Determining modulus of elasticity of full-size plywood panel simply supported on two opposite sides using a vibration method
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摘要:
目的 研究足尺胶合板两个主要方向（即长度和宽度方向）弹性模量的两对边简支振动检测，为足尺胶合板两个主要方向弹性模量的在线无损检测提供一种新方法。 方法 以4种厚度共20块足尺胶合板为研究对象，采用有限元软件COMSOL Multiphysics和PULSE振动测试系统分别对两对边简支的足尺胶合板进行了模态灵敏度分析和试验模态分析；提出了一种两对边简支边界条件下的足尺胶合板弹性模量振动检测试验方法，运用此方法提取出足尺胶合板所需模态的频率，将其带入到编写的弹性模量检测算法中，用以计算足尺胶合板两个主要方向的动态弹性模量值；进行了三点弯曲静态试验检测足尺胶合板两个主要方向的静态弹性模量值，用以验证动态弹性模量检测结果的准确性。 结果 确定了用于计算足尺胶合板两个主要方向弹性模量的频率所对应的模态，分别为其自由振动前9阶模态中的第1阶模态（2, 0）和第7阶模态（2, 2）；厚度变化对足尺胶合板的前9阶模态的阶次排序没有影响；足尺胶合板两个主要方向的动态弹性模量均大于静态弹性模量，且同一厚度足尺胶合板的力学性能存在不均匀性；足尺胶合板长度和宽度方向的动态弹性模量与静态弹性模量间均具有显著的线性关系，决定系数分别为0.907和0.655。 结论 基于两对边简支振动和弹性模量振动检测算法检测足尺胶合板两个主要方向的弹性模量具有可行性。 Abstract:Objective To provide a new method for the online non-destructive determination for modulus of elasticity (MOE) in two main directions of a full-size plywood panel, the vibration determination for MOE in two main directions of a full-size plywood panel simply supported on two opposite sides was studied. Method 20 pieces of full-size plywood panels with four different thicknesses were used as study objects. The modal sensitivity analysis and experimental modal analysis of full-size plywood panels simply supported on two opposite sides were performed using finite element software COMSOL Multiphysics and the PULSE vibration test system. The frequencies of needed modes were obtained by a vibration detection method for MOE of the full-size plywood panel proposed in this study, then incorporated into a vibration detection algorithm of the dynamic MOE in two main directions of a full-size plywood panel. In order to verify the accuracy of the dynamic MOE values measured, the static MOE values in two main directions of a full-size plywood panel were also obtained through three-point bending test. Result The frequencies of free vibration modes (2, 0) and (2, 2) as the first and seventh modes in the first nine modes of the full-size plywood panel were conducted for calculation of MOE in the length and width directions of the panels, respectively. The results revealed that thickness variation had no effect on the ordering of the first nine modes for the full-size plywood panels. The dynamic MOE values were greater than the static ones of the full-size plywood panels, and there existed inhomogeneity of mechanical properties for the full-size plywood panels with the same thickness. Dynamic MOE had a strong correlation with static MOE in both length and width directions of the full-size plywood panels (R2 = 0.907 and 0.655, respectively). Conclusion It is feasible for determining MOE of a full-size plywood panel based on two-opposite-side vibration and vibration detection algorithm for MOE. -
图 1 足尺胶合板两对边简支
Figure 1. Full-size plywood panel simply supported on two opposite sides
图 2 两对边简支的足尺胶合板的前9阶计算模态振型图
Figure 2. Diagrams of the first nine calculated mode shapes of full-size plywood panels simply supported on two opposite sides
图 3 两对边简支足尺胶合板前9阶模态对弹性模量的灵敏度分析结果
Figure 3. Sensitivity analysis results of the first nine modes to MOE of full-size plywood panels simply supported on two opposite sides
图 4 两对边简支足尺胶合板弹性模量的振动检测算法程序流程图
Figure 4. Flowchart of vibration detection algorithm for MOE of the full-size plywood panels simply supported on two opposite sides
图 5 两对边简支足尺胶合板的模态参数振动检测装置图
Figure 5. Diagram of vibration detection equipment for modal parameters of the full-size plywood panel simply supported on two opposite sides
图 6 激振点的频率响应函数幅值图
Figure 6. Amplitude diagram of frequency response function of excitation point
图 7 两对边简支的足尺胶合板的前9阶模态振型图
Figure 7. The first nine mode shapes of full-size plywood panels simply supported on two opposite sides
图 8 运用振动法测得的足尺胶合板两个方向弹性模量与ρ/h2间的关系
Ex d和Ey d分别为足尺胶合板的长度和宽度方向的动态弹性模量，h为板材厚度， ρ为板材密度。Ex d and Ey d represent the dynamic MOE in length and width directions of full-size plywood panel, respectively. h is thickness, and ρ is density of the panel.
Figure 8. Relationship between MOE values in both directions of full-size plywood panels obtained from vibration method and ρ/h2
图 9 运用振动法测得的足尺胶合板两个方向弹性模量与其对应振动模态的f 2ρ/h2间的关系
Figure 9. Relationship between MOE values in both directions of full-size plywood panels obtained from vibration method and f 2ρ/h2 of the corresponding modes
图 10 两种方法测得的足尺胶合板弹性模量间的关系
Figure 10. Relationship between MOE results of full-size plywood panels measured by two methods
表 1 固有频率表达式中的系数值
Table 1. Coefficient values of the natural frequency expression
宽度方向 Width direction 长度方向 Length direction 系数
Coefficient取值 Value 系数
Coefficient取值 Value m = 2 n = 0 n = 2 Pm 1 Pn 0 1.506 Im 1 In 0 1.247 Jm 1 Jn 0 5.013 Km 0 Kn 1.883 5.328 Lm 0 Ln 0 0.182 Mm 0 Mn 0 3.584 P(m+2) 3 P(n+2) 1.506 3.500 I(m+2) 9 I(n+2) 1.247 10.022 J(m+2) 9 J(n+2) 5.013 18.935 K(m+2) 0 K(n+2) 5.328 6.092 L(m+2) 0 L(n+2) 0.182 0.999 M(m+2) 0 M(n+2) 3.584 7.914 注：P、I、J、K、L和M均为与足尺胶合板的模态振型相关的系数，其取值引自参考文献[8]。Notes：P, I, J, K, L and M are the coefficients related to mode shapes of full-size plywood panel, and the values of these coefficients are cited from reference [8]. 
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表 2 两对边简支的足尺胶合板计算模态分析的初始参数
Table 2. Initial parameters for calculated modeanalysis of full-size plywood panels simply supported on two opposite sides
板材种类
Panel type弹性模量
Modulus of elasticity (MOE)/MPa剪切模量
Shear modulus/MPa泊松比
Poisson’s ratio (υxy)密度
Density/(kg·m−3)尺寸
DimensionEx Ey Gxy Gyz Gxz 胶合板 Plywood 5 200 6 950 950 170 220 0.039 574 2 440 mm × 1 221 mm × 18 mm 注：表中数据引自参考文献[10]、[13]和[14]。Ex和Ey分别为足尺胶合板长度和宽度方向的弹性模量，Gxy、 Gyz和Gxz分别为足尺胶合板x-y、y-z和x-z平面内的剪切模量。下同。Notes：data in the table are cited from reference [10], [13] and [14]. Ex and Ey represent the MOE in length and width directions of full-size plywood panel, respectively. Gxy, Gyz and Gxz represent the shear modulus in the x-y, y-z and x-z planes of full-size plywood panels, respectively. Same as below.
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表 3 被测足尺胶合板的基本参数
Table 3. Basic parameters of the measured full-size plywood panels
板材
Panel板材尺寸
Panel dimension平均密度
Average density/(kg·m−3)PW12 2 441 mm × 1 221 mm × 12 mm 541 PW15 2 440 mm × 1 221 mm × 15 mm 534 PW18 2 441 mm × 1 222 mm × 18 mm 532 PW20 2 440 mm × 1 221 mm × 20 mm 524 注：PW12、PW15、PW18和PW20分别代表标称厚度为12、15、18和20 mm的胶合板。Notes: PW12, PW15, PW18 and PW20 represent the full-size plywood panels with nominal thickness of 12, 15, 18 and 20 mm, respectively.
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表 4 两对边简支足尺胶合板的前9阶模态参数
Table 4. The first nine mode parameters of full-size plywood panels simply supported on two opposite sides
Hz 阶次 Order PW12 PW15 PW18 PW20 振型 Vibration mode 1 3.1 3.6 4.3 4.5 (2, 0) 2 6.4 7.5 9.1 9.4 (2, 1) 3 11.3 13.8 16.7 17.2 (3, 0) 4 15.3 17.3 22.2 24.6 (3, 1) 5 24.8 30.2 34.0 35.0 (4, 0) 6 29.8 34.6 42.2 43.0 (4, 1) 7 31.0 38.2 46.7 49.0 (2, 2) 8 35.2 43.2 54.3 56.5 (3, 2) 9 48.5 57.0 70.0 71.0 (4, 2)
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表 5 两种方法测得的足尺胶合板弹性模量结果
Table 5. Results for MOE values of full-size plywood panels measured by two methods
板材 Panel MOE 振动法 Vibration method 静态法 Static method Ex d Ey d Ex s Ey s PW12 平均值 Mean value/MPa 7 057 6 841 5 864 5 964 标准偏差 Standard deviation/MPa 481 210 410 178 变异系数 Coefficient of variation/% 6.8 3.1 7.0 3.0 相对偏差 Relative deviation/% 20.4 14.7 PW15 平均值 Mean value/MPa 5 151 6 954 4 625 6 262 标准偏差 Standard deviation/MPa 470 267 360 211 变异系数 Coefficient of variation/% 9.1 3.8 7.8 3.4 相对偏差 Relative deviation/% 11.4 11.0 PW18 平均值 Mean value/MPa 4 846 6 886 4 383 6 619 标准偏差 Standard deviation/MPa 460 144 362 126 变异系数 Coefficient of variation/% 9.5 2.1 8.3 1.9 相对偏差 Relative deviation/% 10.6 4.0 PW20 平均值 Mean value/MPa 5 358 5 755 5 041 5 644 标准偏差 Standard deviation/MPa 863 260 733 270 变异系数 Coefficient of variation/% 16.1 4.5 14.5 4.8 相对偏差 Relative deviation/% 6.3 2.0 注：变异系数 = 标准偏差/平均值；相对偏差 = (动态弹性模量 − 静态弹性模量)/静态弹性模量。Notes: coefficient of variation is the ratio of standard deviation to mean value; relative deviation is the ratio of difference between dynamic MOE and static MOE to the static MOE.
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表 6 两种方法测得的足尺胶合板弹性模量的一元线性回归及相关参数
Table 6. One-dimensional linear regression and related parameters of MOE of full-size plywood panels measured by two methods
MOE 板材
Panel数量
Numbery = kx + r 相关系数
Correlation coefficientF值
F value显著性
Significancek r Ex PW12 5 0.812 134.225 0.952 29.329 0.012 PW15 5 0.704 996.482 0.921 16.733 0.026 PW18 5 0.754 727.515 0.957 32.957 0.010 PW20 5 0.841 534.570 0.991 157.710 0.001 PW 20 0.691 1 107.289 0.965 246.123 0.000 Ey PW12 5 0.771 690.256 0.911 14.635 0.031 PW15 5 0.693 1 440.095 0.878 10.062 0.050 PW18 5 0.839 842.478 0.961 36.134 0.009 PW20 5 1.028 −272.761 0.993 206.327 0.001 PW 20 0.616 2 052.229 0.809 34.185 0.000 注：y = kx + r，式中x和y分别代表足尺胶合板的动态和静态弹性模量，k为回归系数，r为常数。Notes: in the formula y = kx + r, x and y represent the dynamic and static MOE of full-size plywood panel, respectively, k is the regression coefficient and r is a constant.
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