W. N. Cheng Graduate Research Assistant C. C. Cheng Professor e-mail: imeccc@ccu.edu.tw Advanced Institute of Manufacturing for High-Tech Innovations and Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi, 621, Taiwan, R.O.C. G. H. Koopmann Professor Department of Mechanical and Nuclear Engineering, Pennsylvania State University, University Park, PA 16802 A New Design Strategy for Minimizing Sound Radiation of Vibrating Beam Using Dimples A method is proposed for minimizing the sound radiation of a vibrating beam by pattern- ing the beam with a series of cylindrical dimples such that one or more of the vibration modes have the same shape as the corresponding weak modes. In implementing the proposed approach, the objective is to minimize the shape difference between the vibra- tion mode(s) and the designated weak mode(s) rather than to minimize the radiated sound power at a specific frequency or over a certain bandwidth. The design objective is achieved by calculating the weak modes of the beam using the finite element method and then applying an optimization scheme with the modal assurance criterion (MAC) as the objective function. The optimization results, which cause the vibration mode(s) of the dimpled beam to approach the corresponding weak modes(s), determine the dimple angle and dimple depth. The numerical results show that the radiation efficiency of the opti- mized dimpled beam using MAC as the objective is generally lower than that of a uniform beam. However, the effectiveness of the proposed design strategy depends on the degree of closeness between the shape of the vibration mode(s) of the dimpled beam and that of the designated weak mode(s). DOI: 10.1115/1.4003939 1 Introduction A structure that radiates minimal acoustic energy is generally referred to as a weak radiator or a quiet structure. Traditionally, weak radiators are designed by moving the structural resonances out of the frequency range of interest and/or by minimizing the radiated acoustic power in a limited frequency range 1. In addi- tion, weak radiators can also be achieved by raising the coinci- dence frequency such that the radiated acoustic power is sup- pressed by the effects of hydrodynamic cancellation. The literature contains many proposals for meeting these design objec- tives by changing the material or geometry of the vibrating struc- ture, modifying the boundary conditions, and/or adding auxiliary structures such as masses, ribs, and so on to the structure 2–4. However, although the boundary conditions have a significant ef- fect on the acoustic radiation of a vibrating body 5,6, they can only be adjusted within a limited range due to manufacturing and assembly constraints. Furthermore, while the addition of auxiliary structures such as ribs or masses provides an effective means of moving the structural resonance of the structure out of the fre- quency range of interest, the process of computing the design variables e.g., the rib size, the rib location, and so on is compu- tationally complex and time consuming. Accordingly, the present study proposes a method for altering the vibro-acoustic characteristics of a structure by patterning the structure with a series of cylindrical dimples using a simple press machine, as shown in Fig. 1. The dimension of each dimple lo- cated D from the left end, as illustrated in Fig. 1b, is character- ized by the radius R, the angle , and the uniform thickness h. The proposed method provides a simple and cost-effective means of transforming the vibrating structure into a weak radiator. Cunefare 2 obtained the acoustic radiation mode through op- timizing acoustic power subject to velocity constraints. Subse- quently, surface velocity profiles represented by the linear combi- nations of those radiation modes, i.e., the weak modes, were proposed for a structure that results in minimum acoustic radia- tion. However, these radiation modes did not comply with struc- tural boundary conditions. Motivated by the concept of weak mode and its potential appli- cation in design weak radiators, the objective of the proposed approach is to minimize the shape difference between the desig- nated vibration modes and the corresponding weak modes rather than to minimize the total sound power over a specific frequency bandwidth 7. That is, the idea is to modify the shapes of the vibration modes of a structure to be the same as those of weak modes and thus the sound power is minimized by redistrib- uting the areas of high and low surface vibrations using an appro- priate dimple pattern rather than by reducing the overall vibration level of the structure. In satisfying the design objective, it is nec- essary to determine the dimple size, dimple number, and dimple locations, which ensure that the shapes of the vibration modes of the dimpled structure approach those of the weak modes. Accord- ingly, an optimization method combined with the finite element method and the modal assurance criterion MAC is proposed. The validity of the proposed approach is demonstrated by means of two numerical examples. 2 Acoustic Weak Modes Consider a baffled beam of length L, thickness h, and width b. The total acoustic power W radiated from the beam when vibrat- ing harmonically at a frequency of  is expressed as 3 W = 1 2  r=0 L  x=0 L b 2 2 x sinSx, r Sx, r   rdxdr 1 where  is the velocity of the beam in the normal direction,  denotes complex conjugate,  is the density of the acoustic me- dium,  is the acoustic wavenumber, and S is the distance from the observation point at r o to the acoustic source at r s , i.e., S = r o - r s . The acoustic weak mode of the beam can be obtained by mini- mizing the total radiated acoustic power when the beam velocity is normalized as Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received May 24, 2010; final manuscript received December 23, 2010; published online August 31, 2011. Assoc. Editor: Liang-Wu Cai. Journal of Vibration and Acoustics OCTOBER 2011, Vol. 133 / 051008-1 Copyright © 2011 by ASME Downloaded 21 Sep 2011 to 140.123.121.19. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm