压电实验室团队近期在国际期刊Journal of Sound and Vibration发表了一篇题为An accurate beam theory and its first-order approximation in free vibration analysis的论文，提出了一个精确计算弹性梁的全部振动频率的理论。文章以Peter C.Y. Lee的板理论的位移展开函数为基础，探索了其在一维问题中的应用。文章用一阶近似理论对梁弯曲振动进行了分析，得出了一些新的理论和结果。文章以平面应力问题为出发点，参考Lee的方法，把位移以特殊的三角级数形式展开，然后以级数的系数为广义位移，利用Hamilton变分原理，建立了一维的梁的振动分析的理论体系。理论上，这一维体系可以精确的给出二维平面应力问题的结果。
文章最后，作者期许能够运用Lee梁高阶理论去解决梁的高频振动分析中全部频率的计算问题。论文第一作者为谢龙涛博士，力学专业本科生丁军磊参与了研究工作，王骥教授为通讯作者，论文的合作者包括英国City University of London的J Ranjan Banerjee教授。
Fig. 1. Anaccurate beam theory and its first-order approximation in free vibration analysis
Fig. 2. Mode shapes of beams for different order number n and ratio of length to thickness l/b when both ends are simply supported by using LBT1st, TBT, and PST.
Fig. 3. Mode shapes of beams for different order number n and ratio of length to thickness l/b when both ends are clamped by using LBT1st, TBT, and PST.
Piezo Dev Lab team published a paper in JSV
Recently, Piezoelectric Device Laboratory published a paper entitled anaccurate beam theory and its first order approximation in free vibration analysis in the Journal of Sound and Vibration. Based on Peter C.Y.Lee's plate theory, the paper explores its application in one-dimensional problems. In this paper, the first-order approximation theory is used to analyze the bending vibration of beams, and some new conclusions are obtained. This is part of the effort in the research on the analysis of complete vibration frequencies of an elastic beam.
Based on the plane stress problem and Lee's method, the displacement is expanded in the form of special series oftrigonometric functions, with the coefficients of series as the generalized displacement. After the use of Hamilton’s variational principle,a one-dimensional system is established. The one-dimensional system can give accurate results of two-dimensional plane stress problems.
Furthermore, the first-order approximation of the system is discussed. A new first-order theory of beam is proposed based on the assumption that the deformation is parabolic along the cross-section and there is no transversenormal stress in the beam. The differential equation given by this theory is consistent with Timoshenko’s beam theory with the correction factor being π/12, while the displacement expression given by the new theory is different from that of Timoshenko’s beam theory. The consistent differential equations show that the new first-order theory can give the same results as Timoshenko’s beam theory in frequencies. However, the vibration modes of the beam obtained by the new theory is different from those given by Timoshenko’s beam theory. Compared with the finite element results of plane stress problems, we find that the new theory has some advantages in the prediction of mode shapes. In particular, the new theory does not require the assumption of plane section after deformation, and is closer to the finite element results at higher frequencies.
At the end of the paper, the author hopes to solve the problem of high-frequency vibration of the beam by using this Lee's high-order beam theory. This first author of the paper is Dr. Longtao Xie with participation of Mr.Junlei Ding, an undergraduate student, and Professor Ranjan Banerjee of City, University of London.
Paper website: https://doi.org/10.1016/j.jsv.2020.115567
An accurate beam theory and its first-order approximation in free vibration analysis
Longtao Xie,Shaoyun Wang,Junlei Ding,J Ranjan Banerjee,Ji Wang
An infinite system of one-dimensional differential equations is derived from the two-dimensional theory of elasticity by expanding the displacement field in a series of trigonometrical functions together with a linear term. Since the trigonometrical functions are pure thickness-vibration modes of infinite plates or beams with the top and bottom surfaces being free, the differential equations and the corresponding boundary conditions serve as the basis of an accurate beam theory for vibration analysis, named Lee's beam theory (LBT). Naturally, a high-order set of the infinite system should be quite useful in the analysis of beams at high frequencies. With the objective of vibration analysis of beams, this paper focuses on the first-order approximation, which leads to a first-order shear deformation beam theory for flexural vibrations (LBT1st). The differential equations in LBT1st are equivalent to those in Timoshenko's beam theory (TBT). The most important difference between LBT1st and TBT is the different field displacements. For the assessment of the accuracy of LBT1st, the numerical results of frequencies of free vibrations, frequency spectra and mode shapes of beams with classical boundary conditions are obtained and compared with those by TBT and plane stress problem of elasticity. Considering the plane stress problem as a reference, LBT1st is slightly more accurate in describing the field shapes of beams than TBT. Therefore, LBT1st, as well as LBT, is an addition to the existing beam theories with improved accuracy for the vibration analysis of beams and their combinations.