The Vibration Dynamic Model for Blister Detection in Medium-Density Fiberboard

: Ultrasonic detection is currently used in the industry of medium-density fiberboard to detect blister defects. Due to the small detection area of a single sensor, multiple sensors need to be used, which results in high costs. Starting from elastic thin plate vibration theory, this paper builds a vibration dynamic model to detect blisters. The size and depth of the blister area can be established by determining the natural frequency of the thin plate vibration in the blister area. In this model, if the elastic modulus and density are known, the natural frequency of the thin plate vibration at the blister place is directly proportional to the blister depth and inversely proportional to the square of the blister radius. The size and depth of the blister can be determined by measuring the first third-order natural frequency of this area of research. A total of 25 specimens with blister sizes and depths were simulated, and the natural frequencies of the specimens were measured. Subsequently, the detection model was verified by comparing its experimental data with theoretical values. The theoretical value was highly consistent with the measured data. The measured values of the first, second, and third-order natural frequencies were slightly smaller than the theoretical calculated values, with average relative deviations of − 1.6%, − 1.34%, and − 1.03%, respectively. As the order progressed, the deviation exhibited a downward trend, and the third-order natural frequency displayed the smallest deviation and highest accuracy. The proposed vibration dynamic model can detect larger blister areas by measuring the natural frequency, which can overcome the shortcomings of small ultrasonic detection areas in current actual industries. Thus, the practical online blister detection device is expected to be further developed.


Introduction
Wood fibers or other plant fibers are used as raw materials to make mediumdensity fiberboard (MDF).The dispersed fibers are applied with synthetic resin and compressed under heating and pressure conditions, resulting in a density range of 0.65-0.80g/cm 3 .The structure of MDF is more uniform than that of natural wood, thereby avoiding problems such as decay and insect damage.Moreover, with such advantages as a small expansion and contraction, a simple process, and a flat surface for easily sticking finishes, MDF is an ideal manmade board for making furniture.MDF's superior performance [1] has led to its worldwide development, and global production reached 100 million cubic meters.At the same time, performing the online detection of performance and defects during the production of MDF [2] is becoming increasingly urgent.Blisters are one of the common defects among them."Surface blister" is a term used to describe raised blister-like defects on the surface of the board, which are caused by uneven fiber moisture content, improper hot-pressing technology, and other causes.In the ISO Standard and Chinese National Standard for sanding panels, blisters are prohibited due to mandatory requirements for physical and mechanical properties and defects [3] (ISO 16895:2016 and GB/T 11718-2021).Because blister defects [4] affect the strength, stability, and appearance of the board, they can lead to a decline in performance in terms of load, stress, and processing.The use of MDF in structural applications that require high strength and stability is limited, and it may affect production efficiency.To ensure the overall performance and application range of MDF, detecting blister defects is crucial.Acoustic technology is a common method for detecting internal defects in composite materials.Acoustic technology [5] can detect the physical and mechanical properties of composite materials as well as their defects, such as matrix microcracks, local delamination, or fiber fractures based on mechanical vibration.Two different types of laminated wood samples were tested for quasi-static tensile strength using acoustic emission (AE) technology.The data on the AE rate behavior under quasi-static tensile load conditions to failure conditions were analyzed for the AE signal parameters.The cumulative AE and AE rate curves for various AE signal parameters [6] demonstrated an exponential increase with time or load.To find holes in wood, a frequency sweep method was employed to measure the natural frequency of wood with holes, and a substantial connection between pore size and natural frequency was found.As a combination of acoustic and ultrasound methods, ultrasonic AE [7] utilizes high-frequency (above 20,000 Hz) AEs generated through energy release, which is beyond the range of human hearing.The introduction and development of internal structural changes or defects in composite materials can be detected by using ultrasonic AE.Ultrasonic AE can be used for the characterization of composite materials, which means checking the mechanical properties of composite materials by analyzing their response to ultrasonic waves.Ultrasonic AE [8] can also detect damage, such as cracks, deformations, and other damages, and monitor the structural health status of composite material components in bridges, buildings, and pipelines.Another advantage of ultrasonic AE [9] is that it can be used to inspect the cumulative damage inside composite materials caused by fatigue loads or impact damage.Ultrasonic AE [8] cannot detect a single large defect, such as delamination or voids, which is a disadvantage.GreCon's latest ultrasonic blister detection system UPU6000, with a measuring point diameter of 50 mm, needs to be used in parallel on the production line.The maximum distance between sensors is 100 mm, and more than 10 are needed for a measuring width, so the price is high, limiting its widespread use.
Vibration is a method of detection that utilizes the model between the inherent frequency of a specimen and its mass distribution, elastic modulus, Poisson's ratio, geometric dimensions, and boundary conditions.Vibration is widely used to measure the elastic modulus of materials.F. Bos [10] established a model based on elastic thin plate vibration theory and measured the bending stiffness of man-made boards.Yoshihara [11] used bending vibration technology to measure the elastic modulus and shear modulus of MDF, based on Timoshenko's vibration theory.In 2013, Hunt [12] created a cantilever beam device that can measure the elastic constants of small, thin wood or composite board samples.Hunt's research showed that the dynamic modulus of the cantilever beam and the static elastic modulus have a good linear correlation, but the dynamic elastic modulus is always higher than the static elastic modulus measured by the threepoint bending method.Luis Acuna [13] employed the transverse and longitudinal frequencies of the cantilever beam vibration to predict the modulus of elasticity (MOE) of five different spectral density woods.The dynamic MOE of vibration has a prediction accuracy of 94.9% accuracy, slightly higher than the ultrasonic prediction model's 93.7%.Mirbolouk [14] determined the elastic modulus of the board by conducting a bending vibration test on the full-size panel.Guan [15] proposed a method for detecting the MOE of wood composite panels (WCP) of a full-board size based on plate vibration theory under "free" support conditions.The dynamic MOE of the full-size panel and the static MOE of the small sample had a considerable linear relationship.The above study revealed that the vibration method is different from other methods in that the mechanical properties of large and small specimens can be calculated by measuring their Forests 2024, 15, 1058 3 of 12 natural frequencies.The literature review mentioned earlier indicated that MDF blisters can be detected using AE and ultrasound, but a drawback exists in terms of their small measurement area.To measure a large area, multiple sensors must be arranged in an array, which increases the cost of the detection equipment and decreases the reliability of the system.However, a vibration can establish an elastic specimen model based on the characteristics of the specimen, which can reflect the overall features, and can measure the size of the blister and the depth of the existing position.Vibration can provide information to improve the analysis of the cause of the blister and the rational use of the blister board.In addition, compared with AE and ultrasonic detection, vibration has such advantages as requiring much fewer instruments, simple operation, and high reliability.However, vibration detection research on blister defects is lacking, so this research establishes a vibration dynamic model based on the characteristics of the blister and verifies the rationality of the model through experiments, providing a new idea for blister detection.

Experimental Materials
To study the relationship between the diameter size, depth location of the blisters, and their inherent frequencies, five different thicknesses of MDF were used as faceplates (1, 2, 3, 5, and 8 mm), representing the depth location of the blisters.A 15 mm thick MDF was used as the base plate for the experiments.Each blister specimen consisted of a base plate with a milled groove 2 mm deep and of a certain diameter, and a faceplate bonded to it, as shown in Figure 1.The base plate thickness of each specimen was fixed at 15 mm, with circular grooves of different diameters, namely, 15, 20, 25, 30, and 35 cm (representing the blister diameter sizes).The faceplate thickness varied among the specimens, with five specifications of 1, 2, 3, 5, and 8 mm.Each specimen measured 600 mm in length and width, with 25 pieces of the 15 mm base plate and 5 pieces of each faceplate thickness, for a total of 25 specimens.Before the experiment, the boards were air-dried in a laboratory with stable temperature and humidity to achieve an air-dry moisture content within the range of 10%-12%.
drawback exists in terms of their small measurement area.To measure a large are tiple sensors must be arranged in an array, which increases the cost of the detection ment and decreases the reliability of the system.However, a vibration can esta elastic specimen model based on the characteristics of the specimen, which can re overall features, and can measure the size of the blister and the depth of the exis sition.Vibration can provide information to improve the analysis of the cause of th and the rational use of the blister board.In addition, compared with AE and ul detection, vibration has such advantages as requiring much fewer instruments, operation, and high reliability.However, vibration detection research on blister d lacking, so this research establishes a vibration dynamic model based on the char tics of the blister and verifies the rationality of the model through experiments, pr a new idea for blister detection.

Experimental Materials
To study the relationship between the diameter size, depth location of the and their inherent frequencies, five different thicknesses of MDF were used as fa (1, 2, 3, 5, and 8 mm), representing the depth location of the blisters.A 15 mm thi was used as the base plate for the experiments.Each blister specimen consisted o plate with a milled groove 2 mm deep and of a certain diameter, and a faceplate to it, as shown in Figure 1.The base plate thickness of each specimen was fixed at with circular grooves of different diameters, namely, 15, 20, 25, 30, and 35 cm (rep ing the blister diameter sizes).The faceplate thickness varied among the specime five specifications of 1, 2, 3, 5, and 8 mm.Each specimen measured 600 mm in len width, with 25 pieces of the 15 mm base plate and 5 pieces of each faceplate thickn a total of 25 specimens.Before the experiment, the boards were air-dried in a lab with stable temperature and humidity to achieve an air-dry moisture content wi range of 10%-12%.According to the Chinese National Standard [16], the density and the elastic m of the specimens were measured.The density and the elastic modulus of the spe are shown in Table 1.According to the Chinese National Standard [16], the density and the elastic modulus of the specimens were measured.The density and the elastic modulus of the specimens are shown in Table 1.

Establish a Vibration Model
Blisters are a common defect in medium-density boards, characterized by localized delamination within the board.However, the adhesive around the defect remains intact with gas inside, and the surface is slightly raised, forming an almost circular shape.The blister area is well bonded all around without any crack, requiring careful observation for detection.Based on the characteristics of the blister, the slightly raised area can be considered a circular thin plate, which is inlaid on a thicker base plate and fixed with adhesive around the circumference.The thickness of the circular thin plate corresponds to the depth of the blister.During excitation, the thin, small, blister area vibrates, while the base plate, which is much larger in mass and size than the circular plate of the blister area, is assumed not to deform.Therefore, by detecting the vibrational characteristics of the thin plate in the blister area, the size and depth of the blister can be determined.The vibrational mechanical model for the blister area is the vibration of the circular thin plate in this area, which is assumed to be a circular region.Thus, the thin plate in the blister area can be considered a constrained elastic thin circular plate with fixed boundaries, facilitating the use of elastic circular plate dynamic modeling.The vibration equation is as follows: The fixed boundary conditions surrounding it are as follows: In Formula ( 1), D = Eh 3 12(1−v 2 ) represents the bending stiffness, where ( h) is the thickness of the circular thin plate in the blister area, corresponding to the depth of the blister; (E) is the MOE for bending; (v) is the Poisson's ratio, which is 0.3 for homogeneous materials; (ρ) is the density; (w) describes the displacement of the plate surface in the circulation area as it vibrates over time and is a function of time and coordinates; and (w tt ) is the secondorder partial derivative of (w) with respect to time (t).Using the method of separation of variables under polar coordinates, the vibration equation for the circular thin plate can be obtained.
where (W) is a function of the polar radius (r) and (θ); (α) and (λ) are constants determined by boundary conditions.k 4 = ω 2 ρh D , where (ω) represents the angular vibration frequency, and ( f ) is the natural frequency.(J n ) denotes the Bessel function of the first kind, and (I n ) denotes the Bessel function of the second kind.The formula for the vibration frequency of the circular plate is as follows: where (R) represents the radius of the circular plate, (α) is a constant determined by the node diameter (n) and the number of node circles (s), (E) is MOE, (ρ) is the density, and (v) is the Poisson's ratio, which is taken as 0.3 for isotropic materials.The value of (α) is fixed when the circular plate is fixed on the periphery, and it can be found in the reference [17].

Experimental Apparatus and Methods
The experimental setup, depicted in Figure 2, comprised primarily of a Yunzhihui Signal Acquisition and Analysis Instrument (model INV3062T, developed by the Beijing Oriental Institute of Vibration and Noise Technology) and DASP-V11 software.This combination created a high-performance system for data acquisition and vibration signal analysis processing.The testing method utilized a force hammer to excite the drum blister, and a high-precision microphone was employed to capture the vibration signals generated by the hammer's impact.These signals were then transmitted to the acquisition system for analysis to determine the inherent frequencies of the vibrations.In the experiments, 25 specimens were numbered and named according to the "diameter-thickness" rule.For instance, "20-2" represents a drum blister specimen with a diameter of 20 cm and a faceplate thickness of 2 mm.During the tests, the specimens were placed on a horizontal table, faceplate up.Then, a thin rope was used to secure a highprecision microphone to a fixture and adjusted to hang 3 cm above the specimen.Throughout the testing, the defective area of the drum blister was struck, causing the circular plate at the blister to vibrate.To identify the first, second, and third-order natural frequencies, the system was configured to select the first three peak values from the vibration signal.This approach allowed for the precise determination of the modal frequencies, which are crucial for characterizing the vibrational properties of the specimen.The testing procedure involved striking the defective area of the drum blister to induce vibrations in the circular plate at the blister site.The high-precision microphone, secured 3 cm above the specimen, transmitted the vibration signals to the acquisition system, which automatically collected data based on signal strength.After each impact, the system recorded the data once, and each specimen was subjected to three measurements.The average of these three measurements was considered the final test result for each specimen.The collected signals underwent analysis using FFT by the host computer's signal analysis software, which provided frequency domain information to analyze the inherent frequencies.The vibration data analysis graph is presented in Figure 3.In the experiments, 25 specimens were numbered and named according to the "diameter-thickness" rule.For instance, "20-2" represents a drum blister specimen with a diameter of 20 cm and a faceplate thickness of 2 mm.During the tests, the specimens were placed on a horizontal table, faceplate up.Then, a thin rope was used to secure a high-precision microphone to a fixture and adjusted to hang 3 cm above the specimen.Throughout the testing, the defective area of the drum blister was struck, causing the circular plate at the blister to vibrate.To identify the first, second, and third-order natural frequencies, the system was configured to select the first three peak values from the vibration signal.This approach allowed for the precise determination of the modal frequencies, which are crucial for characterizing the vibrational properties of the specimen.The testing procedure involved striking the defective area of the drum blister to induce vibrations in the circular plate at the blister site.The high-precision microphone, secured 3 cm above the specimen, transmitted the vibration signals to the acquisition system, which automatically collected data based on signal strength.After each impact, the system recorded the data once, and each specimen was subjected to three measurements.The average of these three measurements was considered the final test result for each specimen.The collected signals underwent analysis using FFT by the host computer's signal analysis software, which provided frequency domain information to analyze the inherent frequencies.The vibration data analysis graph is presented in Figure 3.

Error Analysis
In this study, we employed a method to precisely measure the natural frequencies of the samples and compared the results with the theoretical model to validate the model's accuracy.To ensure the reliability of the data, multiple measurements were taken for each sample, and the errors and relative errors were calculated.These error analyses help us to assess the effectiveness of the model in practical applications.

Theoretical Calculation and Experimental Measurement of Natural Frequencies
Based on Formula (4) and the material parameters in Table 1, the natural frequencies of drum blister specimens with varying diameters and thicknesses were calculated.The natural frequencies for nodal circles of 0 and nodal diameters of 0, 1, and 2 were computed; for ease of expression, these were referred to as the first, second, and third order, respectively.The theoretical calculated values of the first three orders of natural frequencies are shown in Table 2.

Error Analysis
In this study, we employed a method to precisely measure the natural frequencies of the samples and compared the results with the theoretical model to validate the model's accuracy.To ensure the reliability of the data, multiple measurements were taken for each sample, and the errors and relative errors were calculated.These error analyses help us to assess the effectiveness of the model in practical applications.

Theoretical Calculation and Experimental Measurement of Natural Frequencies
Based on Formula (4) and the material parameters in Table 1, the natural frequencies of drum blister specimens with varying diameters and thicknesses were calculated.The natural frequencies for nodal circles of 0 and nodal diameters of 0, 1, and 2 were computed; for ease of expression, these were referred to as the first, second, and third order, respectively.The theoretical calculated values of the first three orders of natural frequencies are shown in Table 2. Table 2 reveals that when the drum blister has a greater depth and a smaller diameter, the natural frequency is higher.Conversely, when the faceplate is thinner and the defect diameter is larger, the frequency is lower.
The natural frequency data obtained from the vibration system collection and analysis are as shown in Table 3.Some data could not be excited due to the excessively high natural frequency and large damping.Note: For specimens 15-8 and 20-8, some natural frequency data could not be measured due to the limited excitation by the force hammer.

Variation in Natural Frequency with Different Depths
After organizing the data in Table 3, the effects of thickness on natural frequency under defects of different diameters can be compared.With depth as the variable, the line graphs of the first three orders of experimental natural frequencies for five drum blister diameters were plotted against the depth of the faceplate.In Figure 4, the horizontal axis represents the depth of the drum blister specimen's faceplate, and the vertical axis represents the natural frequency values corresponding to each specimen.Here, a, b, and c represent the theoretical calculations of the first-, second-, and third-order natural frequency, whereas d, e, and f correspond to the actual measured first-, second-, and third-order natural frequency, respectively.
In Figure 4, the horizontal axis represents the blister depth (mm), and the vertical axis represents frequency (Hz).The different colored curves indicate the relationship between the depth and natural frequency for specific blister diameters.Comparing Figures, a, b,  and c with d, e, and f reveals that the trend of the curves in the corresponding figures are consistent.With the same diameter, the natural frequency of each order increases with the increase in the panel depth, resulting in a higher value of the blister's natural frequency.
According to Formula (4) f = αh , with constant radius, elastic modulus, and density, the natural frequency increases with depth.Figure 4 shows that for blister diameters of 15, 20, 25, 30, and 35 cm, the natural frequency of the specimens increases with the increase in blister depth, which is highly consistent with the theoretical model.Taking the partial derivative of Formula (4) with respect to (h) yields , which shows that the trend of frequency change with depth is inversely proportional to the square of the radius.This outcome indicates that the smaller the diameter, the more remarkable this change.Experimental data show that with the smallest blister diameter, when the specimen diameter is 15 cm, the natural frequency obtained by the specimen is much higher than that of other sizes, and its frequency change trend is also the most evident.
remarkable this change.Experimental data show that with the smallest blister diame when the specimen diameter is 15 cm, the natural frequency obtained by the specimen much higher than that of other sizes, and its frequency change trend is also the most e dent.

Analysis of Natural Frequency under Different Blister Diameters
To study the variation law of the natural frequency of blisters with different diam ters better, the blister diameter was taken as the horizontal axis, the natural frequency the vertical axis, and the line charts of the first three orders of experimental natural f quencies under five different panel thicknesses were drawn, as shown in Figure 5. Amo them, a, b, and c are the first-, second-, and third-order natural frequencies calculated t oretically, respectively, whereas d, e, and f are the first-, second-, and third-order natu frequencies measured, respectively.
In Figure 5, the horizontal axis represents the blister diameter (cm) and the verti axis represents the natural frequency (Hz); different colored curves indicate the relati ship between diameter and natural frequency at different depths.When the blister dep is 1, 2, 3, 5, and 8 mm, the first three order natural frequency values of the specimen chan with the diameter in the same pattern, which is, the larger the diameter, the lower natural frequency, showing a negative correlation between natural frequency changes a diameter size.The formula shows that  = ( ) , that is, when the depth, elas modulus, and density remain unchanged, the natural frequency decreases as the diame increases.The actual measurement data and the theoretical calculation curves in Figur show a consistent trend.When the blister depth remains unchanged, the natural f quency of the thin plate gradually decreases as the blister diameter increases.Taking partial derivative of Formulas ( 2) and ( 3) with respect to () yields = ( The trend of natural frequency change is opposite to the change pattern of the radius, a it is inversely proportional to ( ).The smaller the radius, the steeper the change cur

Analysis of Natural Frequency under Different Blister Diameters
To study the variation law of the natural frequency of blisters with different diameters better, the blister diameter was taken as the horizontal axis, the natural frequency as the vertical axis, and the line charts of the first three orders of experimental natural frequencies under five different panel thicknesses were drawn, as shown in Figure 5.Among them, a, b, and c are the first-, second-, and third-order natural frequencies calculated theoretically, respectively, whereas d, e, and f are the first-, second-, and third-order natural frequencies measured, respectively.
In Figure 5, the horizontal axis represents the blister diameter (cm) and the vertical axis represents the natural frequency (Hz); different colored curves indicate the relationship between diameter and natural frequency at different depths.When the blister depth is 1, 2, 3, 5, and 8 mm, the first three order natural frequency values of the specimen change with the diameter in the same pattern, which is, the larger the diameter, the lower the natural frequency, showing a negative correlation between natural frequency changes and diameter size.The formula shows that f = αh , that is, when the depth, elastic modulus, and density remain unchanged, the natural frequency decreases as the diameter increases.The actual measurement data and the theoretical calculation curves in Figure 5 show a consistent trend.When the blister depth remains unchanged, the natural frequency of the thin plate gradually decreases as the blister diameter increases.Taking the partial derivative of Formulas ( 2) and (3) with respect to (R) yields . The trend of natural frequency change is opposite to the change pattern of the radius, and it is inversely proportional to (R 3 ).The smaller the radius, the steeper the change curve, and the larger the radius, the more gradual the trend.Compared with other depths, when the depth is 8 mm, the natural frequency shows the most substantial downward trend.and the larger the radius, the more gradual the trend.Compared with other depths, wh the depth is 8 mm, the natural frequency shows the most substantial downward trend The above analysis shows that the experimental data and the theoretical formula highly consistent.To analyze the gap between the model and the actual measureme further, an error analysis was conducted.

Error Analysis
The calculated natural frequency data of the specimen were compared with the perimental test data to calculate the error and relative error between the actual measu ments and the theoretical calculations.Table 4 lists the errors and relative errors of first-, second-, and third-order natural frequencies.The above analysis shows that the experimental data and the theoretical formula are highly consistent.To analyze the gap between the model and the actual measurements further, an error analysis was conducted.

Error Analysis
The calculated natural frequency data of the specimen were compared with the experimental test data to calculate the error and relative error between the actual measurements and the theoretical calculations.Table 4 lists the errors and relative errors of the first-, second-, and third-order natural frequencies.
The data in Table 3 reveal that the average error of the first-order natural frequency is −9.16, with a standard deviation of 21.85, and the mean relative error is −1.6%.The average error of the second-order natural frequency is −2.58, with a standard deviation of 19.8, and the mean relative error is −1.34%.The average error of the third-order natural frequency is −2.86, with a standard deviation of 19.56, and the mean relative error is −1.03%.The relative error probability density distribution is shown in Figure 6.The actual measured values of the first-, second-, and third-order natural frequencies agree with the theoretical calculated values, and the actual measurements are slightly smaller than the theoretical values.The average relative deviation is within 2%, and this deviation gradually decreases as the order increases.The actual measurements are smaller than the theoretical calculations because the periphery of the blister is bonded with the adhesive and not completely fixed as the model assumes, which does not fully match the theoretical hypothesis.The data in Table 3 reveal that the average error of the first-order natural frequency is −9.16, with a standard deviation of 21.85, and the mean relative error is −1.6%.The average error of the second-order natural frequency is −2.58, with a standard deviation of 19.8, and the mean relative error is −1.34%.The average error of the third-order natural frequency is −2.86, with a standard deviation of 19.56, and the mean relative error is −1.03%.The relative error probability density distribution is shown in Figure 6.The actual measured values of the first-, second-, and third-order natural frequencies agree with the theoretical calculated values, and the actual measurements are slightly smaller than the theoretical values.The average relative deviation is within 2%, and this deviation gradually decreases as the order increases.The actual measurements are smaller than the theoretical calculations because the periphery of the blister is bonded with the adhesive and not completely fixed as the model assumes, which does not fully match the theoretical hypothesis.

Conclusions
This paper presents a model for detecting blister defects in density fiberboards using vibration detection.The model can detect the size and depth of the blister area by

Conclusions
This paper presents a model for detecting blister defects in density fiberboards using vibration detection.The model can detect the size and depth of the blister area by measuring the natural frequency of vibrations in the thin plate at the blister area.The model shows that factors such as blister depth, blister diameter, elastic modulus, and density all affect the natural frequency of blister vibrations.Blister depth is inversely proportional to the natural frequency, meaning deeper blisters result in higher vibrational frequencies; similarly, blister diameter is inversely proportional to the natural frequency, and larger diameters result in lower frequencies.The vibration model shows that each order of natural frequency can be used to detect the size and depth of blisters, and by detecting any one order, the size and depth of the blister area can be calculated.In this paper, theoretical calculations and actual measurements were conducted using blister defect specimens of five different depths and diameters to validate the detection model.The actual measured values of the first-, second-, and third-order natural frequencies were slightly smaller than the theoretical calculations, with a relative deviation within 2%, indicating that the measurement detection of blister defects is effective.The experiments revealed that as the order increases, the relative deviation between the theoretical and actual values becomes smaller, suggesting that using the third-order natural frequency for detecting blisters is more accurate and reliable than the first order.Compared with the ultrasonic detection methods currently used in the industry, the vibration detection method can effectively identify larger blisters, potentially reducing the number of sensors needed for blister detection.
Although the vibration detection model proposed in this study has shown good consistency in theory and experiment, and has potential applications in detecting blister defects in density fiberboards, there are still some limitations.Firstly, the model assumes the homogeneity and isotropy of the plate, which may be difficult to fully satisfy in actual industrial production.Additionally, environmental noise and other types of defects in the plate may interfere with the measurement of vibration frequencies.The current model also does not take into account the impact of changes in plate thickness on vibration characteristics.
Future research could address these issues by, firstly, introducing more complex physical models to simulate the vibration characteristics of non-homogeneous or anisotropic materials.Secondly, more advanced signal processing techniques could be developed to improve the accuracy of natural frequency measurements in noisy environments.Furthermore, future studies could explore the effects of different plate thicknesses on vibration characteristics and how to adjust the model to accommodate these changes.With these improvements, the vibration detection method is expected to find broader industrial application, providing more effective technical support for the quality control of medium-density fiberboards.

Figure 1 .
Figure 1.Schematic diagram of the model.

Figure 1 .
Figure 1.Schematic diagram of the model.

Figure 3 .
Figure 3. Vibration data acquisition and analysis.

Figure 4 .
Figure 4. Relationship between thickness and natural frequency.Notes: The horizontal axis rep sents the blister depth (mm), and the vertical axis represents frequency (Hz).The different colo curves indicate the relationship between the depth and natural frequency for specific blister diam ters.

Figure 4 .
Figure 4. Relationship between thickness and natural frequency.Notes: The horizontal axis represents the blister depth (mm), and the vertical axis represents frequency (Hz).The different colored curves indicate the relationship between the depth and natural frequency for specific blister diameters.

Figure 5 .
Figure 5. Relationship between thickness and various order natural frequencies under different ameters.Notes: The horizontal axis represents the blister diameter (cm) and the vertical axis rep sents the natural frequency (Hz); different colored curves indicate the relationship between dia ter and natural frequency at different depths.

Figure 5 .
Figure 5. Relationship between thickness and various order natural frequencies under different diameters.Notes: The horizontal axis represents the blister diameter (cm) and the vertical axis represents the natural frequency (Hz); different colored curves indicate the relationship between diameter and natural frequency at different depths.

Figure 6 .
Figure 6.Probability density distribution of relative error.Notes: The horizontal axis represents the relative error, and the vertical axis represents the probability density.The graph(a-c) illustrates how the actual measured values of the natural frequencies for the first, second, and third orders compare to the theoretical calculated values, with a noted average relative deviation within 2%.This deviation decreases with higher orders.

Figure 6 .
Figure 6.Probability density distribution of relative error.Notes: The horizontal axis represents the relative error, and the vertical axis represents the probability density.The graph(a-c) illustrates how the actual measured values of the natural frequencies for the first, second, and third orders compare to the theoretical calculated values, with a noted average relative deviation within 2%.This deviation decreases with higher orders.

Table 1 .
Material properties and quantities.
Blisters are a common defect in medium-density boards, characterized by lo

Table 1 .
Material properties and quantities.

Table 2 .
Theoretical calculation values of the first three orders of natural frequencies.

Table 2 .
Theoretical calculation values of the first three orders of natural frequencies.

Table 3 .
Measured natural frequency values.

Table 4 .
Errors and relative errors of various order natural frequencies.

Table 4 .
Errors and relative errors of various order natural frequencies.Note:For specimens No. 15-8 and No. 20-8, some natural frequency data were not measured due to limited excitation by the impact hammer; hence, they are not discussed.
Note: For specimens No. 15-8 and No. 20-8, some natural frequency data were not measured due to limited excitation by the impact hammer; hence, they are not discussed.