﻿ 基于谱方法的矩形薄板自由振动分析 Free Vibration Analysis of Rectangular Thin Plates Based on Spectral Method

International Journal of Mechanics Research
Vol. 08  No. 01 ( 2019 ), Article ID: 29272 , 11 pages
10.12677/IJM.2019.81007

Free Vibration Analysis of Rectangular Thin Plates Based on Spectral Method

Zhongkai Zhao1, Junhua Zhang1, Yanqi Liu2

1College of Mechanical Engineering, Beijing Information and Science Technology University, Beijing

2Beijing Municipal Institute of Labor Protection, Beijing

Received: Feb. 21st, 2019; accepted: Mar. 7th, 2019; published: Mar. 15th, 2019

ABSTRACT

A Chebyshev spectral method without both stiffness and mass matrices is proposed. Free vibrations of rectangular thin plates with clamped boundaries are researched by Chebyshev spectral method. Based on the partial differential equations of thin plates for free vibrations, the equations of eigen frequencies are educed by separating variables. Relationship between frequencies and physical parameters of materials is established. Referring to Chebyshev spectral method, high order derivative matrix is obtained which is a power operation of first order derivative matrix. The eigen frequency matrix equation of free vibration of thin plates is derived, but there is no stiffness matrix and no mass matrix. An example of clamped rectangular thin plate is solving by two methods. Compared with finite element method, efficiency and convergence of spectral method are verified. Comparing results by two methods, the frequencies and modes of the plate are basically consistent.

Keywords:Spectral Method, Free Vibration, Chebyshev Polynomials, Rectangular Thin Plates

1北京信息科技大学，机电工程学院，北京

2北京市劳动保护科学研究所，北京

1. 引言

2. 位移方程和边界条件

Figure 1. Rectangular plate

(1)

(1)式中：为薄板的抗弯刚度，E为薄板材料的弹性模量，为薄板材料的泊松比，为薄板材料的密度，h为薄板的厚度。双调和算子的运算满足式(2)：

(2)

(3)

(4)

(4)式中：为波数，为薄板的自由振动频率。对于四边固支边界，薄板在边界上不发生横向位移，也不产生由位移变化引起的转角。因此，四边固支薄板的边界关系满足式(5)。根据文献 [14] 的研究，薄板的固支边界条件为微分方程的第一类边界条件和第二类边界条件的组合。

(5)

3. 切比雪夫谱方法

(6)

(6)式中：n为非负整数。第一类切比雪夫多项式存在交错点组，使得，在数值分析中被称为切比雪夫插值点，是在区间[−1,1]选取的N + 1个不同的点，切比雪夫插值点的定义如式(7)所示：

(7)

Figure 2. Distribution of Chebyshev points (N = 8)

(8)

(9)

(10)

(10)式中：i、j分别代表1阶切比雪夫谱求导矩阵中第i + 1行、第j + 1列，满足式(11)：

(11)

(12)

(13)

(13)式中：代表克罗内克积 [20] 。

4. 数值算例与结果分析

4.1. 算例

Figure 3. Finite element model

4.2. 频率分析

Table 1. Frequencies and errors calculated by spectral method and FEM

4.3. 振型分析

Figure 4. The relationship between frequencies and the number of interpolation points

(1) (2)
(3) (4)
(5) (6)
(7) (8)
(9) (10)
(11) (12)
(13) (14)
(15) (16)
(17) (18)
(19) (20)
(21) (22)
(23) (24)
(25)

Figure 5. Graph of modes

5. 结论

Free Vibration Analysis of Rectangular Thin Plates Based on Spectral Method[J]. 力学研究, 2019, 08(01): 54-64. https://doi.org/10.12677/IJM.2019.81007

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