拓樸最佳化應用於模態塑造與飛輪轉子設計

Synthesizing Desired Eigenfrequencies and Mode Shapes and Design of Flywheel Rotors Using Topology Optimization

蔡騰德(Teng-Der Tsai)

指導教授 ： 鄭志鈞

國立中正大學/工學院/機械工程學系/博士(2012年)

摘要

本文以拓樸最佳化方法(Topology optimization)的研究為主，主要分成兩個部分：第一個部分，在提升其自然頻率的問題中，塑造結構之模態；第二個部分，針對飛輪儲能系統之需求，設計飛輪轉子的幾何形狀。拓樸最佳化由三個模組所構成：拓樸材料法則使用密度函數法(Solid isotropic method with penalization, SIMP)；結構分析工具為有限元素法(Finite element method, FEM)；最佳化工具則使用移動漸進線法(Method of moving asymptotes, MMA)，三種模組皆利用數學運算軟體Matlab撰寫而成。本文以權重搭配連結法(Bound formulation)，在最佳化的過程中，可預防模態互換(Mode switching)的問題。為了將特定模態塑造成預定之形狀，本文於最佳化中加入MAC(Modal assurance criterion)限制條件並推導其靈敏度。文中列舉數個塑造模態的範例，並應用於機器之解耦合支撐設計。飛輪轉子應用於飛輪儲能系統，其設計需求不外乎增加其儲存能量以及延長其使用壽命，也就是最大化其剛性、轉動慣量與自然頻率。本文以拓樸最佳化方法設計飛輪轉子，在設計靜態轉子的部分，最大化其自然頻率與轉動慣量；動態轉子必須考慮轉子的旋轉效應，也就是其慣性所造成的離心力，因此最佳化的拓樸結構與轉速有關。在設計動態轉子的部分，著重於最大化其動態剛性、靜態剛性與轉動慣量。

關鍵字

拓樸最佳化 ； 模態塑造 ； 飛輪轉子

並列摘要

This thesis focuses on the development of topology optimization for the determination of the material distribution of a structure. It is mainly divided into two parts: the first part is to synthesize a desirable eigenmode shape for problems of maximizing fundamental eigenfrequency; the second part is to design the geometry of rotors for flywheel energy storage system (FESS). In implementing the proposed approach, the design objective is achieved using solid isotropic method with penalization (SIMP), the method of moving asymptotes (MMA) and finite element method (FEM) in topology optimization. Weighted constraints added in bound formulation are utilized in maximizing the fundamental natural frequency, which provides an easy but straightforward way to prevent the occurrence of mode switching in an optimization process. Besides maximizing the fundamental frequency, the topology layout of a structure with a desirable eigenmode is obtained by adding the modal assurance criterion (MAC) as an additional constraint in the formulation. To illustrate the methodology of the present approach, several examples are presented. A potential application of the proposed technique in decoupling a mechanical system is demonstrated. Rotor design issues include maximizing either the natural frequency, stiffness or moment of inertia. For a static rotor, this paper investigates the optimum structural layouts respectively achieved from maximizing the fundamental torsional natural frequency, maximizing moment of inertia and their combinations. For a rotating rotor, the centrifugal force caused by the rotational speed is design-dependent. Therefore the optimal layout of the flywheel rotor changes with the rotational speed. Examples to illustrate the methodology are maximizing the dynamic, static stiffness, the moment of inertia and their combinations.

並列關鍵字

Topology optimization ； Mode shape synthesizing ； Flywheel rotors

參考文獻

[1] Y. W. Kwon, H. Bang, The finite element method using matlab, Second Edition, CRC Press, USA, 2000.

[2] A. G. M. Michell, The limits of economy of material in frame structures, Philos. Mag. Ser. 7, 8 (47) (1904) 589–597.

[3] M. P. Bendsøe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Comput. Meth. Appl. Mech. Eng. 71 (2) (1988) 197–224.

[4] Z. D. Ma, N. Kikuchi, I. Hagiwara, Structural topology and shape optimization for a frequency response problem, Comput. Mech. 13 (3) (1993) 157-174.

[5] Z. D. Ma, N. Kikuchi, H. C. Cheng, I. Hagiwara, Topological optimization technique for free vibration problems, J. Appl. Mech. 62 (1) (1995) 200-207.