen can be optimized to 250 mm × 40 mm × 5 mm (length × width × thickness) to save test materials and to reduce the specimen size. Based on the above method of modal simulation analysis, the first-order bending mode frequencies of three kinds of optimized pine specimens were obtained, and the detection errors were calculated according to Equation (7) and are shown in Table 4. Table 4 illustrates that the maximum relative error is also less than 1.5% using the optimized specimens, which meets the actual detection requirement.

#### 3.2. Verification of the Validity of the Proposed Method Based on the EMI Simulation

_{Ld}detection based on the EMI response. The thickness of the PZT patch should be as thin as possible to reduce its influence on the mode of the specimen. Two shapes of PZT patches were selected for this study. One is a rectangular PZT patch with dimensions of 20 mm × 10 mm × 0.2 mm (length × width × thickness), and the other is a circular PZT patch with dimensions of Φ15 × 0.2 mm. Based on the optimized dimension of the pine specimen, the impedance response simulation model of the PZT coupled with the specimen was established by the two functional modules of the piezoelectric effect and frequency domain analysis. The PZT patch was located at the center of the top surface of the specimen. The connection layer between the PZT and the specimen was ignored in the simulation model, and the PZT and the specimen were set as a combination. The PZT patch is composed of PZT-5H piezoelectric material, and the parameters are listed in Table 6. The polarization direction of the PZT patch was set perpendicular to its surface. A harmonic excitation voltage with an amplitude of 1 V was applied to the top surface of the PZT patch, and its bottom surface was set as the ground. The maximum size of the meshed finite elements was 2 mm, which could meet the requirement of consisting of five nodes per half wavelength [33]. To ensure simulation accuracy and save time, the beam specimen was divided into three parts through the working face during meshing. The two parts without the PZT patch are mapped and swept meshed, and the maximum element size of the mesh is 2 mm. The part with the PZT patch was meshed by free tetrahedral mesh, and the maximum element size of the mesh is 1 mm. The PZT patch was also mapped and swept meshed, and the maximum element size of the mesh is 0.5 mm. One of the finite element meshes of the PZT coupled with the specimen is shown in Figure 4.

_{Ld}values were computed based on Equation (3) for three kinds of pine specimens, and the relative errors were further calculated based on Equation (7). The results are summarized in Table 7. The maximum relative errors of three kinds of specimens with the rectangular PZT patch and circular PZT patch are 1.34% and 1.81%, respectively. The rectangular PZT and the circular PZT have high detection accuracy, which verifies that the proposed method for the E

_{Ld}determination of pine based on the EMI response is feasible and effective.

#### 3.3. Validation of the Effectiveness of the Proposed Method Based on the Test

#### 3.3.1. Elastic Modulus Determination by the EMI Method

_{Ld}values of three kinds of pine were computed and are listed in Table 8.

#### 3.3.2. Verification of the Effectiveness of the EMI Method

_{Ld}determination of pine. The commercial buzzer has nearly the same detection accuracy as the rectangular PZT patch, which further demonstrates that the EMI method is effective, practical, and economical, for the commercial buzzer is very cheap and each buzzer costs approximately $0.4.

## 4. Conclusions

- The results of the modal simulation on the transverse vibration method indicate that the detection accuracy of the dynamic modulus of elasticity is higher when the length to thickness ratio of the pine specimen is larger. When the length to thickness ratio of the pine specimen reaches about 50, the detection accuracy meets the actual demand, with which the size of the pine specimen was optimized for impedance measurement.
- The scanning frequency range of the EMI detection is determined to be 300–600 Hz based on the mode frequencies of three kinds of pine specimens, which nearly cover the first-order bending mode frequencies of all the pine specimens and cannot reach other vibration modes of the pine specimen.
- The EMI simulation results illustrate that a unique and significant formant appears in the real part of each EMI response curve, and the maximum relative errors using the rectangular PZT patch and the circular PZT patch are 1.34% and 1.81%, respectively, which verifies the feasibility and validity of the proposed method.
- The EMI test results indicate that the maximum relative errors using the rectangular PZT patch and the commercial buzzer are 1.41% and 1.68%, respectively, compared with the corresponding results obtained using the traditional transverse vibration method, which verifies the effectiveness and practicality of the EMI method.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Kharrat, W.; Koubaa, A.; Khlif, M.; Bradai, C. Intra-ring Wood Density and Dynamic Modulus of Elasticity Profiles for Black Spruce and Jack Pine from X-ray Densitometry and Ultrasonic Wave Velocity Measurement. Forests
**2019**, 10, 569. [Google Scholar] [CrossRef] - Hearmon, R.F.S. The Elasticity of Wood and Plywood; Forest Products Research, Special Report, No. 7; H.M. Stationery Office: London, UK, 1948. [Google Scholar]
- Radu, A.; Brenndörfer, D. Zur zerstörungsfreien Prüfung des Holzes durch Schwingungsversuche. Holz Roh Werkst.
**1976**, 34, 219–222. [Google Scholar] [CrossRef] - Haines, D.W.; Leban, J.-M.; Herbé, C. Determination of Young’s Modulus for Spruce, Fir and Isotropic Materials by the Resonance Flexure Method with Comparisons to Static Fexure and Other Dynamic Methods. Wood Sci. Technol.
**1996**, 30, 253–263. [Google Scholar] [CrossRef] - Yin, Y.; Lu, J.; Ni, C.; Ren, H. Evaluation of Bending, Tensile and Compressive Strength of Structural Lumber with Transverse Vibration Technique. J. Build. Mater.
**2007**, 10, 543–547. [Google Scholar] - Wang, S.Y.; Chen, J.H.; Tsai, M.J.; Lin, C.J.; Yang, T.H. Grading of Softwood Lumber Using Non-destructive Techniques. J. Mater. Process. Technol.
**2008**, 208, 149–158. [Google Scholar] [CrossRef] - ASTM D6874; Standard Test Methods for Nondestructive Evaluation of Wood-Based Flexural Members Using Transverse Vibration. American Society for Testing and Materials: West Conshohocken, PA, USA, 2012. [CrossRef]
- Li, W.; Liu, T.; Zou, D.; Wang, J.; Yi, T.H. PZT Based Smart Corrosion Coupon Using Electromechanical Impedance. Mech. Syst. Sig. Process.
**2019**, 129, 455–469. [Google Scholar] [CrossRef] - Ritdumrongkul, S.; Fujino, Y. Identification of the Location and Size of Cracks in Beams by a Piezoceramic Actuator–sensor. Struct. Control Health Monit.
**2007**, 14, 931–943. [Google Scholar] [CrossRef] - Ai, D.; Zhu, H.; Luo, H.; Wang, C. Mechanical Impedance Based Embedded Piezoelectric Transducer for Reinforced Concrete Structural Impact Damage Detection: A Comparative Study. Constr. Build. Mater.
**2018**, 165, 472–483. [Google Scholar] [CrossRef] - Fan, X.; Li, J. Damage Identification in Plate Structures Using Sparse Regularization Based Electromechanical Impedance Technique. Sensors
**2020**, 20, 7069. [Google Scholar] [CrossRef] [PubMed] - Huo, L.; Chen, D.; Liang, Y.; Li, H.; Feng, X.; Song, G. Impedance Based Bolt Pre-Load Monitoring Using Piezoceramic Smart Washer. Smart Mater. Struct.
**2017**, 26, 057004. [Google Scholar] [CrossRef] - Du, F.; Wu, S.; Xu, C.; Yang, Z.; Su, Z. Electromechanical Impedance Temperature Compensation and Bolt Loosening Monitoring Based on Modified Unet and Multitask Learning. IEEE Sens. J.
**2023**, 23, 4556–4567. [Google Scholar] [CrossRef] - Ferreira, F.I.; de Aguiar, P.R.; da Silva, R.B.; Jackson, M.J.; Baptista, F.G.; Bianchi, E.C. Monitoring of Cylindrical Plunge Grinding Process by Electromechanical Impedance. IEEE Sens. J.
**2022**, 22, 12314–12322. [Google Scholar] [CrossRef] - Marchi, M.; Baptista, F.G.; de Aguiar, P.R.; Bianchi, E.C. Grinding Process Monitoring Based on Electromechanical Impedance Measurements. Meas. Sci. Technol.
**2015**, 26, 045601. [Google Scholar] [CrossRef] - Çakir, F.H.; Er, Ü.; Tekkalmaz, M. Monitoring the Wear of Turning Tools with the Electromechanical Impedance Technique. J. Intell. Mater. Syst. Struct.
**2023**, 34, 1341–1352. [Google Scholar] [CrossRef] - Wang, J.; Li, W.; Lan, C.; Wei, P.; Luo, W. Electromechanical Impedance Instrumented Piezoelectric Ring for Pipe Corrosion and Bearing Wear Monitoring: A Proof-of-concept Study. Sens. Actuators A Phys.
**2020**, 315, 112276. [Google Scholar] [CrossRef] - Wu, J.; Yang, G.; Wang, X.; Li, W. PZT-Based Soil Compactness Measuring Sheet Using Electromechanical Impedance. IEEE Sens. J.
**2020**, 20, 10240–10250. [Google Scholar] [CrossRef] - Zhang, J.; Zhang, C.; Xiao, J.; Jiang, J. A PZT-Based Electromechanical Impedance Method for Monitoring the Soil Freeze-Thaw Process. Sensors
**2019**, 19, 1107. [Google Scholar] [CrossRef] - Zhu, X.; di Scalea, F.L. Sensitivity to Axial Stress of Electro-Mechanical Impedance Measurements. Exp. Mech.
**2016**, 56, 1599–1610. [Google Scholar] [CrossRef] - Ong, C.-W.; Yang, Y.; Naidu, A.; Lu, Y.; Soh, C. Application of the Electromechanical Impedance Method for the Identification of In-situ Stress in Structures. Proc. SPIE
**2002**, 4935, 503–514. [Google Scholar] - Carbajo, J.; Poveda, P.; Segovia, E.; Rincon, E.; Ramis, J. Determination of Dynamic Elastic Modulus of Materials Under a State of Simple Stresses by Using Electrodynamic Actuators in Beam-Type Mechanical Elements. Mater. Lett.
**2022**, 320, 132383. [Google Scholar] [CrossRef] - Ferreira, D.; Fonseca, E.; Pinto, C.; Borges, P. Tensile Strength of pine and Ash Woods–Experimental and Numerical Study. In Proceedings of the 6th International Conference on Mechanics and Materials in Design (M2D), Ponta Delgada, Portugal, 26–30 July 2015. [Google Scholar]
- Fajdiga, G.; Rajh, D.; Necemer, B.; Glodez, S.; Sraml, M. Experimental and Numerical Determination of the Mechanical Properties of Spruce Wood. Forests
**2019**, 10, 1140. [Google Scholar] [CrossRef] - Carreira, M.R.; Dias, A.A.; Segundinho, P.G.D. Nondestructive Evaluation of Corymbia Citriodora Logs by Means of the Free Transverse Vibration Test. J. Nondestruct. Eval.
**2017**, 36, 26. [Google Scholar] [CrossRef] - Giurgiutiu, V.; Zagrai, A.N. Embedded Self-sensing Piezoelectric Active Sensors for On-line Structural Identification. J. Vib. Acoust.
**2002**, 124, 116–125. [Google Scholar] [CrossRef] - Kouroussis, G.; Ben Fekih, L.; Descamps, T. Assessment of Timber Element Mechanical Properties Using Experimental Modal Analysis. Constr. Build. Mater.
**2017**, 134, 254–261. [Google Scholar] [CrossRef] - Chui, Y.H.; Smith, I. Influence of Rotatory Inertia, Shear Deformation and Support Condition on Natural Frequencies of Wooden Beams. Wood Sci. Technol.
**1990**, 24, 233–245. [Google Scholar] [CrossRef] - Yang, N.; Zhang, L. Investigation of Elastic Constants and Ultimate Strengths of Korean Pine from Compression and Tension Tests. J. Wood Sci.
**2018**, 64, 85–96. [Google Scholar] [CrossRef] - Pencik, J. Modelling of Experimental Tests of Wooden Specimens from Scots Pine (Pinus sylvestris) with the Help of Anisotropic Plasticity Material Model. Drvna Ind.
**2015**, 66, 27–33. [Google Scholar] [CrossRef] - Shahhosseini, S.; Crovella, P.; Smith, W.B. Comparing The Effect of Presence of the Knot and the Size of the Knot on the Rolling Shear Properties in Cross Laminated Timber (CLT) by Modified Planar Shear Test and FEM Analysis. In Proceedings of the 16th World Conference on Timber Engineering (WCTE), Santiago, Chile, 9–12 August 2021. [Google Scholar]
- Merhar, M. Determination of Elastic Properties of Beech Plywood by Analytical, Experimental and Numerical Methods. Forests
**2020**, 11, 1221. [Google Scholar] [CrossRef] - Wang, L.; Yuan, B.; Xu, Z.; Sun, Q. Synchronous Detection of Bolts Looseness Position and Degree Based on Fusing Electro-mechanical Impedance. Mech. Syst. Sig. Process.
**2022**, 174, 109068. [Google Scholar] [CrossRef] - Kubojima, Y.; Tonosaki, M.; Yoshihara, H. Effect of Additional Mass on the Young’s Modulus of a Wooden Beam. J. Test Eval.
**2005**, 33, 278–282. [Google Scholar] [CrossRef] - Islam, M.; Huang, H. Understanding the effects of adhesive layer on the Electromechanical Impedance (EMI) of Bonded Piezoelectric Wafer Transducer. Smart Mater. Struct.
**2014**, 23, 125037. [Google Scholar] [CrossRef] - Abbas, S.; Li, F.; Abbas, Z.; Abbasi, T.U.R.; Tu, X.; Pasha, R.A. Experimental study of effect of temperature variations on the impedance signature of PZT sensors for fatigue crack detection. Sound Vib.
**2021**, 55, 1–18. [Google Scholar] [CrossRef] - Wang, J.; Li, W.; Lan, C.; Wei, P. Effective Determination of Young’s Modulus and Poisson’s Ratio of Metal Using Piezoelectric Ring and Electromechanical Impedance Technique: A Proof-of-concept Study. Sens. Actuators A Phys.
**2021**, 319, 112561. [Google Scholar] [CrossRef]

**Figure 6.**The EMI simulation results of three kinds of pine specimens. (

**a**) The EMI response curve of the rectangular PZT coupled with the specimen. (

**b**) The EMI response curve of the circular PZT coupled with the specimen.

**Figure 8.**The test results of three kinds of pine specimens. (

**a**) The EMI response curve of the rectangular PZT coupled with the specimen. (

**b**) The EMI response curve of the circular PZT coupled with the specimen.

Property | Korean Pine | Scots Pine | Eastern White Pine |
---|---|---|---|

ρ (Kg/m^{3}) | 430 | 505 | 349 |

E_{L} (MPa) | 8856 | 14,300 | 9404 |

E_{R} (MPa) | 986 | 700 | 734 |

E_{T} (MPa) | 429 | 545 | 357 |

G_{LR} (MPa) | 499 | 1230 | 490 |

G_{RT} (MPa) | 42 | 500 | 47 |

G_{LT} (MPa) | 450 | 800 | 451 |

ν_{LR} | 0.43 | 0.30 | 0.30 |

ν_{RT} | 0.65 | 0.38 | 0.40 |

ν_{LT} | 0.51 | 0.40 | 0.30 |

No | Size (mm) | LTR | f_{1} (Hz) | E_{Ld} (MPa) | Relative Error |
---|---|---|---|---|---|

1 | 800 × 40 × 80 | 10 | 516.7 | 6953 | 21.49% |

2 | 800 × 40 × 40 | 20 | 281.9 | 8279 | 6.52% |

3 | 800 × 40 × 25 | 32 | 179.8 | 8622 | 2.64% |

4 | 800 × 40 × 20 | 40 | 144.5 | 8701 | 1.75% |

5 | 800 × 40 × 16 | 50 | 116.0 | 8762 | 1.06% |

No | Size (mm) | LTR | f_{1} (Hz) | E_{Ld} (MPa) | Relative Error |
---|---|---|---|---|---|

1 | 800 × 40 × 80 | 10 | 610.9 | 11,415 | 20.17% |

2 | 800 × 40 × 40 | 20 | 331.4 | 13,437 | 6.03% |

3 | 800 × 40 × 25 | 32 | 211.0 | 13,945 | 2.48% |

4 | 800 × 40 × 20 | 40 | 169.6 | 14,077 | 1.56% |

5 | 800 × 40 × 16 | 50 | 136.1 | 14,165 | 0.94% |

Species of Pine | Korean Pine | Scots Pine | Eastern White Pine |
---|---|---|---|

The first-order bending mode frequency f_{1} (Hz) | 371.3 | 435.4 | 424.4 |

E_{Ld} (MPa) | 8766 | 14,157 | 9296 |

Relative error | 1.02% | 1.00% | 1.15% |

Species of Pine | Korean Pine | Scots Pine | Eastern White Pine |
---|---|---|---|

The first-order bending mode frequency (Hz) | 371.3 | 435.4 | 424.4 |

The second-order bending mode frequency (Hz) | 1010.2 | 1185.7 | 1153.5 |

The first-order longitudinal mode frequency (Hz) | 9053.4 | 10,622.0 | 10,367.0 |

Parameters | Values |
---|---|

$\mathrm{Electric}\mathrm{permittivity}{\epsilon}_{11}^{\mathrm{T}}/{\epsilon}_{22}^{\mathrm{T}}/{\epsilon}_{33}^{\mathrm{T}}$ | 3130/3130/3400 |

Piezoelectric strain coefficients (10^{−10} m/V) | −2.74/−2.74/5.93/7.41/7.41 |

Compliance S_{11}/S_{12}/S_{13}/S_{22}/S_{23}/S_{33}/S_{44}/S_{55}/S_{66} (10^{−12} m^{2}/N) | 16.50/−4.78/−8.45/16.50/−8.45/20.70/43.50/43.50/42.60 |

Density ρ (kg/m^{3}) | 7500 |

Dielectric loss factor tanδ | 0.02 |

Species of Pine | PZT Shape | f_{1} (Hz) | E_{Ld} (MPa) | E_{L} (MPa) | Relative Error |
---|---|---|---|---|---|

Korean pine | Rectangular | 372.0 | 8799 | 8856 | 0.64% |

Circular | 371.0 | 8752 | 1.17% | ||

Scots pine | Rectangular | 436.0 | 14,196 | 14,300 | 0.73% |

Circular | 435.0 | 14,131 | 1.18% | ||

Eastern white pine | Rectangular | 424.0 | 9278 | 9404 | 1.34% |

Circular | 423.0 | 9234 | 1.81% |

Species of Pine | Size (mm) | Density (kg/m^{3}) | PZT Shape | f_{1} (Hz) | E_{Ld} (MPa) |
---|---|---|---|---|---|

Korean pine | 250 × 40 × 5.30 | 454 | Rectangular | 379.0 | 8583 |

Circular | 378.5 | 8560 | |||

Mongolian Scots pine | 250 × 40 × 5.58 | 483 | Rectangular | 481.5 | 13,296 |

Circular | 480.5 | 13,240 | |||

Chinese white pine | 250 × 40 × 5.40 | 432 | Rectangular | 502.5 | 13,829 |

Circular | 500.0 | 13,692 |

Species of Pine | Size (mm) | Density (kg/m^{3}) | f_{1} (Hz) | E_{L} (MPa) |
---|---|---|---|---|

Korean pine | 1000 × 40 × 19.95 | 454 | 89.8 | 8706 |

Mongolian Scots pine | 1000 × 40 × 20.42 | 483 | 110.8 | 13,458 |

Chinese white pine | 1000 × 40 × 20.50 | 432 | 119.3 | 13,846 |

Species of Pine | Results Using the Proposed Method | Results Using the Traditional Transverse Vibration Method | Relative Error | |
---|---|---|---|---|

PZT Shape | E_{Ld} (MPa) | E_{L} (MPa) | ||

Korean pine | Rectangular | 8583 | 8706 | 1.41% |

Circular | 8560 | 1.68% | ||

Mongolian Scots pine | Rectangular | 13,296 | 13,458 | 1.20% |

Circular | 13,240 | 1.62% | ||

Chinese white pine | Rectangular | 13,829 | 13,846 | 0.12% |

Circular | 13,692 | 1.11% |

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**MDPI and ACS Style**

Li, S.; Xu, G.; Jiang, C.; Hu, H.
Determination of the Dynamic Modulus of Elasticity of Pine Based on the PZT Transducer. *Forests* **2024**, *15*, 459.
https://doi.org/10.3390/f15030459

**AMA Style**

Li S, Xu G, Jiang C, Hu H.
Determination of the Dynamic Modulus of Elasticity of Pine Based on the PZT Transducer. *Forests*. 2024; 15(3):459.
https://doi.org/10.3390/f15030459

**Chicago/Turabian Style**

Li, Shaocheng, Guangzhou Xu, Chenkan Jiang, and Hailong Hu.
2024. "Determination of the Dynamic Modulus of Elasticity of Pine Based on the PZT Transducer" *Forests* 15, no. 3: 459.
https://doi.org/10.3390/f15030459