可調整型質量矩陣對樑元素自然頻率之影響

Influence of adjustable mass matrix on the natural frequency of beam element

黃成凱(Cheng-Kai Huang)

指導教授 ： 施延欣

中原大學/工學院/機械工程研究所/碩士(2012年)

https://doi.org/10.6840/cycu201201002

摘要

本研究論文，使用有限元素法計算樑的振動自然頻率，在有限元素法的方程式中，質量矩陣分別使用Consistent-Mass Matrix、HRZ-Mass Matrix、RC diagonal Mass Matrix與Adjustable RC diagonal Mass Matrix，其中Adjustable RC diagonal Mass Matrix是依調整 值的不同來得到不同的質量矩陣。
本文中，先將自然頻率無因次化，再利用Matlab軟體程式分別計算上列多種質量矩陣，進一步求取不同質量矩陣的自然頻率，並分別與標準解的自然頻率比較誤差。
本研究結果，針對懸臂樑的第一至第五模態的自然頻率，選取最接近標準解的 值質量矩陣，計算出無因次自然頻率結果，並比較誤差。
本文中除了提供前五個模態各自建議的 值與最少所需的元素數目外，同時也提供了前兩個、前三個、前四個以及前五個模態同時考慮下所建議的 值與最少所需的元素數目。

關鍵字

質量矩陣 ； 自然頻率 ； 懸臂樑 ； 有限元素法

並列摘要

In this study, use finite element method (FEM) to calculate the natural frequency of cantilever beam. The mass matrix in FEM is set as consistent mass matrix, HRZ mass matrix, RC diagonal mass matrix and adjustable RC diagonal mass matrix. By changing the and the different mass matrices can be obtained, and are called adjustable RC diagonal mass matrices.
In this thesis, the natural frequency is considered to a dimensionless form. The dimensionless natural frequency are calculated by five types of mass matrix with matlab program. The error of dimensionless natural frequency with respect to the close form is also provided. The natural frequencies of the first five modes of a cantilever beam are provided in this study.
Beside, it not only provides the and the minimum elements required of the first、the second、the third、the fourth and the fifth mode, but also suggest the and the minimum elements required of the first two、the first three、the first four and the first five modes.

並列關鍵字

mass matrix ； natural frequency ； cantilever beam ； finite element method

參考文獻

[1] J. S. Przemieniecki, Theory of Matrix Structural Analysis, Dover, New York, 1985.

[2] S. Rao , Mechanical Vibration, Addison-Wesley Publishing Co,Inc.1995

[4] G. C. Archer, T. M. Whalen, Development of rotationally consistent diagonal mass matrices for plate and beam elements, Comput. Methods Appl. Mech. Engrg. 194 (2005) 675-689.

[6] R.D. Cook, D.S. Malkus, M.E. Plesha, Concepts and Applications of Finite Element Analysis, John Wiley & Sons, New York, 1989.

[7] J. Chung, G.M. Hulbert, A family of single-step Houbolt time integration algorithms for structural dynamics, Comput. Methods Appl. Mech. Engrg. 118 (1994) 1–11.