Date of download: 10/28/2017 Copyright © ASME. All rights reserved. From: Multi-Objective Optimization of Elastic Beams for Noise Reduction J. Vib. Acoust. 2017;139(5):051014-051014-10. doi:10.1115/1.4036680 Figure Legend: 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.

Date of download: 10/28/2017 Copyright © ASME. All rights reserved. From: Multi-Objective Optimization of Elastic Beams for Noise Reduction J. Vib. Acoust. 2017;139(5):051014-051014-10. doi:10.1115/1.4036680 Figure Legend: 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.

Date of download: 10/28/2017 Copyright © ASME. All rights reserved. From: Multi-Objective Optimization of Elastic Beams for Noise Reduction J. Vib. Acoust. 2017;139(5):051014-051014-10. doi:10.1115/1.4036680 Figure Legend: The Pareto front of the simply supported Timoshenko beam. Empty circles, the MOPSO + SCM hybrid algorithm; Solid circles, the MOPSO.

Date of download: 10/28/2017 Copyright © ASME. All rights reserved. From: Multi-Objective Optimization of Elastic Beams for Noise Reduction J. Vib. Acoust. 2017;139(5):051014-051014-10. doi:10.1115/1.4036680 Figure Legend: 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.

Date of download: 10/28/2017 Copyright © ASME. All rights reserved. From: Multi-Objective Optimization of Elastic Beams for Noise Reduction J. Vib. Acoust. 2017;139(5):051014-051014-10. doi:10.1115/1.4036680 Figure Legend: 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.

Date of download: 10/28/2017 Copyright © ASME. All rights reserved. From: Multi-Objective Optimization of Elastic Beams for Noise Reduction J. Vib. Acoust. 2017;139(5):051014-051014-10. doi:10.1115/1.4036680 Figure Legend: 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.

Date of download: 10/28/2017 Copyright © ASME. All rights reserved. From: Multi-Objective Optimization of Elastic Beams for Noise Reduction J. Vib. Acoust. 2017;139(5):051014-051014-10. doi:10.1115/1.4036680 Figure Legend: 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.

Date of download: 10/28/2017 Copyright © ASME. All rights reserved. From: Multi-Objective Optimization of Elastic Beams for Noise Reduction J. Vib. Acoust. 2017;139(5):051014-051014-10. doi:10.1115/1.4036680 Figure Legend: 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.

Date of download: 10/28/2017 Copyright © ASME. All rights reserved. From: Multi-Objective Optimization of Elastic Beams for Noise Reduction J. Vib. Acoust. 2017;139(5):051014-051014-10. doi:10.1115/1.4036680 Figure Legend: 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.

Date of download: 10/28/2017 Copyright © ASME. All rights reserved. From: Multi-Objective Optimization of Elastic Beams for Noise Reduction J. Vib. Acoust. 2017;139(5):051014-051014-10. doi:10.1115/1.4036680 Figure Legend: 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.

Date of download: 10/28/2017 Copyright © ASME. All rights reserved. From: Multi-Objective Optimization of Elastic Beams for Noise Reduction J. Vib. Acoust. 2017;139(5):051014-051014-10. doi:10.1115/1.4036680 Figure Legend: 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.