基于悬臂板扭转模态测试材料剪切模量

Testing Shear Modulus of Materials Based on Torsional Mode of Cantilever Plate

【目的】探讨基于悬臂板扭转模态测试正交各向异性材料(木材)和各向同性材料剪切模量的原理和方法。【方法】首先根据ANSYS计算的悬臂板一阶扭转振形,应用优化原理分别得到垂直和平行于板中面位移w和u的一阶扭转振形函数,然后借助能量法导出悬臂板一阶扭转频率与板材弹性模量和剪切模量的关系式,该式中的2个振形系数通过与动能、扭转应变能和拉压应变能相关积分计算,从振形系数计算值的二元线性回归得到振形系数依赖于悬臂板宽长比和厚宽比的相关式。【结果】导出的悬臂板一阶扭转频率与板材弹性模量和剪切模量的关系式,只有在测出悬臂板一阶弯曲频率得到弹性模量后,才能应用这个关系式由悬臂板一阶扭转频率推算出剪切模量。【结论】导出了悬臂板一阶扭转频率与弹性模量和剪切模量之间的关系式,其中振形系数依赖于悬臂板宽长比和厚宽比。该关系式动态测试剪切模量的正确性不仅得到低碳钢、轧制铝、桃花心木、白蜡木和轻木剪切模量仿真计算的验证,还得到低碳钢、西加云杉(径切面和横切面)、蒙古栎(顺纹)和油松(弦切面和径切面)等材料剪切模量的动态和静态试验的验证,且基于悬臂板一阶扭转模态与自由板扭转振形法测试木材或各向同性材料的剪切模量相当吻合。基于悬臂板一阶扭转模态提供了一个应用悬臂板频谱动态测试剪切模量的简便和快速的方法,该方法不仅适用于测试木材的3个主向剪切模量GLT、GLR和GRT,还适用于测试各向同性材料的剪切模量。

【Objective】 The principle and method of testing shear modulus of orthotropic materials （wood） and isotropic materials based on the torsional mode of cantilever plate are discussed.【Method】Firstly,based on the first-order torsional vibration shapes of the cantilever plate according to the ANSYS calculation,the first-order torsional vibration shape function of the displacement w and u,which are perpendicular and parallel to the middle of the plane respectively,are obtained by applying optimization principle.Then,the energy method is used to derive the relational expressions between the first-order torsional frequency and elastic/shear modulus of the cantilever plate,the two vibration shape coefficients are calculated through the integral related with kinetic energy,torsion/tension/compression strain energy,and the binary linear regression of the calculated value of the vibration shape coefficient depends on related expressions of the width-tolength ratio and thickness-to-width ratio.【Result 】 The relational expressions derived between the first-order torsion frequency and elastic/shear modulus of the cantilever plate,it can be used to calculate shear modulus by the first-order torsional frequency only after obtaining elastic modulus through measuring the first-order bending frequency of the cantilever plate.【Conclusion】 The relationship between the first-order torsional frequency and the elastic/shear modulus of the cantilever plate is derived,and the vibration shape coefficient is dependent on the width-to-length ratio and the thickness-to-width ratio.The correctness of relational expressions are verified not only by the dynamic shear modulus determined from low carbon steel,rolling aluminum,Swietenia mahagoni,Fraxinus americana and Ochroma pyramidale,but also by the dynamic and static shear modulus tested from low carbon steel,Picea sitchensis （radial section and transverse section）,Quercus mongolica （parallel to grain）,Pinus tabulaeformis （tangential section and radial section） andso on.Based on the first-order torsional mode of the cantilever plate and the torsional vibration shape method of the free plate,shear modulus of wood or isotropic materials are in good agreement.The first-order torsional mode based on the cantilever plate provides a simple and rapid method to determine the dynamic shear modulus using cantilever plate.This method can be used not only to test wood shear modulus G_（LT）,G_（LR） and G_（RT） in the three main directions,but also be applied in testing shear modulus of isotropic materials.