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Research Papers

Multi-Objective Optimization of Elastic Beams for Noise Reduction

[+] Author and Article Information
Meng-Xin He

Department of Mechanics,
Tianjin University,
Tianjin 300072, China
e-mail: hemengxin309@sina.cn

Fui-Rui Xiong

Science and Technology on Reactor System
Design Technology Laboratory,
Nuclear Power Institute of China,
Chengdu 610041, China
e-mail: xfr90311@gmail.com

Jian-Qiao Sun

Fellow ASME
School of Engineering,
University of California,
Merced, CA 95343;
Honorary Professor
Department of Mechanics,
Tianjin University,
Tianjin 300072, China
e-mail: jqsun@ucmerced.edu

1Corresponding author.

Contributed by the Noise Control and Acoustics Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November 12, 2016; final manuscript received May 1, 2017; published online July 13, 2017. Assoc. Editor: Nicole Kessissoglou.

J. Vib. Acoust 139(5), 051014 (Jul 13, 2017) (10 pages) Paper No: VIB-16-1540; doi: 10.1115/1.4036680 History: Received November 12, 2016; Revised May 01, 2017

This paper presents a study of multi-objective optimization of elastic beams with minimum weight and radiated sound power. The goal of this research is to discover the potentials to design multi-objective optimal elastic structures for better acoustic performance. We discuss various structural-acoustic properties of the Pareto solutions of the multi-objective optimization problem (MOP). We have found that geometrical and dynamic constraints can substantially reduce the volume fraction of feasible solutions in the design space, which can make it difficult to search for the optimal solutions. Several case studies with different boundary conditions are studied to demonstrate the multi-objective optimal designs of the structure.

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Figures

Grahic Jump Location
Fig. 1

The Pareto front of the clamped–clamped Timoshenko beam. The population size of the MOPSO method is 100. Eighty generations are computed. Empty circles, the TMM; solid circles, the FEM.

Grahic Jump Location
Fig. 2

The Pareto front of the simply supported Timoshenko beam. The population size of the MOPSO method is 100. Eighty generations are computed. Empty circles, the TMM; Solid circles, the FEM.

Grahic Jump Location
Fig. 3

The Pareto front of the simply supported Timoshenko beam. Empty circles, the MOPSO + SCM hybrid algorithm; Solid circles, the MOPSO.

Grahic Jump Location
Fig. 4

Examples of optimal beams obtained with the MOPSO (beam 1) and the PSO + SCM hybrid algorithm (beam 2). The mass is 35.13 kg for beam 1 and 35.23 kg for beam 2. The radiated sound power level is 91.13 (dB) from beam 1 and is 92.97 (dB) from beam 2.

Grahic Jump Location
Fig. 5

The optimal thickness profile of the simply supported Timoshenko beam under a concentrated unit magnitude harmonic load which is applied in the interval marked by the red lines. Beam 1 has the minimum mass, while beam 2 has the minimum integrated sound power.

Grahic Jump Location
Fig. 6

Left: The radiated sound power of beam 1 in Fig. 5. Right: The zoomed view of the left figure in the frequency range 200–600 Hz, containing four active modes. Solid lines: The optimal nonuniform beam. Dashed lines: The uniform beam with the same mass. The reduction of the radiated sound power level of the optimal design is 0.91 dB compared with the baseline beam.

Grahic Jump Location
Fig. 7

Left: The radiated sound power of beam 2 in Fig. 5. Right: The zoomed view of the left figure in the frequency range 200–600 Hz, containing two active modes. Solid lines, the optimal nonuniform beam; dashed lines, the uniform beam with the same mass. The reduction of the radiated sound power level of the optimal design is 8.25 dB compared with the baseline beam.

Grahic Jump Location
Fig. 8

The Pareto front for the simply supported beam, consisting of 84 solutions. The horizontal line is the fundamental frequency ω1 of the uniform beam with the constant mass, that is 19.62 Hz.

Grahic Jump Location
Fig. 9

The Pareto front of the clamped–clamped beam, consisting of 35 solutions. The horizontal line is the fundamental frequency ω1 = 44.46 Hz of the uniform beam with the constant mass.

Grahic Jump Location
Fig. 10

Extreme designs of the simply supported beam under a harmonic load with different frequencies. The harmonic load is applied in the interval marked by the vertical lines. Left: The thickness profiles of the minimum sound radiation under the forcing frequency from 50 Hz to 250 Hz, 650 Hz to 850 Hz, 1250 Hz to 1450 Hz, and 1850 Hz to 2050 Hz, from top to bottom, respectively. Right: The thickness profiles of the maximum ω1 in the same frequency ranges as the ones in the left figure.

Grahic Jump Location
Fig. 11

Extreme designs of the clamped–clamped beam under a harmonic load with different frequencies. The harmonic load is applied in the interval marked by the vertical lines. Left: The thickness profiles of the minimum sound radiation under the forcing frequency from 50 Hz to 250 Hz, 650 Hz to 850 Hz, 1250 Hz to 1450 Hz, and 1850 Hz to 2050 Hz, from top to bottom. Right: The thickness profiles of the maximum ω1 in the same frequency ranges as the ones in the left figure.

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