Topology optimization of vibrating structures with frequency band constraints
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RESEARCH PAPER
Topology optimization of vibrating structures with frequency band constraints Quhao Li 1,2 & Qiangbo Wu 1 & Ji Liu 3 & Jingjie He 2 & Shutian Liu 2 Received: 20 May 2020 / Revised: 20 September 2020 / Accepted: 25 September 2020 # Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Engineering structures usually operate in some specific frequency bands. An effective way to avoid resonance is to shift the structure’s natural frequencies out of these frequency bands. However, in the optimization procedure, which frequency orders will fall into these bands are not known a priori. This makes it difficult to use the existing frequency constraint formulations, which require prescribed orders. For solving this issue, a novel formulation of the frequency band constraint based on a modified Heaviside function is proposed in this paper. The new formulation is continuous and differentiable; thus, the sensitivity of the constraint function can be derived and used in a gradient-based optimization method. Topology optimization for maximizing the structural fundamental frequency while circumventing the natural frequencies located in the working frequency bands is studied. For eliminating the frequently happened numerical problems in the natural frequency topology optimization process, including mode switching, checkerboard phenomena, and gray elements, the “bound formulation” and “robust formulation” are applied. Three numerical examples, including 2D and 3D problems, are solved by the proposed method. Frequency band gaps of the optimized results are obtained by considering the frequency band constraints, which validates the effectiveness of the developed method. Keywords Topology optimization . Eigenvalue optimization . Frequency band constraint . Heaviside function
1 Introduction Vibration brings lots of harmful effects for engineering structures in aerospace, high-speed trains, and high precision machinery, etc. Thus, how to reduce structural vibration has been a hot topic in engineering and academics. Lots of engineering structures, such as machine tools, and vibration isolators, usually operate in some frequency bands (Pierson 1972; Sun et al. 2012). When the working or external excitation frequency is equal to or close to the natural frequency of a structure, resonance phenomena will Responsible Editor: Nestor V Queipo * Shutian Liu [email protected] 1
Key Laboratory of High Efficiency and Clean Mechanical Manufacture of MOE, School of Mechanical Engineering, Shandong University, Jinan 250061, People’s Republic of China
2
State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, People’s Republic of China
3
Institute of Systems Engineering, China Academy of Engineering Physics, Mianyang, Sichuan 621900, People’s Republic of China
happen. The system will oscillate at a higher amplitude than when the same force is applied at other, non-resonant frequencies, which can cause large deformations and even fracture. Therefore, by rea
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