摘要在进行轨道交通引起建筑结构振动响应计算时，为了提高高频响应的准确度，采用谱单元法(spectral element method, SEM)求解矩形楼板的动力响应，并针对既有谱单元法中板与梁（柱）单元自由度不相容问题，通过修正板的节点自由度，实现了板与梁、柱的耦合。首先，将四边任意边界条件的矩形板分解为两个可以转化为类一维问题的单向板；其次，基于Kirchhoff薄板的振动控制方程，对每一个单向板进行动力计算，并分析波的传播与横截面的振型。通过最小二乘法优化板的节点自由度，使之与梁、柱单元的自由度对应，实现谱单元法在梁板柱耦合结构中的应用。将谱单元法的计算结果与解析解和有限元法结果对比得出：在1-250 Hz内，谱单元法的计算结果比有限元准确，尤其在高频范围；谱单元模型节点数量较少，建模和计算效率高于有限元。
Abstract：To improve the accuracy of the building vibration in the high frequency range, a modified spectral element method is proposed to analyse the dynamic response of the rectangular plate and the plate coulped with beams and columns. First the plate is split into two sub-problems, each of which is solved as a quasi-one-dimensional problem using spectral element method based on the governing equation of the Kirchhoff plate. Then the nodal dofs is modified to be in accord with the dofs of beams and columns by using the least-squares approach. After that the coupled ‘plate-beam-column’ structure is solved entirely by the spectral element method. Some examples are given to validate the developed method and it shows that the spectral element method gives accurate result and is more computationally efficient than the finite element method.
曹容宁，马蒙，孙晓静，刘维宁. 框架结构中矩形楼板动力响应求解的谱单元法[J]. 振动与冲击, 2021, 40(24): 99-106.
CAO Rongning，MA Meng，SUN Xiaojing，LIU Weining. A spectral element method for modelling a rectangular plate vibration in frame structures. JOURNAL OF VIBRATION AND SHOCK, 2021, 40(24): 99-106.
[1] STANDARD I, ISO B. Mechanical vibration ground borne noise and vibration arising from rail systems—Part 1: general guidance [J]. ISO 14837, 2005, 1.
[2] 徐芝纶. 弹性力学.下册[M]. 高等教育出版社, 2016.
Xu Zhilun. Elastic mechanics Volume II[M]. Higher Education Press, 2006. (in Chinese)
[3] 曹志远. 板壳振动理论[M]. 中国铁道出版社, 1989.
Cao Zhiyuan. Vibration theory of plates and shells[M]. China Railway Publishing House, 1989. (in Chinese)
[4] CHEUNG M S, CHEUNG Y K. Natural vibrations of thin, flat-walled structures with different boundary conditions[J]. Journal of Sound and Vibration, 1971, 18(3): 325–337.
[5] LIN Y K, DONALDSON B K. A brief survey of transfer matrix techniques with special reference to the analysis of aircraft panels[J]. Journal of Sound and Vibration, 1969, 10(1): 103–143.
[6] GAVRIĆ L. Finite Element Computation of Dispersion Properties of Thin-Walled Waveguides[J]. Journal of Sound and Vibration, 1994, 173(1): 113–124.
[7] LANGLEY R S. Application of the dynamic stiffness method to the free and forced vibrations of aircraft panels[J]. Journal of Sound and Vibration, 1989, 135(2): 319–331.
[8] DOYLE J F. Wave Propagation in Structures: Spectral Analysis Using Fast Discrete Fourier Transforms[M]. 2nd edition. New York: Springer, 1997.
[9] LEE U. Vibration analysis of one-dimensional structures using the spectral transfer matrix method[J]. Engineering Structures, 2000, 22(6): 681–690.
[10] LEE U, KIM J. Dynamics of elastic-piezoelectric two-layer beams using spectral element method[J]. International Journal of Solids and Structures, 2000, 37(32): 4403–4417.
[11] LEE U, JANG I, GO H. Stability and dynamic analysis of oil pipelines by using spectral element method[J]. Journal of Loss Prevention in the Process Industries, 2009, 22(6): 873–878.
[12] BIRGERSSON F, FINNVEDEN S, NILSSON C-M. A spectral super element for modelling of plate vibration. Part 1: general theory[J]. Journal of Sound and Vibration, 2005, 287(1): 297–314.
[13] PARK I, KIM T, LEE U. Frequency Domain Spectral Element Model for the Vibration Analysis of a Thin Plate with Arbitrary Boundary Conditions[J]. Mathematical Problems in Engineering, 2016, 2016: 1–20.
[14] CHEUNG B Y K. Finite strip method in structural analysis[M]. Pergamon Press, 1976.
[15] CHAKRAVERTY S. Vibration of Plates[M]. Boca Raton :CRC Press, 2008.
[16] 张元林. 积分变换[M]. 第五版. 北京: 高等教育出版社, 2012.
Zhang Yuanlin. Integral transformation [M]. 5th Edition. Beijing:Higher Education Press, 2012. (in Chinese)
