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Research Papers

# Optimal Design of a Helmholtz Resonator With a Flexible End Plate

[+] Author and Article Information

Pennsylvania State University,
Altoona, PA 16601
e-mail: mhk13@psu.edu

G. Scott Duncan

Department of Mechanical Engineering,
Valparaiso University,
Valparaiso, IN 46383
e-mail: scott.duncan@valpo.edu

Shahin S. Nudehi

Department of Mechanical Engineering,
Valparaiso University,
Valparaiso, IN 46383
e-mail: shahin.nudehi@valpo.edu

1Corresponding author.

Contributed by the Noise Control and Acoustics Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 29, 2012; final manuscript received January 27, 2014; published online March 18, 2014. Assoc. Editor: Lonny Thompson.

J. Vib. Acoust 136(3), 031004 (Mar 18, 2014) (8 pages) Paper No: VIB-12-1365; doi: 10.1115/1.4026849 History: Received December 29, 2012; Revised January 27, 2014

## Abstract

This paper describes a design process that produces a small volume Helmholtz resonator capable of achieving high transmission loss across a desired frequency range. A multiobjective optimization formulation was used to design a Helmholtz resonator with a flexible end plate. The optimization formulation generated a Pareto curve of design solutions that quantify the trade-off between the optimization goals: minimum resonator volume and maximum transmission loss across a specified frequency range. The optimization problem was formulated and solved in the following manner. First, a mathematical formulation for the transmission loss of the Helmholtz resonator with a flexible plate was completed based on the resonator design parameters. Then, the weighted transmission loss across a specified frequency range and a minimum resonator volume were defined as optimization objectives. Finally, the Pareto curve of optimum design solutions was calculated using a gradient-based approach via the $ɛ$-constraint method. The optimization results allow the designer to select resonator design parameters that meet the requirements for both transmission loss and resonator volume. To validate the optimization results, two optimal Helmholtz resonators were manufactured and experimentally confirmed.

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Topics: Design , Optimization

## Figures

Fig. 1

Schematic and subassemblies of a Helmholtz resonator with flexible plate

Fig. 2

Pareto front (square markers present nondominated designs and circle markers represent dominated designs)

Fig. 3

Constraint optimization of weighted TL and chamber volume (α = 1)

Fig. 4

Trade-off curve with objective function adjusted to favor nominal desired frequency (α = 3)

Fig. 5

Trade-off curve with objective function adjusted to favor nominal desired frequency (α = 3) and constraint on diameter ratio

Fig. 6

Resonator experimental setup assembly mounted to an instrumented duct

Fig. 7

Predicted and measured transmission loss for baseline design

Fig. 8

Predicted transmission loss for optimal designs A and B with experimental measurement

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